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On the temperature dependence of the paramagnetic resonance line shape function in the case of the Ising… Yu-yang, Lee 1961

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ON THE TEMPERATURE DEPENDENCE OF THE PARAMAGNETIC RESONANCE LINE SHAPE FUNCTION IN ...:THE- CASE OF THE I SING MODEL by -LEE  YU-YANG  6.Sc., University of British Columbia, 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGEEE OF MASTER OF SCIENCE - in the Department . . • of ' " • • Physics '  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1961  -iii-  ABSTRACT  The adequacy of the approximation method used by McMillan and Opechowski (I960} in their theoretical study of the temperature dependence of the paramagnetic resonance line shape function is very difficult to ascertain for the case of a typical paramagnetic  crystal.  For this reason the approximation method has been investigated, in this thesis, for the very simple case of the one-dimensional Ising model.  Exact expressions for the  line shape function of the model are compared with expressions obtained by the above mentioned approximation method.  The agreement between the two expressions is  found to be very good for all temperatures, and in particular, for extremely low temperatures.  In presenting this thesis in-..partial fulfilment pf> 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^a,gree• 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.  /  ,  Department of (^^^i^fLt^^: The University of British Columbia, Vancouver S, Canada; Date  f].-fj sIaj , / 7 . a  /  ACKNOWLEDGEMENTS  I wish to express ray gratitude to Professor W. Opechowski for suggesting this problem and for his continued interest and valuable advice throughout the performance of this research.  I wish also to thank  the National Research Council of Canada for financial assistance.  -iv-  Table of Contents  Page ACKNOWLEDGEMENTS  .  ABSTRACT  iii  Table of Contents SECTION I  .  . . . . . . . . . . .  iv  Introduction - Magnetic Resonance and Line-Shape Function  SECTION II  ii  1  Statement of the Problem in General Terms and the Description of One-Dimensional Ising Model  SECTION III  ......  One-Dimensional Ising Model (Open-Chain) for N = 4  SECTION IV  One-Dimensional Ising Model  17 (Ring)  for Arbitrary N SECTION V  7  23  Intensity Function in the First Approximation for the Ring Model with. Arbitrary N  SECTION VI  . . . .  • .57  Comparison of the Asymptotically Exact Expressions  (4-61a),(4-61b),  (4-61c) with First Approximation Expressions  (5-la),(5-lb),(5-lc) at  Very Low Temperature and Conclusions  ,  59  -V-  APPENDICES 1.  General Formula for the Number of Component Lines Present in the Absorption Band for an Ising Model with and Arbitrary Form of Lattice  2.  (see p.27}  65  To Prove  ( s e e p  3.  The Exact Evaluation of Q l f  _47)_  67  Q 2 and Qg (see p.48).  4.  To Prove the Existence of the Contour c' (see p.52).  5.  68  To Prove the Relations  (4-58) (see p.54).  71  .  74  . . . . . . . . . . . „ . . ,  20  FIGURES AND TABLE Fig.  (3-1)  Energy Levels and Transition Scheme of  One-Dimensional Ising Model with 4 Spins Table (3-6)  (Open-Chain)  The Values of the Erequency Function,  the first and the second moments about the Zeeman transition energy, in the exact expression, first  calculated approxima-  tion and second approximation at various temperatures. with N = 4 )  (For Open-Chain Ising Model 21  -vx-  Fig. (3-7)  Comparison of Frequency Function  in the Exact and First Approximation. (For Open-Chain Ising Model with N = 4) -Fig. (6-2)  . . . .  22  Comparison of Frequency Function  in the Asymptotically Exact Calculation and First Approximation Calculation . . . . Bibliography  64  78  SECTION I  Introduction  Magnetic Resonance and Line Shape Function  Let us consider a paramagnetic ion to the extent that it is characterized by a total angular momentum vector S and a magnetic moment vector ST satisfying the relation  where the various constants take their usual meaning.  We  shall call such a quantum mechanical model, with arbitrarily fixed values of g and S, a "spin".  We shall call  a system containing an arbitrarily large number N of identical spins, i.e., spins with the same values of g and S, a "spin-system" or simply a "system". When such a spin-system is placed in a constant external magnetic field H, in the absence of interactions among the spins, the energy of the system will then be  quantized into a finite number of distinct Zeeman energy levels.  The number of such levels depends on the  angular momentum quantum number S as well as the number N of spins present in the system.  These levels, except  for the highest and the lowest, are highly degenerate. The degeneracy is in general different for different energy levels, and depends again on the values of S and N. When a weak, sinusoidally varying magnetic field "H^, frequency  V  , is superimposed perpendicularly onto the  constant magnetic field H, and is so weak that its influence on the energy levels of the system is entirely negligible, transitions will take place among those levels if and only if the condition  where  is satisfied, where E and E' are the energy levels of the system among which transition is induced. In the case where there is no interaction among the spins, we have £•'-£  -ffiH(M'-M)  where M* and M are the quantum numbers corresponding to E' and E respectively with the selection rule  \m'-M\ =/  Thus  Hence an absorption line will be observed at the frequency  Vo  and we shall call this process "Magnetic  Resonance". •..•:•. Were it not for the fact that the occupation numbers are different for different energy levels, there would be neither net absorption nor net emission of energy.  Since  according to a general result of quantum mechanics, the transition probability between two levels is the same both ways.  The average absorption and emission of  energy in unit time thus cancel.  But in the actual case,  the system of paramagnetic ions is in a more or less well-defined thermodynamic equilibrium state, to which let us assign an absolute temperature T.  