THE MAGNETIC SUSCEPTIBIUTY OF SOME THlGUREA-COORDIMTED TRANSS'HQN MEFAL HAUDES EN THE PARA-MAGNETIC AND ANTiFERRO - MAGNS‘E‘EC SYAHS Thesis for éhe Degree of Ph. D. MICHIGAN SMR‘E UNEVERSETY Howard 35); Van Till 1965 [finals 0-169 Date This is to certify that the thesis entitled THE MAGNETIC SUSCEPTIBILITY OF SOME THIOUREA-COORDINATED TRANSITION METAL HALIDES IN THE PARA-MAGNETIC AND ANTIFERRO-FRGNETIC STATES presented by Howard Jay Van T111 has been accepted towards fulfillment of the requirements for Ph. D. degree intheiee July 25, 1965 Major protestr ‘3 A . .-' “was! 693.3. ear-w. I .5" 3:3 7e ,4 .12 V l t» ,- '. . -"‘ - .rg'::;1 stare 4. ity . . 1' .‘ . ‘ ‘ I. - ‘1' . ‘.‘Y ;.;. - ~,‘ w-u-t' - ABSTRACT THE MAGNETIC SUSCEPTIBILITY OF SOME THIOUREA-COORDINATED TRANSITION METAL HALIDES IN THE PARA— MAGNETIC AND ANTIFERRO- MAGNETIC STATES by Howard Jay Van Till Magnetic susceptibility measurements were made on single crystal and powdered samples of thiourea-coordinated transition metal halides having the chemical formulas e;;, c 00012 4LKNH2)2 Each of these materials was observed to undergo a transition CS], MnC12‘4[(NH2)2CS], and NiBr2’6[(NH2)2CS]. from the paramagnetic to the antiferromagnetic state at low temperatures: On the basis of the behaviour of the magnetic susceptibility an attempt has been made to identify the directions of the sublattice magnetizations, and a comparison of the measured values of the parallel suscepti- bility has been made with the results of recent calculations based on the Ising model‘ The magnetic susceptibility of all samples was measured by the ac mutual inductance method using a commercial mutual inductance bridge operating at 17 cps. The temperature of the sample was determined by measuring 3 the vapor pressure of the He or He“ bath in which the sample was immersed, HOWARD JAY VAN TILL The results of the measurements on Co(tu)uCl2 indi— cate that it has an antiferromagnetic transition at O.93°K. The sublattice magnetization vectors lie along or near to the c axis of this tetragonal crystal and a simple two- sublattice model is investigated. The behavior of the magnetic susceptibility of Mn(tu)uCl2 shows that this tetragonal crystal has an anti- ferromagnetic transition temperature of O.56°K. However, the sublattice magnetization vectors do not lie near the c axis as was the case for Co(tu)uCl A two-sublattice 2. model in which the magnetization vectors lie in the ab plane is suggested but leads to some difficulties which can only be resolved by measurements at lower temperatures. The thiourea-coordinated nickel bromide, Ni(tu)6Br2, exhibits the most unusual behavior of the three materials here reported. The susceptibility data leads to the con— clusion that this material has two transitions of the antiferromagnetic type at 2 OOK and 2.20K respectively. This conclusion is supported by specific heat data which shows peaks in the specific heat at these temperatures. Furthermore, along none of the principal magnetic axes does the susceptibility approach zero at low temperatures. Thus a simple two-sublattice model can be eliminated immediately and a four—sublattice model, with the magneti— zation vectors lying in the ac plane of this monoclinic crystal, is suggested. THE MAGNETIC SUSCEPTIBILITY OF SOME THIOUREA-COORDINATED TRANSITION METAL HALIDES IN THE PARA- MAGNETIC AND ANTIFERRO- MAGNETIC STATES By Howard Jay Van Till Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1965 DEDICATION This thesis is dedicated to my wife and parents who have provided the help, encouragement, and patience to make this work possible. ii ACKNOWLEDGMENTS The author wishes to express his sincere appreciation and thanks to Dr. J. A. Cowen and Dr. R. D. Spence for their invaluable aid and encouragement in this work; to Dr. H. Forstat and Dr. T. O. Woodruff for their suggestions during the preparation of this thesis; to Richard and Carolyn Au for their x ray work and assistance in taking data; to Charles Taylor and George Johnston for their computer work; to the electronic and machine shop personnel for their assistance in the construction of the apparatus; and to the National Science Foundation and Army Research Office (Durham) for their support of both the author and this project. 111 TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF Chapter I. II. III. APPENDICES INTRODUCTION EXPERIMENTAL APPARATUS AND PROCEDURE ’IIL'UCJOUHD GeneralMethod - The Mutual Inductance Coils and Bridge He Dewar and Sample . . . Experimental Procedure Data Reduction and Experimental Accuracy Sample Preparation and Chemical Analysis BRIEF DISCUSSION OF EXISTING THEORIES. The Scope of This Discussion Direct and Indirect Exchange-—Early Theories . Exchange Interactions--Modern Theory. Approximations to Interaction Potential. 1. Preliminary remarks . 2. The molecular field approach 3. The Ising model Statistical Mechanics of the Ising Model Formulation of the problem . Low temperature series expansions High temperature series expansions Discussion of the results of Sykes and Fisher. 1‘:me iv Page iii vi viii xi [\JOKONEUU W FJH 15 15 21 21 21 29 3O 3O 33 33 3A Chapter Page IV. EXPERIMENTAL RESULTS AND DISCUSSION . . . . A1 A. Co(tu)uCl2 . . . . . . . . . . . 41 B. Mn(tu)uCl2 . . . . . . . . . . . 514 C. Ni(tu)6Br2 . . . . . . . . . . . 61 D. Others . . . . . . . . . . . . 78 V. CONCLUSION . . . . . . . . . . . . 80 REFERENCES . . . . . . . . . . . . . . . 83 APPENDICES . . . . . . . . . . . . . . . 86 Table 10. 11. 12. 13. 14. LIST OF TABLES Results of chemical analysis of samples Comparison of the transition temperatures of the thiourea- and water—coordinated compounds Parallel susceptibility for honeycomb, plane square, simple cubic, and body—centered cubic lattices as computed from the Ising model series expansions. Magnetic susceptibility data for CO(tu) “012 along the [001] axis in units of cc/mole . . Magnetic susceptibility data for Co(tu)u012 along the [110] axis in units of cc/mole . . Magnetic susceptibility data for Co(tu)uCl2 along the [100] axis in units of cc/mole . . . Magnetic susceptibility data for powdered Co(tu)uCl2 in units of cc/mole . Magnetic susceptibility data for Mn(tu)uCl2 along the [001] axis in units of cc/mole. . . Magnetic susceptibility data for Mn(tu)uCl2 along the [100] axis in units of cc/mole. . . . Magnetic susceptibility data for Mn(tu) “C12 along the [110] axis in units of cc/mole. . . Magnetic susceptibility data for Ni(tu)6Br2 along the b axis in units of cc/mole . . Magnetic susceptibility of Ni(tu) Br along the , . . . , 2 a aXlS In units of cc/mole. X = 0.98xa, a! Magnetic susceptibility of Ni(tu)6 Br along the c axis in units of cc/mole. , _ xc - 60.98xC Magnetic susceptibility of Ni(tu) 6Br2 along the a axis in units of cc/mole. , _ xa - 1.02xa vi Page 14 81 93 97 98 99 100 101 102 103 104 105 106 107 Table Page 15. Magnetic susceptibility of Ni(tu)6 Br 2 along the a axis in units of cc/mole. 6X; _ 1'02Xa . 108 16. Magnetic susceptibility of powdered Ni(tu)6Br2 in units of cc/mole. . . . . 109 vii LIST OF FIGURES Figure l. The geometry of the mutual inductance coils and sample 2. The He3 dewar 3. The effect of a perpendicular magnetic field on the sublattice magnetization vectors. 4. Sketch of X vs. T in the molecular field approximation 5. Energy levels of the two interacting electron problem (a) using the Heisenberg model, (b) using the Ising model 6. x/xC vs. T/Tc for two-dimensional honeycomb Ising lattice 7. x/xC vs. T/Tc for two-dimensional plane square Ising lattice 8. x/xC vs. T/TC for three-dimensional simple cubic Ising lattice 9. x/xc vs. T/TC for three—dimensional body centered cubic Ising lattice 10. Magnetic susceptibility of Co(tu)uCl [001], [100], and [ll0] axes 2 along the 11. Magnetic susceptibility of CO(tu)-“Cl2 powder compared to {r(xOOl + X100 + x110). l2. Plot ofcime)/dT for Co(tu)uCl2 13. Possible Spin arrangement for Co(tu)uCl2 having the symmetry PAé/n or PAé/m viii Page 28 28 31 35 36 37 38 A2 “3 AA A6 Figure Page 1A. Possible Spin arrangement for Co(tu)uCl 2 having the symmetry Icul’ . . . . . . . 47 l5. Curie-Weiss behavior of Co(tu)uCl2 “9 16. Comparison Of X" for Co(tu)uCl2 with honeycomb Ising lattice. The data points have been normalized to T = 0.93°K and X = 1.7A2. . 51 c max. 17. Comparison of X" for CO(tu)-“Cl2 with plane square Ising lattice. The data points have = O = been normalized to TC 0.93 K and Xmax. 1.552 . . . . . . . . . . . . . . 52 18. Comparison of X" for Co(tu)uCl2 with honeycomb Ising lattice. The data points have been normalized to T = 0.85°K and X = 1.7A2 . 53 c max. 19. Magnetic susceptibility of Mn(tu)uCl2 along the [001], [110], and [110] axes . . . . . . 55 20. Plot of x"(T) for Mn(tu)uCl2 . . . . . . . 57 21. Comparison of X" for Mn(tu)uCl2 with the three- dimensional Simple cubic Ising lattice . . . 58 22. Comparison of X" for Mn(tu)uCl2 with the plane square Ising lattice. The data has been . = o = normalized to TC 0.56 K and xmax. 1.552 . 59 23. Plot of d(x"T)/dT for Mn(tu)u012 . . . . . 60 2A. Curie-Weiss behavior of Xp for Mn(tu)uCl2. = L 62 Xp 5(x001 + xIOO I X110) 25. Identification Of directions in the ac plane Of Ni(tu)6Br2. The b axis points directly into the plane of this page . . . . . . . . 6“ ix Figure Page 26. Magnetic susceptibility of Ni(tu)6Br2 vs. temperature along the b, a', and c directions . . . . . . . . . . . . 65 27. Magnetic susceptibility of Ni(tu)6Br2 vs. temperature along the a and f directions . . 66 28. Magnetic susceptibility of Ni(tu)6Br2 at l.3°K as a function of direction in the ac plane. . 68 29. Magnetic susceptibility of powdered Ni(tu)6Br2 . 70 30. Possible directions of sublattice magnetization in the ac plane of Ni(tu)6Br2. C is esti- mated to be 33°. . . . . . . . . . . 72 31. Four-sublattice model for susceptibility cal- culations. . . . . . . . . . . . . 73 32. Plot of d(x"T)/dT VS. temperature for Ni(tu)6Br 76 2 33. Molar heat capacity of Ni(tu)6Br2 . . . . . 