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Effect of deuteration on the neel temperature of CoCl2.6H20 Sahri, Darshan Singh 1966

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EFFECT OF DEUTERATION ON THE NEEL TEMPERATURE OF CoCl 2'6H 20 by Darshan Singh Sahri A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS " We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f a r t h e r agree that permission., f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by.the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of" t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GEADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINAHQN ' FOR THE DEGREE OF ' DOCTOR OF PHILOSOPHY of 'EARSHM SINGH SAHRI B.Sc. (lions.)* Pan jab U n i v e r s i t y , 195k M.Sc. (Hons.), Panjab Un i v e r s i t y , 1955 M.Sc., U n i v e r s i t y of. B r i t i s h Columbia, I 9 6 2 FRIDAY, SEPTEMBER 30, I966 A T 3;30 P . M . I l l ROOM 302, HEMINGS BUILDING COMMITTEE IK CHARGE Chairman: N. Epstein M . Bloom C Schwerdtfeger G, M . Volkoff B. A . Dunell B. T u r r e l l D. L. Williams External Examiner; T. Haseda Department of Physics Kyoto U n i v e r s i t y Kyoto, Japan Research Supervisor: M . Bloom EFFECT OF DEUTERATION ON THE HEEL TEMPERATURE OF CoGU'6H o0 ABSTRACT The nuclear magnetic resonance technique has been used to study the e f f e c t of deuteration on the Neel temperature of COC1 2 °6H 2° s i n g l e c r y s t a l s . The Neel temperature varies with a period of l 8 0 ° as the external magnetic f i e l d Is rotated about the b<=axis _L the a<=c plane. On deuteration^ the N6el temperature r i s e s f o r a l l orientations^ the more deuterium being Introduced the higher the t r a n s i t i o n point. The maximum increase of approximately 6$ i n Neel temperature i s obtained with a 92$ concentration of deuterium. In addition^ the anisotropy i n T^  decreases from approximately 0*08°K f o r 0$ deuter-a t i o n to s l i g h t l y l e s s than. 0 .05°K f o r 92$ deuter-a t i o n . Further,,.the orientation-averaged Neel tem-perature seems to vary l i n e a r l y with the cube root of r e l a t i v e concentration of deuterium. In the v i c i n i t y of H = i+s000 gauss, the Keel temperature increases with an increase i n the external magnetic f i e l d H , Ii being J. c~axis. This unusual behaviour i s common to deuterated as w e l l as non=deuterated samples and has not yet been explained. Following Haseda's conjecture, a serai-empirical attempt i s made to e s t a b l i s h a connection among the change i n Neel temperature upon deuteration^ the super-exchange parameter and the p o t e n t i a l of a proton i n a hydrogen bond. The p i c t u r e presented i s that the Neel. temperature r i s e s on deuteration because of a change i n the average of the super=>exchange parameter over the ground vibra= t i o n state of the hydrogen atom. An x=ray an a l y s i s shows that a t room temperature the CoClg°6H 2o and CoCl2°6D20 c r y s t a l s have the same symmetry and t h e i r c e l l dimensions do not d i f f e r by more than 0.2%. The i n f r a - r e d spectrum of CoCl^'oDgO has been used to determine the value of e l e c t r o s t a t i c f i e l d gradient at the deuteron s i t e s . This value i s consistent with the observed qimdrupole s p l i t t i n g of the n.m.r. spectrum of the deuterons. The n.m.r. l i n e s belonging to deuterons i n the water molecules hot forming a square configuration around cobalt ions have been i d e n t i f i e d . GRADUATE STUDIES Electromagnetic T h e o r y Elementary Quantum Mechanics Special R e l a t i v i t y Quantum Mechanics of Molecules Quantum Theory of S o l i d s Nuclear Physics Theory of Measurements Non Linear A n a l y s i s Crys tallography Magnetism and D i e l e c t r i c s G. Mo Volkoff W„ Opechowski W. Opechowski J . A. R 0 Coope Ro Barrie Jo Bo Warren J . R.: Prescott A. Co Soudak Ko B. Harvey Mo Bloom PUBLICATION Sahri, D„. S O J E f f e c t of Deuteration on the Neel Temperature of CoCl »6lIo0, 1966 Physics L e t t e r s i2- 625. ^ Supervisor: Myer Bloom ABSTRACT DARSHAN SINGH SAHRI. EFFECT OF DEUTERATION ON THE NEEL TEMPERATURE OF CoCI -6H20. The nuclear magnetic resonance technique has been used to study the e f f e c t of deuteration. on the Neel temperature of CoCl2'6H 20 sing l e c r y s t a l s . The Neel temperature varies with a period of 180° as the external magnetic f i e l d i s rotated about the b-axisX the a-c plane. On deuteration, the Neel temperature r i s e s f o r a l l o r i e n t a t i o n s , the more deuterium being •, introduced the higher the t r a n s i t i o n point. The maximum increase of approximately 6% i n Neel temperature i s obtained with a 92% concentration of deuterium. In addition, the anisotropy i n T^ decreases from approximately 0.08°K f o r 0% deuteration to s l i g h t l y l e s s than 0.05°K f o r 92% deuteration. Further, the orientation-averaged Neel temperature seems to vary l i n e a r l y with the cube root of r e l a t i v e concentration of deuterium. In the v i c i n i t y of H D = 4,000 gauss, the Neel temperature increases • i f with an increase i n the external magnetic f i e l d H Q, H Q being X c^axisC This unusual behaviour i s common to deuterated as w e l l as non-deuterated samples and has not yet been explained. Following Haseda's conjecture, a semi-empirical attempt i s made to e s t a b l i s h a connection among the change i n Neel temperature upon deuteration, the super-exchange parameter and the p o t e n t i a l of a proton i n a hydrogen bond. The picture presented i s that the Neel temperature r i s e s on deuteration because of a change i n the average of the super-exchange parameter over the ground v i b r a t i o n a l state of the hydrogen atom. An x-ray analysis shows that at room temperature the CoCl2*6H20 and 00012*60^0 c r y s t a l s have the same symmetry and t h e i r c e l l dimensions do not d i f f e r by more than 0.2%. The i n f r a - r e d spectrum of CoCl2'6D20 has been used to determine the value of e l e c t r o s t a t i c f i e l d gradient at the deuteron s i t e s . This value i s consistent with the observed quadrupole s p l i t t i n g of the n.m.r. spectrum of the deuterons. The n.m.r. l i n e s belonging to deuterons i n the water molecules not forming a square configuration around cobalt ions have been i d e n t i f i e d . 1X1 CONTENTS CHAPTER I CHAPTER II CHAPTER I I I CHAPTER IV Page Introduction 1 Apparatus: E l e c t r o n i c s 5 Magnet 8 Low Temperature Equipment 8 Automatic manostat 11 C r y s t a l growing 12 X-ray and Infrared equipment 14 Experimental Results: C r y s t a l structure and Magnetic Symmetry of CoC^-SR^O Measurements of Neel Temperature Deuteron n.m.r. Spectra Infrared Spectrum of CoCl2'6H20 and CoCl2'6D 20 Analysis of X-ray powder Photographs Discussion of the S i g n i f i c a n c e of the S h i f t i n T N: Exchange and super-exchange Interactions 35 The Hydrogen Bond 40 A Model f o r the E f f e c t of Isotopic S u b s t i t u t i o n on the Neel Temperature 44 A p p l i c a t i o n of the Model to C o C ^ ^ ^ O Using Haseda's Conjecture 48 Relationship of the Isotope S h i f t to other Experiments 50 Bibliography 53 Appendix A - Nuclear Magnetic Resonance Spectrum of Deuterium i n CoC^^I^O at 4.2°K 55 15 19 29 31 31 i v TABLES Page 3.1 Dependence of Neel Temperature on Hg 28 V FIGURES Page 2.1 Block diagram of the n.m.r. apparatus 6 2.2 Design of the marginal o s c i l l a t o r 9 2.3 A t y p i c a l trace of the proton spectrum i n " CoCl 2-6H 20 (30% deuterated) 10 2.4 Automatic Manostat " 9 3.1 Atomic p o s i t i o n s i n CoCl 2*6H 20 c r y s t a l 16 3.2 Spin alignments of CoCO^^R^O i n the a n t i -ferromagnetic phase 17 3.3 Proton n.m.r. spectra of CoC^'SR^O i n paramagnetic and antiferromagnetic phases 20 3.4 A t y p i c a l trace of the amplitude of the d e r i v a t i v e of the absorption n.m.r. s i g n a l of protons v/s temperature ' 2 2 3.5 Neel temperature v/s o r i e n t a t i o n of external magnetic f i e l d with respect to c r y s t a l axes . 24 3.6 Orientation-averaged Neel temperature v/s (Relativ e concentration of deuterium)-*-/^ 1 25 3.7 Deuteron n.m.r. spectra of CoCl 2'6D 20 (92% deuterated) at 4.2°K 30 3.8 Infrared spectrum of CoCl 2«6H 20 and CoCl 2-6D 20 32 v i ACKNOWLEDGMENTS The author wishes to thank h i s supervisor, Dr. Myer Bloom, fo r h i s help and support. Dr.. K. Gray a s s i s t e d i n some of the measurements. Mr. William Morrison was r e a d i l y a v a i l a b l e f o r r e p a i r i n g the low temperature plumbing and.Mr. John Lees helped with the glass blowing. Dr. N. K. Jha as s i s t e d i n the x-ray work. Thanks are due to a l l these f r i e n d s . The author also thanks the Chemistry department f o r the permission to use t h e i r x-ray and i n f r a r e d f a c i l i t i e s . The research was f i n a n c i a l l y supported by the National • • \ Research Council of Canada. 1 CHAPTER I  Introduction The nuclear magnetic resonance (n.m.r.) technique has proven to be a powerful t o o l i n the study of the magnetic properties of matter. In the simplest type of n.m.r. experiment, a d-c magnetic f i e l d H 0 k and -=? a r - f f i e l d 2 Hj c o s w t i at r i g h t angles to each other are applied to a system of nuclear spins. When the frequency i s var i e d through the Larmor frequency tVJjj of the n u c l e i , a resonant absorption• of energy from .the r - f f i e l d takes place which can be detected by a method such as the one described i n Chapter 2. By measuring one can get information on the l o c a l magnetic f i e l d at the nuclear s i t e H-j_0^  and hence le a r n something about the d i s t r i b u t i o n of magnetic moments i n the c r y s t a l •= 7 % o c -5 - • ; where % n - ( - i s the co n t r i b u t i o n to the l o c a l magnetic f i e l d due to . magnetic dipoles i n the c r y s t a l and T i s the gyromagnetic r a t i o of the roxc.1e.us. For protons t y p i c a l values of the Larmor frequency range up to 50 Mc. In t h i s work the highest frequency used was 21 Mc. In s a l t s such as the one studied here, C o C ^ ^ ^ O , the magnetic ions i n t e r a c t by means of so - c a l l e d "exchange i n t e r a c t i o n s " , which w i l l be discussed i n d e t a i l i n Chapter 4, so that the spin of an i n d i v i d u a l magnetic ion i s not a constant of the motion because i t does not commute with the exchange i n t e r a c t i o n Hamiltonian. Therefore, the i n t e r n a l f i e l d near a p a r t i c u l a r magnetic i on i s a r a p i d l y varying function of time. Since the exchange frequency f o r CoCl2*6H20 i s of order 1 0 ^ sec 1, the proton spins are unable to follow the rapid f l u c t u a t i o n s i n the i n t e r n a l f i e l d , and hence — ^ Hint i n equation (1.1) i s to be interpreted as the time average of the i n t e r n a l f i e l d at the nuclear s i t e . Let <fA/^7 be the time averaged value of the magnetic moment of the i ! t h magnetic ion located at a distance r ^ f from a p a r t i c u l a r nucleus. For such l o c a l i z e d spins, we can write the d i p o l a r f i e l d at the nucleus as r L i r L i I f the c r y s t a l i s in.the paramagnetic.state, then the behaviour of the n.m.r. spectrum as the o r i e n t a t i o n of H 0 i s varied i s as follows: (1) For each nuclear s i t e i s symmetric with respect to rot a t i o n s o f ~ $ Q by 180°. This can be seen from equations.