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Comparison of hexagonal close packed and cubic close packed rare gas solids Hurst, Michael James 1975

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COMPARISON OF HEXAGONAL CLOSE PACKED AND CUBIC CLOSE PACKED RARE GAS SOLIDS by MICHAEL JAMES HURST B. S c , U n i v e r s i t y of Calgary, 1967 H.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as ccmforming to the required standard THE UNIVERSITY OF BRITISH MAY, 1975 COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 CHAIRPERSON: PROF. ROGER HOWARD i i ABSTRACT Hexagonal close packed and cubic close packed structures of heavy rare gas atoms are compared. L a t t i c e expansion ef f e c t s are examined for a model i n which atoms int e r a c t v i a Lennard-Jones (12-6) potentials, l a t t i c e motion i s according to the quasiharmonic approxi-mation, and the model c r y s t a l s are in equilibrium at constant temp-erature and pressure. Phonon frequencies are found to depend sensi-t i v e l y upon l a t t i c e parameters. In the cubic (hexagonal) structure the phonon modes most strongly affected by expansion are long (short) wavelength, longitudinal (transverse), acoustic (optic) modes. Phonon frequencies for the cubic structure a l l remain re a l for expansions considerably greater than would cause imaginary frequencies for the hexagonal structure. Equilibrium l a t t i c e parameters are found to deviate from t h e i r s t a t i c equilibrium values by amounts depending upon the De Boer parameter, the deviations being s l i g h t l y greater for the hexagonal structure, which exhibits nonideal stacking. The prob-lem of explaining the observed s t a b i l i t y of the cubic structures of re a l rare gas so l i d s i s discussed. i i i TABLE DF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS • i i i LIST OF TABLES v;'i LIST OF FIGURES ...... v i i i ACKNOWLEDGEMENTS x CHAPTER I INTRODUCTION : 1 II THEORETICAL BACKGROUND 5 THE LAW OF CORRESPONDING STATES 6 INTERATOMIC INTERACTIONS 10 Lang-Range Interactions 10 Short-Range Interactions 11 The Central Pair Potential • 12 The Total Interaction Potential 15 FCC AND HCP LATTICES 17 STRUCTURAL STABILITY OF CRYSTALS 23 Absolute S t a b i l i t y 23 Relative S t a b i l i t y 2k Relative S t a b i l i t y of St a t i c Structures . 26 THE QUASIHARMONIC APPROXIMATION 2B Quasiharmonic Equations of Motion '28 Phonon Frequency Dis t r i b u t i o n s 30 Quasiharmonic Gibbs Function '31 iv III DESCRIPTION OF CALCULATIONS 32 MODEL 32 REDUCTION OF VARIABLES 33 CALCULATION OF PHONON FREQUENCIES 35 CRUDE MODEL 36 Pressure of the Zero Point Motion 38 Zero Point Expansion 3B IV PHONON FREQUENCY DISTRIBUTIONS AS FUNCTIONS OF LATTICE PARAMETERS V i SLIGHT EXPANSION kZ MORE EXTENSIVE EXPANSION k6 INSTABILITY OF FCC AND HCP LATTICES 69 SUMMARY 71 V EQUILIBRIUM CONFIGURATIONS OF THE RARE GAS SOLIDS 73 EQUILIBRIUM LATTICE PARAMETERS 73 THERMAL EXPANSIVITIES 77 ISOTHERMAL COMPRESSIBILITIES 79 SUMMARY 81 VI RELATIVE STABILITIES OF THE TWO STRUCTURES 82 HISTORY OF THE RGS STABILITY PROBLEM B2 Central Pair Force Calculations 83 Noncentral Pair Farce Calculations 87 Three-Body Force Calculations 88 PRECISION REQUIREMENTS OF THE STABILITY PROBLEM 89 V VII CONCLUSION 93 SUMMARY OF RESULTS 93 DISCUSSION OF MODEL ASSUMPTIONS 96 Interatomic I n t e r a c t i o n s 96 L a t t i c e Dynamics 97 The RGS S t a b i l i t y Problem 101 BIBLIOGRAPHY 103 APPENDIX A CALCULATION OF FCC AND HCP LATTICE SUMS 106 B DETERMINATION OF FCC AND HCP PHONON FREQUENCY DISTRIBUTIONS 150 C PRECISION REQUIREMENTS OF THE RGS STABILITY PROBLEM 222 v i LIST OF TABLES TABLE Page II-1 Quantities pertinent to the Law of Corre-sponding States for the RGS 9 I I - 2 Fee and hep l a t t i c e and reciprocal l a t t i c e vectors 20 I I I - 1 Zero point expansion properties of the RGS i n terms of the crude model of section III-if .... 39 IV- 1 C o e f f i c i e n t s for the interpolation formulas C4—1> for various moments of fee and hep phonon frequency d i s t r i b u t i o n s kk V- 1 : Expansion c o e f f i c i e n t s for fee and hep RGS with long-range (12-6) interactions and at zero pressure 75 VI- 1 Estimates of permissible r e l a t i v e uncertain-t i e s i n quantities required to calculate the fec-hep ground state energy difference, A Cc *H £ 91 A-1 Values of S ( O ) f o r m=6,12,16 for various ranges of interaction i n fee and i d e a l hep l a t t i c e s 109 A-2 Estimates of costs and execution times for single c a l l s to the l a t t i c e sum subroutine packages, and for enough c a l l s to construct 5 a phonon spectrum of ~3x10 phonon frequencies. 126 v i i B-1 Variation of with fineness of the sample net 155 v i i i LIST OF FIGURES FIGURE Page 2-1 Comparison of the potential wells of (m-6), (m-8-6), and Bobetic-Barker potentials for krypton 16 2-2 Portion of a hexagonal planar array used to construct fee and hep l a t t i c e s 17 2-3 Unit c e l l and B r i l l o u i n zone for the fee l a t t i c e 21 Z-k Unit c e l l and B r i l l o u i n zone for the hep l a t t i c e .. 22 if-1 Scaled phonon frequency d i s t r i b u t i o n s for fee l a t t i c e s bound by f i r s t neighbor (12-6) interactions <t7-51 U-Z Scaled phonon frequency d i s t r i b u t i o n s for hep l a t t i c e s bound by f i r s t neighbor (12-6) inter a c t i o n s . The d i s t r i b u t i o n s represent i s o t r o p i c expansion with S^=0 53-56 Scaled phonon frequency d i s t r i b u t i o n s for hep l a t t i c e s bound by f i r s t neighbor (12-6) int e r a c t i o n s . The d i s t r i b u t i o n s represent a x i a l expansion with ^=0 58-63 k-k Scaled phonon frequency d i s t r i b u t i o n s for hep l a t t i c e s bound by f i r s t neighbor (12-6) o i n t e r a c t i o n s . The d i s t r i b u t i o n s represent mixed expansion 65-67 i x k-5 The i n s t a b i l i t y l i n e for the quasiharmonic hep l a t t i c e 68 5-1 Thermal expansivities at one atmosphere for fee and hep xenon, compared with experimental data 78 5-2 Isothermal compressibilities of fee and hep RGS 80 5-3 Dependence of the stacking parameter of the hep l a t t i c e upon pressure at T=0K and T=10K .. 80 7-1 Comparison of phonon frequency d i s t r i b u t i o n s for s o l i d hydrogen and the RGS 100 A-1 Schematic diagram of EUALDSUMS 118 A-2 Schematic diagram D f SHELLSUMS 119 A-3 Schematic diagram of IMIMSUMS 120 B-1 Variation of <£>>_ and <<^ >. upon the fee hep p number of phonon frequencies i n the sample ... 156 B-2 Development of the shape of the fee frequency d i s t r i b u t i o n with increasing sample size 157-158 B-3 Development of the shape of the hep frequency d i s t r i b u t i o n with increasing sample size 159-160 B-^ Polarization branches corresponding to the K=32 fee frequency d i s t r i b u t i o n i n Figure B-2d 163 B-5 Polarization branches corresponding to the H=6 hep spectrum i n F i g . B-3d I6*t B-6 Schematic diagram of FCCFRQ 168 B-7 Schematic diagram of HCPFRQ, 169 X ACKNOWLEDGEMENTS The author acknowledges the kind assistance of: Dr. Roger Howard Dr. Betty Howard Dr. Laurens Jansen University of B r i t i s h Columbia Department of Physics S t a f f University of B r i t i s h Columbia Computing Centre S t a f f National Research Council of Canada 1 CHAPTER I INTRODUCTION The heavy rare gases - neon, argon, krypton, xenon, and radon -c r y s t a l l i z e i n cubic close packed s t r u c t u r e s , the only trace c f any other s t r u c t u r e being hexagonal close packed regions that have been observed i n h i g h l y f a u l t e d , cubic close packed argon. Although extensive hexagonal close packing of rare gas atoms, other than helium atoms, i s not encountered i n nature, such s t r u c t u r e s have a t t r a c t e d abundant a t t e n t i o n over the past quarter century. The reason f a r t h i s i n t e r e s t i s that i t has been very much ea s i e r to give a theoret-i c a l explanation f o r the s t a b i l i t y of the f i c t i t i o u s hexagonal s t r u c -tures than f o r the observed cubic s t r u c t u r e s . At present, the general opinion i s that the cubic s t r u c t u r e s are s t a b i l i z e d by short-range, many-body forces and/..'tlo some extent, by anharmonic e f f e c t s . Unfortunately, c a l c u l a t i o n s adequate to elevate t h i s opinion to accepted f a c t have not been performed. Such c a l c u l a -t i o n s would be extremely d i f f i c u l t , since the cohesive energies of the two close packed s t r u c t u r e s probably d i f f e r by l e s s than .5%. The purpose of t h i s t h e s i s i s to compare hexagonal and cubic close packed s t r u c t u r e s of rare gas atoms, i n order to gain some i n s i g h t i n t o d i f f e r e n c e s i n the p r o p e r t i e s of the two s t r u c t u r e s . For b r e v i t y i n the f o l l o w i n g d i s c u s s i o n , the rare gas s o l i d s are r e f e r r e d to as "RGS", with "hexagonal cl o s e packed" shortened to "hep" and "cubic close packed" shortened to "fee" ( r a t h e r than "ccp", since "face centered cubic" i s equivalent to "cubic c l o s e packed", 2 but i s more commonly encountered i n the l i t e r a t u r e ) . The comparison of the two structures i s concerned with three problems: 1. Hou are phonon frequency d i s t r i b u t i o n s affected by changes in l a t t i c e parameters? 2. Hou do l a t t i c e parameters depend upon atomic motions? 3. Which structure i s more stable? The f i r s t two problems are examined in terms of a model in which rare gas atoms interact via Lennard-Cones (12-6) potentials and move according to the quasiharmonic approximation of l a t t i c e dynamics. For the f i r s t problem, no attempt i s made to relate l a t t i c e parameters to the physical environments of the model c r y s t a l s , since t h i s allows the e ffects to be analyzed independently of most model parameters. Calculated phonon frequency d i s t r i b u t i o n s , corresponding to a wide variety of l a t t i c e parameters for the two structures, show a strong reduction'in frequencies for small increases i n interatomic spacing. The frequencies are not affected uniformly, but d i f f e r e n t i a l effects are found to be weak compared with o v e r a l l e f f e c t s . In the fee (hep) l a t t i c e the modes most strongly affected by l a t t i c e expansion are long (short) wavelength, longitudinal (transverse) modes. This difference suggests di f f e r e n t natures for the dynamical i n s t a b i l i t i e s of the two structures at catastrophically large s p e c i f i c volumes. For the second problem i t i s assumed that the model c r y s t a l s are in thermodynamic equilibrium at constant temperature and pressure, 3 with l a t t i c e parameters which minimize the quasiharmonic Gibbs free energies. These equilibrium conditions provide a 'means of r e l a t i n g l a t t i c e parameters to atomic species and environmental conditions. La t t i c e parameters, calculated for a variety of low temperatures and pressures, are found to deviate from t h e i r static, equilibrium values by amounts proportional to the De Boer parameter. For i n t e r -actions r e s t r i c t e d to f i r s t neighbors, the effe c t s i n fee and hep are i d e n t i c a l ( in lowest order) and hep remains i d e a l l y stacked. For longer-range interactions, the expansion of the fee l a t t i c e i s some-what weaker than that of hep, which deviates s l i g h t l y from i d e a l stacking. Thermal expansivities and isothermal comp r e s s i b i l i t i e s , calculated by numerical d i f f e r e n t i a t i o n , are i n rough agreement with experimental values. The t h i r d problem, that of determining the r e l a t i v e s t a b i l i t i e s of the two structures, i s discussed mostly i n terms of previous calculations of the fec-hep cohesive energy difference. IM a results are presented which would a l t e r the conclusions of previous studies. It i s suggested, however, that the near i d e n t i t y of the cohesive energies should lead to cautious interpretations of s t a b i l i t y con-clusions. The remainder of this thesis i s organized as follows: Chapter II gives a b r i e f review of RGS theory. The reader who i s f a m i l i a r with t h i s material may wish to refer to t h i s chapter only to establish notation•conventions. if Chapter III describes the model.chosen for the analysis of l a t -t i c e expansion effects and presents some estimates of the results presented i n subsequent chapters. Chapter 11/ considers the dependence of phonon frequency d i s t r i -butions upon l a t t i c e parameters. Chapter V relates l a t t i c e parameters to model parameters and environmental conditions of temperature and pressure. Chapter UI examines the d i f f i c u l t y of distinguishing the r e l a -tive s t a b i l i t i e s of fee and hep RGS. Chapter VII summarizes the re s u l t s of the thesis and suggests some problems far future consideration. Appendix A describes the l a t t i c e sums required for the calc u l a -tions and documents three, interchangeable, F0RTRA(\! subroutine packages uhich calculate the l a t t i c e sums. Appendix B describes the phonon frequency d i s t r i b u t i o n s of fee and hep RGS and documents several F0RTRAIM program segments uhich determine the d i s t r i b u t i o n s . Appendix C discusses reasonable precision requirements for c a l -culations ta determine the fec-hep cohesive energy difference. 5 CHAPTER II THEORETICAL BACKGROUND Crystals of neon, argon, krypton, xenon, and radon are of special interest i n l a t t i c e dynamics because: 1. The electron clouds of the ground state atoms are t i g h t l y bound, so the atoms are e l e c t r i c a l l y neutral, weakly inte r a c t i n g , and singly occurring. 2. The ground state atoms are nearly spherical, so the i n t e r -atomic forces are e s s e n t i a l l y central, pairwise forces and the cry s t a l s have close packed structures. 3. The atomic masses are large, so dynamical'problems can be treated c l a s s i c a l l y , with quantum effects included as perturba-tions. Because of these simplifying features, the theory of the heavier rare 1-6 gas soli d s - RGS - i s well-developed and widely described. 1 E. R. Dobbs and G. 0. Jones, Rept. Progr. Phys. 20, 516 (1957). Argon, Helium and the Rare Gases, edited by G. A. Cook (Inter-science Publishers,, Inc., New York, 1961). 3 G. K. Horton and J . Id. Leech, Proc. Phys. Soc. 82, 816 (1963). G. L. Pollack, Rev. Mod. Phys.. 36,7it8 (196*0. 5 G. K. Horton, Am. J . Phys. 36, 93 (1968). ^ Rare Gas Solids, edited by M. L. Klein and J . A. Uenables (Academic Press, Inc., New York, 197^). 6 S o l i d helium, which i s ignored i n t h i s t h e s i s , must be treated by d i f f e r e n t methods than the other RGS, because i t s atomic mass i s 7 so small that quantum e f f e c t s are of overwhelming importance. S o l i d radon should e x h i b i t the strongest c l a s s i c a l behavior of the RGS, because i t s atomic mass i s enormous, but i s i t u s u a l l y ignored be-cause i t i s not w e l l - i n v e s t i g a t e d . In p r i n c i p l e , a great deal of valuable information about s o l i d radon should be a v a i l a b l e from the Law of Corresponding S t a t e s , which i s most u s e f u l when atomic masses are l a r g e . II- 1 THE LAW OF CORRESPONDING STATES The equations of state of systems f a l l o w i n g the Law of Corre-sponding States are i d e n t i c a l when expressed i n a p p r o p r i a t e l y reduced p h y s i c a l v a r i a b l e s . An exact b a s i s f o r the Law i s p o s s i b l e provided that the p a r t i c l e s i n t e r a c t v i a c e n t r a l , pairwise p o t e n t i a l s of the form: cpcv-)= ei (V/<jO (2-1) where: i s the separation of the i n t e r a c t i n g p a i r of p a r t i c l e s , <£ and <y are, r e s p e c t i v e l y , c h a r a c t e r i s t i c energy and length parameters f o r each species of p a r t i c l e , and the dimensionless f u n c t i o n 8 9 i s the same f o r a l l substances to which the Law i s to be a p p l i e d . ' The C l a s s i c a l Law of Corresponding States i s expressed by w r i t i n g 7 H. R. Glyde, chapter 8 i n reference. 6. j? 3. De Boer, Physica V*, 139 (194B). 3. De Boer'>and B. S. B l a i s s e , Physica Jj+, 149 (1948). 7 the reduced equation of state i n the form: P*= r f v V * ) , (z.z) uihere r i s a universal function determined by The reduced pressure P , the reduced temperature T , and the reduced s p e c i f i c volume V are defined (by De Boer ) as follows: P** P<T 3/£ (2-3a) KT/a ( 2 .3b) V/*- V / / / V C T 3 , ( 2 . 3 c ) where /C i s Boltzmann's constant and M i s the number of p a r t i c l e s . The c l a s s i c a l form of the Law i s followed closely only by those condensed gases with large molecular weights, because zero point and thermal motion ef f e c t s are ignored. The quantum mechanical general-i z a t i o n of the Law requires the additional (De Boer) parameter: ft ff = yftT <T ' cz-**) where ft i s Planck's constant and M i s the mass of the molecules. The new reduced equation of state i s expressed as: V* - F ( V*, T * /\*), The magnitude of the De Boer parameter determines the strength of quantum e f f e c t s . In the l i m i t as A ~^ O , equation (2-5) reduces to equation (2-2). The occurrence of the De Boer parameter in equation (2-5) spoi l s A * the usefulness of the Law,-because the /\ -dependence of equation a (2-5) i s nonlinear for large ranges of values of A . Since the reduced equation of state must usually be determined empirically, i t may be necessary to consider nearly as many values of ^t*as there are substances to uihich the Lam i s to be applied. For small enough A * ranges of / l , li n e a r interpolation may be successful. For massive enough p a r t i c l e s , the c l a s s i c a l form of the Law may be s a t i s f a c t o r y . Table H-1 l i s t s some quantities pertinent to the Law of Corre-sponding States for the RGS. Experimental values for £ and <J" depend upon the type of data from which they are derived. The values g l i s t e d i n Table II-1 were obtained from data for the gaseous state. Of p a r t i c u l a r interest here are the l a s t three rows, which demon-strate the r e l a t i v e weakness of the zero point motion for small values of A . The l a t t i c e potential energy $ and zero point energy £g were calculated i n terms of a model for which the c l a s s i c a l form of 10 the Law i s exact. It should be noted that the magnitude of the it zero point energy of s o l i d He exceeds that of the potential energy, it indicating the i n s t a b i l i t y of s o l i d He i n the absence of external pressure. Roger Howard, Phys. Letters 29A, 53 (1969). 9 TABLE H-1. Quantities pertinent to the Laui of Corresponding States for the RGS. Unless otherwise stated, the data were obtained from reference 8. Ne Ar Kr Xe heavy element M (amu) 4. CD 20.2 40.0 83.8 131.3 CO £ (x10~ 2 3 J.) 14.03 48.82 165. 230. : 318. cr (S) 2.87 3.07 3.83 4.10 4.40 E/K (K) 10.22 35.6 120. 166. 230. (cm /mole) 14.3 17.7 33.9 42.0 51.5 e/<r3 (atm) 59.3 166. 292. 326. 365. A* 0.379 0.085 0.026 0.014 0.009 -1.379 -0.309 -0.096 -0.052 -0.033 -0. e#A5b -3.6 -3.6 -3.6 -3.6 -3.6 -3.6 Quantities in t h i s column obtained by extrapolating to A =0. b ^ i s the l a t t i c e potential energy and £jz i s the zero point energy. Reference 10. 1D I1 - 2 INTERATOMIC INTERACTIONS The v a l i d i t y of the concept of interaction potential for the RGS i s guaranteed by the a p p l i c a b i l i t y of the Born-Oppenheimer approx-imation and the adiabatic p r i n c i p l e to the ground state of rare gas i 1 atoms. One would wish, of course, to obtain a functional form for the interaction potential from f i r s t p r i n c i p l e s , as has been done 12 for the H 2 molecule. This would be p r o h i b i t i v e l y d i f f i c u l t at present, so one must be content with a more humble sta r t i n g point. One must simply guess what the solution of the electronic problem must be and proceed with that guess u n t i l i t proves unsatisfactory. II - 2 - 1 Lang—Range Interactions The long-range, a t t r a c t i v e part of the interaction potential arises c h i e f l y from the interaction of induced instantaneous multi-pale moments, resulting from rapidly fluctuating electronic charge d i s t r i b u t i o n s in the ground state atoms. These fluctuation i n t e r -actions appear f i r s t i n second-arder perturbation theory as central pair interactions, described by a potential of the form: (2-6) 1 1 . M; Born and H. Huang, Dynamical Theory of Crystal L a t t i c e s (Oxford University Press, London, 1964), chapter IV and appendix VIII. 1 2 Y. Sugiura, Z. Phys. 45, 484 (1927). 11 The r~k term represents dipole-dipole i n t e r a c t i o n s . The remain-ing terms represent higher-order, multipole-multipole interactions, which are neg l i g i b l e in d i l u t e gases. For pairs of atoms separated —Q — 10 by RGS f i r s t neighbor distances, the r~ , r~ terms contribute about 16%, 1.3%,... to the t o t a l interaction energy. Three-body, fluctuation interactions appear i n third-order per-13 14 turbation theory. ' For a t r i p l e t of atoms at the corners of a triangle uith sides r ^ . r ^ r ^ and angles & n the Axilrod-T e l l e r , t r i p l e - d i p o l e potential i s : These interactions contribute roughly 10% to the cohesive energies of the RGS. 1 5 II-2-2 Short-Range Interactions When atoms are in close proximity, t h e i r electronic clouds over-lap and they repel each other - in accordance u i t h the Exclusion P r i n c i p l e . The strongest component of th i s interaction i s central and pairuise. Molecular, beam scattering experiments indicate an 16 17 exponential form for the potential: ' 1 3 B. M. Axilrod and E. T e l l e r , 3. Chem. Phys. 11, 299 (1943). U B. M. Axilrod, 3. Chem. Phys. 19, 719,724 (1951). 1 5 wVGdtze and H. Schmidt, Physik 19j=.» ^ (1966). 16 R. B. Bernstein, i n Molecular Forces (North Holland Publish-ing Co., Amsterdam, 1967), pp. 23-44. 1 7 M. V. Bobetic and 3. A. Barker, Phys. Rev. B2, 4169 (1970). 12 c p R ( v ) = aR(y) e ~ ~ y (2-8) where <3.^ (Y*) i s a slowly varying polynomial i n the interatomic separ-ation. At very close range powerful many-body interactions arise as a 18 result of electron exchange. Jansen and his collaborators have examined single, e f f e c t i v e , gaussian electron exchange i n the RGS and a l k a l i - h a l i d e c r y s t a l s . A detailed account of t h i s interaction mechanism i s beyond the scope of t h i s t h e s i s . Some of the interesting res u l t s are that four-body interactions are weaker than three-body interactions, which are weaker than pairwise interactions, and that the three-body interactions make important contributions to the cohe-sive energy differences of observed and alternate structures of the RGS and a l k a l i - h a l i d e c r y s t a l s - i n the d i r e c t i o n of s t a b i l i z i n g the observed structures. II-2-3 The Central Pair Potential Thus far, t h i s discussion has been r e s t r i c t e d to interactions which dominate at very large and very small interatomic separations. What i s required, however, are functional forms for the potentials which could be used over the entire range of interatomic separations occurring i n the c r y s t a l . Since l i t t l e i s known concerning the i n t e r -action mechanisms that operate at intermediate separations, i t i s 18 L. Jansen, in Modern Quantum Chemistry, Part 2 (Academic Press, Inc., New York, 1965), pp. 239-263. 13 customary to approximate true interaction potentials by a superpo-s i t i o n of long-range and short-range potentials. The simplest and most widely used potentials for rare gas atoms are central pair potentials, consisting of a single short-range, repulsive term and a single long-range,attractive term. The Buckingham potential i s just a l i n e a r combination of the leading terms i n formulas (2-6) and (2-8): _ C. cp(rO=/)<£-~ ~ T & i (2-9) which, after s l i g h t algebraic manipulation, yields the Modified  Buckingham or (exp-6) potentials: The (exp-6) potentials are f a i r l y s a t i s f a c t o r y , because, i n the absence of three-body forces, they s a t i s f y the requirements for the v a l i d i t y of the Law of Corresponding States, and because they give f a i r l y good f i t s to second v i r i a l c o e f f i c i e n t and s o l i d state . spe-19 c i f i c heat data for 12 5«<<15. However, the (exp-6) potentials are not superior to the Mie-Lennard-Jones or (m-6) potentials: C p O ) = ^ - ( ^ ( W - S ( W / . (2-11) Besides giving f a i r l y good f i t s to certain experimental data, 19 Id. G. Schneider and E. Uhalley, J . Chem. Phys. 23, 1644 (1955). 14 the (12-6) potential has important algebraic advantages over the other (m-6) potentials and the (exp-6) potentials. For t h i s reason, the (12-6) potential i s the potential that has been most commonly chosen to represent central pair interactions between rare gas atoms. Although these simple central pair potentials behave generally l i k e the actual central potentials, they have too few parameters to be f l e x i b l e enough to represent the true potentials accurately. 17 20-22 Barker and his collaborators ' have developed twelve parameter generalizations of the (exp-6) potentials for argon and krypton, with the parameters f i t t e d to data for the s o l i d and gaseous states. These potentials, i n conjunction with the A x i l r o d - T e l l e r potential, y i e l d excellent f i t s to experimental data, so they may be useful for i d e n t i f y i n g the defects of the simpler potentials. Figure 2-1 compares the shapes of the potential wells of the Bobetic-Barker potential for krypton and those of (m-6) and (m-B-6); m=12 and 30. The (m-8-6) potentials, , (2-12) and the (m-6) potentials are given the proper long-range behavior and the correct location for the potential minimum, but the depths and curvatures of the simple potentials cannot both be correct for J . A. Barker and A. Pompe, Australian 3. Phys. 21, .1683 (1968). 21 M. V. Bobetic, 3. A. Barker, and M. L. Klein, Phys. Rev. B5, 3185 (1972). 22 3. A. Barker, R. A. Fisher, and R. 0. Watts, Moi. Phys. 21, 657 (1971). 15 any reasonable choice of Wi or Cg'/C^. In figure 2-1 the choice i s cg/c6=Q.2, which i s a b i t too large - U.M^^-S/^ < 0.19. Perhaps, the f i t to the Bobetic-Barker potential could be improved by including more terms in the generalized (m-6) potentials, but i t seems l i k e l y that t h i s procedure would lead to potentials with nearly as many parameters as the Bobetic-Barker potentials. Therefore, unless one i s prepared to use very elaborate potentials, there i s good reason to r e s t r i c t interactions to close neighbors and adopt an appropriate (m-6) p o t e n t i a l . The (12-6) potential i s adopted for the calculations i n t h i s thesis and, i n most cases, interactions are r e s t r i c t e d to f i r s t neighbors. II-2-4 The Total Interaction Potential In p r i n c i p l e , the t o t a l interaction potential for rare gas atoms has contributions from a l l orders of many-body in t e r a c t i o n s . As i n the case of the central pair potentials, the mechanisms for many-body interactions are f a i r l y - w e l l understood for very large and very small interatomic separations, but hardly at a l l otherwise. Since i n t e r -atomic separations in the RGS range from small to large, i t would be desirable to construct an all-range, e f f e c t i v e , many-body potential by superimposing long- and short-range potentials. The Axi l r o d -T e l l e r potential could be used for the long-range part, but there i s presently no candidate suitable for the short-range part. This leaves the A x i l r o d - T e l l e r potential to account for a l l many-body effects in the RGS. This thesis considers only central pair forces. 1G I n t e r a t o m i c S e p a r a t i o n (V/O F i g u r e 2-1. Comparison a f the p o t e n t i a l w e l l s o f ( I T I - D ) , (m-3-S), and B a b e t i c - G a r k e r p o t e n t i a l s f o r k r y p t o n . 17 II-3 FCC AND HCP LATTICES The cohesive energies of fee and hep RGS are nearly identical because the two lattice structures are very closely related. The common building block for both structures i s a hexagonal planar array of rare gas atoms, as illustrated in figure 2-2. Figure 2-2. A portion of a hexagonal planar array used to construct fee and hep lattices. The lattice points are at the intersections of the lines. Perfect fee and hep lattices can be constructed by stacking these planes, parallel ta one another, at equal intervals along the z-axis. To minimize the unoccupied volume, every lattice paint is placed directly above and below the centers of voids in adjacent planes. This allows three types of plane to be defined. Planes of 13 type A are obtained from the reference plane, which has the o r i g i n of the coordinate system at one of i t s l a t t i c e s i t e s , by t r a n s l a t i n g the reference plane along the z - a x i s . Planes of type B are obtained by t r a n s l a t i n g type A planes normal to the z - a x i s , so that the l a t -t i c e s i t e s move i n t o what mere voids of type . Planes of type C are s i m i l a r to type B planes, except that the l a t t i c e s i t e s occupy what were voids of type r . In p r i n c i p l e , any stacking pattern s a t i s f y i n g the above condi-t i o n s would represent the c l o s e s t packing of i d e n t i c a l hard spheres. However, the stacking patterns of i n t e r e s t here are the p e r i o d i c , s t acking patterns, ...ABABAB... and .. .ABCABC.... I f the i n t r a - and i n t e r - p l a n a r f i r s t neighbor separations are equal, these stacking patterns correspond to i d e a l hep and fee, r e s p e c t i v e l y . For any other stacking i n t e r v a l the former stacking pattern r e t a i n s the same symmetry p r o p e r t i e s , so the l a t t i c e remains hep. However, the l a t t e r s t a cking pattern would no longer have cubic symmetry. This case i s not considered i n t h i s t h e s i s because experimental observations r e v e a l that the RGS have fee s t r u c t u r e s . The fee l a t t i c e i s a Bravais l a t t i c e with the symmetry of the point group D^. I t s r e c i p r o c a l l a t t i c e i s body centered cubic, and •v. also has the symmetry of 0^. The hep l a t t i c e has two atoms per u n i t c e l l and the symmetry of the point group fi-^. Each hep s u b l a t t i c e and the r e c i p r o c a l l a t t i c e are simple hexagonal l a t t i c e s - stacking pattern ...AAAA... - with the symmetry of t>gh. L a t t i c e s i t e s are l a b e l l e d (xi, where £ i d e n t i f i e s one of the N u n i t c e l l s and A i d e n t i f i e s one of the 5 s u b l a t t i c e s . The vector 19 separating the l a t t i c e p a i n t s (^)and {/s-)is: uhere and (2-15) 3 f a r any i n t e g e r s •l-^.t A N D -^3* The vectors {<-U}t-„( are the l a t t i c e S generating vectors and the vectors describe the separation of the s u b l a t t i c e s from the reference s u b l a t t i c e . Corresponding to each l a t t i c e i s a r e c i p r o c a l l a t t i c e , which i s a Bravais l a t t i c e , generated by the vectors { 1 s a t i s f y i n g : 0.1 ' b-j = *S,c-j . (2-16) The s i t e s of the r e c i p r o c a l l a t t i c e are 3 "B (k) = 2Z ^ fe/ v (2-17) — f a r any i n t e g e r s k, , hj,. and H3. The vectors { ^ j ^ , , » { ^ ' j ; ^ , and f ^ j ^ - i f o r fee and hep l a t t i c e s are l i s t e d i n Table I I - 2 . The i n t r a p l a n a r f i r s t neighbor separations are a^ , and a^, r e s p e c t i v e l y , and Y = ^>5 3 - i n i d e a l hep. Sketches of fee and hep u n i t c e l l s and the u n i t c e l l s af t h e i r r e c i p r o c a l l a t t i c e s - that i s , t h e i r B r i l l o u i n zones or BZ's - are given i n Figures 2-3 and 2-4. Also i n d i c a t e d i n these diagrams are those parts 20 Table I I - 2 . Fee and hep l a t t i c e and reciprocal l a t t i c e vectors. The intraplanar f i r s t neighbor separations are a- and a., respectively, and IT a %, ~ -^ f in i d e a l hep. FCC HCP f (o I I) a: - ^= < (I o f ) [ ( l o o ) I ( o o r ) R, = o [ ( • i n ) Df the BZ's which, by symmetry, are equivalent to the rest of the BZ's. These are called the i r r e d u c i b l e parts of the BZ's, The fee l a t t i c e vectors i n Table II - 2 correspond to rectangular cartesian coordinates with axes aligned uith the three four-fold axes of the fee structure. In t h i s system the stacking axis for the ' hexagonal planes i s any of the three-fold axes along the body diag-onals, for instance, the (111) d i r e c t i o n . Figure 2-3. Unit c e l l (above) and Brillouin zone (below) For the fee l a t t i c e . The irreducible part of the brillouin zone i s indicated by dotted lines. 