77 LIST OF APPENDICES Appendix A. B. Series expansions for the parallel suscepti- bility of some antiferromagnetic Ising lattices Susceptibility data in tabular form xi Page 87 96 I. INTRODUCTION Measurements of the magnetic susceptibilities of metals and alloys of the transition elements first led Neell to introduce the concept of antiferromagnetism. He was able to explain the qualitative features of the experim ental results by postulating that the magnetic moments of the magnetic ions were arranged in an alternating antiparallel pattern. Since that time this concept has been shown to be valid for many metals and insulators and has been the subject of much experimental and theoretical work. Information concerning the nature of the antiferro- magnetic state can be obtained from a variety Of experiments, including magnetic susceptibility, neutron diffraction, x ray diffraction, electron spin resonance, nuclear magnetic resonance, specific heat, thermal and electrical conduc- tivity, and others. Each of these measurements contributes to the description of the antiferromagnetic state of a material, and a complete description must include all of these prOperties. The work described in this thesis has a three-fold purpose. 1. The construction of a system capable of measuring the magnetic.susceptibility of antiferromagnetic materials in the He3 and He“ temperature range (O.AO°K to A.2°K). 1 2. The measurement of the magnetic susceptibility of materials not previously studied or known to be antiferro- magnetic with transition temperatures in this temperature range. 3. The discovery of new antiferromagnetic materials suitable for further investigation by methods already existing in this laboratory, such as proton magnetic resonance and specific heat. The main body of this report consists of a description of the experimental apparatus and procedure, a brief dis— cussion of some of the existing theories of antiferro— magnetism, and a presentation and discussion of the behavior of the magnetic susceptibility of three antiferromagnetic materials discovered in this work. II. EXPERIMENTAL APPARATUS AND PROCEDURE A. General Method Magnetic susceptibility was measured by the ac mutual inductance method. This method utilizes the fact that the mutual inductance of two concentric solenoids is propor- tional to the magnetic permeability of the space or material within the solenoids. If we let AM be the difference in the mutual inductance Of a set Of coils with and without a sample in some specified volume then AM will be propor- tional to the difference in the permeability of that volume and hence proportional to the susceptibility Of the sample. AM = Cx, (2.1) where X = the susceptibility of the sample. The proportionality constant C in (2.1) can be evaluated in two ways. (1) A theoretical value can be calculated using the geometry of the system of coils and sample. (2) An experimental value can be obtained by measuring AM for a sample with known susceptibility. Method (1) is ideal when the geometry of the system is simple and known to a high degree of accuracy. However, in the experiments reported here method (2) has been used since it relaxes the requirements on coil construction and since standard samples with reasonably well known susceptibilities are readily available. Standards used here were single crystals Of ferric ammonium alum and potassium chrome alum with susceptibilities of 9.02 x 10-3 cc/gm T and 3°66 X 10-3 CC/gm in their Curie Law regionsc2’3 T B. The Mutual Inductance Coils and Bridge Figure 1 Shows the geometry of the coils and sample. The primary consists Of two concentric solenoids which are wound such that the condition N_l = 3232 (2.2) N2 D1/ is approximately satisfied, where N1 = 6450 = number of turns on inner solenoid, N2 = 3840 = number of turns on outer solenoid, D2 = 3.5 cm = diameter of outer solenoid, D1 = 2.6 cm = diameter of inner solenoid. These two solenoids are connected in series so as to oppose each other. If (2.2) is exactly satisfied, then in the absence of a sample the net magnetic flux through the entire primary is zero. Thus the primary has no magnetic dipole moment and will not interact strongly with nearby magnetic materials. However, the magnetic field intensity at the sample position is not zero and in fact has the value C 25" N1 — N2 I oersteds, (2.3) L H where H II current through primary (amperes), pressure measuring tube >— He3 dewar liquid He3\ L measuring secondary sample—/ , : . . 5.0cm prumanes< compensating secondary l" on o 3 2.6 cm .n. . ~o-a... rat-.n... .......... aunou -.-...o..-- a .o-n...-.-.. ......... ........- 2.6 cm l" on o 3 [$2.6 cm -al 3 6—3.5cm Fig.|. The geometry of the mutual inductance coils and sample. t" I! length of primary (cm), 0 ll constant, slightly less than one due to the finite length of the solenoids. Therefore, the presence of a sample with x E 0 will introduce a change in the magnetic flux which can be measured by the secondary. An additional secondary coil identical to the measuring secondary and connected in Opposition to it is used to compensate for any flux resulting from the fact that (2.2) may not be exactly satisfied. The coils were wound With #36 Nylclad coated copper wire. The method of construction is similar to that described by Abel, Anderson and Wheatley.“ The mutual inductance was measured with a Cryotronics model 17B electronic mutual inductance bridge5 Operating at 17 cps. When the bridge is balanced the mutual induc- tance being measured is equal to the product of the course and fine dial readings multiplied by 50 x 10-5 bh: The sensitivity of the coils was such that the difference in dial readings with and without sample had to be multiplied by 2.2 x 10-8 in order to obtain the susceptibility. Thus: X (cc/mole) = (P - P ) M.Wi\ x 2.2 x 10-8, (2.A) s e W ./ where PS = Product of course and fine dial settings with sample in coils, P = Product of course and fine dial settings without sample in coils (empty), M.W. molecular weight of sample, w 5 actual weight of sample in grams. The mutual inductance bridge was most easily balanced by applyingifluaprimary coil voltage to the x axis of an oscilloscOpe and the output of the bridge amplifier to the y axis. An Offbalance condition appears as an ellipse on the SCOpe, while a perfectly balanced condition appears as a single horizontal line. The resistive balance primarily controls the tilt of the ellipse and the inductive balance primarily controls the Openness Of the ellipse. Thus it was quite easy to Judge what adjustments were necessary to obtain balance. 0. He3 Dewar and Sample As shown in Fig. l the coils are mounted on the out- side of a glass He3 dewar. This dewar is Shown in more detail in Fig. 2. As can be seen, the design is very simple and only very crude attempts were made to minimize radiation heat leaks. However, this system worked very well and temperatures as low as o.A”K could be reached. For lower temperatures one should provide a more efficient radiation trap and possibly provide larger pumping lines. The sample, ground in the shape of a Short cylinder with rounded ends, was of such a Size as to slide freely down the inner tube, yet maintain its orientation with respect to the axis of the coils. It was held in place «exchange 3 gas tube He pumping g A 6 liquid l-le4 < 55" glass tube e 31,-" glass tube 1w liquid H93 U sample Fig. 2. The Hes dewar. at the bottom of the dewar by three or four drops of Dow Corning 70A fluid which freezes upon cooling. For measurements below 1.19K the sample was immersed in liquid He3. The temperature was measured by noting the 3 as sampled by a l/8" teflon tube vapor pressure of the He which extended from an NRC Alphatron pressure gauge to a point just above the surface of the liquid He3. The guage was calibrated by measuring the susceptibility of ferric ammonium alum and potassium chrome alum, which are known to obey the Curie law in the He3 temperature range, as a function of the pressure gauge readings. The value of the susceptibility gives the temperature corresponding to the indicated pressure. D. Experimental Procedure Listed below is a typical sequence of Operations during a single experimental run 1. Pre-cool to liquid nitrogen temperature and evacuate both sections of the He dewar to a pressure C1 . —5 - about 10 mm. Hg, 2. Take a reading of the mutual inductance, making sure that thermal equilibrium has been reached. LU 3. Isolate the inner and outer chambers of the He 14 dewar; introduce He gas into the exchange gas section a: a pressure of about 200 U and He3 gas into the He3 section at a pressure of about 1 mm. 10 A. Transfer liquid Heu into the large He“ dewar and immediately begin to pump on the He“. 5. While pumping on the He”, introduce the remainder of the He3 gas into the He3 dewar and allow the He3 to liquify. 6. Allow the system to come to thermal equilibrium at the lowest Hel1l bath temperature (% l.l°K). 3 7. Remove exchange gas from He dewar. 8. By pumping on the He3, lower the temperature of the He3 bath and crystal in a stepwise manner, maintaining each different pressure until M ceases to change. Thermal equilibrium was attained much more quickly when decreasing the temperature than when allowing the temperature to drift up. 9. When finished with measurements below l.l°K admit some exchange gas again (m 100 u) and pump out the remaining He3. 10. Allow the HeLl bath to warm up to A.2°K. ll. Decrease the He“ bath temperature and again measure M at apprOpriate temperature intervals down to l.l°K. E. Data Reduction and Experimental Accuracy The experimental procedure above describes how M(T) was measured. But what we really want to know is AM(T). Theoretically one should be able to simply take the difference of M(T) measured with and without a sample. 11 But in actual practice it was found that an additional constant had to be included in this difference in order to account for small changes in the coil characteristics from run to run. Therefore AM(T) = M(T) with — M(T) without + Mo , (2.5) sample sample where Mo = lim M(l - M I l + 0 \T without T with T sample sample . (2.6) This method does have the disadvantage, however, of placing too much emphasis on the measurements of M at liquid nitrogen temperature. If thermal equilibrium has not been attained a constant error will be introduced. An alternative method is to construct the system such that the sample can be physically moved in and out of the coils. This was not done in the He3 cryostat but for many of the measurements in the He“ temperature range the He3 dewar was removed and replaced by a sample holder which could be easily moved in and out of the coil region. This method was very helpful incfluaflchngearlier data, and in the case of Ni(tu)6Br where the susceptibility is 2) small, it proved to be the only reliable method. The overall accuracy of the data presented in this report is estimated at i 10%. 