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LI BRA; 1-" Michigan 3:15;: . $5 Universrty j"; V..- T— This is to certify that the thesis entitled THE MAGNETIC BEHAVIOR OF SEVERAL TRANSITION METAL SALTS - SOME LOW-DIMENSIONAL EFFECTS presented by Charles Ralph Stirrat has been accepted towards fulfillment of the requirements for PhoDo degree in PhYSiCS ’ f J Major professor Date August 21 ,1974 0-7639 751W '6 If HUM? 8 SUNS' . “'5 BOOK BINDERY INC. 4 LIBRARY BINDERS r--n-“-o um IIIII 44 5—.___. rvfis ———. ‘5‘ t ‘4 ABSTRACT THE MAGNETIC BEHAVIOR OF SEVERAL TRANSITION METAL SALTS- SOME LOW-DIMENSIONAL EFFECTS By Charles Ralph Stirrat Low-temperature magnetic measurements are reported on three transition metal salts, ((CH3)3NH)CuC1 °2H O, ((CH NH) 3 2 3):. S-ZHZO, and NiIZ-GHZO. These measurements include elec- CoCl tron spin resonance, nuclear magnetic resonance, and magnetic susceptibility in zero and applied field. The experimental results have been interpreted in terms of existing theories. Trimethylamine capper chloride orders magnetically at TC=0.157i0.003 K. Above the transition the susceptibility is given by the high-temperature expansion for a two-dimensional square Heisenberg lattice with JF/k=0.28t0.02 K. Below Tc the crystal exhibits behavior characteristic of weak anti- ferromagnetic coupling between the ferromagnetic layers. The successive spins along a are canted slightly away from a due to the presence of two inequivalent anisotropic g-tensors. The magnetic susceptibility of trimethylamine cobalt chloride in the ordered state(T0(doub1et lowest). Page 96 INTRODUCTION The magnetic properties of transition metal salts have been studied by experimentalists and theorists for a long time. Historically the interplay between experiment and theory has been varied and complex. Sometimes an experiment will produce new and unexpected results which require the de- velopment of a new theory to explain them. At other times theorists have predicted behavior long before it was experi- mentally observed. In this thesis experiments designed to substantiate existing theories are reported. Failure of the theories to predict experimental results indicates the inap- propriateness of a particular model or the need for expanded theoretical work. An exact theoretical description of the magnetic prOper- ties of any real crystal is an impossibility because of the number of particles involved. Therefore most theories project out of the exact description those terms which adequately des- cribe the situation of interest and yet are amenable to cal- culation. One assumption, currently the subject of much theo- retical work, is that the exchange interaction in one or more directions is negligible. This leads to a "low-dimensional" model in which a linear(one-dimensional) or planar(two-dimen- sional) lattice, rather than a three-dimensional lattice, is considered. Experiments on three salts; trimethylamine copper chlo- ride, trimethylamine cobalt chloride, and nickel iodide; are 1 2 reported in this thesis. The interpretation of the results involves models with varied assumptions as to what interac- tions are important. The trimethylamine salts exhibit low- dimensionality while nickel iodide does not. A short summary of the theoretical models and some of the assumptions they imply is given in Section I. Section 11 describes the experimental apparatus and the techniques used for the measurements reported in this thesis. The experiments include magnetic susceptibility, electron spin resonance(ESR), and nuclear magnetic resonance(NMR). The experimental results and interpretation of measure- ments on ((CH3)3NH)CuC13-2HZO are given in Section III. A two-dimensional-square Heisenberg lattice is appropriate for trimethylamine copper chloride. In Section IV results of field dependent magnetic suscep- tibility measurements on ((CH3)3NH)C0C13-2HZO are reported. Prior work has shown this salt to be described by a two-dimen- sional rectangular Ising latticel. The salt exhibits unusual metamagnetic behavior in applied field2 and the measurements included here reflect this behavior. In Section V ESR and susceptibility experiments are des- cribed which show that the exchange interaction in NiIz'6H20 is less important than the electrostatic crystal field inter- action. NMR measurements are also discussed. In addition, for each of the three salts, three dimen- sional magnetic order was observed at sufficiently low tem- peratures. I. THEORETICAL BACKGROUND The magnetic properties of transition metal salts are due to the orbital and intrinsic spin angular momenta of un- paired electrons. Therefore any theoretical model, other than a purely phenomenological one, must describe this system of 3,4 electrons. A general Hamiltonian can be written: H=HINTRA " “c1: H"so +7"z +7"1m +7'IHF (1'1) The first term, NINTRA’ includes the intraatomic Coulomb interactions between the electronic and nuclear charges. Most theories treat this term using a product of single electron wave functions as a first approximation for the intraatomic wave functions. These one-electron functions are the solu- tions for the electron moving in the Coulomb potential of the nucleus. The ground state of the ion is determined by ”INTRA but its configuration is summarized by Hund's rule. The rule states that the minimum energy configuration will have the maximum possible spin S, and that the orbital angular momen- tum, L, will have the largest value that can be associated with that maximum 8. The Coulomb interactions between each electron and the charges external to the ion are included in the second term, NEF' This crystal field term treats the potential V(r) seen by the electron as being due to point charges at the sites of other ions. A more complete calculation involves treating the spatial distribution of charge on the neighboring sites. Such a ligand field theory can be emplOyed when the 3 4 neighboring ion orbitals overlap the electron orbitals of interest. It is easier and usually sufficient to use the crystal field approach. The symmetry of the lattice is re- flected in V(?) so that group theory provides a means of de- termining the degeneracy of the levels resulting after'fl-CF splits the degenerate ground state of the isolated ion. The combination of ”INTRA and ”CF allows the characteri- zation of the ion's ground state by a total orbital angular momentum, f, and total Spin angular momentum, S. The spin-orbit term,1H%0, represents the interaction between the orbital angular momentum and the intrinsic spin angular momentum. The contribution for each ion can be writ- ten 'Héo = xii-Si (1.2) where I and S are defined above and A is the spin-orbit para- meter. Although ”80 couples 1 and S, for the iron group tran- sition metals,‘7«l'cF normally dominates N50 and the characteri- zation of the ground state in terms of t and S is still appro- priate. 74%, the Zeeman term, involves the interaction between the . . + . magnet1c moment of the ion, "i’ and a un1form external magne- tic field, H. Thus ‘14; = Iii-ii = uB(Ii+2§i)-fi (1.3) where "B is the Bohr magneton. The magnetic dipole-dipole contribution,14%g, for the magnetic moments at sites i and j separated by Iii is given by + + + A + A .. p.°u.-3(u.'r..)(u.°r..) 13= 1 J 1 1) j 13 7+DD r3 (1.4) 11 The hyperfine term,74HF, involves the interaction between the nuclear magnetic moment and the electronic magnetic mo- ment. For most problems it is only a small perturbation on the electronic energy levels, but it can be very important in considering the nuclear energy levels(Section V). It consists of an orbital term, a contact term and a dipolar term. The last two are included in the magnetic hyperfine coupling ten- OI. sor, A. Thus NHP = '“BW11 111% I T33 (1.5) where YN is the nuclear gyromagnetic ratio and I is the nu- clear spin.'HHF is considered negligible in treating the elec- tronic magnetic properties. The use of one-electron wave functions for ”INTRA fails to predict behavior resulting from the fact that the solution should be antisymmetric. Adding a term of the form 13' 7"lax = -2JijSi-Sj (1.6) will include some of the effects lost by failing to use anti- symmetric solutions. This is known as the Heisenberg exchange Hamiltonian. Jij’ the exchange energy between sites i and j, is greater than zero for ferromagnetic exchange and less than zero for antiferromagnetic exchange. It is also 6 possible to have "superexchange" where the magnetic ion wave- functions overlap the wavefunctions of an intervening anion, thus increasing the effective exchange between the two sites. Because of its many-body nature, the