9 9 Z • F i g u r e 2-4. U n i t c e l l (above) and B r i l l o u i n zona (belou) f o r the hep l a t t i c e . The i r r e d u c i b l e p a r t of the B r i l l o u i n zone i s i n d i c a t e d by d o t t e d l i n e s . 23 II-4 STRUCTURAL STABILITY OF CRYSTALS Whatever happens during c r y s t a l l i z a t i o n cannot be described rigorously because too many p a r t i c l e s are involved. Therefore, one cannot solve for the observed structures of r e a l c r y s t a l s . One can only show that certain structures would not be stable at a l l , and that the possibly stable structures can be ordered according to t h e i r r e l a t i v e s t a b i l i t i e s . 11-4-1 Absolute S t a b i l i t y A stable structure with every atom in i t s s t a t i c equilibrium position must be stable under small homogeneous deformations, since otherwise a perturbation to the structure could cause i t to change 23 permanently. Born and his collaborators have applied t h i s condi-tion to various cubic and hexagonal l a t t i c e s occupied by atoms which interact via central pair potentials with the general properties expected far interacting rare gas atoms. St a t i c fee and hep structures, i n which every atom has twelve f i r s t neighbors, are stable for a l l reasonable choices of the poten-t i a l function. The s t a t i c body centered cubic structure, i n which every atom has eight f i r s t neighbors, i s stable only for unreasonable choices of the potential function. The s t a t i c simple:cubic and , simple hexagonal structures are not stable for any choice of the potential function unless, possibly, the structures are subject to very high pressures. In these structures each atom has six f i r s t neighbors. 23 ' ~ Born and Huang, reference -11, chapter I I I . 2k L a t t i c e s occupied by rea l p a r t i c l e s are never i n s t a t i c e q u i l -ibrium. Even at absolute zero the p a r t i c l e s make excursions from their mean positions. In order for dynamic l a t t i c e s to be stable, the frequency of every normal mode must be r e a l , or else the assump-tion of time-periodic motion uould f a i l . This matter i s considered in d e t a i l for fee and hep RGS i n chapter IV. Il-k-2 Relative S t a b i l i t y The r e l a t i v e s t a b i l i t i e s of alternative s t a t i c configurations of atoms are determined by the configurational potential energies: the lower the potential energy the greater the s t a b i l i t y . For dynamic structures the same c r i t e r i o n i s used, except that i t i s applied to the thermodynamic potential appropriate to the environment of the system. In t h i s thesis i t i s assumed that the model c r y s t a l s are i n equilibrium at constant temperature T and constant pressure P. The pertinent thermodynamic potential in t h i s case i s the Gibbs free energy G. The equilibrium of a. p a r t i c u l a r configuration i s achieved by : v a r i a t i o n of the unconstrained system parameters to minimize the con-fig u r a t i o n a l Gibbs free energy. The s t a b i l i t y of a p a r t i c u l a r e q u i l -ibrium configuration i s guaranteed i f the value of i t s Gibbs free energy minimizes the Gibbs free energy over the set of possible equilibrium configurations. Unless there i s extensive s t r u c t u r a l reorganization, the struc-t u r a l changes which accompany a change of physical conditions (P,T) 25 can usually be described in terms of very feu unconstrained para-meters. I f an fee c r y s t a l remains fee throughout such a change, the only st r u c t u r a l change, i f surfaces can be ignored, i s in the f i r s t neighbor separation a^. Therefore, i t i s useful to describe the achievement of equilibrium by an fee c r y s t a l i n terms of uniform l a t t i c e expansion, depending upon the single, continuous variable a f . This leads to the equilibrium condition: \ " 3 a f /P,T S i m i l a r l y , since an hep l a t t i c e can change i t s intraplanar f i r s t neighbor separation a^ and i t s interplanar stacking parameter )f, the equilibrium conditions for an hep l a t t i c e would be: ("T^ W ' ^  r- M-r°-I f only fee and hep structures need be considered, the sign of the cohesive energy difference A & ( P , T ) - 6 k c f > | V V i m W -^c,XP>T) determines which i s the stable structure. I f A^>Othe fee structure i s stable. I f A£^«cothe hep structure i s stable. I f A c r - O b o t h structures may coexist in the same specimen. 26 II-4-3 Relative S t a b i l i t y of S t a t i c Structures In s t a t i c l a t t i c e s held together by (12-6) potentials the poten-t i a l energy of each atom i s (2-21) where the summation extends over a l l neighbors of the reference atom, at R(i)*0. I f the structure i s constrained so that only i s o t r o p i c changes may occur, the only unconstrained parameter i s the f i r s t neighbor separation a. The equilibrium condition ~? CL. leads to the r e l a t i o n C 5 (2-22) CK. / 6 s - s Z'lifc) 1 /I (2-23) Hence, the s t a t i c equilibrium potential energy per atom i s 12. (2-24) The dominant contributions to the l a t t i c e sums S are from the f i r s t neighbors, of which there are , say. Therefore, i f inter-actions are r e s t r i c t e d to f i r s t neighbors : «-E b u y . V, (2-25) Clearly, the number of f i r s t neighbors strongly influences the r e l a t i v e s t a b i l i t i e s of structures - see the second paragraph of section II-4-1. The structures which maximize the number of f i r s t 27 neighbors are the clos e packed s t r u c t u r e s , f o r which N (=12. I t i s also c l e a r that great d i f f i c u l t y i s encountered i n any attempt to i d i s t i n g u i s h the r e l a t i v e s t a b i l i t i e s of various close packed s t r u c -t u r e s . The p o s s i b i l i t y of changing the stacking parameter "if does not a f f e c t equation (2-25), since close packed s t r u c t u r e s with only f i r s t neighbor i n t e r a c t i o n s r e t a i n i d e a l s t a c k i n g . The e f f e c t s of i n t e r a c t i o n s i n v o l v i n g d i s t a n t neighbors are to make <X< <f by about 3% and J F u p , wv'tw. < ? f c C ( D V roughly .01%. Allowance f o r v a r i a t i o n s i n the hep stacking parameter leads to a value of ~f which i s .01% smaller than i n i d e a l s t a c k i n g , but t h i s •T- 7 2 k reduces the value of J ^ ^ p , vwix °y only one part i n 10 . Roger Howard, Phys. L e t t e r s 32A, 37 (1970). 28 II-5 THE QUASIHARMONIC APPROXIMATION The quasiharmonic approximation - QHA - i s a very simple and u s e f u l improvement upon the t r a d i t i o n a l form o f the harmonic approx-11 25-27 imation of l a t t i c e dynamics. ' I t r e t a i n s the same mathemat-i c a l formalism, but allows the interatomic force constants to be evaluated i n terms of the l a t t i c e of mean p o s i t i o n s , rather than the s t a t i c l a t t i c e . This almost t r i v i a l m o d i f i c a t i o n permits l a t t i c e expansion, because the phonon frequencies become dependent upon the l a t t i c e parameters and, hence, become capable of s h i f t i n g the minimum of the Gibbs f u n c t i o n . II-5-1 Quasiharmonic Equations of Motion I1) The c l a s s i c a l equation of motion of atom [XI at r ( x l t ) * R C a K i W ^ ) ( Z. Z 6 ) I S \A UIAIO = - ^ (2-27) PA i s the atomic mass, assumed l a r g e , and $ i s the l a t t i c e where p o t e n t i a l energy, which has the form: 25 A. A. Maradudin, E. Id. M o n t r a l l , G. H. Weiss, and I . P. Ipatova, Theory of L a t t i c e Dynamics i n the Quasiharmonic Approxi-mation (Academic Press, Inc., New York, 1971). G. L e i b f r i e d and W. Ludwig, S o l i d State Phys. 12, 275 (1961), 27 G. L e i b f r i e d , i n L a t t i c e Dynamics, edited by R. F. W a l l i s (Pergamon Press, Inc., New York, 1965), pp. 237-45. 29 (2-28) f o r c e n t r a l p a i r f o r c e s . The QHA assumes small atomic excursions £ £ ^ A [ t ) a n d expands $ i n a Taylor s e r i e s about the mean l a t t i c e p o s i t i o n s , up to second-order i n the atomic excursions. Upon the assumption of time-periodic s o l u t i o n s (2-29) the equations of motion of the l a t t i c e can be w r i t t e n as a 3si\lx3siM eigenvalue problem: ^ i S ' B ^ ' (2-30) Since macroscopic c r y s t a l s have an enormous number of atoms, the dynamical matrix ^ I /i*=2 cannot be d i a g o n a l i z e d , i n general. However, i t can be used to obtain expressions f o r the even moments of the frequency d i s t r i b u t i o n : / " 3 5 W fr- ' r V 2: (2-31) These even moments are exact i n terms of the QHA. The equations of motion can be s i m p l i f i e d by i n t r o d u c i n g c y c l i c boundary co n d i t i o n s ~ U " ( A ) < S . C 2 . 3 2 ) 30 The s i m p l i f i e d equations of motion are where , ; 3 7 T ^ . £Y<0,>*O i s the Fourier transformed dynamical matrix. Equations (2-33) represent N d i f f e r e n t 3sx3s eigenvalue prob-lems, each corresponding to a d i f f e r e n t wave vector r\ . The eigen-values of the Fourier transformed dynamical matrices are the squared frequencies of phonons which da not i n t e r a c t with ane another, and which belong to one of the 3s p o l a r i z a t i o n branches. II-5-2 Phonon Frequency D i s t r i b u t i o n s The frequency d i s t r i b u t i o n f u n c t i o n f o r p o l a r i z a t i o n branch j is (2-35) where i s the number of frequencies i n branch j l e s s than or equal to ^O . These d i s t r i b u t i o n functions s a t i s f y the normalization convention: o . , •Sl3 - - ' ' (2-36) The t o t a l frequency d i s t r i b u t i o n f u n c t i o n i s 3 S „ 31 The q u a n t i t i e s j=1»2,...,3s and &t*<u( are the maximum frequencies i n each c f the branches and f a r the t o t a l d i s t r i b u t i o n . Since macroscopic c r y s t a l s contain too many atoms f o r i t to be f e a s i b l e to examine a l l of the phonons i n the c r y s t a l , i t i s custom-ary to construct frequency d i s t r i b u t i o n s by sampling a large number 28 of phonons. This approach i s c a l l e d the root sampling method. I t i s described i n more d e t a i l i n appendix B. II-5-3 Quasiharmonic Gibbs Function The quasiharmonic Gibbs function i s the Gibbs funct i o n f o r a system of independent harmonic o s c i l l a t o r s : where i s the l a t t i c e p o t e n t i a l energy with every atom i n i t s mean p o s i t i o n and M i s the c r y s t a l volume. At zero temperature the temper-ature term reduces to the l a t t i c e zero point energy The quasiharmonic Gibbs func t i o n i s a function of l a t t i c e para-meters. These are chosen to minimize the Gibbs f u n c t i o n , when the model system i s i n e q u i l i b r i u m at constant temperature and pressure. The e q u i l i b r i u m c o n d i t i o n s f o r fee and hep l a t t i c e s i n t h i s t h e s i s are equations (2-18) and (2-19), with the Gibbs funct i o n given by equation (2-38). 2 8 G. G i l a t , J . Comp. Phys. 10, 432 (1972). CHAPTER I I I 32 DESCRIPTION OF CALCULATIONS This chapter presents a model of the RGS and describes c a l c u -l a t i o n s based on the model. The r e s u l t s of the described c a l c u l a -t i o n s are discussed i n chapters IV and V. I I I - 1 MODEL The model f o r the c a l c u l a t i o n s assumes: 1. Rare gas atoms i n t e r a c t s o l e l y v i a Mie-Lennard-Jones (12-6) p o t e n t i a l s . 2. The l a t t i c e s are s t r u c t u r a l l y p e r f e c t , i s o t o p i c a l l y pure, and i n v a r i a n t under t r a n s l a t i o n s which carry any l a t t i c e s i t e i n t o any other l a t t i c e s i t e . 3. The l a t t i c e s are permitted uniform expansion. For symmetry reasons the fee l a t t i c e must expand i s o t r o p i c a l l y , so only i t s f i r s t neighbor separation may change i n an expansion: The hep l a t t i c e i s allowed a x i a l l y a n i s o t r o p i c expansion. I t may vary both i t s i n t r a p l a n a r f i r s t neighbor separation and i t s stacking parameter: (3-1) (3-2) 33 The expansion parameters £ f , , and £ a r e measured uiith respect to l a t t i c e parameters i n s t a t i c , i d e a l l y stacked reference l a t t i c e s . k. L a t t i c e motion i s i n accordance u i t h the QHA. 5. Thermodynamic e q u i l i b r i u m i s achieved at constant temperature and pressure, u i t h l a t t i c e parameters minimizing the quasihar-monic Gibbs funct i o n (equation (2-38)). I I I - 2 REDUCTION OF VARIABLES The assumption of the QHA means that the equations of st a t e f o r the RGS f o l l o u a quantum l a u of corresponding s t a t e s . Although t h i s f a c t does not g r e a t l y s i m p l i f y c a l c u l a t i o n s of e q u i l i b r i u m l a t t i c e parameters, i t i n d i c a t e s the value of uorking i n terms of reduced v a r i a b l e s . The reduced length parameter of the (12-6) p o t e n t i a l i s uhere l + S ' - ' (3-3) b ° = ( £ ) = 5°"'VV (3-fc) and the l a t t i c e sums 5 ( 2 ' f o r fee and hep l a t t i c e s are described i n appendix A. In equation (3-4) the sums f o r the hep l a t t i c e have S l ^ - 0 . The values of b_ depend upon the range of i n t e r a c t i o n : bo Jo =1 f o r f i r s t neighbor i n t e r a c t i o n s . (3-5) >1 f o r longer range i n t e r a c t i o n s . 3k The reduced l a t t i c e vector the reduced r e c i p r o c a l l a t t i c e vector the reduced wave vector ^ , and the reduced phonon frequency are defined, r e s p e c t i v e l y , as f o l l o w s : t ( k ) = ^ ' g a ) ( 3 - 7 ) - HlVM ^  , ( 3 - 9 ) There i s a s l i g h t a r b i t r a r i n e s s i n the d e f i n i t i o n of the reduced frequencies. The numerical constant i n ( 3 - 9 ) was chosen to give con-venient magnitudes to the reduced frequencies. For f i r s t neighbor i n t e r a c t i o n s j d e f i n i t i o n ( 3 - 9 ) gives f o r both fee and i d e a l hep reference l a t t i c e s . The reduced ( F o u r i e r transformed) dynamical matrices are defined C ^ ( / J = | 6 £ V«p^«h (3-11) In terms of (12-6) p o t e n t i a l s the e x p l i c i t forms are: (3-12) 35 The phase f a c t o r f o r fee i s (3-13) and f o r hep i t i s - L s ' m (i)J f.(3-U) Expressions (3-13) and (3-14) are i n v a r i a n t under changes i n l a t t i c e parameters s a t i s f y i n g (3-1) and (3-2). I I I - 3 CALCULATION OF PHONON FREQUENCIES D i s t r i b u t i o n s of reduced phonon frequencies depend only upon the p a r t i c u l a r l a t t i c e of i n t e r e s t and the values of i t s l a t t i c e para-meters. Since l a t t i c e parameter values are a r b i t r a r y , up to the r point at which e q u i l i b r i u m c o n d i t i o n s are imposed, i t i s p o s s i b l e to examine the dependence of reduced frequency d i s t r i b u t i o n s upon l a t t i c e parameters, independent of any p a r t i c u l a r RGS and any p a r t i c u l a r environmental c o n d i t i o n s (P,T). This matter i s discussed i n d e t a i l i n appendix B and chapter IV. Upon the i m p o s i t i o n of e q u i l i b r i u m c o n d i t i o n s , l a t t i c e para-meters and l a t t i c e motion become r e l a t e d to each other and to p a r t i c -u l a r choices of £ , tf", M, P, and T. This interdependence would represent a serious obstacle to the use of the QHA, i f one had no idea of what would be reasonable ranges of l a t t i c e parameters, since e q u i l i b r i u m l a t t i c e parameters and phonon frequency d i s t r i b u t i o n s 36 must be determined r e c u r s i v e l y , with a new frequency d i s t r i b u t i o n c a l c u l a t e d at each i t e r a t i o n . At the root of the problem i s the d i f f i c u l t y of determining the frequency d i s t r i b u t i o n s a c c u r a t e l y . (See appendix B.) L u c k i l y , even with atoms as l i g h t as neon atoms, l a t t i c e para-meters at low temperatures and pressures are w i t h i n a few percent of t h e i r values i n the reference, s t a t i c l a t t i c e s . This information can be used to s i m p l i f y e q u i l i b r i u m c a l c u l a t i o n s g r e a t l y . The dependence of phonon frequencies upon such narrow ranges of l a t t i c e parameters can be s a t i s f a c t o r i l y described by i n t e r p o l a t i o n formulas, which can be constructed from phonon frequencies corresponding to r e l a t i v e l y few sets of l a t t i c e parameters. Once such i n t e r p o l a t i o n formulas have been set up, i t i s a f a i r l y simple matter to determine phonon frequency d i s t r i b u t i o n s f o r any set of l a t t i c e parameters occ u r r i n g i n the r e c u r s i v e s o l u t i o n of the e q u i l i b r i u m c o n d i t i o n s . I I I - 4 CRUDE MODEL To introduce the nature of the QHA and i t s i m p l i c a t i o n s f o r the RGS, some crude estimates of e f f e c t s are obtained here from a s i m p l i -f i e d version of the model described i n s e c t i o n I I I - 1 . The s i m p l i f i -c a t i o n s are the following.: 1. I n t e r a c t i o n s are r e s t r i c t e d to f i r s t neighbors. 2. The model c r y s t a l s are i n e q u i l i b r i u m at zero temperature, so the temperature dependent term of the Gibbs function reduces to the l a t t i c e zero point energy. This i s p r o p o r t i o n a l to the 37 mean frequency, but the rDot-mean-square frequency i s used i n s t e a d . 3. A l l q u a n t i t i e s depending upon the expansion parameters S and S are expanded about S = S =0 to lowest-order c o r r e c t i o n s . As mentioned, the fee l a t t i c e must always have i d e a l s t acking to r e t a i n cubic symmetry. This point i s ignored, f o r a moment, since p e r m i t t i n g nonideal stacking i n fee y i e l d s the same Gibbs funct i o n f o r both fee and hep. The l a t t i c e p o t e n t i a l energy i s j i ! ! * [ S°''*(Q)/C ns) ,x - a. t*S)i} § o [ h 3 6 Z * ~ ^ S S ' - 8 S ' * ] . (3-15) The zero point energy i s ^ n 7 ) ^ The a p p l i e d pressure term i s , (3-17) With these expressions the Gibbs function can be w r i t t e n as The l a t t i c e sums o (.£; are described i n appendix A. 38 II1-4-1 Pressure of the Zero Point Motion The zero point motion depends upon the l a t t i c e expansion para-meters S and . This dependence gives r i s e to a zero point pres-sure, which causes the l a t t i c e s to expand. The pressure P^ required to r e s t r a i n the l a t t i c e s to t h e i r s t a t i c e q u i l i b r i u m l a t t i c e para-meters, that i s S = S =0, i s r* ^ V lo (3-19) Rough values of P^ f o r the RGS are l i s t e d i n Table I I I - 1 , with values o f A , J i o b t a i n e d from Table I I - 1 . III-4-2 Zero Point Expansion The zero point l a t t i c e parameters i n t h i s crude model are ob-tained by s o l v i n g the f o l l o w i n g equations, which minimize equation (3-18): , ,* The s o l u t i o n i s (3-21) The f a c t that the zero point stacking parameter remains the i d e a l s t acking parameter, means th a t , i n t h i s model, the hep l a t t i c e does not take advantage of the p o s s i b i l i t y f o r a n i s o t r o p i c expansion, and t h a t , i n a more accurate model, i d e a l s t acking i s l i k e l y a good f i r s t approximation. I t also means that equations (3-21) repre-39 sent the zero point l a t t i c e parameters f o r the fee l a t t i c e , and, i n f a c t , any close packed s t r u c t u r e . This i l l u s t r a t e s the d i f f i c u l t y of ex p l a i n i n g the observed s t a b i l i t y of only fee RGS. At vanishing pressure, — 0 . 56 , compared with the value 29 0.565 obtained by Broun from a s i m i l a r model. These values of S are l i s t e d i n Table I I I - 1 f o r the RGS. Also l i s t e d are the co r r e -sponding r e l a t i v e increases i n $ , £^ , and V. Table I I I - 1 . Zero point expansion p r o p e r t i e s of the RGS i n terms of the crude model of section I I I - 4 . ELEMENT P z (atm) % increase i n : § £* V ^He 2600 0.21 130. - 2 0 0 . 77 . Ne 1600 0.047 8 . - 5 0 . 15. Ar 860 0.015 0.8 - 1 5 . 5 . Hr 530 0.008 0.2 - a . 2.6 Xe 380 0.005 0.08 - 5 . 1.7 The numbers l i s t e d i n Table I I I - 1 i n d i c a t e : U 1. He cannot survive i t s zero point expansion, unless r e s t r a i n e d by pressure, because i t s zero paint motion becomes unstable before the expansion i s complete. 2. The zero point expansion of s o l i d Ne halves i t s zero point energy, suggesting that the QHA i s i n a p p l i c a b l e f o r such small 3. S. Broun, 3. Chem. Phys. 50, 5229 ( 1 9 6 9 ) . 40 atomic masses. 3. The zero point expansions of s o l i d Ar, Hr, and Xe are r e l a -t i v e l y s l i g h t , the zero point energies being a f f e c t e d much more str o n g l y than the p o t e n t i a l energies. These observations are confirmed by the r e s u l t s of s i m i l a r c a l c u l a t i o n s based upon the f u l l model described i n sec t i o n I I I - 1 . In f a c t , t h i s crude model turns out to be remarkably good. See chapters IV and V. 41 CHAPTER Iv PHONON FREQUENCY DISTRIBUTIONS AS FUNCTIONS OF LATTICE PARAMETERS This chapter i s concerned with the e f f e c t s of quasiharmonic l a t t i c e expansion upon the phonon frequency d i s t r i b u t i o n s of fee and hep l a t t i c e s held together by (12-6) p o t e n t i a l s . Many d i s t r i b u t i o n s , corresponding to various fee and hep l a t t i c e parameters, have been c a l c u l a t e d and are discussed. At t h i s stage, no attempt i s made to associate the c a l c u l a t e d d i s t r i b u t i o n s with atomic species or p h y s i c a l c o n d i t i o n s (P,T). Tt i s assumed only t h a t , f o r each atomic species, some set of p h y s i c a l c o n d i t i o n s could produce l a t t i c e parameters with the values chosen. This assumption may imply negative and a n i s o t r o p i c pressures, p a r t i c -u l a r l y f o r hep. The reader, f i n d i n g such c o n d i t i o n s somewhat u n r e a l -i s t i c , may choose to regard the a n a l y s i s i n t h i s chapter as a mathe-mat i c a l e x e r c i s e . P h y s i c a l l y reasonable ranges of l a t t i c e parameters are determined i n chapter V. The reference l a t t i c e s i n t h i s d i s c u s s i o n are s t a t i c , i d e a l l y stacked l a t t i c e s - those f o r which £p = Sk,= ^=0. Phonon frequency d i s t r i b u t i o n s f o r l a t t i c e s with the reference l a t t i c e parameters are described i n appendix B. L a t t i c e parameters corresponding to high compression of the reference l a t t i c e s are avoided because many-body i n t e r a c t i o n s , which are ignored here, would be powerful i n such l a t t i c e s . The l a t t i c e parameters that are considered range from very s l i g h t to c a t a s t r o p h i c a l l y large expansions of the reference l a t t i c e s . 42 The phonon frequency d i s t r i b u t i o n s discussed i n t h i s chapter represent roughly 3x10"' phonon modes and correspond to f i r s t neighbor i n t e r a c t i o n s . C a l c u l a t i o n s u i t h more d i s t a n t neighbor i n t e r a c t i o n s i n d i c a t e that the range of i n t e r a c t i o n a f f e c t s the d i s t r i b u t i o n s mostly by a uniform s c a l i n g of the frequencies, at l e a s t f o r s l i g h t expansions of the reference l a t t i c e s . The r e s t r i c t i o n to f i r s t neighbor i n t e r a c t i o n s i n t h i s chapter i s the r e s u l t of economic consider a t i o n s - see-appendices A and B. IV-1 SLIGHT EXPANSION De Uette and Nijboer have shown that very small changes i n the f i r s t neighbor separation i n an fee l a t t i c e with (12-6) i n t e r a c t i o n s produce s t r i k i n g changes i n i t s phonon frequency d i s t r i b u t i o n , i n the QHA.3'~'~32 S i m i l a r c a l c u l a t i o n s f o r t h i s t h e s i s v e r i f y t h i s r e s u l t f o r fee and hep l a t t i c e s . However, i t i s not necessary to perform elaborate c a l c u l a t i o n s to a r r i v e at the same con c l u s i o n . The crude model of sect i o n I I I - 4 p r e d i c t s a 10% change i n <c«s> f o r 1% changes i n the average f i r s t neighbor separations. The regime of s l i g h t expansion f o r the RGS corresponds to l a t -t i c e parameters w i t h i n about 1% of t h e i r values i n the reference l a t t i c e s . Idithin such narrow ranges of l a t t i c e parameters, the 3 0 B. R. A. Nijboer and F. U. De Uette, Phys. L e t t e r s _17, 256 (1965). 3 1 F. U. De Uette and B. R. A. Ni j b o e r , Phys. L e t t e r s 18, 19 (1965). 32 F. U. De Uette, L. H. Fowler, and B. R. A. Ni j b o e r , Physica 54, 292 (1971). 43 phonon frequency d i s t r i b u t i o n s experience mainly o v e r a l l e f f e c t s , since d i f f e r e n t i a l e f f e c t s on i n d i v i d u a l phonon frequencies are r e l a t i v e l y quite weak. This f a c t i s demonstrated i n Table IV-1, which l i s t s i n t e r p o l a t i o n formula c o e f f i c i e n t s f o r a few moments of fee and hep phonon frequency d i s t r i b u t i o n s . The i n t e r p o l a t i o n formulas are of the form: (4-1) and the c o e f f i c i e n t s i n Table IV-1 were determined from c a l c u l a t e d d i s t r i b u t i o n s f o r which ISf|<1Cf 2 f IS^I-SIO" 2, and I&V | < 10" 2. The values of the c o e f f i c i e n t s , except f Q , are l i t t l e a f f e c t e d i f i n t e r -a c t i o n s are extended beyond f i r s t neighbors. The f a c t that the c o e f f i c i e n t s CL i n the i n t e r p o l a t i o n formulas f o r are nearly p r o p o r t i o n a l to & i m p l i e s that d i f f e r e n t i a l e f f e c t s on phonon frequencies,for s l i g h t expansionsjare weak, and that the e f f e c t s on the d i s t r i b u t i o n s can nearly be eliminated by uniform s c a l i n g , subject to the normalization c o n d i t i o n f a r the d i s t -r i b u t i o n functions - equation (2-37): (4-2) where the s c a l i n g f a c t o r i s Table IU-1. C o e f f i c i e n t s f o r the i n t e r p o l a t i o n formulas (4-1) f o r various moments of fee and hep phonon frequency d i s t r i b u t i o n s . The c o e f f i c i e n t s mere determined f o r s l i g h t l y expanded l a t t i c e s held together by f i r s t neighbor (12-6) i n t e r a c t i o n s f L a t t i c e f0 a b c d e hep 0.272 11.6 7.72 0.326 1.08 -6. fee 0.270 18.3 19.3 hep fee 6.00 6.-00 -12.2 -11.0 -8.50 -4.27 0.418 -18.5 -20. <<£> hep fee 4.09 4.09 -11.9 -10.2 -4.08 -1.7 0.327 -0.935 -2.4 hep fee 4.24 4.24 -12.0 -10.5 -4.15 -3.76 0.326 -1.28 -2.5 hep fee 18.0 18.0 -24.2 -20.9 -10.1 -8.76 0.325 -1.92 -6. hep fee 83.6 83.8 -36.8 -40.0 -16.5 -15.4 0.322 -2.94 -9. hep fee 405. 405. -49.9 -43.4 -22.1 -21.4 0.317 -3.75 -11. 45 The i n t e r p o l a t i o n formulas (4-1) are u s e f u l only f o r s l i g h t l y expanded l a t t i c e s . 'Linen l a t t i c e parameters deviate from t h e i r reference values by more than about 1%, d i f f e r e n t i a l e f f e c t s become noti c e a b l e and the s c a l i n g procedure (4-2) i s no longer capable of accounting f o r the changes i n the frequency d i s t r i b u t i o n s . The s c a l i n g procedure i s s t i l l u s e f u l , though, since i t provides a means of representing a l l d i s t r i b u t i o n s as graphs u i t h the same h o r i z o n t a l dimension and nearly the same v e r t i c a l dimension. This makes d i f f e r -e n t i a l e f f e c t s on i n d i v i d u a l frequencies much more obvious than i n graphs of unsealed d i s t r i b u t i o n s . Figures 4-1a,b, 4-2a-c, 4-3a,b, and 4-4a,b i n d i c a t e that t h i s s c a l i n g procedure el i m i n a t e s most of the changes i n d i s t r i b u t i o n s of s l i g h t l y expanded l a t t i c e s . The subsequent d i s t r i b u t i o n s i n these f i g u r e s i n d i c a t e the usefulness of the s c a l i n g procedure i n d i s p l a y i n g d i f f e r e n t i a l expansion e f f e c t s i n more e x t e n s i v e l y expanded l a t t i c e s . Before proceeding to discuss the frequency d i s t r i b u t i o n s of -'< l a t t i c e s that are more than s l i g h t l y expanded, some comments should be made on the i n t e r p o l a t i o n c o e f f i c i e n t s i n Table IV/— 1 • F i r s t , i t i s noted that the c o e f f i c i e n t s <^  tend to have s l i g h t l y greater mag-nitudes f o r hep than f o r fee. This suggests that expansion e f f e c t s i n hep d i s t r i b u t i o n s are sDmeuhat stronger than f o r fee. This ob-serv a t i o n i s borne out i n the remainder of t h i s chapter. Second, the c o e f f i c i e n t s C f o r hep d i s t r i b u t i o n s are nearly constant and agree u i t h the p r e d i c t i o n s of the crude model of sec t i o n I I I - 4 -Thi r d , the quadratic c o e f f i c i e n t s - ^ , andC- do not behave nearly as s e n s i b l y as do the l i n e a r c o e f f i c i e n t s - A- and C- . Fourth, the 46 v a r i a t i o n s of the mean frequencies are f a i r l y wall-represented by the v a r i a t i o n s of the root-mean-square frequencies, so the zero point energy approximation i n the crude model of s e c t i o n I I I - 4 i s a f a i r l y good approximation, i n terms of the more elaborate model described i n s e c t i o n I I I - 1 . IU-2 MORE EXTENSIVE EXPANSION The changes in scaled frequency d i s t r i b u t i o n s which accompany expansions beyond the regime of s l i g h t expansion cause the scaled fee and hep d i s t r i b u t i o n s to lose t h e i r s i m i l a r i t i e s to the reference d i s t r i b u t i o n s , and to each other. For i n c r e a s i n g S.r the scaled fee d i s t r i b u t i o n s h i f t s toward the high frequency c u t o f f . The narrow, high frequency peak, representing the l o n g i t u d i n a l branch, hardly changes p o s i t i o n , but i t becomes much narrower and t a l l e r . The broad, low frequency peak, representing the transverse branches, also becomes narrower and t a l l e r . I t s h i f t s appreciably toward the c u t o f f . See Figure 4-1. The s h i f t of the fee d i s t r i b u t i o n toward r e l a t i v e l y higher frequencies i n d i c a t e s a weakening of long wavelength motion r e l a t i v e to the short wavelength motion. As the s h i f t progresses, the group 30 o v e l o c i t i e s of long wavelength phonons tend to zero. Near o^=.10 the two peaks of the scaled d i s t r i b u t i o n merge i n t o a s i n g l e peak j u s t below ^ ° m a x ' At £>j!