12 Strictly speaking, the experimental values of the susceptibility should be corrected according to the relation x = x exp , (2.7) [l + (Aw/3 - N) xvl where N is the demagnetization factor along the axis of measurement and xv is the volume susceptibility. Since this correction would amount to 1% or less for this work it was ignored. F. Sample Preparation and Chemical Analysis All samples used in this study were grown from aqueous solutions of the transition metal halide and thiourea. Although good, large single crystals of these materials wererfiflldifficult to obtain, a brief description of the procedure is given here. Large, dark—blue, bipyramidal crystals of Co(tu)uCl2 were easily grown at room temperature by slow evaporation of an aqueous solution containing 0001 6H20 and 2 (NH2)2CS in the ratio of approximately 2:1 by weight. The exact composition of the solution was not very critical and good samples could be grown with a wide range of constituent ratios. 13 Large, dark yellow-green, prismatic crystals of Ni(tu)6Br2 were readily grown at room temperature by slow evaporation of an aqueous solution of NiBr and (NH2)2CS 2 in the ratio of approximately 1:1 by weight. It was found, however, that thiourea tends to act as a reducing agent and small amounts of nickel would collect on the sides ofthe beaker. Therefore the solution was filtered every few days and placed in a clean beaker. Mn(tu)u012 required a little more care and patience than needed for the other materials described above. It was found that the weight ratio of MnCl AH20 to 2 (NH2)2CS had to be approximately 20:1 at room temperature. Good crystals would grow at room temperature but very slowly. It was found that crystals could be grown faster by preparing a solution with a 5:1 ratio of constituents and allowing the solution to cool Slowly from 50°C. The chemical constitution of our samples was determined by subjecting them to a chemical analysis for their nitrogen, sulfur, and metal content. Below is a list of the theoretical and measured compositions. The agreement between theoretical and measured values is sufficiently close to assure us that we are using the correct chemical formulas. 14 Table 1.--Results of chemical analysis of samples. Sample Per Cent N Per Cent S Per Cent Metal theor. meas. theor. meas. theor. meas. Co(tu)uC12 25.8 24.5 29.5 29.3 13.6 12.4 Mn(tu)uCl2 26.0 24.5 29.8 28.7 12.8 12.1 Ni(tu)xBr2 24.9 24.2 28.5 28.4 8.7 8.9 6 gig; III. BRIEF DISCUSSION OF EXISTING THEORIES A. The Scope of This Discussion A complete discussion of all theories pertaining to the behaVIor of antiferromagnetic materials is both im- practical and unnecessary. There exist many comprehensive reviews6m8 of this subJect to which the reader is directed for further discussion and references. As a basis for the discussion of Chapter IV, I shall review here the origin of the spin—dependent interactions, two common approximations to this interaction potential (molecular field approach and Ising model), and the results of series expansion calcu— lations on the magnetic susceptibility of the Ising anti- ferromagnet. B, Direct and Indirect Exchange-- Early Theories In order to account for the ordered state of spin systems it is necessary to know the nature and origin of Spin-dependent forces. Why do neighboring spins prefer a parallel or antiparallel arrangement? In the early work of Heisenberg, Dirac, and others, perturbation theory was applied to the atomic orbitals of neighboring ions, each with one or more unpaired spins outside of a nonmagnetic 15 16 core. Consider the case of one unpaired electron on each ion. If the orbital wave functions vi and $3 of the unpaired (magnetic) electrons on atoms k and J are orthogonal, the perturbation caused by the Coulomb repulsion between these electrons Splits the original ground state, which had a two- fold exchange degeneracy, into two states separated in energy by an amount 2Jij’ where JiJ - S;V1 dv2 w? (Fl) A: (r2) 8 WJ (Fl) Wi(;2), (3.1) and F and F2 are the position vectors of electrons l and 2 l in this two—electron formulation of the problem. J is 13 positive so that the upper state is one with a symmetric orbital function and an antisymmetric (S = 0, spins tot antiparallel) spin function, while the ground state has an antisymmetric orbital function and a symmetric (Stot = l, Spins parallel) spin function. This splitting is caused by the Coulomb interaction between orbitals and the restrictions of the Pauli exclusion principle. Although the splitting is orbital in origin the energy can be expressed formally in terms of the relative orientation of the spins. Let ‘>be the expectation value of the spin-dependent part of the electron-electron interaction. The proper energy levels can be obtained by letting /sp _ \ViJ - 45 J13 (1+4 - 2((éé0;>>- \E;\\- <§J>N , it can be seen that results of the perturbation calculation are retained. Generalizing the problem to a macroscopic crystal instead of an isolated pair of atoms, the spin-dependent part of the electron-electron interaction has been written as sp + (1+AS ' S ) , (3 3) where the summation is to be carried over all pairs of atoms in the crystal. This form is often called the Heisenberg interaction. One problem with this formulation is that Jij is always positive (ferromagnetic) and cannot account for an antiferromagnetic Interaction. This can In some cases be overcome by using nonorthogonal orbitals in which case 2 e r 12 for J13. If the electrons are in an attractive potential, L\ is replaced by the total Hamiltonian in the expressl n 7' J13 may become negative. In either ca e di ‘ssed aOOve toe (f) l.) U) effect depends on the direct contact k) r”S O < (D '3 ’_ j a: ”Cl 0 t”) 5-1 Q) (I: functions and has traditionally been C) 91 'r. H l) L C. l '3 (I. t) (‘f (D f (\ .1) (W 0. (D Most antiferromagnetic materials are insulators in which the magnetic ions are separated by non—magnetic ions or groups of ions. Early thinking on this matter led to tn LI) idea that direct exchange played an unimportant role in interactions between magnetic ions in such cases and a dizfer— ent effect was sought By virtue of the fact that this interaction acts over large distances it was given the name superexchange (or indirect exchange). Without going into the mathematical details, which may be found elsewhereé’g’lo, let us describe qualitatively the origin of this phenomenon. Consider a three-ion system such as Mn20++. The zero—order ground state can be described as Mn++ 0" Mn++ dl p d 2 where we have concentrated our attention on one of the d electrons in each Mn atom and on two p electrons of the O atom. One possible excited state can be described by Mn+ 0‘ Mn++ V dldl p d 2 where one of the p electrons from the O atom has gone over to one of the Mn atoms. If one now carries out a perturba— tion calculation and considers the contributions of the excited states to the true ground state one finds some interesting results. The first-order correction to the energy is zero. The second order correction is independent of the spins of d and d The third order correction to l 2’ /+ -.+ the energy can be written in the form J \51 ’ 82>>. Thus, once again we have an interaction which can formally be written as in (3.3). In this case, however, the interaction is transmitted by an intermediate non—magnetic ion such as O, S, F, Cl, Br, or I. 19 C. Exchange Interactions—-Modern Theory The early theories discussed above seemed to have many good qualities but ran into difficulty on many points. The perturbation calculation used in the superexchange problem converges quite slowly and has many large non—magnetic terms. Other excitations such as double electron transfer lead to similar results, suggesting that the real mechanism is simpler than assumed: Finally, one does not really know the prOper wave functions to use in calculating Jij’ The modern theory of exchange interactionsll, now in its early stages, attempts to treat the problem in a more realistic way. The program consists of two parts: 1. In order to assure a more rapidly converging perturbation series, calculate by means of modern ligand field theory the wave functions of the magnetic and non- magnetic ions in the crystal, neglecting all exchange effects. 2. Using the results of part (1) include now the exchange effects and calculate the interaction characteris- tics of the magnetic ions. This program has the advantage that the perturbations are weaker and more rapidly converging. In addition, all interactions are now directly between magnetic ions, the intermediate non-magnetic ions being taken care of by the ligand field calculations. Thus the early distinction between direct and indirect exchange is broken down and new meaning is given to these terms. 20 Direct exchange now refers only to the effect provided by the electrostatic Coulomb repulsion of electrons. This applies to any pair of magnetic ions regardless of the presence of non—magnetic ions. Such an interaction is always ferromagnetic. Superexchange is now the second-order effect of a virtual transfer of electrons between magnetic ions. It is a kinetic energy effect related to the fact that when two neighboring spins are parallel their orbital functions are orthogonal, while for antiparallel neighboring spins the spin functions are orthogonal so that their orbital functions may overlap. Such an interaction is always anti- ferromagnetic and the splitting between parallel and anti— parallel states is of the order ”b2 , where U is the Coulomb U energy gained by an electron in going from one magnetic ion to another, both of which originally contained the proper number of electrons, and b is the ”transfer integral” connecting these states. This new superexchange mechanism depends on a direct contact between wave functions of neighboring magnetic electrons as opposed to the early theory in which the intermediate non—magnetic ion provided the connecting link. These two effects, direct exchange related to the Coulomb repulsion of electrons, and superexchange related to electron transfer, are the major contributors to 21 spin—dependent forces in the modern theory of exchange in insulators. In each case the interaction potential can be put in the form (3.3). D. Approximations to Interaction Potential 1. Preliminary Remarks From the previous discussion we are led to solve the problem of a system of magnetic ions with an interaction potential of the form (1 + “Si ' SJ) , (3.3) t ij where the summation is over all pairs of magnetic ions. This has proved to be an extremely difficult problem and many approximations to (3.3) have been made. A first simplification can be made by ignoring all but nearest neighbor pairs in the summation of (3.3). In many cases this is a good approximation, but by itself does not lend much simplification to the problem. Thus we are forced to go to a greater degree of approximation. In the following sections I shall describe the principal features of two such approximations. 