solution of this general Hamiltonian (1.1) is impossible; however, it is often possible to project out of'N the terms that adequately de- scribe the properties of interest. This is done by using a Spin Hamiltonian3 in which the orbital contribution does not appear explicitly. A spin treatment of the 3d transition metal salts is especially appropriate because the expectation value of each component of L for these ions in a crystal field is zero. This "quenching of the orbital angular momentum" by the crystal field greatly reduces the contribution of the orbital angular momentum to the magnetic moment. Although (Lx)= (LY)=(Lz)-0, the magnitude of L may be nonzero. If the crystal has a symmetry such that the orbital de- generacy of the ground state is completely lifted, then to first order in A, (E)=0. The higher order orbital contribu- tions can be included as an anisotropic g-tensor: 131 = "B .g'gi: (1'7) where g does not differ greatly from the free electron value, g=2.0023. The spin Hamiltonian corresponding to ”Z'IJSO' and- the zero-field splitting of the spin states contained inWCF can be written, for a single ion spin, Si, as 14; = uBfi-‘E-S‘i + 93:1 + usii-sii). (1.8) 7 It is often possible to use the spin Hamiltonian forma- lism even if the orbital degeneracy of the ground state is not lifted by "CF and L40. S must be replaced by an effective spin, 3, which is chosen so that the number of states into which the ground state splits in an applied field is (25+l). In such cases the spin-orbit interaction produces first order shifts of the energy, and thus the g-factors will differ sig- nificantly from the free electron value. For example in a cubic crystal field Co++ has a ground state with L=l, S=3/2. The spin-orbit interaction splits this lZ-fold degenerate level so that a doublet lies lowest. Thus at low temperatures Co++ can be treated as having effective spin s=1/2 with a very anisotropic g-tensor. Using an effective spin Hamiltonian implies no zero-field splitting so + «9+ Hi " "B 8'51 (1'9) and H: = "B fi-‘g-Ei (1.10) Thus the spin Hamiltonian formalism can be used either with S or s depending on the orbital degeneracy of the ground state. In the rest of this section the spin is denoted by S but if L40 then E should be used along with equations (1.9) and (1.10) instead of (1.7) and (1.8). In addition to the single spin terms considered above, one may include two-spin interactions, exchange and dipole- dipole in the spin Hamiltonian formalism. In the term N113?) one uses ii determined by equation (1.7). The exchange contribution is HEX = 1/2 F9413 (1.11) 11 where the sum is over all pairs of spins, ifij, but it is usually sufficient to consider nearest neighbor spins only. Although a simple form for the exchange term is given in equation (1.6), the most general form is written in terms of the exchange dyadic,tT: 1j’ as 74%)} =-z§i-‘.'1"ij-§j (1.12) If Sij has only symmetric elements, then (1.12) reduces to the Heisenberg Hamiltonian, 11 = - ii ii 13' )IHEIS 2(‘Jxxsixij + Jnyinjy + Jzzsizsjz) (1°13) The case J;;=J;;=O is known as the Ising Hamiltonian. If the antisymmetric elements ofAUZj are nonzero, the antisymmetric part of equation (1.12) can be written in the form 1' _ x HA; -§ij-§i s). (1.14) which may lead to canting of the spins. In many problems the antisymmetric terms may be neglected. Thus the total spin Hamiltonian can be written as = i ’ ij 1 13 ii = ii 11' where'NS “EX +‘HDD' The eigenvalues of‘HS can, in principle, be found and used to calculate the statistical average for such magnetic quantities as magnetization, magnetic specific heat, and 9 magnetic susceptibility. This is done using the total density operator p. The thermal equilibrium value of an operator Q is given by5 -74 /k =Tr(pQ) = “(Qe S J. (1.16) In practice the eigenvalues for?)S can be found only for special cases since the problem is still a many-body problem. A number of methods will be used here to handle this problem including single ion treatments, molecular fields, high-tem- perature expansions, and low-dimensionality. If Néj<<flé it is often possible to consider the case of noninteracting spins. Then the eigenvalues of'Né can be found since ”S: 2?; and the density operator becomes p- 291 so i i i (Q) = Z Tr(Qe-NS/kT) . - i 1 Tr(e-NS/kT) (1.17) A system of noninteracting magnetic moments describable by'NS with N§J=0 will have a magnetic susceptibility, xuvs that is temperature dependent. Such behavior is described as 3.9 paramagnetic. At high temperature the zero-field suscepti- bility of a system of spins, each described by equation (1.8), is given by the Curie-Weiss law, C "V xvv = 1C3“' (1°13) uv where Cuv=NogfiquS(Sf1)/3k and 9 contains terms proportional to D and E. If NE§ is not negligible one may assume that the exchange 10 produces an effective field at site i proportional to the av- erage magnetization of the rest of the sample. Thus in the molecular field approximations for isotropic exchange 11' _ . 74m A, 211]. $3.) 131, (1.19) whereIQSj) is determined from M=NguB>J2zJ3, can be approximated by a linear chain with nearest neighbor exchange, J1. A "low-dimensional" model may apply for structures in which the magnetic ions are separated by nonmagnetic organic complexes in one or two directions but not in the others. Conversely the presence of certain anions between the magnetic ions in one direction can enhance the exchange in that direc- tion through the mechanism of "superexchange." For spins situated on a primitive orthogonal lattice, there will be a one-to-one correspondence between the Jij's and J1, J2, and J3. If the spins are not on such a lattice, the connection will not be as simple but J1, J2, and J3 can often still be used. Consider a three dimensional lattice composed of layers in which the spins form a square lattice but the spins in successive layers are not directly above their neighbors. Then J1=J2 for spin i would equal Jij where j refers to the four nearest neighbor spins in the plane. J3 might correspond to an effective exchange between spin i and several near spins in the plane above or below spin i. In conclusion, although the complete description of the magnetic properties of a transition metal salt must involve a description of the orbital and spin angular momentum of the unpaired electrons, it is often possible to obtain important information using only a spin Hamiltonian. Even the solution 13 of this problem is often too dificult if two-spin processes, like exchange, are important. In these cases it is necessary to use one or more of the methods outlined above including molecular fields, high-temperature expansions, or low-dimen- sionality to make the problem tractable. II. EXPERIMENTAL TECHNIQUES AND APPARATUS A. Magnetic Susceptibility Magnetic susceptibility was measured by a mutual induc- tance technique 7’8 at 17 Hertz using a Cryotronics Model 17B9 mutual inductance bridge. Mutual inductance coils were wound to fit three different cryostats. In each case the coils were wound with a single primary and multiple secondary windings with sections wound oppositely so that the total mutual inductance of the empty system was approximately zero. The measuring field was less than 5 0e. Absolute measurements of x(T,6) were made in an immersion 4He cryostat in which the sample could be removed from the coil at each temperature. The sample was mounted on an epoxy rotor which had a relative accuracy of 1°, and an absolute accuracy of 13°. This apparatus was calibrated with a single crystal of Ferric Ammonium Alum to give an accuracy of approx- imately 10'6 emu/gm for a 0.1 gm sample. Sample masses were between 0.1 and 0.3 gm. Temperatures from 0.025 K to 1.5 K were achieved with a 3He-4He dilution refrigerator. The sample is fixed so that only relative values were obtained. More-or-less absolute values were achieved by normalizing the results obtained with this apparatus near 1.5 K to those obtained with the 4He appa- ratus in the same temperature region. The temperature was determined with a carbon resistance thermometer which has been 14 15 calibrated using the temperature dependence of the magnetic susceptibility of a pellet of Cerium Magnesium Nitrate powder. This calibration is good to 10.005 K. Thus, although the numerical value of X is not very accurate, the temperature at which, for example, magnetic ordering takes place is well determined. The field dependence of the susceptibility was measured in a second 4He immersion cryostat which consisted of a single horizontal axis rotor surrounded by a set of mutual inductance coils whose axis was, as in all three