The occupation  number of the energy level E of a system of spins will be proportional to the Boltzmann factor namely exp where k is the usual Boltzmann constant. When then  E exp  (- g r )  > >  E' exp (-  so there will be more spins at lower energy levels than at higher ones.  Consequently there will be more transitions  from lower states into higher ones resulting in the net absorption of energy from the oscillating magnetic field H^. When weak exchange as well as dipole interactions  exist among the spins, each energy level of the system will be decomposed into a large number of closely spaced sublevels, each of which may or may not be degenerate. As a result of this, the magnetic resonance will take place at a large number of frequencies in the neighbourhood of V0  .  Instead of a single monochromatic  absorption line at  , we have a large number of  absorption lines with different intensities distributed in the close neighbourhood of >£ band.  forming an absorption  Each component line of this band, say a line at'  frequency  *, is characterized by an intensity  function I( )£ ) (p = 1,2, ... ©, the number of component lines in the band).  The number © of component lines  present in the band depends on the type and the magnitude of the interactions as v/ell as on the geometrical properties of the system. We shall define a function ^ (>p J in terms of the intensity function as follows  We shall call  normalized discrete frequency  function at frequency  corresponding to a component  * See note at the end of this section.  line in the band. When the interaction is very weak, the band is very narrow.  If, in addition, the interactions are such that  the number of component lines present in the band is very large, we can approximate the discrete distribution of intensity function with a continuous one, i.e.,  6 Z p-i  Ibf)  « J I (r)  dr  (1-2)  On the other hand, when the absorption band is very narrow, we can still call it a line but it now has a "shape".  The "shape" of the line is then characterized  by a normalized continuous frequency function known as "shape function" defined as follows  =  i  (r)  (1-3)  JlOrJdV  Note ••.•••• We shall use the following notations consistently:  f  t0  correspond to Zeeman transition frequency and transition energy ••  yj> > tp  correspond to a particular component in the absorption band ,  y, t  correspond to any component in the  -6-  absorption band without specification} used when we want to speak about the frequency distribution as a whole.  also  SECTION II  Statements of the Problem in General Terms and the Description of One-Dimensional Ising Model  Let us describe more explicitly the spin-system which is discussed in McMillan and Opechowski *s paper  (McMillan  and Opechowski I960 ). The spin-system consists of an arbitrarily large number N of identical spins forming a three-dimensional rigid lattice.  The properties of each spin are determined ft?  by a spin-Hamiltonian  , which incorporates the effect  of the external constant magnetic field II and of the crystalline electrostatic field. The total Hamiltonian of the system is then given by  K  ft*  +  K'  *  K  a  (2-1)  where N K  (2-2)  The Hamiltonian K  describes the spin-spin inter-  actions which include magnetic dipole-dipole interactions as well as exchange interactions.  ft"  describes the  effect of the high frequency oscillating magnetic field  H,  on the system. We shall assume that the spin-spin interactions are  .weak, i.e.,  K°  »  ft'  »  K'  (2-3)  and also  K"  (2-4)  That is, the weak oscillating magnetic field  H^  only  induces transitions, its effect on the energy levels of the system is entirely negligible (see p.2). Thus the Hamiltonian of the system can be written approximately as  a  We shall call  =  + K  l  (2-5)  the unperturbed Hamiltonian and "ft the  perturbation. In this case, due to the presence of the weak perturbation stfj and the huge number N of spins in the  -9-  system, the discrete frequency function (1-1) can be taken, to a very good approximation, as a continuous one (1-3). The calculation of  for this system is a very  difficult and still unsolved problem.  Since we know  from mathematical statistics that a distribution function, such as the line shape function  -f(y) in the present  case, is uniquely determined by its moment generating ,• function, instead of trying to obtain directly we calculate its moments which are defined as follows  rlfcr)dr  f  where of  Jl = 1,2 ...  f (y) .  (2-6)  give the first, second ... moments  The moments about the frequency  Zeeman transition energy  £0  fare  >£  of  given by  The formulae for the first and second moments have been obtained by various authors (see Van Vleck 1948, Pryce and Stevens 1950, Kambe and Usui 1952 and McMillan and Opechowski 1960).  They are of the following forms  (see McMillan and Opechowski)  h  -  c1,)/b  (2-8a)  -10-  h <A^y =  d ' / B  (2-8b)  where B , ^ 1 ' and D^ 1 ' are expressions containing the Boltzmann factor  JZ/Xfl (-  )  where  is a  projection operator defined as follows  S  and  = f\/M  M>, k )>  (2-9)  satisfies the eigenvalue equation  K°  (k  \KK>  U k =  > =  ly 2p «* # .  j^K}  (2-io)  $ the degeneracy of E ^ )  These moments characterize sufficiently the position and shape of a line shape function  -fcrj  of a para-  magnetic resonance to make possible a comparison of the theory with experimental data. In order to evaluate these moments, McMillan and Opechowski employed the following method of approximation. Since  a°>yfC  (2-3), they expanded the Boltzmann  factors appearing in (2-8a) and (2-8b) in the following way •••  -11-  e  RKR KT  KT  = e  (2-11)  We shall call the approximation consisting of keep-  ing only the first term in the square bracket "the first approximation" and so on.  In this way the expressions  for the first and second moments in both the first and the second approximation have been obtained (see McMillan and Opechowski I960). The calculation based on this method of approximation may seem, at first sight, to be entirely incorrect at very low temperature. zero. ~ ' kT  In equation (2-11), as T tends to  tends to infinity, hence the terms in the square  bracket containing this factor goes to infinity and thus the method of approximation seems to be inappropriate at very low temperature. still be good.  