=.109 the group v e l o c i t i e s of the longest wavelength, l o n g i t u d i n a l phonon modes vanish. Further expansion causes the longest wavelength, l o n g i t u d i n a l mode frequencies to go imaginary and the scaled d i s t r i b u t i o n of r e a l frequencies to topple 47 0 X 4 H Figure 4-1. Scaled phonon Frequency d i s t r i b u t i o n s For Fee l a t t i c e s bound by F i r s t neighbor (12-6) i n t e r a c t i o n s : a) b ) % = . 0 1 f c) Sf=.Q2, d) %=.D4, e) £f = .D6, F) $c=.0S, g) S.c=.ia, h) S'f =.12, and i ) the v a r i a t i o n oF the mean, roo mean-square, and maximum Frequencies with £ p . or cn 50 52 over backwards, g i v i n g greater representation to r e l a t i v e l y lower frequencies. The turning point f o r t h i s rather dramatic change, E>£ = .109, i s the i n f l e c t i o n point of the (12-6) p o t e n t i a l and c o r r e -sponds to the l i m i t of dynamical s t a b i l i t y f o r the quasiharmonic fee l a t t i c e . This r e s u l t has been obtained p r e v i o u s l y by De Uette and N i j b o e r . ' Expansion e f f e c t s i n the scaled hep d i s t r i b u t i o n are more complicated because the hep l a t t i c e i s described by two l a t t i c e para-meters, rather than one, and i t s frequency d i s t r i b u t i o n i s composed of s i x , rather than three, p o l a r i z a t i o n branches. Figure U-2 shows how the scaled d i s t r i b u t i o n of an i d e a l hep l a t t i c e changes with S^. C l e a r l y , i s o t r o p i c expansion does not a f f e c t hep as i t does fee. The s h i f t i n the d i s t r i b u t i o n i s away from the high frequency c u t o f f , toward mid-range frequencies. The s h i f t does not proceed f a r enough to be as dramatic as i n the fee case, because imaginary frequencies appear for =.036, one-third of the c r i t i c a l value i n fee. Uhere-as the quasiharmonic fee l a t t i c e becomes unstable because of long wavelength, l o n g i t u d i n a l , a c o u s t i c modes, the quasiharmonic i d e a l hep l a t t i c e becomes unstable because of short wavelength, transverse, o p t i c modes with wave vectors (0,0,k ) on the BZ surface. 53 Figure 4-2. Scaled phonon frequency d i s t r i b u t i o n s f o r hep l a t t i c e s bound by f i r s t neighbor (12-6) i n t e r a c t i o n s . The d i s t r i b u t i o n s represent i s o t r o p i c expansion u i t h S ^ = 0 : a) S/\=-.01, b) 8"K=0., C) ^=.D1, d) S^ = .04, e) $^=.07, and f ) the v a r i a t i o n of the mean, root-mean-square, and maximum frequencies u i t h ^j^. T — — : 1 1 1 r J I I I L U3 in j - m c\j <- . o U l U l 57 Figure 4-3 shows haw the scaled hep d i s t r i b u t i o n changes i n a x i a l expansions - i n c r e a s i n g , constant. As i n i s o t r o p i c expansion the s h i f t i s toward lower frequencies. Far S^=0, as i n Figure 4-3, the frequencies remain r e a l up to .066, beyond which the transverse, o p t i c modes mentioned i n the preceding paragraph became imaginary. A x i a l expansion i s d i f f e r e n t from i s o t r o p i c expansion i n that only those force constants which couple atoms i n d i f f e r e n t hexagonal planes are a f f e c t e d . ; As a r e s u l t , o n e-third of the frequencies are af f e c t e d much more than the other twa-thirds. This can be seen i n Figure 4-3 by f o l l o w i n g the changes i n the scaled d i s t r i b u t i o n f a r beyond the l i m i t of quasiharmonic s t a b i l i t y . For S^=«09, there are two very t a l l and narrow peaks near C O /3. As O L increases f u r t h e r , these peaks move to the l e f t and max lose height as more and more frequencies vanish and become imaginary. Meanwhile, the r e s t of the d i s t r i b u t i o n i s hardly a f f e c t e d . For S^.13 roughly 16% of the frequencies are imaginary, This i s nearly one e n t i r e p o l a r i z a t i o n branch. A second branch disappears from the d i s t r i b u t i o n by the time the expansion reaches S^^.17. The d i s t r i -bution of frequencies that remain r e a l r i s e s l i n e a r l y from <^=0, 25 i n d i c a t i n g two dimensional behavior. The O K =.17 d i s t r i b u t i o n i s not the d i s t r i b u t i o n f o r i s o l a t e d hexagonal planes, however, since f u r t h e r increases i n 8 ^ lead to changes i n the d i s t r i b u t i o n . Nor can the d i s t r i b u t i o n f o r i s o l a t e d planes be obtained from the program segments described i n appendix B j u s t by s e t t i n g very l a r g e , since then the low frequency l i n e a r 58 6 r 1 1 ' 1 L 5 J 1 2 3 4 5 60 Figure 4-3. Scaled phonon frequency d i s t r i b u t i o n s f o r hep l a t t i c e s bound by f i r s t neighbor (12-S) i n t e r a c t i o n s . The d i s t r i b u t i o n s represent a x i a l expansion u i t h £^=0: a) £^=0, b) $K=.01, c) Sv[=.D3, d) =.0G, e) S\'=.09, f ) ^ ( = .13, g) ^ = - 1 7 , h) $^=.205, i ) $^=.363, j ) S|/=2.f and k) the v a r i a t i o n of the mean, raat-maan-square, and maximum frequancias u i t h . 6 3 H f ) ^ k = D . 5^=.13 HCP UNSTABLE 7 i 1 1 - i r 1 r 2 3 4 X X. 8k =.363 HCP UNSTABLE -XI V J 6 0 ~> r 2 3 64 behavior of the d i s t r i b u t i o n becomes obscured by spikes due to numer-i c a l e r r o r i n d i a g o n a l i z i n g the dynamical matrices - see Figure 4-3j. Figure 4-4 gives four examples of how the scaled hep d i s t r i b u t i o n changes when both l a t t i c e parameters deviate from t h e i r reference values. A d e t a i l e d d i s c u s s i o n of such mixed expansion e f f e c t s i s not given here, since such a d i s c u s s i o n would e n t a i l c o n s i d e r a t i o n of a lar g e number of a r b i t r a r i l y chosen l a t t i c e parameter s e t s , most of which uould not be a t t a i n a b l e using current experimental methods. Instead, two general observations are mentioned. F i r s t , the e f f e c t s of i s o t r o p i c and a x i a l expansion, discussed above, can be p a r t i a l l y reversed by reversing the signs o f S ^ a n d S " ^ . Therefore, i t : i s p o s s i b l e to compensate, to some extent, f o r i s o t r o p i c expansion e f f e c t s by i n c l u d i n g a x i a l compressions, or v i c e versa. This compensation does not occur i f e i t h e r l a t t i c e parameter deviates from i t s reference value by more than a few percent. Second, ho matter how the l a t t i c e parameters are v a r i e d , as long as the v a r i a t i o n s increase the average interatomic spacing, the f i r s t phonon frequencies to vanish correspond to transverse, o p t i c modes with wave vectors (0,0,k z) on the BZ surface. The curve along which these frequencies vanish i s nearly l i n e a r , f o r -.05 5 o|^- .10, and i s f a i r l y w e l l approximated by Figure 4-5 shows t h i s l i n e . The opposing arrow heads enclose regions, determined from d i s p e r s i o n curves, i n which the frequencies vanish. The l i n e , S \ + .6 S ^ = .05, i s a f i r s t approximation to 1 (4-4), 1 2 3 ^ 4 5 6 Figure 4-4. Scaled phancn Frequency d i s t r i b u t i o n s For hep l a t t i c e s bound by F i r s t neighbor (12-6) i n t e r a c t i o n s . The d i s t r i b u t i o n s represent mixed expansion: a) S^ = .GQ58, SK. =-.0018, b) £\=.01S4, &k =-.0049, c) S"k=.02, S\.'=-.01, and d) Sk = .QG, S\[ =-.02. LO 63 as d e t e r m i n e d from f r e q u e n c y d i s t r i b u t i o n c a l c u l a t i o n s which i n c l u d e d msny phonon modes whose f r e q u e n c i e s remain veal an bat h s i d e s D f the i n s t a b i l i t y l i n e . How f a r balow the l i n e (4-4) the hep l a t t i c e remains s t a b l e was not d e t e r m i n e d . However, i t i s s t a b l e i n the p o r t i o n o f F i g u r e 4-5 t h a t i s below the i n s t a b i l i t y l i n e . ^ \ 4 _ HCP STABLE 2 " 1 1 o HCP UNSTABLE " X - c f 0 0 1 1 -4 -2 0 1 1 T TL — r ^ s r 2 c , 4 6 & \ ^ F i g u r e 4-5. The i n s t a b i l i t y l i n e f o r the q u a s i h a r m o n i c hep l a t t i c e . The e q u a t i o n o f the l o w e r l i n e i s ( 4 - 4 ) -and t h a t o f the upper l i n e i s $\+.6S^=,G5. The o p p o s i n g arrow heads e n c l o s e r e g i o n s i n which the u n s t a b l e phonon mode f r e q u e n c i e s v a n i s h . 69 IV-3 INSTABILITY DF FCC AND HCP LATTICES In the QHA the atomic displacements from mean p o s i t i o n s depend upon time through l i n e a r combinations of terms of the form e1U)^. When a phonon frequency vanishes the corresponding exponentials lose t h e i r time dependence and each atom receives a s t a t i c displacement equal to the c o e f f i c i e n t of the a f f e c t e d exponential i n the appro-p r i a t e l i n e a r combination. L a t t i c e i n s t a b i l i t y a r i s e s when the f i r s t phonon frequency vanishes and a phase t r a n s i t i o n ensues, the new st r u c t u r e being determined by the array of s t a t i c displacements 33 3U suffered by the atoms of the c r y s t a l . ' Since completely d i f f e r -ent types of phonon modes are responsible f o r dynamical i n s t a b i l i t y i n quasiharmonic fee and hep l a t t i c e s , i t i s to be expected that the phase t r a n s i t i o n s experienced by the two s t r u c t u r e s would be d i f f e r e n t . The long wavelength, l o n g i t u d i n a l , acoustic waves, which cause i n s t a b i l i t y i n the fee l a t t i c e , are compressional waves with wave-lengths comparable to the longest l i n e a r dimension of the c r y s t a l . When the frequencies of these waves vanish, the s t a t i c displacements of the atoms create macroscopic regions of s l i g h t l y lower density, and other regions of s l i g h t l y greater d e n s i t y , than the mean density of the c r y s t a l . Since these regions are macroscopic, d r a s t i c s t r u c -t u r a l r e o r g a n i z a t i o n seems l i k e l y . Simple hexagonal, simple cubic, and body centered cubic l a t t i c e s 3 3 P. A. Fleury, Comments on S o l i d State Phys. k, 149;167 (1972). 3 £ f E. P y t t e , Comments on S o l i d State Phys. 5, 41;57 (1973). 70 are r e l a t e d t a the Fee l a t t i c e , but these s t r u c t u r e s would not be 23 s t a b l e , even as s t a t i c s t r u c t u r e s , except under high pressure. Hep bears a very close r e l a t i o n s h i p to Fee, and i t i s very s t a b l e , but not at the low d e n s i t i e s For which Fee becomes unstable. These Facts suggest that the h i g h l y expanded, unstable Fee l a t t i c e might melt. IF long wavelength, l o n g i t u d i n a l acoustic modes were responsible For the i n s t a b i l i t y oF the hep l a t t i c e , the Foregoing remerks For the Fee l a t t i c e would also apply to the hep l a t t i c e . The only diFFerence between the behaviors oF the two s t r u c t u r e s would be that the hep l a t t i c e would be s l i g h t l y more s t a b l e , since i t could survive an expansion to SK. =-115 (with 5^=0), whereas the Fee l a t t i c e would S 31 ~ if = .109. ThereFore, i n t h i s case an unstable Fee l a t t i c e might convert to hep s t r u c t u r e . However, according to the c a l c u l a t i o n s For t h i s t h e s i s , t h i s i s not the case. The i n s t a b i l i t y oF the hep l a t t i c e i s caused by short wavelength, transverse o p t i c modes with wave vectors (0,0,k ) on the BZ surFace. When the Frequencies oF these modes vanish, see Figure 4-5, the two hep s u b l a t t i c e s are dis p l a c e d i n opposite d i r e c t i o n s normal to the z - a x i s , thus separating the stacking axes oF the two simple hexagonal l a t t i c e s oF which hep i s composed. The soFt mode 33 theory oF phase t r a n s i t i o n s proposes that the s t r u c t u r e evolving From t h i s s i t u a t i o n would be a Bra v a i s , i F i t i s c r y s t a l l i n e , with the wave vector oF the unstable hep mode at the center oF a BZ i n the new s t r u c t u r e . The new s t r u c t u r e could be Fee, which appears more st a b l e with respect to expansion than i s hep. 71 The hcp-fcc t r a n s i t i o n has not been observed i n the RGS. In martensite t h i s t r a n s i t i o n proceeds by long-range c o l l e c t i v e motion: an i n t e r f a c e between fee and hep regions moves disc o n t i n u o u s l y along the stacking d i r e c t i o n with e n t i r e hexagonal planes moving i n t o place 35 at each step of the i n t e r f a c e . Such a process can be v i s u a l i z e d by noting how a plane of one type can be changed i n t o a plane of the other two types. The t r a n s l a t i o n which c a r r i e s a type A plane i n t o a type B plane also c a r r i e s a type B plane i n t o a type C plane. An equal and oppo-s i t e t r a n s l a t i o n c a r r i e s type A i n t o type C and type C i n t o type B. I f these t r a n s l a t i o n s be denoted by + and r e s p e c t i v e l y , and zero t r a n s l a t i o n by 0, the p e r i o d i c stacking pattern ...ABABAB... can be transformed i n t o the p e r i o d i c stacking pattern ...ABCABC... by the p e r i o d i c sequence of t r a n s l a t i o n s ...00++—.... This transformation leaves i n d i v i d u a l hep u n i t c e l l s i n t a c t . T r a n s i t i o n s to close packed s t r u c t u r e s , other than fee, would be excluded because they would not be Bravais s t r u c t u r e s and, presumably, would be roughly as unstable as hep. IlMf SUMMARY In t h i s chapter the v a r i a t i o n of fee and hep phonon frequency d i s t r i b u t i o n s with changes i n l a t t i c e parameters has been examined. D. Coates, Department of M e t a l l u r g i c a l Engineering, Univer-s i t y of B r i t i s h Columbia, p r i v a t e d i s c u s s i o n . 72 The r e s u l t s obtained i n d i c a t e : 1. For s l i g h t expansions the d i s t r i b u t i o n s experience s l i g h t changes, which can be l a r g e l y eliminated by a simple uniform s c a l i n g procedure - equation (4-2). 2. For more extensive expansions the scaled d i s t r i b u t i o n s d i s p l a y d i f f e r e n t i a l e f f e c t s upon i n d i v i d u a l frequencies. These d i f f e r e n t i a l e f f e c t s cause the scaled d i s t r i b u t i o n s to lose t h e i r s i m i l a r i t i e s to the reference d i s t r i b u t i o n s and to each other. 3. The fee l a t t i c e becomes unstable due to long wavelength, l o n g i t u d i n a l , acoustic waves at $^=.109. The hep l a t t i c e i s much l e s s s t a b l e with respect to expansion. I t becomes unstable due to short wavelength, transverse, o p t i c waves along a curve given approximately by equation (4-4). 4. The phase t r a n s i t i o n s experienced by c a t a s t r o p h i c a l l y ex-panded fee and hep l a t t i c e s are l i k e l y quite d i f f e r e n t . 73 CHAPTER V EQUILIBRIUM CONFIGURATIONS OF THE RARE GAS SOLIDS In t h i s chapter the l a t t i c e parameters of fee and hep RGS are r e l a t e d to atomic species and environmental c o n d i t i o n s . I t i s assumed that the model c r y s t a l s are i n thermodynamic e q u i l i b r i u m at constant temperature and pressure, and that the e q u i l i b r i u m l a t t i c e parameters minimize the quasiharmonic Gibbs functions of fee and hep l a t t i c e s - see equations (2-1B), (2-19), and (2-3B). V-1 EQUILIBRIUM LATTICE PARAMETERS Eq u i l i b r i u m l a t t i c e parameters at zero temperature mere deter-mined for a crude, f i r s t neighbor i n t e r a c t i o n model i n s e c t i o n I I I - 4 . In the crude model l a t t i c e expansion to compensate f o r the zero point motion does not depend upon stacking pattern because of the s i m p l i c i t y of the model assumptions. The r e s u l t s presented i n t h i s chapter represent a more accurate treatment. The improvements are as f o l l o w s : 1. Even the s t a t i c hep l a t t i c e i s not i d e a l l y stacked when 24 i n t e r a c t i o n s extend to d i s t a n t neighbors. In the r e s u l t s presented below every atom was allowed to i n t e r a c t with neighbors out to |? =27 - about 850 l a t t i c e s i t e s i n both l a t t i c e s . 2. The mean frequencies and thermal averages were c a l c u l a t e d it from phonon frequency d i s t r i b u t i o n s of roughly 3x10 frequencies. Second-order i n t e r p o l a t i o n formulas of the form (4-1) were used to approximate the dependence upon l a t t i c e parameters of i n d i -Ik v i d u a l phonon frequencies. The i n t e r p o l a t i o n formulas were accurate to one part i n 10 f o r l a t t i c e parameters w i t h i n a percent or so of t h e i r reference values. 3. The Gibbs functions were expanded to second-order i n the expansion c o e f f i c i e n t s - the S's - and minimized i t e r a t i v e l y , using Newton's method of s a l v i n g nonlinear e q u a t i o n s . 3 ^ Conver-7 gence of l a t t i c e parameters to one part i n 10 was achieved i n l e s s than 15 i t e r a t i o n s i n nearly a l l cases. D i f f i c u l t i e s were encountered f o r hep Ne, f o r which the r e s u l t s were ge n e r a l l y poor, except at zero temperature and high pressure. As i n the crude model, the zero paint expansions of both l a t t i c e s at zero pressure are p r o p o r t i o n a l to the De Boer parameter. Unlike the crude model, t h i s p r o p o r t i o n a l i t y f a i l s f o r atoms as l i g h t as neon atoms, and the p r o p o r t i o n a l i t y constants d i f f e r between the two l a t t i c e s . Table V-1 l i s t s the p r o p o r t i o n a l i t y constants f a r the zero point expansions and, a l s o , f o r expansions corresponding to temper-atures up to 30H. The r e l a t i o n 6"jp//l*=.456 has been obtained p r e v i o u s l y f o r the zero point expansion of the fee l a t t i c e with long-range (12-6) i n t e r -29 a c t i o n s . The 20% reduction i n t h i s p r o p o r t i o n a l i t y constant i s the r e s u l t of i n c l u d i n g long-range i n t e r a c t i o n s , which weaken the zero point energy by 20% with respect to the l a t t i c e p o t e n t i a l energy. I f the hep l a t t i c e were r e s t r a i n e d to expand i s o t r o p i c a l l y , with S^=0, 35 v"' A. Ralston, A F i r s t Course i n Numerical A n a l y s i s (M Graw-H i l l , Inc., New York, 1965), s e c t i o n B-k. 75 Table V - ' l . Expansion c o e f f i c i e n t s f o r fee and hep RGB u i t h long-range (12-6) i n t e r a c t i o n s and at zero pressure. Results f o r neon are not incl u d e d . The quantity = i s the average f r a c t i o n a l " increase i n interatomic spacing i n the hep l a t t i c e . DH 10K 2 OH 30K Sjr/A* .46 .47 .49 .63 .55 .56 .58 .75 sL/A* -.16 -.16 -.17 -.21 s^/A* .50 .51 .52 .68 .92 .92 .94 .93 s k / r k ' -.32 -.31 • -.33. -.31 i t s zero paint expansion uould, again, be the same as that of the fee l a t t i c e . Houever, i n t e r a c t i o n s u i t h d i s t a n t neighbors induce the hep l a t t i c e to take advantage of the p o s s i b i l i t y of a n i s o t r o p i c expansion. For a l l temperatures l i s t e d i n Table V/-1 the hep l a t t i c e has i t s hexagonal planes s l i g h t l y too c l o s e l y spaced to be i d e a l l y stacked -that i s , 0 - and i t s average interatomic spacing s l i g h t l y greater than i n the fee l a t t i c e - that i s , SKJ= S^4 S^/j >S^ ? • The f a c t that &}[•< 0 i s s l i g h t l y advantageous to the s t a b i l i t y 76 of the quasiharmonic hep l a t t i c e . Whereas the i d e a l hep l a t t i c e would became unstable f o r S V * ^ =. .036, a nonideal hep l a t t i c e with Sl\-~S k/3 could survive an expansion to k =.04. The fee l a t t i c e , which expands l e s s e x t e n s i v e l y and i s s t a b l e up to Slf =.109, appears to be considerably more st a b l e than hep, i n the long run. This does not imply, however, that fee would be more st a b l e than hep under conditions f o r which both s t r u c t u r e s would be a b s o l u t e l y s t a b l e . The p r o p o r t i o n a l i t y between the expansion c o e f f i c i e n t s and the De Boer parameter f a i l s f o r atoms as l i g h t as neon. I t i s i n t e r e s t -i n g , though, to ignore t h i s f a r a moment and use the zero point expan-sion information i n Table V-1 to get an approximate answer to the question: What i s the l i g h t e s t RGB that can survive i t s zero point expansion without the a i d of e x t e r n a l pressure? For the fee RGS, the r e l a t i o n ?6/l*S./Of must be s a t i s f i e d . Therefore, only those fee RGS with / I * * £* '238 c a n e x i s t at T=0K and P=D atm. In other words, only fee helium would not e x i s t under these c o n d i t i o n s . For the hep RGS the r e l a t i o n to be s a t i s f i e d , Sl^-<°- SoA*so.oyt i m p i i B S that A ^ .08, so hep neon may not survive i t s zero point expansion. This observation i s supported by the c a l c u l a t i o n s of the zero point l a t -t i c e parameters of hep neon. These i n d i c a t e d that hep neon i s dynam-i c a l l y unstable at OK, except under pressure. 77 (5-2) U-2 THERMAL EXPANSIVITIES Three i s o b a r i c , thermal e x p a n s i v i t i e s uere c a l c u l a t e d by numer-i c a l d i f f e r e n t i a t i o n of the S ' s : These are l i n e a r e x p a n s i v i t i e s . The volume e x p a n s i v i t i e s are: Curves of these e x p a n s i v i t i e s f o r s o l i d xenon at P=1 atm. are shown i n Figure 5-1. The stronger expansion e f f e c t s i n the hep l a t t i c e are r e f l e c t e d i n the r e l a t i v e magnitudes of the e x p a n s i v i t i e s : _ 3 7 3 g o C ^ > Vc x ^ . The experimental points ' do not agree w e l l with the c a l c u l a t e d curves, since the QHA i s not r e l i a b l e above about T=1QK,k and since the bowl of the (12-6) p o t e n t i a l w e l l i s too weakly curved - see Figure 2-1. C l e a r l y the model c r y s t a l s are more respon-si v e to thermal expansion than are r e a l c r y s t a l s of xenon. Expan s i v i t y curves f o r the other RGS and f o r higher pressures are q u a l i t a t i v e l y the same as those shown i n Figure 5-1. For the l i g h t e r RGS the e x p a n s i v i t i e s are greater, roughly by the f a c t o r £^ e/£ . For higher pressures the e x p a n s i v i t i e s are reduced, but the 2 e f f e c t i s weak unless P>10 atm. A. 3. Eatwell and B. L. Smith, P h i l . Mag. 6, 461 (1961), ( S e r i e s B). 3 8 C. R. T i l f o r d and C. A. Swenson, Phys. Rev. B5, 719 (1972). 70 40 30 A in a > •H U) C ro a. LU 20 ra B 10 T ;—i : — I 1 r 0 10 20 30 40 50 Temperature (K) Figure 5-1. Thermal e x p a n s i v i t i e s at one atmosphere f o r fee and hep xenon, compared with experimental data (x - r e f . 37., o - r e f . 33). 79 V-3 ISOTHERMAL COMPRESSIBILITIES Three l i n e a r , isothermal c o m p r e s s i b i l i t i e s were c a l c u l a t e d by numerical d i f f e r e n t i a t i o n of the S 's: with volume c o m p r e s s i b i l i t i e s given by: (5-4) Curves of the r e s u l t s obtained f o r s o l i d Ar, Hr, and Xe at T=0H are shown i n Figure 5-2. Curves f o r are not shown, since they nearly coincide with those f o r Kj^. The agreement with experimental data . . . . . 39,40 improves with i n c r e a s i n g atomic mass. ' 2 The c o m p r e s s i b i l i t i e s are almost constant below 10 atm. and t h e i r r e l a t i v e magnitudes, ^ K *K K > K.£ , i n d i c a t e that the fee l a t t i c e 2 i s more r e s i s t a n t to compression than i s hep. Above 10 atm. the c o m p r e s s i b i l i t i e s f a l l o f f s t r o n g l y , and near 10 3 atm. the fee and hep curves cr o s s , i n d i c a t i n g a r e l a t i v e hardening of the hep l a t t i c e . As temperature i n c r e a s e s , so do the c o m p r e s s i b i l i t i e s , but the pressure-dependence remains q u a l i t a t i v e l y the same as i s Figure 5-2. Figure 5-3 shows how changes with pressure f o r hep Ar, Hr, and Xe at T=0K and 10H. Below 10"5 atm. the curves are f l a t . Above 3 ^ / 10 atm. the curves turn upward, presumably approaching the o^-axis as the s p h e r i c a l hard-cores are squeezed together. 19 J . R. Barker and E. R. Dobbs, P h i l . Mag. 46, 1069 (1955). 40 P. A. B e z u g l y i , L. M. Tarasenko, and 0. I . B a r y s h e v s k i i , Soviet P h y s i c s - S o l i d State 13, 2003 (1972). m 14 c >-TJ I C M cr U CM D >: 12 H E p-i Q J a n S i — i ^ J 3 •r-l P -I CL C a CJ 10" r i q u r c b-c •Ar *f.Ar I ? : * „ J k & L _Xfi»_-_AXe J l t K r n 1 v 10 i o ' 10*-F r e s s u r e (atm.) IO" -1 •2 J o X -3 -f -4 1 10 -1 XG Hr 7 1 0 " : i i 1 0 J 1E31 10' P r e s s u r e (atm.) F i g u r e 5 - 3 . Dependence of the st a c ' parameter of the hep l a t t i c e upon pressure at T=0K ( s o l i d ) and T=10K (dashed). i n g 03 • Icothermal c o m p r e s s i b i l i t i e s of fee Experimental p o i n t s from re-fs. 39 and 40 . 81 SUMMARY The r e s u l t s presented i n t h i s chapter i n d i c a t e that the i n c l u -sion of d i s t a n t neighbor i n t e r a c t i o n s d i f f e r e n t i a t e s between the l a t t i c e parameters of quasiharmonic fee and hep l a t t i c e i n e q u i l i -brium at constant temperature and pressure. Except at very high pressures, the hep l a t t i c e i s more amenable to expansion and compres,-; sion than i s the fee l a t t i c e . A l s o , the e q u i l i b r i u m hep l a t t i c e deviates from i d e a l s t a c k i n g . I t s hexagonal planes are s l i g h t l y too close f o r i d e a l s t a c k i n g , by an amount depending upon atomic species 3 2 but not pressure, unless P >10 atm. Pressures l e s s than ~ 10 atm. hardly a f f e c t the l a t t i c e parameters of e i t h e r l a t t i c e . 82 CHAPTER Ul RELATIVE STABILITIES DF THE TWO STRUCTURES This chapter considers the problem of g i v i n g a t h e o r e t i c a l explanation f o r the observed s t a b i l i t y of fee, but not hep, s t r u c t u r e s of the RGS. The di s c u s s i o n c o n s i s t s of a review of the h i s t o r y of the s t a b i l i t y problem and an i l l u s t r a t i o n of the s t r i n g e n t p r e c i s i o n requirements to be s a t i s f i e d by c a l c u l a t i o n s attempting to resolve i t . VI-1 HISTORY OF THE RGS STABILITY PROBLEM Real RGS have fee s t r u c t u r e s from t h e i r melting points down to the lowest temperatures f o r which t h e i r s t r u c t u r e s have been deter-1 —6 mined - about 2K. ~ The only evidence of hep s t r u c t u r e i s i n the 41 form of stacking f a u l t s i n fee c r y s t a l s of high p u r i t y argon. The term, "RGS s t a b i l i t y problem", r e f e r s to the problem of e x p l a i n i n g the fee s t r u c t u r e s . I t i s c l e a r form f i r s t neighbor arguments - s e c t i o n II-4 - that the only s t r u c t u r e s that could be more st a b l e than fee s t r u c t u r e s at low temperatures and pressures would also have to be close packed. Complicated p e r i o d i c stacking patterns' are not of i n t e r e s t , since they are of low symmetry and are not commonly observed i n nature. Therefore, the RGS s t a b i l i t y problem reduces to a comparison of fee and hep s t r u c t u r e s . For the purposes of t h i s d i s c u s s i o n , i t i s assumed that the 41 L. Meyer, C. S. B a r r e t t , and P. Haasen, J . Chem. Phys. 40, 2744 (1969). 33 r e l a t i v e s t a b i l i t i e s of these two s t r u c t u r e s , i n e q u i l i b r i u m at con-stant temperature and pressure, are determined by the d i f f e r e n c e between t h e i r Gibbs functions or cohesive energies - see equation (2-20). The problem of d i s t i n g u i s h i n g the signs of the cohesive energy d i f f e r e n c e s i s d i f f i c u l t because the c o n t r i b u t i o n s of the dominant, c e n t r a l p a i r forces are extremely i n s e n s i t i v e to the d i f f e r -ences i n stacking patterns. Indeed, i f c e n t r a l p a i r i n t e r a c t i o n s are r e s t r i c t e d to f i r s t neighbors, the fcc-hcp cohesive energy d i f f e r e n c e s vanish f o r the s t a t i c l a t t i c e s and at T=0K for crude models of the harmonic and quasiharmonic approximations - see sections Il-k and Ill-k. F i n i t e cohesive energy d i f f e r e n c e s a r i s e only i f c e n t r a l p a i r i n t e r a c t i o n s extend at l e a s t as f a r as t h i r d neighbors. Unfortunately, the cohe-siv e energies then favor hep s t r u c t u r e s since t h i r d neighbors are s l i g h t l y c l o s e r i n hep - *\Jz^'a i n i d e a l hep, compared with VT a i n fee, where a i s the f i r s t neighbor separation. Attempts to reverse the signs of the fcc-hcp cohesive energy d i f f e r e n c e s have been numerous and generally unsuccessful. Several of these attempts are described below. VI-1-1 C e n t r a l P a i r Force C a l c u l a t i o n s Kihara and Koba have c a l c u l a t e d the p o t e n t i a l energies of s t a t i c fee and hep l a t t i c e s of atoms i n t e r a c t i n g v i a long-range (exp-6) and (m-6) p o t e n t i a l s . The p o t e n t i a l energies were found to 4 2 T. Kihara and S. Koba, 3. Phys. Soc.(Japan) 7, 343 (1952). Qk be nearly i d e n t i c a l , and t h e i r d i f f e r e n c e favored the hep l a t t i c e f o r (m-6) p o t e n t i a l s with 7Srn f 18 and f o r (exp-6) p o t e n t i a l s with 8.765ic<518. For-oc< 8.765 the fee l a t t i c e would be mare s t a b l e , however, such values would be unreasonable since (exp-6) p o t e n t i a l s with o<<8.5 have no p o t e n t i a l w e l l . k3 Barron and Domb have c a l c u l a t e d the p o t e n t i a l energies of s t a t i c fee and hep l a t t i c e s held together by long-range (m-n) poten-t i a l s - g e n e r a l i z a t i o n s of the (m-6) p o t e n t i a l s . Their c a l c u l a t e d p o t e n t i a l energy d i f f e r e n c e s favored the hep l a t t i c e by roughly .01% for a l l reasonable choices of m and n. The s t a t i c fee l a t t i c e at zero pressure i s more s t a b l e only f o r unreasonable choices of m and n - m=6,n=5; 5<mi8,n=4. Barron and Domb also considered the e f f e c t s of ex t e r n a l pressure and thermal motion. For the (12-6) p o t e n t i a l , the s t a t i c fee l a t t i c e i s s t a b i l i z e d at pressures which reduce the s p e c i f i c volumes by a 5 fa c t o r of two - about 10 atm. f a r s o l i d argon. L a t t i c e motion was included by adding the thermal energies of Debye s o l i d s to the l a t t i c e p o t e n t i a l energies. This reduced the energy excess of the fee l a t t i c e but d i d not s t a b i l i z e i t , except f o r temperatures comparable to the melting temperatures. These r e s u l t s are not s a t i s f a c t o r y f o r e x p l a i n -ing the fee s t r u c t u r e s of the RGS at low temperatures and pressures. kk Oansen and Dawson have c a l c u l a t e d the zero pressure ground state energies of quasiharmonic fee and hep l a t t i c e s bound by (m-6) _ T. H. H. Barron and C. Domb, Proc. Roy. Soc.(London) A227, kkl, (1955). kk L. Oansen and 3. Dawson, J . Chem. Phys. 23, ^ 82 (1955). 85 and (exp-6) p o t e n t i a l s . The zero point energies ware approximated by 45 Corner's method, i n which every atom moves as though every other atom were f i x e d i n i t s mean p o s i t i o n . For 75m <16 and 10i«»<f:S16, and with values of A*in the range corresponding to the RGS, the hep l a t t i c e was favored by .01% of the ground state energies. 46 Wallace has considered the e f f e c t of using r e a l i s t i c d i s t r i -butions of phonon frequencies to c a l c u l a t e the zero point energies. F D T (12-6) p o t e n t i a l s and motion i n the harmonic approximation, the ground state energy d i f f e r e n c e s favored the hep s t r u c t u r e by .01%. The i n c l u s i o n of anharmonic frequency s h i f t s could j u s t s t a b i l i z e the fee s t r u c t u r e s of neon and argon, but not krypton or xenon. 