cryostats, vertical. These were in turn surrounded by and coaxial to a supercon- ducting solenoid. Although it was possible to remove the sample and rotor from the measuring coil, the balance and sensitivity of the mutual inductance coils was a strong func- tion of the field produced by the solenoid so that only rela- tive values of X could be determined. The solenoid produced approximately 17 kOe with an excitation current of 10 amperes. From rather crude measurements, the homogeneity of the field was estimated to be 0.3% over a 1 cm sphere, which is slightly larger than the size of samples used. This produced some broadening of the field dependent susceptibilities determined in Section V10. B. Electron Spin Resonance (ESR) A variety of ESR spectrometers was employed to obtain the data reported on in Section III and V. Simple reflection type spectrometers Operating from 8 to ll, 13 to 15, and 22 to 25 16 GHz were used in conjunction with a large electromagnet capable of producing 20 kOe. For those cases where frequency dependence was of interest, the samples were mounted on the narrow wall of a shorted piece of rectangular waveguide. For those cases where more accurate values of g or stronger sig- nals were desired, either rectangular TE102 or cylindrical TE011 cavities were used. Measurements were made at 300, 77, and from 1-4 K in a 4He immersion cryostat. C. Nuclear Magnetic Resonance (NMR) Here too a variety of techniques were used. Marginal . . . . 4 osc1llator and free-1nduct1on spectrometers were used 1n He, 3 4 He, and 3He- He dilution refrigerators with and without applied magnetic fields. Frequencies between 2 and 20 MHz and fields from zero to 20 kOe were used with different cryo- stats at different times. For example, pure quadrupole reso- 127 nances of I in NiIZ-GHZO were observed in an AC modulation 4 field using the marginal oscillator and the He immersion cryostat at temperatures between 1 and 4 K. No pure quad- 3 rupole resonances were observed below 1 K in either the He 3 cryostat or the He-4He cryostat for different reasons. In the 3He cryostat the sample is immersed in the liquid 3He which is contained in a small dewar that is in turn immersed 4 in the He. The NMR coil is wound on the outside of the small dewar giving a filling factor less than 0.2. The dilu- tion refrigerator also has a very small filling factor - the 3 sample is immersed in the He-4He mixture and the coil is l7 wound on an epoxy form which is far from the sample chamber. In addition one may not use an AC modulation field because of eddy current heating; thus the marginal oscillator, which is the more sensitive of the two NMR spectrometers, may not be used. 27 . 1 I resonances was observed 1n Zeeman splitting of the all cryostats using both spectrometers, but quantitative data was only obtained above the transition (Tc=°120 K) due to presumed rf heating. The free induction spectrometer is a "mini pulser" 11. It consists of a low level rf developed by S. I. Parks oscillator, an rf pulse amplifier, a rf receiver and detector operating with a single coil. The apparatus is unique and operates as follows. A small rf signal at frequency no is continually applied to the rf receiver. A pulse sequence triggers the rf pulse amplifier which applies an rf pulse (of frequency mo) to the coil. If the external magnetic field is adjusted so that the NMR frequency, w, is approximately equal to “o the magnetization is tipped through an angleei and precesses at w. The signal induced in the coil is fed to the rf receiver-detector combination producing a beat frequency at w—wo which decays in time as the net transverse magnetiza- tion decays. This repetitive pattern is normally presented on an oscilloscope. The amplitude of the free induction beat pattern is a complicated function of w-wo but when the field is adjusted to be at the center of the resonance line, w-mo=0, the beat frequency is zero. 18 Figure 1. Block diagram of pulsed NMR spectrometer. 19 POWER SUPPLIES RF COIL RF MINI-PULSER OSCILLATOR MAGNET FREQUENCY HALL ._av COUNTER PROBE OSCILLO- x-Y SCOPE RECORDER BOXCAR INTEGRATOR Figure l 20 Figure 2. Time sequence of mini-pulser and boxcar inte- grator operation. 21 (J 1 "mini" OUIPUI W —t ”Nb“, __/\./\_ fireh- boxcar "window" .1 n n Figure 2 22 Figure 3. Sample recording using the pulsed NMR spectrometer. 23 ~12 90.»; 7.5.: : Figure 3 *~ 6%-... =2. 24 The mini pulser as described is relatively sensitive and needs no modulation field. Its disadvantages, as compared to the marginal oscillator, are complexity of tuning (there are several rf stages to tune) and the relative difficulty of determining from the scope the exact field at which zero beat occurs. By comparison the marginal oscillator is normally used with first or second derivative detection and for well resolved lines the resonant field is quite accurately deter- mined. Mr. Paul Newman of our laboratory suggested a very nice scheme for overcoming this last limitation which uses a Boxcar integrator. A block diagram of the system is shown in Figure 1. Each time the mini-pulser produces an rf pulse it also produces a "sync" pulse. The Boxcar(PAR model 160)12 was used in a mode where each "sync" pulse initiated a time base for the integrator. With the model 160 set on "HOLD" the inte- grator sampled the mini pulser output over a narrow time aper- ture(~.l usec) at a fixed time, TD’ after each "sync" pulse. The output of the boxcar was then proportional to the average voltage level at some fixed time after the rf pulse (TD:>pulse width). The various time bases and voltages are shown in Figure 2. The boxcar output was displayed on the Y axis and a Hall probe output, proportional to the applied field, was displayed on the X axis Of an X—Y recorder. Figure 3 shows a sample recording made using this technique. The field at which zero 127 beats occurred for each of the five I resonances is marked 25 by an arrow. Several unresolved proton resonances are also seen between 2 and 3 kOe. D. Crystal Orientation All three crystals exhibited well developed faces with consistent morphology. Orientation was done by optical goni- ometry or simply by looking at the faces. As a check on the morphologies, at least one sample of each salt was checked by X-ray crystallography to confirm the correct labeling of axes . 111. ((CH3)3NH)CuC13-2H O - 2 A TWO DIMENSIONAL SQUARE HEISENBERG FERROMAGNET A. Crystal Structure and Preparation The crystal structure and some magnetic properties of ((CH3)3NH)CuCl3-ZHZO have recently been reported by Losee et. al.13 The compound is monoclinic and belongs to the space group P21/c with a=7.479 A, b=7.864 A, c=16.730 A, and B=91.28° with four chemical-formula units per unit cell. The most significant feature of the structure is the chains of edge-sharing CuCl4(OH2)2 octahedra running along the a-axis. These chains may be pictured as bonded into layers of compo- sition CuCl3-ZH O situated at heights 0 and 1/2 along the 2 c-axis. These layers are separated by layers composed of (CH3)3NH. The prOperties of the analogous cobalt salt have been well explained using a two-dimensional rectangular Ising model. Although the cobalt salt is orthorhombic and the copper is monoclinic, the monoclinic salt can be derived from the orthorhombic by a 2° macroscopic distortion. Thus the description of their microscopic properties may be similar. Green single crystals were grown from a 1:1 mixture of ((CH3)3NH)C1 and CuClz-ZHZO in water at room temperature. Following the convention of Losee et.al.12, the a and c axes have been interchanged relative to Groth's morphological ' description14. 26 27 Figure 4. Angle dependence of the ESR of ((CH3)3NH)CuC13° ZHZO in the ac' plane plotted as g2 vs. a. The curves are the least squares result used in the principal axis determination. I 11.