However, the approximation may  The presence of the Boltzmann factors in  both the numerators and the denominators of (2-Sa) and (2-8b) as well as in (1-1) or (1-3) indicates that at very low temperature, the terms contained in the square bracket in (2-11) in the numerator and denominator may to a good approximation cancel each other. This thesis is primarily concerned with the investigation of this rather queer, low temperature behaviour of  the line shape function To investigate the low temperature behaviour of <f(v) for the system discussed by McMillan and Opechowski is by no means easy.  For this reason it may be useful to  discuss a much simpler system such as a one-dimensional Ising model (Ising 1925) in the hope that we can throw some light on the low temperature behaviour of / 6 0  in  the actual problem. Apart from the simplification of the three-dimensional to -one-dimensional lattice of the system of N spins, the employment of a one-dimensional Ising model consists of the following modifications to the model discussed by McMillan and Opechowski  (see p. 'J of this thesis):  (a) the angular momentum quantum number S of a spin is taken as S = 1/2 , (b) the crystalline electrostatic field is omitted , (c) the dipole-dipole interactions are omitted , (d) the isotropic exchange interactions Jj  J  =G w W : • L  2L  Sj.  S  ( £ ,m labels the spins in the system andTLi  . t,m •  means summation over all neighbouring pairs • of: spins )• are replaced by highly anisotropic exchange interaction J  A  -13-  Ja - ComX^nX.  Since we are dealing with one-dimensional Ising model, an adjacent pair of spins can be labelled as £ respectively.  Therefore  Ja ~ OmA^tcunX  where  I  f  and £ + 1  ^y-/, y  means summation over all adjacent pairs of  spins. In order to write down the general expression of the energy levels for this one-dimensional Ising model, we shall use the following terminology:  We shall call a  spin positive if its magnetic moment vector /CO is parallel to the constant external magnetic field H, negative if it is antiparallel.  We shall assume that  the exchange energy for an adjacent pair of spins is positive if the spins are antiparallel, and negative if otherwise. We introduce the following notations: N = total number of spins in the system external constant magnetic field \T1 = number of positive spins in the system V£ = number of negative spins in the system  -14-  Q + _ = number of adjacent pairs of antiparallel spins Q «= total number of adjacent pairs of spins J = positive constant of exchange interaction M = magnetic moment of individual spins The general expression for the energy levels of the system is then given by t=suH(v,-v,)  +  jQ+_  -J(Q-Q^)  or  (2-12)  are respectively the Zeeman and exchange energy of the system. Due to the frequent appearance of the Boltzmann factor in the following work, we shall consistently use the abbreviations J_ and  e  "KT  The intensity function at the transition energy £ f can be written as follows  (2-13) (p = 1 ... 9)  -15-  where  W(n,n' ) = number of permissible transitions from the level E„ to the level E„. where the Zeeman energy of level E n is greater than the Zeeman energy of level En' and L = summation over all pairs of n,n' for n.n' which | - En/ I = <£p > 0 The frequency function (1-1} at Yj>  can then  be written as i ( n  fttp) = e  '  Z  P'=I1 r  I ®  . .  (2-14 j  ikf) R  (p = i  e)  Before proceeding, we must say a few words about a special feature of the one-dimensional Ising model.  In  this case the only interactions present among the spins are the exchange interactions.  The number © of the  component lines in the discrete absorption band is indeed very much limited.  The reason for this will be explained  in detail in Section IV.  It is due to this fact that we  must not replace the summation over an integration over <£  in (2-14) with  in contrast to what we did at the  end of Section I. The JL  moment of  about the Zeeman transition  -16-  nergy  i s given by (see (2-6) and (2-7) ) 0  -17-  SECTION III  One-Dimensional Ising Model (Open-Chain) for ,M«=4  As a sample calculation, we shall take a one-dimensional open-chain Ising model consisting of only four spins and calculate the frequency function first and the second moments about £e  "f^p)  > "the  , the Zeeman  transition energy, in the exact expression, first approximation and second approximation at various temperatures. In this case, since N is small, we can construct the energy spectrum explicitly as shown in Fig. (3-1).  Simply  by counting all the permissible transitions amoncj, the various energy levels, we see that there exist only five distinct values of transition energy, namely:  Zp-H±4J  ,  2jlH .  (3-2)  The reason for this will be explained in general terms in the next section. The McMillan and Opechowski's expansion (2-11) now takes the following form:  -18-  +  L  We shall assume  KT  f  "'J  (3-3)  = 20 J.  By using the equations (2-12), (2-13), (2-14) for '•f(£p)  and equation (2-15) for the moments of ^fCt) ,  the exact values of frequency function ^Ctp)  at the  transition energies (3-2) and the first and second moments of --^(t) about  to = £M>H  (3-4 )  are calculated at the temperatures  T - + O O ,  T - Z O J / K ,  T ~ Z T / K ,  T - + O  {3*5)  The values in the first and second approximation are also calculated by making use of the equation (3-3). The results were tabulated in Table (3-6) for comparison.  A graph (3-7) was plotted to show the general  features of the temperature dependence of the function in the exact calculation, the first approximation as well as the second approximation. Our calculation shows that for a one-dimensional open-chain Ising model consisting of four spins, McMillan  -19-  ancl Opechowski's method of approximation, in general, holds satisfactorily.  At very low temperature, that is, when  KT is of the order of exchange constant J, the approximate values become so close to the exact values that the deviation is really entirely negligible.  -20-  Arrange- Zeeman inent Energy  tm  Total Energy  Exchange Energy  M L  mt  tw  ttlf  2xi H  M  M  ZMJi-J  tm utt  wt tm  3j  Mt  tut m  J  ttu  !M  -J  tHf  tm mt  3i_  mt m m t t t i f m M  W  mt mi  mi  M  M  un  J  Hit  dL -JL  M  M g  Wi m ^  -zu-H+J  JttJW  WH  {Ill  'JMM P  -4M>H-3J  Fig. (3-1) Energy levels and transition scheme of one-dimensional Ising model (open-chain) with 4 spins.  yU,ll - 20 J  >J> — o O tr  Exact  T=2J/k  Exact  2nd app'n  T —  0  Exact 1st & 2nd app'ns*  2/<H~4j  9/80  1/8  9/80  .0053  .0071  .0053  0  0  - 2J  19/80  1/4  19/80  .0490  .0596  .0495  0  0  20/80  1/4  20/80  .0932  .1050  .0939  0  0  21/80  1/4  21/80  .4463  .4404  .4459  1/2  1/2  11/80  1/8  11/80  .4062  .3879  .4055  1/2  1/2  0.15J  0  0.15J  2.3982J  2.2846J  2.3948J  3J  3J  6J 2  6 J2  6J 2  8.5652J2  8.3196J2  8.5496J2  10J 2  10J 2  2jUE  h<*y>  2nd 1st app' n app' n  T=20J/k 1st app'n  Table (3-6) Results show that in general, McMillan and Opechowski's method of approximation holds satisfactorily at all temperatures. At temperatures below say , the differences between approximate, and exact values are too small to be shown here.  -22-  Fig. (3-7) Comparison of frequency function in the exact and first approximation. The 2nd approximation values nearly coincide the exact values and are not shown.  SECTION IV  One-Dimensional Ising Model (Ring) for Arbitrary N  For the model discussed in the previous section, we have seen that  McMillan and Opechowski's method of  approximation holds sufficiently we.l I at all temperatures, and particularl y so at very low temperatures.  In order to  investigate the low temperature behaviour of the frequency function  in more general terms, we shall take a  one-dimensional Ising model consisting of an arbitrarily large number N of spins arranged into a ring.  We shall  call this model a "ring model" or simply a "ring".  When  N is sufficiently large, the results obtained for the ring model can be applied to the one-dimensional open-chain Ising model as well,  since if N is very large, the end  effect associated with an open-chain is indeed negligible. To calculate the general expression for the frequency function  model, we shall use the following  terminology introduced by Ising:  By a "negative (positive)  gap", we shall mean a series of negative (positive) spins v/ith positive (negative) spins situated at the two ends.  Since the arrangement is a ring the number of negative gaps is always the same as the number of positive gaps in the same arrangement.  The gap is single if it consists of  only one spin, multiple otherwise J as shown diagramatically as follows: double positive gap  + • +  single negative gap  -  double negative gap +  +  double positive gap  Biagle negative gap  multiple positive gap  Let 3 be the number of negative (positive) gaps in the ring, then (see  )  *• 2S (4-1) Q  =  N  Equation (2-12) then becomes  E=  H-4S)  Hence the Zeeman energy of the system is determined by S,  -25-  the number of negative gaps in the ring. Let us define  4 V* =  - Vz  (4-3)  AS * S'-S AE  =  (4-4)  E'-E  (4-5)  then for a transition of the type  (4-6)  i.e.  = -/  the change in energy  (4-7)  -a E can be obtained as follows:  "Since • •.  E ** ju,H (zVt-N) -  (N-4S)j  (4-2)  (4-2)  by using (4-4 J, (4-5) and (4-7), we have  A E = -ZJU.H +  (4-8)  -1604-  Hence for each transition of the type (4-6), the Zeeman energy decreases by a fixed amount  of  2/*H  while  the exchange energy variation depends on the change in S. Thus we can label A E as  A E  = - 2/cH  -4S  * * (4S) J  (4-9)  For any transition of the type (4-6), only one of the three values:  f  -1  a S can have  -1, 0, +1, that is  if the transition takes place at a single negative gap;  (4-9)  AS =  /  0  if the transition takes place at either end of a multiple negative gap;  +1  if the transition takes place elsewhere.  This is shown diagramatically as follows:  + + + -- - - - + + - - +- + 1 t tt f It t A s =  0,1,1,1,0,  0,0,  -1  Here for convenience, we have cut up the ring into an open chain where the two end spins should be taken as adjacent  -27-  spins in the original ring.  The arrows point to where the  transition of the type (4-6) is to occur with the value AS associated with .such a transition. Combining (4-8) and (4-9), we shall have the following three cases:  AS--/  i  fat-H+47)  (4-10a)  ^ 5 - 0  = -ZjU,H  (4-10b)  AS=+I  =-(^H-47)  (4-10c)  Note that in the case of a one-dimensional open-chain Ising model, there exist five distinct values of transition energy  A E, two among which, namely 2/mH tZJ  , arise  from the end effect of the open-chain (see Section III). By choosing a ring instead, we have discarded the end effect; thus only three values of transition energy are left, namely:  Z&H+47  , &>H  ,  ZJ^H-  4J  .  In  general, for an Ising model with an arbitrary form of lattice, if the number of closest neighbours of a spin is K which is a constant, then the number of transition energy values is K + 1 provided that the number of spins in the system is sufficiently large that we can neglect the effect of the boundaries. be found in Appendix I.  A proOf of this statement will  -28-  From equation (4-2), the energy levels are uniquely defined by giving Yg and S, therefore we can replace the quantum number "n" and "n"' by v 2 , S and v 2 ',S'. Equation (2-13) thus becomes  -1Ek-s)wWs*.s)  xfe;-iL  (4-II  Since a transition is permissible if and only if condition (4-6), namely  AV*  ~l  =  is satisfied, this imposes a relation between V 2 ' and V 2 in equation (4-11). and  Moreover (see (4-5), (4-lOa), (4-^lOb)  (4-lOc)),  r  '  1  (4-12)  Hence for an intensity function corresponding to a particular component line in the absorption band, the values A V 2 and  A S are fixed.  