10 Howard repeated Wallace's harmonic approximation c a l c u l a t i o n s , but did not consider anharmonic e f f e c t s . With frequency d i s t r i b u t i o n s composed of roughly three and s i x times as many frequencies as those used by Wallace f o r fee and hep l a t t i c e s , r e s p e c t i v e l y , Howard obtained the same r e s u l t : the hep l a t t i c e was favored by .01% of the ground state energies. The c a l c u l a t i o n s mentioned so f a r considered only i d e a l hep 24 l a t t i c e s * Howard has shown that the p o t e n t i a l energy minimum f o r s t a t i c hep l a t t i c e s bound by long-range (12-6) i n t e r a c t i o n s c o r r e -sponds to a value f o r the stacking parameter that i s smaller than the value f o r i d e a l stacking by one part i n 10 . This reduces the poten-7 t i a l energy by one part i n 1D . The only other c o n s i d e r a t i o n of d e v i a t i o n s from i d e a l s t a c k i n g , 45 3. Corner, Trans. Faraday Soc. 44, 914 (1948). 46 D. C. Wallace, Phys. Rev. A133, 153 (1954). 86 to date, i s t h i s t h e s i s . As mentioned i n chapter I/, d e v i a t i o n s from i d e a l stacking i n dynamic hep l a t t i c e s are i n the same d i r e c t i o n as i n the s t a t i c l a t t i c e s , but they are much l a r g e r . These d e v i a t i o n s , along u i t h i s o t r o p i c changes i n interatomic spacing, increase the cohesive energy of the quasiharmonic hep l a t t i c e by about .01%. However, the corresponding changes i n the quasiharmonic fee l a t t i c e increase i t s cohesive energy by nearly the same amount, so the cohe-s i v e energy d i f f e r e n c e s s t i l l favor the hep l a t t i c e by .01% f o r low temperatures and pressures. Two conclusions can be drawn from the c a l c u l a t i o n s described above: that c e n t r a l p a i r forces and (quasi)harmonic motion are i n -adequate f o r r e s o l v i n g the s t a b i l i t y problem, and that models based upon such assumptions y i e l d cohesive energy d i f f e r e n c e s on the order of .01%. The most promising of the c a l c u l a t i o n s i s that of Wallace. However, the success i n e x p l a i n i n g the fee s t r u c t u r e s of neon and argon, but not krypton or xenon, i s somewhat pe r p l e x i n g . Anharmonic e f f e c t s may, indeed, be s u f f i c i e n t to s t a b i l i z e the fee s t r u c t u r e s of a l l RGS, but i t could also be the case that the success with neon and argon was the r e s u l t of the r e l a t i v e smallness of t h e i r atomic masses and that anharmonic e f f e c t s play no e s s e n t i a l r o l e i n the RGS s t a b i l i t y problem. A c a l c u l a t i o n based on a more accurate treatment of anharmonic e f f e c t s i s needed to c l a r i f y t h i s matter. 87 VI-1-2 IMoncentral P a i r Force C a l c u l a t i o n s The e l e c t r o n i c charge d i s t r i b u t i o n s of rare gas atoms i n t h e i r ground st a t e s are not s t r i c t l y s p h e r i c a l , p a r t i c u l a r l y i f the atoms experience a nonspherical p o t e n t i a l f i e l d , as they would i n a c r y s t a l . As a r e s u l t , the p a i r p o t e n t i a l can be expected to e x h i b i t a weak angular dependence. kl Cuthbert and L i n n e t t have found that spin c o r r e l a t i o n s among the eight outer e l e c t r o n s of i s o l a t e d rare gas atoms cause the elec t r o n s to p a i r o f f at the v e r t i c e s of regular tetrahedra. Since the patterns of neighbors are d i f f e r e n t i n fee and hep l a t t i c e s , the i n t e r a c t i o n s of these t e t r a h e d r a l charge d i s t r i b u t i o n s may a f f e c t the d i f f e r e n c e i n s t a t i c l a t t i c e p o t e n t i a l energies. C a l c u l a t i o n s of the i n t e r a c t i o n energies f o r sev e r a l r e l a t i v e o r i e n t a t i o n s of two r i g i d t e t r a h e d r a l charge d i s t r i b u t i o n s showed that the pattern of, e s p e c i a l l y , f i r s t , second, and t h i r d neighbors may be to favor the fee s t r u c t u r e . kB Kihara has suggested that the pattern of f i r s t neighbors i n hep l a t t i c e s favors the in d u c t i o n of an octupole moment i n each atom, and that the r e p u l s i o n of these octupole moments would decrease hep s t a -b i l i t y . In the fee l a t t i c e , the lowest order induced moments would be hexadecapales with much weaker mutual r e p u l s i o n s . Mo q u a n t i t a t i v e r e s u l t s have been published f o r e i t h e r of these models. 3. •Cuthbe'rt. and J.. UJ. L i n n e t t , Trans. Faraday Soc. 5k_, 617 (195S). ^ 8 T. Kiha r a , J . Phys. Soc.(Japan) 15, 1120 (1960). aa VI-1-3 Three-Body Force C a l c u l a t i o n s A t h i r d l i n e of approach to the RGS s t a b i l i t y problem has been the i n c l u s i o n of three-body i n t e r a c t i o n s . The A x i l r o d - T e l l e r , t r i p l e d i p o l e p o t e n t i a l c o n t r i b u t e s roughly 10% of the t o t a l energies of the RGS. The d i f f e r e n c e between the c o n t r i b u t i o n s to the energies of s t a t i c fee and hep l a t t i c e s i s only .08% of the t o t a l t r i p l e d i p o l e c o n t r i b u t i o n s . This d i f f e r e n c e reduces the energy excess of the fee l a t t i c e , but cannot s t a b i l i z e i t because of the weakness of the i n t e r -. . 14 a c t i o n . Jansen's a n a l y s i s of the e f f e c t s of three-body exchange forces 18 i s the most su c c e s s f u l treatment of the s t a b i l i t y problem, to date. These i n t e r a c t i o n s are very s e n s i t i v e to c o n f i g u r a t i o n a l d i f f e r e n c e s and are much stronger f o r f i r s t neighbors than are t r i p l e d i p o l e i n t e r a c t i o n s . U i t h three-body exchange forces among f i r s t neighbors, the s t a t i c p o t e n t i a l energy d i f f e r e n c e f o r fee and hep l a t t i c e s appears to be about .4% i n favor Df the fee l a t t i c e . This i s f o r t y times greater than the cohesive energy d i f f e r e n c e s obtained from c e n t r a l p a i r force c a l c u l a t i o n s . Apparently, three-body forces play a s i g n i f i c a n t r o l e i n the s t a b i l i t y problem. However, Hansen's success should not be accepted without q u a l i f i c a t i o n ^ since the complexities af h i s model permit only crude treatments. The strong r a d i a l and angular dependences of the many-body fo r c e s , which are responsible f o r the c o r r e c t s t a b i l i t y c onclusion, may cause problems i f l a t t i c e motion and d e v i a t i o n s from i d e a l stacking are considered. 89 VI-2 PRECISION REQUIREMENTS OF THE STABILITY PROBLEM The d i s c r i m i n a n t f o r the RGS s t a b i l i t y problem i s the cohssive energy d i f f e r e n c e : which cannot be determined experimentally because the hep s t r u c t u r e s of the RGS are not observed. However, two estimates are suggested by t h e o r e t i c a l c a l c u l a t i o n s : 1. C e n t r a l p a i r force c a l c u l a t i o n s suggest that li|l*'10~ £ t. 2. Jansen's c a l c u l a t i o n s suggest that l 7 j l ~ 4 x 1 0 ~ 3 . This e s t i -mate agrees with one derived from an a n a l y s i s of stacking f a u l t 49 energies i n s o l i d argon. Estimates of dubious r e l i a b i l i t y can also be obtained from systems which e x h i b i t both s t r u c t u r e s . Cobalt, f o r instance, exper-iences an fec-hep t r a n s i t i o n at 720K. The energy inv o l v e d i n the t r a n s i t i o n i s .06% of the sublimation energy, and the fee s t r u c t u r e 50 occupies .3% more volume than the hep s t r u c t u r e . These f i g u r e s cannot be applied r e l i a b l y to the fec-hep cohesive energy d i f f e r e n c e fo r the RGS, since cobalt i s a metal. However, i t i s encouraging that the f r a c t i o n a l energy change i n the cobalt t r a n s i t i o n i s the geometric mean of the two t h e o r e t i c a l estimates of f o r the RGS. R. Bullaugh, H. R. Glyde, and J . A. V/enables, Phys. Rev. L e t t e r s 17, 249 (1966). 50 0. T u r n b u l l , S o l i d State Phys. 3, 225 (1957). (6-1) 90 The expected smallness of the cohesive energy d i f f e r e n c e places s t r i n g e n t p r e c i s i o n requirements on c a l c u l a t i o n s which seek only the sign of I f one c a l c u l a t e s &fcc and to equal p r e c i s i o n , the maximum p e r m i s s i b l e , r e l a t i v e u n c e r t a i n t y i n each of the cohesive energies i s ^ ^ j l . I f computational e r r o r s exceed t h i s l i m i t , they w i l l mask the sign of and prevent any s t a b i l i t y conclusion from being drawn. Uncertainty l i m i t s can also be obtained f o r other q u a n t i t i e s to be c a l c u l a t e d . Table l i s t s some estimates of per m i s s i b l e uncer-t a i n t i e s f o r cohesive energy d i f f e r e n c e c a l c u l a t i o n s i n terms of a model i n which atoms i n t e r a c t v i a (12-6) p o t e n t i a l s and l a t t i c e s move according to the harmonic approximation - f o r T=0K and P=0 atm. D e t a i l s of the considerations upon which these estimates are based are given i n appendix C. I t should be noted that the estimates f o r dynamical matrix elements and l a t t i c e sums are very crude. For t h i s reason, no attempt was made to account f o r zero point expansion. The p e r m i s s i b l e u n c e r t a i n t i e s l i s t e d i n Table VI-1 are excep-t i o n a l l y s m a l l . They point out that the RGS s t a b i l i t y problem i s extremely d i f f i c u l t , even i n terms of an inadequate model. Experimental e r r o r s i n the model parameters £ , <5~, and M are s u b s t a n t i a l l y l a r g e r than the estimates l i s t e d i n Table VI-1, but these e r r o r s should not a f f e c t the cohesive energy d i f f e r e n c e s s i g -n i f i c a n t l y , since such e r r o r s represent small p e r t u r b a t i o n s i n the parameters, compared with much more extensive v a r i a t i o n s i n the same parameters throughout the RGS - see Table I I - 1 . Jansen has argued that the RGS s t a b i l i t y problem depends only upon the general proper-Table 1 / 1-1. Estimates of perm i s s i b l e r e l a t i v e u n c e r t a i n t i e s i n q u a n t i t i e s required to c a l c u l a t e the fcc-hcp ground s t a t e energy d i f f e r e n c e , &(x =">] 6c . See appendix C. Quantity Symbol Perm i s s i b l e R e l a t i v e Uncertainty u n i t s of l ^ l =10"^ \r\[ =4x10~ 3 Ground state energy Cc 1/2 5x10" 5 2x10* 3 L a t t i c e p o t e n t i a l energy $ 1/5 2x10" 5 8x10'^ Zero point energy and mean frequency 1/2 5x10" 5 2x10~ 3 L a t t i c e sums f o r $ 1/10 _5 10 4x10"^ Phonon frequencies ^ x 1/2 5x10"5 2x10~ 3 Dynamical matrix elements 1 1/10 io" 5 4x10*"3 4x10"^ L a t t i c e sums f o r V^(j:J) 1 1/10 1/100 -4 10 _5 10 10" 6 4x10~ 3 -4 4x10 4x10~ 5 92 t i e s of the proper p h y s i c a l model, rather than upon p r e c i s e numerical 1fl r e l a t i o n s h i p s among model parameters. I f t h i s i s not the case, there seems l i t t l e chance of r e s o l v i n g the s t a b i l i t y problem s a t i s -f a c t o r i l y without d r a s t i c improvements i n experimental and theoret-i c a l techniques. I f i t i s the case, which seems more l i k e l y , s a t i s -f a c t o r y c a l c u l a t i o n s do not have to be concerned with small e x p e r i -mental e r r o r s i n model parameters, but they s t i l l have the very d i f f i c u l t problems of developing a s a t i s f a c t o r y t h e o r e t i c a l model and performing s u f f i c i e n t l y accurate c a l c u l a t i o n s . F i n a l l y , i t should be noted that the experimental s i t u a t i o n may be more complex than i s c u r r e n t l y b e l i e v e d . Perhaps there i s an fcc-hcp t r a n s i t i o n between zero temperature and 2K, but t h i s seems 43 u n l i k e l y since the a n a l y s i s of Barron and Domb i n d i c a t e s that cen-t r a l p a i r forces can s t a b i l i z e the fee s t r u c t u r e Dnly at temperatures comparable with the melting temperatures. A l s o , the discovery of 41 small hep regions i n fee argon i n d i c a t e s that the range of p o s s i b l e s t r u c t u r e s f o r condensed rare gases i s only incompletely explored. 93 CHAPTER VII CONCLUSION This t h e s i s has compared fee and hep s t r u c t u r e s of heavy rare gas atoms. Most of the r e s u l t s f o r the fee s t r u c t u r e s are supported by p r e v i o u s l y published m a t e r i a l , but the r e s u l t s f o r the hep st r u c t u r e s are mainly new r e s u l t s , p a r t i c u l a r l y those i n v o l v i n g non-i d e a l s t a c k i n g . VII-1 SUMMARY OF RESULTS Chapter IV examined the v a r i a t i o n of f i r s t neighbor, phonon frequency d i s t r i b u t i o n s u i t h changes i n l a t t i c e parameters. The r e s u l t s obtained i n d i c a t e : 1. For s l i g h t expansions u i t h respect to the s t a t i c l a t t i c e , the d i s t r i b u t i o n s experience s l i g h t changes, uhich can be l a r g e l y e l iminated by a simple, uniform s c a l i n g procedure - equation (4-2). 2. For mare extensive expansions the scaled frequency d i s t r i -butions d i s p l a y d i f f e r e n t i a l e f f e c t s on i n d i v i d u a l frequencies. These d i f f e r e n t i a l e f f e c t s cause the scaled fee and hep d i s t r i -butions to lose t h e i r s i m i l a r i t i e s u i t h the reference d i s t r i b u -t i o n s and u i t h each other. 3. The fee l a t t i c e becomes unstable due to long uavelength, l o n g i t u d i n a l , a c o u s t i c uaves, at £.=.109, uhich corresponds to 94 the i n f l e c t i o n point of the (12-6) p o t e n t i a l . The hep l a t t i c e i s much l e s s s t a b l e u i t h respect to expansion. I t becomes dynamically unstable due to short wavelength, transverse, o p t i c uaves, along a curve approximated by equation (4-4). 4. The phase t r a n s i t i o n s experienced by c a t a s t r o p h i c a l l y expanded fee and hep l a t t i c e s are l i k e l y quite d i f f e r e n t . Chapter V presented r e s u l t s f o r the l a t t i c e parameters D f quasi-harmonic fee and hep l a t t i c e s i n thermodynamic e q u i l i b r i u m at constant temperature and pressure. The r e s u l t s i n d i c a t e : 1. I n c l u s i o n of d i s t a n t neighbor i n t e r a c t i o n s i s necessary to obtain d i f f e r e n c e s i n l a t t i c e parameters f o r the tuo s t r u c t u r e s . 2. At l o u temperatures and pressures, the l a t t i c e parameters of s o l i d argon, krypton, and xenon deviate from t h e i r s t a t i c l a t t i c e values by amounts p r o p o r t i o n a l to the De Boer parameter. These dev i a t i o n s are s l i g h t , so the QHA i s l i k e l y a p p l i c a b l e i n these cases. S o l i d neon experiences much stronger d e v i a t i o n s : hep neon appears unstable at the loue s t temperatures, unless high pressures are a p p l i e d . 3. Deviations from s t a t i c l a t t i c e parameters are sameuhat smaller i n fee than i n hep. The hep sta c k i n g parameter i s smaller than the i d e a l s t acking parameter by an amount propor-t i o n a l to the De Boer parameter. This should be s l i g h t l y advan-tageous to hep s t a b i l i t y f o r lar g e expansions. 95 k. Thermal e x p a n s i v i t i e s and isothermal c o m p r e s s i b i l i t i e s are s l i g h t l y l a r g e r For the hep l a t t i c e than f o r the fee l a t t i c e . The c a l c u l a t e d values f o r these q u a n t i t i e s are i n rough agree-ment u i t h experimental values. 2 5. Pressures l e s s than ~10 atm. hardly a f f e c t the l a t t i c e parameters of e i t h e r l a t t i c e . Chapter V/I discussed the r e l a t i v e s t a b i l i t i e s of fee and hep RGS. The f a l l o w i n g conclusions can be drawn from the d i s c u s s i o n : 1. C a l c u l a t i o n s based upon c e n t r a l p a i r forces and (quasi) har-monic l a t t i c e motion y i e l d r e s u l t s which favor the hep l a t t i c e by ~ . 0 1 % of the cohesive energies. I n c l u s i o n of anharmonic e f f e c t s may be s u f f i c i e n t to reverse the sign of the cohesive energy d i f f e r e n c e . 2. Moncentral forces are s e n s i t i v e to d i f f e r e n c e s i n l a t t i c e s t r u c t u r e and seem to favor the fee l a t t i c e . Short-range, three-body, exchange farces can s t a b i l i z e the s t a t i c fee l a t t i c e and y i e l d cohesive energy d i f f e r e n c e s an the order of 3. T h e o r e t i c a l explanations f o r the observed s t a b i l i t y of the fee s t r u c t u r e s of the RGS are persuasive, but incomplete. The expected small magnitude of the fec-hep cohesive energy d i f f e r -ence makes a f i n a l r e s o l u t i o n of t h i s problem very d i f f i c u l t . 96 VI1-2 DISCUSSION OF MODEL ASSUMPTIONS Ths a n a l y s i s of l a t t i c e expansion e f f e c t s i n chapters IV/ and V i s based upon the p h y s i c a l model described i n se c t i o n I I I - 1 . L i m i -t a t i o n s on the v a l i d i t y of r e s u l t s obtained from t h i s model are discussed beloui. VII-2-1 Interatomic I n t e r a c t i o n s To represent the i n t e r a c t i o n s between rare gas atoms as c e n t r a l p a i r i n t e r a c t i o n s described by (m-6) p o t e n t i a l s i s a gross approxi-mation, because many-body i n t e r a c t i o n s , though r e l a t i v e l y weak, can have s i g n i f i c a n t e f f e c t s upon accurate c a l c u l a t i o n s , and because the (m-6) p o t e n t i a l s are not f l e x i b l e enough to account a c c u r a t e l y f o r the t r u e , c e n t r a l p a i r component of i n t e r a c t i o n s i n v o l v i n g rare gas atoms. Despite these f a u l t s , the (m-6) p o t e n t i a l s are very u s e f u l f o r c a l c u l a t i o n s which do not intend to compare computed r e s u l t s with experimental r e s u l t s , because they are e a s i l y manipulated, and because they d i s p l a y the general p r o p e r t i e s of more accurate c e n t r a l p a i r p o t e n t i a l s . Therefore, there i s no need to abandon them completely, provided that t h e i r l i m i t a t i o n s are recognized. Since the (m-6) p o t e n t i a l s are bound to be d e f e c t i v e , e i t h e r at short-range or at long-range, there i s no sense i n going to great t r o u b l e to ensure that a l l i n t e r a c t i o n s are considered. For many problems, i n c l u s i o n of i n t e r a c t i o n s between close neighbors should be s a t i s f a c t o r y . The defectiveness of the (m-6) p o t e n t i a l s should lead one to 97 suspect the r e s u l t s presented i n chapters IV and V f a r two reasons. F i r s t , the bottom of the p o t e n t i a l w e l l can be given the c o r r e c t depth, p o s i t i o n , and curvature by c a r e f u l choice of the parameters £ , CT, and m. The choice m=12 i s a f a i r l y good choice. However, more parameters would be needed to obtain the c o r r e c t v a r i a t i o n of the curvature near the p o t e n t i a l minimum. Therefore, the f i r s t neighbor force constants do not have the co r r e c t dependence upon l a t -t i c e parameters, and that i s p a r t l y why the thermal e x p a n s i v i t i e s and isothermal c o m p r e s s i b i l i t i e s agreed poorly with experimental values. Second, the i n s t a b i l i t y of quasiharmonic l a t t i c e s depends upon the i n f l e c t i o n point i n the <p(r) versus r curve. Therefore, even b a r r i n g the l i m i t a t i o n s of the QHA, the ranges of l a t t i c e parameters f o r which the fee and hep l a t t i c e s would be st a b l e would not be c o r r e c t , unless the (12-6) p o t e n t i a l happens, f o r t u i t o u s l y , to have i t s i n f l e c -t i o n point i n the proper p l a c e . VII-2-2 L a t t i c e Dynamics The QHA gives good r e s u l t s f o r thermodynamic p r o p e r t i e s of the RGS, other than s o l i d neon, up to about T=1QH.^ For higher temper-atures, b e t t e r l a t t i c e dynamical models are necessary. The zero point motions i n the RGS are s l i g h t , so the atoms remain much c l o s e r to t h e i r own mean p o s i t i o n s than they ever get to the mean p o s i t i o n s of t h e i r neighbors. However, the atomic displacements are not i n f i n i t e s i m a l , so the interat o m i c force constants f l u c t u a t e r a p i d l y with the l a t t i c e motion. In s o l i d xenon, f o r i n s t a n c e , the harmonic approximation p r e d i c t s rrns displacements at T=DK of roughly 98 3% of the mean separation of f i r s t neighbors. This means that f i r s t neighbors are frequ e n t l y ^6% c l o s e r or more d i s t a n t than on the average. As a r e s u l t , the force constants coupling f i r s t neighbors can f l u c t u a t e by ~ 6 0 % . Such extensive force constant f l u c t u a t i o n s i n d i c a t e strong anharmonicity, even i n the heaviest RGS at the lowest temperatures. The f a c t that harmonic approximations, which ignore these f l u c t u a t i o n s , , do not y i e l d r e a l l y poor r e s u l t s f o r the proper-t i e s of the RGS i s quite remarkable. The changes i n phonon frequency d i s t r i b u t i o n s with changes i n l a t t i c e parameters, as noted i n chapter IV/, can be a t t r i b u t e d d i r e c t l y to a general weakening of interatomic force constants with i n c r e a s i n g interatomic separations. For l a t t i c e parameters not much d i f f e r e n t from t h e i r s t a t i c l a t t i c e values, the frequency d i s t r i b u t i o n s change mostly by a uniform s c a l i n g , and the QHA probably gives a f a i r l y good account of anharmonic e f f e c t s . For l a t t i c e parameters with values cl o s e to the l i m i t i n g values f o r dynamical s t a b i l i t y , changes i n the frequency d i s t r i b u t i o n s are much more extensive, and the p r e d i c t i o n s of the QHA cannot be t r u s t e d . In such l a t t i c e s the atomic motion would be so extensive that the forc e constants would f l u c t u a t e w i l d l y , and the atoms would frequently encounter the hard cores of t h e i r neighbors. While one might f e e l confident that quasiharmonic phonon f r e -quency d i s t r i b u t i o n s f o r s o l i d argon, krypton, and xenon would be f a i r l y r e a l i s t i c f o r low enough temperatures, one would not expect such d i s t r i b u t i o n s to be at a l l a p p l i c a b l e to quantum s o l i d s . Such expectations are c u r i o u s l y o v e r - p e s s i m i s t i c . 99 Figure 7-1 shows phonon frequency d i s t r i b u t i o n s c a l c u l a t e d by a quantum theory of l a t t i c e dynamics f o r fee and hep s t r u c t u r e s of 51 hydrogen and deuterium at T=0K. Due to the large zero point expan-sions i n these systems, the corresponding quasiharmonic d i s t r i b u t i o n s contain imaginary frequencies. In the quantum model, however, a l l frequencies are r e a l and t h e i r d i s t r i b u t i o n s are not much d i f f e r e n t from those determined f o r the RGS i n the harmonic approximation. The low l y i n g d i s t r i b u t i o n s i n Figure 7-1 represent the d i f f e r e n c e s between the frequency d i s t r i b u t i o n s f o r hydrogen and the frequency d i s t r i b u t i o n s f o r the RGS with s t a t i c e q u i l i b r i u m l a t t i c e parameters and long-range (12-6) i n t e r a c t i o n s . The d i s t r i b u t i o n s have been scaled to have equal c u t o f f frequencies - c f . equation (4-2). C l e a r l y , the ground st a t e motions do not depend s e n s i t i v e l y upon whether the c r y s t a l i s a quantum c r y s t a l or a c l a s s i c a l c r y s t a l . What, then, i s the s i g n i f i c a n c e of the extensive changes i n frequency d i s t r i b u t i o n s described i n chapter IV? I f dynamical i n s t a b i l i t y i s the r e s u l t of the vanishing of p a r t i c u l a r phonon frequencies, such extensive changes should be expected. Does the QHA i n d i c a t e c o r r e c t -l y which frequencies are the f i r s t to vanish or, even approximately, what are the upper l i m i t s on the expansion of st a b l e l a t t i c e s ? F. G. Mertens and W. Biem, Z. Phys. 250, 273 (1972). J I I , \ , 1 L I , _ , |l , , r-0 1/4 1/2 3/4 1 0 1/4 1/2 3/4 Figure 7-1. Comparison of phonon frequency d i s t r i b u t i o n s f o r s o l i d hydrogen and the RGS. The s o l i d hydrogen d i s t r i b u t i o n s correspond to T=0.K. The low l y i n g d i s t r i b u t i o n s represent the d i f f e r e n c e s (g u -a-.--) • where the d i s t r i b u t i o n s have been scaled to have equal c u t o f f frequencies. 101 VII-2-3 The RGS S t a b i l i t y Problem The RGS s t a b i l i t y problem, as i t has been customary to discuss 52, i t , i s a problem of c a t a s t r o p h i c c a n c e l l a t i o n . - That i s , whichever s t r u c t u r e - fee or hep - i s the more st a b l e i s determined by the d i f f e r e n c e between two nearly i d e n t i c a l cohesive energies. Unless the sum of the u n c e r t a i n t i e s i n the two c a l c u l a t e d cohesive energies i s l e s s than t h e i r d i f f e r e n c e , the sign of the d i f f e r e n c e i s i n d e t e r -minate - see s e c t i o n VI-2. Catastrophic c a n c e l l a t i o n problems req u i r e s p e c i a l a t t e n t i o n . I f i t i s p o s s i b l e to reorder the terms i n the expressions f o r the q u a n t i t i e s to be compared, i t may be p o s s i b l e to e l i m i n a t e much of the d i f f i c u l t y . In the RGS s t a b i l i t y problem, f o r i n s t a n c e , i f i t were p a s s i b l e ta subtract away f i r s t neighbor c o n t r i b u t i o n s before comparing the cohesive energies, i t would be s a t i s f a c t o r y to deter-mine the remaining c o n t r i b u t i o n s an order of magnitude l e s s p r e c i s e l y than otherwise. This would be f a i r l y easy to accomplish i f there were no atomic motions, but d i f f i c u l t otherwise because the dynamical matrices are only dominated by f i r s t neighbor c o n t r i b u t i o n s . A l s o , the dynamical matrices f o r the two s t r u c t u r e s are of d i f f e r e n t orders, so a p e r t u r b a t i o n treatment would l i k e l y be as d i f f i c u l t as the usual treatment. However, i f both l a t t i c e s were treated as having s i x atoms per _____ The author i s g r a t e f u l to Mr. A. Fowler of the UBC Computing Centre f o r s t r e s s i n g the s i g n i f i c a n c e of c a t a s t r o p h i c c a n c e l l a t i o n i n connection with the RGS s t a b i l i t y problem. 102 u n i t c e l l , a per t u r b a t i o n treatment might be r e l a t i v e l y easy. This would e n t a i l a formal transformation to fee and hep l a t t i c e s i n which the u n i t c e l l s would be c h a r a c t e r i z e d by the s t a c k i n g p e r i o d , ABCABC and ABABAB, r e s p e c t i v e l y . In such a formulation the frequency d i s t r i -butions f o r bath l a t t i c e s would be composed of eighteen p o l a r i z a t i o n • branches. 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Phys. 40, 38 (1968). 106 APPENDIX A CALCULATION OF FCC AND HCP LATTICE SUMS TABLE OF CONTENTS Page Introduction 106 La t t i c e Sums by Direct Summation 108 La t t i c e Sums by Euald's Method 110 Dependence of the L a t t i c e Sums Upon L a t t i c e Parameters 115 A Note an Shells of Hep L a t t i c e Sites 116 Documentation for the L a t t i c e Sum Subroutine Packages 117 L i s t i n g of ELJALDSUMS 127 L i s t i n g of SHELLSUMS 140 L i s t i n g of NNSUMS 146 Introduction In t h i s appendix the l a t t i c e sums required for the c a l c u l a t i o n s in t h i s thesis are described. Three F0RTRAN subroutine packages are described and l i s t e d . These subroutine packages calculate l a t t i c e sums and are interchangeable i n terms of use, though each performs a s l i g h t l y d i f f e r e n t task. 107 A l l of the l a t t i c e sums required i n the calculations for th i s thesis are of.the form: , » r jt where the summation extends to n^ h neighbors of either an fee or an hep l a t t i c e , and the notation i s explained i n Chapter I I . By trans-forming to the reduced variables introduced i n Chapter I I I , i t i s passible to remove the dependence upon the intraplanar f i r s t neighbor separation, a, and define.reduced l a t t i c e sums: sl?(^,V sa ^ p i Z ' A i 'ZZ-qm^ - cn- 2) For fee l a t t i c e s the reduced l a t t i c e sums depend only upon the indices shown. Far hep l a t t i c e s there i s an additional dependence upon the stacking parameter, ^ . Two sorts of abbreviations are used here. I f ^ = 0 the cartesian indices e< and (5 are dropped. When there i s only one sublattice and, also, when the l a t t i c e sums have been accumulated over a l l sublattices the sublattice index A i s dropped. Not a l l of these reduced l a t t i c e sums are independent. Some obviou9 relationships are: = 2- $ o< B ( ¥•» "» )> (A-3c) 108 Also, as a result of inversion symmetry i n the fee l a t t i c e and i n each of the hep sublat t i c e s : <V C s V f , k i f ( ^ , 0] » o - (A-^b) Lat t i c e Sums by Direct Summation The evaluation of the reduced l a t t i c e sums by direct summation i s straightforward. The l a t t i c e points are grouped i n s h e l l s centered on the reference l a t t i c e point, and contributions from s h e l l s of suc-cessively increasing radius are accumulated u n t i l some sort of trunca-tion c r i t e r i o n i s s a t i s f i e d . In d i r e c t summation formulas (A-1,2) are treated l i t e r a l l y as written. Two sorts of truncation c r i t e r i a are useful. F i r s t , a l l l a t t i c e sums may be truncated after the inclusion of n s h e l l s . Since the l a t t i c e sums come out of the interaction potential, t h i s c r i t e r i o n amounts to l i m i t i n g interactions ta n^ n neighbors, that i s , to adopt-ing an nl\l model. The l a t t i c e sums are then exact nl\l l a t t i c e sums. Alt e r n a t i v e l y , one may regard the range of interaction as i n f i n i t e and the truncated l a t t i c e sums as approximations to the all-neighbor, or AW, l a t t i c e sums. Since the rate of convergence of AIM l a t t i c e sums depends strongly on (*i-2^), a di f f e r e n t truncation c r i t e r i o n must be adopted i f a l l l a t t i c e sums are to be evaluated to equal accuracy. One such c r i t e r i o n i s to truncate once the s h e l l contributions f a l l 109 below some small fraction £. of the contributions already accumulated. The r e l a t i v e truncation error in t h i s case may often exceed £, but by less than a factor of 10, even for the most slowly converging l a t t i c e sums required i n t h i s thesis - namely those with (»n-2^ .)=6 and corresponding to the long-range t a i l of the interaction p o t e n t i a l . For S °*Y_>) i t i s possible to replace the truncated terms with an in t e g r a l correction, but in most cases the truncation terror must be estimated by comparing values based upon diff e r e n t values of £. The dependence of some l a t t i c e sums upon the number of s h e l l s included i n th e i r evaluation i s i l l u s t r a t e d in Table A-1. The sums Table A-1. Values of S^Cq) for wt=6,12,16 for various ranges of interaction i n fee and id e a l hep l a t t i c e s . - fee id e a l hep n of sh e l l s 5oyK(o) # of sh e l l s 6 1 2 a oo 12. 