2. The Molecular Field Approach The Weiss molecular field approach to antiferromagnetic materials proposed by Neel in 1932 has been responsible for a considerable advance in the qualitative understanding of many properties of antiferromagnetic systems. Let us assume 22 that we can divide the lattice of magnetic ions into two interpenetrating sublattices A and B. We conceive of the ground state of the system to be one in which all the spins on sublattice A are parallel to each other and anti- parallel to the spins on sublattice B, each sublattice having the same number of spins so that the net magnetic moment is zero. Consider one ion 1 on one of these sublattices, say A. To simplify the summation in (3.3) let us assume that this ion interacts with z neighbors on sublattice B with an exchange constant J and z' neighbors l, on sublattice A with exchange constant J2. Now we can express the interaction potential of atom i on sublattice A as 181 + A . Z v1 = -2J s. -2J s. - Y s , (3.4) l R J 2 LL. 'C‘/JN where the first summation is over the z neighbors on sublattice B and the second summation is over the 2' neighbors on sublattice A. Equations (3.4) and (3.3) can be related by U) L. i (1.1. VSp = constant + i V? + Z V. . (3-5) J The next approximation consists in replacing the summations in (3.4) by Zgg and Z'gA §A and §B are to be considered as statistical averages of respectively, where the spin on each sublattice. Doing this, (3.A) becomes A _ _ = — l . i _ . ' ' i i 2lesi g8 22 J2§i sA "\ U.) 0 OK \./ V 23 Remembering the relation between magnetic moment and spin for an electron, 3 = -gB§ , where g = the spectroscopic splitting factor, B = the Bohr magneton, we can write A —— y —— V = -2ZJ1 p- “ u - 22 J2 u L (3.7) i - i B i A 2 2 2 2 g 8 g 8 Now define +A — — 2zJ. é 2z'J é . Heff ’ 1 “B I -2 “A , (3.8) 2 2 2 g 8 g B A so that wecan write Vi in the form 4A A _ j . g Vi ’ —"i Heff ’ ‘3°9) B with a similar expression for V the interaction potential :13 for atom j on sublattice B. Thus we have now defined an effective magnetic field A,B .."~ . H which acts on the magnetic moment of each ion. This eff is called the molecular or exchange field. It must be emphasized that this is not a real magnetic field but merely a convenient way to express the energy of a magnetic ion coupled to its neighbors by the exchange effect. With the interactions considered thus far the spin system may assume an antiparallel arrangement but has no preferred directions with respect to crystallographic axes. This preferred direction can be introduced phenomenologically in the form of an "anisotropy energy" which is a function of the direction of the sublattice magnetization with respect to the crystallographic axes. Good discussions of this subject can be found in references 6 and 12. Now let us calculate the magnetic susceptibility in the molecular field approximation which we have introduced. For the purposesof this discussion let us write Heff in a slightly different form. Note that .+ ' ? MA,B ‘ 1/2 N ”A,B , where MA B = the magnetization of sublattice A or B, 3 the total number of magnetic ions per unit 2: fl volume- Then we can write A Heff =”MB — fl%i , B + , } (3:10) HEff '—JMA - {MB , where a = -_2_ 2chl , N 3 2 c 1 g B ; (3 ll) Y = -% 2z'J;\ 2 2 g B/ 13 From the theory of paramagnetic susceptibility we know that for a spin only paramagnet (J = S) 25 M = NgBSBS(y) , (3.12) where BS(y) is the Brillouin function, \ / B(y)==2s+i ______?y)-_;_coth"_x. \ —— _/ 2s Kgs), (3'13" S /’r ) 28 = / H = applied magnetic field. For an antiferromagnet let us calculate MA or MB by replacing H with H + Heff. (where H and Heff are assumed to be parallel). Consider first the case H = 0. Then MA = aNgBSBgffiaM + (M AlMgBS 1) (3.1“) S \I B kT In the absence of an applied field MA = -MB and lMAI= ‘MB| = M, where :1 1/ “._- * S M 2Ng§SBS ('2 LL FEB > (3.15) M is a decreasing function of temperature and vanishes for T > TN, where TN can be evaluated in the following manner. We assume that y : ia-yngBS is small and replace SBS(y) TikT by the leading term in the series expansion, 1/3 (S + l)y. Thus, for M small we have M = 1/2NgB ° 1/3 (S + 1,)Id-Yl IVISgB/kT . (3.16) This yields for the temperature at which the expansion is most valid Ng252S(s + l) o-v , (3. 3k 2 TN F.) \1 \/ 01" TN = % Ia-vl , (3.18) 26 where C is the Curie constant 2 2 C = Mg 8 S(S+l) . 3k Next consider the case T > IN and H e 0. Again yA and yB will be small and MA B can be approximated by 3 MA = gwgsi/B (S + 1) (H - anB ~ .MA) SgB/KT , C . = — . — (F, — .fl MA 2T H “a i.) ’ . w I (3.1y) C = — H - A — - MB 2T ( aIA YMB) = + . ( .2 But xH MA MB 3 20) Putting (3.19) into (3.20) we have xH = C {2s — (w + MD) (a + v) } , (3.21) — A u 2T so that x = C_ , (3 9:) T+O where 9 = Q (a + Y) (j 93) 2 For T < TN we consider two cases. a. H parallel to the sublattice. m::;.:;:r1etiEation. To calculate X we must expand the Brillcoin function H in powers of H and solve for -|f? X -|1A + B IL ) f. 5..)- l The mathematical details are not particularly illumina. and we give here only the general features. v vaniercg at . C — T — O and rises to T776 at T _ TN” 27 b. H perpendicular to the sublattice magnetization. From Fig. 3 we can see that XLH = (MA + MB) sin Q = 2M sin H. To obtain sin G as a function of H we note that at the equilibrium value of Q the torque on each sublattice must vanish, or | MA x (a + fi:ff)| = o . (3.2u) This leads to 2Msin Q = H , (3~25) so that a xi = % = constant . (3.26) Figure A summarizes the results of these susceptibility calculations. The principal feature of the molecular field approxima- tion lies in equation (3.6) where the summation over neigh- bors is replaced by a statistical average over the whole sublattice. This in effect removes the concept of local interactions and considers each ion to be interacting equally with each member of a given sublattice. Although the constants a and Y defined by (3.11) still contain the factors 2 and 2', these factors only occur in products such as le and z'J2 and hence merely allow us to characterize the strength oftflueinteraction by a two-particle parameter J. The result of this averaging over the sublattice as done in (3.6) is that all effects of short range order disappear. 28 M. MA 'Tr\/n" Fig.3. The effect of a perpendicular magnetic field on the sublattice magnetization vectors. T-—> Fig. 4. Sketch of X vs T in the molecular field approximation. 29 Only long range ordering is considered. At high temperatures short range ordering is thermally reduced and may certainly be neglected. Some long range ordering is introduced by the application of a magnetic field. Thus in the high tem- perature limit the molecular field approximation should be quite reasonable. However, near or below T short range N correlations could surely be expected to influence such properties as the magnetic susceptibility and one must not put too much faith in the details of results obtained above. This has been demonstrated repeatedly by experiment where in general the results agree with the molecular field theory with respect to qualitative features but disagree with respect to quantitative details, except possibly in the high tempera- ture limit. 3. The Isinngodel The Ising model of the exchange interaction simply replaces Si . SJ by 821 823’ thus allowing the state of the system to be described by assigning a value of SZ to each magnetic ion- Equation (3.3) is then replaced by v5p = —A S (3.27) g3 Jig ziSzJ ’ where A is a constant which has been given various values by different authors. We will use A = A so that for spin % atoms the values of VSp in (3.27) range from —J to +J as they would in (3 3) 30 Let us go back to the two electron problem discussed earlier and compare the energy levels resulting from (3.3) and (3.27). Figure 5 shows the energy level scheme for both potentials and for both signs of J. As can be seen, the level scheme is different for the Heisenberg exchange and the Ising model potentials. The Ising model gives rise to two doublets separated by 2 [Ji for either sign of J. On the other hand, the Heisenberg model gives rise to singlet and triplet levels separated by 2 {J} and inverts the order of these levels upon a sign change of J. In spite of these difficulties the Ising model has been very useful in treating magnetic properties as well ascflfimfl°problems in which the state of the system is described by assigning values of a single-particle parameter to each member of the system. In the next section we shall make use of this Ising model in a statistical treatment of the properties, with emphasis on the magnetic susceptibility, of a system of magnetic ions which undergoes an antiferro- magnetic transition. E. Statistical Mechanics of the Ising Model I. Formulation of the problem In the statistical treatment of a problem it is very useful to know the partition function of the system under consideration because from this function many thermodyngmic 3| 2lJl 2 MI J’ singlet ‘l' triplet J < O J > O (0) ZIJI 2IJI ‘l' doublet ‘L doublet J < O J > O (b) Fig. 5. Energy levels of the two Inter- ootlng electron problem (0) using the Heisenberg model, (b) using the Ising model. 32 properties can be calculated. The partition function is defined as Ce—Ec/kT , s (3.28) N ll (.3 NA where EC is an energy state of the system, gc is the number of states having that energy, and the summation is over all possible energy states. In particular, the magnetic susceptibility of the system is X(T) z kT —3§ [1n 2 8 \ l0 - >< 0.8 - 4 0.6 - d 0.4 .. - 0.2 - - o , l l I l O l 2 3 4 5 171}; Fig.7. X/Xc vs. T/Tc for two-dimensional plane square Ising lattice. 