Thus equation  (4-11)  reduces to the following form  -L(b'-'^-b™;w* where  V V ^ (Vr,5)  (*,s)  <4  denotes the number of permissible  -29-  transitions from the level E v and JEj : vZ/S "  g  to the level Ev a -1,5+^5  ;  means summation over all V 2 and S such that  F  - E  The subscript ^ S  v  *  = a E  C  (4-14)  appearing in both ^E^s  and  ^ s  specifies the particular component line we are interested in in the absorption band. In order to calculate  (VZ,S)  in (4-13), we have  first to know the following quantities:  A)  G ( V tu 0,S)  «=  the number of different arrangements of spins for given values of V 2 and 3, i.e., the degeneracy of a given energy level  B)  D(V 2 ,S)  =  ,g  (4-14)  the total number of single negative gaps appearing in all possible arrangements for given values of V 2 and S  To find (Ising 1925).  G(V2,S)  (4-15)  we essentially follow Ising's method  In order to have S negative gaps, we need  at least S positive ana S negative spins arranged into an alternating series in the form of a ring.  The remaining  Vi-S positive spins and V2~S negative spins are to be  -30-  distributed arbitrarily among the S positive and S negative gaps respectively.  This leads to the following expression  for G ( V 2 , S )  [v,,vz]  5=0  (4-16b j  where the notation [a,b] is defined as follows  a M ]  Xj.  =  (4-17)  b  Ij. (X >b  The factor 1 arises from the cyclic symmetry property of the arrangement. To find D(V 2 »S), we proceed as follows: Let "i" be the number of multiple negative gaps in a given arrangement of spins, let di be the number of arrangements belonging to G(V2S), each of these d ± arrangements has "i" multiple negative gaps.  D(VZ/5)  where  t. •  Then it is clear that  (4-18)  means summation over all values of i present in  the G(V2,S ) arrangements. To evaluate di, we must distinguish the following  cases which give the relations among V 2 , S and i: (a)  Vz>S>2.  implies  l ^/  Since there are more negative spins than negative gaps, some of the negative gaps must contain more than one negative spin. (b)  Vz >5 = I  implies  t =»/  Since there is only one negative gap, all the V 2 negative spins must be contained in this gap. (c)  V% =5  implies  1 ~o  This means that all the negative gaps are single.  We  shall label the three cases of D as D a , Db and D c and shall deal with each of them as follows: Case  (a) Since  number of ways to form S negative gaps with V^ positive spins; see  (4-16a)  = number of ways to form S negative gaps with V 2 negative spins, S-i gaps being single; see  (4-16a)  number of ways to choose i multiple negative gaps out of S negative gaps, hence we have the expression for d  i  (4-19)  -32-  where again, the factor £ arises from the cyclic symmetry  S  property of the arrangement. Substituting  (4-19) into (4-18), vie obtain  Ck(K,»-Zo-o*m:WZ-')  (4-20)  where the lower limit of summation is certainly i = 1 as required by the condition the upper limit is  [v 2 -S, S-lJ  which implies as can be seen from the  equation (4-20) remembering that S  ^  [v*,V 2 ]  (see 4-16a),  Hence we have  Da <*, 5) f f Z o m S M W ) ««=/•  •• Case (b) :•••. . Since there exist no single negative gaps,  D  Case  b  U , S ) ~ 0  (4-21b)  (c) Since all the negative gaps are single, it is clear  from the discussion of case (a) that:  %  (*.S) - N {tfi  With the help of equations (4-9)(410a)(4-lOb) and  (4 21oi  '  (4-10c),  -33-  the factor  W^  tyz,S)  in  the  equation (4-13) can be  expressed as follows:  (!)  ^5  |  <= - (Z^tt+4-j)  Since this type of transition takes place only at single negative gaps,  • Wi, (vif 5) = D ( 5 )  (II)  (4-22a)  as=O  Since this type of transition takes place only at either end of a multiple negative gap,  W0te,s)=2(S$-D)  <^-22b)  where SG(V2,S) is the total number of negative gaps and SG(V2,S) - D(V2»S) is the total number of multiple negative gaps for given V2 and S.  ( i n )  =  Since this type of transition takes place "elsewhere" and there are altogether V2G(V2»S) possible transitions for given V2 and S,  -34-  w,, {v2/ s ) = Vz (k - w . , - W 0 = {yt-ZS)Q-  (4-22C )  tp  where  D(V2,S) =  ' Da(V2,S)  if  V2> S^ 2  Db(V2,S)  if  V2 > S = 1  DC(V2,S)  if  V2 = S  (4-23)  Combining (4-lOa,b,c)(4-13)(4-14) and (4-22a,b,c) we have the following set of expressions for the intensity function;  I {zmH+^j) =/_Jb VI,S  -b  J  D  (4 -24 a) AE0  - - ZyU-H  I(z/iH)  -  xZx(S(k-D)  = -  [ZM'4-J)  x[(vz-25)<t + D]  (4~*fcJ  -35-  Since  S  = [fV-V2  , the upper limits of  summation over S in the above set of equations depend on whether V 2 > | or V 2 < | . where V£ =  If N is even, there is a case  hence the problem is slightly more complica-  ted than in the case of N odd.  For sufficiently large N,  the intensity function will not depend on whether N is even or odd.  Therefore we will assume, in the remainder  of this thesis, that N is odd. By substituting equations (4-23)(4-21a)(4-21b) and (4-21c ) into equations (4-22a)(4-22b) and (4-22c) remembering that N is odd, we obtain the following set of expressions :•••••••••••••.•••  xN x  JJ-I vi-i  Bi-ss-il  rr'j ^ M ? ) ^ 1 . )  •Nr  + Z,  Vz = |  b  ( w  J  J  (4-*fa.)  -36-  =(l-t> 2 A H J  i b M ** 2  ttYz  (N+ZX-UH-NT  4  xzbl*  .zvzMH+45jiN-vt-r \  S=2  M-l  i  z^z.icH +4J  b  +  ZYzMH + 45 J (N-Vz-f 2 L Z L H S- | Vz =3 5=2 fltL z.  b  Z L  i  (4-25 b) (H+2)MH -a1-4)7  N-2  N-V2  =  S=2  L  s-ij+Acs-  N-l  + L  b  Wz-Z)  V  ZNjlhl  Mzl 2Vz/lH +4SJ 1 /N-Vg-l + Z Z, V^=3 5=2  b  t  2 Vi^w-H -f-4 J {4-25 C )  -37-  In order to determine the upper limit of summation over i, we proceed as follows: Let V 2 =  2j  for even V 2 , then  zj-S >5-1  ^  (4-26a)  - 7  and V 2 s 2j+l for odd V 2 , then  /2/-5 + K 5 - I  JLj.  S^-^^^+Z (4-26b) 1  2.1-Sfl =5-I Zj.-S+\>5-\  .If. s = -^f - ® JLj.  No confusion will arise by using the same letter "j" to express both even and odd V 2 , since we shall replace the summation over V 2 by a summation over even V 2 and a summation over odd V2-  This is necessary because the  upper limits of summation over i are different In the two cases as seen from equations (2~26a) and (2-26b}.  In  carrying out this replacement, however, we have to specify whether VVj = N  ~Z Y vfrkL  , the lower limit of summation  N-l , as well as V 2 = —s— , the upper limit of z,.  Vc =3  . is even or odd.  