—In Om. om. m o... 8. oo. om om ow om o q q d .Né Figure 4 29 B. Experimental Results In order to confirm low-dimensional behavior as suggested by Losee et.a113, magnetic susceptibility was measured down to .025 K. This permitted determination of the ordering tempera- ture as well. In addition ESR measurements were made in the paramagnetic state at T=300, 77, and 4 K. 1. ESR - Principal Axis Determination Although this is a magnetically concentrated salt, the ESR lines at 77 K were only 250 Oe wide, so that the signals from nonequivalent coppers were resolvable and g-value accu- racy of 1/2% could be Obtained. The principal g values and axis orientations relative to the crystal were determined from measurements of the g-value variation in the ab, bc', and c'a planes. The c' axis is defined as the axis perpendicular to the ab plane. It differs from the c-axis by less than 2°. Using the method of Schon- 15 land , these rotation patterns were fit by a least squares technique to g2 = a + BCOSZO + ysin26 (3.1) Figure 4 shows the rotation pattern for g2 in the ac' plane and the resulting fit. From a, 8, and 7 determined for each rotation, the g-tensor elements were determined. The prin- cipal values were obtained by diagonalizing this tensor. There are four chemical-formula units, and therefore four copper atoms per unit cell, but no more than two resonance lines were observed in any rotation. Thus only two 30 nonequivalent paramagnetic complexes could be resolved. The Observed rotation patterns correspond to two principal- axis g-tensors having essentially the same principal values but different principal axes. The principal values and polar and azimuthal angles specifying these two tensors are given in Table 1. Although the principal axes do not coincide with the crystal axes, the g-values parallel to the crystal axes are well determined, being the g-value extremes in the ab rotation, where only one resonance line was Observed, and the values where the two observed lines cross for the other two rotations. Near the crossing where two lines overlap, the absorption maxima were not resolved so that the fit to the rotation patterns gives a better determination of the crystal axes g-values than a single measurement on each axis. The values obtained are ga=2.23810.010, gb=2.037:0.010, and gc=2.l9510.010. The data used in this determination was taken at T=77 K, but no difference in the rotation patterns was observed in data taken at 300 and 4 K except for a signal to noise enhance- ment at lower temperatures. 2. Magnetic Susceptibility The magnetic susceptibility data is displayed in Figure 5 where x-1 is plotted for the a, b, and c axes over the entire temperature range for which data was taken. In Figure 6 the magnetic susceptibility, x, parallel to these three axes at temperatures below 0.45 K is shown. 31 Table I Principal g-tensor Values and Orientations e is measured from the c' axis toward the ab plane and 4 is measured from the a-axis toward the b-axis. NNN NNN .343 .027 .095 .333 .029 .101 00 |+ H- H- 0Q |+ I+ H- .010 .010 .010 .010 .010 .010 Tensor 6 49.8 74.4 44.4 Tensor 6 140.5 76.9 42.4 H- 2. 0 170.1 273.6 20.3 176.9 103.7 151.6 2.0 32 Figure 5. Inverse magnetic susceptibilities of ((CH3)3NH) CuCl3-2H20 measured parallel to the a, b, and c axes . IO 0) A X“'(mOIe/emu) 33 .- . a #235" 3“." {’0' :2. c". I 2 - T( K) Figure 5 34 Figure 6. Magnetic susceptibilities of ((CH3)3NH)CuCl3- ZHZO measured parallel to the a, b, and c axes near the transition temperature. 3S II II I I I IA 0 b C II I X X x In I I I A I A I. I I I I A IIIII I II I OIII II LA I lfl I I A.“ ’I AAIAAWIIII ‘5 I. I At II ’ I. II b ’0 u - fl . I b b n h I b I . m 2 8 4 l 3.08}...on 0.4 0.3 36 There are three interesting features in the results. First, the system does order magnetically at Tc=0.15710.003 K, where the ordering temperature is taken as the point of maximum dxa/dT. Second, the behavior of X well below Tc appears to be antiferromagnetic in nature, yet Xa’ which most resembles x", does have a nonzero value as T+0. Experimental errors due to misalignment and empty-sample-holder contributions are smaller than this deviation. In addition, xb and xc exhibit strong temperature dependence below Tc’ unlike XL for an antiferromagnet. The third feature is the large value of the peak susceptibility for all three axes. Such a large peak in the susceptibility could arise from antiferromagnetic order with the spins canted away from an antiparallel alignment; or it could be the result of ferromagnetic planes weakly coupled antiferromagnetically. C. Interpretation of Resu1ts The ESR results do not lend themselves to a simple inter- pretation. Attempts at relating the principal axes orienta- tions to the Cu-Cl and Cu-O directions of the distorted octa- hedron surrounding the coppers were not successful. The angle dependence of the line widths was compared with the Dietz et.al.16 result for exchange narrowing in a one dimensional system, and no correlation was found. The main value of the ESR results is in the determina- tion of g values to be used to calculate Curie constants for the susceptibility. In addition, the presence Of two 37 different g tensors provides a possible explanation for the behavior of x in the ordered state. I The susceptibility results above 1.5 K show no unusual behavior. Thus Losee et.al.13 found they could fit both ‘their susceptibility and specific heat results to a variety of models with no conclusive results. They fit their results to Curie-Weiss, Ising linear chain, and isotropic Heisenberg ferromagnetic chain models. The data in Figure 5 Shows that x'1 varies from linear Curie-Weiss behavior below 1 K. Fitting x in this temperature region should distinguish between various models. The cobalt saltl’Z is best described as an Ising lattice with Jl/k=7°7 K, JZ/k=0.09 K, and J3=-.01 K giving rise to ferromagnetic sheets very weakly coupled antiferromagnetically with a net moment. (See Section IV) The lack of deviations from Curie-Weiss behavior above 1 K for the copper salt indi- cates that J1 is much smaller than it was in the cobalt salt. Since copper has rather small g-factor anisotropy, and (being S=l/2) no zero field splitting, it is unlikely to be Ising- like. In the light of these factors a fit of the susceptibility data above TC to a two-dimensional ferromagnetic Heisenberg model with an anisotropic g-tensor was performed. Since there are no exact solutions available, the high temperature series expansion, due to Baker et.al.17, for a square lattice was used. In order to best display deviations from Curie-Weiss behavior one normally plots C/xT vs. J/kT. Figure 7 is a Figure 7. 38 The magnetic susceptibility of ((CH3)3NH)CuC13- ZHZO measured parallel to the c-axis plotted as C/xT vs. J/kT. Solid curve (1) is the Curie- Weiss result with 6a2J/k,. (2) is the square Heisenberg ferromagnet, and (3) is the Heisen- berg ferromagnetic chain(abscissa scaled down by a faCtor of 2). (J/kT=0.S corresponds to T=0.56 K) 0.8 0.6 C/XT 0.4 0.2 39 0.2 0.4 ‘J/kT Figure 7 . 0.6 40 graph of C/XCT vs. J/kT, where C is calculated from C=Nog2u§S(S+l)/3k using S=l/2 and gC from the ESR results. The solid line (2) is obtained using the first ten terms of Baker's expansion. The best fit is Obtained for Jl/kEJ2/k= 0.28:0.02 K. For comparison, curve (1) is the Curie-Weiss law with 9=22JS(S+l)/3k, which reduces to (C/xT)=(l-2J/kT) for the two-dimensional square lattice; and curve (3) is the Bonner and Fisher18 result for a Heisenberg linear chain with S=l/2. Note that the abscissa for the Bonner and Fisher18 result has been scaled down by a factor of 2. There is no value of J/k for which either of these expressions gives as good an overall fit as does the two-dimensional square ferro- magnetic Heisenberg interaction. The b-axis data is fit equally well using gb from the ESR results with the same value of J/k. The interpretation of xa is not as simple. Using the value of ga=2.24 found by ESR, no value of J/k gives anything resembling a fit to any of the proposed models. 'However, if we make a two parameter fit to the square Heisenberg lattice high temperature expansion allowing g and J/k to vary, a fit comparable to that obtained for the b and c axes is obtained. This fit corresponds to ga=2.36t.02 and J/k=0.2810.02 K. It is striking that although there is a discrepancy between the ESR and susceptibility g values, the same J values are obtained from all three fits. None of the models considered offered any resolution of this discrepancy in the value of ga. The susceptibility results above the transition 41 temperature indicate that trimethylamine copper chloride behaves as a two-dimensional Heisenberg ferromagnet. Fitting the results to a square lattice in which J/kT=0.28 K may well be an over-simplification, since a rectangular lattice with two different exchange interactions would no doubt be more realistic, but this problem has not been solved. Below Tc’ the data seems to indicate that the ferromagnetic sheets are antiferromagnetically ordered with successive ab planes oriented oppositely. The antiferromagnetic interaction may arise from dipole-dipole interactions or from a weak exchange interaction. In any event, the interlayer interaction, J3, will be denoted by JAF and the intralayer ferromagnetic inter- action, J1=J2, by JF' The properties of a layered structure with ferromagnetic intralayer exchange and a weaker antiferromagnetic interlayer coupling have been studied by a number of authorslg'zs; both theoretically, using a variety of two sublattice models, and experimentally. Attempts have been made to apply several of these models to the experimental results. 