12.75 14.453922 1 2 coa 12. 12.75 14.454898 12 1 2 20 oo 12. 12.094 12.131878 12.131880 12.1318802 1 2 10 19 a oo 12. 12.094 12.13219 12.13229 12.1322938 16 1 2 20 40 12. 12.023 12.027354840 12.027354844 1 2 10 19 12. 12.023 12.0274786 12.0274794 Direct summation over the f i r s t 450 s h e l l s and truncated terms replaced by an i n t e g r a l correction. Reference 1. Roger Howard, Phys. Letters 32A,37 (1970). 110 0 £Oi*t0 were chosen because every term i s positive, so there i s no for-tuitous cancellation to enhance convergence, and because these sums are nearly equal i n fee and i d e a l hep l a t t i c e s . Some of the points i l l u s t r a t e d are: 1. For 11M the values of the sums does not depend upon the value of *>*U This i s generally true i f ^ = 0 . 2. Since the rate of convergence depends upon K, 1X< S°^(0) £ S °~'(0) j (A-5) 3. The 1N and 2IM values of 3 (2) are equal for fee and id e a l hep, but i f more s h e l l s are included, S I T ? ( - ) , (A-6) because t h i r d neighbors are closer i n i d e a l hep than i n fee. La t t i c e Sums by Ewald's Method Often the truncation error i n direct summation poses a serious problem. In such cases i t i s desirable to transform the sums so that they converge more rapidly. Ewald's generalized theta function trans-2-4 c^"1/ - \ N formation method transforms the l a t t i c e sums ^xpiiL,*) into much mare complicated expressions than (A-1), but the rate of convergence i s improved to such an extent that truncation error can usually be ignored. 2 P. P. Ewald, Ann. Phys., Leipzig 64,253 (1921). 3 M. Born and M. Bradburn, Proc. Camb. P h i l . S o c , 39,104 (1943). M. Born and K. Huang, Dynamical Theory of Crystal L a t t i c e s (Oxford University Press, London, 1964), appendix I I I . The s t a r t i n g paint i n Ewald's method i s to replace the denom-inator i n the sum with ^ L " ' e f (A-8) which i s obtained from one af the d e f i n i t i o n s of the gamma function. Then, the i n t e g r a l i s s p l i t a r b i t r a r i l y , at u=uQ, and (A-7) takes the (A-9) The f i r s t i n t e g r a l i n (A-9) has the required farm, the second i n t e g r a l i s t r i v i a l , but the t h i r d requires further manipulation. A function with the p e r i o d i c i t y of the l a t t i c e can be written as a Fourier transform involving a reciprocal l a t t i c e sum: = Z J (k(M) '<£.  f (A-1D) where ^ ^ ^ _ t* tir h ( 3 (A-11) and v„ i s the reduced unit c e l l volume. Now, i f a ' 112 (A-12) l i e * f ^ T , the Fourier c o e f f i c i e n t s can be evaluated in closed form. In three dimensions: 77-*-,. . - i g ( b ( M ) = u 3 / z ^ ^ ' (A-13) Therefore, , _up^d) -f i Z 7 i f i d ) The t h i r d i n t e g r a l i n (A-9) i s brought to the desired form by i n t e r -it changing the order of integration and summation ^and u t i l i z i n g (A-14): TAJ-2<> . 5-L H e + X (A-15) X Ji uihere the substitutions, ui=u/Ug and _i=uQ/u, have been made i n the f i r s t and t h i r d i n t e g r a l s , respectively. The f i n a l form i s obtained by using the functions: 00 ' 1 ( A - 1 E 113 With these functions the zero-order sums i n (A-15) take the form: where <f ~ Uo/lT , T f a = a 5 A / / J (A-18) and S = ' A " C/ + ° ~ W T " (A-19) The symbol S i s the anisotropic expansion c o e f f i c i e n t used i n the body of the t h e s i s . The is o t r o p i c expansion c o e f f i c i e n t S i s contained i n the symbol a=aQ(1+8). I t should be noted that l a t t i c e s with S=S=0 need not be i d e a l l y stacked l a t t i c e s with s t a t i c equilibrium f i r s t neighbor separation. That prescription was chosen i n the body of the thesis as a convenient standard. Higher-order Ewald-transformed sums are obtained from (A-17) by d i f f e r e n t i a t i o n , according to (A-3a). The phonon spectrum calcula - r j,-m. tions i n t h i s thesis required only S (f^A) and S«p(!i>l) , but the minimization of the Gibbs function by Newton's method required S«/3 (S,%\ The e x p l i c i t forms of the f i r s t - and second-order sums are: 114 + i (A-20) The e f f e c t of Ewald summation i s to s p l i t the l a t t i c e sums i n t o d i r e c t and r e c i p r o c a l l a t t i c e components. The prime on the d i r e c t l a t t i c e summation i n d i c a t e s that the term with ^^J=Cj i s to be ex-cluded. However, i n the r e c i p r o c a l l a t t i c e summation, the term with =JJ must be inclu d e d , e s p e c i a l l y f o r small wavevectors f o r which i t i s the most important c o n t r i b u t i o n . Just as i n d i r e c t summation the evaluation of Ewald sums proceeds by accumulating c o n t r i b u t i o n s s h e l l - b y - s h e l l . For the d i r e c t l a t t i c e components there i s no problem, but f o r the r e c i p r o c a l l a t t i c e compo-115 nents the appearance of (b/k)+ )^ i n the arguments of the <f>-functions leads to ambiguity i n the d e f i n i t i o n of s h e l l s , except for^=p_. Problems associated with t h i s ambiguity can be l a r g e l y ignored i f ^_ i s chosen to remain always w i t h i n the B r i l l u o i n zone. The parameter i s a r b i t r a r y . I t s value determines the r a t e s of convergence of the two components of the transformed l a t t i c e sums: the l a r g e r i t s value the f a s t e r i s the convergence of the d i r e c t l a t t i c e component and the slower i s the convergence of the r e c i p r o c a l l a t t i c e component. Born and Bradburn have used the value f = f c , but t h i s choice does not seem superior to the choice ^=3/8, which hastens the convergence of the more time consuming r e c i p r o c a l l a t t i c e component. A l s o , i t should be noted that the Ewald transformation a l t e r s the meaning of s h e l l c o n t r i b u t i o n s . To r e s t r i c t to c o n t r i b u t i o n s from the s h e l l s of f i r s t neighbors does not have the e f f e c t of adopting a 1I\1 i n t e r a c t i o n , but only of accepting very crude approximations to the AN l a t t i c e sums. For t h i s reason i t i s probably advisable to truncate Ewald sums on the b a s i s of the magnitude of the c o n t r i b u t i o n s , rather than on the b a s i s of a c e r t a i n number of c o n t r i b u t i n g s h e l l s . Dependence of the L a t t i c e Sums upon L a t t i c e Parameters I t has already been noted that the reduced l a t t i c e sums have no dependence upon i s o t r o p i c l a t t i c e expansions. That dependence has been removed by the e x t r a c t i o n of the f a c t o r G\ from the l a t t i c e sums i n ( A - 1 ) . Since fee l.atticesVaire assumed to expand i s o t r o p i c a l l y , fee l a t t i c e sums warrant no f u r t h e r comment i n t h i s s e c t i o n . The reduced hep l a t t i c e sums depend upon the stacking parameter in a very complicated fashion, since some contributions depend upon while others do not. For f . 5 % deviations o f " ^ from i t s value i n i d e a l stacking, long-range values of S (2) and (°) vary by and +2%, 1 respectively. These resul t s can be explained on t h B basis of f i r s t neighbors only. Of twelve f i r s t neighbors of each s i t e i n an hep la t t i c e , : t h e r e are six for which =1 and six for which ^ a 3 3 <•' K ' where 9^ i s measured from i d e a l stacking. Therefore, ' C ? (A-22) For £/;=+.005 the deviations from the S^=0 values are exactly +1% for >*=6 and +2% for"»t=12. For substantially greater deviations from i d e a l stacking, second and further neighbors must be included. A Note on Shells of Hep L a t t i c e Sites In i d e a l hep the two groups of f i r s t neighbors mentioned above, in connection with (A-22), c l e a r l y belong to the same s h e l l , whereas in nonideal hep they constitute two s h e l l s . A s p l i t t i n g of f i r s t neighbors into two s h e l l s has computational advantages since half of them are affected by the stacking parameter. However, t h i s s p l i t t i n g should be regarded as an a r t i f i c e to a s s i s t computation, and the two groups of f i r s t neighbors should be regarded as belonging to the same physical s h e l l . The same comment applies to any set of s h e l l s which have the same radius i n i d e a l hep. 117 Documentation for the L a t t i c e Sum Subroutine Packages Three interchangeable subroutine packages have been developed to evaluate l a t t i c e sums. The programming language i s F0RTRAIM and a l l arithmetic i s double precisi o n . EUALDSUMS uses Ewald's method, so i t i s the slowest but most accurate for infinite-range l a t t i c e sums. The l a t t i c e sums available are SjpCM) for -j =0,1,2 and7*'=6,8,10, l«,m+2,-afrit :*>6. (For-^=2,^=0 i s assumed.) The sums are truncated once the s h e l l contributions amount to only a small f r a c t i o n , RELERR, of the contributions already accumulated. SHELLSUMS evaluates l a t t i c e sums d i r e c t l y , so i t can be used in nIM interaction models. It i s e s s e n t i a l l y EUALDSUMS with the Ewald transformation,removed. Shells beyond those for which £ v>/=RELERR are ignored. NIMSUMS evaluates 1f\l sums. It i s the simplest and fastest of the three subroutine packages. The-^=2 sums are not av a i l a b l e . Although EUALDSUMS and SHELLSUMS have been tested only for fee and hep l a t t i c e s , they should, i n p r i n c i p l e , be applicable to a wide variety of l a t t i c e structures i f the input l a t t i c e data i s changed. IMIMSUMS, on the other hand, i s r e s t r i c t e d by i t s s i m p l i c i t y to fee and hep. The structures of these three subroutine packages are shown schematically i n Figures A-1,2,3. The subroutine names are i n upper case and a cryptic description i s given i n lower case. The arrows go from the CALLing subroutine to the CALLed subroutine. The MAIN pro-grams are not parts of the packages. 118 EWALD l a t t i c e sums SQ s"YW) Figure A-1. Schematic diagram cf EWALDSUMS  EWIIMIT LTLOAD i n i t i a l i z a t i o n — * loads l a t t i c e data FAC GHLF rem) AVALD sublattice sums SM INDIX supervises indexing S1 S2 PHIS calculation of +n*(X) EH EI REKUR recurrence r e l a t i o n 119 MAIN EWALD l a t t i c e sums SO Figure A-2. Schematic diagram af SHELLSUMS  EWINIT LTLOAD i n i t i a l i z a t i o n — * loads l a t t i c e data Al/ALD s u b l a t t i c e sums SM z IIMDIX supervises indexing S1 SZ Input L a t t i c e Data. The s p e c i f i c a t i o n s of the input l a t t i c e data f i l e can be determined by examining the subroutine LTLOAD i n e i t h e r EWALD5UMS or SHELLSUMS. The meanings of the important v a r i a b l e names i n LTLOAD are as f o l l o u i s : NDLAT: the number of d i r e c t s u b l a t t i c e s (1P„ +2. =3). tec hep IUDS: the number of s h e l l s i n each d i r e c t s u b l a t t i c e . For hep i t i s necessary at t h i s stage to so r t l a t t i c e s i t e s i n t o s h e l l s according to both the radius of the s h e l l and the 120 MAIN Figure A-3. Schematic diagram of NNSUMS EtiilNIT i n i t i a l i z a t i o n EWALD l a t t i c e sums AV/ALD sublattice sums (hep) HCPNNI HCPNNO strPcu) ARGS t r i g , function arguments FCCNNO FCCNN1 magnitudes of the z-components of the l a t t i c e vectors. This additional d i s t i n c t i o n i s ignored at the stage of determining what constitutes a s h e l l contribution, but i t i s important i f nonideal stacking i s to be considered. ND: indices for the endpoints of each set of sublattice paints in DL. DL: a one dimensional array uith d i r e c t l a t t i c e points l i s t e d 121 sequentially. NDL: indices for the endpoints of dir e c t l a t t i c e s h e l l i n DL. DR: the squared reduced radius af each direct l a t t i c e s h e l l . DZ: the squared reduced z-component of the,'direct l a t t i c e s i t e p ositions. Space i s provided for fee, but only data far hep i s used. The same variable names ui t h "R" replacing "D" have the same meanings in connection u i t h the reciprocal l a t t i c e s . The input l a t t i c e data i s read from unit 5. COMMON /SUMS/. To f a c i l i t a t e interchangeability of the three l a t t i c e sum subroutine packages, a l l u t i l i z e the same COMMON /SUMS/ f i e l d , or a subset of i t . The f i e l d i s set up by: COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z(3),D(3),C0EFF(6),PAR(13), 1 DL(5060),RL(4350),DR(95),DZ(95),RR(72),RZ(72), 2 NDLC95),NRL(72),NDS(4),NRS(3),ND(4),NR(3),J3D(6) The meanings of these variable names are as follows: * Y: the reduced wave vector (set to 0 by EUINIT). * DF=1+S^ (set to 1 by EUINIT). * E=-&^(2+SK) (set to 0 by EUINIT). * F=-E/DF/DF (set to 0 by EUINIT). * RELERR: the truncation parameter. In EUALDSUMS the sums are truncated when the s h e l l contributions f a l l short af the current value of the sum by the small f r a c t i o n , RELERR. 122 In SHELLSUMS the sums are truncated mhen the squared reduced r a d i i exceed RELERR. ** Z: i s the zero vector. ** D: i s the vector separating hep s u b l a t t i c e s . *** COEFF: an array u i t h the c o e f f i c i e n t s outside the braces i n equations (A-17,20,21). *** PAR: an array uith other c o e f f i c i e n t s required for Euaid summation. * 33D: an array uith the minimum numbers of s h e l l s to be included i n the various d i r e c t and r e c i p r o c a l l a t t i c e sum contribu-tions i n EUIALDSUMS. (ElilINIT sets those for the d i r e c t l a t t i c e to 10 and those for the r e c i p r o c a l l a t t i c e to 6). * May be changed at any time before or a f t e r a l a t t i c e sum i s required. DF,E, and F must s a t i s f y the r e l a t i o n s l i s t e d . ** Must, not be changed at a l l during execution. *** Parameters required for EUIALDSUMS. They may be changed only by subroutine EkllNIT. I n i t i a l i z a t i o n . Each of the l a t t i c e sum subroutine packages has an i n i t i a l i z a t i o n stage that must be executed before any l a t t i c e sums are calculated. The i n i t i a l i z a t i o n sets up arrays and reads l a t t i c e data, i f necessary. EUALDSUMS was designed s p e c i f i c a l l y for use i n see previous section on input l a t t i c e data. 123 calculations involving (m-6) potentials. The repulsive exponent, m, determines what i s put into COEFF and PAR by the i n i t i a l i z a t i o n , and i t s value cannot be changed i n EWALDSUMS without r e - i n i t i a l i z a t i o n . In SHELLSUMS and IMIMSUMS the value of m i s not fixed by i n i t i a l i z a t i o n . To i n i t i a l i z e : CALL EUJINIKIIMIT,A,XI ,PELERR ,M,M0DD) where f =0 La t t i c e data read and arrays i n i t i a l i z e d . #0 Arrays i n i t i a l i z e d , but no l a t t i c e data read. A: The f i r s t neighbor separation. Set to 1 i f input value i s not p o s i t i v e . A=1 yields reduced l a t t i c e sums. Xl=jf : The convergence parameter. Set to 3/8 i f input value i s l ess than 0.01. PELERR: The value to be used for RELERR, the truncation parameter. M: The (m-6) repulsive exponent. Must exceed 6. (=0 i f M even.l MOOD i f M Q d d f Required only for EUIALDSUMS. Obtaining L a t t i c e Sums. To obtain l a t t i c e sums after i n i t i a l i z -ation : CALL Al/ALD(J,M>MOOD,POEFF,ZUMfDUM) for hep sublattice sums, or CALL EWALD(_t,M,NS,MOOD,POEFF,SUM) for l a t t i c e sums. The input parameters are: 3: The order of the l a t t i c e sum. For EWALDSUMS and SHELLSUMS 3=0,1,2. For IMIMSUMS J=0,1 only. 124 M: The exponent i n the denominator of the l a t t i c e summand. For calculations u i t h (m-6) potentials the only values permitted i n EUALDSUMS are M=6,8,10,n\, 7K+2, and rfi. +4. {=1 for fee. =2 for hep. MOOD: As i n previous section. P0EFF=C0EFF(L), where L=1,2,3,4,5,6 for M=6,8,10,7n ,7n+2,^+4. Needed only i n EUALDSUMS. ZUM: I [ sVp($,l) DUM SUM contain ^ 5 * / 3 ^ , 3 - / In ZUM(I,J), DUM(I,3), and SUM(I,J) the r e a l and imaginary parts are stored i n the 1=1 and 2 locations, respectively. 3 labels the cartesian coordinate p a i r s : 3=1,2,3,4,5,6 for (e<,^ )=(1,1),(2,2), (3,3),(2,3),(3,1),(1,2), respectively. E f f i c i e n c y . These subroutine packages form a sequence: EUALDSUMS, SHELLSUMS, and NNSUMS, where the order i s that of decreasing complex-i t y , accuracy (for AN sums), and execution times. A t h e o r e t i c a l evaluation of errors i n EUALDSUMS i s very d i f f i c u l t . I t i s possible, houever, to obtain good error estimates by noting the dependence of l a t t i c e sums upon the arbit r a r y parameter ^ and the accuracy of the input l a t t i c e data. I f the summation were performed exactly, there would be no ^-dependence, despite the strong f-depend-ence of the direct and rec i p r o c a l l a t t i c e components separately. However, computers use f i n i t e d i g i t representations of rea l numbers, so 125 round-off error should introduce some £-dependence. Also, the sums are truncated, and t h i s should introduce some ^-dependence. In a test with RELERR=3x10~8 and 1A-Jf\5 1 f l a t t i c e sums were 7 reproducible to roughly 3 parts i n 10 uiith l a t t i c e data accurate to the f u l l double word length and, also, u i t h l a t t i c e data accurate to 10 1 part i n 10 . The only difference between the two sets of re s u l t s was that more s h e l l s of the le s s accurate data were required to s a t i s f y the truncation c r i t e r i o n . Smaller values of RELERR and values of f outside the indicated range could not be considered with the amount of l a t t i c e data currently handled by EUIALDSUMS. In setting up a dynamical matrix, two c a l l s to the chosen subroutine package are necessary. I f the sums are high precision sums from EUIALDSUMS, the cost i s quite high - about 20$ per dynamical matrix. Sums from SHELLSUMS, involving a comparable amount of l a t t i c e data, would be much less expensive, as a result of not having to go through the complexities of the Ewald transformation. I f the sums were obtained from IMIMSUMS, the costs would be yet lower because much le s s l a t t i c e data and programming structure would be involved. The tests of these subroutine packages were performed on the IBM 360/167 at the University of B r i t i s h Columbia. Costs and execution times were estimated from the re s u l t s of the tests and are l i s t e d i n Table A-2. Listed are the costs and times for single c a l l s for l a t t i c e sums and for enough c a l l s to construct phonon spectra composed of ~3x10^ phonon frequencies. Since these tests, UBC has received a new 370/168 for which the cost per unit time of the CPU i s roughly t r i p l e d , but the execution time i s cut by a factor of about s i x . Therefore, the costs l i s t e d below may be halved on the new system. Table A-2. Estimates of casts and execution times for single c a l l s to the l a t t i c e sum subroutine packages, and for enough c a l l s to construct a phonon spectrum of 5 /x-'3x10 phonon frequencies. Costs ($) (Rate factor=1.0) Execution Time (sec) Single C a l l Spectrum Single C a l l Spectrum •EUIALDSUMS „ (RELERR=3x10 ) .1 2400 .6 15,000 SHELLSUMS 1N 10l\l 20N 3 ON .0017 .0024 .0036 .0048 44 60 80 120 .012 .015 .024 .06 300 400 600 700 NNSUMS .001 24 .006 150 L i s t i n g . The F0RTRAIM coding for the subroutine packages EUIALDSUMS, SHELLSUMS, and NNSUMS i s l i s t e d below. ******************************************************* ****** LISTING OF EWALDSUMS ************************ ******************************************************* 127 SUBROUTINE EWIN IT{INIT,A,XI,PELERR»N,MOOD) IMPLICIT REAL*8 (A-H,Q-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z(3),D<3),COEFF<6),PA R ( I 3 ) , 1 DL(5060),RL{4350),DR(95),DZ{95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) 10 FORMAT(* STOP BECAUSE ( N.LT.6)*) 20 FORMAK•01NITIALIZATION FOR EWALD SUMS COMPLETE * t/» * A = .,»F10.5, l» XI=',F10.5,» RELERR-*.F10.8,' N=«,I3) R2=1.41421356237309500 R3=1.7320508083D0 PI=3.14159265358979D0 RELERR=PELERR IFIN.GE.6) GO TO 100 WRITE(7, 10) STOP 100 IF(INIT.NE.O.AND.NRS(l).NE.l) CALL LTLOAD IF(A.LE.O.DO) A=1.D0 I F ( X I . L E . l . D - 2 ) XI=0.375D0 IF(RELERR.LT.l.D-lO) RELERR=3.D-8 D(1)=0.D0 D(2)=-1.D0/R3 D(3)=R2/R3 DO 200 1=1,3 Y(I)=0.D0 Z(I)=0.D0 JJD(I+I)=10 J J D ( I + I - l ) = 6 200 CONTINUE DF=1.D0 E=O.DO F=O.DO PAR(1)=PI+PI PAR(2)=PI*XI X=PAR(2)+PAR(2) PAR(3)=PI/XI PAR(4)=PAR(3)+PAR(3) PAR(5)=PAR(4)*PAR(4) PAR(6)=A*A PAR(7)=PAR(6)*PAR(6) PAR(8)=R2/XI/DSQRT(XI) PAR(9)=0.5D0*PAR(8) PAR(10)=PAR(8)/X PAR(11)=0.5DO*PAR(10) PAR(12)=PAR(10)/X PAR(13)=0.5D0*PAR(12) CI=PAR(2)/PAR(6) L = N/2 LL=L+2 M00D=M0D(N,2) C0=1.D0 DO 300 K=i,LL CO=CO*CI IFtK.EQ.3) C0EFF(1)=C0 IF(K.EQ.4) C0EFF(2)=C0 IF(K.EQ.5) C0EFF(3)=C0 IF(K.EQ.L) C0EFF(4)=C0 IF(K.EQ.(L + l ) ) C0EFF(5)=C0 IF(K.EQ.LL) C0EFF(6)=C0 300 CONTINUE DO 400 1=1,3 COEFF(I)=COEFF(I)/FAC(1+1 ) 400 CONTINUE DCI=DSQRT(CI> DO 600 1=4,6 IF(MOOD.EQ.l) GO TO 500 C0EFF(I)=C0EFF(I)/FAC(L+I-5) GO TO 600 5 00 C0EFF(I)=C0EFF( I )*DCI/GHLF(L +1-4) 600 CONTINUE WRITE(7,20) A,XI,RELERR,N RETURN END SUBROUTINE LTLOAD IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z(3),D(3),COEFFl6),PA R ( 1 3 ) , 1 DL(5060),RL{4350),DR<95),DZ(95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) 10 FORMAT('OLATTICE DATA LOADED',///) RE AD ( 2) NDL AT, (IMDS( I ) , I = 1,NDLAT) ,(ND( I ) ,1=1, NDL AT) ,ND( NDLAT+1) READ(2) NRLAT,(NRS(I),I=1,NRLAT),(NR(I),1=1,NRLAT),NR( NRLAT+1) NN=3*ND(NDLAT+1) DO 100 I=1,NN,24 L=I+23 IF(L.GT.NN) L=NN READ(2) (DL(J ),J=I,L) 100 CONTINUE NN=3*NR(NRLAT+1) DO 200 I=1,NN,24 L=I+23 IF(L.GT.NN) L=NN READ(2) (RL(J ),J=I,L) 200 CONTINUE NS=0 DO 300 I = l t NDLAT NS=NS+NDS(I) 300 CONTINUE READ(2) (NDL(J),J=1,NS) READ(2) (DR(J),DZ(J),J=1,NS) NS=0 ^ DO 400 1=1,NRLAT NS=NS+NRS(I ) 400 CONTINUE l 2 g READ(2) (NRL(J),J=1,NS) READ(2) (RR(J),RZ(J),J=1,NS) NDS(4)=NDS(3)+NDS(2)+NDS( NDS(3)=N DS(2)+NDS(1} + 1 NDS(2)=NDS(1)+1 NDS(1)=1 NRS(3)=NRS(2)+NRS(1)+1 NRS(2)=NRS(1)+1 NRS(L)=1 WRITE(7,10) RETURN END FUNCTION FAC(N) REAL*8 F A C X I F ( N . L T . l ) STOP FAC=1.D0 DO 100 1 = 1,N X=I*FAC FAC = X 100 CONTINUE RETURN END FUNCTION GHLF(N) REAL*8 GHLF fX I F ( N . L T . l ) STOP GHLF=1.772453850905516D0 DO 100 I = l t N X=(I+I-l)*GHLF/2.D0 GHLF=X 100 CONTINUE RETURN END SUBROUTINE AVALD(M,N,MOOD,COEFF,ZUM,DUM) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z(3),D(3),POEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL(4350),DR(95),DZ(95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION ZUM(2,6),DUM(2,6),S(2) IF(M.NE.O) GO TO 100 CALL SO(N,2,MOOD,COEFF,Z,S) ZUM(1,1)=S(1) ZUM(2,1)=S(2) CALL S0(N,3,MOOD,COEFF,D,S) DUM(1»I)=S(1) DUM(2,1)=S(2) GO TO 200 100 CALL SM(M,N,2,MOOD,COEFF,Z,ZUM) CALL SM(M,N,3,MOOD,COEFF,D,DUM) 200 RETURN END SUBROUTINE EWALDIM,N,NS,MOOD,COEFF,SUM) IMPLICIT REAL*8 (A-H,0-Z) 1 3 0 COMMON / SUMS/ Y(3),DF,E,F,RELERR,Z(3),D(3),POEFF(6),PA R113), 1 DL(5060), RL (4350),DR(95),DZ(95),RR(72 ) ,R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDSI 4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION SUM(2,6),TUM(2,6),S(2) IF(M.NE.O) GO TO 100 CALL S0(N,NS,MOOD,COEFF,Z,S ) SUM(1,1)=S(1 ) SUM(2,1)=S(2) IF(NS.EQ.l) GO TO 400 CALL SO(N,3,MOOD,COEFF,D,S) SUMl1,1)=SUM(1,1)+S(1) SUM(2,1)=SUM(2,1)+S(2) GO TO 400 100 CALL SM<M,N,NS,MOOD.COEFF,Z,SUM) IF(NS.EQ.l) GO TO 400 CALL SM(M,N»3,MOOD,COEFF,D,TUM) DO 300 1=1,2 DO 200 J=l, 6 SUM(I,J) = SUM(11 J)+TUM ( I , J) 200 CONTINUE 300 CONTINUE 400 RETURN END SUBROUTINE SM(M,N,NS,MOOD,COEFF,D,SUM) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y{3),DF,E,F,RELERR,V<3),W(3>,POEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL(4350),DR{95),DZ495),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION SU(2),TU(2),SUM(2,6),D(3) DO 300 1 = 1,6 CALL IND I X ( I , I A , I B ) IF(M.EQ.2) GO TO 100 CALL S1(N,NS,MOOD,D,IA,IB,TU) SUU ) = TU(1)*PAR{6) SU(2)=TU(2)*PAR(6) GO TO 200 100 CALL S2(N,NS,M00D,D,IA,IB,TU) SU(1)=TU(1)*PAR(7) SU(2)=TU(2)*PAR(7) 200 SUM(1,I)=COEFF*SU(1) SUM(2,I)=COEFF*SU(2) 300 CONTINUE RETURN END SUBROUTINE I N D I X ( I , I A , I B ) IFU.GT.3) GO TO 100 IA=I IB=I RETURN 100 I F ( I . G T . 4 ) GO TO 200 IA=2 I B = 3 RETURN 200 I F ( I . G T . 5 ) GO TO 300 IA=3 I B = 1 RETURN 300 I A = 1 IB=2 RETURN END SUBROUTINE PHIS < X , N , N S I G , M O O D , N 0 , P H I } I M P L I C I T RE AL *8 ( A - H , 0 - Z ) A = D E X P ( - X ) / X MN=N/2 I F ( N S I G . L T . O ) GO TO 200 NDIR=1 I F ( M O O D . E Q . 1 ) GO TO 100 XMO=O.DO F I = A M=MN-1 GO TD 600 100 XMO=-0 .5D0 F I = E H ( X ) M=MN GO TO 600 200 I F ( N O . L T . N ) GO TO 400 NDIR=1 IF (MOOD.EQ.l) GO TO 300 X M 0 = - 0 . 5 D 0 F I = E H ( X ) M=NO-MN+1 GO TO 600 300 XMO=0.D0 F I=A M=NO-MN GO TO 600 4 0 0 N D I R = - 1 M=MN-N0-1 I F ( M O O D . E Q.l) GO TO 500 X M 0 = - 0 . 5 D 0 F I = E H ( X ) GO TO 600 500 X M 0 = - 1 . D 0 F I = E I ( X , A ) 600 I F ( M . E Q . O ) GO TO 700 CALL R E K U R ( M » X M O , N D I R , F I , A , X , PH I ) GO TO 800 700 P H I = F I 800 RETURN END SUBROUTINE R E K U R ( M t X M O , N D I R , F I , A , X , F F I M P L I C I T REAL*8 ( A - H , 0 - Z ) GG=F I I F ( N D I R . L T . O ) GO TO 200 DO 100 1=1,M XI=XMO+I „ B=XI/X G=A+B*GG GG=G 100 CONTINUE GO TO 400 200 DO 300 1=1,M XI=XM0+1-I B=X/XI G=(GG-A)*B GG=G 300 CONTINUE 400 FF=GG RETURN END FUNCTION EI(X,Y) IMPLICIT REAL *8 (A-H,0-Z) IF(X.GT.l.DO) GO TO 100 EI = ( ( ( ( 1 . 0 7 8 5 7D-03*X-9.76004D-03)*X+5.519968D-02)*X-2. 4991055D-01) l*X+9.9999193D-0 1)*X-5.7721566D-01-DL0G(X) GO TO 1000 100 A=( UX+8.5733287401D0)*X+18.0590169730D0)*X+8.63476089 25D0)*X+ 10.2677737343D0 B=(((X+9.5733223454D0)*X+25.6329561486D0)*X+21.0996530 827D0)*X+ 1 3.958496922800 EI=Y*A/B 1000 RETURN END FUNCTION EH(X) IMPLICIT REAL *8 (A-H,0-Z) Y=DSQRT(X) EH=1.772453850905516D0/Y*(1.DO-DERF(Y)) RETURN END SUBROUTINE SO{N,NS,MOOD,COEFF,D,SUM) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,VI 3),W(3),POEFF( 6),PA R U 3 ) , 1 DL(5060),RL(4350),DR(95),DZ(95),RR(72 ) ,R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD<6) DIMENSION SUM(2),D(3) INITIALIZATION MS=NS IF(NS.EQ.3) MS=2 VOEFF=PAR(7+MS)/DF SDR=0.D0 SDI=O.D0 SRR=O.DO SRI=O.DO DIRECT LATTICE SUM MD=NDS(NS)-1 KD=MD+JJD(1) JD=NDS(NS+1)-l LIMH=NOL(MD + 1)-5 IF(MD.NE.O) LIMH=NDL(MD)-2 RERR=O.DO RERI=O.DO 100 MD=MD+1 IF(MD.GT.JD) GO TO 600 SKR=O.DO SKI=O.DO LIML= LIMH+3 LIMH=NDL(MD)-2 DO 500 I=LIML,LIMH,3 IF(I.NE.LIML) GO TO 300 DD=DR(MD ) IF(NS.NE.l) DD=DD-DZ(MD)*E DDI=DD*DFL0AT{LIMH-LIML+l)*3.D-9 DDR=DDI*i.D4 IF(DD.LT.l.D-lO) GO TO 100 ARG=DD*PAR(2) CALL PHIStARG,N,l,MOOD,0,PHI) 300 YR=0.D0 DO 400 J = l , 3 YR=YR+DL(I+J-l)*Y(J) 400 CONTINUE ARG=YR*PAR(1) SKR=SKR+DCOS< ARG) SKI=SKI+DSIN(ARG) 500 CONTINUE SCR=SKR*PHI IF(DABS(SKR).LT.DDR.AND.SDR.NE.0.DO) GO TO 533 SCR=SKR*PHI SDR=SDR+SCR 533 IF(DABS(SKI).LT.DDI.AND.SOI.NE.0.DO) GO TO 567 SCI=SKI*PHI SDI=SDI+SCI 567 IF(MD.LE.KD) GO TO 100 IF(DABS(DR(MD)-DR(MD+1)).LE.1.D-4) GO TO 100 IF(DABS(SDR).GT.1.D-10) RERR=DABS(SCR/SDR) IF(DABS(SDIJ.GT.l.D-10) RERI=DABS(SCI/SDI) IF(RERR.LT.RELERR.AND.RERI.LT.RELERR) GO TO 600 GO TO 100 RECIPROCAL LATTICE SUM 600 MD=NRS(MS)-1 KD=MD+JJD(2) JD=NRS(MS+1)-l RERR=0.DO RERI=0.D0 LIMH=NRL(MD+1)-5 IF(MD.NE.O) LIMH=NRL(MD)-2 700 MD=MD+i IF(MD.GT.JD) GO TO 1200 SCR=O.DO SCI=O.DO LIML=LIMH+3 LIMH=NRL(MD )-2 DO 1100 I=LIML,LIMH,3 1 3 A DD=O.DO DO 800 J = l , 3 P=RL(I+J-l)+Y(J ) IFtMS.NE.l.AND.J.EQ.3) P=P/DF DD=DD+P*P 800 CONTINUE IF(DD.GT.l.D-lO) GO TO 900 DEN=N-3 PHI=2.D0/DEN C=1.D0 S=O.DO GO TO 1050 900 ARG=DD#PAR(3) CALL PHIStARG,N,-l,MOOD,0,PHI) YR=O.DO DO 1000 J = 2,3 YR=YR+RL(I+J-1)*D(J) 1000 CONTINUE ARG=-YR* PAR(1 ) C=DCOS(ARG) S=DSIN(ARG) 1050 SCR=SCR+C*PHI SCI=SCI+S*PHI 1100 CONTINUE IF((DABS(SCI).LT.l.D-12).OR.(DABS(SCI ) . LT.1.D-9.AND. 1SRI.EQ.0.D0)) SCI=O.DO SRR=SRR+SCR SRI=SRI+SCI IF(MD.LT.KD) GO TO 700 IF(DABS(RR(MD)-RR(MD+1)).LE.l.D-4) GO TO 700 IF ( DABS'* SRR).GT.1.D-L0) RERR= DABS ( SCR/SRR ) IF(DABS(SRI).GT.l.D-10) RERI=DABS(SCI/SRI) IF(RERR.LT.RELERR.AND.RERI.LT.RELERR) GO TO 1200 GO TO 700 1200 IF(NS.NE.3) SDR=SDR-2.DO/DFLOAT(N) SUM(1)=C0EFF*(SDR+V0EFF*SRR) SUM(2)=C0EFF*(SDI+V0EFF*SRI ) RETURN END SUBROUTINE S1(N,NS,MOOD,D,IA,IB,TU) IMPLICIT REAL*8 (A-H.O-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,V i3),W(3),POEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL(4350),OR(95),DZ!95),RR(72),R Z(72) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION TU(2),0(3) C INITIALIZATION MS=NS IF(NS.EQ.3) MS=2 V0EFF=PAR(9+MS)/DF SDR=0.D0 SDI=O.DO SRR=O.DO SRI=O.DO C DIRECT LATTICE SUM MD=NDS(NS)-1 KD=MD+JJD(3) JD=NDS(NS+1 )-L LIMH=NDL(MD+1)-5 IF(MD.NE.O) LIMH=NDL(MD)-2 RERR=O.DO RERI=O.DO FAC=l.DO IF(NS.EQ.l ) GO TO 100 IF(IA.EQ.3) FAC=DF IFUB.EQ.3) FAC=FAC*DF 100 MD=MD+1 IF(MD.GT.JD) GO TO 600 SKR=O.DO SKI=O.DO LIML=LIMH+3 LIMH=NDL(MD)-2 DO 500- I=LIML,LIMH,3 IF(I.NE.LIML) GO TO 300 DD=DR(MD) IF(NS.NE.l) DD=DD-DZ(MD)*E DDR=DD*DFLOAT(LIMH-LIML+1)*3.D-5 DDI=DDR*1.D-2 IF(DD.LT.l.D-lO) GO TO 100 ARG=DD*P AR(2) CALL PHIS!ARG,N,1,MOOD,0,PHI) 300 YR=O.DO DO 400 J = l , 3 YR=YR+DL(I+J-1)*Y{J) 400 CONTINUE ARG=YR*PAR{1) C=DCOS(ARG) S=DSIN(ARG) RRAB=DL(I+IA-1)*DL(I+IB-1) SKR=SKR+C*RRAB SKI=SKI+S*RRAB 500 CONTINUE NU=0 IF(SKR.EQ.O.DO.ANID.SDR.NE.O.DO) GO TO 520 IF(DABS(SKR).LT.DDR) GO TO 540 SCR=SKR*PHI*FAC SDR=SDR+SCR GO TO 540 520 NU=1 540 IF(SKI.EQ.O.DO.AND.SDI.NE.O.DO) GO TO 560 IF(DABS(SKIJ.LT.DDI) GO TO 580 SCI=SKI*PHI*FAC SDI=SDI+SCI GO TO 580 560 NU=2 580 IF(NU.NE.O) GO TO 100 IF(MD.LE.KD) GO TO 100 IF(DABS(DR(MD)-DR(MD+I)J.LE.l.D-4) GO TO 100 IF(DABS(SDRJ.GT.l.D-10) RERR=DABS(SCR/SDR) IF(DABS(SDI).GT.1.D-10) RERI=DABS<SCI/SDI) 136 IFIRERR.LT.RELERR.AND.RERI.LT.RELERR) GO TO 600 GO TO 100 C RECIPROCAL LATTICE SUM 600 MD=NRS(MS)-1 KD=MD+JJD(4) JD=NRS(MS+1)-1 RERR=0.D0 RERI=0.D0 LIMH=NRL(MD+1)-5 IF(MD.NE.O) LIMH=NRL(MD)-2 YMO=2.D0/(DFLOAT(N)-5.D0) 700 MD=MD+1 IF(MD.GT.JD) GO TO 1200 SUR=O.DO SUI=O.DO SKR=O.DO SKI=0.D0 LIML = LIMH+3 LIMH=NRL(MD)-2 DO 1100 I=LIML,LIMH,3 DD=O.DO DO 800. J = l , 3 C = RL( I + J - l ) + Y ( J ) IFtMS.NE.l.AND.J.EQ.3) C=C/DF DD=DD+C*C 800 CONTINUE IF(DD.GT.l.D-10) GO TO 900 CHI=O.DO IF(IA.EQ.IB) CHI=YMO SKR=SKR-CHI GO TO 1100 900 ARG=DD*PAR(3) CALL PHIS(ARGtN»-l» MOOD,2» PHI) CHI=0.D0 IF(IA.NE.IB) GO TO 950 AA=DEXP(-ARGJ/ARG CHI=ARG*(AA-PHI)*YMO 950 YR=0.D0 DO 1000 J=2,3 YR=YR+RL(I+J-1)*D(J) 1000 CONTINUE ARG=-YR*PAR ( 1) C=DCOS(ARG) S=DSIN(ARG) P=RL(I+IA-1)+Y(IA) Q=RL(I + IB-1) + Y( IB) IF(MS.EQ.l) GO TO 1050 IF(IA.EQ.3) P=P/DF IF(IB.EQ.3) Q=Q/DF 1050 PQ=P*Q*PHI SUR=SUR+C*PQ SUI=SUI+S*PQ I F l I A . N E . I B ) GO TO 1100 SKR=SKR-C*CHI SKI=SKI-S#CHI 1100 CONTINUE 1 3 7 SCR=PAR{4)*SUR+SKR SCI=PAR(4)*SUI+SKI NU=0 IF(SCR.EQ.O.DO.AND.SRR.NE.O.DO) GO TO 1120 IF((DABS(SCR).LT.l.D-12).OR.(DABS(SCR).LT.1.0-9.AND. 1SRR.EQ.0.D0)) GO TO 1140 SRR=SRR+SCR GO TO 1140 1120 NU=i 1140 IFISCI.EQ.O.DO.AND.SRI.NE.O.DO) GO TO 1160 1F((DABS(SCI).LT.l.D-12).OR.