37 |.2 I I I I l.0 - 0.9 - 0.8 .- 0.7 - 0.6 - x xxc 0.5 - 0.4 - - 0.3 - .4 0.2 - - 00' " cl T/Tc Fig.8. X/Xc vs. T/Tc for three-dimensional simple cubic lsing lattice. 38 l.2 I I l I Ll - - l.0 - 0.9 - 0.8 '- xxxc 17 TC Fig.9. X/Xc vs. T/Tc for three-dimensional body centered cublc Ising lattice. 39 some interesting differences. The molecular field approxi- mation, which considers only long range ordering, results in a parallel susceptibility which has a very sharp peak at the transition temperature, with a discontinuity in the derivative of x with respect to T. On the other hand, the Ising model, which considers both short and long-range ordering, gives rise to a rounded peak in the susceptibility at a temperature slightly above the transition temperature, which is characterized by a vertical inflection point in the susceptibility. The difference in the nature of the peak in X" is, I feel, mainly a result of the way in which short—range order is treated. In the molecular field approach each spin is considered to be interacting with the average magnetization of each sublattice, ignoring the effects of local fluctuations, while the Ising model con- siders each spin to be interacting with its nearest neighbors, which allows both short and long-range effects to be con- sidered- Short—range effects must be considered near the transition temperature where fluctuations are large. A comparison of Figs. 6—9 among themselves shows that the differences within the two—dimensional and three— dimensional lattices is small, while there is a more striking difference between the two and three-dimensional lattices. Both two—dimensional lattices have their transition tempera— ture considerably below a very broad peak in the parallel susceptibility while both three-dimensional lattices have HO their transition temperature only slightly below a more sharply rounded peak in x": Thus dimensionality plays a more important role than do differences among lattices of the same dimension. IV. EXPERIMENTAL RESULTS AND DISCUSSION A. Co(tu)4012 Co(tu)uCl2 forms crystals with tetragonal point symmetry A/m. From the x ray data available it is found that the unit cell dimensions are a = 13.52 A and c = 9.10 A with four formula units per unit cell, and that the space group is PAZ/n. Magnetic susceptibility measurements were made on several single crystals and powdered samples of Co(tu)uCl2 in the temperature range 0.459K to 4 20K. The sample weight was in the range of 1.0 to 1.5 grams. The experi- mental data is given numerically in Tables 4-7 in Appendix B and shown graphically in Figs. 10 and ll. Figure 10 shows the susceptibility data taken for the [001], [100], and [110] axes. From this data it is deduced that this material is antiferromagnetically ordered below T = 0.931K and that the sublattice magnetization vectors lie along or C) very near to the axis. The antiferromagnetic transition temperature is defined as the temperature at which d (an)/dT assumes its maximum value.19 In Fig. 12 this quantity has been plotted as a function of temperature. There is an asymmetrical but well- defined peak at T = O 93°K indicating that this is indeed 41 42 0.7lilllllrilliiill'r‘1 0.6r- oo 00 00000 o 00 o 0o 0.5- o °o[om] o 00 000000 00‘ % 0.4- 00. ......O 0 2 §' '. 0.. [I00] \ o. 0. be 0 o. ..e 000 303- 09g 0. 0 ><’ @0043 "' ”0 ,o [1 0.2- 0 o o o o 0.l- 0 o o o ollllllllllLlllllllll 0 LG 2.0 3.0 4.0 T(deg.K) Fia.l0. Magnetic susceptibility of CO(tu)4C|2 along the [oon].[loo] and [no] axes. 43 0'6 I I I I I I I I I I I I I I I I I I I I 0.5" '- A 0.4- fl 3 E E 0.3- . 3 x 0.2- .1 00 DJ - 3(XOOI+XIOO+XIIO) - o Xp ollJlIllllIllllIlllJI 0 LG 2.0 3.0 4.0 T (deg. K) Fig.ll. Magnetic susceptibility of Comma2 powder compared to {-(xom-o-x'oo-i-xm). 3.0 2.5 2.0 le..Tl/d1’ 'o 0.5 'au I I l 1 O l T =- 0.93°K O ITTTT ITI—rl I l r 000 (bod) O llllIlllLl o000000 l.0 T (deg. K) 3.0 4.0 Fig.l2. Plot of d(X"T)/dT tor 00(tul4Cl2. 45 the correct value for the transition temperature of Co(tu)uCl This is confirmed by proton magnetic resonance 2. experiments which indicate that this is the temperature at which local internal magnetic fields begin to appear at the proton sites. The conclusion that the sublattice magnetization vec- tors 1ie along or near to the c axis is consistent with the results of the proton resonance investigation carried out by Spence et al. An exhaustive analysis of the proton resonance data leaves three possibilities for the magnetic symmetry group, two of which give the same sublattice structure for Co(tu)uC12. The possible spin arrangements are shown in Figs— 13 and l“. The cobalt ions are located at (t, t, t), (4, §, t), (f, %, fi), and (§, %, $3 respectively in each chemical unit cell. In Fig. 13 the magnetic unit cell is identical to the chemical unit cell, while in Fig. 14 the magnetic unit cell is twice the size of the chemical unit cell, being elongated along the c axis. If one could make some predictions, based on the details of the crystal structure and the nature of indirect exchange interactions, about the magnitude and sign of the exchange integral between various neighboring cobalt ions one could possibly decide which of these spin arrangements is most likely correct. The program of calculation is fairly straightforward. 46 Fig.l3. Possible spin arrangement Coltul4C|2 or P 4’2 /m . having the symmetry for P 4', In 47 Fig. l4. Possible spin arrangement for CO(IU)4C|2 having the symmety Ic4,. A8 1. For each spin arrangement calculate the ground state energy E. 2. If one of these ground state energies is obviously lower than the other the problem is solved. If there is uncertainty or if a check is desired go to step (3). 3. For each spin arrangement calculate the suscepti- bility at T = O°K by minimizing the quantity E - N.fi with respect to deviations in the direction of the sublattice magnetizations. This calculated susceptibility will be a function of the values of the various exchange integrals. At From the measured values of the susceptibility calculate the values of the exchange integrals by requiring agreement between measured and calculated values of the susceptibility for each spin arrangement. 5. Compare the sets of exchange integrals calculated from (A) with the initial predictions and choose the spin arrangement which leads to the most consistent results. Unfortunately, at this time there is not yet enough information available upon which to base predictions con- cerning the values of the various exchange integrals. Hence, the program outlined above cannot yet be carried out. It has long been customary to compare the behavior of the magnetic susceptibility above the transition temperature with the Curie-Weiss law, X = C/(T + 0), based on the molecular field approximation, Figure 15 shows that within the limited temperature range of 20K to AOK the powder 49 4.0 I I I I r j I T I T (deg. K) Fig. l5. Curie-Weiss behavior of Coltul4Cl2. 50 susceptibility obeys the relation Xp = 2 57/(T + 4°K). If one were interested in a more critical comparison with the Curie-Weiss law more data above 4.2CK would be needed. In Figs. 16—18 the c axis susceptibility of Co(tu)uCl2 is compared to some of the results of the Ising model calculations. Figures 16 and 17 show the comparison with the two—dimensional honeycomb and plane square lattices where the data has been normalized to TC = 0.93OK and the maximum susceptibility equal to that of the Ising lattices. In each case there is agreement on general features but some differences in detail. The broad nature of the peak suggests a two-dimensional character of the exchange interactions. Possibly the exchange interactions within a set of planes in the lattice are large compared to interactions between planes, although this is not obvious from available structural information. In an attempt to obtain a better fit in the region of the transition temperature Fig. 18 shows a comparison with the honeycomb lattice where the data has now been normalized to TC = 0.850K. In the temperature range below 2°K the fit is indeed better, but this is somewhat illusory since TC = O.93°K, not O.85°K. A more serious difficulty brought out by these com- parisons is the fact that at the low temperature extreme the experimental data begins to differ significantly from the theoretical curve and seems to indicate a non-zero value for the susceptibility at T = 00K. This descrepancy may be 5| 2.0 l l I T/Tc Fig. l6. Comparison of X" for Coltul4Cl2 with honeycomb Ising lattice. The data points have been normalized to Tc= 0.93°K and Xmaxf l.742. 52 X (rel. units) T/Tc Fig.l7. Comparison of X“ for CO(IU)4C|2 with plane square Ising lattice. The data points have been normalized to Tcso.93°K and Xmaxf l.552 . 53 T/ To Fig.l8. Comparison of X" for Coliui4C|2 with honeycomb Ising lattice. The data points have been normalized to Tc-O.85°K and XmOX. 3 L742 . 