22 in fitting x vs. T The success of Berger and Freidberg for the layered structure, Ni(N03)2-2H20, using a simple mole- cular field calculation suggested such a calculation might describe the ordered state susceptibility of ((CH3)NH)CuC13- ZHZO. No meaningful results were Obtained using this approach. From measurements made on several metallo-organic salts exhibiting two-dimensional ferromagnetic Heisenberg behavior, de Jongh et.al.23 have drawn a series of empirical conclu- sions concerning the magnetic properties of such salts. They 42 define Té to be the temperature at which x for such a two dimensional lattice diverges. They observe that above TE x drops steeply. They deduce that the relationship between Té and JF is kTé/JF~0.44. In one compound, (CZHSNH3)2CuCl4, they observed the effect Of a weak antiferromagnetic coupling between the layers. In this compound three dimensional order occured at a temperature, Tc’ approximately 25% greater than T; determined from JP. Also they found that x“ drOpped steeply on both sides of Tc from a large peak value given by xumax Tc/C ~85. xl_showed only slight temperature dependence below Tc. They concluded that the effect of JAF is to shift the ordering temperature up from Té determined by JF’ and to cause x" to fall steeply below Tc. They also conclude that the ferromagnetic layers explain the large peak value and the steep fall of x above Tc‘ Inserting JF/k=0.28 K into the expression they deduced for Té gives TE~0.12 K. The ordering temperature for ((CH3)3NH)CuCl ~2H20, TC=0.157 K, is approx- 3 imately 25% greater than TE. x T /C~5 for this salt max c is much less than that for (CZHSNH3)zCuCl4, but it is still larger than the molecular field prediction, xmaxTN/Csl. Thus the behavior of x agrees with the empirical results de Jongh et.al.23 deduced for a two dimensional Heisenberg ferromagnet with weak antiferromagnetic interplanar coupling. As noted previously, the susceptibility in the a direc- tion, which we would like to associate with x“, has a nonzero value as T+0. This could be due to canting of spins on inequivalent ions. Spin resonance results indicate there are 43 two anisotropic g-tensor orientations for the copper ion sites. Silvera et.al.24 have shown that two anisotropic g-tensors tilted with respect to each other are sufficient to produce canting of the spins on the inequivalent sites, even with isotropic exchange. Assume as a first approximation that tensor 2 can be obtained from 1 by a rotation of 26 around the b-axis, as in Figure 8. The x, y, z coordinate system is the crystal system and x3, ya, 23 coincide with the principal axes of the ith g-tensor. In order to consider anisotropic effects due to the g tensors the exchange is described as an isotropic exchange between two "true" spins. 1+= ~21FSl-SZ (3.2) Although the exchange written in this form is isotropic, the true Spin magnitudes will be anisotropic if the ions' envi- ronment is a distorted octahedron. The problem is handled more simply by using isotrOpic effective spin variables S1 and S2, and including the anisotropy in effective exchange constants. In the case of Cu++ the orbital contribution is zero to first order and the anisotropic g values relate the true spin components to the effective spin components in the principal axis coordinate system as where i=x', y', 2'. Referring to Figure 8, the Hamiltonian, equation (3.2), can be written in terms of the effective spins as 44 Figure 8. The various coordinate systems used to show canting due to g-tensor inequivalence. xyz is the crystal system; x'y'z' is the system in which the g-tensor is diagonal; x"y"z" is the system in which S lies along y" in equilibrium. 45 46 74= -J[( )Zs s + ( )chS 265 s 8x' lx' 2x' gy' ly' Zy' + (gz.)2cOSZGSlz.szz. (3.4) ’ gyvgzISinze(Slyvszzv ‘ SIZ'SZY')] Thus the symmetric elements of the effective exchange dyadic (See Section I) are 2 JX'X' = '(gxl) J Jy,y, = - (gy,)2Jc0526 (3.5) Jz'z' = - (gz,)2Jc0526 and the nonzero antisymmetric elements are J = gy,gz.Jsin26 (3.6) yvzc = JZ'Y' It is this antisymmetric exchange which leads to a canting 24 Of the effective spins. Silvera et.a1. then define the x" i’ y'i', z'.' coordinate axes, such that in equilibrium (SI) 1 and (S2) are parallel to y'l' and y'z'. In this frame Sy" is a constant 0f the motion, therefore 111 Y / [5), “I ( 7) Solving this for the equilibrium angle, 3 , S makes with the s y' axis gives -2gy,gz,tan26 (3.8) (2y.17+(gz.)2 tan2¢S = or in terms of the effective exchange constants, ZJ 1 1 tan2¢s = y z - (3-9) 47 Figure 9. The relationship between the effective spin S1 and magnetic moment 51. 48 Figure 9 49 The values obtained from the principal axes determination, gy1=2-34, gz,=2.10 and e~40°, gives ¢S = -39.97° Therefore, the effective spins are canted less than .l° from the y(i.e. a) axis. The quantity of real interest, however, is the orientation of the magnetic moment. Defining the angle between the magnetic moment equilibrium position and the y' axis, Om, the situation for spin 1 is shown in Figure 9. Since fi=u§§~§ it follows that _ gz. tan¢m - 'g—YT tan¢ s (3.10) Substituting the appropriate values one obtains ¢m=-37°. The canting angle of the moment relative to the y axis is given by e + Om. Therefore moment 1 is canted 3° from a towards (-c') while moment 2 is canted 3° towards +C'- Although the rotational inequivalence of the two tensors is more complicated than assumed above, the conclusion that the magnetic moments are canted a few degrees from the a-axis is probably valid. Other factors such as anisotropic exchange may contribute to the canting, but canting is a probable explanation for the nonzero value of xa as T+O. Canting also explains the strong temperature dependence of xb and xC below Tc' An upper limit on the magnitude of JAF can be obtained by assuming that the value of xb and xc extrapolated to T=0 is given by the molecular field expression for x;_of a two sublattice layered antiferromagnet23, 50 NguBS Ngzug ( ) x = = 3.11 ‘- 2HA1: ZZJAF Using xc extrapolated to zero and z=2 one obtains JAF/k m -0.023 K. This is an upper limit on the magnitude of JAF since xc(T+0) will be less than the molecular field predic- tion if canting is present. Therefore the magnitude of the intraplanar exchange is at least a factor of 10 greater than the interplanar exchange. The probable ordered state spin configuration is shown in Figure 10. The spins lie in the ac plane mostly along a. Successive spins along a are canted towards c and -c. Spins in adjacent ab planes are antiparallel. There are five members of the family25 of magnetic space groups belonging to P21/c. Since the copper atoms are located at inversion centers in P21/c, and since no magnetic moment can exist at an antiinversion center, the two members con- taining anti-centers can be eliminated. The three remaining groups are PZaZl/C’ Pazl/C, and PZi/c'. If the group were PZi/c' then the a component of all spins would be equal and parallel. The assumption that the a components of spins on successive layers are antiparallel eliminates this group. PZazl/C’ obtained by doubling the unit cell along a, is pos- sible but not necessary to explain the observed data. Thus the probable magnetic space group is Pazl/c which is consis- tent with Figure 10. In conclusion, ((CH3)3NH)CuC13-2HZO displays the proper- ties of a two-dimensional Heisenberg ferromagnet above 51 Figure 10. Proposed ordered state spin arrangement for ((CH NH)CuCl3'2HZO. The Spins lie in the ac 3):, plane. 