(1.1) and (1.2) because when H0-=jr - H Q, f'jJ —^"-^Miy and the dependence on ,-> r ^ of % n - t - i s governed by Y2rn( L i ) w n e r e L i ^ s t n e o r i e n t a t i o n of r e l a t i v e to <r^7 . (2) There w i l l be a c e r t a i n number of d i s t i n c t ' —> - v values n. of H-j^ f o r each o r i e n t a t i o n of H 0; the number n can be calc u l a t e d from the c r y s t a l symmetry. • When the c r y s t a l becomes antiferromagnetic the n.m.r. spectrum changes ( P o u l i s & Hardeman, 1952) (1) The n.m.r. spectrum becomes symmetric with respect to r o t a t i o n s of H 0 by 360 instead of 180 as i n the paramagnetic phase. The. reason f o r t h i s Is that there i s a preferred d i r e c t i o n of magnetization i n the antiferromagnetic phase so that /^*^ "> does not follow H 0 and, i n p a r t i c u l a r ^P"^? does not reverse i t s e l f —> ^ > when H0—> - H 0. (2) For a simple two s u b - l a t t i c e antiferromagnet, the number of d i s t i n c t values of % n t changes from n to 2n, because f o r each c r y s t a l l o g r a p h i c a l l y equivalent p o s i t i o n , there e x i s t two magnetically non-equivalent p o s i t i o n s . Therefore, the temperature T^ at which the paramagnetism to antiferromagnetism phase t r a n s i t i o n takes place can be very accurately, measured by monitoring the n.m.r. spectrum. In the case of CoC^^H^O, Sawatzky and Bloom (1962) were able to measure T^ to within a few mi l l i d e g r e e s . I t may be mentioned that an important feature of the n.m.r. spectrum very close to TJJ i s the anomalous broadening of the resonance, l i n e s . This phenomenon i s analagous to the f l u c t u a t i o n s i n the density of f l u i d s near the c r i t i c a l point. In n.m.r., the anomalous l i n e widths are- associated with f l u c t u a t i o n s i n short-range magnetic order (Moriya 1962). Many systems which undergo paramagnetic-antiferromagnetic phase t r a n s i t i o n s have been studied using n.m.r., though the number of studies-i n which the s e n s i t i v i t y . o f the technique has been f u l l y exploited i s s t i l l small. In t h i s t h e s i s a new type of observation i s reported. I t has been found that when the protons are replaced by deuterons, a large (approximately 6%) change i n T^ occurs. A f t e r t h i s observation was made, i t was discovered that Haseda (1960) had already predicted that TJJ might change appreciably i n CoCl2'6H20 when the protons are replaced by deuterons. In f a c t , Haseda t r i e d unsuccessfully to observe t h i s e f f e c t by monitoring the magnetic s u s c e p t i b i l i t y ' X of C 0 C I 2 ' 6 ^ 0 and CoCl2*6D20 as a function of temperature. 4. The s u s c e p t i b i l i t y i s a much l e s s s e n s i t i v e measure of TJJ than i s n.m.r. and- that i s why Haseda did not succeed i n v e r i f y i n g h i s conjecture. According to molecular f i e l d theory (Nagamiya, Yosida and Kubo.1955) the slope of the "X versus T curve does have a d i s c o n t i n u i t y at T^. However, because of the short range magnetic order e x i s t i n g i n r e a l systems, the "Xversus T curve usually.has a rounded maximum at a temperature above T^ which makes i t d i f f i c u l t to determine T-^  d i r e c t l y from a "X versus T curve. (Nakamura 1962) Haseda based h i s conjecture on the f a c t that small changes i n the energy of hydrogen bonding do occur upon deuteration and that there might be some connection between hydrogen bonding and super-exchange. He pointed out that some of the possible super-exchange paths contain hydrogen bonds. In Chapter 4 a semi-empirical attempt i s made-to e s t a b l i s h a connection between the change i n T^ upon deuteration, the super-exchange parameter and the p o t e n t i a l of.a proton i n a hydrogen bond. The picture presented i s that T^ increases on deuteration because of the change i n the average of the super-exchange parameter over the ground v i b r a t i o n a l state of the hydrogen atom. I t i s suggested that t h i s type of isotope e f f e c t may be found i n other systems such as the t r a n s i t i o n element oxides i n which 0 ^ i s replaced by O-^ .-' ' 5 CHAPTER II  Apparatus E l e c t r o n i c s The n.m.r. sign a l s of the protons and deuterons i n our c r y s t a l s were detected with a Pound-Knight spectrometer of the design described by Sawatzky (1962). This device i s b a s i c a l l y a marginal o s c i l l a t o r i n which the l e v e l of o s c i l l a t i o n s decreases when i t s load (represented by a shunt conductance across i t s tuned c i r c u i t ) i s increased. We place the sample containing the nuclear spins i n the c o i l of the tuned c i r c u i t of the o s c i l l a t o r . When the frequency of the o s c i l l a t o r i s equal to the Larmor frequency of nuclear spins, the spin system absorbs energy from the r . f . f i e l d of the o s c i l l a t o r and thus provides an a d d i t i o n a l r e sistance. • The decrease i n l e v e l of o s c i l l a t i o n s caused by t h i s a d d i t i o n a l load i s used to detect nuclear magnetic resonance. The s i g n a l to be observed i s , however, weak; therefore the usual technique of narrow banding by l o c k - i n detection was used. The block diagram of the apparatus i s shown i n fig'. 2.1. The commercial equipment used has been indicated In the diagram. The o s c i l l a t o r section of the Pound-Knight spectrometer had to be redesigned to improve the s i g n a l to noise r a t i o . Most of the.noise came from the o s c i l l a t o r tube i t s e l f . Replacing the 6J6 o s c i l l a t o r tube by a QDS4 nuvistor t r i o d e improved the s i g n a l to noise r a t i o by a f a c t o r of two. A design su i t a b l e f o r t h i s purpose i s given i n f i g . 2.2. For maximum s e n s i t i v i t y the marginal o s c i l l a t o r has to be operated at the lowest possible r . f . l e v e l . The o s c i l l a t o r tube, under these conditions works in"a non-linear region. Because of t h i s non-Freq. C o u n Y o r Digita Recorder Narrow Band AmpI. Phase Ssns. Det. Record Millia. Scope Broad Band AmpI. Phase Shift Power Phase 30 cps. AmpI. Shift Source 2 n d Oscil. Detector for Field determination FIG. 21 Block Diagram of Apparatus. 7 l i n e a r i t y , the r . f . generated by the o s c i l l a t o r i s e a s i l y modulated by the noise picked from external sources. For example, the modulation c o i l s pose the following problem. Since they are not exactly orthogonal to the. sample c o i l , the l a t t e r picks up small audio-voltages of the modulation frequency along with the f l u c t u a t i o n s of these voltages. This produces a modulation of the r . f . output as mentioned above. On detection t h i s modulated r . f . produces a s i g n a l which s h i f t s the zero l i n e of the recorder, c o n t r i b u t i n g also to the general background noise. Most of the background noise was eliminated by using a stable a u d i o - o s c i l l a t o r f o r the modulation and by avoiding the use of fluorescent l i g h t s . The s h i f t i n g of the zero l i n e of the recorder was : a b i t d i f f i c u l t to cure. The usual method recommended against such trouble i s to induce an e.m.f. equal i n amplitude but opposite i n phase to that received from the modulation c o i l s . This requires the winding of a compensating c o i l on the sample holder but the s i z e of the cryostat-didn't permit i t . So the sample c o i l had to be very c a r e f u l l y aligned normal to the modulation c o i l s i n order to get r i d of the s h i f t s i n the zero l i n e of the recorder. Another source of malfunctioning i n the c i r c u i t was the r . f . leakage between marginal o s c i l l a t o r , d i f f e r e n t stages of the r . f . a m p l i f i e r and the detector-unit of the Pound-Knight spectrometer. The use of 'Ferox beads' and feed through capacitors was found very e f f e c t i v e i n removing t h i s trouble. The filaments of the c i r c u i t were fed by a d.c. battery and the Hewlett Packard model 712B power supply was used f o r high voltage because noise free low and high voltage supplies are e s s e n t i a l f o r detecting weak s i g n a l s . The Magnet —> The external f i e l d H 0 i s supplied by an ai r - c o o l e d iron-core electromagnet (Newport Instruments L t d . , S e r i a l No. 6010/3). I t has plane pole t i p s of four inch diameter and an adjustable air-gap. With an a i r gap of 3.2 cm an inhomogeneity of about 0.3 gauss per cm e x i s t s near the centre of the pole faces.at a magnetic f i e l d of 5.0 K gauss. The magnet i s mounted on a r o t a t i n g platform so that ~H^  can take d i f f e r e n t o r i e n t a t i o n s with respect to the c r y s t a l axes. The power supply of the magnet follows the design of Garwin (1959) as modified by Sawatzky (1962). The t r a n s i s t o r s i n t h i s design are water cooled and the v a r i a t i o n s i n the reference r e s i s t o r are minimized by the use of a water cooled o i l bath. The magnetic f i e l d produced by t h i s power supply has been found stable to 1 part i n 10^ over several hours as checked by an n.m.r. probe. The nuclear magnetic resonance si g n a l s were studied by varying the marginal o s c i l l a t o r frequency f o r a constant external magnetic f i e l d A t y p i c a l trace of proton "spectrum i n CoCI^'G^O at 4.2°K has been shown i n f i g . 