54 due to experimental error, either in the value of the sample temperature or, more likely, in the value of Mo as calculated from Eq. (2.6). It might also be a real effect showing that a two-sublattice model is inadequate. In either case, measurements below 0 U5’K would help resolve the difficulty. It is hardly appropriate to try to draw any further conclusions from these comparisons since the Ising model, like any other model, is based on a set of simplifying assumptions which are not in general satisfied by any real crystal. However, the more satisfactory treatment of the statistical properties in the Ising model is certainly a great improvement over the molecular field approach. Further improvements in the theoretical treatment of an antiferromagnet will require a more realistic and complete Hamiltonian whose range of interaction extends over several lattice parameters, while retaining a rigorous statistical treatment. B. Mn ‘u‘ C Mn(tu)uCl2 forms crystals with tetragonal point symmetry A/m similar in habit to CO(tuluC12. X ray data indicates that it has the space group FAB/n with four chemical formula units per unit cell of dimensions a = 13.76 A and c = 5.L7 A. Figure 19 displays the results of measurements of the magnetic susceptibility along the [001], [100], and [llCl X (cc/mole) 2.6 2.4 2.2 2.0 LG LG l.4 l.2 l.0 0.8 0.6 0.4 0.2 0 0 55 IIjT'TIITrIII IllTlrl llJlIllllIllljllllll l.0 2.0 T (deg. K) 3.0 4.0 Fig.l9. Magnetic susceptibility of Mn(tu)4Cl2 along the [00!] , [I00], and [no] axes. 56 axes. The numerical data is listed in Tables 8-lO in Appendix B. The general features of Fig. 19 indicate that Mn(tu)uCl2 undergoes an antiferromagnetic transition at T = 0.56QK. However, the sublattice magnetization vectors do not appear to lie near the c axis as in CO(tuluCle. In fact, the c axis data indicates that the sublattice magneti— zation vectors probably lie close to the ab plane. If one considers a simple two-sublattice model with the sublattice magnetization vectors lying in the ab plane it must be noted that these vectors could lie along either of two equivalent axes which are at right angles to each other. Thus, the possibility of domains would have to be considered. If the volume of each type of domain were equal then the susceptibility in the ab plane would, in the first approximation, be isotropic and equal to % (Xu + XL). The susceptibility along the c axis would be Xi' Following this model a little further, Fig. 20 shows a plot of xniT) where x" = {x100 + x110) - x00 . Figures 21 and 22 show a comparison of this x with the three- dimensional simple cubic and two-dimensional plane square lattiCes. In each case the fit is poor but the three- dimensional case seems to be preferred, in contrast with Co(tu)uCl2 which fit reasonably well with the two-dimensional results. Figure 23 shows a plot of d(x" }/iT to demonstrate that'TC = O.56°K and to show the expected behavior of the magnetic contribution to the specific heat. X (cc/mole) Fig. 20 3.2 3.0 2.8 2.6 2.4 P 0 LG LG L4 l.2 l.0 0.8 0.6 0.4 0.2 J J 1111 111' lf'l 1111 l.0 2.0 T (deg. K) .Plot of X"(T) for Mn(tu)4CI2. 3.0 4.0 58 I.2 I T I I L! L - 1.0 - 0.9 - A 0.8 - 0’ .2: 0.7 _ C 3 _- 0.6 - m b v 0.5 - x 0.4 . - 0.3 - .. 0.2 *- . 0.1 b - o I I I I 0 l 2 3 4 5 171}; Fig.2l. Comparison of X" for Mnitu)4C|2 with the three-dimensional simple cubic Ising lattice . 59 2.0 r . . . LB l- - Le b 1; 1.4 - .2: § l.2 - x 0.0 - . 0.6 p . 0.4 - . 0.2 .. - o r . . . . 0 l 2 3 4 5 T/Tc Fig.22.Comporison of X" for Mn(tu)4Cl2 with the plane square Ising lattice. The data has been normalized to Tc . 0.56°K and Xmax. 8 Lou . 7.0 6.0 d(X..T)/dT :- O 9' o LO 0 60 lleTfilIUrlerII'III' lT=0.se°K o 0 o o 00 II 0 - Q) <9 - (2,0 - o o 000 L 0o .. 0 0° 0 0 0 0000 00 IILILIIIJJIIIIIIIJLI 0 LG 2.0 3.0 4.0 T (deg. K) Fig.23. Plot of d(X.,T)/dT for llfln(tu)4i.‘.l2 . 61 There is, however, ar annoying pro lem with the y" as caloulated. It appears that our x" will have a non- zero value at T : 03K. This ooald be the result of errors in the temperatire measurements th suoh large errors are not expecte- More likely possibilities are that a simple two-sublattice mode; is inadequate or that k”0‘ is signifi- vii. cantly different from the r. in the ab plane. In either A. case measurements at mu:h lower temperatures are required to resolve the roblem. No sasoeptibility measurements were made on powdered samples. However, ip is estimated to be equal to ¥~{' + 'u.. + - F‘ ‘I"=.29 s ews - comrariS"v"“’ X100 5.1.;0 XOOL)- $gd v . he. a. w 1 1.4.2- .J L .1- the ip With the Curie—Weiss law X a C. NigtaléBr Of the three materials investigated in this pr~ {i} \ (1 Cl :3” 3)) (II + ‘1‘ (I) thiourea-ooor inated n;:Ke; bromide, Ni(tul Brg, “H A '5 .f‘ V A Y'\ AP- - ' r\ h ‘ fl " A.“ r most interest-ng behavior. The two Ieatires or speo-a- “rt: st r thvr- r*‘ We no v4: 4-“: t‘c ~ra»»+;a L L‘v'w-Peflu— ate ‘ a- 0 ‘. -) akvng 61V a>L$Ku VHFTK.’ {1‘ l]l;b[.d~_4 13 D” 8... t I‘ -~ . .-. 1 ..-x A - .' 1“\ ’ 1 ‘ ... a ’i A. 2‘ 1‘. 3 cisceptibility a ze.o at low temperatares, a; .iln 7‘5 "(1 H O (‘1 A CT ., 'fi . 1’ —+- “new 4.3 3851. Naia the, i ll) (.3 the magnetic sus ibil! ta and O (D yd Q) (I) P p i indicate that there are two distinct transitions at 2.05K and 2.2:K respectively. These features Will be demon- strated in the allowing graphic and tabular presentation of the experimental data. 62 L4 (.3 - l.2 '- l.0 r 0.9 P 0.6 ‘- I xxp 0.6 '- 0.5 - 0.4 - 0.3 b 0.2 *- 0.| *- T (deg. K) Fig.24. Curie-Weiss behavior of Xp for 63 2. Presentation of Data A preliminary x ray diffraction study of Ni(tu)6Br2 indicates that it has monoclinic point symmetry 2/m, space group C2/c, with four formula units per unit cell of dimensions a = 2A.3OA, b = 8.91A, c = 16.79A, and B = l37.02°. Figure 25 shows the appearance of a typical single crystal as viewed along the b axis, and indicates the other directions of interest in the ac plane. Single- crystal susceptibility measurements were made along the a,ea$ f, c, and b directions. Susceptibility data was also taken on powdered samples. Tables 11—13 of Appendix B give the data for measure- ments of the susceptibility along the b, a3 and 0 directions. This is shown in graphic form in Fig. 26. A very careful observation of the c axis data will show inflection points in X vs. T at T = 2.0°K and T = 2.2GK reSpectively. This feature will be more clearly seen in Fig. 32 to be described later. Note also that the b axis, which by symmetry considerations must be a magnetic principal axis, exhibits a magnetic susceptibility which increases below T = 2.2°K and approaches a maximum value of approximately 0.092 cc/mole as T + 00K. Tables 1“ and 15 list the experimental points showr graphically in Fig. 27 which exhibits the behavior of the magnetic susceptibility along the f and a directions of Ni(tu)6Br Again, a careful study of the behavior of 2. 64 i 44° 46° 43° l 47° l C \ $ 43° 0 44° —h‘ Fig.25. Identification of directions in the ac plane of Ni(tu)38r2 . The b axis points directly into the plane of this page. 65 .3238? c ace .0 .n 9: 98.0 822352 .3 «5.332 do 3232883 0:232 .86: A v. .33 F 04V o.m o.~ 0.. A _ . u . — d d a d _ 1 . . 4 O n .0. 1 No. 6) (9 O G G) O 00 1 l 4 ID ‘3' m . 0. 0. 0. (slow/as) X G) G) l (D O O O O I p 0. m @ 00000000 ~c 0.. 66 .mcotomtu u. can a 23.0.0063 .m> «52.5.2 .0 Izszamomzm A v. .003 ._. o... a 0.» od 9: 93.0 0:832 Km .2“. 0.. . . d . . _ . . . . _ . 6%: O O O O 4.. l '5. . q d O n .0. l N 0. I '0 0 0 % I I 8. 3. . (snow/o0) x l ‘D Q 0.. 67 xa vs. T will reveal the presence of inflection points at T = 2.0°K and T = 2.2°K respectively. One should also note at this point that along none of the directions investigated does the magnetic susceptibility approach zero as T + 0°K. The behavior of the susceptibility in the ac plane is seen more clearly in Fig. 28 which shows the experimental values of X along the f, a',a, and c directions at T = l.3°K, and a cosine curve fitted to these values. The directions of the magnetic principal axes in the ac plane are labelled x and 2 respectively in this drawing. By . extrapolation it is estimated that at T = 0°K we would have the following values for the susceptibility along the principal magnetic axes: xX = x260 = 0.0310 cc/mole Xy = Xb = 0.0920 cc/mole XZ - X—6M° = 0.0755 cc/mole In Tables 9—l2 there are listed values of X and x' = (l i 0.02) x for the susceptibility in the ac plane. The corrected values x' are used for the following reason: The x and z axes are the principal magnetic axes above and below the transition temperatures. At h.2°K Xx > Xz while at l.3°K Xx < Xz' If these axes remain principal axes at all temperatures between these points, then at some temperature Xx = X and X is isotropic in the ac plane. Z 68 (Slow /03) X man... on of :. c2386 .o c2823 0 mo xom. B «5:322 00 3:33.383 020.322 .mméfi. 00 00 0¢ 0m 0 0w: 0.1. 00.. 00: — d u q q —- q q q — q q 4 J ‘d - - u d— % 0| fix 0 \O b. [moo o o o .3 .30. -00 O O -h0. .ommlmvmmoomhomo.lmh¢mo. .l o w $0. 32. .oEmE..maxm 0 69 The 2% corrections listed are such as to allow X; = Xa' = = X; = 0.0775 cc/mole at T = 2.7°K. It is felt that ! xa this correction provides more consistent values for the susceptibility in the ac plane. Figure 29 shows the experimental data listed in Table 16 for the susceptibility of a powdered sample and compares this data with a curve which represents §(xa,+ Xb + Xc)’ The a' and c axes are not principal magnetic axes but are reasonably close° 3. Discussion of Sublattice Structure It has already been noted that the susceptibility along each of the principle magnetic axes approaches a non-zero value as T + 00K. Thus, a simple two-sublattice model can be eliminated immediately. The next model to be considered is a four-sublattice model in which pairs of antiparallel sub- lattices lie along different directions and the total magneti- zation still vanishes. The latter restriction is imposed because none of the susceptibility data indicates a spontane- ous net magnetization, which is usually characterized by large values of the susceptibility with a sharp peak at the transition temperature. Unless prohibited by the symmetry properties of the lattice, these sublattice magnetization vectors could lie in any one of the three planes determined by the principal magnetic axes. This implies that alone one of these axes only Xi will be observed while along the other two both Xi and X" will contribute to the observed suscepti- bility. The most likely candidate for the perpendicular 7O 0.¢ 43335.2 3338 .0 323.383... 0:232 33.2“. .x.amu. h 0.» 0.N 0.. 1 a q _ _ . _ . J _ . d _ . O Anx + 0X +~0XYW I L N0. Eon 3335.313 . .(8|Ow/OO) X 0.. C O . INC. 0.. 71 direction is the b axis since its low temperature suscepti— bility is significantly higher than long either of the other two principal axes. Thus we are led to propose a four sublattice model consisting of two pairs of sublattice magnetization vectors lying in the ac plane. Figure 30 shows the directions of the sublattice magnetization vectors based on this simple model. All vectors lie in the ac plane and make an angle of approximately 33° with the principal axis labelled x. This sublattice arrangement is based on the following elementary considerations. Assume that the sublattice mag- netization vectors lie in the xz (ac) plane. This is the most reasonable choice since Xy has the largest value at T = 0°K. Let the magnetization vectors make an angle 0 with the axis as shown in Fig. 31. Let the measuring field be repre— sented by 3, making an anglee with the x axis. If X" and X1 are the parallel and perpendicular susceptibilities for either pair of opposing sublattices l and L or£2and U,then in the x plane the susceptibility is x(0) = x"0082(0—0) + Xi 8102(9-0) +X"0082(0+0) + XL sin2(e+0) , (U. which reduces to x(9) = (xn + xi) + (xn - xi) 00820 00820 . (A. At T = 0°K , x" = 0 and x(0) becomes x(0) = X (1 — 00820 00820) . (A. Thus: L Xx =X(0) = XL<1 — 00820) = 0.0310 , } (b XZ =x(%) = xL(l + 00820) = 0.0755 x Z 2) 3) 72 52 \\ \\ s, \\ 4—? O \\ $3 ‘\ 4’ /2s° \ 4> \\ \ \ ;\ 84 Fig. 30. Possible directions of sublattice magnetization in the ac plane of Ni(tulsBrZ. 