52 10 F1 53 Tc=0'157 K with JF/k=0.28 K. Below Tc it exhibits behavior characteristic of antiferromagnetic coupling between the ferromagnetic layers with JAF m —0.023 K. Apparently the spins are canted slightly away from the a axis in the ordered state. The configuration has no net moment and corre- sponds to the magnetic space group Pazl/C' IV. ((CH NH)COC1 '2H 0 - 333 3 2 METAMAGNETIC BEHAVIOR A. Crystal Structure and Preparation 1 In a recently published paper, Losee et.al. reported 3)3 3-2H20 1n addItIOn to detailed magnetic susceptibility and specific heat measurements in the structure of ((CH NH)COC1 zero field. The compound is orthorhombic and belongs to the space group ana with a=16.67 A, b=7.27 A, and c=8.ll A with four chemical formula units per unit cell. The most signi- ficant feature of the structure is the chains of edge-sharing COC14(0H2)2 octahedra running along the b-axis. These chains are bonded into layers of composition CoClS'ZHZO situated at heights 0 and 1/2 along the a-axis. These layers are sepa: rated by layers composed of (CH3)3NH. The structure is similar to that of the monoclinic copper salt but the axes are labeled differently. Single crystals were grown from equimolar solutions of ((CH NH)Cl and COC12°6H20. The crystals are violet/red in 3)3 color but appear blue when viewed along the a-axis with light polarized parallel to c. This optical dichroism aided orien- tation of the samples. B. Experimental Results 1 Losee et.al. concluded from their measurements that above TN=4.135 K the magnetic properties could be described by a two-dimensional rectangular lattice with Ising-like 54 55 exchange. They found good agreement with such a model with J1/k=7.7 K and Jz/k=0.09 K. Below TN the compound ordered three dimensionally as a canted antiferromagnet. The anti- ferromagnetic order indicates a small negative value for J3. From the crystal structure and their measurements Losee et.al.1 proposed a spin arrangement for the ordered state. 2 were able to confirm this arrangement Spence and Botterman using NMR and found that it could be described by the magnetic space group an'a'. In this configuration the spins lie nearly parallel to c, but are canted towards the a-axis. Spence found that the application of a magnetic field greater than 64 Oe along c drastically modified the magnetization of the sample. (In order to explain this metamagnetic behavior they proposed a phenomenological model. The model success- fully explained the NMR and magnetization measurements. In this section measurements of the magnetic susceptibility in applied field are compared with the predictions of the model. The model proposed by Spence and Botterman has no temper- ature dependence; it predicts results for the applied field only in the ac plane and only for temperatures near 2 K where they performed their‘NMR and magnetization experiments. Although experiments have been performed at a variety of temperatures, only the results obtained near T=2 K and with H in the ac plane will be discussed here. The data was normalized so that the experimental and theoretical values of xc were approximately equal in an applied field, Hc=450 0e. In Figure 11 the magnetic susceptibility at T=2.0S K Figure 11. 56 The field dependence of the magnetic suscep- tibility of ((CH NH)COC1 -2H 0. (a) The 3):. 3 2 open squares are experimental results for Hflc. The solid line is the theoretical prediction for 6=0°; the dashed line is the theoretical prediction for e=3°. (b) Data and theory for H“a(e=90°). 3) X(emu/cm X(emu/cm3) 0.2 r .. n ‘ 0 ""1: a" Hll c ‘ g T=2.05K ‘ O.l - .. . 0.0 l l 1 in L O 0.4 0.8 1.2. H (kOe) Figure ll(a) 0.2 - ~ Hll O :—-l T=2.05K " o..- : . ' l 1 COM 0 0.4 0.8 l.2 H (kOe) Figure ll(b) 58 is plotted as a function of magnetic field, H, applied parallel to the c-axis(e=0°) and to the a-axis(e=90°). The theoretical O predictions for 6=0°, 3 , amd 90° are also shown. The data is plotted only for fields up to 1400 De since for higher fields the susceptibility is approximately zero. The rotation data taken at T=2.32 K is plotted in Figures 12-15. Each successive figure shows x vs 9 for a larger value of H than the preceding figure. In addition the theoretical fits are displayed. C. Interpretation of Results In order to compare the results for x(H) with the Spence model it is necessary to summarize the model. He assumes that the sample consists of regions in which the spin orientations correspond to one of the configurations C1, C2, C3 shown in Figure 16. It should be noted that although the model does not use an explicit Hamiltonian, this is equivalent to assuming the exchange is Ising-like; that is, the spins on a given sublattice can only point parallel or antiparallel to an axis 2'. z' is approximately 10° from 2 toward +x for one sublattice and towards -x for the other. In this model satu- ration corresponds to the entire sample being in one of the three spin configurations. It does not refer to the rotation of spins away from 2' at very high fields. Spence denotes by f1, f2, and f3 the fractions of the total sample volume occu- pied by each of the corresponding spin configurations, thus f1 + f2 + £3 = 1 (4.1) 59 Figure 12. x vs. 9 with H=300 Oe for ((CH3)3NH)COC13- ZHZO. Figure 13. x vs. 6 with H=438 Oe for ((CH3)3NH)COC13- ZHZO. 6O 0.24 3) .° 8 X (emu/cm o 'o m A 0.24 X (emu/c1113) a°°° T=2.32K 1 1 1 U I r I 1 H=438 06 Figure 13 61 Figure 14. x vs. 9 with H=976 Oe for ((CH3)3NH)COC13' 2HZO. Figure 15. x vs. 6 with H=1075 Oe for ((CH NH)COC1 - 3):, 3 2H20. X '(emu/cm3) X(Omulcm3) 62 0.24 ~ .0 m 0.08 ’ 0.24 U .0 m 0.08 H=|075 Oe T=2.32K Figure 15 63 Figure 16. The Spin configurations C1, C2, and C3 assumed by the Spence2 model. 64 x Figure 16 65 The magnetic energy of such a system will then be the usual magnetostatic term plus the energy to create configuration CZ from spins in configurations C1 and C3. This creation term involves the energy to overcome the weak antiferromagnetic exchange J3, which is characterized by a critical field, Hk=64 Oe. Spence minimizes the magnetic energy subject to the restriction of equation (4.1). In order to do this he assumes a linear relation between the applied field, H, and the inter- nal field, Hi, given by H. = H -‘N-F’4 (4.2) where N’is an effective demagnetizing tensor. The associa- tion ofN with the normal demagnetizing tensor26 is, as Spence points out, probably invalid because the magnetization, M, is spatially inhomogeneous. Nevertheless, such a linear rela- tionship appears to be valid experimentally. In order to make use of Spence's results the following definitions and assumptions are needed: a is the angle between applied field and c-axis, Mx and M2 are the components of magnetization along a and c, Ma and MC are the saturation values of these components, N is diagonalized by x and z, (4.3) Nxx and N22 are the diagonal elements of N, Ha = NxxMa’ Hc = szMc’ p = Ha/HC. g = Ma/Mc' 66 With these definitions and referring to Figure 16, it is clear equation (4.1) can be rewritten as M M x z = M— + M— + 2f3 1 (4.4) a C Five regions of the H-6 plane can be distinguished depending on the form of equation (4.4). The components of M for the five regions are given by: Region A (Mx/Ma+Mz/MC<1) M =N'1 x xx H sine (4.5) M 2 -1 N22 (HcosB-Hk) Reg1on B (Mx/Ma+Mz/Mc=l) _ -l -l . _ _ _ Mx - Nxx (1+;p) [cpHSIne p(Hcose Hk HC)] (4 6) _ -l -l _ _ . _ Mz - sz (1+§p) [(Hcose Hk) c(HSIne Ha)] Re Ion a (Mz/Mc=1) M = 0 x (4.7) M = M 2 c Region 8 (Mx/Ma=l) M = M x a (4.8) M2 = 0 Region 7 (Mz=0, and Mx/Ma<1) -1 _ M = N H SInB x xx (4.9) M = 0. Z 67 Figure 17. Regions of magnetization in the H6 plane. 68 Figure 17 69 Regions a and 8 correspond to saturation of M2 and Mx’ The H-e boundary between any two of these regions is found by determinimg the values of H and 6 for which Mx and M2 in both regions become identical. Figure 17 shows the five regions of magnetization in the H-6 plane for the following choice of parameters: Hk=64 Oe, Ha=150 Oe, HC=650 Oe, p=.231, and c=.166. The parameters used to fit the susceptibility data are M M Ha, and He. The critical field, Hk’ is deter- a’ c’ Hk’ mined very accurately by NMR to be 64 Oe and since the susceptibility measurements only indicate that it is less than 100 Oe, the NMR value will be used. The saturation magnetizations are well determined from the magnetization measurements to be Ma=21.2 emu/cm3 and MC=127.4 emu/ems. Thus these values determined by Spence and Botterman are con- sidered fixed and the shape dependent parameters Ha and HC are the only parameters to be fit in this experiment. The calculation of x from the magnetization expression can be done numerically using, (6,H (e,H+AH - (9,11) x(H,9) 5 d-E-H—l ‘-'- MH )MH (4.10) dH AH where AH is small and MH(6,H) is the component of M(6,H) para- llel to H. Then MH(e,H) = Mx51n8 + Mzcose (4.11) where Mx and MZ are given by the expressions for the 70 appropriate region of the H-6 plane. Using such a numerical differentiation gives values of x even at boundaries between regions where M vs H is discontin- uous and x is really undefined. The model correctly predicts x only on either side of the boundary. A model to explain the behavior of x at these boundaries would have to include the spin dynamics. Since all measurements were made on a single sample, one set of shape dependent parameters, H and Hc’ should fit the a data if the model is to be further confirmed. The effect of demagnetizing factors is included in the model so normal