For this reason we  -38-  shall assume that N has the following particular form  N  S  4P-1  CP=t 2 '"")  (4-27)  This particular choice of N will certainly not affect the generality of our results thus obtained provided that N is sufficiently large. This choice of N leads to the following set of simple yet important relationships:  J L2t L - m  even  V£ a Zj.  V^^-  ^ p (4-28a)  odd  w=i 3 zp-i  y^Hg  odd  vfc-J^l  =>  j-p  ^u-p-l (4-28b)  By using (4-26a ) (4-26b)(4-28a) and (4-28b) we obtain the following set of replacements; a formula a sum ti-z H-yft bt-S,S-i] L L L  whenever we have in  -39-  Y/e shall r e p l a c e  Lx L^Li  it by  +  Vp  v2  even  odd (4-29a)  Similarly v/e shall replace  v-J ~~2T~ vi-f [Vj-5^5 -1J 5-2  i=l  by  p-| / j- S-|  2LZ.IL  p-/ / j- S-l  +2L  s=i+\  ZL  j  +  L  L,  2-j+  ZL  2/-S-/  zL  P-J  + L  j-2 \ s-z (•=) S^t2 i=i /  V 2 even  j-  ZL  i m  V2 odd (4-29b)  After making use of expressions (4-29a) and (4-29b) equations (4-25a,b,c j become:  -40-  ZP-2  +LL  /-2S«2«1  ^ps^+2  2/-S P-l  Zl-1  L + L L  !=/  j.*Z  s*j+z  2C-S  L  i-1  /WW//  1  2P-Z  2p-|  •z, Vi®/  Of-30 a)  IL L*LL t r =P 5=h-| ,'=/ >=2 S=>t| i=J  H  *  w  e  ^  u  -  r  J y^i r' h  ir-z +L Vz=Z  zVtJiH+4 J a  I  (4-30b)  n  m  ^  -41-  I (M-4ll)  (  = (l-[>^H'4J)xUy:b'(^)MH'(N'4)7  •  ^ry. r fp-Z-1 p-1 Zt-J Zi-S zc-s . ,, .. Jzprz J:o-i 5-1 zp-z jp-x-i zjhs P-1 zj-\ t= i LL+L n L Li + r Lz Ls=r t ^ m ZP-l r 5-1 ^ zpz «_ ,^ [s+, . 5t CI* 4P-2J.-Z < y-5 t, /p r. -l Zl.2l'-2Si+-  .=1  f  1  r  L  r  m  i  d  (  t  J-  Vi»3 jZN/IH I b  (4-30c)  The summations over i appearing in the above set of expressions can be reduced by applying combinatorial analysis. The fundamental equalities we are going to use are the following:  (r) = ("r ') +C?-i)  (4-31a)  and •..•••  z m j - n v  <«»>  ;  f  -42-  We observe that:  then we have, for example:  L  2  t-i  «=/  '(rnrw-w-'jj-s&MTfc •  (tv  j  J  Following the same procedure, all the other summations over i can be performed.  After some simple manipulations  we arrive at the following forms:  : M  +4T)-(i-r H" J)*b'UtVMH-M [zp-2  jr  2P-2  ll+L  s=2 /-P ^  4P-Z}-I  L  P~l  Zi-\  . . „,]  „.  ,  *it  /«* 5'jtz t 4 Q i - O T ^ f P ^ s ^ ^ l J  f Yz«=j  (4-3ZCL)  -43-  ' * z z r z 5=2  p-) ai * l l - * L L r S=/r2 S=fu  V  M  H  ,, " 5r ; r x r , )  |  ti  £=2  }  i  .  J  (4-32 b) i(1^1-47!)-b-iT"-47) (2P-Z  *  y  ZP-2 4P-ZC-]  , p-l  2j-z  '  L L * L L ' i L L  z aZ Z J  £  J-P  •• 2P-2  2 /  ^z H  £  tf^mmm  b ^ ^ n ' x r ) ' / ' V ' ""  r2  Vi=3  z  N X  •• • •  (4-32C)  -44-  Now by rearranging the summations, applying repeatedly the equalities (4-31a) and (4-31b), and after some algebraic manipulation, it is possible to reduce the above set of expressions to the following forms:  f  ' :1" v I M J  I  j / :^ ^ X - l ) )  . f=2 S-2.  (4.33a) >  - ( / - r t x t - ^ " " a . 33b)  r  Z L -3 5=1  b ^ ^ T s - r T s  2  ) I 1 {4-33c)  As an independent check of these expressions we have calculated the values of intensity function for the ring model with N « 7 (or p = 2, N = 4P-1) by constructing the energy spectrum explicitly and counting all the permissible transitions among the various energy levels, as we once did for the N = 4 linear-chain model (see Section III). Results are compared with those obtained by using the expressions (4~33a)(4-33b} and (4-33c ), and found to be identical.  In order to evaluate the double summations in the expressions (4-33a ) (4-33b) and (4-33c ),  we introduce the  following transformation:  T  t - 2  r  S  R  N - 3  -  1  (4-34)  A  3  ,4j  then we have,  (4-3fa)  xwu) U  l(kuti-4Jl)-(l-l>  J*b  x  N*lh  -35b)  Uji-b (4-35'c)  J  -46-  where R-l  r+i B r ( R r T )(r-i)  0, = fg Z^'A* Z L T=O  (4-36a}  P.  d  -  L  Qs-IbL  ts/  A  t  a  t  L  <4-36b>  B'P-rVIT)  L b-(R;T)(rl) r=o  <4-36°)  To evaluate Q's let us consider a generating function:  fur (},b) = (l+}Id)R~r (l + fj5)T  T*R  (4-37 J  which is analytic except at the origin. Using binomial expansion, we have R- 7  f..r<}.B)~L u  LL—O  (  W  f  -  U  -v ( z m y U  L  ' L L C Z K M ' " } " " U.-0 V"0 V R-T i „  R-TT "  yzz-f  ' tts=0 VfeO  ~  . . t/lt-irj f U (4-38)  Let R-T  fl RfT  T  u-o r=o  u,r ) Q  ( J e n -XT r  (4-39 J  -47-  then • •  R-r K-T  fB,T (J, B ) = L s "T  f R T (r, B ; ^  (4-40)  and ••. •  .. ,  //  1  •  U.BO  U.= | .  R-T  r=o,  (4-410.)  T  ZU  a . - (qB) =ZL(V)(I)(/b) * ' . : iL—o •• '  '=  r H s ; - ZL (V ) & ) ( J B )  1  zZ.I V X D b " ' ••• • tu=»ii^o o (-f-4/ W  - I b L (VXa+Jd"'  Comparing (4-41a), (4-41)3) and (4-4lc) with (4-36a), (4-36b) and (4~36c), we have R-1  Q, = Z J ^ ^ T - O / S ) r-o v'  (4-42a)  R  6?3 - Z L /\T  <4~42c)  Hs)  By applying Cauchy integral theorem to (4-40), we have  If,®)  j r+l  (4-43)  -48-  where contour c denotes any closed path of integration around the origin in the counter-clockwise sense (see Appendix 2 for proof).  Substituting (4-37) into (4-43),  we have  • .  f ^ f o ) « zvtJ  o  + ( I  T  +f/d)  ^  (4-44)  Substituting (4-44) into (4-42a), (4-42b) and (4-42c), after interchanging the finite summation with integration, we obtain  (4-45a) If. R  (4-45b)  1  (4-45c)  rfffiff Substituting these expressions for Q's into expressions (4-35a), (4-35b) and (4-35c ) respectively, we finally obtain: .  *  /  A  VIH+ZT  J_  { D£(  ^lf(l^fe)  R-)*R  Z^A  ALL^LI  (4-46a)  ^  -49-  (4-46b)  (4-46c)  Let us summarize what we have done so far.  We have  calculated the set of three exact expressions (see pages 27-29) for the intensity function as defined by (4-13) for a ring Ising model consisting of an arbitrary number N of spins, where N is of the form N = 4P-1 (P = 1,2,  ).  For large N, this restriction on N will be of no importance, so that the above set of expressions can be assumed to hold quite generally for large N.  However once we  assume that N is very large, we can greatly simplify these expressions.  This point will become clear in the next  step when we try to derive an explicit expression for the intensity function. Let us resume our calculation:  -50-  Since  ZZ  T-O  [ao* [A  mi]  R  hi  /t/B^J.  1-  A d rfflf) /1-  equation (4-45a) can then be written as '  zml  {B  L H-I&y  J  }  (4-47)  where c denotes any contour around origin, and z 2 and Zg are the roots of the equation  1 namely  (4-48a) >  -ikh'l+V  +  °<J  - oi  where  c< -W-  A)z  t4AB  (4-48b)  (4-49)  -51-  exact  evaluation of (4-47) leads to a combinatorial  sum (see Appendix 3) which is no easier to use than the equation (4-33a) which we have started from.  