(DABSlSCI).LT.1.D-9.AND. 1SRI.EQ.0.D0)) GO TO 1180 SRI=SRI+SCI GO TO 1180 1160 NU=2 1180 IF(NU.NE.O) GO TO 700 IF(MD.LT.KD) GO TO 700 IF(DABS(RR(MD)-RR(MD+1)).LE.l.D-4) GO TO 700 IF(DABS(SRR).GT.1.D-10) RERR=DABS(SCR/SRR) IF(DA8S(SRI).GT.1.D-10) RERI=DABS(SCI/SRI) IF(RERR.LT.RELERR.AND.RERI.LT.RELERR) GO TO 1200 GO TO 700 1200 TU(1)=SDR-SRR*VOEFF TU(2)=SDI-V0EFF*SRI RETURN END SUBROUTINE S2(N,NS,MOOD,D,I A,IB,TU) IMPLICIT REAL*8 <A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,V(3),W(3),POEFFI6),PA R(13) , 1 DL(5060),RL(4350),DR(95),DZ195),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION TU(2),D(3) C INITIALIZATION MS=NS . IF(NS.EQ.3) MS=2 VOEFF=PAR(11+MS)/DF SD=O.DO SRR=O.DO C DIRECT LATTICE SUM MD=NDS(NS)-1 KD=MD+JJD(5) JD=NDS(NS+1)-1 LIMH=NDL(MD+1)-5 IF(MD.NE.O) LIMH=NDL(MD)-2 RERR=O.DO FAC=1.D0 IF(NS.EQ.l) GO TO 100 IFUA.EQ.3) FAC=DF IF(IB.EQ.3) FAC=FAC*DF FAC=FAC*FAC 100 MD=MD+1 IF(MD.GT.JD) GO TO 500 SC=O.DO LIML=LIMH+3 LIMH=NDL(MD)-2 DO 400 I=LIML,LIMH.3 IFU.NE.LIML) GO TO 300 DD=DR(MD ) IF(NS.NE.l) DD=DD-DZ(MD)*E IF(DD.LT.l.D-lO) GO TO 100 ARG=DD*PAR(2) CALL.PHIS(ARG,N,1,MOOD,0,PHI) 300 RAB=DL(I+IA-1)*DL(I+IB-1) RRAB=RAB*RAB SC=SC+RRA8 400 CONTINUE SCR=SC*PHI*FAC IF(SC.EQ.O.DO) GO TO 100 SD=SD+SCR IF(MD.LE.KD) GO TO 100 IF(DABSJDRtMD)-DR(MD+1)).LE.l.D-4) GO TO 100 IF(DABS(SD).GT.l.D-10) RERR=DABS(SCR/SO) IF(RERR.LT.RELERR) GO TO 500 GO TO 100 C RECIPROCAL LATTICE SUM 500 MD=NRS(MS)-1 KD=MD+JJD(6) JD=NRS(MS+1)-l RERR=O.DO LIMH=NRL(MD+l)-5 IF(MD.NE.O) LIMH=NRL(MD)-2 AZ=1.D0 AY=O.DO IF(IA.NE.IB) GO TO 600 AZ=3.D0 AY=4.D0 600 MD=MD+1 IF(MD.GT.JD) GO TO 1100 SKR=0.D0 SUR=O.DO SYR=O.DO LIML=LIMH+3 LIMH=NRL(MD)-2 YMO=(9.D0-N)/2.DO DO 1000 I=LIML,LIMH,3 IF(I.NE.LIML) GO TO 800 DD=RR(MD ) IF(MS.NE.l) DD=DD-RZ(MD)*F IF(DD.GT.1.D-10) GO TO 750 DEN=N-7 PHI=2.DO/DEN SKR=SKR+PHI GO TO 1000 750 ARG=DD*PAR(3) CALL PHIS(ARGfN,-l»MOOD,4,TRY) AA=DEXP(-ARG)/ARG XMO=YMO CH1 = ARG*(TRY-AA ) /XMO XM0=XM0-1.00 PHI=ARG*(CHI-AA)/XMO 800 P = RL.< I+IA-1 ) Q=RL(I + IB-1 ) IF(MS.EQ.l ) GO TO 850 IFUA.EQ.3) P = P/DF IF(IB.EQ.3) Q=Q/DF 850 PQ=P*Q. PP=P*P QQ=Q*Q PQPQ=PP*QQ YR=0.00 DO 900 J = 2,3 YR=YR+RL(I+J-1)*D(J) 900 CONTINUE ARG=-YR*PAR(1) C=DCOS(ARG) SKR=SKR+PHI*C SUT=PQ*AY+PP+QQ SUR=SUR-SUT*CHI*C SYR=SYR+PQPQ*TRY*C 1000 CONTINUE SCR=AZ*SKR+PAR(4)*SUR+PAR(5)*SYR SRR=SRR+SCR IF(MD.LE.KD) GO TO 600 IF(DABS(RR(MD)-RR(MD+1)KLE.l.D-4) GO TO 600 IF(DABS(SRR).GT.1.D-10) RERR=DABS(SCR/SRR) IF(RERR.LT.RELERR) GO TO 1100 GO TO 600 1100 TU<1)=SD+SRR*VOEFF TU(2)=0.D0 RETURN END ******************************************************* ****** LISTING OF SHELLSUMS ************************ ******************************************************* SUBROUTINE EWINIT(INIT,A,XI,PELERR,N,MOOD) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z<3),D(3),COEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL(4350)»DR(95),DZ(95)»RR(72)» R Z ( 7 2 ) , 2 NDL195),NRL(72),NDS(4),NRS T3),ND(4),NR(3 ),JJD(6) 20 FORMAT('01NITIALIZAT ION FOR SHELL SUMS COMPLETE*,/, 1» RELERR=*,F10.8,* NO. OF SHELLS=*,3110) PI=3.14159265358979D0 RELERR=PELERR IF(INIT.NE.0.AND.NRS(1).NE.1) CALL LTLOAD DF=1.D0 E=0.D0 F=O.DO PAR(1)=PI+PI DO 200 NS=1,3 JJD(NS)=0 LL=NDS(NS) LH=NDS(NS + 1 DO 100 L=LL,LH IF(RELERR.LT.DR(L)) GO TO 200 JJD(NS)=JJD(NS)+1 100 CONTINUE . 200 CONTINUE WRITE(7,20) REL ERR,IJJD(I ),1=1,3) RETURN END SUBROUTINE LTLOAD IMPLICIT REAL *8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z{3),D(3),COEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL(4350),DR(95),DZf95),RR(72),R Z(72) , 2 NDL(95),NRL(72),NDS14),NRS(3),ND(4),NR(3 ),JJD(6) 10 FORMAT(•OLATTICE DATA LOADED',///) READ(2) NOLAT,(NDS(I),I=1,NDLAT),{ND(I),1=1,NDLAT),NDI NDLAT+1) READ(2) NRLAT,(NRS(I),1=1,NRLAT),(MR(I),I=1,NRLAT),NR( NRLAT+1) NN=3*ND(NDLAT+1) DO 100 I=1,NN,24 L=I+23 IF(L.GT.NN) L=NN READ(2) ( D L ( J ) , J = I , L ) 100 CONTINUE NN=3*NR(NRLAT+1) DO 200 I=1,NN,24 L=I+23 IF(L.GT.NN) L=NN READ(2) ( R L ( J ) , J = I , L ) 200 CONTINUE NS=0 k DO 300 1=1,NDLAT NS=NS+NDS( I ) 300 CONTINUE READ(2) (NDL(J),J=1,NS) READ(2) (DR(J)» DZ(J)» J=l» NS) NS=0 DO 400 1=1,NRLAT NS=NS+NRS( I ) 400 CONTINUE READ(2) (NRL(J),J=l,NS) RE AD ( 2 ) (RR(J),RZU),J=1,NS) NDS(4)=NDS(3)+NDS(2)+NDS(1)+l NDS(3)=NDS(2)+NDS(1)+1 NDS(2)=NDS(1)+l NDS(1)=1 NRS(3)=NRS(2)+NRS(1)+1 NRS(2)=NRS(1)+l NRS(1)=1 WRITE(7,10) RETURN END SUBROUTINE AVALD(M,N,MOOD,COEFF,ZUM,DUM) IMPLICIT REAL *8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z(3},D(3),POEFF(6),PA R l 1 3 ) , 1 DL(5060),RL(4350),DR(95),DZ(95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION ZUM(2,6),DUM(2,6),S(2) IF(M.NE.O) GO TO 100 CALL SO(N,2,MOOD,COEFF,Z,S) ZUM(1,1)=S(1) ZUM(2,1)=S(2) CALL SOIN,3,MOOD,COEFF,D,S) DUM(1,1)=S(1) DUM(2,1)=S(2) GO TO 200 100 CALL SM(M,N,2,MOOD,COEFF,Z,ZUM) CALL SM(M,N,3,MOOD,COEFF,D,DUM) 200 RETURN END . SUBROUTINE EWALD(M,N,NS,MOOD,COEFF,SUM) IMPLICIT REAL *8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,Z(3),D(3),POEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL(4350),DR(95),DZI95),RR(72),R Z(72) , 2 NDL(95),NRL(72),NDS(4),NRS i3),ND(4),NR(3 ),JJD(6) DIMENSION SUM(2,6),TUM(2,6),S(2) IF(M.NE.O) GO TO 100 CALL S0(N,NS,MOOD,COEFF,Z,S) SUM(1,1)=S(1) SUM(2,1)=S(2) IF(NS.EQ.l) GO TO 400 CALL SO(N,3,MOOD,COEFF,D,S ) SUM(1,1)=SUM(1,1)+S(1) SUM(2,1)=SUM(2,1)+S(2) GO TO 400 100 CALL SM(M,N,NS,MOOD,COEFF,Z,SUM) IF(NS.EQ.l) GO TO 400 CALL SM< M,N,3,MOOD,COEFF,D,TUM) DO 300 1=1,2 DO 200 J= l , 6 SUM( I,J)=SUM(I,J)+TUM(I,J) 200 CONTINUE 300 CONTINUE 400 RETURN END SUBROUTINE SM(M,N,NS,MOOD,COEFF,D,SUM) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,V(3),Wi3),POEFFI6),PA R ( 1 3 ) , 1 DL(5060),RL(4350),DR{95),DZ(95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 ),JJD(6) DIMENSION SU(2),TU(2),SUM(2,6),0(3) DO 300 1 = 1,6 CALL IND I X ( I , IA , I B ) IF(M.EQ.2) GO TO 100 CALL S1(N,NS,MOOD,D,I A,IB,TU) GO TO 200 100 CALL S2(N,NS,M00D,D,IA,IB,TU) 200 SUM(1,1)=TU(1) SUM(2,I)=TU(2) 300 CONTINUE RETURN END SUBROUTINE INDI X ( I , I A,IB) IFU.GT.3) GO TO 100 IA = I IB=I RETURN 100 IF(I.GT.4) GO TO 200 IA = 2 IB = 3 RETURN 200 IF(I.GT.5) GO TO 300 IA=3 IB=1 RETURN 300 IA=1 IB = 2 RETURN END SUBROUTINE S0(N,NS,MOOD,COEFF,D,SUM) IMPLICIT REAL #8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,V(3),W(3),POEFF(6),PA R ( 1 3 ) , ^ 3 1 DL(5060),RL(4350),DR{95),DZI 95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL<72),NDS(4),NRS(3),ND(4),NR<3 ),JJD16) DIMENSION SUM(2),D(3) SDR=O.DO SDI=O.DO MD=NDS(NS )-1 KD=MD+JJD(NS) JD=NDS(NS+1)-l IF(KD.GT.JD) KD=JD LIMH=NDL(MD+1 )-5 IF(MD.NE.O) LIMH=NDL(MD)-2 100 MD=MD+1 IF(MD.GT.KD) GO TO 600 SKR=0.D0 SKI=O.DO LIML=LIMH+3 LIMH=NDL(MD )-2 DO 500 I= LIML» LIMH» 3 IF(I.NE.LIML) GO TO 300 DD=DR(MD ) IF(NS.NE.l) DD=DD-DZ(MD)*E IF(DD.LT.l.D-lO) GO TO 100 DEN=DSQRT(DD)**(~N) 300 YR=O.DO DO 400 J=l,3 YR=YR+DL(I+J-1)*Y(J) 400 CONTINUE ARG=YR*PAR(1) SKR=SKR+DCOS(ARG) SKI=SKI+DSIN(ARG) 500 CONTINUE SCR=SKR*DEN SDR=SDR+SCR SCI=SKI*DEN SDI=SDI+SCI GO TO 100 600 SUMl1)=SDR SUM(2)=SDI RETURN END SUBROUTINE SI(N»NS,MOOD,D,I A,IB,TU) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,V(3),W{3),POEFF(6),PA R ( 1 3 ) , 1 DL(5060),RL14350),DR{95),DZ(95),RR{72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS{3),ND(4),NR(3 ),JJD(6) DIMENSION TU(2),D(3) SDR=O.DO SDI=O.DO MD=NDS(NS)-1 KD=MD+JJD(NS) JD=NDS(NS+1 ) - l IF(KD.GT.JD) KD=JD LIMH=NDL(MD+1)-5 IF(MD.NE.O) LIMH=NDL(MD)-2 F A O l .DO IF(NS.EQ.l) GO TO 100 IFUA.EQ.3) F AC=DF IFUB.EQ.3) FAC=FAC*DF 100 MD=MD+1 IF(MD.GT.KD) GO TO 600 SKR=0.D0 SKI=0.D0 LIML = LIMH+3 LIMH=NDL(MD)-2 DO 500 I=LIML,LIMH,3 IF(I.NE.LIML) GO TO 300 DD=DR(MD ) IF(NS.NE.l) DD=DD-DZ(MD)*E IF(DD.LT.l.D-lO) GO TO 100 DEN=DSQRT(DD)**(-N) 300 YR=O.DO DO 400 J = l , 3 YR=YR+DL(I+J-1)*Y(J) 400 CONTINUE ARG=YR*PAR(1) C=DCOS(ARG) S=DSIN(ARG) RRAB=DL(I+IA-1)*DL(I+IB-1) SKR= SKR+ C*RRAB SKI=SKI+S*RRAB 500 CONTINUE SCR=SKR*DEN*FAC SDR=SDR+SCR SCI=SKI*DEN*FAC SDI=SDI+SCI GO TO 100 600 TU(1)=SDR TU(2) = SD I RETURN. END SUBROUTINE S2(N,NS,MOOD,D,I A,I 8,TU) IMPLICIT REAL #8 (A-H,0-Z) COMMON /SUMS/ Y(3),DF,E,F,RELERR,V(3),W(3),POEFF(6),PA R ( 1 3 ) , 1 DLI5060),RL(4350),DR(95),DZ(95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRSI3),ND(4),NR{3 ),JJD(6) DIMENSION TU(2),D(3) SD=O.DO MD=NDS(NS)-1 KD=MD+JJD(NS) JD=NDS(NS+1)-l IF(KD.GT.JD) KD=JD LIMH=NDL(MD+1)-5 IF(MD.NE.O) LIMH=NDL(M0)-2 FAC=L.DO IF(NS.EQ.l) GO TO 100 IF(IA.EQ.3) FAC=DF IFUB.EQ.3) FAC=FAC*DF FAC=FAC*FAC 100 MD=MD+1 IF(MD.GT.KD) GO TO 500 SC=O.DO LIML=LIMH+3 LIMH=NDL(MD)-2 DO 400 I=LIML,LIMH,3 IF(I.NE.LIML) GO TO 300 DD=DR{MD ) IF(NS.NE.l) DD=DD-DZ(MD)*E IF(DD.LT.l.D-lO) GO TO 100 DEN=DSQRT(DD)**(-N) 300 RAB=DL(I+IA-1)*DL(I+IB-1) RRAB=RAB*RAB SC=SC+RRAB 400 CONTINUE SCR=SC*DEN*FAC SD=SD+SCR GO TD 100 500 TU(1)=SD TU(2)=0.D0 RETURN END *********************** ************************** ****** ****** LISTING OF NNSUMS *************************** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 11+6 SUBROUTINE EWINIT(11,AA,X,R,NN,MM) -IMPLICIT REAL*8 (A-H.O-Z) COMMON /SUMS/ Q(3),DF,E,F,REL,V(3),W(3),C(6),P(13),DL( 5060) R2=1.4142135623730950488D0 R3=1.7320508075688772935D0 DO 100 1=1,72 DL(I)=0.DO 100 CONTINUE DL(1)=1.D0 DL(2)=1.D0 S=1.D0 DO 200 L=3,6 DL(L)=0.25D0 DL(6+L) = 0.75D0 IF(L.GT.4) S=-1.D0 DL(30+L)= 0.25 D0*R3#S 200 CONTINUE DO 400 L = l , 6 DL(48+L)=0.6666666666666666700 IF(L.GT.4) GO TO 300 DL(36+L)=0.25D0 DL(42+L)=0.08333333333333333D0 DL(54+L)=.33333333333333333D0/R2 DL(60+L)=l.D0/R2/R3 DL(66+L)=0.25D0/R3 GO TO 400 300 DH42+L) =0. 3333333333333333300 400 CONTINUE DL(47)=.3333333 333333333300 DL(48)=DL(47) DL(60)=R2*0.3333333333333333300 DL(59)=-DL(60) DL(56)=-DL(56) DL(58)=-DL(58) DL(62)=-DL(62) DL(63) = -DL(63 ) DL(69)=-DL(69) DL(70)=-DL(70) Q(1)=0.D0 Q(2)=0.D0 Q(3)=0.D0 DF=1.D0 E=O.DO F=O.DO RETURN END SUBROUTINE ARGS(Q,X,Y) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION Q ( 3 ) , X ( 6 ) , Y ( 6 ) PI 1 = 6.28 31853 0717958647692528700 C60=0.866025403784439D0 R23=l.4142135623730950488D0*C60 R12=3.464101615137754587000 A=PII*0.5D0*Q(1 ) B=PII*C60*Q(2) C=PII*Q(2)/R12 D=PII/R23*Q(3 ) X(1)=PII*Q(1) X(2)=-X( 1) X(3)=A+B X(4)=-A-B X(5)=-A+B X(6)=A-B Y(1)=A+C+D Y(2)=A+C-0 Y(3)=-A+C+D Y(4)=-A+C-D Y(5)=-C-C+D Y(6)=-C-C-D RETURN END SUBROUTINE EWALDt M,N,NS,MO,C,SUM) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,REL,V(3),W{3),CF(6),P(13),DL (5060) DIMENSION SUM(2,6),TUM(2,6) IF(NS.EQ.2) GO TO 200 IF(M.EQ.O) GO TO 100 CALL FCCNN1(SUM) GO TO 500 100 CALL FCCNNO(SUM) GO TO 500 200 CALL AVALD(M,N,MO,C,SUM,TUM) DO 400 1=1,2 DO 300 J = l , 6 SUM(I,J)=SUM(I,J)+TUM(I,J ) 300 CONTINUE 400 CONTINUE 500 RETURN END SUBROUTINE AVALD(M,N,MM,C,ZUM,DUM) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION ZUM(2,6),DUM(2,6),S(2),TI2). IF(M.NE.O) GO TO 100 CALL HCPNNO(S,T,N ) ZUM(1,1)=S(1) ZUM(2,1)=S(2) DUM(1,1)=T(1) DUM(2,1)=T(2) RETURN 100 CALL HCPNNI(ZUM,DUM,N) RETURN END SUBROUTINE FCCNNO(S) IMPLICIT'REAL * 8 JA-H,0-Z) COMMON /SUMS/ Q(3) DIMENSION S(2,6 ) PIIR=4.44288293815838D0 S (2,1)=0.DO A=2.D0*DC0S(PIIR*Q(1)) B=2.D0*DCOS(PIIR*Q(2)) 1 0 C=2.D0*DC0S(PIIR*Q(3)) S(1,1)=A*B+A*C+B*C RETURN END SUBROUTINE FCCNN1(SUM) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q13) DIMENSION SUM(2,6),S(3),C(3) PIIR=4.44288293815838D0 DO 100 1=1,3 SUM(2,I)=0.D0 SUM(2,1+3)=0.D0 A = PIIR*Q(I ) S(I)=DSIN(A) C ( I ) = DCOS(A ) 100 CONTINUE SUM(1,1)=2.D0*C(1)*<C(2)+C(3)) SUM(1,2)=2.D0*C(2)*(C<1>+C(3)) SUM(i,3)=2.D0*C13)*(C(1)+C(2)) SUM(1,4)=-2.D0*S(2)*S(3) SUM(1,5)=-2.D0*S(3)*S(1) SUM(1,6)=-2.D0*S(1)*S(2) RETURN END SUBROUTINE HCPNNO(S,T,N) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,REL,V<3),W(3),CF<6),P<13),DL . (5060) DIMENSION X ( 6 ) , Y ( 6 ) , S ( 2 ) , T ( 2 ) CALL ARGS(Q,X,Y ) CCC=(l.DO/(l.DO-O.666666666666666667D0*E))**(DFLOAT(N) *0.5DO) DO 100 1 = 1,2 S(I)=0.DO T(I)=0.D0 100 CONTINUE DO 200 1=1,6 S(1)=S(1)+DCOS(X(I)) T(1)=T(1)+DCOS(Y( I) ) T(2)=T(2)+DSIN(Y(I)) 200 CONTINUE T(1)=T(1)*CCC T(2)=T(2)*CCC RETURN END SUBROUTINE HCPNNI(S,T) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,REL,V(3),W(3),CF(6),P(13),DL (5060) DIMENSION S ( 2 , 6 ) , T ( 2 , 6 ) , X ( 6 ) , Y ( 6 ) CALL ARGS(Q,X,Y) CCC=(l.DO/(1.DO-O.666666666666666667D0*E))**(DFLOAT(N) *0.5D0) DDF=DF*DF DO 200 1=1,2 DO 100 J = l , 6 SCI,J)=0.DO T(I,J)=O.DO 100 CONTINUE 200 CONTINUE DO 400 1 = 1,6 I 1=6*( 1-1 ) JJ=I1+36 FF=CCC IFII.EQ.3) FF=CCC*DDF IF(I.EQ.4.0R.I.EQ.5) FF=CCC*DF DO 300 J = l , 6 S ( 1 , I ) = S(1 , I ) + D L ( I I + J ) * D C O S ( X ( J ) ) T(1,I)=T (1 , I)+DL(JJ+J)*DCOS(Y<J))*FF T ( 2 , I ) = T ( 2 , I ) + D L ( J J + J ) * D S I N ( Y ( J ) ) * F F 300 CONTINUE 400 CONTINUE RETURN END 150 APPENDIX B DETERMINATION OF FCC AND HCP PHONON FREQUENCY DISTRIBUTIONS TABLE OF CONTENTS Page Int r o d u c t i o n 151 De s c r i p t i o n of fee and hep Phonon Spectra .... 152 Documentation f o r FCCFRQ and HCPFRQ 166 Documentation f o r BRANCH and HSTGRM 174 Documentation f o r HISTPRINTER 179 Sample Output from BRANCH 181 Sample Output from HISTPRINTER 185 L i s t i n g of FCCFRQ 189 L i s t i n g of HCPFRQ 197 L i s t i n g of BRANCH 212 L i s t i n g of HSTGRM 216 L i s t i n g of HISTPRINTER » 220 151 Introduction This appendix discusses the phonon frequency d i s t r i b u t i o n s used in t h i s thesis and describes a set of F0RTRAN program segments developed to calculate phonon frequencies and to set up d i s t r i b u t i o n s of them. Attention i s li m i t e d to f e e and hep l a t t i c e s held together by (m-S) potentials. L a t t i c e sums, when required, are assumed to be obtained from one of the l a t t i c e sum subroutine packages described i n Appendix A. Phonon frequencies can be determined by diagonalizing the Fourier transformed dynamical matrices, obtained from the l i n e a r i z e d equations of motion by introducing c y c l i c boundary conditions - see Chapter I I . For a l a t t i c e with s l a t t i c e s i t e s per unit c e l l and N unit c e l l s , there are l\l of these dynamical matrices, each corresponding to a dif f e r e n t wave vector i n the f i r s t B r i l l o u i n zone, and yielding 3s phonon frequencies. Since IM i s enormous for macroscopic c r y s t a l s , i t i s more convenient to deal with the frequency d i s t r i b u t i o n function, or frequency spectrum, than with the i n d i v i d u a l frequencies. The phonon spectrum function, gC^ 5), i s defined such that g(^0)doJ i s the f r a c t i o n of frequencies i n the i n t e r v a l ( ^ j ^ + d ^ ) . The term "spectrum" i s used here to denote the d i s t r i b u t i o n of the square roots of the dynamical matrix eigenvalues, rather than the d i s t r i b u t i o n of excited phonons. Also, i t i s not the frequencies themselves that are discussed here, but the reduced frequencies defined i n Chapter I I I . 23 Actually, i t i s not essential to diagonalize l\l~1D dynamical matrices i n order to obtain the phonon spectrum. A few thousand to a few m i l l i o n frequencies s u f f i c e to establish the c h a r a c t e r i s t i c s Df 152 the spectrum, i f the frequencies correspond to a uniformly, d i s t r i -buted net of wave vectors. The method of determining spectra, by sampling dynamical matrices at a r e l a t i v e l y small number of points i n the B r i l l o u i n zone i s c a l l e d the simple root sampling method . Since each dynamical matrix yields 3s frequencies, and since most wave vectors are equivalent by symmetry to many other wave vectors, many more frequencies are generated than dynamical matrices diagonal-ize d . For fee and hep l a t t i c e s each dynamical matrix yields e f f e c t i v e -l y Ikk phonon frequencies, except for wave vectors at symmetry points of the B r i l l o u i n zone. Description of fee and hep Phonon Spectra The quality of phonon spectra obtained from the simple root sampling method depends upon the number wave vectors considered. Since the spectra must be determined i n terms of histograms, often smoothed, the greater the number of sample wave vectors, the smaller may be the frequency i n t e r v a l s used to construct the histograms, and the greater may be the degree of s i g n i f i c a n t d e t a i l i n the resultant spectra. However, only a very small f r a c t i o n of possible wave vectors may ever be considered, so spectra obtained by root sampling are prone to two types of inaccuracy: G. G i l a t , 3. Comp. Phys., _10,432 (1972). 153 1. A sample net uith enough wave vectors to determine the bulk of the spectrum generally under-samples the center of the B r i l l o u i n zone and cannot reproduce the proper low frequency behavior: J (u>) ^  , O J O (B-1) uhich i s known from the theory of Debye. 2. The root sampling method also cannot reproduce s i n g u l a r i t i e s in the spectra, because regions very close to c r i t i c a l wave vec-tors are under-sampled. Both of these flaws can be corrected by resampling the appropriate regions of the B r i l l o u i n zone, determining the correct behavior, and then superimposing these d e t a i l s on the bulk of the spectrum. Neither of these matters i s considered here, since they did not affect the calculations for t h i s t h e s i s . What i s of concern here i s the determination of the bulk of the spectra. For the fee l a t t i c e the sample wave vectors form a simple cubic l a t t i c e i n the f i r s t B r i l l o u i n zone. The fineness of t h i s sample net i s described by an integer, K, which should be even i f the surfaces of the B r i l l o u i n zone are to be sampled. The number of phonon frequen-cies obtained from such a net i s 3(4K 3+3K). For the hep l a t t i c e the sample wave vectors form a simple hexagonal l a t t i c e , whose fineness i s described by the integer, H* The number of frequencies i s 6(6H)"3. The dependence of fee and hep phonon spectra upon the fineness parameters i s discussed below. Interactions are assumed to be 1N (m-6) 154 inter a c t i o n s . L a t t i c e parameters are those for s t a t i c equilibrium uith i d e a l stacking. For t h i s model the reduced frequencies defined in Chapter III are such that: ^max=^t<^V=1B, and <£>%405, for both fee and hep. Convergence of <(^jThe zero point energy of a c r y s t a l at OH i s determined by <o$)t whose correct value i s not known, but whose variation with the fineness parameters, K and H, can be used as an indi c a t i o n of how well-determined a root sampled spectrum i s . Table B-1 contains a l i s t of values of <co> for various values of the fineness parameters. Also l i s t e d are the errors i n <.°° S and < ^ > obtained from the corresponding spectra. C l e a r l y , the values of<£>> converge as the size of the sample increases. The errors i n <^ > ^  are too -\_. *• 'J/^f small to account for the variation of < ^ > , but the errors in s are large enough. I f these are taken as the errors in <w)> f then < ^ > -Fee ~ 4-0907 ± O.O OOZL , < ^ > k t p = t - o f J l l ± o. oof 6. It should be noted that such errors make i t impossible to determine which l a t t i c e has the smaller value o f < c o > . However, i f the values of < u , > are plotted against the number of frequencies i n the samples, as in Figure B-1, there i s reason to suspect that error estimates based upon the error i n <*o > are too conservative and that The reason for the errors i n <£> /> being much smaller than those i n <^>t> i n Table B-2 i s that the method of diagonalizing the dynam-i c a l matrices conserves th e i r traces. Therefore, errors in the squared 155 Table B-1. V a r i a t i o n D f u i t h fineness of the sample net. Results based upon 1N (.12-6) i n t e r a c t i o n s and s t a t i c e q u i l i b r i u m l a t t i c e parameters. fee K Number of frequencies <UJ> % e r r o r i n 2 4 6 10 14 1B 22 28 32 114 804 2,646 12,090 33,054 70,146 127,974 263,676 393,504 3.98 4.09332 4.09238 4.09211 4.09148 4.09114 4.09094 4.09079 4.09073 -2.8 .6 .6 .3 .02 .01 .006 .001 .00002 -3.4 -.002 .6 .5 .05 .03 .01 .006 .005 hep H 1 2 3 4 5 6 1,296 10,368 34,992 82,944 162,000 279,336 4.09454 4.09364 4.09289 4.09248 4.09222 4.09205 5x10 3x10 10 10 5x10 5x10 -7 -9 -7 -7 -8 -8 -.2 -.1 -.07 -.06 -.05 -.04 reduced frequencies would not a f f e c t the values of <£>x>nearly as much as they would a f f e c t the values of < <^* v^ ^  n^2. In f a c t , whatever e r r o r s do show up i n <£>° , 'y^reflect e r r o r i n c u r r e d i n the averaging process, as w e l l as those a t t r i b u t a b l e to the d i a g o n a l i z a t i o n process. 4. 100 4.093 V 4.096 4.094 4.092 4.090 156 i i i 1 FCC i \ \ \ \ \ HCP 10 20 30 ,4, Number of modes i n sample (x10 ) Figure B-1. V a r i a t i o n of < w > ^ and < « ^ \ t p upon the number of phonon frequencies i n the sample. Values obtained from Table B-1: column 3 p l a t t e d against column 2. Shapes of fee and hep Phonon Spectra. Corresponding to the convergence o f < ^ > u i i t h i n c r e a s i n g sample s i z e i s a development of a d e f i n i t e shape i n the histogram r e p r e s e n t a t i o n s of This development o f shape i s shown i n Figures B-2 and B-3 f o r fee and hep, r e s p e c t i v e l y . A l l of the histograms shown have the same i n t e r v a l s i z e , OJ /100, to f a c i l i t a t e comparisons. ' max 5 For samples of l e s s than--^10 frequencies, the histograms d i s p l a y abundant, spurious s t r u c t u r e because the i n t e r v a l s i z e i s too s m a l l . However, once ^ 1 0 ^ frequencies are i n c l u d e d , the histograms become r e l a t i v e l y smooth and t h e i r c h a r a c t e r i s t i c shapes become d i s c e r n i b l e . 5 A 4 4 ^ 3 H ^ 2 i a) K=1D JL4 — r ~ 3 b) H=14 ,n/lrT 6 0 1 i l l ru OO Figure B-2. Development of the shape D f the fee frequency d i s t r i b u t i o n u i t h i n c r e a s i n g sample s i z e : a) K=10, b) K=14, c) K=2<+, and d) K=32. 3 6 0 3 to Figure B-3. Development of the shape of the hep frequency d i s t r i b u t i o n u i t h sample s i z e : a) H=2, b) H=3, c) H=5, and d) H=6. i n c r e a s i n g UD LEAF 161 OMITTED IN PAGE NUMBERING. 162 The frequency spectra of both fee and hep l a t t i c e s are very s i m i l a r . The low frequency ends are nearly i d e n t i c a l up to co /2. Each has 7 * max a broad peak j u s t above CO /2 and a narrow peak j u s t below cO J max max The main d i f f e r e n c e s are the shapes of the broad peaks and the r e l a t i v e heights of the two peaks i n each spectrum. P o l a r i z a t i o n Branches. More extensive d i f f e r e n c e s between fee and hep phonon spectra are apparent i f they are resolved i n t o p o l a r -i z a t i o n branches. This can be accomplished approximately by assigning the 3s phonon frequencies corresponding to each wave vector to p o l a r i z a t i o n branches on the b a s i s of t h e i r r e l a t i v e magnitudes: * r A l l $ ) * * T A l C $ U ^ A C & * Z T * & S * T * ^ * Z L O C $ \ (B-3) where the s u b s c r i p t s T, L, A, and 0 are the i n i t i a l l e t t e r s of the words "transverse", " l o n g i t u d i n a l " , " a c o u s t i c " , and " o p t i c " . The assignment of these a d j e c t i v e s to the r e s u l t a n t p o l a r i z a t i o n branches i s very approximate, since t h e i r usual meanings are s a t i s f i e d r i g o r -ously by only a small f r a c t i o n of phonon modes and since the p r e s c r i p -t i o n (B-3) takes no account of degeneracies and crossovers. Fee and hep p o l a r i z a t i o n branches determined according to (B-3) are shown i n Figures B-k and B-5, r e s p e c t i v e l y . The branches of the 2 fee spectrum f i t q u a n t i t a t i v e l y those determined by Leighton f o r a s i m i l a r l a t t i c e motion model, but a rigorous d e f i n i t i o n of branches. 2 R. B. Leighton, Rev. Mod. Phys. 20,165 (1948), 163 0 '3 <cr> Figure 3-4. P o l a r i z a t i o n branches corresponding to the K=32 fee frequency d i s t r i b u t i o n i n Figure B-2d. In p a r t i c u l a r , the high frequency c u t o f f s correspond w i t h i n an i n -t e r v a l width to Leiqhton's r e s u l t s : ' to //I, J3"> /2, and VO m a x m a x m a x P o l a r i z a t i o n branches f o r hep spectra have not been published p r e v i o u s l y . Those shown i n Figure 3-5 i n d i c a t e that the ac o u s t i c modes are responsible f o r the broad, low frequency peak and that the op t i c modes determine the narrow, high frequency peak. Furthermore, the obvious s i m i l a r i t i e s betueen fee and hep t o t a l spectra are not r e f l e c t e d i n the p o l a r i z a t i o n branches. 164 Figure B-5. P o l a r i z a t i o n branches corresponding to the H=6 hep spectrum i n F i g . B-3d. 165 Range of Interaction. A l l (m-6) interactions involving second and further neighbors are a t t r a c t i v e , so the ef f e c t of increasing the range of i n t e r a c t i o n , i n a c a l c u l a t i o n , i s to reduce the spacing between atoms and to increase the magnitudes of the l a t t i c e potential and k i n e t i c energies. In going from 1N to AN (12-6) interactions, the f i r s t neighbor separations i n fee and i d e a l hep l a t t i c e s are reduced by only 3%, whereas the magnitudes of the potential and zero point energies are increased by k0% and 20%, respectively. Hence, the zero point energy i s reduced r e l a t i v e to the potential energy. The 20% increase i n the zero point energy i s the result of the phonon frequencies a l l being enhanced by 20%. Therefore, an AN spec-trum d i f f e r s from -the corresponding !1N spectrum mostly by uniform scaling - the scales on the ordinates and abscissae are adjusted so that the cutoffs and.the areas under the curves are equal. The fact that the scaled spectra are p r a c t i c a l l y independent of range of i n t e r -action can be put to good use i n calculations which require reasonable, but not excellent, representations of the phonon spectrum. One may simply calculate a 1N spectrum and scale i t so that the cutoff i s cor-rect for AN c a l c u l a t i o n s . Thus, one may avoid the expense of calcu-l a t i n g numerous AN l a t t i c e sums. The reader should bear i n mind that the foregoing discussion of the properties of fee and hep phonon spectra applies only when the l a t t i c e s have s t a t i c equilibrium l a t t i c e parameters. Modifications i n the spectra as a result of l a t t i c e expansion are discussed at length i n Chapter IV. 166 Documentation for FCCFRQ and HCPFRQ FCCFRQ and HCPFRQ are F0RTRAN subroutine packages which calculate reduced phonon frequencies for fee and hep l a t t i c e s held together by (m-6) potentials. Reduced l a t t i c e sums from one of the l a t t i c e sum subroutine packages described i n Appendix A are used to evaluate the reduced dynamical matrices defined i n Chapter I I I , and these are diagonalized by algebraic means. Two execution modes are available: 1. In "dispersion curve mode" the calculated frequencies correspond to wave vectors uniformly distributed along l i n e segments i n rec i p r o c a l space. The frequencies are not stored in core but are written on output unit 7 as they are calculated. 2. In "spectrum*mode" the wave vectors are uniformly dis t r i b u t e d throughout the ir r e d u c i b l e part of the B r i l l o u i n zone. The squared frequencies are stored i n core for l a t e r use. In bath execution modes FCCFRQ and HCPFRQ can handle more than one set of l a t t i c e parameters: up to 3 and 6 sets, respectively. For each wave vector every set of l a t t i c e parameters i s treated before a new wave vector i s considered. The l a t t i c e parameters are defined as follows. The user's program s p e c i f i e s a value of the (intraplanar) f i r s t neighbor separation a Q , i n units of the (m-6) potential para-meter CT. Thereafter, fee ".and i d e a l hep l a t t i c e s with f i r s t neighbor separation a^ are treated as reference l a t t i c e s . The l a t t i c e para-meters of other l a t t i c e to be considered are assumed to be: 167 where the expansion parameters , and S^ . a r e given by the user. Regardless of the s p e c i f i e d value.of a Q , the value to be used i n the i n i t i a l i z a t i o n stage of the l a t t i c e sum subroutine packages i s ag=1i since only reduced l a t t i c e sums are required i n FCCFRQ and HCPFRQ. St r u c t u r e s . HCPFRQ i s a modified and expanded version of a s i m i -l a r program by Dr. Roger Howard of the Department of Phy s i c s , Univer-s i t y of B r i t i s h Columbia. The m o d i f i c a t i o n s to Howard's o r i g i n a l program make i t able to handle (m-6), rathe r than j u s t (12-6), p o t e n t i a l s , l a t t i c e expansion, and the l a t t i c e sum subroutine subroutine packages described i n Appendix A. FCCFRQ i s a somewhat simpler analog of HCPFRQ. The s t r u c t u r e s of FCCFRQ and HCPFRQ are shown sch e m a t i c a l l y i n Figures B-6 and B-7, r e s p e c t i v e l y . The subroutine names are i n upper case and a c r y p t i c d e s c r i p t i o n i s given i n lower case. The arrows go from the CALLing subroutine to the CALLed subroutine. The MAIN programs and l a t t i c e sum subroutines are not pa r t s of the packages. Use of FCCFRQ and HCPFRQ. In order to use FCCFRQ and HCPFRQ i n a program, i t i s necessary to observe c e r t a i n conventions. F i r s t , the COMMON /SUMS/ f i e l d and a l a b e l l e d COMMON f i e l d f o r the frequency c a l c u l a t i o n s are to be set up, and the l a t t i c e sum c a l c u l a t i o n s are i n i t i a l i z e d by a CALL to EliJINIT with a n=1. The l a b e l l e d COMMON f i e l d 158 Figure B - 6 . Schematic diagram • f FCCFRQ. IMUMPTS degeneracy of k ORDER orders frequencies FREAQS c o n t r o l s c a l c u l a t i o n s CUBICS solves cubics 5L SQUAD salves quadratics FCCFRQ >i SUMO coordin a t i o n k=0 sums t BRYld YIM KLINES "spectrum mode" "di s p e r s i o n curve made" L a t t i c e sum subroutine package SUMK k#3 sums 169 MAIN Figure 8-7. Schematic diagram of HCPFRQ;. HCPFRQ coordination ORIGIN k=0 sums BRILLO "spectrum mode" LINING "d i s p e r s i o n curve mode" DEGEN degeneracy of k ORDER orders frequencies MASTER c o n t r o l s c a l c u l a t i o n s SECULA sets up 6x6 dyn. matrices QUADRA salves quadratics SEXTIC solves s e x t i c s L a t t i c e sum subroutine package SHELLS k*0 sums PHASES transforms dyn. matrix CUBICS salves cubics QUARTI salves q u a r t i c s FOURTH f a c t o r s dyn. matrices 170 f a r the frequency c a l c u l a t i o n s i s : C O M M O N /FR/ I J J ( 1 4 5 0 0 , 6 ) , S ( 2 6 , 6 ) , B ( 5 ) , A ( 3 , 6 ) , N K ( 3 0 0 0 , 6 ) where the v a r i a b l e names have the f o l l o w i n g meanings: U J : Contains the squared reduced frequencies c a l c u l a t e d i n "spec-trum mode", arranged i n order o f - i n c r e a s i n g magnitude f o r each wave ve c t o r . The dimensions represent upper l i m i t s on the number of frequencies that can be handled - s i x sets of 14500 frequencies. These l i m i t s correspond to fineness parameters H=32 and K=6. S: Contains the k=0 l a t t i c e sums required f o r each set of l a t t i c e parameters. B: Contains information about the reference l a t t i c e . This array i s to be f i l l e d by the user's program. For (m-6) p o t e n t i a l s and f i r s t neighbor separation a^= er/b, B(I)= b m + 2 , b m + 2/(m+2), b 8 , b 8/8, and 9/(2m-12), f o r 1=1,2,3,4,5. A : Used to set up the reduced dynamical matrices. N K : Contains the weight assigned to each wave vector, namely the number of wave vectors equivalent, by symmetry, to the wave vectors i n the i r r e d u c i b l e part of the B r i l l o u i n zone. Next, the reference l a t t i c e s must be s p e c i f i e d , through a^, and the expanded l a t t i c e s s p e c i f i e d by expansion c o e f f i c i e n t s , ar S^ and c*^. (In t h e i r present forms FCCFRQ and HCPFRQ each occupy nearly as much' core space as i s a v a i l a b l e to the i n d i v i d u a l user. Therefore, bath cannot be used i n the same program.) F i n a l l y , frequency c a l c u -171 l a t i o n s may be requested. Appropriate CALL statements are of the form: CALL FCCFRQ(M,MOOD,ND,DEL,K,NUMKS) CALL HCPFRQ(M,MOOD,ND,DELTA,K,NUMKS) where M i s the r e p u l s i v e exponent of the (m-6) p o t e n t i a l s . M00D=0 or 1 f o r M even or odd, r e s p e c t i v e l y . Needed only f o r EUALDSUMS. ND i s the number of l a t t i c e parameter sets to be considered: f o r fee ND S 3 and f o r hep ND<6. DEL(3) contains the values of S-f to be used by FCCFRQ. DELTA(2,6) contains the values of Sk and S k to be used by HCPFRQ. The p o s i t i o n s DELTA(1,I) contain the values of , and the p o s i t i o n s DELTA(2,I) contain the corresponding values of o^. K i s the sample net fineness parameter. The choice K>0 invokes "spectrum mode": f o r fee H^32 and should be even, f o r hep The choice K=0 invokes " d i s p e r s i o n curve mode" and the choice K<0 r e s u l t s i n garbage. NUMKS(6) contains the number of phonon frequencies represented i n each of the ND sets of "spectrum mode" frequencies. In t h e i r present forms, FCCFRQ and HCPFRQ produce i d e n t i c a l numbers of frequencies i n each s e t , so f i v e of the p o s i t i o n s i n NUMKS are redundant. 