demagnetizing corrections of the data are not necessary. The qualitative nature Of the model suggests that a quantitative fitting procedure is inappropriate. A qualita- tive "best fit" was found by varying the parameters to deter- mine if an "eyeball" fit with the appropriate angle and field dependence could be found. The best values found in this manner were Ha=150125 Oe and Hc=650125 Oe. The predicted susceptibility obtained using these values is shown as the solid and dashed lines on Figure 11 and the solid lines on Figures 12-15. The model predicts(Figure 11) that for H parallel to C (e=0°), solid line, the susceptibility is zero for H(Hc+Hk) (Region a). For HkHa(Region B). The experimental susceptibility for H>Ha fluctuates but is quite small and approaches zero at high fields. The low value of xa’ compared to the model prediction, for HHk+HC(Figures 14 and 15) three regions, a, B, and B, are Observed. The fits are only quali- tatively correct but three distinct angle dependences are observed for each rotation. The distinction between regions A and a appearing as H is increased is quite striking. For completeness, note that Figure 11 shows evidence of the 7 region where xC(H)~0 for H3100 0e. Using equations (4.3), the values of Ma and MC given by Spence and Botterman, and the values of Ha and Hc given above, one obtains for the Shape dependent demagnetizing factors Nxx=7.lil.0 and sz=5.lt.2. These values fall between the results which Spence and Botterman obtained using NMR on ellipsoidal samples and those obtained from their magnetiza- tion measurements on unshaped samples. Since the samples used for the susceptibility measurements were unshaped, these are not unreasonable values of Nxx and N22. In conclusion the magnetic susceptibility measurements in applied field further confirm the validity of the 73 assumptions made by Spence in describing ((CH3)3NH)COC13-2HZO. The model predicts qualitatively the dependence of x on H and e. The inclusion of temperature .in a model for the behavior of this system would be interesting but difficult. If one could design such a model, experimental investigations below 2 K would be worthwhile. At present nothing is known of the spatial distribution of the various spin configurations as a function of field, orientation, and temperature, nor of domain boundary conditions. Optical and electron spin resonance experiments, in addition to further NMR, magnetization, and susceptibility experiments, might answer some of the questions. V. N112'6HZO - CRYSTAL FIELD SPLITTING WITH WEAK EXCHANGE A. Crystal Structure and Preparation Gaudin-Louer and Weigel27 have reported the X-ray deter- mination of the unit cell parameters and Ni and I positions in NiIZ'OHZO. At room temperature the unit cell is hexagonal with a=7.67 A and c=4.87 A. There is one molecule per unit cell and the density is 2.825 gm/cms. The fractional coor- dinates are: N1**: 0,0,0 I- : l/3,2/3,u and 2/3,l/3,u where u40.22. The 0 positions were not determined but they probably form a regular octahedron about the Ni. Depending on the symmetry Of this octahedron, Gaudin-Louer et.al. con- clude the chemical space group is P3 or P3nl. The magnetic measurements reported in this section indi- cate that at low temperatures there are two different Ni++ complexes, each having axial symmetry about the c-axis. The appearance of two different Ni environments implies that a crystallographic phase change occurs between room temperature and 4.2 K. This phase change must result in a doubling of the unit cell while the axial symmetry suggests the change may involve an axial distortion of the octahedron. Single crystals were grown from a saturated aqueous solution of NiIz-OHZO. The green hexagonal crystals have a 74 75 cleavage plane perpendicular to the c axis and are quite hygro- scopic. B. Experimental Results ESR, zero-field magnetic susceptibility and NMR measure- ments were made to study the relative magnitudes of crystal field splitting and magnetic exchange in nickel iodide. l. ESR At 1 K the resonance lines in this magnetically concen- trated salt were between 1 and 2 kOe wide, thus the center of the lines could be determined to 1100 De. For a rotation about the c-axis only one broad line, exhibiting no anisotropy, was observed. Figures 18 and 19 Show the rotation pattern data for resonant fields in the ac plane for v=10 and 25 GHz. 0 is the angle between the applied field and the c-axis. The resonant frequencies plotted as a function of field applied parallel to c are given in Figure 20. Negative fre- quencies correspond to transitions observed at fields greater than the field at which the energy levels cross(Figure 26). 2. Magnetic Susceptibility Figure 21 shows the reciprocal magnetic susceptibility measured parallel to the c and a axes plotted for T<4.2 K. No anisotrOpy was Observed in x for rotations about c between 1&4 K. The magnetic susceptibility measured parallel to c and a for T<1.4 K is plotted in Figure 22. 76 Figure 18. ESR rotation patterns in the ac plane(e-0° is the c-axis) for NiIZ-OHZO at v=10.01 GHz and T=l.l K. The solid lines are the theoretical predictions. 207 u. 1” N I (D 5 ‘ 2 II in 1- ‘77 ISLr- I S! I (D (9011“‘H Figure 18 -20 -4O 78 Figure 19. ESR rotation patterns in the ac plane (e=0° is the c-axis) for NiIz-OHZO at v=24.45 GHz and T=l.l K. The solid lines are the theoretical predictions. 00. 00 om O¢ ON 0 ON. O¢u ? T a 4 4 d J.— o H T. nu “w w s . . M . 0 . e r 210 n¢.¢N u a F b r 0N Figure 19 80 Figure 20. Resonant frequency vs. field applied parallel to the c-axis for NiIZ-OHZO. The solid and dashed lines are theoretical predictions. 81 H(k0e) ' o c: (2H9) a Figure 20 I O “3 82 Figure 21. Inverse magnetic susceptibility of NiIz'OHZO measured parallel to the a and c axes. The solid lines are the theoretical results using equations (5.3)-(5.8). 83 Assoxméoev T x T(K) Figure 21 84 Figure 22. Magnetic susceptibility of NiIz-OHZO measured parallel to the a and c axes below 1.4 K. 85 32 I I T ' ' fiF—I b AP' 1 OO O 24 - 2. A 8 £2 .- I0 0 o o 0 E O \ o EIG - 3 ° .2: 2 ° x o o -3 3 '- e - 2 “i gag: :- %o a )(Cl QIDO 0 ‘, 0 Macao loooooqmoooiooo op OO 01 00 a 0 0.4 0.8 l.2 T(K) Figure 22 86 It is apparent that magnetic order occurs somewhere below 0.3 K. Defining the ordering temperature as the temperature with maximum positive dxC/dT gives TC=0.120:.005 K. In con- sidering xa the detailed behavior is best displayed in Figure 21 because the scale for Figure 22 necessary to display the peak in Xc does not Show the dip in xa at approximately 0.2 K. 3. NMR Two pure quadrupole resonances were observed at vl=5.70: .01 MHz and vz=11.39:.01 MHz near 1 K. The line widths were approximately 250 kHz. The results of NMR measurements near 1 K in applied field are summarized in Figures 23 and 24. In Figure 23 the data for resonant frequency versus field applied perpendicular to c is plotted. In this orientation the lines were sufficiently intense to permit a mapping of 0 vs H over the entire range of experimental variables. Figure 24 shows v vs H for Hllc. In this orientation the lines were much less intense and were not observed below 8 MHz. Both the intensity and position of the NMR lines were strongly temperature dependent. Although the temperature dependence of the position is important in determining whether a transferred hyperfine term is present, the rapidly decreasing intensity of the lines allowed measurement of the temperature dependence only over a narrow temperature range. Figure 25 shows two data points and the theoretical calculation for the temperature dependence of the mI=+l/2+-+mI=-1/2 resonance line. 87 127 Figure 23. I NMR frequency vs. magnetic field applied perpendicular to c for NiIZ'OHZO. 88 89 127 Figure 24. I NMR frequency vs. magnetic field applied parallel to c for NiIZ-OHZO. y(MHz) 90 91 Figure 25. Temperature dependence of the mI=+l/2++mI=-1/2 resonance line of 127I in Ni12°6H20 for 0'13.l MHz. 92 N I Z «5 u fin l5)- J ¢ ('9 011—9“ H Figure 25 l3- IO. 30. l00. T ( K) 3.0 L0 0.3 0.1 93 Although quantitative results were not obtained below Tc=0.12 K, two zero-field NMR lines were observed. The fre- quency splitting varied with time(temperature) due to rf heating. The Observed extremes of frequency were approximately 6.1 and 5.3 MHz. C. Interpretation of Results 28-31 A number of authors have successfully interpreted the magnetic properties of NiH salts in terms of a spin Hamiltonian(eq. 1.8) of the form 2 i _ _ _ 2 _ 2 , HS - DSzi E(Sxi Syi) + guBH Si (5.1) with S=l and an isotropic g'v2.25. Schlapp and Penney32 have shown theoretically that a cubic crystal field, produced