2.3. Frequency measurements were made with the help of a Hewlett Packard E l e c t r o n i c counter model 524G coupled to a model 561B D i g i t a l Recorder. Low Temperature Equipment The low temperatures were produced using a common double dewar glass cryostat as described by Sawatzky (1962). -Temperatures lower than 4.2°K were obtained by pumping on the helium vapour. The dependence of temperature on vapour pressure of helium i s known very accurately and th Fe All Capacitances in fiF FIG. 2-2 A To Vacuum Pump B = Brass Bellows Automatic Manostat FIG. 2-4 FIG. 2-3 A typical trace of Proton Spectrum in 30% deuterated Co c6 2 .6H 2 0. T= 2.4 eK H 0 * 5.5 kgauss H o 11 dependence was used i n the measurement of temperatures. The vapour pressure dn a manometer connected with the cryostat was measured using a cathetometer and the corresponding value of temperature was found from vapour pressure v/s temperature t a b l e s . While studying the changes i n symmetry of the proton spectrum associated with para- to antiferromagnetic phase t r a n s i t i o n , the temperature had to be varied by very small amounts and maintained constant t o l O - ^ °C over several minutes. . To achieve t h i s an automatic manostat was connected i n s e r i e s with the pumping l i n e ; The temperature was s t a b i l i z e d by s t a b i l i z i n g the vapour pressure with the help of t h i s manostat whose working i s described below. The Automatic Manostat As shown i n f i g . 2.4, the manostat works on the p r i n c i p l e of negative feed-back. A standard pressure i s maintained i n a brass bellows the pumping l i n e . An the bellows and thereby which controls the cross section of a tube In] excess of pressure i n the cryostat compresses! enlarges the cross s e c t i o n a l area through which pumping i s taking place. This increases the pumping rate and the pressure i n the cryostat s t a r t s f a l l i n g t i l l i t becomes equal to the standard pressure i n the bellows. The reverse happens when the pressure i n s i d e the cryostat f a l l s below the pressure i n the bellows. Using t h i s device the temperature could be maintained constant within l O - ^ °C over a few minutes. There were however slow v a r i a t i o n s of vapour pressure which could be c o n t r o l l e d manually by manipulating the needle valve shown i n f i g . 2.4. As described previously, i n order to measure the ,Neel temperature, the n.m.r. s i g n a l amplitude had to be studied as a function 12 of•temperature. The temperature had to be varied continuously and we had to make sure that the c r y s t a l was i n thermal eq u i l i b r i u m with the helium bath a l l the time. The sweep speed of 3 m i l l i d e g r e e s per minute was found s a t i s f a c t o r y f o r t h i s purpose. However, to be sure that e q u i l i b r i u m r e a l l y existed, the temperature v a r i a t i o n was often stopped at some a r b i t r a r y temperature, and i t was found that the s i g n a l amplitude remained at whatever value the temperature v a r i a t i o n was.stopped. This indi c a t e d that the sample temperature followed the helium temperature very c l o s e l y . C r y s t a l Growing As we had to grow c r y s t a l s from heavy water s o l u t i o n s , the . technique described by Sawatzky wasn't s u i t a b l e f o r our purpose. The solutions i n Sawatzky's method remained open to the atmosphere, and under these conditions the heavy water molecules exchange t h e i r deuterium very r a p i d l y with the hydrogen atoms of the atmospheric water vapour. To avoid t h i s we grew c r y s t a l s by seeding super saturated s o l u t i o n s i n sealed j a r s thus avoiding contact with the atmospheric moisture. Quite often i t was found that the s o l i d deposited i n a poly-c r y s t a l l i n e mass on the walls of the j a r or on the seed i t s e l f . This happens because the s o l i d s t a r t s depositing on other microscopic seeds which are present i n the v e s s e l and on the main c r y s t a l seed. Two precautions are necessary i n order to get r i d of these unwanted micro-scopic seeds. a) The j a r s should be washed thoroughly and then heated under an i n f r a - r e d lamp or on a flame. 13 b) At the s t a r t the s o l u t i o n should be kept s l i g h t l y under-saturated. This di s s o l v e s any microscopic seeds present on the main c r y s t a l seeds. The temperature of the j a r should be then lowered quickly by 1°C, so that the main seed doesn't completely dissolv e i n the s o l u t i o n . As the c r y s t a l grows the excess of s a l t i n the s o l u t i o n diminishes and the growth rate decreases. To prevent the growth from stopping completely, the s o l u t i o n has to be cooled so that i t becomes supersaturated at a lower temperature and thus capable of depositing more s o l i d on the seed. To have a smooth and uninterrupted growth the '., so l u t i o n should be cooled slowly and uniformly. For C o C L ^ K v j O a cooling rate of 4°C per day was found s a t i s f a c t o r y . To make sure that the specimens grown were.single c r y s t a l s we took some Laue patterns of these specimens on an x-ray machine. The doubtful specimens were rejec t e d . As described by Groth (1906) the c-axis of C o C ^ ^ K ^ O s i n g l e c r y s t a l can be e a s i l y i d e n t i f i e d . This information along with the p o s i t i o n of cleavage plane (001) was used to f i n d the o r i e n t a t i o n of other c r y s t a l axes. The c r y s t a l s were grown from the reagents supplied by " A l l i e d Chemical's L t d . " C r y s t a l Mounting In order to t e s t that the coexistence of two phases mentioned previously wasn't due to s t r a i n s i n the c r y s t a l , the c r y s t a l s were c a r e f u l l y handled avoiding a l l types of shocks. The c y l i n d r i c a l specimens were prepared, not by machining but by d i s s o l v i n g the c r y s t a l with a wet brush. Various t e s t s showed that the n.m.r. proton spectra of specimens 14 prepared by the above method and of those prepared by machining were I d e n t i c a l at l i q u i d nitrogen temperatures. The specimens were cooled very slowly i n these t e s t s . Later i t was found that the coexistence region was about s i x m i l l i d e g r e e s as e a r l i e r reported by Sawatzky (1962) f o r a l l these c r y s t a l s . We conclude from t h i s that i f the coexistence region i s due to the e f f e c t of s t r a i n , c a r e f u l handling of the c r y s t a l as described has not reduced these s t r a i n s . The p o s i t i o n of c r y s t a l axes was etched on the specimens and they were then.mounted i n t e f l o n holders of Sawatzky's design. X-ray and Infrared Apparatus For i n f r a r e d work a commercial double beam spectrometer, model 21 manufactured by. Perkin Elmer, was used. The x-ray powder photographs were taken on a General E l e c t r i c Debye Scherrer camera. 15 CHAPTER III Experimental Results 3.1 Review of the C r y s t a l Structure and Magnetic Symmetry Properties of CoCl 2-6H 20 Before giving the experimental r e s u l t s , i t would be appropriate to describe the system studied i n some d e t a i l : C o C l 2 - 6 ^ 0 forms dark purple monoclinic c r y s t a l s as described by Groth (1906). The u n i t c e l l edges are a = 10,34 A, ' b = 7.06 1 and c = 6.67 & and according to x-ray analysis of Mizuno et a l (1959) the monoclinic l a t t i c e has the symmetry of space group C 2/m with j3 = 122° 19' Perfect cleavage occurs along the c (001) face. Each u n i t c e l l has two formula u n i t s with C o + + metal ions at the corners and the centres of i t s a-c faces. The structure i s based on an octahedral co-ordination about the metal ions with four water oxygens l y i n g at the corners of a s l i g h t l y d i s t o r t e d square and two c h l o r i n e s symmetrically disposed above and below the plane of the square. The '" oxygens of the remaining waters of hydration are located i n the a-c mirror plane. F i g . 3.1 shows the p o s i t i o n of various atoms i n the c r y s t a l s t r u c t u r e . The x-ray data gives us no information about the l o c a t i o n of protons. The co-ordinates of other atoms are given below: Atomic Parameters Type of atom X Y Z Co 0 0 0 CI 0.278 0 0.175 Oj 0.0288 - 0.221 0.255 On 0.275 0 0.700 FIG. 3-1 Atomic positions in the crystal. 17 not to scale • \ FIG. 3*2 Alignment of C o + + spins in C o c £ 2 . 6 H 2 0 in the antiferromagnetic phase. 18 E l S a f f a r (1962) derived the proton p o s i t i o n s from the n.m.r. studies of the proton spectrum of C0CI2. 6H 20, using the observed HOH o angle and assuming the 0-H distance to be 0.97 A. His r e s u l t s are consistent with those obtained by Sawatzky and Bloom (1962) i n which they assumed the distance between two protons of a water molecule to be 0 . 1.595 A. The proton p o s i t i o n s as given by E l Saffar are as follows. Atomic parameters Atom . X Y Z °I °II H^ 0.097 0.313 0.268 H^ .0.427 0.205 0.205 H J X 0 . 1 9 1 0 . 0 0 0 0 . 5 4 9 0 . 2 2 6 0 . 5 0 0 0 . 1 6 2 Haseda and Kanda (1957) reported the antiferromagnetic behaviour of 00012*6^0. Since then, the Neel temperature of t h i s substance has been measured with i n c r e a s i n g accuracy by seve r a l people. Spence et a l (1964) have investigated the magnetic symmetry of CoC^^K^O i n the antiferromagnetic phase. The magnetic l a t t i c e can be described by the space group Pc 2 j ^ i n Belov's notation. Layers of spins p a r a l l e l to the b-c plane are stacked together so that i f i n one l a y e r a l l the spins point p a r a l l e l to the c-axis then the spins i n the next l a y e r are a l l a n t i p a r a l l e l to the c-axis. Thus the c r y s t a l has an antiferromagnetism of oppositely dir e c t e d ferromagnetic l a y e r s stacked side by side. The spin o r i e n t a t i o n s are shown i n f i g . 