4’ is estimated to be 33° . 73 M2 Ml . 9 ¢ 4: M3 M4 Fig. 3| . Four-sublattice model for susceptibility calculations. 7A Solving these equations for KL and 0 we get 2x = 0.1065 _L 3 } (4.5) Q = 33° The value 2Ki.= 0.1065 compares reasonably well with X = 0.0920 although there is no exact agreement. Crystal- b line anisotropy could certainly account for such a difference. The sublattice model here presented is only a suggested model based on susceptibility data alone and should be subjected to further tests such as proton magnetic resonance or neutron diffraction correlated with a detailed crystal structure determination. A. Discussion of Transition Temperatures As stated earlier, the magnetic susceptibility along the a and c axes exhibits inflection points at 2.00K and 2.2°K respectively. This behavior is more clearly seen if 19 the quantity x(T) is differentiated. As Fisher has shown, of most interest is the temperature derivative of the product (Xn T). This quantity has been calculated from Xa and X0 by assuming that x¢_remain8 constant with tempera- ture and subtracting its contribution in accordance with the proposed sublattice model. This yields: 1.25 x" = xa - 0.039 3 } (14.6) 1.34 x". = XC — 0.035 75 Note that X" still applies to only one pair of sublattices, or half of the nickel ions. Figure 32 shows graphically the behavior of d(xH T) as a function of the temperature. dT The peaks in this curve correspond to magnetic transitions of the antiferromagnetic type. Since this derivative is directly proportional to the magnetic contribution to the specific heat it is quite appr0priate to compare Fig. 32 to Fig. 33 which shows the specific heat data taken by Forstat, Love, and McElearney. Because of the presence of two peaks in both the susceptibility data and the specific heat data it is concluded that there are indeed two transi- tions, both magnetic in character. The entropy change calculated by Dr. Forstat from the magnetic contribution to the Specific heat is R 1n 3, which is consistent with S = l for nickel and all spins ordered at T = 00K. Possible explanations of the presence of two transi- tions are that there are two spin systems which order at different temperatures or that there are two types of ordered states which are stable in different temperature ranges. However, without a detailed knowledge of the crystal structure or the spin arrangement it is impossible at this time to make any definite statements concerning these transitions except that they are of the antiferro- magnetic type. X ray diffraction and proton magnetic resonance experiments are being carried out by Dr. Spence and his group and should shed considerable light on this problem. 76 33.4.5.2 0.¢ .8 832352 .2 .2533“. S ta .35.“. 5.33 e lP/(J."X)P 0N. 77 rfiIIIIitI[OTIII1IITIT1III 00 009° 0 0 080 0o 4.0 3.0 2.0 T (deg. K) Ni(tulsBr2 . I I I I I I I I I J I I I I I I I I I 14 I L ‘0. °. '2 0. ‘0. 0 N N "" "' o ()lo slow/100) do heat capacity of LG Fig. 33. Molar 78 D. Others In addition to the three materials already discussed, some of the other members of this series of materials were briefly investigated, or should be. 1. Ni(tu)u01 Hare and Ballhausen2O have reported 2. that the magnetic susceptibility of this material obeys the relation x = 1.1A/(T + 6C) in the temperature range from 77°K to 3705K. It was because of this information that the low-temperature properties of this series of compounds was investigated. The proton resonance of Ni(tu)UCl2 was observed down to 0.40‘K- The resonance lines attained a splitting of about 100 gauss but showed no indication of an antiferromagnetic transition. Low— temperature susceptibility measurements should yet be made on this material. 2. Fe(tu)uCl The proton resonance investigation 2. of this material gave no evidence for a magnetic transition above 0 403K. The resonance lines showed a small and rather constant splitting. No susceptibility measurements have yet been made. 3. Co(tu)uBr2: No magnetic transition above 0.409K was indicated by proton resonance. This material has not yet been subjected to magnetic susceptibility measurements. A. Fe(tu)6Br Magnetic susceptibility measurements 2: on a powdered sample indicates an anomaly near 1.03K which should be investigated further. 79 5. An unsuccessful attempt was made to grow a thiourea-coordinated manganous bromide. Perhaps this could be pursued further. 6. In addition to the chlorides and bromides con- sidered, the iodides are also worthy of investigation. In particular, Ni(tu)6I2 grows crystals similar in appear- ance to Ni(tu)6Br2. V. CONCLUSION The experimental results discussed in Chapter IV indicate that this work has met the objectives set forth in Chapter I. With the apparatus constructed as described in Chapter II the magnetic susceptibility of three new antiferromagnetic materials was measured. Each of these materials is now the subject of further Study for the purpose of providing a more complete description of its antiferromagnetic state, which will add to the body of knowledge concerning antiferromagnetic materials upon which a more satisfactory theory of anti- ferromagnetism can be based. One unique feature of these materials is the abundance of protons. This makes the analysis of proton resonance data much more difficult than for hydrated materials but allows a more complete mapping of the internal magnetic fields, which may be useful in estimating the distribution of magnetic moment within the crystal giving valuable information concerning the wave functions of the magnetic electrons. It is of some interest to compare the transition tem- peratures 0f the thiourea—coordinated materials with those of the corresponding water-coordinated compounds. Table 2 80 81 lists the transition temperatures of the hydrated materials and the thiourea—coordinated materials, along with their ratios. Table 2.--Comparison of the transition temperatures of the thiourea- and water-coordinated compounds. Compound TN(3K) Compound TN(3K) Ratio COCl2 - 6H20 2 28 Coftu)uCl2 0.93 2 “5 MnCl2 ° UH20 1.68 MH(tu)u012 0.56 3.0 NiBr2 . 6H20 6.5 Ni(tu)6Br2 2.2 2.9 NiCl2 6H20 6.2 Ni(tu)uCl2 < O.“ b 15.5 F8012 “ 5H2O 1.0 F€(L0)u012 “ 0.4 ‘ 2.5 CoBr2 ° 6H20 3.08 Coitu)uBr2 < 0.3 : 7.7 For the three compounds studied in detail the ratio of the transition temperature of the hydrated to the thiourea- coordinated materials is quite uniform. Thus for each pair of materials the ratio of exchange parameters must also be approximately the same. This might also be true for the ferrous chloride type materials. However, for the nickel chloride and cobalt bromide compounds this similarity breaks down. For Ni(tu)uC1 the difference probably results from 2 the fact that its crystal structure is different from [’11 Co(tu)uCl and Mn(tu)uCl The space group of Ni(tU)buL2 2 2’ is IA while the space group of the others is Pug/n. 82 Structural differences will also most likely account for the fact that Co(tu)uBr behaves differently than 2 Ni(tu)6Brp. No further statements should be made until the crystal structure analysis of these materials is completed. REFERENCES 83 10. ll. 12. 13. 1A. 15. REFERENCES L. Neel, Ann. phys. 11, 6A (1932). H. B. G. Casimir, w. J. de Haas, and D. de Klerk, Physica g, 241 (1939). w. J. de Haas, and C. J. Gorter, Proc. K. Akad. Amsterdam 1, 676 (1930). w. R. Abel, A. C. Anderson, and J. C. Wheatley, Rev. Sci. Instr. 3;, AAA (196A). W. L. Pillinger, P. S. Jastrum, and J. G. Daunt, Rev. Sci. Instr. 29, 159 (1958). T. Nagamiya, K. Yosida, and R. Kubo, Advances in Physics A, l (1955). A. B. Lidiard, Reports on Progress in Physics 11, 201 (195A). C. Domb, Advances in Physics 9, 1A9, 2A5 (1960). P. w. Anderson, Phys. Rev. _9, 350 (1950). J. H. Van Vleck, Jour. de Phys. et le Radium 12, 262 (1951). P. w. Anderson, "Exchange in Insulators" in Magnetism edited by George T. Rado and Harry Suhl (Academic Press Inc., New York, 1963) Vol. 1, pp. 25—83. Junjiro Kanamori, "Anisotropy and Magnetostriction of Ferromagnetic and Antiferromagnetic Materials” in Magnetism edited by George T. Rado and Harry Suhl (Academic Press Inc., New York, 1963), Vol. 1, pp. 127— 203. L. F. Bates, Modern Magnetism, (Cambridge University Press, 1961) Ath ed., p. A3. T. Oguchi, J. Phys. Soc. Japan 6, 27 (1951). M. F. Sykes and Michael E. Fisher, Physica 22) 919 (1962). 8A 16. 17. 18. 19. 20. 85 Michael E. Fisher and M. F. Sykes, Physica 28, 939 (1962). T. Oguchi, J. Phys. Soc. Japan 6, 31 (1951). M. F. Sykes, J. Math. Phys. 2, 52 (1961). Michael E. Fisher, Phil. Mag. 1, 1731 (1962). C. R. Hare and C. J. Ballhausen, J. Chem. Phys. 59. 