It is for  this reason that we shall assume that N is large, and we shall examine the asymptotic behaviour of the function (4-47) with respect to R (=N-3).  Expression (4-47) can  be rewritten as follows:  (4-50) Note that the integrand in (4-47) has one and only one singularity, namely at the origin} the two integrands in (4-50) however, have each three singularities, namely at z = o, z = z 2 and z = z 3 >  This does not lead to any  difficulty so long as the two contours in the two integrations in (4-50) are the same; in that case, the contributions due to the singularity z = z 2 and to the singularity z = z 3 in the two integrations cancel.  We can now choose  a contour c' such that for R tending to infinity,  Hence the contribution from the second term in (4-50) i  -52-  arbitrarily small in comparison with that from the first term and is consequently dropped (see appendix 4 for the choice of contour c* ).  Equation (4-50) can then be  written, to a good approximation, as  h U  I M l M f  -  - ZTTHR I I 7o~7V7r"V\  *  (4-52)  where contour c' and R satisfy the condition (4-51) and the symbol "A " denotes the asymptotic value. Now Cauchy Residue Theorem states that  zkJ  f(p (Ly  Rju5 iffy), fc}  <4"53 >  and  (4-54)  for simple poles.  A  =  x f_L  By using (4-53) and (4-54), we obtain  , J2M&2^  , SHJEk^L  ]  (4-as.)  Similarly we obtain  S ^ O f j f L . o f ^ L ]  $  ^  . x r M l i m ^ , M l U M Z L ] >-b h-%  (4_55c,  -53-  By using (4-48a) and (4-48b), we have the following set of relationships:  h  c</rs  -W:  --A I* is it  — ~z (l-rA -I-**)  (4-56)  Combining (4-35a), (4-35b), (4-35eJ, (4-55a), (4-55b) (4-55c ) and (4.-56), we arrive at the following set of expressions:  ,.R+Zr  'HtJ  R+l  ft-fi -j)  LI-li-A-«)(li-A+C<) -(-I+A+°0(h-A-x) JJ (4-57a)  i(ZJU,H)  )*b  'xzti*  (4Sjb) a,.  ,  I\  I UwH'4jO~(hb  r .7.^-4-7 \ i-H(M#-tZ)+(2/lHl-+J)  )xb  X fiS:?*Z [(-1 tA +«Xl+A t4* ]-  (-l+A-«)(l+A  (4S]c)  If we assume that B ? i\, i.e.,  ^  4j, then we  have the following relations (see appendix 5 for proof):  1 > • 1 + A ~<x > 4 > 2  >  1 + A +<* >  0 2  | - 1 + A -ocj >  1  I 2 ^ - 1 + A +cx  (4-58)  When R approaches infinity as required by the condition (4-51), R+l »  (l+A-«)  hence we can omit the term consisting of (1+A-  (4-59)  ) ^  in  each of the expressions (4-57a), (4-57b) and (4~57c). Moreover, since we are interested only in the relative magnitude of these expressions, a common factor  can be  extracted, where  0 Let  1 = 1*®  » remembering that R = N-3, we finally  arrive at the following set of expressions:  -55-  i  V  ^  j  (4-6 la)  i'UMH)  (4-6ib) =(l-b*»H-4T)b2»H'%+A-0  where  (4-eicj  K T  ^  =  ^  A  «= e  *  = LO-c  *T £  /fr  It is clear from the expressions (4-61a), (4-61b) and (4-61c) that if  2^112* 4j, then the intensity function  is independent of N, the number of spins present in the system, provided that the conditions imposed on N are satisfied.• If 2yUE < 4J, we have to use the expressions (4-57a ), (4-57b) and (4-57c ) which hold for any values of ^ Let us summarize what we have done.  II and J.  We have obtained  the set of expressions (4-61a), (4~6lb) and (4-61c) for the intensity function as defined by (4-13) for a onedimensional Ising  model consisting of an arbitrarily large  number N of spins arranged into a ring, where N is of the  form N = 4p-l (p = 1,2,  }.  The assumption that N is  large enters our calculation for the expressions (4-61a), (4-61b) and (4-6ic) at the following points: 1) that N must be sufficiently large such that (see (4-51))  where z satisfies the condition imposed on the contour c' (see appendix 4 ) , 2) that N must be sufficiently large such that (see 4-59)  However for large N, the restriction that N takes the particular form N = 4p-l will be of no importance, so that the expressions (4-61a) (4-61b) and. (4-61c ) can be regarded as quite generally valid.  Moreover, when N is  large these expressions can be used for a one-dimensional open-chain Ising model as well.  -57-  SECTION V  Intensity Function in the First Approximation for the Ring Model with Arbitrary N  The first approximation calculation (2-11) amounts to putting j = o (i.e. B = 1, see (4-34)) in our previous calculation for the intensity function of the ring model. By so doing, the set of expressions (4-32a), (4-32b), (4-32c )  (4-33a ), (4-33b), (4-33c ); (4-57a), (4-57b),  (4-57c) and (4-61aJ, (4-61b), (4-61c), as well as the expressions (5-a), (5-b) and (5-c) in Appendix 3, are all reduced, to the same set of expressions which is as follows:  i * (2/^47.),  I*(ZM,H)  =d-b 2MjH)z  ~{}-bmH)4lf¥M  (5-la)  (5-lb)  (5-lc) and (5-2)  where  •  (gj  is given by (4-60) with J = 0, i.e.,  / ® - ± F,  ,2AHSN-3  W, = £(/ + b  J  , -NAOH  Kb  KN  (5-3)  and the subscript "1" at the lower right corner indicates that the various expressions are in the first approximation. The frequency function in the first approximation can be easily obtained by substituting (5-la), (5-lb) and (5-lc ) into (1-1):  'i  f (zM>H-i-4T), =  f(2AH) r * f (IZAH'+JQ, =  I  (5-4a)  (i+b^y r+M'H * {L BZM,HT  (5-4b)  (5-4C )  -59-  SECTION VI  Comparison of the Asymptotically Exact Expressions (4-61a ),(4-Slb),(4-Slc) with First Approximation Expressions (5-la),(5-lb),(5-lc) at Very Low Temperature and Conclusions  i - W  +P**  4 7  1*0Zjrii-nQ H i '  (4-61C )  i*{zj*H+4T),-0-b*Ul)Z  (5-la)  iVH), f w - f i O r d - r t ^  (5-lbJ 4  * " (5-lc )  Nov; let us resume our original question about the  -60-  goodness of McMillan and Opechowski *s .. tion at very low temperature.  :  approxima-  Since ( oi - A •+ 1J and  ( o( + A - 1) are finite at all^temperatures (4-58), it is clear that at very low temperature the factor in (4-61b) and (4-61c), and b 2 J l E in (5-lb) and (5-lc), tend to zero; hence the frequency function calculated from both sets of expressions tends to the same limit, namely /A  f0  -0  (6-1)  where we have used a subscript "o" to denote the temperature T = 0°K. 1  ;  A graph (6-2) is plotted by putting jub H = 20J for the frequency function  ^(L)  calculated from both the  expressions (4-61a), (4-61b), (4-61c), and the expressions (5-la), (5-lb), (5-lc) at various temperatures.  