3.2. The neighbouring spins which are directed opposite to each other are li n k e d with the super-exchange 19 i n t e r a c t i o n . As mentioned before, some paths of super-exchange probably contain hydrogen bonds according to Haseda (i960). The r e l a t i o n s h i p of these bonds with the super-exchange i n t e r a c t i o n w i l l be discussed i n the next chapter. The symmetry properties of the proton n.m.r. spectra i n the paramagnetic and antiferromagnetic phases of C o C ^ ^ ^ O have been exhaustively studied by Sawatzky and Bloom1 (1962). As discussed e a r l i e r i n Chapter I, the frequencies of proton resonance change- when we vary the o r i e n t a t i o n of the external magnetic f i e l d H Q with respect to the c r y s t a l l i n e axes. There iSjhowever, a p e r i o d i c i t y i n t h i s v a r i a t i o n . In the paramagnetic phase the spectrum repeats i t s e l f when H 0 changes i t s o r i e n t a t i o n by 180°, whereas i n the antiferromagnetic case t h i s r e p i t i o n occurs a f t e r a 360° i n t e r v a l . F i g . 3.3 shows a p l o t of the resonance frequencies of protons i n C o C ^ ^ ^ O f o r d i f f e r e n t o r i e n t a t i o n s of H Q as H Q i s rotated i n the a-c plane of the c r y s t a l . The change i n symmetry of the proton spectra i n d i c a t e s the phase t r a n s i t i o n i n a very s t r i k i n g manner. 3.2 Measurements of Neel temperature ^ Beside the change i n symmetry, there are other e f f e c t s connected with the phase t r a n s i t i o n which were mentioned i n the Introduction. ( i ) Doubling of the number of l i n e s i n going from para- to antiferromagnetic phase and ( i i ) Fading away of the proton l i n e by broadening (and consequent decrease i n the s i g n a l amplitude) with the approach of antiferromagnetic phase. RESONANCE FREQUENCY Mc FIG. 3-3 PROTON N.M.R. SPECTRA OF Co cZz 6 H 2 0 FROM SAWATZKY AND BLOOM. 21 H i s t o r i c a l l y , Van der Lugt and Pau l i s (i960) were the f i r s t to use the phenomenon ( i i ) , f o r the purpose of measuring the Neel temperature. They observed a proton l i n e from CuQ^^H^O on an o s c i l l o s c o p e as the . temperature of the sample was being changed and the a r r i v a l of the Neel temperature was s i g n a l l e d by the disappearance of the l i n e . Sawatzky and Bloom (1962) used the same p r i n c i p l e but an improved monitoring technique. The strength of the technique used by Sawatzky and Bloom depends on improved s i g n a l to noise r a t i o achieved by using narrow banding and l o c k - i n detection. In both these techniques, one cannot t e l l whether the s i g n a l has a c t u a l l y disappeared or i s being merely masked by the background noise. However, the smaller the background noise the b e t t e r would be the accuracy of monitoring the disappearance of the l i n e and hence the s u p e r i o r i t y of Sawatzky and Blooms' technique. F i g . 3.4 shows our t y p i c a l recorded trace of the amplitude of the d e r i v a t i v e of the n.m.r. absorption s i g n a l v/s temperature. The temperature was varied smoothly and uniformly at an approximate rate of 2 x l O - ^ °K per minute by varying the vapour pressure i n the helium dewar. The i n i t i a l vapour pressure (and temperature) was c a r e f u l l y set with the help of a mercury manometer connected with the helium dewar and the vapour pressure (therefore temperature) was slowly varied by manipulating a coarse and a f i n e needle valve i n s e r i e s with the pumping l i n e . The vapour pressure d i f f e r e n c e s during the sweep were read on a d i f f e r e n t i a l o i l manometer. Marks corresponding to the known values of vapour pressure were e l e c t r i c a l l y t i c k e d o f f on the margin of the paper 23 on which the s i g n a l was being continuously recorded. The recorder trace, thus served as a graph between the amplitude of the d e r i v a t i v e of proton absorption s i g n a l v/s vapour pressure. The temperature corresponding to the vapour pressure at which the s i g n a l disappeared.-was found from the standard t a b l e s . The measurements were repeated using d i f f e r e n t i n i t i a l temperatures and sweep speeds and, as described i n Chapter I I , the conditions of continuous thermal equilibrium were maintained during the temperature sweep. Also several times during the measurements, the accuracy of the apparatus was checked by measuring the ^  - point of helium which served as a b u i l t i n standard. Measurements were made f o r three d i f f e r e n t r e l a t i v e concentrations of deuterium. In each case two samples were prepared from d i f f e r e n t solutions and the r e s u l t s compared. The r e s u l t s from d i f f e r e n t samples of the same percentage of deuteration agreed very w e l l within the accuracy of the experiment. F i g . 3.5 shows that the Neel temperature v a r i e s with a period • of 180° as the external magnetic f i e l d - i s rotated about the b-axis of the c r y s t a l . On deuteration the Neel temperature r i s e s f o r a l l or i e n t a t i o n s and the more deuterium we introduce the higher t h i s t r a n s i t i o n point goes. The maximum increase i n Neel temperature as the r e l a t i v e concentration of deuterium i s changed from 0% to 92% i s approximately 6%. Measurements were made i n samples containing 0%, (30 + 2 ) % and (92 ± 5)% deuterium r e s p e c t i v e l y . In addition, the anisotropy i n T^ decreases from 0.08 °K f o r 0% deuteration to s l i g h t l y l e s s than 0.05 °K f o r 92% deuteration. Further as shown i n f i g . 3.6 the orientation-averaged Neel temperature seems to vary l i n e a r l y with the cube root of r e l a t i v e concentration of deuterium. 2 15' 1 1 1 1 1 1 1 1 I I 1 1 i 1 • 1 ' 1 0 50 100 150 H IIC AXIS H 0 X C AXIS Orientation of H 0 in degrees F I G 3-6 26 The external magnetic f i e l d was kept at 5,000 gauss f o r the measurements reported i n f i g . 3.5 and 3.6. In the case of 30% deuterated c r y s t a l s the r e l a t i v e concentration of deuterium was estimated from the r e l a t i v e proportion of K^O and D2O i n the parent s o l u t i o n from which the c r y s t a l s were grown. For the 92% deuterated samples, the value of r e l a t i v e concentration obtained by the above mentioned method was checked against that obtained from the r e l a t i v e strength of deuteron and proton n.m.r. si g n a l s . These, gave r e s u l t s agreeing within the accuracy quoted above. In these measurements, the vanishing of n.m.r. sign a l s has to be int e r p r e t e d rather c a r e f u l l y . We have to be sure that the s i g n a l disappears because of broadening and not due to the s h i f t s i n the p o s i t i o n of the l i n e i n the frequency spectrum. This wasn't a serious problem f o r us because the proton and deuteron l i n e s didn't s h i f t by more than a tenth of t h e i r l i n e width during a temperature sweep of 20 m i l l i d e g r e e s above, the Neel temperature. The coexistence region poses another problem i n t h i s technique. For some proton l i n e s , as the temperature i s being va r i e d , an a n t i -ferromagnetic l i n e appears before the disappearance of a paramagnetic l i n e . The frequencies of these antiferromagnetic l i n e s , as studied by Sawatzky (1962), are very s e n s i t i v e to temperature. These l i n e s usually disappear about 20 m i l l i d e g r e e below TJJ because of the s h i f t i n t h e i r frequencies and cannot be r e l i e d upon f o r the measurement of Neel temperature. However, on c l o s e r examination i t i s found that the recorder trace shows a s l i g h t turning at the point where the antiferromagnetic l i n e tunes i n . Such traces usually give a f a l s e Neel temperature and should be reje c t e d . We can, then, r e s o r t to other a v a i l a b l e paramagnetic l i n e s f o r the detection of phase change. In our experiment we used more than one proton l i n e as an i n d i c a t o r of phase t r a n s i t i o n f o r each o r i e n t a t i o n of H Q. These l i n e s disappear at d i f f e r e n t points i n the coexistence region. The 6 m i l l i d e g r e e s s c a t t e r i n the measured values of Tj^, f o r a f i x e d o r i e n t a t i o n , i s thus merely a measure of the width of the coexistence region. This was confirmed by studying-the f u l l proton spectrum as a function of temperature i n the v i c i n i t y of T^ f o r one o r i e n t a t i o n . The uncertainty of 6 m i l l i d e g r e e caused by the coexistence of two phases i s much bigger than the e r r o r s caused by background noise and serves as an upper l i m i t on the errors i n these measurements of Tj^. For the purpose of comparison, the Neel temperature of the samples having d i f f e r e n t percentages of deuteration.were measured at the same magnetic f i e l d . This i s necessary i n order to i s o l a t e the e f f e c t of deuteration on T^ from the dependence of the Neel temperature on H Q. . A study of T^ v/s H Q with H Q X c-axis was undertaken to show that the : e f f e c t s reported i n f i g . 3.4 are a c t u a l l y due to the changes i n the percentage of deuteration and not due to d i f f i c u l t i e s i n the r e p r o d u c i b i l i t y of H Q. For 92% r e l a t i v e Concentration of deuterium, the r e s u l t s are as shown below: 28 Table 3.1 Dependence of T^ on H Q with HQ_L c-axis f o r 92% deuterated s i n g l e c r y s t a l of CoCl 2-6H 20 Magnetic f i e l d Neel Temperature i n .'. Neel Temperature i n Ki l l o g a u s s 92% Deuterated non-Deuterated CoCl 2-6H 20 CoCl 2-6H 20 (Sawatzky & Bloom) 3.5 2.236 ' 2.251 4.2 2.363 2.268 5.00 2.370 2.280 Table 3.1 shows that even s u b s t a n t i a l changes of H Q (3.5 to 5 Ki l l o g a u s s ) produce only small changes i n T N (2.236 °K - 2.370 °K). As H Q could be e a s i l y reproduced to within one part i n 10^ the changes i n T^ i n f i g . 3.5 can be considered as a leg i t i m a t e e f f e c t of . deuteration. Table 3.1 shows that i n the neighbourhood of H 0 = 4000 gauss, the Neel temperature increases with an increase i n H 0, H Q being J- the c-axis. This unusual behaviour i s common to deuterated as w e l l as non-deuterated s i n g l e c r y s t a l s and has not.yet been explained (Sawatzky and Bloom, 1962). F i g . 