172 Input to "Dispersion Curve Mode". I f " d i s p e r s i o n curve mode" i s to be used, a d d i t i o n a l data must be made a v a i l a b l e to s p e c i f y the l i n e segments to be considered. A f t e r a CALL to FCCFRQ or HCPFRQ u i t h K=0, the l i n e segment data i s read from u n i t 5 according to the format: 10 FORMAT(714) READ(5,10,END=600) L,M,N,LA,MA,IMA,IMSP uhere the statement l a b e l l e d "600" causes a RETURN to the CALLing subroutine and: (L,M,I\!) and (LA,MA,IMA) s p e c i f y the endpoints of the l i n e segments, and IMSP i s the number of i n t e r v a l s i n t o uhich the l i n e i s to be d i v i d e d . In FCCFRQ the wave vectors are: k - £ { ( L ) < v \ ) M ) + - ^ ( L A - 4 ) M ^ M , ^ - W ) l -- * S P J (B-5) In HCPFRQ the uave vectors are a b i t more complicated: For each point on the l i n e segments, the frequencies corresponding to every set of l a t t i c e parameters are c a l c u l a t e d and w r i t t e n on u n i t 7 before the next point i s considered. The output i s produced by statements of the form: 173 20 FORMAT(10',I4,9D12.5) IJJRITE(7,20) NK,(Q(J),J=1,3),(LJ(J)f3=1,NB) where NK i s the number of wave vectors i n the B r i l l o u i n zone that are equivalent the the current wave ve c t o r . Q i s the current wave v e c t o r . Ul contains the frequencies of the 1MB branches corresponding to the current wave ve c t o r . IMB= 3 and 6 f o r fee and hep, r e s p e c t i v e l y . E f f i c i e n c y . The costs and execution times f o r frequency c a l c u -l a t i o n s by FCCFRQ and HCPFRQ depend upon the sample net fineness parameters and the range of l a t t i c e sums. For 1N sums the cost and execution time f o r the evaluation and d i a g o n a l i z a t i o n of a s i n g l e dynamical matrix are roughly 1$ and .1 seconds f o r FCCFRQ and about twice these amounts f o r HCPFRQ. These estimates are based upon t e s t s c a r r i e d out on UBC's IBM 360/167. The costs may be halved on the new 370/168 system. The most d e t a i l e d spectra a v a i l a b l e from FCCFRQ and HCPFRQ con-5 t a i n roughly 3x10 phonon frequencies. I f 1IM sums are obtained from NPJSUM5, the costs and times f o r the frequency c a l c u l a t i o n s are about $32 and 200 seconds. Since the cost and times f o r the 1N sums alone are $2k and 150 seconds (see Table A-2 i n Appendix Av), the cost and time;due to FCCFRQ or HCPFRQ are about $8 and 50 seconds. Therefore, 5 the cost and time f o r (MD spectra of ^3x10 frequencies, c a l c u l a t e d with n(M sums, are: 174 Cost ~ S ^ND(8 + C n N ) } (B-7) Time ^ ND(50 + Tnftj) seconds, where C „, and T n, are costs and times f o r the nl\l l a t t i c e sums alone. ni\l nN Some values of C ., and T can be obtained from the columns headed nl\l ni\l "Spectrum" i n Table A-2. The quantity ND i n (B-7) i s the number of sets of l a t t i c e parameters to be considered. L i s t i n g . The F0RTRAN coding f o r FCCFRQ and HCPFRQ i s l i s t e d at the end of t h i s appendix. Documentation f o r BRANCH and HSTGRM BRANCH and HSTGRM are F0RTRAN subroutines which produce histogram representations of fee and hep phonon spectra by s o r t i n g through the arrays 01(14500,6) and NK(3000,6) a f t e r they have been f i l l e d with frequency data by FCCFRQ and HCPFRQ, operating i n "spectrum mode." BRANCH t r e a t s frequency data f o r one set of l a t t i c e parameters at a time. I t resolves the data i n t o p o l a r i z a t i o n branches, according to (B-3), and constructs a histogram f o r each branch and f o r the t o t a l spectrum. BRANCH must be executed ND times, i f the data f o r ND sets of l a t t i c e parameters i s to be analyzed. HSTGRM c a l c u l a t e s quadratic i n t e r p o l a t i o n formula c o e f f i c i e n t s f o r the volume dependence of each phonon mode frequency, so i t t r e a t s three ( s i x ) sets of fee (hep) frequency data, corresponding to as many sets of l a t t i c e parameters, at a time. The set of l a t t i c e parameters placed f i r s t i n the l i s t i s taken as the reference s e t , about which the i n t e r p o l a t i o n fdr,mulas:;are expansions, and to which the histogram corresponds. As HSTGRM determines the i n t e r p o l a t i o n c o e f f i c i e n t s , i t 175 s u b s t i t u t e s them For the frequencies i n l_(14500,6) Since only the l o c a t i o n s Ul(3,1); 3=1,2,3,... s t i l l contain a c t u a l reduced frequencies a f t e r execution of HSTGRM terminates, BRANCH cannot be used a f t e r HSTGRM, except on the data i n UI(J,1); 3=1,2,3,..., which corresponds to the reference l a t t i c e f o r the i n t e r p o l a t i o n . (Note that the reference l a t t i c e f o r i n t e r p o l a t i o n i s not the reference l a t t i c e i n FCCFRQ or HCPFRQ, unless the f i r s t set of l a t t i c e parameters i n the l i s t are such that 8^=0 or SL^= S\ = Q . ) The form of the i n t e r p o l a t i o n formulas considered by HSTGRM i s : The c o e f f i c i e n t s are obtained by f i t t i n g formula (B-8) to three ( s i x ) values'''of each fee (hep) phonon mode frequency. For fee the three values of S\£ must a l l be d i f f e r e n t . For hep the values of Sj^ and O^ must be chosen i n a p a r t i c u l a r manner. HSTGRM expects to f i n d frequency data f o r l a t t i c e parameters corresponding to the sequence: ( s » , S o l i % , * A (B-9) where the choice of SA , S { f ST , , 5Z , and i s a r b i -t r a r y . In a d d i t i o n to histograms, BRANCH and HSTGRM also determine co max and n=+1,2,3,4. BRANCH determines these q u a n t i t i e s f o r each of the branches and f o r the t o t a l spectrum under c o n s i d e r a t i o n . HSTGRM determines i n t e r p o l a t i o n formula c o e f f i c i e n t s f o r these q u a n t i t i e s . Spectra may be produced i n e i t h e r scaled or unsealed form. I f scal e d , the reduced frequency a x i s of the p r i n t e d histogram i s scaled 176 by the f a c t o r 6 / ^ m a x a n c l the ^ (<^)-axis i s scaled r e c i p r o c a l l y . This s c a l i n g does not a f f e c t the data stored i n U(14500,6). This s c a l i n g e l i m i n a t e s much of the l a t t i c e parameter dependence of and f a c i l -i t a t e s comparisons between s p e c t r a . The s t r u c t u r e s of BRANCH and HSTGRM are very simple. BRANCH i s a s i n g l e subroutine. HSTGRM i s a set of three subroutines, HSTGRM, INTERF, and INTERH. The l a t t e r p a i r perform the i n t e r p o l a t i o n c a l c u -l a t i o n s . The user must be sure to hav/e INTERF and INTERH a v a i l a b l e at load time, but need have no f u r t h e r concern f o r them. Use of BRANCH and HSTGRM. Once FCCFRQ.or..HCPFRQ.have terminated execution i n "spectrum mode" BRANCH and/or HSTGRM may be requested to analyze the frequency data by i n c l u d i n g statements of the f a l l o w i n g s o r t i n the MAIN program: DO 100 N=1,NL CALL BRANCH(NUMKS,ND,NfNSCALE,D,DI,UIMN) 100 CONTINUE CALL HSTGRM(NUMK,ND,NTAPE,NSCALE,D,DD,WMN) The input parameters are: NUMKS(6): The same as the array of the same name i n FCCFRQ and HCPFRQ. NUMK=NUMKS(1). ND=3 f o r fee and 6 f o r hep. NL: The number of l a t t i c e parameter sets to be analyzed by BRANCH (1 < NL 5ND). 177 !<0: No tape f i l e w r i t t e n . >•: The i n t e r p o l a t i o n formula c o e f f i c i e n t s are w r i t t e n on u n i t 9, as they are determined. The r e s u l t i n g f i l e d i f -f e r s from the array hl( 14500,6) only i n the order of the e n t r i e s : the order i n the f i l e on u n i t 9 i s such that the phonon frequencies are grouped according to the histogram i n t e r v a l to which they belong. !< 0: The frequencies are l e f t as they are. >0: Each entry i n LI(14500,6) i s replaced by i t s p o s i -t i v e square r o o t . In the CALL statements on the previous page, the same value of IMSCALE would not be used i n both BRANCH and HSTGRM. Al s o , i f NSCALE j >1: The histograms are not sc a l e d . (-1: The histograms are scaled so that — > 6. max WMN: The smallest value of to be included i n the c a l c u l a -t i o n of <°° /. I f the low frequency flaws i n the h i s t o -grams are not of concern, LIMN may be chosen as small as de s i r e d , as long as i t i s p o s i t i v e . D,DI: In the CALL to BRANCH,D= £ and D I = S / . D(12),DD(3): These arrays contain information about the sets of l a t t i c e parameters to be considered by HSTGRM i n s e t t i n g up i n t e r p o l a t i o n formulas f o r the phonon frequencies. These arrays must be set up by the user. 178 For fee the l a t t i c e parameters are s p e c i f i e d , i n terms of a^, by So .f'^l i and c5^. Qnly the f i r s t f i v e l o c a t i o n s i n D(12) are used. The contents should be: 0(1)= £, ~&o D(2)=D(1)*D(1) D(3)= ^ i - So D(4)=D(3)*D(3) D(5)=D(1)*D(3)*(D(3)-D(1)). For hep the l a t t i c e parameters are s p e c i f i e d , i n terms of a^ and i d e a l s t a c k i n g , by { ^ 5 o and £ & £ t o . The contents of D(12) and DD(3) expected to be: D(I)= ST-S0 DD(1)=D(1)*D(3)*(D(3)-D(1)) D(I+3)=D(I)*D(I) 0 ' DD(2)=D(7)*D(9)*(D(9)-D(7)) D(I+S)= 0 _ r - 6 0 DD(3)=D(5)*D(11) D(I+9)=D(I+6)*D(I+6) where 1=1,2,3. Output. The output from BRANCH and HSTGRM i s e s s e n t i a l l y a synopsis of what has been produced by FCCFRQ or HCPFRQ i n "spectrum mode." The output u n i t s involved are numbered 7,8, and 9. Unit 9 i s used to record a tape f i l e of i n t e r p o l a t i o n formula c o e f f i c i e n t s f o r each of the phonon mods'/frequencies, and i s required only f o r HSTGRM. A l l other output from HSTGRM i s w r i t t e n on u n i t 7, which i s expected to be a p r i n t e r or a p r i v a t e l i n e f i l e . The output from BRANCH i s w r i t t e n on u n i t s 7 and 8» For fee only u n i t 7 i s re q u i r e d . For hep u n i t 7 i s used to record histogram i n f o r -mation about the p o l a r i z a t i o n branches. Unit 8 i s used f o r information about the a c o u s t i c , o p t i c , and t o t a l s p e c t r a . Both u n i t s are expected to be p r i n t e r s or p r i v a t e l i n e f i l e s . 179 A sample of output from BRANCH i s included i n t h i s appendix immediately a f t e r the documentation f o r HISTPRINTER. The output i s fo r hep. Output f o r fee i s s i m i l a r , but l e s s extensive. No output from HSTGRM i s inclu d e d , because i t i s s i m i l a r to that of BRANCH, but contains considerably l e s s i n f o r m a t i o n . L i s t i n g . The F0RTRAN coding f o r BRANCH and HSTGRM i s l i s t e d at the end of t h i s appendix. Documentation f o r HISTPRINTER HISTPRINTER i s a simple F0RTRAN program which t r a n s l a t e s the numerical output from BRANCH i n t o histogram sketches p r i n t e d on a standard p r i n t e r . I f u n i t 7 i s a p r i v a t e l i n e f i l e f o r the execution of BRANCH, HISTPRINTER can use that f i l e as input data. The f i l e i s read as u n i t 5, each histogram datum i s t r a n s l a t e d i n t o a histogram bar f i l l e d with the number assigned to the appropriate p o l a r i z a t i o n branch, and the histogram bar i s p r i n t e d an u n i t 7, i f i t i s part of a branch spectrum, or on u n i t 8, i f i t i s part of the t o t a l spectrum. Numbers are as-signed to the p o l a r i z a t i o n branches according to ( c f . (B-3)): ± £)A(g) 2e33Cj)* ^ 9 ^ £ ^ ^ ) . ( B - i o ) HISTPRINTER cannot be used, because of i t s s i m p l i c i t y , f o r h i s t o -grams with more than 62 i n t e r v a l s . The s c a l i n g option ensures that there are always e x a c t l y 62 i n t e r v a l s . Without s c a l i n g , care must ^-be taken to ensure the only those spectra with UJ 5 6.2 be processed max by HISTPRINTER. Sample output from HISTPRINTER i s included u i t h output from BRANCH on the next feu pages. The l i s t i n g of the F0RTRAN coding f a r HISTPRINTER i s included at the end of t h i s appendix. r - r ry j r - r r , m i ' i ' i n n <;prr.Ttii|u r A l r i P l 1 T I (Hi', TOP HCP PCS f 41 C H AT IONS H » S f - 0 " N ( 1 2 - 6 ) UFAOFST NFIOHUOUR PnTFNTIAI. C H T i r p r f « = . 6 r n p WAVF V F r . TO " NFT SAMPLE OUTPUT FROM BRANCH (UNIT 7) HCP PHOTON F O ^ O U C N C I F S C A I C I I L A T F O p r o PHTNCM SPFr.TPIJM H I S T T G 3 A"S f j . j M O C H r p Mf lnFS IM F » r M ~ l ? » V l C . H « ~ 46656~W I f n^i ' ir.HFS AHP O F F INFO »»»iiR«»iiY AS r n u n w S ; M l TA? H 2 7 9 9 7 6 Wt TA1 l< I => W( L A "MOOFS" IN A L L BRANCHFS TA? 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I 2 4 4 5 4 O 0 0 0 . 9 2 7 3 1 0 0 - 0 1 0 . 6 6 8 0 2 8 0 - 0 1 m i 4 6 7 6 5 4 6 5 4 5 4 6 6 5 5 F O F T i F N r T F S > 0 . 0 6 0 0 0 0 0 0 0 0 HISTCOOA-S A9F SCAIFO SO THAT WtMAX)=6 HISTOGRAM VAI.UFS ARF WPITTFN AS PFSCFNTAGFS . 5 5 6 8 7 9 0 - 0 1 4 6 6 5 6 . 4 4 0 8 5 2 0 - 0 1 4 6 6 5 6 )W I LD) I 0 34 6 2 5 5 6 9 0 9 7 2 7 9 7 4 P 6 5 4 31 8 8 5 9 8 5 3 7 8 2 1 7 7 3 8 2 0 7 " ™ 1 5 5 6 7 3 0 02 744?P .6n 02 4 0 7 0 4 0 0 - 0 1 4 6 6 5 6 TOT 5 . 3 4 6255690: ) " 2 . 9 3 5 7 0 1 6 : 1 5 1 0 . 2 8 57 37 374 1 " ' 3 . 2 0 7 1 3 7 9 5 6 9 " 0 . 4 0 6 2 3 0 O 0? 0 . 1 7 7 0 7 5 Q 07 0 . 4 2 " 9 4 6 5 0 0 0 2 7 9 5 3 7 I NT E PV Al t 0 . 0 , 0 . 1 0 ) I 0 . 1 0 / 0 . 2 3 ) ' I 0 . 2 0 , 0.70) I 0 . 7 0 . 0.401 GIT A H N t T M ) GITA21 NIT 421 G(I.A) N(LA) G t T O l ) NITOl) G(T02) N I H ! ) G(IO) VMI.nj 0 . 1 0 5 0 . 105 0 . 0 4 O _ 0 . 0 4 0 0^000 0 . 0 0 0 0 . 0 (KO 0 . 0 0 . 0 0 . 0 0 . 0 ~ 0 . 0 1 6 0.120 0.001 6.041 0.001 0.001 0 . 0 0 . 0 0 . 0 0.3 0 . 0 0 . 0 0 . 0 1 4 0.174 0 . 0 0 3 0 . 0 4 4 0.001 0 . 0 0 2 0 . 0 0 . 0 0 . 0 0 . 0 O . n 3 . 0 0 . 0 6 1 0 . 1 9 5 0 . 0 1 9 0.063 0 . 0 0 . 0 0 ? 0 . 0 0^0 0^0 0^0 0^0 0.0 or - i i i m I ' H O ' W I ST-CIOUM CALCULATIONS FOR HC.P KGS C M . r i J I AT i r i ' i l BASFf) W I 1 7 - M N F A 7 F S T NCir.HnOUH p o TFNT [Al runf.r rr- K= I. TOP WAVF V F C T O P NET SAMPLE OUTPUT FROM BRANCH (UNIT 8) nrt> OHn>,r)*l S O F C T K I ' " H I S T O G R A M S M i - M F u I T MOMFS I N F A C H BSANCH= 4665* WITH 279936 MOOES IN ALL BRANCHES " V I ' . H C A?F nF F INTP AoniTPA ' J ll.v AS FOLLOWS: W( TA1 |<(» I W ( TA2 l< (»>W( LA) <(«IWITOl )<! = IWIT02 K l . " IH (LOI H I S T ^ O A X S ADC SCALFn S" THAT W(MAX)»«. l « J ' R " T . H A M VAI UFS A9F VIPITTFN AS PERC.FNT AGES 184 i c e r- N o « U H ^ (7> O O* u" C rv Cr , r - o If— i r oo t * N p c * r P i - r s ^ | i r . c N ^ r c IT c x rv rs. (M cr >r v~ r -c • r. — c N <C N O , !<• U* (V tN (J — — C N \ ? ! N f . C N l f CO c c c c o -J c l * r f ' . u ' cr C i T — f** CC c c c. r» c r--c — c o « r w r- f- :r- co r> • • • } • • • — C U" c o e t c O O O o c ^ U c c c i o c c o o o K t r p O - c c j C O O i O O O c c - c y o c ! c o c lo c c •tf -tf >tf IT' U" itf" tf" UVtfi IT tf" C C IT < r- I c v.* c c ! r . — — O C C ; O O C O O C C O O O O O i O O 0;0 o o •— c 'o •4" c o o o c*- C O « r « o a : O M T O O • • # • -« o o c f - ~ o,c <!• O C O c c o ,c • • * • c o o *r co oi tr o; r\. c O O o o o?c o c o ; c o c- o e o O O O O O 0;0 o c r ^ o c o o c o c o i o o o o If I f |f. I f If ( f ; IT I f i , * tf. ^ ^ < o r o o c c o o c ; o c o o ^ i i ' - IT II ** If |»" tr ^ i l f IT i ' 185 cr UJ i— rr a. LT) X o rr n a i— • UJ _J Q. c t t r t r r- ro o r- o •c c ^ " * m rn <r — t r o C >» r -C — rs. m j t r . ^3 p -r v r s t tc * - | t r — -o *-« r v o r»*, tra- o-\>c .o t / \ !op aa tr. c — J O o r t o r . r r , o ^ O- C P~ r.*|r* o J B t c — : m 0- r u c» C i ^ — c c i o pg o r t r - r - . f r - o - — c * c - —1*1 r c t r c iPy <c c r-o •© r-> (\i t r . c j c . e r - c - c r - | n c - — ~* r - n t r *o o r» r y IN r\j i s . r g r n f\' f\; r - (\ ! ! < ° * •* f r - o | o tf c c <M r » r - [ t f <\, <x> f - — r - . - ' t f o •*< tf r \ N M c i a •£ M M oc s k ' p " - c <^ M K I ^ N C O> O cn ~ ~ • ~ - — _ i O tf I : . o c c V ) CC -> I (V O i — C (NI & tf tf I u_ — tf , I00 tf tf -J" ~* x tf m LT co tf [m tf o tf —• Irt 0" rtjtf rt <J tf * r ^- i f cc tf r v «»- c * r - i c r v tf r - c r~ t~« cc m i * - r>-^ o ID K IT, «• L / i M n ^ K - I f , A / . ^ r. ,- k O t o ^  I , c l » c f\j o C r— I O ^ r - h~ r—' '"" tf r~ c O — J O O O C O O C C — LU CP t v : — rv o o c r — f - C o o o o cc —« C r-tf r-r v r \ tf. ' tf e o cc r v vT a c o — tf tf tf c* c c r v <r O o O tf rt> rt c r <r r - — r c c | n tf r - tf tf f [<-„• o r v j e e tf c tf tf u- — i c U" r v ,* — cc n r . c u> ! c m i \ tf O tf N i o c r v rn r v c ' o — * r > - - « n i f >l m i tf m tf o o c IPJ o <r ~- c - * c i f . »j- ~* o *?• a : — S M O o a - f r-- ; o »\> <. o- m r". ^ ir> r g ^ i o c O ^- C\t~- rr\ >o'-r~ o-O O C * c o O j O — O j ' 1 " : t n e-.jrN, <M " • j ' " ' * I I T N « o i n I T o | o i n o ^ ^ ^ | o ^ K i t o c i — j r v i u* c o ' c c f\t o o n co r v c o ; t r . m r. %r f\; jir. co (V »r 0 ' ;•— fsi i % N s * c c c - « i ' c p i<* m ir- - \ J — i c x c - c «J A w IT « i*^ c c - c - r c « I T C I M T - c r o s f c i - f » fN.-tf c o — r v i m t r . tf cc 0 - o Irsj r- ^ tfr-jo o m (*" 1*1 •* o . < ir. IT- c • o — r-• o o c • tf ec > c c cc • o t r IT* « ^ IT- »J-(C1 m C Cf c r - rvj o tf ^- r v r» <— *••> r-0 c o o o c 1 1 I I I 0" cc co rv rv O •! o- i n IT. N — tf m tf r - ^> u~ cr Cv IT- c f\; M i " . < i ^ . ' 1 o c ^ ^ ^ - , — r * * c — o * " - < \ O f * W m c u - r r r-.' c* . O t f O r - I ^ O t f r -c - r^ —. — tf <r r v « i r . r - c c t f i t f . ^ r (\, a N v L >j c c r c r m ^ -tf t r i t f r c r .j- S L c c tf tf c o o c i c c o o c c o o o c c o o o o o c o o o c c c o o o o c :> — < — i_ a CC ILL <. ; — i " C 1 r l / " u > 'M •—' ,«— C w > !*-' ! i X < i u . c f ; 3 O . C C C C - u . a; w C c L. u . tf lc u < > CL 1 * • « r L. :7 1/" i; > J CM !(M IM I f s ; r ^ r v i r n f * - t r ! j - c c o l t f r \ , • J - r v ^ i T ' t f C \*0 t>~ '.st **-C >f CO «H f c ' r * «tf ^ * c in 0 «tf r~ «tf r- c i " ^ r- fvr- o tr. c •-" c P - r^i^r m o : — r-, c o o o o o : o o c ' o o 1 I i i I III I t ! t I Ic » - O 'rv fv. i*- tn o tf rr- (»-, |c c?1 tf o •» o - t n r i —' m cc |CC tf « H r , - ' C J C C ! C M f i f \ tf (M <*• O P - ° " j r - c C tf c e - i i h * C co 1 r. cr ; c e c r- m cc c cc — r- r\ —• c a c cc c m t r tf c o o o c e tf tr-• • • • • f . tf c : o c O O C O C O o o c e o < — i N m t f l t f K e o r y f ' " tftfr-«tftff>-jcri>crf*>tf tf!r-cctfr»cu > — c m . r*. c tf tr tf rv c c - m r v i O r v c r ^ c r if txcr tr r v c r : (\i f. |tf cc C r-v ir r* a. r " o r*-:ec r* ec tvj i * - tf CC tf r - X J rv r V j r s , r\' r\j rv f\j r\ f% rw r\j r\( r>j t v j j r v rv rs. r v rs, r v ; i \ ; rv r \ rs. r\ > C o j c O C O C C C C O O O c " | o O c - c C C I C O C O C c % ec cc cc cc or u". IT, u*. i f m r r I c o c o c c ~ — — !-* r v r v r v rv. r^ r v r v tf tf tf ^ C C C O C C C C C O O C | c C C • © C O ! C O C : ' (T "it 0* rv, 0* tf C C r ; N (f . i t f tf CT. O C C l ' i r ft r » « i f r ; f ^ rv r - r - —« rv c . |-C < i f t f r » tf , tf f e? o c N 1 C r ; r j ^ — . ! l f N C (i O O I i0* tf f - r** rs] r v m 0 IT C O O O I O tfl 'CT tf u- " ' t f o. co cc c . 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I f lf> !4f. i f i ^ X ******************************************************* ****** LISTING OF FCCFRQ *************************** ******************************************************* 1Q9 IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,RELERR,VVV(3),WWW(3),COEFF(6 ) ,PAR(13), 1 DL(5060),RL(4350),DR(95)» DZ(95),RR(72),R ZI72)» 2 NDLI 95),NRL(72),NDS(4),NRS(3),ND(4),NR(3 )tJJ0(6) COMMON /FR/ W(14500 ,6),S(26,6),B(5),A(3,6),NK(3000,6) DIMENSION DEL(3),NUMKS(6),SUM(2,6),D(12),DD(3) 10 FORMAT(414) 20 FORMATP1REDUCED PHONON SPECTRUM CALCULATIONS FOR FCC RGS*,/,'0CAL 1CULATI0NS BASED ON (•,I3, ,-6) NEAREST NEIGHBOUR POTENT IAL ' t / t'OCHO 2ICE OF K=*,I4,' FOR WAVE VECTOR NET») 30 F0RMAT(3F10.5) READ(5,10) M M00D=M0D(M,2) Q( l)=0.D0 Q(2)=0.D0 Q(3)=0.D0 CALL EWALDt 0,M,1,MOOD,COEFF(4),SUM) DEN=SUM(1,1) CALL EWALD(0,6,1,0,COEFF(1),SUM) BB=SUM(1,1)/DEN BBB=BB**(l.D0/DFL0AT(M-6)) B(1)=BBB**M+2 B(3)=BBB**8 READ (5,30) WMN B(2)=B(1)*DFL0AT(M+2)/2.D0 B(4)=B(3)*4.D0 B(5)=9.D0/DFL0AT(M-6) 90 READ(5»10»END=4C0) K,KM,NSCALE IF(ND.GT.3) STOP 0 DO 300 L L = l,KM READ(5,30) (DEL(JK),JK=1,ND) WRITE(7,20) M,K CALL FCCFRQ(M,MOOD,ND,DEL,K,NUMKS) LP=IABS(NSCALE) LN=-LP DO 100 NL=1,ND CALL BRANCH(NUMKS,ND,NL,LP,WMN,DEL(NL),DEL(NL ) ) 100 CONTINUE NTAPE=-1 NUMK=NUMKS<1) IF(ND.NE.3) GO TO 300 D(1)=DEL(2)-DEL(1) D(2)=D(1)#D(l) D(3)=DEL(3)-DEL(l ) D(4)=D(3)*D(3 ) D(5)=D(1)*D(3)*(D(3)-D(1)) CALL HSTGRM(NUMK,ND,NTAPE,LN,D,DD,WMN) 300 CONTINUE GO TO 90 400 STOP q n END ^ U SUBROUTINE FCCFRQ(N,MOOD,ND,DEL,K,NUMKS) IMPLICIT RE AL*8 (A-H,0-Z) COMMON /FR/ WM4500 ,6),S(26,6),T(5),U(3,6),NK(3000,6) DIMENSION DEL(3),NUMKS(6) 10 FORMAT(' FCC PHONON FREQUENCIES CALCULATED*) CALL SUMO(N,MOOD,ND,DEL) IF(K.EQ.O) GO TO 100 CALL BRYWYN(K,N,MOOD,ND,NUMKS) WRITE(7,10) GO TO 200 100 CALL KLINES(K,N,MOOD,ND,NUMKS) 400 RETURN END SUBROUTINE SUMO(M,MOOD,ND,DEL) IMPLICIT REAL*8 (A-H,0-Z) COMMON /FR/ W(14500 ,6),S(26,6),T<5),UI 3,6),NK(3000,6) COMMON /SUMS/ Q(3),DF,E,F,RELERR,V(3),Z(3),COEFF(6) DIMENSION SUM(2,6),TUMI 2,6),DEL(3) Q(1)=0.D0 Q(2)=0.D0 Q(3)=0.D0 DF=1.D0 E=O.DO F=0.D0 CALL EWALD(l,M+4,1,MOOD,C0EFF(6),SUM) CALL EWALD(1,10,1,0,COEFF(3 ),TUM) DO 100 1=1,ND S(1, I )=DEL(I) DF=1.D0+DEL(I) A=DF**8 AA=DF**(M+2) DDF=DF*DF S(2,I)=DF S(3,I)=DDF S(4,I)=DDF*DF S(5,I)=SUM(1,1)/AA S(6,I)=SUM(1,1)*3.D0/AA S(7, I) = TUM(1,1)/A S(8,I)=TUM(1,1)*3.D0/A S(9,I)=A S ( 1 0 , I ) = AA 100 CONTINUE RETURN-END SUBROUTINE BRYWYN(K,MN,MOOD,ND,NUMK) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3) COMMON /FR/ W(14500 ,6 ) ,S(26,6),B(5),A(3,6),NK ( 3000,6) DIMENSION WW(6),NUMK(6) WRITE(7,900) 900 FORMAT('ONEGAT IVE SQUARED FREQUENCIES(IF ANY)»,/,» KA SE» ,4X, 1-WAVE VECTOR COMPONENTS',9X,'Q**2',5X,* SQUARED FREQUEN CIES',//) NUM=0 L=-l M=0 N=0 LIML=0 NH=3*K/2 KK=K#K RD= L.DO / DFLOAT{K)/1.41421356237309505D0 100 NUM=NUM+I 150 IF(((N+1).GT.M).OR.((L+M+N+l).GT.NH)} GO TO 200 N=N+1 NN=N*N GO TO 600 200 N=0 IF(((M+l).GT.L).OR.((L+M+N+l).GT.NH)) GO TO 300 M=M+l MM=M#M GO TO 600 3 00 M=0 IF(((L+l).GT.K).OR.((L+M+N+l).GT.NH)) GO TO 400 L = L+1 LL=L*L GO TO 600 400 DO 500 1=1,ND NUMK(I)=NUM-1 500 CONTINUE RETURN 600 Q(1)=RD*DFL0AT(L) Q(2)=RD*DFL0AT(M) Q(3)=RD*DFLOAT(N) I 1=0 IF(N.EQ.O) 11=1 12=0 IF(L.EQ.K) 12=1 13=0 IF(M.EQ.N) 13=1 14=0 IF(L.EQ.M) 14=1 15=0 IF(L+M+N.EQ.NH) 15=1 KASE=KASENQ{II,12,13,14,15) IF(KASE.NE.17) GO TO 680 WW(1 )=0.D0 WW(2)=0.D0 WW(3)=0.D0 NUMPT=1 GO TO 690 680 CALL NUMPTS(K,L,M,N,NH,NUMPT) 690 DO 800 1=1,ND I-FIKASE.EQ.17) GO TO 695 CALL FREAQS(MN,MOOD,I,WW,KASE) CALL 0RDER(WW,3) 695 DO 700 J = l , 3 W(LIML+J,I)=WW(J) 700 CONTINUE NK(NUMtI)=NUMPT 800 CONTINUE i q p LIML=LIML+3 GO TO 100 END SUBROUTINE KLINES(K,MN,MOOD,ND,NUMK) IMPLICIT REAL *8 (A-H,0-Z) COMMON /SUMS/ Q(3) COMMON /FR/ WI14500 ,6),S(26,6),B(5),A(3,6),NK(3000,6) DIMENSION.WW(6),NUMKI6) 10 F0RMAT(7I4> 20 FORMAT(* 0*,I4»2X,6D12.5) DO 100 1=1,ND NUMK( I )= 0 100 CONTINUE 200 READt5,10,END=600) L,M,N,LA,MA,NA,NSP NPT=NSP+1 LA=LA-L MA=MA-M NA=NA-N L=NSP*L M=NSP*M N=NSP*N QS=1.DO/DFLOAT!NSP) NH=3*NSP/2 11=0 12=0 13=0 14=0 15=0 IF(N.EQ.O) 11=1 IF(L.EQ.NSP) 12=1 IF(M.EQ.N) 13=1 IF(L.EQ.M) 14=1 IF(L+M+N.EQ.NH) 15=1 KASE=KASENO<11,12,13,14,15) DO 500 1 1=1,NPT Q(1)=QS*DFL0AT(L) Q(2)=QS*DFLOAT!M) Q(3)=QS*DFL0AT(N) CALL NUMPTS(K,L,M,N,NH,NUM) DO 400 I=1 ,ND CALL FREAQS(MN,MOOD,I,WW,KASE) CALL 0RDER(WW,3) NUMK(I)=NUMK(I)+1 LIML = 3*(NUMK(I)-l) DO 300 J = l , 3 W(L IML+J,I)-DSQRT(WW(J)) 300 CONTINUE NK(NUMK(I),I)=NUM WRITE(7,20) NUM,Q(1),Qt2),Q(3),IW(LIML+J,I),J=1,3) 400 CONTINUE L=L+LA M=M+MA N=N+NA 500 CONTINUE GO TO 200 600 RETURN 193 END SUBROUTINE FREAQS(MN,MOOD,ND,WW,KASE> IMPLICIT REAL*8 <A-H,0-Z) C0MPLEX*16 X{6) COMMON /SUMS/ Q(3) COMMON /FR/ WM4500 ,6),S(26,6),B(5),A(3,6),NK(3000,6) DIMENSION WW(6) 10 FORMAT I• IMAGINARY FCC FREQUENCIES,Q=•,3D20.10, 1 KAS E=«,I3) CALL SUM K(MN,MOOD,ND) IFIKASE.EQ.21) GO TO 110 IF(KASE.GT.5) GO TO 20 GO TO (11,12,13,14,110),KASE 11 X(1)=A(3,3) C=-A(1,1)-A(2,2) D=A(1,1)*A(2,2)-A(1,2)*A(1,2) GO TO 16 12 X(1)=A(1,1) C=-A(2,2)-A(3,3) D=A(2,2)*A(3,3)-A(2,3)*A(2,3) GO TO 16 13 X(1)=A(2,2)-A(2,3) C = - A ( l , l )-A(2,2)-A(2,3) D=A(1,1)*(A(2,2 )+A(2,3))-2.D0*A{1,2)*A(1,2) GO TO 16 14 X(1)=A(1,1)-A(1,2) C=-A(1,1)-A(2,2)-A(1,2) D=A(2,2)*(A(1,1)+A(1,2))-2.DO*A(1,3)*A(1,3) 16 CALL SQUAD(C,D,X) GO TO 400 20 MASE=KASE-5 GO TO (21,21,22,11,24,12,26,13,14,21,24,700,21,26,22), MASE 21 X(1)=A(1,1) X(2)=A(2,2) X(3)=A(3,3) GO TO 400 22 X(1)=A(3,3) X(2)=A(1,1)+A(1,2) X{3)=A(1,1)-A(1,2) GO TO 400 24 X(1)=A(1,1) X(2)=A(2,2)+A(2,3) X(3)=A(2,2)-A(2,3) GO TO 400 26 X(1)=A(1,1)-A(1,2) X(2)=X(1 ) X( 3 ) = A ( i , 1 )+A(1,2)+A(1,2) GO TO 400 110 V = - A ( l , l ) - A ( 2 , 2 ) - A ( 3 , 3 ) C=A(1,1)*A(2,2)+A(1,1)*A(3,3)+A(2,2)*A(3,3)-A(1,2)*A( 1 ,2) 1-A(1,3)*A(1,3)-A(2,3)*A(2,3) D=-A(1,1)*A(2,2)*A(3,3)-2.D0*A(1,2)*A(2,3)*A(1,3)+ 1 A ( 3 , 3 ) * A ( 1 , 2 ) * A ( 1 , 2 ) + A ( 2 , 2 ) * A ( 1 , 3 ) * A { 1 , 3 ) + A U , l ) * A ( 2 , 3 )*A(2,3) i g , CALL CUBICS(VtCtOfXffl) 400 KOMP=0 DO 500 1 = 1,3 WW{I)=DREAL(X(I)) IF(WW(I).LT.0.00) K0MP=K0MP+1 500 CONTINUE IF(KOMP.EQ.O) RETURN QQQ=Q(1)*Q(I)+Q(2)*Q(2)+Q(3)*Q(3) WRITE(7,600) KASEi(Q(I),I=1»3),QQQ,(WW(I),1=1,3) 600 FORM AT ( ' • , I 5 , 7F10. 5 ) 700 RETURN END SUBROUTINE SUMK(M,MOOD,ND) IMPLICIT REAL *8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,RELERR,VV(3),VW(3),COEFF(6) COMMON /FR/ WU4500 ,6),S(26,6),B(5),A{3,6),NK(3000,6) DIMENSION SUM{2,6),SAM(2,6) DF=1.D0 E=0.D0 F=O.DO IF(ND.NE.l) GO TO 100 CALL EWALD(l,M+4,1,MOOD,COEFF(6),SUM) CALL EWALD<1,10,1,0,COEFF(3),SAM) SS=S<10,1) TT=S(9,1) GO TO 200 100 SS=SS/S(10,ND-1)*S(10,ND) TT=TT/S(9,ND-1)*S(9,ND) 200 SU=0.D0 SA=0.D0 DO 300 1 = 1,3 SUM(1,I)=SUM(1,I)/SS SUMC.lt 1+3 ) = SUM( l,I+3)/SS SAM(1,I)=SAM{1,I)/TT SAM(1,1+3)=SAM(1,I+3)/TT SU=SU+SUM{1,1) SA=SA+SAM11,1) 300 CONTINUE Y=B(2)*S(5,ND)-B(4)*S(7,ND) Z=B(1)*(S{6,ND)-SU)-B(3)*(S(8,ND)-SA) DO 400 1=1,6 CALL I ND IX ( I , I A , I B ) A( IA,IB)=-B(2)*SUM(1,1)+B(4)*SAM{ 1, I) I F U . L E . 3 ) A( IA, IB)=A( IA, IB)+Y-Z A ( I A , I B ) = B ( 5 ) * A ( I A , I B ) A( IB,IA)=A(IA,IB) 400 CONTINUE RETURN END FUNCTION KASENOlI,J,K,L,M) N=I+J+K+L+M+1 KASEN0=21 GO TO (500,100,200,300,400,400),N 100 N=I+2*J+3*K+4*L+5*M KASENO=N GO TO 500 200 IF(I.EQ.O) GO TO 210 KASEN0=4+2*J+3*K+4*L+5*M GO TO 500 210 IF(J.EQ.O) GO TO 230 IF(K.EQ.O) GO TO 220 KASEN0=10 GO TO 500 220 KASEN0=11 GO TO 500 230 KASEN0=5+3*K+4*L+5*M GO TO 500 300 IF(I.EQ.O) GO TO 310 KASEN0=21-I-2*J-3*K GO TO 500 310 IF(J.EQ.O) GO TO 320 KASEN0=16 GO TO 500 320 KASEN0=19 GO TO 500 400 STOP 9 500 RETURN END SUBROUTINE NUMPTS(K,L,M,N,NH,NUM> NE = 1 NZ = 0 NB=1 IF(L.EQ.O) NZ=NZ+1 IF(M.EQ.O) NZ=NZ+1 IF(N.EQ.O) NZ=NZ+1 NUMZ=2**NZ IF(L.EQ.M.OR.L.EQ.N.OR.M.EQ.N) NE= IF(L.EQ.M.AND.M.EQ.N) NE=3 NUME=1 IF(NE.NE.l) NUME=NE*(NE-1) IF((L.EQ.K).OR.((L+M+N).EQ.NH)) NB NUM=48/NUME/NUMZ/NB RETURN END SUBROUTINE ORDER! W t N) REAL*8 W ( 6),V ( 6 ) DO 200 I=1,N WMIN=1.D10 DO 100 J=1,N IF(W(J).GE.WMIN) GO TO 100 WMIN=W(J) IMIN=J 100 CONTINUE V ( I )=W(IMIN) W( IMIN) = 1.D10 200 CONTINUE DO 300 1 = 1,N W( I )=V(I ) 300 CONTINUE RETURN END SUBROUTINE CUBICS(B,C,D,X,N) i g s IMPLICIT REAL *8 (A-H,0-Z) C0MPLEX*16 X(6) T=-0.3333333333333333300*8 Q. = T*T R = -.5D0*(D + T*(C - Q - Q) ) Q=Q-0.33333333333333333D0*C U = Q**3 - R*R IF(U.LT.O.DO) GO TO 2 P = 2.DO *DSQRT(Q) H=0.3333 3333333333333DO*DATAN2(DSQRT(U»,R ) IF(N.NE.O) GO TO 1 B = T + P#DCOS(H) RETURN 1 X(N) = T + P*DCOS(H) X(N+1)=T+P*DCOS(H+2.0943951023931955D0) X(N+2)=T+P*DC0S(H-2.0943951023931955D0) RETURN 2 U = DSQRT(-U) H = R + U H=DSIGN(DABS(H)**0.33333333333333333D0,H) R = R - U R=DSIGN(DABS(R)**0.33333333333333333D0»R) IF(N.NE.O) GO TO 3 B = T + H + R RETURN 3 X(N) = T + H + R X(N + 1)=DCMPLX(T-0.5D0*(H+R),0.86602540378443 865DO*(R-H ) ) X(N+2) = DC0NJG(X(N+1)) RETURN END SUBROUTINE SQUAD(C,D,X) IMPLICIT REAL*8 (A-H,0-Z) C0MPLEX*16 X(6) Y=-C/2.D0 DISC=Y*Y-D IF(DABS(DISC).LT.l.D-7*DABS(D)) DISC=O.DO IF(DISC.LT.O.DO) GO TO 100 Z=DSQRT(DISC) X(2)=Y+Z X(3)=Y-Z RETURN 100 Z=DSQRT(-DISC) X(2)=DCMPLX(Y fZ) X(3) = DC0NJG(X(2) ) RETURN END ****************************************** ****** LISTING OF HCPFRQ *************************** ******************************************************* 197 IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,RELERR,VVV(3),WWW(3),COEFF( 6 ),PAR(13), 1 DL(5060),RL(4350),DR(95),DZ(95),RR(72),R Z ( 7 2 ) , 2 NDL(95),NRL(72),NDS(4),NRS<3),ND(4),NR(3 ),JJD(6) COMMON /FR/ W(14500,6),S(26,6),B(5),C(3,6),NK(3000,6) DIMENSION DEL(2,6),SUM(2,6),NUMKS(6) 10 F0RMATI3I4) 20 FORMAT(*IREDUCED PHONON SPECTRUM CALCULATIONS FOR HCP RGS» ,/,'-OCAL 1CULATI0NS BASED ON (',14,'-6) NEAREST NEIGHBOUR POTENT IAL',/,-OCHO 2ICE OF K^-,14, 1 FOR WAVE VECTOR NET") 30 F0RMAT(2F10.5) READ(5,10) M DO 200 1=1,2 Q( I )=0.D0 DO 100 J = l , 6 DEL(I,J)=0.DO 100 CONTINUE 200 CONTINUE Q(3)=0.D0 M00D=M0D(M,2) CALL EWINIK1,1.DO,l.DO,l.DO,M,MOOD) CALL EWALD(0,M,2,M00D,C0EFF(4),SUM) DEN=SUM(1,1) CALL EWALD(0,6,2,0,C0EFF(1),SUM) BB=SUM(1,1)/DEN BBB=BB**(1.DO/DFLOAT(M-6)) B(l)=BBB**M+2 B(3)=BBB**8 B(2) = B(1 )*DFL0AT(M + 2) B(4)=B(3)*8.D0 B(5)=4.5D0/DFL0AT(M-6) READ(5,30) WMN 300 READ(5,10,END=600) K,KM,NSCALE IF(NSCALE.LT.O) NSCALE=-NSCALE DO 500 LL=1,KM READ(5,30) DEL(1,1),DEL(2,1) WRITE(7,20) M,K WRITE(8,20) M,K CALL HCPFRQ(M,MOOD,1,DEL,K,NUMKS) CALL BRANCH(NUMKS,6,1,NSCALE,WMN,DELI 1,1),DEL(2,1)) 500 CONTINUE GO TO 300 600 STOP END SUBROUTINE HCPFRQ(N,MOOD,ND,DEL,K,NUMKS) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION DEL(2,6),NUMKS(6) 10 FORMAT (*. OHCP PHONON FREQUENCIES CALCULATED") N0=1 CALL ORIGIN(N,MOOD,ND,DEL) i g f l IF(K.EQ.O) GO TO 100 LL=0 MM=0 NN = 0 CALL BRILLO<K,LL,MM,NN«J,N,MOOD,ND,NUMKS) K=K/3 GO TO 200 100 CALL LINING(K,J,NO,N,MOOD,ND,NUMKS) 200 RETURN END SUBROUTINE ORIGINIM,MOOD,ND,DEL) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,RELERR,Z{3),V(3),COEFFl6) COMMON /FR/ W{14500,6),S(26,6),B(5),A(3,6),NK(3000, DIMENSION ZUM(2,6),DUM(2,6),DEL(2,6) Q( 1 )=0.D0 Q(2)=0.D0 Q(3)=0.00 DO 400 I = 1,ND S(1,I)=DEL(1,I) S(2,I)=DEL(2,I ) AF=1.D0+S(1,I) DF=1.D0+SI2,I)/AF S(3,I)=DF DDF=DF*DF E=1.D0-DDF F=1.D0-1.DO/DDF S(4,I)=E S(5,I)=F S(8, I )=AF**M+2 S(9,I)=AF**8 S(10,1)=AF*AF S ( 6 , I ) = S ( 8 , I ) * S ( 1 0 , 1 ) S ( 7 , I ) = S ( 9 , I ) * S ( 1 0 , I ) S(14,I)=0.D0 S(22,I)=0.D0 S(26, I) = O.DO S(18,I)=0.D0 CALL AVALD(1,M+4,MOOD,COEFF(6),ZUM,DUM) DO 200 J = l , 3 S(10 + J,I ) = ZUM(1,J)/S(6, I ) S(14+J,I)=DUM(1,J)/S(6,I) S(14,I)=S(14,I)+S(10+J,I) S (18, I ) = S( 18, I ). + S ( 14+J, I ) 200 CONTINUE CALL AVALD(1,10,0,COEFF(3),ZUM,DUM) DO 300 J = l , 3 S(18 + J,I ) = ZUM(1 , J )/S(7, I ) S(22+J,I)=DUM(1,J)/S(7,1) S(22,I)=S(22,I)+S(18+J,I) S(26,I)=S(26,I)+S(22+J,I) 300 CONTINUE 400 CONTINUE RETURN END SUBROUTINE SHELLS(M»MOOD»ND) i g g IMPLICIT REAL *8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F,REL ERR,P<3),V(3),COEFF(6} COMMON /FR/ W(14500,6),S(26,6),B(5),A(3,6),NK{3000,6) DIMENSION DUM(2,6),DAM(2,6),ZUM(2,6),ZAM{2,6) DF=S(3,ND) E =S(4,ND) F =S(5,ND) CALL AVALD(1,M+4,MOOD,COEFF(6),ZUM,DUM) CALL AVALD(l,10,0,COEFF(3),ZAM,DAM) ZU =O.DO DU =O.DO ZA =O.D0 DA =O.DO DUN=O.DO DAN=O.DO DO 200 1=1,6 DO 100 J=l , 2 ZUM(J,I)=ZUM( J ,I)/S(6,ND) ZAM(J,I)=ZAM(J,I)/S(7,ND) DUM(J,I)=DUM(J,I)/S(6,ND) DAM(J,I)=DAM(J,I)/S(7,ND) 100 CONTINUE IF(I.GT.3) GO TO 200 DU=DU+DUM(1,1) DA=DA+DAM{1,1) ZU = ZU+ZUM(1,1 ) ZA=ZA+ZAM(1,I) DUN = DUN+DUM(2,I ) DAN=DAN+DAM(2,I ) 200 CONTINUE X=B(1)*(S(14,ND)-ZU)-B(3)*(S(22,ND)-ZA) XX=B(1)*(S(14,ND)+S(18,ND))-B(3)*(S(22,ND)+S(26,ND)) Y=B(1)*(S(18,ND)-DU)-B(3)*(S(26,ND)-DA) Z=B(1)*DUN-B(3)*DAN DO 700 1=1,6 T=-ZUM(1,I) TT=-DUM(1,1) TTT=O.DO U=-ZAM(1,I) UU=-DAM(1,I) UUU=O.DO IF(I.GT.3) GO TO 400 T=T+S(10+I,ND) TT=TT+S(14+1,ND) TTT=S(14+I,ND)+S(14+I,ND> U=U+S(18+I,ND) UU=UU+S(22+1,ND) UUU=S(22+I,ND)+S(22+I,ND) 400 A(1,I)=B(2)*(T+TT)-B(4)*(U+UU) A(2,I)=B(2)*(T-TT+TTT)-B(4)*(U-UU+UUU) A(3,I)=-B(2)*DUM(2,I)+B(4)*DAM(2,I) IF(I.GT.3) GO TO 500 UU=UU+S(22+I,ND) A(1,I)=A(1,I)-(X+Y) A(2,I)=A(2,I)-(X-Y)+XX A(3,I)=A(3,I)+Z ? n o 500 DO 600 J = l, 3 A { J , I ) = B ( 5 ) * A ( J , I ) 600 CONTINUE 700 CONTINUE RETURN ENO SUBROUTINE LINING1/K/,/JN/,NO,MN,MOOD,ND, NUMK) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3) COMMON /FR/ W(14500,6),S<26,6),B(5),A(3,6),NK(3000,6) DATA QB / Z4093CD3A2C8198E2 / DATA QC / Z409CC470A0490973 / DIMENSION WW(6),NUMK(6) 10 FORMAT( * 0*,I4»2X,9D11.4) QA=0.16666666666666667D0 DO 100 1=1,ND NUMK(I)=0 100 CONTINUE 1 READ(5,2,END=4) L,M,N,LA,MA,NA,NSP NPT=NSP+l 2 F0RMAT(7I4) IF(NO.NE.l) NO = 2 LA = LA - L MA = MA - M NA = NA - N L = NSP*L M = NSP*M N = NSP*N K = NSP*K KK = 2*K QS=QA/DFLOAT(K) QT = QS*QB QU = QS*QC DO 3 11=1,NPT IF(L.EQ.O) J = 1 IF(L.NE.O) J = 10 IF(M.EQ.L) J = J + 3 IF(L+2*M.EQ.KK) J = J + 6 JN = J IF(N.EQ.O) JN = JN + 1 IF(N.EQ.K) JN = JN + 2 Q(3) = QU*N IF(J.EQ.13) GO TO 21 Q ( l ) = QS*L Q(2) = QT*(L + 2*M) GO TO 22 21 Q( 1) = -QS*(L + L ) Q(2) = O.DO 22 DO 300 1=1,ND CALL MASTER!JN,MN,MOOD,1,WW) CALL 0RDER(WW,6) NUMK( I ) = NUMK( I ) +1 LIML=6*(NUMK{I)-1) DO 200 J = l , 6 W(LIML+J,I)=DSQRT(WW(J)) 200 CONTINUE 201 NK(NUMK(I),I)=NDEGEN(JN»1) WRITE(7,10) NK(NUMK(I),I),Q(1),Q(2),Q(3),IW(LIML + J , I ) , J=l,6) 300 CONTINUE L = L + LA M = M + MA 3 N = N + NA GO TO 1 4 RETURN END SUBROUTINE BRILLO(/K/,/L/•/M/,/N/,/JN/,MN,MOOD,ND,NUMK ) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3),DF,E,F COMMON /FR/ W(14500,6),S(26,6),B(5),A(3,6),NK(3000,6) DATA QB / Z4093CD3A2C8198E2 / DATA QC / Z409CC470A0490973 / DIMENSION WW(6),NUMK(6),NQ(3) LOGICAL LZERO,LGREAT,LEVEN,MZERO,MLEAST,MGREAT,NZERO,N GREAT QA=0.16666666666666667D0 DO 100 1=1,ND NUMK ( I ) = 0 100 CONTINUE WRITE(7,600) 600 FORMATI* ONEGATIVE SQUARED FREQUENCIES(IF ANY)*,/,* JN«, 1 4X,«WAVE VECTOR COMPONENTS9X,'Q**2•,21X, 2 1 SQUARED FREQUENCIES 1,/) LIML=0 NI=0 NJ = 0 LMAX = 2*K NMAX = 3*K QS=QA/DFLOAT(K) QT = QS*QB QU = QS*QC K = NMAX L ZERO = L.EQ.O LEVEN = L.EQ.2*(L/2) MMAX = (2*NMAX-L)/2 MZERO = M.EQ.O ML EAST = M.EQ.L NZERO = N.EQ.O JL = 1 IF(.NOT.LZERO) JL = 10 ASSIGN 12 TO I J ASSIGN 14 TO IK 11 LGREAT = L.EQ.LMAX 12 MGREAT = M.EQ.MMAX JM = JL IF(MLEAST) JM = JM + 3 IF(LEVEN.AND.MGREAT) JM = JM + 6 13 NGREAT = N.EQ.NMAX JN = JM IF(NZERO) JN = JN + 1 202 IF(NGREAT) JN = JN + 2 GO TO IK, (14,15) 14 ASSIGN 15 TO IK IF(LZERO) ASSIGN 28 TO IL IFILEVEN.AND..NOT.LZERO) ASSIGN 29 TO IL IF(.NOT.LEVEN) ASSIGN 30 TO IL IF(MZERO) ASSIGN 24 TO IM IFtMLEAST.AND..NOT.MZERO) ASSIGN 25 TO IM IF(.NOT.MLEAST) ASSIGN 26 TO IM IF(NZERO) ASSIGN 21 TO IN IF(.NOT.NZERO) ASSIGN 13 TO IN IF(.NOT.LZERO.OR..NOT.MZERO.OR..NOT.NZERO) GO TO 20 15 NQ(3)=N IFUM.EQ.13) GO TO 16 NQ(2)=L+2*M NQ(1)=L GO TO 19 16 NQ(1)=-L-L NQ(2)=0 19 Q(1)=QS*DFL0AT(NQ{1)) Q(2)=QT*DFL0AT(NQ(2)) Q(3)=QU*DFL0AT(NQ(3)) DO 400 1=1,ND CALL MASTERlJN,MN,MOOD,I,WW) CALL 0RDER(WW,6) NUMK(I) = NUMK(I)+1 DO 300 J = l , 6 W(LIML+J,I)=WW(J) 300 CONTINUE NK(NUMKlI),I)=NDEGEN(JN) 400 CONTINUE LIML=LIML+6 IF ( N I . E Q . l ) JN=JN+2 NI=0 IF ( N J . E Q . l ) JN=JN+6 NJ=0 20 IF(NGREAT) GO TO 23 N- = N + 1 GO TO IN, (13,21) 21 NZERO = .FALSE. ASSIGN 13 TO IN GO TO 13 23 IF(MGREAT) GO TO 27 M = M + 1 GO TO IM, (24,25,26) 24 MZERO = .FALSE. 