It is  seen from the graph that the method of approximation is quite good in general and extraordinarily good at tempera4J ture below, say, T = — . Note that the expressions (4-33a), (4-33b) and (4-33c) can be written as follows:  -61-  (6-3)  i = i * <f> where  (6-4)  and  t"2  •H-  , , z wH \  v  I (zaH)  ~(l-b  )  I 4  S=2  fH-l M  L  t=2  L 3-1 (6-Sb)  N-J  i  W  -  f  i  -  O  '  t'Z  L  f  ^  M  (6-5c)  ^  i  then  r b 47  and  -2MH  DO  b  b  -4T  (6-6 J  'O  —  0  (6-7)  0  hence  •Hf I0 O / t H - M j ) = I rt Ia (ZyUCH) IoG^l-4-Tl)  = O =0  (6-8)  It is easy to see that the frequency function tends to the same limit as (6-1), namely:  f 0 (2/lH+4J)  =I  f0 (ZAH) f0(\ZM>H-4Tl)  —O (6-9)  -63-  This shows that the frequency function calculated from the set of expressions (4-33a), (4-33b) and (4-33c ) tends to the same limit (6-9) in the following two processes: (1)  We first let N tend to infinity and then let T tend  to zero, (2) on N.  We let T tend to zero and impose no new condition  Kr/r Fig. (6-2) Comparison of the asymptotic values of frequency function in the exact (4-61a,b,c) and first approximation (5-la,b,c}.  APPENDIX I  General Formula of the Number of Component Lines Present in the Absorption Band for an Ising Model with an Arbitrary Form of Lattice  (see p.27J  Since the Zeeman transition energy is always the same (2/tH), the number of transition lines in the absorption band is given by the number of different values the transition exchange energy can take Let K = number of closest neighbouring spins (constant) V^ = number of positive spins among the K spins V 2 = number of negative spins among the K spins then  Vi + V 2 = IC  [Vlfy2]  £  0  The number of different values that the transition exchange energy can take is equal to the number of different values that Vi and V 2 can take such that equation (1) is satisfied.  It is obvious that this  (1)  -66-  number is K + 1.  Hence the number of transition lines in  the absorption band is K + 1 , provided that we neglect the effect of the boundaries.  -67-  APPENDIX 2 (see p.47)  To prove  i n ^r-t-l  3c,r  (4-43)  where R-T (4-40)  Proof.  Substituting  (4-40)  into  R.H.S.  of  we have, R-T  R-H.S.  1 X  &  t  (KB) c  o  ••• 1 ?  r'+l  R.H.S.  r A-r  2  .ZTfi v'^r  L.H.5. Q. E.D.  (4-43)  -68-  APPENDIX 3  The Exact Evaluation of Q-p Qp and Qg (see p.4S)  Since  ^-zirilfB  f  (4-47)  by taking an arbitrary contour around the only singularity of the integrand, namely the origin, and applying Cauchy's Residue Theorem for both simple pole and pole of higher order, we obtain  In order to calculate the higher order  derivative  in the above expression, we proceed as follows:  -69-  and the coefficient of z R in the above expression is  &  L . m i t J d f  r, R-r + S + t, R  (2)  where  7  '/j, t  means summation over all possible combinations  of r , s and; t. •• . •• Since.  AL  ft- -  .w  CLf  tf-w  -  I  (3) R<W  0  substituting (2)(3) into (1), we thus have  -m-£\£-h  L  Combining (4-35a)t we have  v  i  u  L, „  J  (4-48a}, (4-48b), (4-56) and (4)  (4)  -70-  T/  U  ,-r\  !  -ti(M>Ht J)  Similarly we obtain  i m j ) - (i - r  x  +  1  2  W  K t* . . • • •  ,  ^ (5-bj  fi/z, (7unhA*4%hA-«j%ti •. • K t + t  •e.f/V-z;^/.]' J  (5-c)  APPENDIX I  To Prove the Existence of the Contour c' (see p.52)  The contour c' in (4-52) must satisfy the condition  lA(i + f&/})l <1 l+/aj-1  (1)  wnere -IMbL I > f\ s e  I > 6  -  e  KT  K T  >0  (2)  & o  <3)  i.e. A and B are positive quantities independent of z but dependent on the absolute temperature T. Let  and  J ~  XA  i f ®  * f  (5)  -7 2-  then (1) can be written as follows:  A I l-r ^.(x-if)!  < I I +JQ (x+.y;/  A2 I I t f x t ^  < j \tzJEx t 6  (6) This is the condition that the contour c 1 must satisfy. Let us choose x and y such that  f-X*+f  >I  (7)  If T = 0, we have, by (2) and (3), A = 0 and 3 « 0, hence condition (6) is satisfied. If T > 0, by (2j and (3), A > 0 and B > 0, hence by  (7) we always have  (X + yz-A  I.-A*  )6  >0  (3)  > o (9)  However, we must distinguish the case X > 0 and the case X < 0.  If X ^ 0, then  JE ( l ' f ) x  (10)  hence, by (S), (9) and (10), condition (6) is satisfied. If X < 0, we can choose y sufficiently large such that  > M  hence condition (6) is again satisfied. Therefore the contour c' can always be found and the approximation (4-52) always valid.  -74-  APPENDIX 5  To Prove the Inequalities (4-58)  (see p.54)  Given  cx! ~ j(l-A)z+4Ab  (i)  f * B & A (2)  first we need to prove the inequality  (3)  Proof:  the inequality (3) can be rewritten as  (!-A)Z or  /-  +4A&  > A*  2A(l-28)  >0  (4)  -75-  If  & >-t  then  •  l f  If  o< B <i  ••  /-2B < O  /-2B=0  (4)  .-.(4)  holds  holds  A * B <£  hence /-2B<I ZA<\ if  then  8=0  he/ICC  (4) holds (4) holds  . (3) is always true.  Q.E.D.  V.'e always need to prove the inequality  * < 2 •  (5)  Proof:  7  0 «6 S I  = ](/-A)*+4AB <]0-A)z-t-4A  but  -  !+A  o ^ A ^ ) o< ^ Z  Q.E.D.  (6)  -76-  Since  >  *  BJ(l-A)z  I-A  y  ?  O  />.  r ^ o o  (7)  +4A6 = I  J j . A,B  i.e. 7 = CO  = O  18)  We now have the following set of relationships:  by  (2) ,  Z  (6) ,  It A &  (3)  1+ A & I  (9 (10  cK ^ A  (11  (7)  U > \-A  if A,B * 0  (12  (8)  c< -  if A ,B <= 0  (13  J  It is then easy to see that by (2), (5) and (11)  (2), (5) and (12)  by (14)  Z  / > ! + A - <x 5*0  (11) and (10)  (10), (11)  -4 ** I + A t c<  -/ > -/ i-A  2 > -/ + A +  &.-1+A-  (f+A)  >0 = -2  ^ 2*1-1+A-«/ > /  (14)  -7 7-  Kenee the set of inequalities (4-58) is proved.  i  Bibliography  Ising, E., Zeitschift fur Physik 31, 253 (1925-1). Kara be, K., and Usui, T., Pr egress of Theoretical Physics 8, 302 (1952J. McMillan, M. , and Opechowski, W. , Canadian Journal Physics 38, 1168 (I960). Pryce, M. Ii. L. , and Stevens, K. W. H. , Proceedings of the Physical Society A33, 36 (1950). Van Vleck, J. II. , Physical Review 74, I1G8 (1948).  

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