3.5 shows that when H 0 i s rotated around the b-axis of the c r y s t a l , the maximum of T^ occurs when "3^  // c-axis. This i s true f o r a l l the samples having, d i f f e r e n t r e l a t i v e concentrations of deuterium and can be used to check the alignment of the c r y s t a l axes with respect to H Q. 29 For each sample the b-axis arid the cleavage plane (001) were used to a l i g n the c r y s t a l . This alignment was then checked against r o t a t i o n diagram of the proton spectrum shown i n f i g . 3.1 f o r the 0% deuterated sample and was also found to be consistent with the Neel temperature.diagram of f i g . 3.5. A s i m i l a r check f o r consistency was made f o r the 30% deuterated sample. However, because of the quadrupole i n t e r a c t i o n , the. deuteron n.m.r. spectrum f o r the sample with 92% concentration of deuterium d i f f e r s q u a l i t a t i v e l y from the proton n.m.r. spectrum of CoCl^'SJ^O. since the p r i n c i p a l axes of the e l e c t r i c f i e l d gradient at the deuterium s i t e and those of the magnetic s u s c e p t i b i l i t y tensor are not n e c e s s a r i l y correlated, the r o t a t i o n diagram of the deuteron n.m.r. frequencies cannot be used as a check f o r the c r y s t a l alignment. For the 92% samples, the morphological information about the c-axis and the (001) cleavage plane together with the Neel temperature diagram o f F i g . 3.5 was;used to a s c e r t a i n the p o s i t i o n . ' . J of c r y s t a l axes. The erro r i n the alignment \was l e s s than 5% i n each case. 3.3 Deuteron n.m.r. Spectra F i g . 3.7 shows the po s i t i o n s of various deuteron n.m.r. l i n e s as H Q i s rotated i n the a-c plane _L the b-axis of the si n g l e c r y s t a l of 92% deuterated CoCl2*6H 20 at 4.2 °K. Two sets of spectra were taken ( i ) at H Q = 4.914 K gauss and ( i i ) at H Q = 1.828 K gauss. I t i s i n t e r e s t i n g to see that whereas H 0 has been reduced by a f a c t o r of 0.37 the maximum s p l i t t i n g between the l i n e s has been decreased only by a f a c t o r of 0.75. O r d i n a r i l y , i i n the paramagnetic phase the i n t e r n a l f i e l d s are prop o r t i o n a l to the external f i e l d and the maximum s p l i t t i n g between the proton l i n e s should vary by the same f r a c t i o n as the external f i e l d does. The lack of p r o p o r t i o n a l i t y between the maximum s p l i t t i n g and the external f i e l d s suggests B. Ho • 1.828 KILOGAUSS FIG. 3-7 Deuteron n. m.F. spectra of Coce26D20 at 4.2 K° as H 0 is rotated in the a—c plane. 31 that there are other mechanisms present which give r i s e to some a d d i t i o n a l s p l i t t i n g of l i n e s over that caused by the i n t e r n a l magnetic f i e l d s . We ascribe t h i s a d d i t i o n a l s p l i t t i n g to the quadrupole i n t e r -action of deuterons with the e l e c t r o s t a t i c f i e l d gradients present i n s i d e the c r y s t a l . In the region of magnetic f i e l d 4.9 k i l l o g a u s s the two mechanisms seem to give s p l i t t i n g s of the same order of magnitude. A more d e t a i l e d discussion of the spectrum w i l l be given i n Appendix A. 3.4 The Infrared Spectrum The i n f r a r e d spectrum of CoC^^K^O powder taken at room temperature with the KBr d i s c technique i s shown i n f i g . 3.8. I t shows the absorption due to s t r e t c h i n g of 0-H bond at 3450 cm""-*-' and due to bending of the 0^ molecule at 1605 cm - . These frequencies are quite close to those f o r water i n l i q u i d s tate. F i g . 3.8 also shows the spectrum due to deuterated CoC^^K^O powderj. For the deuterated specimen we have two s t r e t c h i n g frequencies - 3450 cm~l corresponding to 0-H and 2530 cm--'- corresponding to 0-D. However, the deuterated specimen gives three bending frequencies as compared to one given by the non-deuterated sample. These are at 1625 cm~l, 1425 cm~l.and 1190 cm~l. They H yH are possibly associated with 0 ^ , 0 ^ and 0^ configurations. 3.5 The Analysis of X-ray Powder Photographs The x-ray powder photographs f o r 00012* 6H20 a n < ^ CoCl 2*6D 20 were, studied at room temperature, i n order to f i n d any possible changes i n c r y s t a l symmetry or u n i t c e l l dimensions caused by deuteration. The shortest distances between the neighbouring p a r a l l e l h k l m i l l e r planes Abtorbanca (Optical dentHy) 33 (dhkl) were calculated from the observed pattern. The symmetry of the c r y s t a l remains unchanged. Assuming the same c e l l dimensions f o r both specimens, the calculated values do not d i f f e r by more than 0.2% fo r deuterated and non-deuterated C o C l ^ l ^ O . Moreover, the 0.2% differences are not systematic therefore we can conclude that the c e l l dimensions i n the two cases are equal within 0.2%. The r e s u l t s are as follows: Plane d h k l h k l CoCl 2-6H 20 CoCl 2-6D 20 001 5.55 5.54 020 4.77 . 4.78 040 3.51 3.50 102 2.91 2.909 301 2.699 2.711 311 2.553 2.555 501 2.398 2.398 571 2.202 2.200 133 2.070 2.074. 312 1.980 1.975 105 1.894 •• 1.891 402 1.858 1.857 204 1.757 1.763 214 1.699 1.696 333 1.604 1.606 The wavelengths of the x-rays are known to an accuracy part i n 10^ which- i s the t h e o r e t i c a l accuracy of the x-ray methods. This accuracy i s not experimentally r e a l i z e d because of the subjective random errors i n l o c a t i n g the true centre of symmetry. Further errors are caused because the p o s i t i o n of the maximum i n t e n s i t y i n a l i n e may not be the 34 centre of gravity of the l i n e . This causes an erro r of 0.003%. The f i n i t e s i z e of the specimen, the lack of c o l l i m a t i o n , etc. cause s t i l l more e r r o r s . In p r a c t i c e the accuracy i s determined experimentally from the photographs. I f the camera can resolve the doublet the accuracy i s 0.005%, i f they are fuzzy the accuracy f a l l s to 0.02%. A furt h e r check i s provided by the back r e f l e c t i o n l i n e s . I f they are fuzzy the accuracy i s about 1%. In our photographs we couldn't see the back r e f l e c t i o n l i n e s and the accuracy determined roughly by comparing various d ^ ^ distances i n the two cases i s about 0.2% which i s between 1% and 0.02% quoted above. 35 . CHAPTER IV Discussion of the Si g n i f i c a n c e of the S h i f t i n TJJ 4.1 Exchange and Super Exchange Interactions  Exchange In t h i s s ection we give a general discussion of the i n t e r -actions, which go by the generic name of exchange i n t e r a c t i o n s , between the unpaired electron spins. No attempt i s made to give a f u l l review of the subject. However, the complexity of the problem i s pointed out. and some recent r e s u l t s relevant to our experiments are discussed. We can get a good idea of the mechanism by considering two unpaired electrons - 1 and 2, belonging j o i n t l y to a.pair of i d e n t i c a l n u c l e i A and B. In the absence of nucleus B and ele c t r o n 2, ele c t r o n I can be described by a s p a t i a l wave function < 9 ^ ( l ) such that ?tk J*A(1) = E ^ A ( l ) S i m i l a r l y f o r an i s o l a t e d system of nucleus B and e l e c t r o n 2 1 B ^ ( 2 ) = E ^ B ( 2 ) ' The b i c e n t r i c system of n u c l e i B and A and electrons 1 and 2 can then be constructed by br i n g i n g the two atoms together. I f the i n t e r a c t i o n between the two atoms i s small, the system would be described by a wavefunction Y / = ^AC 1) S*"B(2) f o r which c f A + 7 r B ) ^ = 2 E ^ Now according to quantum mechanics the electrons are i n d i s t i n g u i s h a b l e ; therefore, interchanging electrons should leave the system undisturbed. The n u c l e i A and B being i d e n t i c a l , V = ^ A ( 2 ) ^ B ( I ) 36 should be as good a so l u t i o n a s " ^ = )^ ^ ( l ) ^ B ( 2 ) . The functions \js and'y/ have the same eigen value;therefore they have what i s commonly known as 'exchange degeneracy'. As the two atoms are brought c l o s e r , the e l e c t r o s t a t i c forces between electrons and n u c l e i perturb the system. The correct wave functions of the system w i l l now be a l i n e a r combination of Y'and yy i n which ^ and are equally weighted because of the symmetry of the problem. The electrons, however, are fermions, therefore the wave function of a system of electrons should be antisymmetric. Keeping t h i i n mind two l i n e a r combinations of the s p a t i a l functions and""^ are possible. V \ = ^ A ( l ) ^ B(2) +. A ( 2 ) ^ B ( l ) (4.1) a n d ^ . = < ^ A ( l ) <^ B(2) - ^ A ( 2 ) y ^ B ( l ) (4.2) 'Y'+ being symmetric the spin function °)C (1, 2) associated with i t w i l l have to be antisymmetric. An antisymmetric^*(1, 2) f o r two electrons requires the spins to be a n t i p a r a l l e l to each other, hence / >j/ / + describe the s i t u a t i o n when the two spins are oppositely d i r e c t e d . For the same reason represents the system when both spins point i n the same d i r e c t i o n . A s o l u t i o n of the problem = E+ + , gives eigen values E, = 2E + Q - J (4.3) . 1 1 ± S A B 2 where the Hamiltonian of the composite system i s Jl = ^ ( v 2 + v 2 } _(_!_•+ i + J L + J : — - i ) (4.3a) </ 2K i 2.J v r A 1 r B 1 r A 2 r B 2 r 1 2 R ' i n Hartree u n i t s . 37 = ^ A ( i ) | ^ i ) > ^ (4.4) Q = .<&(!) <j> *W)\7t- 2E| ^ A ( 1 ) ^ B ( 2 ) > (4.5) J = ^ _ 2 S a b < f A ( 1 ) | r - l j f ^ d ^ +<7*A(1) 7^B(2)| ^ | ^ A ( 2 ) ^ B(D> ( 4 . 