by octahedral coordination of the NiH ion lifts the orbital degeneracy leaving an orbital singlet lowest. A rhombic dis- tortion of the field produces zero field splitting of the spin states described by the D and E terms in equation (5.1). The ESR transition frequencies, vi, are given by lEi-E.l v = 1 (5.2) h where Ei and Ej are any two of the three eigenvalues of Ifé. The symmetry of the ESR rotation patterns indicates that the c axis corresponds to z. The absence of any anisotrOpy in the plane normal to c is conclusive evidence that the x and y axes are equivalent; indicating E=0. The ESR patterns also give conclusive evidence that there are two nonequivalent Ni sites each having 2 parallel to c and E=0 but with different 94 magnitudes for D. Figure 26 shows the energy levels for the Hamiltonian (5.1) with H applied along c. The ESR results do not reveal the Sign of D but the figure is drawn assuming D>0; implying the doublet lies lowest in zero field. Selection rules allow only AmS=1 transitions for this orientation. The two "eggs" Observed in the rotation patterns(Figures l8 and 19) and the two lines in the frequency vs field data (Figure 20) can only be explained by assuming two different Ni sites. Using equations (5.1) and (5.2) the theoretical results shown in Figures 18 and 19 are obtained using g1=g2=2.22, El-E2=0, D1/k=l.6 K and Dz/k=2.3 K. The same parameters give 0 vs Hc shown as the solid lines in Figure 20. The magnetic susceptibility can be calculated from the Hamiltonian (5.1) using the density operator defined in Sec- tion I. Doing this one obtains: _ 2 2 xi — ZNOg "B61 (5.3) where i=x,y,z, eD/kT Z kT 2 57“»«1 ’ e (5.4) and 1 D/kT 1 5 = 5 = _ e ' - x Y D 2 DH?!” e (5.5) Assuming 50% of the sites are characterized by D1 and 50% by D2, the total susceptibility is given by (1) (2) xi *‘X i x. = (5.6) 1 2 95 Figure 26. Energy level diagram of Ni++(S=l) as a function of magnetic field, H, parallel to z with D>0 (doublet lowest). 96 U Figure 26 97 Equations (5.3) - (5.5) give xz>xx= xy if D>0, and xzxa; therefore, since the ESR showed 2 parallel to c, both D1 and D2 are assumed positive. If the values of D1 and D2 found from the ESR are used in fitting x,the agreement is quite good but the predicted magnitudes of xa and xc are somewhat low. Since the salt does order magnetically, a contribution due to exchange may be re- quired. A simple way to include exchange in the molecular field approximation is to modify the Hamiltonian (5.1) as follows: “pi = -DS:i + guBH'Si - 22J(§)'Si (5.7) Equation (5.3) becomes 2Nogzu§5i Xi = (5-8) l-ZzJGi Using this expression in equation (5.6) with g, D1, D2 given above andeJ/k=+0.05 K gives the solid lines shown in Figure 21. The inclusion of a ferromagnetic exchange makes the agreement between experiment and theory for xa quite good but the theoretical results for xc are still less than the experi- mental data. Including exchange in the description of x suggests that it should be included in the interpretation of the ESR. McMillan and Opechowski33 have calculated the shift in resonant 98 frequency away from the value determined by'Né due to an exchange term, "EX’ with NEX<0(doub1et lowest) and zllc. The presence of ferromagnetic exchange(coupling the 99 moments) is also in agreement with the large value of xc' The susceptibility could not be measured to sufficiently low tem- perature to determine the nature of the ordered state. If xC continues to decrease so that xc=xu+0 as T+0 then the ordered state is probably antiferromagnetic due to a weak antiferro- magnetic exchange as in the case of ((CH3)3NH)CuC13°2HZO. If xC does not continue to decrease the ordered state is prqbably consistent with the ferromagnetic exchange observed in the paramagnetic state being three dimensional. The fact that xa is increasing at the lowest temperatures Obtained is incon- sistent with simple antiferromagnetic order but is not con- clusive. The transition temperature predicted by the MF result, = ZzJS(S+1) C 3E , T (5.10) with 22J/k=0.05 K is Tc~0.03 K, considerably below the observed Tc=0.12 K. Although molecular field estimates of transition temperatures are notoriously unreliable, the low value of Tc predicted may indicate that additional exchange, raising the value of Tc’ is present. The interpretation of the NMR measurements in the para- magnetic state does not expand greatly on the understanding of the magnetic properties of the system, but the 1271 resonances do include some points of intrinsic interest. A nuclear spin Hamiltonian to describe a nucleus of spin I=5/2 127 like I is analogous in many ways to the electron spin Hamiltonian which describes the Ni++ ion, equation (5.1). 100 The Hamiltonian can be written RN =NQ +741), (5.11) where'I-lQ is the nuclear electric quadrupole term and ”D is the nuclear magnetic dipole term. ”Q involves the interaction of the nuclear quadrupole moment with the electric field gradient at the nucleus. Dropping constant terms,')lQ can be written, _13 2 2 2 NQ- 2 12+ 79(1x-Iy) (5.12) where “Q s3equ/21(ZI-l)h. Q is the scalar nuclear quadrupole moment, eq is the axial component of the electric field gra- dient, and n is a measure of the nonaxial nature of the field gradient. The similarity between these two terms and the D and E terms in equation (5.1) is quite apparent. The dipole term,‘ND, includes the interaction of the nuclear magnetic dipole moment with the external magnetic field and all internally produced magnetic fields. Therefore I‘D = -yNnIoHi (5.15) where YN is the nuclear gyromagnetic ratio. It is assumed that the contributions to Hi can be separated into those due to external fields, those due to electrons in I- orbitals, and those due to electrons external to the I-. Thus, a + - -1«. -3 Hi Ho (YNh) A S + ZuB1 + Hd (5.14) H0 is the applied field. In this case the magnetic hyperfine coupling tensor, ‘A’, represents the transferred hyperfine 101 contribution due to the unpaired spin present in I' orbitals because they overlap the Ni++ 3d wavefunctions. The orbital hyperfine term is assumed to be small. The term Hd includes fields due to all the other electronic magnetic dipoles in the crystal including Shape dependent demagnetizing fields. In this case it is quite small and can be neglected as it was for NS. Since the correlation times for the electrons are usually much shorter than the nuclear precession time, the nuclear magnetization responds to the average electronic field so 5+ (8). Neglecting the last two terms equation (5.14) becomes H. = Ho + (yNn)‘1°A‘~<§) (5.15) 1 Often the nuclear spin Hamiltonian is solved for either the low field (NDNQ) case using a 34,36 perturbation calculation Since the NMR data presented here was taken over a range of fields covering both cases it is necessary to solve for the eigenvalues of.”N exactly37. Since Hi=0 in the paramagnetic state when H080, the two Amfl resonances observed in zero field uniquely determine “Q and n as vQ=5.7O MHz and n=0. With n=0 equation (5.12) is axially symmetric and no anisotropy is expected or observed in the xy plane. This is an unusually low value of vQ for 127I. With Q=-0.75><10'24cmz it is customary to find the pure quadrupole resonances at frequencies greater than 100 MHz. This implies that the electric field gradient at the I site is small with correspondingly high symmetry. It should be 102 noted that only one set of I resonances was observed. The two I sites predicted by the room temperature X-ray structure have the same symmetry and thus should have the same value for 0Q and n. However the apparent doubling of the unit cell indicated by ESR might produce two pairs of I sites. Since only one set of lines were observed, either the I sites still all have the same electric field gradient, or the other site 127I and is beyond the has a more typical value of vQ for experimental range of frequencies available. Assuming‘Aeo as a first approximation, then fiigfio and the resonant frequencies can be determined as a function of field from the eigenvalues of‘NN. The solutions, for Holc, are shown as the solid lines in Figure 23. The results for Hollc are shown in Figure 24. At high fields the Zeeman line is split by the quadrupole interaction into five lines with the splitting for H parallel to c much larger than that for H perpendicular to c. Although the theory is qualitatively correct, the data appears at lower fields than predicted. The agreement between theory and experiment can be improved by including the hyperfine term in equation (5.15). Assume firto be axially symmetric with diagonal elements denoted AxstyygAl-and Azz'A"' Then for Huc,‘