788 (196A). APPENDICES APPENDIX A SERIES EXPANSIONS FOR THE PARALLEL SUSCEFTIBILITY OF SOME ANTIFERRCMAGNEIIC ISING LATTICES a. 88 l. Two-dimensional honeycomb lattice T < To = 1.5186519 111 k From Eq. (8.7), Ref. 15: x(T) = mg {14313 + 12y5 + By6 + 148327 + 963:8 + 320319 kT + 888y10 + 27A8yll + mm12 + 2631403!l3 + 0 1213 Z (y/yc)n+2 . n=12 n(n + 1) exp (—2 IJl/kT) E: D“ (D "S (D t<1 ll 0.2679492 ‘< II T > Tc = 1.5186519 111 k From Eqs. (7.6) and (7.7), Ref. 15: - 2A x(T) = NEE (l-JR v)-% (1 + 3v2)-é exp 2 fntn kT n=1 + R2u(t)]3 _ 2 3 R2U(t) — - [0.18ut + O.U2t + 0.150t —o.07t“] Z t“? r=6 r(r+1) Where: v = tanh (J/kT) t = v/vC V = -O-5773503 C 89 and the fn are given by: f1 = +0.0179U9 f9 = —0.0UO772 f17 = -0.009535 f2 = —O.2SOOOO f10 = -0.05000O f18 = -0.0192U3 f3 = +0.005983 fll = -0.01262u f19 = -0.00716O fu = -0.083333 f12 = +0.005194 f20 = +0.002A36 f5 = -0.07339O f13 = -0.0l7627 f21 = -0.006252 f6 = -0.083333 fl“ = -0.030227 f22 = -0.0132UU f7 = _0.015769 f15 = -0.010208 f23 = —0.009891 f8 = -0.00A63O f16 = +0.003629 f2“ = +0.001889 2. Two—Dimensional Plane Square Lattice a. T < T0 = 2.2621852 [J] k From Eq. (6.7), Ref. 15: x(T) = Nm2 {:Ayu + 16y8 + 32ylO + 156yl2 + 608y1u kT + 0.08“ E (y/yc)2n+2 n= n(n + l) where: y = exp (-2 IJI/kT) yC = 0.u1u2136 b. T > TC = 2.2621852 Iii Eq (5.18), Ref. 15, should be corrected to read: 16 w 2 2 -% n (-t) x(T) = Nm (l-2v-v ) exp 2 d t —2.o22 kT n=1 “ n217 “(n+15 where: v = tanh (J/kT) t = v/[vcl v = -0.AlA2l36 90 and the dn are given by: d1 = +0.20711 d9 = +0.02203 d2 = -0.21AA7 le = -0.01788 d3 = +0.08291 dll = +0.01UBA dA = -0.08A63 d12 = -0 01268 d5 = +0.0A999 d13 = +0.01095 d6 = -0.04251 dlu = -0 00959 d7 = +0.0357l d15 = +0 008A0 d8 = -0.02811 dl6 = -0.007A3 3. Three-Dimensional Simple Cubic Lattice < = . l '2 | l a. T TC 4.5 03 J k From Eqs. (A.3) and (A A), Ref. 16: x(T) = Nm2 {1:516 .83:12 + 37.72uylu/(1 + 3.23312) + 22.28571“ kT 1 1 -22 15yl6 + 22.33y‘8 + 23 23y2O + 93.925722 49.6143)”1 + 98.00;:26 + 295.88y28 + 0.329 E ~<.y/y,,)‘r1 n=15 n(n+l) where: y = exp (-2!JI/kT) yC = 0.6A183 b. T > TC = 9.51032 ii! Eq. (3.9), Ref. 16, must be corrected to read: 2 -i 11 n x(T) = Nm (1 + t) * exp "- 1 f t -R (t) ___ i n 11 kT l—n- 91 where: v = tanh (J/kT) t = v/vc v = -0.21815 C Eq. (3.7) must be corrected to read: 11 R (t) = 0.338 1 + (l-t) 1n(1-t) - t“ 11 [. t n21 n(n+1):| and the fn are given by: fl = 0.058901 f7 = 0.0059701 f2 = 0.053926 f8 = 0.0046536 f3 = 0.019362 f9 = 0.0037844 f“ = 0.013552 f10= 0.0030197 f5 = 0.011461 f11= 0.0026244 f6 = 0.0078652 4. Three-Dimensional BodyeCentered Cubic Lattice a. T < T0 = 6.35080 LgL k From Eq. (4.7), Ref. 16: x(T) = gm: 4y8 -8y16 + 112y2O -256y22 + 96y2“ + R2u(y;} kT Eq. (4.8) must be corrected to read: R2u = 1.22. z (y/yc)2“ n=l3 n(n+1) exp (-2 IJl/kT) where: y y 0-72985 C 92 b. T > TC = 6.35080 Ii! Eq. (3.9), Ref. 16 must be corrected to read: 1‘1 1 fnt -R9(t):] Eq. (3.7), Ref. 16 must be corrected to read: kT ll (\wa c; X(T)= Nm2 (l+t)* exp [:- n 9 R9(t) = 0.30 1 + (l—t) ln(1-t) — Z t“ t n=l n(n+1) where: t = V/Vc v = tanh (J/kT) vC = -0.15617 and the fn are given by: fl = —0.000625 f6 = +0.006726 f2 = +0.039648 f7 = +0.005124 f3 = +0.020097 f8 = +0.003954 fa = +0.012693 f9 = +0.003400 f = +0.008782 93 Table 3.——Para11e1 susceptibility for honeycomb, plane square, simple cubic, and body-centered cubic lattices as computed from the Ising model series expansions. T/Tc x/xC Body— Plane Simple centered Honeycomb Square Cubic Cubic .00 .000 .000 .000 .000 .05 .000 .000 .000 .000 .10 .000 .000 .000 .000 .15 .000 .000 .000 .000 .20 .000 .000 .000 .000 .25 .000 .000 .001 .002 .30 .000 .001 .005 .008 ' .35 .001 .003 .017 .023 .40 .004 .009 .038 .050 .45 .011 .022 .071 .088 .50 .025 .043 .116 .139 .55 .046 .076 .170 .200 .60 .079 .119 .232 .268 .65 .123 .174 .304 .340 .70 .182 .240 .382 .413 .75 .255 .321 .463 .489 .80 .343 .410 .551 .564 .85 .450 .516 .643 .642 .90 .581 .636 .740 .723 .95 .741 .782 .850 .819 1.00 1.000 .1000 1.000 1.000 1.05 1.256 1.201 1.024 1.017 1.10 1.382 1.314 1.028 1.015 1.15 1.485 1.390 1.025 1.009 1.20 1.556 1.441 1.019 1.000 Table 3.--Continued. 94 T/TC x/xC Body- Plane Simple centered Honeycomb Square Cubic Cubic 1.3 1 643 1.507 1.001 .976 1.4 1 697 1.539 .980 .950 1.5 1.726 1 550 .956 .926 1.6 1.740 1.550 .932 .898 1.7 1.742 1 540 .908 .873 1.8 1.738 1.526 .886 .848 1.9 1 728 1.509 .863 .826 2.0 1.716 1.488 .842 .802 2.1 1.698 1 467 .819 .781 2.2 1 681 1.444 .799 .761 2.3 1.660 1.420 .781 .740 2.4 1.640 1.397 .761 .721 2.5 1 619 1.373 .744 .704 2.6 1 596 1.350 .726 .687 2.7 1.575 1 327 .710 .669 2.8 1.552 1.303 .693 .654 2.9 1 531 1 282 .679 .638 3.0 1 507 1.260 .664 .625 3.1 1 486 1.238 .649 .611 3.2 1 463 1.217 .636 .599 3.3 1.443 1 196 .623 .585 3.4 1.423 1 176 .611 .573 3.5 1.400 1.156 .599 .563 3.6 1.382 1.137 .587 .551 3.7 1.362 1.118 .576 .540 3.8 1.341 1.101 .564 .530 3.9 1.323 1.084 .555 .520 Table 3.--Continued 95 T/TC x/xC Body- Plane Simple Centered Honeycomb Square Cubic Cubic 4.0 1.304 1.068 .544 .510 4.2 1.267 1 035 .526 .492 4.4 1.233 1.004 .507 .475 4.6 1.201 .975 .491 .460 4.8 1.169 .946 .475 .444 5.0 1.138 .919 .461 .430 APPENDIX B SUSCEPTIBILITY DATA IN TABULAR FORM 96 97 Table 4 --Magnetic susceptibility data for Co(tu)uC12 along the [001] axis in units of cc,/mole TOK X001 TOK x001 0.459 .0630 1.212 .566 .478 .0661 1.31 .574 .512 .0676 1 393 .577 .547 .0718 1.495 .579 .581 .0850 1.604 .579 .613 .101 1.732 .575 .638 .115 1.814 .571 .666 .135 1.899 .567 .700 .162 1.999 .562 .727 .185 2.120 .555 .750 .208 2.220 .547 .776 .236 2.290 .544 .800 .263 2.410 .532 .830 .296 2.470 .530 .843 .331 2.602 .519 .861 .354 2.786 .505 .875 .376 3.000 .487 .895 .412 3.194 .472 .915 .432 3.406 .459 .942 .495 3.595 .446 .979 .421 3.786 .436 1.050 .543 3.998 .425 1.120 .555 4.194 .418 98 Table 5. --Magnetic susceptibility data for CO(tu) “C12 along the [110] axis in units of cc/mole. T°K x110 T°K X110 .455 .222 1.694 .377 .500 .224 1.814 .369 .549 .225 1.938 .360 .595 .229 2.083 .348 .653 .236 2.107 .348 .700 .245 2.195 .342 .750 .257 2.326 .333 .800 .272 2 470 .322 .842 .292 2 602 .314 .888 .317 2.782 .304 .919 .338 3.004 .292 .955 .366 3.221 .283 1 007 .381 3.411 .275 1.194 .394 3.608 .270 1.292 .391 3.802 .264 1.391 .388 4.012 .257 1.560 .382 4.2 .251 99 Table 6. --Magnetic susceptibility data for Co)tu) “Cl2 along the [100] axis in units of cc/mole. T°K x100 T°K X100 0.450 .266 1.571 .423 .515 .267 1.810 ‘.411 .613 .274 2.001 .398 .700 .289 2 197 .384 .750 .302 2.398 .371 .800 .318 2 602 .359 .851 .347 2.800 .348 .880 .361 3.000 .337 .911 .386 3.200 .328 .939 .404 3.396 ~3l9 .970 .414 3.608 .310 1 000 .424 3 802 .303 1.085 .432 4.006 .295 1 201 .433 4.2 .288 1.404 .430 Table 7.—-Magnetic susceptibility data for powdered Co(tu)uCl 100 in units of cc/mole. 2 T°K Xp T°K Xp 0.452 0172 1.812 .439 .555 .180 2.000 .428 .643 .202 2.200 .414 .720 .228 2.420 .399 .800 .271 2.602 .388 .855 .320 2.800 .376 .904 .366 3.000 .365 .951 .420 3.193 .355 1.007 .435 3.398 .346 1.090 .448 3.610 .337 1.170 .451 3.804 .329 1.318 .455 4.017 .321 1.401 .454 4.185 .314 1.592 .449 Table 8. —-Magnetic susceptibility data for Mn(tu) l4Cl [001] axis in units of cc/mole. 101 2 along T°K x001 T°K X001 .422 1.36 1.506 1.14 .472 1.39 1.658 1.09 .513 1.41 1.685 1.08 .543 1.43 1.810 1.03 .573 1.46 1.893 1.008 .621 1.49 2.003 .971 .672 1.48 2.095 .944 .722 1.47 2.290 .885 .770 1.46 2.493 .831 .832 1.44 2.694 .783 .880 1.42 2.900 .737 .938 1.39 3.162 .684 1.014 1.35 3.396 .642 1.107 1.31 3.608 .606 1.182 1.27 3.806 .577 1.327 1.21 4.009 .550 1.352 1.20 4.2 .528 1.480 1.16 102 Table 9 —-Magnetic susceptibility data for Mn(tu) ”012 along the [100] axis in units of cc/mole. T°K T°K X100 X100 .477 1.904 1.663 1.688 .498 1.998 1.715 1.654 .532 2 043 1 770 1.634 .558 2.126 1.858 1.590 .583 2.217 1.873 1.568 .625 2.258 1.986 1.532 .670 2.262 2.012 1.511 .720 2.257 2.090 1.473 .770 2.244 2.090 1.464 0837 2.213 2.216 1.414 .911 2.163 2.309 1.379 .961 2.110 2.371 1.355 1.050 2.042 2.529 1 300 1.072 2.032 2.789 1.220 1.226 1.941 2.989 1.167 1.347 1.861 3.299 1.090 1.453 1.798 3.510 1.045 1.458 1.800 3.665 1.015 1.556 1.744 4.016 . .951 1.605 1.714 4.2 .922 103 Table 10.-—Magnetic susceptibility data for Mn(tu)u012 along the [110] axis in units of cc/mole. T°K X110 T°K X110 .433 1.73 1.208 1.85 .472 1.82 1.383 1.75 .512 1.91 1.395 1.74 .519 1.94 1.515 1.67 .533 1.95 1.596 1.63 .552 1.99 1.701 1.56 .560 2.04 1.794 1.53 .565 2.06 1 855 1.48 .581 2.13 2.007 1.43 .606 2.17 2.125 1.38 .637 2.18 2.375 1.27 .656 2.18 2.650 1.18 .703 2.17 2.891 1.11 .746 2.16 3.143 1.04 .777 2.14 3.414 .982 .851 2.11 3.444 .980 .926 2.05 3.633 .937 1.012 2.00 3.904 .880 1.105 1.93 4.20 .835 104 Table 11.--Magnetic susceptibility of Ni(tu)6Br2 along the b axis in units of cc/mole. T°K 104xb T°K 104xb 1.191 910 2.264 857 1.398 904 2.327 859 1.508 897 2.401 863 1.596 893 2.497 867 1.700 885 2.590 872 1.802 881 2.809 885 1.875 877 2.992 896 1.909 875 3.196 907 1.955 873 3.393 916 1.993 870 3.575 923 2.047 866 3.769 932 2.094 864 4.007 940 2.139 863 4.188 943 2.178 857 2.220 858 105 Table l2.—~Magnetic susceptibility of Ni(tu)6Br2 along the a' axis in units of cc/mole. , _ x a, - 0.98 xa,. T°K louxa' lOux'a. T°K 10uxa. lOux'av 1.216 742 727 2.278 775 760 1.404 741 726 2.327 778 763 1 504 741 726 2.405 781 765 1.535 743 728 2-590 787 771 1.750 744 729 2.780 791 776 1.837 746 731 2.996 795 779 1.913 748 733 3.190 798 782 1.977 752 737 3.385 799 783 2.036 756 741 3.585 800 784 2.085 759 744 3.791 799 783 2.137 762 747 3.994 798 782 2.181 767 752 4.200 795 779 2.222 771 755 106 Table 13.--Magnetic susceptibility of Ni(tu)6Br2 along the c axis in units of cc/mole. X0 = 0.98 X0- T°K 104xC 104x; T°K 10uxC 104x; 1.20 376 370 2.226 667 654 1.306 387 379 2.255 687 673 1.427 402 394 2.294 700 686 1.548 422 413 2.332 712 697 1.638 437 428 2.375 725 710 1.726 458 448 2.477 751 736 1.801 474 464 2.602 776 760 1.919 507 497 2.985 831 814 1.966 523 512 3.184 851 834 2.009 551 540 3.398 866 849 2.061 576 565 3.597 879 861 2.102 593 582 3.802 889 871 2.149 613 601 3.988 894 876 2.185 649 636 4.195 900 882 107 Table 14. --Magnetic susceptibility of Ni(tu) 6Br2 along direction f in units of cc/mole. X% = 1 02xf- T°K 10“).f 10”x% T°K iouxf 104x; 1.299 633 646 2.205 721 736 1.382 637 650 2.254 731 746 1.479 641 654 2.326 737 751 1.582 647 660 2.405 743 758 1.682 654 667 2.503 750 765 1.787 663 676 2.602 755 770 1.831 670 683 2.805 766 781 1.884 672 686 3.003 775 791 1.933 676 690 3.178 781 797 1.983 683 696 3.393 787 802 2.026 691 705 3.590 791 806 2.071 698 712 3.795 796 812 2.114 704 718 3.965 802 818 2.158 709 724 4.191 806 825 108 Table 15 --Magnetic susceptibility of Ni(tu)6Br2 along the a axis in units of cc/mole. X' = 1.02 X . a a T°K 104xa 104x; T°K 10“).a 104x; 1.235 401 409 2.230 651 665 1.296 406 414 2.280 672 685 1.417 418 427 2.313 684 697 1.510 430 438 2.400 706 720 1.606 444 453 2.494 726 741 1.700 461 470 2.601 744 759 1.811 483 493 2.700 760 775 1.854 492 502 2.802 _773 788 1.911 505 516 3.000 795 811 1.950 518 529 3.200 815 831 1.994 532 543 3.400 830 847 2.041 560 571 3.604 841 857 2.080 574 585 3.804 850 866 2.115 589 600 4.000 856 873 2.161 606 618 4.20 859 876 2.196 633 646 Table 16.--Magnetic susceptibility of powdered Ni(tu)6Br in units of cc/mole. 109 2 4 4 T°K 10 Xp T°K 10 Xp 1.257 700 2.232 762 1.387 701 2.290 771 1.507 701 2 325 774 1.625 704 2 394 783 1.735 708 2.607 800 1.823 711 2.793 813 1.898 715 2.993 821 1.958 721 3.190 829 2.015 729 3.395 835 2 061 735 3.591 839 2.106 741 3.797 843 2 150 747 3.987 844 2 190 757 4.200 844 "I7444444144