6 ) . The distances R, r-^ 2 and r-jjj etc. between the d i f f e r e n t members of the system are as shown below In most cases, even when S^g i s very small, the term 2 S A B ^ ^ ( 1 ) I r~:j-1 ^ ( l ) ^ i n equation 4.6 predominates making J ^O. Under these conditions E ± = ' 2E + Q + J (4.7) and E_ f o r p a r a l l e l s p i n s ^ E+ f o r a n t i p a r a l l e l spins. Thus the a n t i -p a r a l l e l arrangement of spins i s e n e r g e t i c a l l y preferred over the p a r a l l e l arrangement. I t i s seen from 4.7 that d i f f e r e n c e between the energies of states having p a r a l l e l and a n t i p a r a l l e l spin i s E + - E_ = - 2J Hence a bigger value of J would mean stronger spin a l i g n i n g i n t e r a c t i o n s . I f we make E_ the zero of the energy scale, the state i n which spins Sj_ and S 2 are p a r a l l e l (S = S^_ + S 2 = 1) w i l l have zero energy and the state i n which spins are a n t i p a r a l l e l (S = Si - S 2 = 0) w i l l have energy = - 2J. Using spin operators Sj_ and S 2 the energies of the two 38 states can be expressed by the r e l a t i o n E = - J (-\ + 2S 1-S 2) " . (4.8) y\ 3 Eigen values of 2Si'S2 being +^ f o r p a r a l l e l spins and - f o r a n t i -p a r a l l e l spins, the state with a n t i p a r a l l e l arrangement w i l l be lower than the state with p a r a l l e l arrangement by 2J. The. constant -\ i n the above expression doesn't influence the r e l a t i v e energy of the two s t a t e s . We can s h i f t the zero of the.energy such that E = - 2 J Si-Is 2 (4.9) This i s the celebrated Heisenberg Dirac Van-Vleck Int e r a c t i o n which has been extensively used i n the theory of f e r r o - and antiferromagnetism. From the foregoing.discussion i t i s c l e a r that t h i s i n t e r -action i s c l o s e l y r e l a t e d to the i n t e r a c t i o n s associated with chemical bonding. The d i f f e r e n c e E_ - E + = - 2J a r i s e s because there i s a c o r r e l a t i o n between the spin configuration and the e l e c t r o s t a t i c energy of the system r e s u l t i n g from the P a u l i Exclusion p r i n c i p l e . I t i s probably t h i s intimate connection between the chemical bond energies and the spin configurations which l e d Haseda to guess a r e l a t i o n between Hydrogen bond energies and the antiferromagnetic exchange. Super Exchange A c t u a l l y , there i s no fundamental d i f f e r e n c e between exchange and super-exchange i n t e r a c t i o n s . The p r e f i x "super" was applied to the large exchange i n t e r a c t i o n which e x i s t s between unpaired spins i n c r y s t a l s i n s p i t e of the large distance between spins. Because of the large 39 separation between neighbouring spins there would seem to be no v i o l a t i o n of the P a u l i exclusion p r i n c i p l e even i f the nearest neighbour spins were p a r a l l e l to each other. However, as pointed out by Kramers (1934), the extension of the range of exchange i n t e r a c t i o n s i n s o l i d s seems reasonable i f one takes in t o account the diamagnetic ligands present i n the space between the spins. The ligands influence the unpaired electrons i n such a way that t h e i r wave functions get spread over rather long distances, g i v i n g a f i n i t e S^g s i m i l a r to the one i n equation (4.4). There i s a f i n i t e J between the nearest neighbouring spins and the a n t i p a r a l l e l configuration r e s u l t s . Thus, from t h i s point of view there i s no d i s t i n c t i o n between exchange and super-exchange. For the a c t u a l c a l c u l a t i o n the formulation has to be generalized from the two-body problem discussed e a r l i e r !to the many body problem of a c r y s t a l . A scheme f o r the s o l u t i o n of the problem i n two stages has been presented by Anderson (1963). a) One e l e c t r o n wave function of the e l e c t r o n are c a l c u l a t e d using the Hartree Fock s e l f consistent f i e l d technique. The e l e c t r o n i s supposed to be subjected to a p e r i o d i c p o t e n t i a l of ligands as w e l l as the core electrons of the whole c r y s t a l l a t t i c e . The e f f e c t of the ligands i n t h i s stage i s absorbed i n the v i r t u a l spreading of the wave functions of the magnetic electrons. b) Using the wave functions obtained i n stage a, the d i f f e r e n c e between the energies of the states with p a r a l l e l and a n t i - p a r a l l e l nearest 40 neighbour spins i s c a l c u l a t e d . This d i f f e r e n c e according t o Anderson (1963) i s where n, n +f describe the l a t t i c e s i t e s separated by displacement 7*, and m, m constitute a p a i r of magnetic electrons situated at s i t e s n and n + f . The quantity b i s the matrix element of the s e l f consistent Hartree-Fock Hamiltonian of the c r y s t a l connecting the two one e l e c t r o n states f o r which the i n t e r a c t i o n i s considered. (J denotes the energy d i f f e r e n c e of the e l e c t r o n i n the state l o c a l i z e d at n with the one i n which the e l e c t r o n i s t r a n s f e r r e d to state l o c a l i z e d at n +.*y . The procedure may seem simple i n p r i n c i p l e but the a c t u a l c a l c u l a t i o n s i n the stage (a) are very cumbersome. The theory only gives an order of magnitude agreement with the experiment. In r e p l a c i n g hydrogen by deuterium we have changed the l i g a n d therefore within Anderson's scheme, the e f f e c t of deuteration has to be introduced i n the stage (a) of the c a l c u l a t i o n s . We have not attempted such a computation i n t h i s work. 4.2 The Hydrogen Bond According to Pimentel and McClellan (1960), a hydrogen bond e x i s t s between a f u n c t i o n a l group A-H and an atom or a group of atoms B, i n the same or d i f f e r e n t molecule, when . ° (a), there i s evidence of bond formation, and 41 (b) there i s evidence that t h i s new bond l i n k i n g A-H and B s p e c i a l l y involves the hydrogen atom already bonded to A. The r e s u l t i n g structure i s symbolically represented as A-H — — B The evidence mentioned i n (a) i s provided by the bond energy (2f 6 K cal/mole) which i s usually needed i n order to detach B from the structure A-H B. T y p i c a l values of the energy associated with these bonds l i e midway between the energy of an ordinary chemical bond and that of the Van-der-Waals i n t e r a c t i o n . There are ex c e l l e n t chances f o r the formation of t h i s bond whenever an e l e c t r o - p o s i t i v e hydrogen i s situated between two electro-negative atoms A and B, such that the distance between A and B Is l e s s than the sum of t h e i r Van-der-Waal r a d i i . A and B can be electro-negative atoms l i k e oxygen, c h l o r i n e , F l u o r i n e , e t c . The condition (b) above i s usually s a t i s f i e d i f the i n f r a r e d frequency of the st r e t c h i n g mode i n the f u n c t i o n a l group A-H doesn't change r a d i c a l l y when A-H i s involved i n a hydrogen bond. Hydrogen i n such cases i s usually c l o s e r to A. There are some hydrogens i n CoCl2*6H20 which are present i n the paths of super exchange and at the same time q u a l i f y to be part of a hydrogen bond according to the above c r i t e r i o n . According to Haseda (1962), some super exchange paths are as follo w s : C Q + 2 CI CI. C0+2 . (I) 2.47 A° 4.10 A 0 2.48 A 0 . C 0 + 2 CI H-0' C 0 + 2 . ( I I ) 2.48 A 0 3.08 A 0 2.2 A° 42 C 0 + 2 0-H2•_ . H-O-H , _ H9-0 C 0 + 2 ( I I I ) 2.47 A 0 2.72 A 0 . 2.72 A 0 2.2 A 0 and C 0 + 2 0-H2_ _ _ _ H 20 C 0 + 2 (IV) 2.21 A 0 3.62 A 0 2.21 A°-The hydrogen bonds i n these are shown by the notations • A-H B As mentioned by Pimentel (1960) i n Chapter 8 of "The Hydrogen Bond', the theory of t h i s bond i s i n a very u n s a t i s f a c t o r y p o s i t i o n . The quantum mechanical c a l c u l a t i o n s based on the resonance method and the molecular o r b i t a l technique have u n c e r t a i n t i e s bigger than the entire, energy of the hydrogen bond. Such theories have only a l i m i t e d q u a l i t a t i v e s i g n i f i c a n c e and at the most provide a language f o r d i s c u s s i o n . An a l t e r n a t i v e approach i s to consider the hydrogen atom i n the p o t e n t i a l of the electro-negative atoms. A and B i n a semi-empirical fashion. At times combinations of these methods have been used to 'explain' some phenomena q u a l i t a t i v e l y . The e f f e c t s of deuteration on the f e r r o e l e c t r i c c u rie point and on the distances between A and B atoms have been summarized by Ubbelohde and Gallagher (1955). These deuterium i s o t o p i c e f f e c t s are generally explained on the b a s i s of changes i n H-bond energy and amplitude of v i b r a t i o n r e s u l t i n g from d i f f e r e n t zero point energies of 0-H and 0-D bonds. This i s presumed to r e s u l t i n a greater 'overlap' i n the H-bond. Anderson's theory as mentioned i n section 4.1, then, throws some l i g h t on the r o l e of the deuterated hydrogen bond i n the antiferromagnetic state 43 of CoCl 2'6H 20. In stage (a) of Anderson's scheme, the net e f f e c t of the ligands i s to extend the s p a t i a l range of the wave functions of the magnetic electrons. The deuterated hydrogen-bonded ligands could, p o s s i b l y , extend t h i s range a l i t t l e f u r t h e r because of t h e i r increased 'overlap' caused by a reduction i n t h e i r zero point v i b r a t i o n a l energy. This could then r e s u l t i n a stronger super-exchange. However, i n Section 4.3 we have t r i e d to understand the e f f e c t of reduction i n the zero point energy on the super-exchange q u a n t i t a t i v e l y i n a semi-empirical way. We approximate the e f f e c t of the electro-negative atoms of the hydrogen bond by a p o t e n t i a l i n which the hydrogen atom moves and assume that the super-exchange has the same f u n c t i o n a l dependence on the p o s i t i o n of the H-atom as the p o t e n t i a l energy. The e f f e c t of deuterating the hydrogen bond on the super-exchange has been hinted e a r l i e r i n l i t e r a t u r e (Meisenheimer and \ . . \ Swalen, 1961). I t has been observed that both HCr0 2 and DCr0 2 follow a curie-weiss law 9 being - 276°K f o r H C r 0 2 and - 214°K f o r DCr0 2. This requires a strengthening of the super-exchange on deuteration. The authors have proposed a super-exchange linkage of the type Cr — 0 — H — 0 — Cr However, t h i s i n t e r p r e t a t i o n can be held i n doubt because Ibers and Hamilton (1961) have f a i l e d to observe the antiferromagnetic order i n 44 HCrO^ by,neutron d i f f r a c t i o n , even at 4.2°K. A lower Neel. temperature of HCrC^ would warrant i j r - ^ 50 which seems very improbable f o r a structure l i k e HCM^. Our experiment on CoCL^H^O shows t h i s e f f e c t . more conclusively because we have measured the changes i n super-exchange by observing the changes i n Neel temperature at which the para- to antiferromagnetic phase t r a n s i t i o n a c t u a l l y takes place. 