25 MLEAST = .FALSE. ASSIGN 26 TO IM ASSIGN 12 TO I J 26 N = 0 NZERO = .TRUE. ASSIGN 21 TO IN GO TO I J , (11,12) 27 IF(LGREAT) RETURN L = L + 1 203 GO TO I L , (28,29,30) 28 L ZERO = .FALSE. JL = 10 29 LEVEN = .FALSE. ASSIGN 30 TO IL MMAX = MMAX - 1 GO TO 31 30 LEVEN = .TRUE. ASSIGN 29 TO IL 31 MLEAST = .TRUE. M = L ASSIGN 25 TO IM ASSIGN 11 TO I J GO TO 26 END FUNCTION NDEGEN(J) GO TO (2,3,3,6,7,7,3,5,5,1,2,2,2,3,3,2,3,3,4,6,6) , J 1 NDEGEN=24 GO TO 8 2 NDEGEN=12 GO TO 8 3 NDEGEN=6 GO TO 8 4 NDEGEN=4 GO TO 8 5 NDEGEN=3 GO TO 8 6 NDEGEN=2 GO TO 8 7 NDEGEN=1 8 RETURN END SUBROUTINE ORDER(W,N) REAL#8 W(6),V(6) DO 200 1=1,N WMIN=1.D10 DO 100 J=1,N IF(W( JKGE.WMIN) GO TO 100 WMIN=W(J) IMIN=J 100 CONTINUE V(I)=W(IMIN) W(IMIN)=1.D10 200 CONTINUE DO 300 1=1,N W ( I ) =V( I ) 300 CONTINUE RETURN END SUBROUTINE MASTER(N,MN,MOOD,ND,WW) IMPLICIT REAL*8 (A-H,0-Z) COMMON /SUMS/ Q(3) COMMON /FR/ W(14500,6),S(26,6),AB(5),Z(3,6),NK(3000,6) DIMENSION WW(6) C0MPLEX*16 X(6) 2 Q k CALL SHELLS(MN,MOOD,ND) IF(N.EQ.7.0R.N.EQ.8.0R.N.EQ.9.0R.M.EQ.16) CALL PHASESl Z) GO TO (1,2,3,4,5,6,7,4,9,10,11,12,13,14,15,13,14,15,19 ,20,21), N 1 1 = 1 GO TO 111 2 00 22 1=1,3 AA=0.5D0*(Z(1,I)+Z(2,I) ) BB=0.5D0*(Z(1,I)-Z(2,I)) A=DSQRT(BB*BB+Z(3,I)**2) X(I)=AA-A 22 X(I+3)=AA+A GO TO 300 3 X(1)=0.5D0*(Z(1,1 ) + Z(2,1 ) ) A=0.5D0*(Z(1,2)+Z(2,2)) B=0.5D0*(Z(1,3)+Z(2,3)) C=0.5D0*(Z(1,4)-Z(2,4)) CALL QUADRA(-A-B,A*B-C*C-Z(3,4)**2,X,2) GO TO 211 4 DO 41 1=1,3 X ( I ) = Z (1 , I ) 41 X(I+3)=Z(2,I) GO,TO 300 5 DO 51 1=1,3 X(I ) = O.DO 51 X ( 1 + 3) =Z ( 2, I ) GO TO 300 6 DO 61 1=1,3 61 X ( I ) = 0 . 5 D 0 * ( Z ( 1 , I ) + Z ( 2 , I ) ) GO TO 211 7 X(1)=Z(1,1) X(2)=Z(2,1) A=Z(2,2) B=Z<1,3) C = Z( 3,4)**2 CALL QUADRA(-A-B,A*B-C,X,3) A=Z(1,2) B=Z(2,3) CALL QUADRA(-A-B,A*B-C,X,5) GO TO 300 9 X(1)=0.5D0*(Z(1,1)+Z(2,1)) A=0.5D0*(Z(1,2)+Z(2,2)) B=0.5D0*(Z(l,3)+Z(2,3)) CALL QUADRA(-A-B,A*B-Z(3,4)**2,X,2) GO TO 211 10 CALL SECULA(Z,X) GO TO 300 11 1 = 3 111 A=DSQRT(0.25D0*(Z(1,I)-Z(2,I))**2+Z13,1)**2) B=0.5D0*(Z(1,I)+Z(2,I)) X(1)=B+A X(2)=B-A CALL FOURTH(Z,I,X) GO TO 300 A=0.5D0*(Z(1,1)+Z(2,1)+Z(1,2)+Z(2,2)) B=0.2 5D0*(Z(1,4)-Z(2,4))**2+Z(3,4)**2 2 0 5 C=0.25D0*(Z(1,5 )-Z(2,5))**2+Z(3,5)**2 D=0.2 5D0*((Z(1,6)+Z(2,6))**2-(Z(1,1)+Z12,1))*(Z(1,2)+Z ( 2 , 2 ) ) ) E=0.5D0*(Z(1,3)+Z(2,3)) F=0.5D0*(Z(1,1)+Z(2,1) ) G=0.5D0*(Z(1,2)+Z(2,2)) H=0.5D0*(Z(1,6)+Z(2,6)) AA=0.5D0*(Z(1,4)-Z(2,4)) BB=0.5D0*(Z(1,5)-Z(2,5)) CALL CUBICS(-A-E,A*E-B-C-D,F*B+G*C+E*D 1 -2.D0*H*(AA*BB~Z(3,4)*Z{3,5)),X,1) GO TO 211 ASSIGN 132 TO K I = 1 E=Z(1,1) F=Z(2,2) G = Z(1,3) H=Z(1,5) B = Z(3,4)**2 0 = Z(3,6)**2 0 = 2.D0*Z(3,4)*Z(3,6) A = E + F C = H*H P = D - E*F CALL CUBICS(-A-G,A*G-B-C-P,E*B+F*C+G*P+0,X,I ) GO TO K, (132,300) ASSIGN 300 TO K 1 = 4 E=Z(2,1) F=Z(1,2) G=Z(2,3) H=Z( 2,5) GO TD 131 X(1)=Z(1,3) X(2)=Z(2,3) A=Z(1,1) B=Z(2,2) C = Z(3,6)**2 CALL QUADRA(-A-B,A*B-C,X,3 ) A=Z(2,1) B=Z(1,2) CALL QUADRA(-A-B,A*B-C,X, 5) GO TO 30 0 A=0.5D0*(Z(1,1)+Z(2,l)+Z(1,2)+Z(2,2)) B = Z(3,4)**2 C=0.25D0*(Z(1,5)-Z(2,5))**2 D = - 0 . 2 5 D 0 * ( Z ( 1 , l ) + Z ( 2 , l ) ) * ( Z ( 1 , 2 ) + Z ( 2 , 2 ) ) Z(1,3)=0.500*(Z(l,3)+Z(2,3) ) Z(1,1)=0.5DO*(Z(1,1)+Z(2,1) ) Z(1,2)=0.5D0*(Z(1,2)+Z(2,2)) CALL CUBICS(-A-Z(1,3),A*Z(1,3)-B-C-D,B*Z11,1)+C*Z(1,2) +D*Z(1,3), 1 X, 1) GO TO 211 19 ASSIGN 201 TO K 191 A=0.500*(Z(1,1)+Z(2,1)) 2 0 S B=0.5D0*(Z(l,3)+Z(2,3) ) C=0.5D0*(Z(1,5)-Z(2,5)) CALL QUADRA(-A-B,A*B-2.DO*C*C,X,2) GO TO K, (201,211) 20 X(2)=0.5D0*(Z(1,1)+Z(2,l) ) X(3)=0.5D0*(Z(1,3)+Z(2,3)) 201 A=Z(1,1)-Z(2,1) B = 0 . 5 D 0 * ( Z ( l , l ) + Z ( 2 , l ) ) X(1)=B+A X(4)=B-A GO TO 212 21 X(1)=0.5D0*(Z(1,1)+Z(2,1)) ASSIGN 211 TO K GO TO 191 211 X(4) = X ( l ) 212 X(5) = X(2) X(6) = X(3 ) 300 KOMP=0 DO 400 1=1,6 WW(I)=DREAL(X(I)) IF(WW(I).LT.O.DO) K0MP=K0MP+1 400 CONTINUE IF(KOMP.EQ.O) RETURN QQQ=Q ( 1) *Q ( 1) +Q ( 2 ) *Q ( 2 ) +Q ( 3 ) *Q( 3) WRITE(7,1000) N,(Q(I),1=1,3),QQQ,(WW(I),1=1,6) 1000 FORMAT(• *,15,1 OF 10.5) RETURN END SUBROUTINE FOURTH(Z,N,Y) IMPLICIT REAL*8 (A-H,0-Z) DATA SB / Z4080000000000000 / DATA SC / Z4055555555555555 / DATA SD / Z4040000000000000 / DIMENSION Z ( 3 , 6 ) , R ( 4 ) , A ( 2 , 2 ) , B ( 2 , 2 ) , C ( 2 , 2 ) , A 2 ( 2 , 2 ) , B 2 ( 2,2),C2(2,2) DIMENSION L(2) C0MPLEX*16 Y(6) DO 1 1=1,2 M = N + I IF(M.GT.3) M = M - 3 1 L ( I ) = M DO 6 1=1,2 DO 6 J=l,2 I F ( J - I ) 2,3,4 2 A ( I , J ) = A ( J , I ) B( I,J) = B ( J , I ) C( I,J) = C ( J , I ) GO TO 6 3 K = L ( I ) GO TO 5 4 K = 9 - L ( I ) - L ( J ) 5 A(I,J)=Z(1,K) B( I , J ) = Z(3,K) C( I,J)=Z(2,K) 6 CONTINUE DO 12 1=1,2 2 0 7 DO 12 J=l,2 I F ( I . L E . J ) GO TO 11 A 2 ( I , J ) = A 2 ( J , I ) C 2 ( I , J ) = C 2 ( J , I) GO TO 12 11 X = B ( I , 1 ) * B ( L t J ) + B(I» 2 ) * B ( 2 , J ) A 2 ( I , J ) = A ( I , 1 ) * A ( 1 , J ) + A (I » 2)*AI 2,J) + X C 2 U , J ) = C ( I , 1 ) * C ( 1 , J ) + C<I,2)*C12,J) + X 12 B 2 ( I , J ) = A l l , 1 ) * B ( 1 , J ) + A ( I , 2 ) * B ( 2 , J ) + B ( I , 1 ) *C I 1,JJ+BI I , 2 ) * C ( 2 , J ) TA = A ( l , l ) + A(2,2) + C ( l , l ) + C12,2) TB = A 2 ( l , l ) + A2(2,2) + C 2 ( l , l ) + C2(2,2) TC = O.DO TD = O.DO DO 15 1=1,2 DO 15 J=l,2 TC = TC + A(I,J)« A 2 ( J , I ) + 2.D0*B( I , J ) * B 2 ( I , J ) + C ( I , J ) * C 2 ( J , I ) 15 TD = TD + A 2 ( I , J ) * * 2 + 2•DO*B 2 ( I , J ) # * 2 + C 2 ( I , J ) * * 2 R(1) = -TA R(2) = -SB*IR(1)*TA + TB) R(3) = -SC*(R(2)*TA + R(1)*TB + TC) R(4) = -SD*(R(3)*TA + R(2)*TB + RI1)*TC + TD) CALL QUARTI(R(1),R(2),R(3),R(4)»Y) RETURN END SUBROUTINE PHASES!Z) REAL*8 Z ( 3 , 6 ) , CS / Z4080000000000000 /, SN / Z40DDB3D 742C26553 / DO 100 1=1,5 IF(I.EQ.4) GO TO 100 A=Z(3,I)*SN*2.D0+(l.D0-CS)*Z{l,I)+(l.D0+CS)*Z(2 tI) B=-Z(3,1)*SN*2.D0+(l.D0+CS)*Z(1,1)+t1.DO-CS)*Z(2,I) Z(1,1)=A*0.5D0 Z(2,I)=B*0.5D0 100 CONTINUE Z(3,4)=-Z(3,4)*CS-(Z(1,4)-Z(2,4))*0.5D0*SN Z(3,6)=-Z(3,6)*CS-(Z(1,6)-Z(2,6))*0.5D0*SN RETURN END. SUBROUTINE QUADRA(B,C,X,N) IMPLICIT REAL*8 (A-H,0-Z) C0MPLEX*16 X(6) Q = -.5D0*B U = Q*Q - C IF(U.LT.O.DO) GO TO 1 U = DSQRT(U) X(N) = Q + U XI N+1.) = Q - U RETURN 1 U = DSQRT(-U) X(N) = DCMPLXlQ,U) XIN+1) = DCONJGlX(N)) RETURN END SUBROUTINE CUB ICS(B,C,D,X,N> 2 0 Q IMPLICIT REAL*8 (A-H,0-Z) C0MPLEX*16 X(6) T = -.33333333333333333*8 Q = T*T R = -.5D0*(D + T*(C - Q - Q)) 0 = 0 - .33333333333333333*C U = Q**3 - R*R IF(U.LT.O.DO) GO TO 2 P = 2.D0*DSQRT(Q) H = .33333333333333333*DATAN21DSQRT{U),R) IF(N.NE.O) GO TO 1 B = T + P*DCOStH) RETURN 1 X(N) = T + P*DC0S<H) X(N+1) = T + P*DC0S(H+2.0943951023931955) X(N+2) = T + P*DC0S(H-2.0943951023931955) RETURN 2 U = DSQRT(-U) H = R + U H = DSIGN(DABS(H)**.33333333333333333,H) R = R - U R = DSIGN(DABS(R)**.33333333333333333,R) IF(N.NE.O) GO TO 3 B = T + H + R RETURN 3 X(N) = T + H + R X(N+1) = DCMPLX(T-.5D0*(H+R),.86602540378443865*(R-H)) X(N+2) = DC0NJG(X(N+1)) RETURN END SUBROUTINE QUART I(B,C,D,E,X) IMPLICIT REAL*8 (A-H,0-Z) C0MPLEX*16 X(6) T = -.25D0*B S = T*T Q = C - 6.D0*S R = D + 2.D0*T*(C - 4.D0*S) P = Q + Q CALL CUBICS(P,Q*Q-4.D0*(E+T*(D+T*(C-3.D0*S))),-R*R,X,0 ) Q = ,5D0*(P + Q) + S S = DSQRTlP) R = .5D0*R/S + S*T T = -T - T CALL QUADRA(T +S,Q-R,X,3) CALL QUADRA(T-S,Q+R,X,5) RETURN END SUBROUTINE SECULA(Z,Y) IMPLICIT REAL*8 (A-H,0-Z) C0MPLEX*16 Y(6) DATA SB / Z4080000000000000 / DATA SC / Z4055555555555555 / DATA SD / Z4040000000000000 / DATA SE / Z4033333333333333 / DATA SF / Z402AAAAAAAAAAAAB / 209 DIMENSION Z ( 3 , 6 ) , R(6) DIMENSION A ( 3 , 3 ) , B ( 3 , 3 ) , C ( 3 , 3 ) , A 2 ( 3 , 3 ) t B2C3,3), C2 ( 3 , 3 ) , 1 A 4 ( 3 , 3 ) , B 4 ( 3 , 3 ) , C4C3,3) DO 5 1=1•3 DO 5 J=i,3 I F ( J - I ) 1 ,2,3 1 A ( I , J ) = ACJ,I) B( I,J) = B ( J , I ) C( I,J) = C ( J , I ) GO TO 5 2 K = I GO TO 4 3 K = 9 - I - J 4 A(I,J)=Z(1,K) B( I, J) = Z(3,K) C( I , J )=Z(2,K) 5 CONTINUE DO 12 1=1,3 DO 12 J = l , 3 I F ( I . L E . J ) GO TO 11 A 2 ( I , J ) = A 2 ( J , I) C 2 ( I , J ) = C 2 ( J , I) GO TO 12 11 X = B ( 1 , 1 ) * B ( I , J ) + B ( I , 2 ) * B ( 2 , J ) + BC1,3 )*B(3,J) A 2 ( I , J ) = A ( I , 1 ) * A ( 1 , J ) + A ( I * 2 ) * A ( 2 t J ) + A ( I , 3 ) * A ( 3 , J ) + X C 2 ( I , J ) = C ( I , 1 ) * C ( 1 , J ) + C ( I , 2 ) * C ( 2 . J ) + C C I , 3 ) * C ( 3 , J ) + X 12 B 2 ( I , J ) = A ( I , 1 ) * B ( 1 , J ) + ACI ,2>*B(2, J ) + A M , 3 ) * B ( 3 , J ) 1 + B(I,1 ) *C(1,J) + B(I,2)*C(2»J) + B ( I , 3 ) * C ( 3 , J ) DO 14 1=1,3 DO 14 J= l , 3 I F ( I . LE. J ) GO TO 13 A4( I , J) = A 4 ( J , I) C4( I , J) C 4 ( J , I ) GO TO 14 A4( I , J) = A 2 ( I , 1 ) * A 2 ( I t A2(3,J) J ) + A2( I ,2)*A2(2, J ) + A2C I , 3 ) * 1 + B 2 ( I , 1 ) * B 2 ( J , B2(J,3) 1) + B2( I , 2 ) * B 2 ( J , 2) + B2C I , 3 ) * C4( I , J) = C 2 ( 1 , 1 ) * C 2 ( 1 , C2 ( 3, J ) J) + C2( I ,2)*C2 C2, J) + C2C I , 3 ) * 1 + B2(1,1)*B2(1, B2(3,J) J) + B2( 2 «I } * B 2 ( ' 2 « J ) + B2( 3 , I >* B4( I , J) = A 2 ( I , 1 ) * B 2 ( 1 , B2(3,J ) J) + A 2 U ,2)*B2<2, J) + A2C I , 3 ) * 1 + B 2 ( I , 1 ) * C 2 ( 1 , C2( 3, J) J) + B2( I ,2)*C2(2, J ) + B2C I , 3 ) * TA = A l l 1) + A(2,2) + 3 ) A(3 1, 3) + C C I t l ) + CC2 2) + C ( 3 , TB = A 2 l l , l ) + A2C2,2) + A2(3,3) + C2C1,1) + C2C2,2) + 15 2 3 TC TD TE TF DO DO TC = O.DO = A4( 1, 1 ) C2(3,3) A4(2,2) C4(3,3) 210 + A4(3,3) + C 4 U , 1 ) + C4(2,2) + = O.DO = O.DO 15 1=1,3 15 J=l,3 = TC + A ( I , J ) * A 2 ( J , I ) ) *C 2 ( J , I ) A ( I , J ) * A 4 ( J , I ) ) * C 4 ( J , I ) A 2 ( I , J ) * A 4 ( J , I ) + I , J ) * C 4 ( J , I ) -TA -SB*(R(1)*TA + TB) -SC*(R(2)*TA + R(1)*TB -SD*(R(3)*TA + R(2)*TB -SE*(R(4)*TA + R(3)*TB ) R(6) = -SF*(RI5)*TA + R(4)*TB 1 )*TE + TF) CALL SEXTIC(R,Y) RETURN END SUBROUTINE SEXTIC(A,X) IMPLICIT REAL*8 (A-H,0-Z) C0MPLEX*16 X(6) DIMENSION A(6) + 2 . D 0 * B ( I , J ) * B 2 ( I , J ) + C ( I , J TE = TE + TF = TF R( 1 ) R( 2) R( 3) R(4) R(5) + 2.DO*B(I,J)*B4{I,J) + C C I t J 2.D0*B2( I , J ) * 8 4 U , J ) + C2I TC) R{ l ) * T C R(2)*TC TD) R(1)*TD TE + R(3)*TC + R(2)*TD + R( N U V S Q BA BB BC BD BE BF = 0 = O.DO = O.DO = .16666666666666667*A(1) = DABS(S*A(5)) + DABS(AI6)) - V - V*BA - V*BB - V*BC DABS(BF) 2 A l l ) - U A (2 ) - U*BA A(3) - U*BB A(4) - U*BC A(5) - U*BD AI6) - V*8D R = DABS(S*BE) + IF(R.GE.Q) GO TO Q = R N = .0 FA = BA FB = BB FC = BC FD = BD GA = U GB = V GO TO 3 N = N + 1 IFIN.EQ.5) GO TO 4 CA = BA - U CB = BB - U*CA - V CC = BC - U*CB - V*CA CD = BD - V*CB CE = CD - U*CC CF = -V*CC D = l.DO/(CE*CD - CC*CF) U = U + D*(CD*BE - CC*BF) V = V + D*(CE*BF - CF*BE) GO TO 1 CALL QUADRA(GA,GB,X,1) CALL QUARTI(FA,FB,FC,FD,X ) RETURN END ************** ***************************************** ****** LISTING OF BRANCH *************************** * ****************************************************** 212 SUBROUTINE BRANCH(NUMKS,ND,NL,NSCALE,WMN,D,DI) IMPLICIT REAL*8 (A-H,0-Z) COMMON /FR/ W(14500,6),S(26,6),B(5),A(3,6),NK(3000,6) DIMENSION WW(7),WMEAN(7),WWM(7),WM<7),WMI<7 ) ,W3(7),W4( 7 ) , 1 N0DES(7),YM0DES(7),NUMKS(6) 2 FORMAT(» OFCC PHONON SPECTRUM HISTOGRAMS•,//) 4 FORMAT(*OHCP PHONON SPECTRUM HISTOGRAMS',//) 6 FORMAT(* NUMBER OF MODES IN EACH BRANCH=•,I 10,• WITH', 110,' MODES U N ALL BRANCHES',/,' BRANCHES ARE DEFINED ARBITRARILY AS FOLLOWS; 2W(TA1)<l=)W(TA2)<<=)W{LA)») 8 FORMAT(»+«,70X,•<(=)W(TOI)<(=)W(T02)<I=)WILO)' ) 10 FORMAT{'OHISTOGRAMS ARE SCALED SO THAT W(MAX)=6«) 12 FORMAT!* HISTOGRAM VALUES ARE WRITTEN AS PERCENTAGES') 1 4 F 0 R M A T C 0 INTERVAL G(TA1) NITA1) G (TA2) N(TA2) G( 1LA) N ( LA )') 16 F0RMAT('+»,72X,'G(TOl) N(T01) G(T02) N(T02) G (LO) N(LO 1) ' ,/ ) 18 F0RMAT(*+',72X»'G(TOT) N(TOT)*,/) 20 FORMAT(* 0 INTERVAL G(AC.) NCAC.) G(OP.) N(OP.) G(T 10T) N(TOT)*,/) 22 FORMAT(• ( •,F5.2,',•,F5.2, ' ) ,7F16.10) ,7F16.10) ,7F16.10) ,7F16.10) ,7016.6) ,7D16.6) ,7D16.6) 24 FORMAT(' WMAX 26 FORMAT(• <W> 28 FORMATC <WW> 30 FORMAT(' RMS(W)=' 32 FORMATC <W**3>=" 34 FORMAT(* <W**4>=' 36 FORMATC <1/W> =' 38 FORMAT(' FOR ',7116) 40 FORMATC FREQUENC I E S> • , F 16 . 10 ) 42 FORMATC «,13X,•TAl•,13X,'TA2• 44 FORMATC +',62X, 'TOI', 13X, 'T02', 14X, 'LO B , 13X , • TOT' , / ) 46 FORMATC+•,62X,'TOT',/) 48 FORMATCOEXPANSION COEFFICIENT (W.R.T. STAT. EQ.)=',F1 0.6) 50 FORMATCOEXPANSION COEFFICIENTS (W.R.T. STAT. EQ.)=',2 F10.6) NUMK=NUMKS(NL) MD=ND+1 IF(ND.EQ.6) GO TO 100 WRITE(7,2) GO TO 200 100 WRITE(7,4) WRITE(8,4) 200 DO 300 1=1,MD WMEAN(I)=0.DO WW(I)=0.DO WWM(I )=0.DO WMI(I)=0.DO W3U) = O.DO W4II) = O.DO NODES(I)=0 300 CONTINUE MODES=0 DO 600 1=1,NUMK II=ND*(I-1) MODES=MODES+NK(I,NL) DO 500 J=l,ND BB=W( I I + J,NL) IF(BB.LT.O.DO) BB=O.DO IF(NSCALE.GT.O) BB=DSQRT(BB) W( II + J,NL) = BB SC=DFLOAT(NK(I,NL))*BB BC=BB*BB WMEAN(J)=WMEAN(J)+SC WWM(J)=WWM(J)+SC*BB W3(J)=W3(J)+SC*BC W4(J)=W4(J)+SC*BC*BB IF(BB.LT.WMN) GO TO 400 NODES(J)=NODES{J)+NK(I,NL) WMI(J)=WMI{J)+DFLOAT!NK(I,NL))/BB 400 IF(BB.LE.WW(J)) GO TO 500 WW (J)=BB IFIWW(J).GT.WW(MD)) WW(MD ) = WW(J) 500 CONTINUE 600 CONTINUE LODES=MODES*ND XMODE S=DFLOAT t LODES) WRITE(7,6) MODES,LODES IFIND.EQ.3) GO TO 700 WRITE17,8) WRITE(8,6) MODES,LODES WRITE(8,8) 700 WRITE(7,42) IF(ND.EQ.6) WRITE(7,44) IFIND.EQ.3) WRITE(7,46) WRITE(7,24) (WW(I),1=1,MD) DO 800 1=1,ND NODES(MD)=NODES(MD)+NODES(I ) YMODES(I)=DFLOAT(NODES(I)) WMEAN(I)=WMEAN( I ) /XMODES WMEANlMD)=WMEAN(MD)+WMEAN(I) WWM(I ) = WWM(I)/XMODES WWM{MD) = WWM(MD)+WWM(I) W3(I)=W3(IJ/XMODES W4(I)=W4(I)/XMODES W3(MD)=W3(MD)+W3(I) W4(MD)=W4(MD)+W4(I) 800 CONTINUE YMODES(MD)=DFLOAT(NODES(MD)) DO 900 1=1,ND WM(I)=DSQRT(WWM(I)) IF(NODES(MD).NE.O) WMI(I)=WMI(I)/YMODES(MD) WMI(MD)=WMI{MD)+WMI(I) 900 CONTINUE WM(MD)=DSQRT(WWM(MD)) WRITE(7,26) ( WMEANl I ),1=1,MD) WRITE(7,28) (WWM(I) ,I=1,MD) WRITE(7,30) (WM(I) ,1=1,MD) WRITE(7,32) (W3(I) ,I=1,MD) WRITE(7,34) (W4(I) ,I=1,MD) WRITE(7,36) (WMKI) ,I = 1,MD) WRITE(7,38) (NODES(I)•1 = 1, MD) WRITE(7,40) WMN YH=0.D0 XH=0.D0 FRAC=0.1D0 LH=10.D0*WW(MD)+2.D0 IF(IABS(NSCAL E ) .NE . 1) GO TO 1000 FRAC=WW(MD)/60.D0 LH=62 WRITE(7, 10) IF(ND.EQ.6) WRITE(8,10) 1000 WRITE(7,12 ) IF(ND.EQ.6) WRITE(8,12) DO 1100 1=1,MD WM(I)=0.DO WW(I)=0.DO W3(I)=0.D0 W4( I)=0.D0 1100 CONTINUE XMODES=XMODES*l.D-2 WRITE(7,14) IF(ND.EQ.6) GO TO 1200 WRITE(7,18) GO TO 1300 1200 WRITE(7,16) WRITE(8,20 ) 1300 DO 1800 1=1,LH XL=XH YL=YH XH=XH+0.1D0 YH=YH+FR AC DO 1400 J=1,MD WM(J)=O.DO W3(J)=0.DO 1400 CONTINUE DO 1600 J=1,NUMK JJ=ND*(J-l) DO 1500 K=1,ND WWW=W(JJ+K,NL ) IF(WWW.LT.YL.OR.WWW.GE.YH) GO TO 1500 WM(K)=WM(K)+DFLOAT(NK(J,NL)) 1500 CONTINUE 1600 CONTINUE DO 1700 K=1,ND WM(K)=WM<K)/XMODES IF(K.LE.3) W3(1)=W3(1)+WM(K) IF(K.GT.3) W3(2)=W3(2>+WM(K) WW(K)=WW(K)+WM(K) 1700 CONTINUE 2 1 5 WM(MD)=W3(1)+W3(2) W4(1)=W4(1)+W3(1) W4(2)=W4(2)+W3(2) WW(MD)=WW{MD)+WM(MD) IF(ND.EQ.3) WRITE(7,22) XL,XH,(WM(L),WW(L)•L=l,MD) IFIND.EQ.3) GO TO 1800 WRITE(7t22) XL,XH,(WM(L),WW(L),L=1,ND) WRITE(8,22) XL,XH,W3(1),W4ll),W3(2),W41 2),WMIMD),WW IMD ) 1800 CONTINUE IFIND.EQ.6) GO TO 1900 WRITE(7t48) D GO TO 2000 1900 WRITE(7,50) D.DI WRITE(8,50) D,DI 2000 RETURN END ******************************************************* ****** LISTING OF HSTGRM *************************** ******************************************************* 216 SUBROUTINE HSTGRM(NUMK,ND,NTAPE,NSCALE,D,DO,WMN) IMPLICIT REAL*8 (A-H,0-Z) COMMON /FR/ WU4500 ,6),S(26,6),B(5),A(3,6>,NK{3000,6) DIMENSION WW(6),WMEAN(6),WWM(6),WMI 6),WMII 6)»W3(6),W4( 6 ) , 1N0DESI6),YMODES(6),D(12) ,DD(3) 10 FORMATl-OFCC PHONON SPECTRUM HISTOGRAM',//) 20 FORMAT('1HCP PHONON SPECTRUM HISTOGRAM',//) 30 FORMAT{* TOTAL NUMBER OF M0DES=',I10) 40 FORMAT(* WMAX =',6F18.10) 50 FORMAT(' <W> =',6F18.10) 60 FORMAT(*OREFERENCE HISTOGRAM SCALED SO THAT WMAX=6«,) 65 FORMAT(•0 INTERVAL',18X,'G(W)',16X, ,N(W)/NT0T',/) 70 FORMAT(* («,F5 . 2,•,•,F5 .2, • ) •,6X,2F20.10) 93 FORMAT(* <1/W> =',6018.8 ) 97 FORMAT('OTHESE ARE INTERPOLATION FORMULAS ABOUT D=', 1 F10.5,' DI=',F10.5) 94 FORMAT(• FOR • ,6118 ) 95 FORMAT(• FREQUENCIES >',F10.8) 80 FORMAT(• <WW> =',6F18.10) 90 FORMAT(' RMS(W)=',6F18.10) 91 FORMAT!* <W**3>=«,6D18.8 ) 92 FORMAT(• <W**4>=•,6D18.8 ) IF..IND.NE.6) WRITE(7,10) IFIND.EQ.6) WRITE(7,20) WMAX=0.D0 DO 100 1=1,ND WMEAN(I)=0.DO WWM(I)=0.DO NODES(I)=0 WMI(I)=0.DO W3(I) =0.D0 W4(I) =0.D0 100 CONTINUE MODES=0 DO 500 1=1,NUMK I I=ND*(I-1) MODES=MODES+NK(I,1) DO 400 J=l,ND DO 200 K=1,ND IF(NSCALE.GT.O) W(11 + J,K)=DSQRT(W(II + J,K)) SC=DFLOAT(NK(I,1))*W(II+J,K) WMEAN(K)=WMEAN(K)+SC WWM(K)=WWM(K)+SC*W(II+J,K) BC=W(II+J,K)*W(II+J,K) W3(K)=W3(K)+SC*BC W4(K)=W4(K)+SC*BC*W( II + J,K) IF(W(II+J,K).LE.WMN) GO TO 200 NODES(K)=NODES(K)+NK(1,1) WMI(K)=WMI(K)+DFLOAT(NK(I,1))/W(II+J.K) WW(K)=W(II+J,K) 200 CONTINUE IF(W( II + J,1).LE.WMAX) GO TO 400 WMAX=W(II+J, 1 ) IF(ND.EQ.3) CALL INTERF(WW,D) IF(ND.EQ.6) CALL INTERH(WW,D,OD) DO 300 K=1,ND W(11+J,K) = WW(K ) 300 CONTINUE 400 CONTINUE 500 CONTINUE MODES=M0DES*ND XMODES=DFLOAT(MODES) WRITE(7,30) MODES IF(ND.EQ.3) CALL INTERF(WW,D) IF(ND.EQ.6) CALL INTERH(WW,D,DD) WRITE(7,40) (WW(I),I=1,ND) DO 600 1=1,ND YMODES(I) = DFLOAT(NODES( I ) ) IF(NODES(I) .NE.O) WMI(I) = WMI(I)/YMODES(I ) W3(I)=W3(IJ/XMODES W4(I) = W4(I)/XMODES WMEAN ( I)=WMEAN( D/XMODES WWM(I)=WWM(I)/XMODES WM(I)=DSQRT(WWM(I)) 600 CONTINUE IF(ND.EQ.6) GO TO 625 CALL INTERF(WMEAN,D) CALL I NT ERF(WWM ,D) CALL I NT ERF(WM ,D) CALL I NT ERF(W3 ,D) CALL INTERF(W4 ,D) CALL INTERF(WMI ,D) GO TO 650 625 CALL INTERH(WMEAN,D,DD) CALL INTERH(WWM ,D,DD) CALL INTERH(WM ,D,DD} CALL INTERH(W3 ,D,DD) CALL INTERH(W4 ,D,DD) CALL I NT ERH(WMI ,D,DD) 650 WRITE(7,50) (WMEAN(I},1=1,ND) WRITE(7,80) {WWM(I),1=1,ND) WRITE(7,90) (WM(I),1=1,ND) WRITE(7,91) (W3(I ) , 1 = 1,ND) WRITE(7,92) (W4(I),1=1,ND) WRITE(7,93) (WMI(I)»1=1,ND) WRITE(7,94) (NODEStI),1=1,ND) WRITE(7,97) D(1),D(2) WRITE(7,95) WMN LH=10.D0*WMAX+2.D0 FRAC=1.DO IF(IABS(NSCALE).NE.l) GO TO 675 WRITE(7,60) LH=63 FRAC=WMAX/6.D0 675 GRAC=FRAC*1.D-1 XM0DES=XM0DES/FRAC*l.D-2 WRITE(7,65) YH=0.D0 XH=O.DO SF=O.DO DO 900 1=1,LH XNFR=0 XL = XH XH=XH+0.1DO YL = YH YH=YH+GRAC DO 800 J=1,NUMK JJ=ND*(J-1) DO 700 K=1,ND WWW=W(JJ+K,1) IF(WWW.LT.YL.OR.WWW.GE.YH) GO TO 700 XNFR=XNFR+DFLOAT(NK(J,1)) IFINTAPE.LE.O ) GO TO 700 WRITE(9) NK{J,I),{WtII+J,L),L=1,ND) 700 CONTINUE 800 CONTINUE FR=XNFR/XMODES SF=SF+FR IF(NPRINT.GT.O) WRITE(7,70) XL,XH,FR,SF 900 CONTINUE WMN=WMEAN(1) RETURN END SUBROUTINE INTERF(OM, D) IMPLICIT REAL *8 <A-H,0-Z) DIMENSION 0M(6),D(12) IF(OM(1).LT.l.D-20) GO TO 200 IF(D(5).EQ.0.D0) GO TO 200 A=0M(2)-0M(1) B=0M(3)-0M(1) 0M(2)=(A*D(4)-B*D(2))/D(5) 0M(3)=(B*D(1)-A*D(2))/D(5) 0M(3)=0M(3)/0M(2) 0M(2)=0M(2)/0M(1) 200 RETURN END SUBROUTINE INTERH(OM,D,DD) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION 0M(6),D(12),DD(3) IF(0M(1).LT.l.D-20) GO TO 200 IF(DD(1).EQ.O.DO) GO TO 200 A=0M(2)-0M(1) B=OM(3)-0M(1) C=0M(4)-0M(1) E=0M(5)-0M(1) F=0M(6)-0M{1) 0M(4)=(A*D(10)-B*D(8))/DD(2) 0M(5)=(B*D(7)-A*D(9))/DD(2) 0M(2)=(C*D(4)-E*D(2))/DD(l) 0M(3)=(E*D(1)-C*D(3))/DD(l) 0M(6)={F-V*D(5)-VV*D(6)-W*D(11)-WW*D(12))/DD(3) 0M(6)=OM(6)/0M{A) 0M(5)=0M(5)/0M(4) 0M(4)=0M(4)/0M{2) •M(3)=0M(3)/0M(2) 0M(2)=0M(2)/0M{1) 200 RETURN END ************************************** ****** LISTING OF HISTPRINTER ********************** ******************************************************* 220 IMPLICIT INTEGER*4 <A-B,I-N) DIMENSION A(32),ASYM(7),ATEST(5),G(6),N(7),B(100) DATA ASYM / ' 1 1 1 1 ' , ' 2 2 2 2 • , • 3 3 3 3 ' , - 4 4 4 4 » 5 5 5 5 6 6 6 6 * ****«/ OATA ATEST /•OHCP•,•OFCC»,• BRA•.•INTE• t• ' / 10 FORMAT!32A4) 20 F0RMAT14A4,6(F9.3,9X)) 30 F0RMAT(4A4,6X,F20.10) 40 FORMAT(* 0****** NO BRANCH INFORMATION GIVEN 1,/) 50 F0RMAT(4A4,4X,100A1) MD=0 100 READ(5,10,END=9000) ( A ( I ) , I = 1, 32 ) WRITE(7,10) (A(I),1=1,32) WRITE(8,10) (A{ I ), 1 = 1,32) I 1=0 IF(A( 1).NE.ATEST( 1) ) GO TO 200 ND=6 GO TO 100 200 IF(A(1).NE.ATEST(2)) GO TO 300 ND=3 GO TO 100 300 IF(A<1).NE.ATEST(3)) GO TO 400 MD='l GO TO 100 400 IF(A(2).NE.ATEST(4)) GO TO 100 IF(MD.EQ.O) ND=1 MD=0 READ(5,10,END=9000) ( A l l ) , 1 = 1 , 4 ) WRITE(7,10) ( A ( I ) , I = 1 , 4 ) WRITE (8, 10) ( A U ) ,1 = 1,4) 500 11=11+1 IF(ND.EQ.1) GO TO 600 READ(5,20) (A(I),1=1,4),(G(I),I=1,ND) GO TO 700 600 READ(5,30) (A(I ),1 = 1,4),G(1) G(1)=G(1)*100. 700 IF(ND.GT.6) GO TO 9000 IF(ND.NE.l) GO TO 1100 WRITE(7,40) GO TO 1500 1100 DO 1200 1=1,100 B( I )=ATEST(5) 1200 CONTINUE B( 1) = ASYM(7) B(31)=B(1) B(61)=Bl1) DO 1300 1=1,ND,3 N( I )= G ( I ) * 1 0 . + 1 N( 1 + 1 )=G(1 + 1)*10.+31 N(I+2)=G(I+2)*10.+61 1300 CONTINUE DO 1400 1=1,ND,3 DO 1390 M=l,3 LX=I+3-M IF(N(LX).GT.100) N<LX)=100 221 LL=61 IF(LX.EQ.2.0R.LX.EQ.5) LL=31 IF(LX.EQ.1.0R.LX.EQ.4) LL=1 LH=N(LX) OO 1310 MM=LL,LH IF(MM.NE.LL) B(MM)=ASYM(LX) 1310 CONTINUE 1390 CONTINUE 1400 CONTINUE WRITE(7»50) (A(I),1=1,4),(B(I),1=1,100) 1500 DO 1600 1=1,100 B(I)=ATEST(5) 1600 CONTINUE B(1)=ASYM(7) DO 1700 1=1,ND I F ( I . N E . l ) G ( I ) = G ( I l + G I I - l ) N ( I ) = 1 0 . * G ( I ) + l IF(N(I).GT.100) N(I)=100 1700 CONTINUE LH=0 DO 1900 1=1,ND LL=LH+1 IF(LL.GT.N(I) ) GO TO 1900 LH=N(I) IF(I.NE.l.AND.LH.EQ.Nt1-1)) GO TO 1900 DD 1800 L=LL,LH I F ( L . N E . l ) B(L)=ASYM(I) 1800 CONTINUE 1900 CONTINUE WRITE (8, 50) ( A H ) ,1 = 1,4), (B(I ) ,1 = 1,100) I F ( I I.EQ.62) GO TO 100 GO TO 500 9000 STOP END 222 APPENDIX C PRECISION REQUIREMENTS OF THE RGS STABILITY PROBLEM The purpose of t h i s appendix i s to determine reasonable p r e c i s i o n requirements f o r a c a l c u l a t i o n of the ground s t a t e energy d i f f e r e n c e : I t i s assumed that the atoms i n t e r a c t v i a (12-6) p o t e n t i a l s , that the l a t t i c e s move according to the harmonic approximation, and that no ext e r n a l pressure i s applied to the l a t t i c e s . The r e s u l t s of these con s i d e r a t i o n s are l i s t e d i n Table UT-1 i n the body of the t h e s i s . For the purposes of t h i s a n a l y s i s , the ac t u a l value of a phy-s i c a l v a r i a b l e i s denoted by i t s usual symbol, the c a l c u l a t e d value i s denoted by a s u b s c r i p t c, and the r e l a t i v e e r r o r i n the c a l c u l a t e d value i s denoted by the corresponding small Greek l e t t e r , unless that choice would confuse symbols. Thus, fi. = G : C ( KT). (c-2) A l s o , i t i s assumed that comparable accuracy can be obtained f a r both l a t t i c e s , so i r f | * H U ( c o ) P r e c i s i o n of G. Using (C-1) and (C-2), one obtains f o r the c a l c u l a t e d grcund s t a t e energy d i f f e r e n c e 223 Since I-7]! < ^ 1, the d i f f e r e n c e s between G's u i t h d i f f e r e n t s u b s c r i p t s can be ignored, so where the RHS must have the same sign as 7| , i f computational e r r o r i s to be small enough. Therefore, or, using (C-3), P r e c i s i o n of <j> and E-y. The ground state energies are of the form: (c-5) where § * $ c C H < T ^ > a ^ c O ^ ) , f i ^ - J T ^ c . (C-6) Using (C-5) and (C-6), one obtains so, with (C-2) and (C-4), i r l - (C-7) Since 5 i s generally much ea s i e r to c a l c u l a t e to high p r e c i s i o n than i s E , assume f o r a moment that cf> =0. Then (C-7) y i e l d s iZZi r ^ 1 Now, i f one performed a l l c a l c u l a t i o n s of E z to the same accuracy, that accuracy must be s u f f i c i e n t f o r a l l of the RGS, and, i f j°f^ D, to b e t t e r accuracy than i n d i c a t e d above f a r neon. Therefore, assume U | < I ^ I A Z , (c-s) and use (C-7) to f i n d out how large can be: or, since ^£1/3 f o r neon 2 Therefors, i t would be safe ta c a l c u l a t e such that | c f l < l-yjl/s. (c-9) P r e c i s i o n of L a t t i c e Sums f o r ^  . The l a t t i c e p o t e n t i a l f o r the model considered here i s «S_ (C-10) where (C-11) and the l a t t i c e sums are described i n appendix A. I f (C-10) can be w r i t t e n as which y i e l d s , with (C-6) and (C-9): 225 (C-12) The maximum p o s s i b l e values of and <J}^can be determined by comparing f i r s t neighbor and a l l neighbor values f a r the l a t t i c e sums i n Table A-1: i * ~ 6 < r , x * , \ < r , z l £ 0.0/. Since $6 and (SJ^are of the same sign and 4 1 z < < < r £ » f Q r equal numbers of c o n t r i b u t i n g s h e l l s , (C-12) i s s a t i s f i e d i f : I <rfe | < ITJ I //o . Cc-13) P r e c i s i o n af Except f a r a p r o p o r t i o n a l i t y constant, the zero point energy i s the same as the mean frequency so (C-6) and (C-8) i n d i c a t e : (C-14) I f one also w r i t e s (C-15) (C-U) y i e l d s (C-1G) A I f £.m^ i s the maximum value af I £x | f a r a l l phonon-modes considered, max then (C-16) i s s a t i s f i e d i f I €\ I < < l / f e . CC-17) 226 provided that the number of frequencies i n the summations i s small enough that round-off e r r o r does not propagate i n t o the s i g n i f i c a n t 1 p o r t i o n of the computer representation of the required summations. For summations c a l c u l a t e d i n REAL*8 a r i t h m e t i c and l**|l'v'10~it, round-g O f f e r r o r would not be important unless more than 10 frequencies were inc l u d e d . P r e c i s i o n of Dynamical Matrix Elements. A c a r e f u l evaluation of p r e c i s i o n 'requirements f o r the dynamical matrix elements i s d i f f i -c u l t , because i t i n v o l v e s rather general perturbations on rather general hermitian matrices. However, a couple of statements are z. _ c l e a r . F i r s t , the r e l a t i v e e r r o r i n i s about 2£^ and, according to (C-17), t h i s must be smaller i n magnitude than l">*)l. Second, i f every dynamical matrix element ^^(/Tv^had the same r e l a t i v e e r r o r S ; such that lSi<lTjl, s u f f i c i e n t accuracy would have been obtained to s a t i s f y (C-17). I t would als o be s a t i s f a c t o r y i f the r e l a t i v e e r r o r s S^jj^v) s a t i s f i e d ( S ^ W J - • - [ • - (C-18) The r e s u l t (C-18) cannot be accepted without f u r t h e r comment. In the BZ there are s p e c i a l p o i n t s , axes, and planes f o r which the 2 dynamical matrix elements must s a t i s f y c e r t a i n symmetry r e l a t i o n s . I f these s p e c i a l regions are not i d e n t i f i e d before d i a g o n a l i z i n g the dynamical matrices, the symmetry r e l a t i o n s cannot be used and the 1 G. E. Forsythe, Amer. Math. Monthly 77, 931 (1970). 2 3. L. Warren, Rev. Mod. Phys. kO, 38 (1968). 227 d i a g o n a l i z a t i o n problem i s needlessly cumbersome. A l s o , even though the r e l a t i v e e r r o r s i n the dynamical matrix elements might be on the order of S t the r e l a t i v e e r r o r s i n the squared frequencies might be on the order of because a d i s c r i m i n a n t that should vanish ex-a c t l y does not. These e r r o r s would not n e c e s s a r i l y a f f e c t because matrix d i a g o n a l i z a t i o n g e n e r a l l y conserves t r a c e s . These e r r o r s would a f f e c t the values of < &^) > c , though. In the c a l c u l a t i o n s f o r t h i s t h e s i s , care was taken to t r e a t a l l s p e c i a l symmetries as e f f i c i e n t l y as p a s s i b l e , so t h i s problem was not encountered. P r e c i s i o n of L a t t i c e Sums f o r fi^jg(>v) . The dynamical matrix elements are composed of complicated combinations of the l a t t i c e sums, \¥'7T^ a n d ^l3 < J'71'» S Q i t i s d i f f i c u l t to determine the e f f e c t s of l a t t i c e sum e r r o r s upon the dynamical matrix elements f o r a gen-e r a l wave vector. I t i s quite c e r t a i n that the frequencies of long wavelength, acoustic phonon modes would be s u s c e p t i b l e to c a t a s t r o p h i c c a n c e l l a t i o n , since the l a t t i c e sums appear as d i f f e r e n c e s of the s%(s)-S«7<*>-- , ^ ' < < | -The frequencies of these modes would be s t r o n g l y a f f e c t e d by e r r o r s i n l a t t i c e sums. In a large sample of frequencies, only a small f r a c t i o n would be a f f e c t e d by t h i s e r r o r . Moreover, these frequencies are small and hardly c o n t r i b u t e to At best, one can expect only that the r e l a t i v e e r r o r s i n the l a t t i c e sums would be comparable to the r e l a t i v e e r r o r s i n the dynam-i c a l matrix elements. In f a c t , the author has obtained t h i s r e s u l t i n s e veral crude estimates of the e f f e c t s of l a t t i c e sum e r r o r . Haw-228 ever, because the author has not discovered a s a t i s f a c t o r y method of analyzing the problem, he i s r e l u c t a n t to accept r e l a t i v e l a t t i c e sum e r r o r s comparable u i t h those allowed f o r the dynamical matrix elements. He would be mare conservative and expect l a t t i c e sum e r r o r s to be at l e a s t an order of magnitude smaller. 

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