4.3 A Model f o r the E f f e c t of Isotopic S u b s t i t u t i o n on the Neel Temperature  In the experimental r e s u l t s described i n the previous chapter, i t was shown that when the protons i n CoC^^H^O are replaced by deuterons, an observable change i n the Neel temperature TJJ occurs. This change was a c t u a l l y predicted by Haseda, as discussed i n the previous section, on the basis of the known influence of deuteration on hydrogen bonding and the existence of a possible close r e l a t i o n s h i p between hydrogen bonding and super-exchange. Some of the super-exchange paths between neighbouring C 0 + + ions i n C o C L ^ ^ O do, i n f a c t , involve Hydrogen atoms as shown i n section 4.3. I t i s obviously of great i n t e r e s t to t r y to account f o r the very large magnitude of the observed change i n T^ upon deuteration. I t i s not c l e a r to us, however, how the postulated r e l a t i o n s h i p between hydrogen bonding and super-exchange can be used d i r e c t l y to make such a qua n t i t a t i v e p r e d i c t i o n . Dr. David Pink has proposed i n a private communication that the e f f e c t of deuteration on T^ can be understood i n terms of the change i n the amplitude of l a t t i c e v i b r a t i o n s as the i s o t o p i c mass of an intervening atom i n the path of super-exchange i s v a r i e d . We 45 now describe a crude model which enables us to estimate the magnitude of t h i s e f f e c t . The antiferromagnetic alignment i s assumed to a r i s e from the Heisenberg-Dirac Hamiltonian f o r the exchange i n t e r a c t i o n between p a i r s of spins which, aside from an additive constant, i s V e x = " 2 J i j S i ' S j (4.11) As discussed i n section 4.1.the presence of non-magnetic X atoms between magnetic atoms and i n a s o l i d gives r i s e to so c a l l e d "super-exchange"' contributions to. -j. In order to make some quantitative predictions about the e f f e c t of v a r i a t i o n of the i s o t o p i c mass of the X atoms on T^ we now make some s i m p l i f y i n g assumptions. 1. I t i s often assumed t h a t . J ^ j = 0, except f o r nearest neighbours, f o r which J- • = J - - | J / . For t h i s case, the spin system i s antiferromagnetic below a temperature Tjj.and paramagnetic above T^, where t ' k T N = C l J i = - CJ (4.12) and.. C i s a constant. . • * Although the c r y s t a l structure of CoGL^^^O i s too complicated f o r the above assumption about J . - to hold, we s h a l l use equation (4.12) and i n t e r p r e t J lo o s e l y f o r complicated systems as a measure pf the t o t a l exchange i n t e r a c t i o n between any spin i on one sub-l a t t i c e with the nearest neighbour spins j on the other s u b - l a t t i c e . 2. The exchange i n t e r a c t i o n i s assumed to be due s o l e l y to super-exchange (Anderson, 1963) which i s assumed to a r i s e from a second l a t t i c e of non-magnetic atoms X superimposed upon the l a t t i c e of magnetic atoms M. 46 Since the d i r e c t exchange between any p a i r of M atoms i s assumed to be n e g l i g i b l e , the exchange i n t e g r a l J]_ 2 between'M^ and M 2 i s a function of the distances between M-j_ and M 2 and the intervening atom X as shown below J 12 = J 1 2 ( 5 > , ' (4.13) of X 3. The mass of an M atom i s assumed to.be much l a r g e r than that of an X atom. Thus M-j_ and M 2 may be taken to be stationary and J^_ 2 i s a function only of the displacement of the X atom from equilibrium J 12 = J12(<D (4-14) 4. The treatment of the l a t t i c e v i b r a t i o n s i s s i m p l i f i e d by assuming that the X atoms v i b r a t e independently of each other i n a simple harmonic motion and only along the l i n e j o i n i n g the two nearest neighbour M atoms. The con t r i b u t i o n of J-^ 2 to J i n equation (4.12) i s obtained by averaging J j _ 2 over the v i b r a t i o n s of the X ' atoms at the temperature of the s o l i d . J ^ 2 ( q ) i s expanded i n a Taylor s e r i e s i n q J l 2 ( ( l ) = J 1 2 ( ° ) + 1 j[l\o) +p j g ) ( 0 ) + - _ _ (4.15) . where 47 J 1 2 ( n ) ( 0 ) . = - d n j 1 2 ^ ) f (4.16) Averaging over q, the odd terms vanish by symmetry, and the leading terms are 1 J 1 2 ( 0 ) = J 1 2 ( 0 ) + J l 2 ( 2 ) ( 0 ) < q 2 > + (4.17) Low temperature approximation: At low temperatures, X i s i n i t s ground v i b r a t i o n a l state so that 4 q 2 > = ~T~~I (4.18) £ 2 mS where ^ = m co2 i s the force constant associated with the p o t e n t i a l w e l l f o r the X atom near the equilibrium p o s i t i o n (q = 0), m i s the mass of X and oJ i s the angular frequency of v i b r a t i o n of the X atom. I f the p o t e n t i a l i s unaffected by changes i n i s o t o p i c mass of X, y, i s unchanged. Therefore, (2) < J i 2 ( 0 > m - J 1 2 ( o ) ( i + i ^ ^ . + — I ( 4 . 1 9 ) ; I J12(°) ^ 2 m 2 j I f X atoms of two d i f f e r e n t i s o t o p i c modifications have masses m]_. and m2 r e s p e c t i v e l y , the differe n c e i n t h e i r Neel temperatures i s given by the following, f o r small changes A T v . g . < J 1 2 ( q ) ) m a - < . J i 2 ( q ) > ^ , ^ • "... < J i 2 ^ L J 1 2 2 ) ( ° ) ^ • f/m,.^ - - (4-2o) J 1 9 ( 0 ) 1 2 ^ ' ; " ^ - 2 mj-2 L> m2 48 4.4 A p p l i c a t i o n of the Model to CoC^^H^O Using Haseda's Conjecture ' We now apply the model described i n the preceding section to a c a l c u l a t i o n of the f r a c t i o n a l change i n i n C o C ^ ^ ^ O when the protons are replaced by deuterons. To i l l u s t r a t e the e f f e c t of deuteration consider one of the super-exchange paths described i n section 4.2 Co — CI H 0 — Co M l X M 2 We pi c t u r e the Co, CI and 0 atoms as being stationary and the H-atom as v i b r a t i n g along the 0-C1 axis. - In order to c a l c u l a t e the value of J-^CQ.)? "the e f f e c t of the CI and 0 atoms on the magnetic electrons would have to be taken i n t o account e x p l i c i t l y using the ligan d f i e l d theory method (Anderson, 1963, page 34). Once Ji2(Q.) n a s been ca l c u l a t e d , the change i n the average value of J i 2 ( q ) with replacement of the proton by a deuteron i s given by equation (4.19) with m^/n^ =0.5 and m-^  = 1.67 x 1 0 - 2 4 gms. -For CoCl2*6H20, the str e t c h i n g frequency associated with the 0-H band was measured (Figure 3.8) to be CO = 2T c = 6.5 x 1 0 1 4 s e c - 1 I f the above path i s s o l e l y responsible f o r the exchange i n t e r -action between Mj_ and M2, then the f r a c t i o n a l change given i n Tj^ i s re l a t e d to J]_2(q) by equation (4.20) using the parameters given above. Haseda's Conjecture: As discussed e a r l i e r , Haseda conjectured that there i s a r e l a t i o n s h i p between the strength of the Hydrogen bond and the strength of the super-exchange i n t e r a c t i o n . The simplest such r e l a t i o n s h i p i s that 49 J12^ = constant (4.21) V(q) where V(q) i s the p o t e n t i a l energy of the proton as a f u n c t i o n of i t s displacement from equilibrium. I f t h i s r e l a t i o n s h i p holds, equation (4.20) gives A % , , v ( 2 ) ( ° ) * - j r . - (-0.293) f V ( 0 ) • = - (0.293) k^ff (4.22) e v ( o ) since , g , v( 2)(0) = (Appendix A). This expression i s simply the f r a c t i o n a l change i n the zero point energy. A t y p i c a l value of the minimum p o t e n t i a l energy of the proton i n a hydrogen bond (Reed, 1959) i s eV(0) = - 1 e.v. = - 1.6 x 1 0 1 2 ergs. Using the measured value of and t h i s value of eV(0), equation (12) gives A T N & + o . 06 This exact agreement with experiment must be f o r t u i t o u s since we have neglected super-exchange paths which do not involve Hydrogen atoms and we have neglected v i b r a t i o n s perpendicular to the d i r e c t i o n of the Hydrogen bond. However, the f a c t that the corre c t sign of A T^ i s obtained i s s i g n i f i c a n t as i s the f a c t that the model c e r t a i n l y gives > order of magnitude numerical agreement even i f a d d i t i o n a l f a c t o r s , neglected 50 here, are taken i n t o account. The t h e o r e t i c a l problem which i s suggested by t h i s agreement i s to j u s t i f y the use of equation (4.21) f o r small values of q. 4.5 Relationship of the Isotope S h i f t to Other Experiments We have seen i n the previous section that the s h i f t i n T^ with change i n i s o t o p i c mass i s s e n s i t i v e to the f u n c t i o n a l dependence of J on inter-atomic separations. The dependence of J on inter-atomic distance influences other experimental parameters (Bloch, 1966), e.g. the pressure dependence of T^ or of the magnetization, s t a t i c and dynamic magnetoelastic e f f e c t s , the influence of thermal expansion on magnetization or s u s c e p t i b i l i t y . I f J i s assumed to have the form | j | = Aa" n (4.23) where 2a i s the equilibrium separation between magnetic ions on a cubic l a t t i c e , then ^2g_£ = n . (4.24) d l o g V 3 where V. i s the volume of the c r y s t a l (Vc< a ). Using the f a c t that T N * J , j i m 3 d l o g Tw -n = 3 d l Q g N a d P § . ( 4 . 2 5 ) d l o g V i / dV \ V M P / Using a v a i l a b l e data f o r the com p r e s s i b i l i t y (Kaminov Jones, 1961; Bridgeman, 1932; Birc h , 1952) and i n order to obtain an expression f o r the isotope s h i f t i n T^j i n MO, i t i s necessary to c a l c u l a t e b /U i n 51 equation (4.10) (Anderson, 1963) as a function of the displacement of 0 from i t s equilibrium p o s i t i o n . For the NaCl structure, the eq u i l i b r i u m p o s i t i o n i s midway between two M atoms M-^  and M2. C a l c u l a t i o n of the pressure dependence of T^, on the other hand, involves the evaluation of b 2/U when the M-^ 0 and M20 equilibrium distances are each changed by the same amount. This program has not yet been c a r r i e d out. In order to obtain a crude estimate of the isotope s h i f t i n MO, we make the following ad hoc approximation f o r J j 2 , J 1 2 ( r x , r 2 ) = K ( r x ) K ( r 2 ) (4.26) Then following the same procedure as i n section 4.3 the following r e s u l t i s obtained A S ( K ( 2 ) ( a ) _ ( K ( P ( a ) j 2 ? , 2 > O m ^ _ ±1 (4>2?) V I K ( a ) V K ( a ) / P ( W ) - j I f Jc< a - 1 0 , then K ( r ) e < r - 5 . For mx/m2 = 16/17, the f r a c t i o n a l change i n TJJ becomes ^ = - 0.15 —f- (4.28) T N a I t i s anticipated that <^q2> / a ^ ^ ^ l O - 2 , i n which case AT^/Tjj-^.10 - 3, Since t y p i c a l values of T^ f o r the t r a n s i t i o n element oxides are of order 200°K, t h i s would p r e d i c t a downward s h i f t of Tj^ of approximately 0.2°K, which would be easy to detect by s p e c i f i c heat measurements. To obtain a b e t t e r estimate of A T^, i t i s necessary to make a more r e a l i s t i c estimate of J i 2 ( r ] _ , r 2 ) than given by equation (4.26) and 52 to estimate ^ q 2 ^ t h e o r e t i c a l l y . Since the values of Tjyf i n these oxides are not much smaller than the Debye temperatures, i s temperature dependent and would have to be evaluated at T^. I t should also be emphasized here that we have assumed that the M atoms are stationary and that the 0 atoms v i b r a t e independently of each other along the M-j_ - 0 - l i n e . These approximations are not adequate f o r the systems under consideration here. One should develop a formula equivalent to equation (4.27) allowing the atoms to v i b r a t e i n 3 dimensions and taking account of the c o r r e l a t i o n between the v i b r a t i o n s of d i f f e r e n t 0 atoms and between M and 0 atoms. 53 BIBLIOGRAPHY Anderson P.W., 1963, Magnetism, V o l . 1, eds. G.T. Rado and H. Suhl (Academic Press). B i r c h F., 1952, J . Geophys. Res., V o l . 57, p. 227. Bloch D., 1966, J . Phys. Chem. S o l i d s , V o l . 27, p. 881. Brldgman P.W., 1932, Proc. Am. Acad. A r t s . S c i . V o l . 67, p. 365. Chiba T., 1964, J . Chem. Phys., V o l . 41, p. 1352. E l S a f f a r Z., 1962, J . Phys. Soc. Japan, Vol. 17, p. 1334. Garwin R.L., 1959, Rev. S c i . In s t r . V o l . 30, p. 105. Groth P., 1906, Chemische Krystallographie 1, T e i l p. 248, (Wilhelm Engleman Verlag, L e i p z i g ) . Haseda T., 1960, J . Phys. Soc. Japan, Vol. 15, p. 483. Haseda T., and Kanda E.,1957, J . Phys. Soc. Japan, V o l . 12, p. 1051. Ibers J.A. and Hamilton W.C., (quoted by Mexseheimer and Swalen as priva t e communication). Kaminov L.P. and Jones R.V., 1961, Phys. Rev., V o l . 123, p. 112. Megaw H.D., 1954, Acta, Cryst. V o l . 7, p. 187. Meiseheimer R.G. and Swalen J.D., 1961, Phys. Rev. V o l . 123, p. 831. Moriya T., 1962, Progress of T h e o r e t i c a l Physics, V o l . 28, p. 371. Mizuno J . , Ukai K. and Sugawara, T., 1959, J . Phys. Soc. Japan, V o l . 14, p. 383. Nagamiya T., Yoshida K. and Kubo R., 1955, Advances i n Physics, V o l . 4, No. 1. Nakamura T., 1962, Phys. Rev. V o l . 128, p. 2500. Pink, D., 1966, Private communication. Poulis N.J. and Hardeman G.E.G., 1952, Physica, V o l . 18, p. 201. Salem L., 1963, J . Chem. Phys., V o l . 38, p. 1227. 54 Sawatzky E. and BloomM., 1964, Can. J . Phys., V o l . 42, p. 657. Sl a t e r J.C. 1941, J. Chem. Phys. V o l . 9, p. 16. Snyder R.G. and Ibers J.A., 1962, J . Chem. Phys. V o l . 36, p. 1356. Spence R.D., Middents P., E l Saffar Z. and Kleinberg R., 1964, J. App. Phys. V o l . 35, p. 854. Van-der-Lugt W. and Poulis N.J., 1960, Physica, V o l . 26, p. 917. i \ 55 APPENDIX A Nuclear Magnetic Resonance Spectrum of Deuterium i n CoCl2'6D2U at 4.2°K F i g . 3.7 shows the change i n deuteron frequencies as the external magnetic f i e l d H Q i s rotated i n the a-c plane perpendicular to b-axis. The main use of t h i s study i s i n s e l e c t i n g s u i t a b l e l i n e s f o r the Neel temperature measurements. However, with some a d d i t i o n a l information a v a i l a b l e , the data i n f i g . 3.7 can be used to ( i ) compare the d i r e c t i o n s of some 0-D bonds with 0-H d i r e c t i o n s as reported by E l S a f f a r (1962) and ( i i ) to determine the quadrupole coupling constant of some of the deuterons i n CoCl2*6D20. The Hamiltonian of a nucleus of Spin I and e l e c t r i c quadrupole' moment Q cm2 i n an external f i e l d H 0 i s % = - A H 0 li + 4 & x ^ _ l } ^ 3 I 2 Z - I ( I + 1 ) J (A-l) • where q i s the value of the f i e l d gradient " 0 at the nucleus and~f i s the gyromagnetic r a t i o of the nucleus. In ( A - l ) , (x,y,z) are along the p r i n c i p a l axes of the . f i e l d gradient and the magnetic f i e l d i s along the d i r e c t i o n z' making an angle Q with the z-axis. We have also assumed the f i e l d gradient to be a x i a l l y symmetric, which i s not generally true. When'T^H^ I^^eqQ, we can consider the nuclear spin to be quantized i n the z' d i r e c t i o n . Defining the x' axis to l i e i n the z-z' plane . • I z = I z , cos a + I x t sin© (A-2) Using t h i s transformation i n (A - l ) and solving the problem by f i r s t order perturbation theory, we get the eigen values 56 m being the p r o j e c t i o n of the spin along the z'-axis. However, i n our c r y s t a l there are also i n t e r n a l magnetic f i e l d s produced by the magnetic moments and the current d i s t r i b u t i o n s i n the environment, hence H Q above has to be replaced by Hj_ o c. For spin I = 1 of. deuterons. the frequencies corresponding to the t r a n s i t i o n s , between the eigen state of w i l l be v* - - TH 1 O C 1 M J L ( » - y - n ( A ^ j IT? - *u 1 ' In case the perpendicular to the plane of r o t a t i o n of makes an angle C< with the z-axis, the equation (A-4) becomes V + = " Z H l o c + 3 s i n V c o s 2 0O) - l j j A _ 5 ) ^ 4h ^ - 2 J where$ = 0Qi.s the d i f f e r e n c e between angles which H Q and the p r o j e c t i o n of the z-axis i n the plane of' r o t a t i o n , make with a f i x e d d i r e c t i o n i n the plane of r o t a t i o n . A d d i t i o n a l complications a r i s e i n the case, of an asymmetrical f i e l d gradient. To solve the problem completely would require r o t a t i o n diagrams l i k e f i g . 3.7 about other axes too. However, we can get some information about Q, q, R i o c and the d i r e c t i o n of the axes of maximum f i e l d gradient from other sources. Using these data i n (A-5), spectra f o r some deuterons can be c a l c u l a t e d and the r e s u l t s compared with the experiment. The quadrupole moment Q of deuteron i s known to be 2.77 x 10~27 c m 2 _ f h e values of other parameters have been obtained as follows: ( i ) *f ^ l o c i s ava i l a b l e from the work of Sawatzky and Bloom (1962) on the proton n.m.r. of CoCl 2*6H 20 sing l e c r y s t a l s ( f i g . 3.3). The frequencies i n f i g . (3.3) w i l l be reduced i n our case by a r a t i o 1 deuteron (H„) deuteron /» , N = -=c ^-2l ( A- 6) J p r o t o n ( H q ) p r o t o n where (H 0) proton = 4970 gauss. As i n the• paramagnetic phase the i n t e r n a l magnetic f i e l d s increase l i n e a r l y with, the external magnetic f i e l d s H 0, (^loc) deuteron ^ g app r 0 Xxmated w e l l by (Hioc) Proton (H 0) deuteron (H 0) proton These reduced frequencies correspond to dotted curves i n f i g . 3.7 and would constitute the n.m.r. spectrum of deuterons i f t h e i r quadrupole moment were zero. In these c a l c u l a t i o n s the small nuclear dipole s p l i t t i n g has been neglected, because i t would be f ^"deuteron \^ 2 1 reduced by a f a c t o r — ~ J £5 "42" a n d hence would not be resolved. ' 1 ^ 2 V ( i i ) Maximum f i e l d gradient q = - : The value of t h i s parameter can be deduced from the i n f r a r e d 0 - D stretching frequency (4.765 x 10^ 4 sec--'-). This value has been obtained from the observed i n f r a r e d spectrum of CoCl2*6D20 i n f i g . 3.8. As established by Chiba (1964), the axis of maximum f i e l d gradient i n the 0 - H group usually l i e s close to the l i n e j o i n i n g 0 and H. This i s also the d i r e c t i o n i n which the str e t c h i n g v i b r a t i o n takes place. I f the nucleus moves by distance A z from i t s mean p o s i t i o n , i t s p o t e n t i a l energy at the new p o s i t i o n w i l l be ' 58 U ( A z ) =eV(0) + ^f^ ( A z ) 2 V — - (A-7) The term l i n e a r in.A z vanishes at the equilibrium p o s i t i o n . I f the p o t e n t i a l (A-5) has to account f o r the simple harmonic motion of the i n f r a r e d s t r e t c h i n g mode then . 2 e l ' 2 ) = $ = m ^ 2 (the Hooke's Law constant of the (A-8) yfz ) 0 motion) From the more exact quantum mechanical r e s u l t obtained by Salem (1963), the c l a s s i c a l r e l a t i o n (A-8) has been shown to be quite accurate. From the observed value of t*> i n f i g . 3.7 f 3 2V\ -T - , .2 e f - ^ — 1 = v = mcj = (3.34 x 1 0 - 2 4 ) ( 2 77x 3 x 1 0 1 0 x 2520) 2 = 0.758 x 10 6 dynes cm - 1. ( i i i ) The d i r e c t i o n s of axes of maximum f i e l d gradient. -> '2 (a) The vector A j o i n i n g OJ-J- and '%•]• l i e s i n the a-c plane making an angle 32°.with the c-axis. (b) According to E l S a f f a r the vector j o i n i n g the protons H-^ 2 • ' and HJ J belonging" to 0-j-j l i e s i n the a-c plane. Hence —> q_ the vector B j o i n i n g OJ-J- with H ^ should also be i n the o -=> a-c plane making an angle of^/103 with A. As already mentioned, according to Chiba (1964), A and should point i n t h e . d i r e c t i o n s of maximum f i e l d gradient f o r deuterons l y i n g at the p o s i t i o n of HJ-J- and H^ -j- . Sawatzky and Bloom (1962) have assigned the doublet no. 1 i n f i g . 3.2 f o r these s i t e s on the basis of i n t e n s i t y a n a l y s i s . Therefore • 59 ^ l o c J ^ o r t ^ l e s e s i t e s c a n D e c a l c u l a t e d from doublet no. 1 of f i g . 3.2. The deuteron has a quadrupole moment Q = 2.77 x 10-27. c m 2 a n ( j •e ( TT) = 0.758 x 10 6 dynes cm - 1. Using, these data i n (A-5), we get curves marked I I , f o r deuterons at the s i t e HJ-J-2 and I I 2 f o r the deuterons at the s i t e Hjl"'". These curves are consistent with the experimental values of the frequencies p l o t t e d i n f i g . 3.7. The frequencies near 3.04 Mc when H 0J_ c-axis were f i r s t neglected because of poor s i g n a l to noise r a t i o . They were l a t e r retained because ( i ) they are strongly suggested by theory and ( i i ) they re-occur i n spectra f o r d i f f e r e n t o r i e n t a t i o n s over a neighbourhood of 30° around the d i r e c t i o n X c-axis. These frequencies are in d i c a t e d by small t r i a n g l e s i n f i g . 3.7. I t might also be mentioned, that the c r y s t a l alignment i s accurate only up to 5° and there might be a small asymmetry i n the f i e l d gradients. S t i l l the agreement with the experiment can be considered good because the information obtained from various sources i s consistent with the po s i t i o n s of deuterons i n the c r y s t a l . 

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