an .33.». a. in. 9,... :1: a ; ....,.....: L: . Erie-tr. 3 Z}. 24. a. . .. ? s 5, 3.6,“ . . v , $32. ... w .3me h... 4 .23 a zawwmur . a «a . . , virvr g 2%? Sm??? «fiwvrjww 1.1! .15.! a. H a ‘ a: g... VIE}; (w :3: 1 fair . {a a: £25 damn.» 3. a??? 1...: . a : ;.:¢.a:.n.fi,m.. girfiflwyaawr. . .1? . :.........:.a r. A a. , . a V1...? . Er r .9 w. .fiurkmmmmflwfldm 4.4.5: 2i. a a; 250. . . It viral. é u '1... $31.5”): a. 4.3 a 1.. #9.: v d '5! ways... , A. W... 13%.. 04;. . ll! fl! d 2:. J25...» , ' 1.1! . . ’Jy . 47 43.9! .,. .. .o .: . .5; . I A: 4 . , h. . . 3. .-n¢ Crystallography Electron.Spin Resonance Nuclear Magnetic Resonance Magnetic Susceptibility Bulk Magnetization III. Discussion A. Evidence for a Proposed Crystal Structure 5 B. Magnetism of 3d S=5/2 C. The Ordered State: Metamagnetism D. The Ordered State: The Magnetic Space Group 4 COPPER L-ISOLEUCINE MONOHYDRATE I. Introduction II. Experimental A. B. C. Magnetic Susceptibility - Zero Field Magnetic Susceptibility - Applied Field Electron Spin Resonance III. Discussion A. B. Magnetism of 3d9 electrons in a 4mm Crystal Field Magnetic Susceptibility 5 CONCLUSIONS AND RECOMMENDATIONS REFERENCES APPENDIX A THE METHOD OF HIGH TEMPERATURE EXPANSION B TWO SUBLATTICE CANTED ANTIFERROMAGNET Page 66 66 69 7O 76 85 90 90 103 105 114 121 121 126 129 136 141 154 154 164 171 175 179 183 TABLE II. III. IV. VI. VII. VIII. LIST OF TABLES Restrictions of special spin positions Zero field NMR frequencies and line widths in MBTMA (T=l.lK) Comparison of unit cell parameters of MBTMA and CCTMA Assignment of proton special positions in MBTMA Magnetic space groups resulting from le/m Demagnetizing corrections for various samples of copper L-Isoleucine Principal values and direction cosines of the principal axes of the two g tensors Irreducible representations of the point group 4mm vi Page 33 71 96 99 115 131 153 157 FIGURE 10a. 10b. 11. 12. 13. 14. 15. 16. LIST OF FIGURES PED for a two sublattice uniaxial Heisenberg antiferromagnet (J>J). PED for the two sublattice canted anti- ferromagnet with no anisotropy. PED for the two sublattice canted antiferromagnet with anisotropy. The mirror operation for a polar vector. The mirror operation for an axial vector. Symmetry operations for axial vectors. The hydrogen bond. Octahedral coordination of ligands. Spatial extent of the d orbitals. Splitting of the d orbitals in an octahedral environment. The Brillouin function. The magnetization apparatus. Circuit diagram for the amplifier and electronic integrator. Schematic showing electrical connections of components. Magnetization of ferric ammonium sulfate. Solid curve is the theory. vii Page 12 15 17 21 24 29 29 31 36 40 40 41 45 57 59 59 63. FIGURE Page 17. The morphology of manganese bromide trimethyl amine dihydrate. 68 18. Temperature dependence of a proton nuclear magnetic resonance frequency in MBTMA. 73 19. Rotation diagram for proton nuclear magnetic resonance in MBTMA with an applied field of 300 Oe. (T=l.lK) 75 20. Zero field magnetic susceptibility of MBTMA. (After J. N. McElearney et al.). 78 21. Magnetic specific heat of MBTMA. (After J. N. McElearney 25 a1.). 80 22. The magnetic susceptibility of MBTMA with the external magnetic field parallel to the b axis. 82 23. Magnetic phase diagram for MBTMA. 84 24. Magnetization of MBTMA with the external magnetic field parallel to the b axis. 87 25. Magnetization of MBTMA below T (T=l.lK) with the external field along the crystallographic axes. 89 26. The crystal structure of cobalt chloride trimethyl amine dihydrate (After J. N. McElearney 33 31,). 92 27. Comparison of the proposed unit cell of MBTMA (lower) with the unit cell of cobalt chloride trimethyl amine (upper). 94 28. Bonding of a water molecule to a manga- nese ion. . 100 29. Magnetic dipole arrangement used in computer calculation. 102 30a. Zero temperature sublattice model for MBTMA with H=0. 107 viii FIGURE 30b. 31. 32. 33. 34. 35. '36. 37. 38. 39. 40. 41. 42. 43. Zero temperature sublattice model for ' 2 MBTMA w1th H- Hcrit PED for MBTMA with H along the b axis. PED for MBTMA with H along the c axis. The magnetic point groups formed from 2 m. The square pyramidal coordination of copper L-Isoleucine showing the derived principal axes of the g tensor. The crystal structure of copper L-Isoleucine. (After Weeks 33 21.). The morphology of copper L-Isoleucine. Zero field susceptibility of copper L-Isoleucine. Curve "a" is a Curie Weiss law and curve "b" is a high tem- perature expansion for a two dimen- sional ferromagnetic Heisenberg exchange. Inverse susceptibility of copper L-Isoleucine. Solid line is a Curie Weiss law. Magnetic susceptibility of copper L-Isoleucine with H parallel to the c axis. Composite H-T phase diagram for copper L-Isoleucine. ESR signals with H in the bc plane of c0pper L-Isoleucine. Curve a is with H parallel to c. Curve b is with 50° from c. ESR rotation about the a axis in copper L-Isoleucine. ESR rotation about the b axis in c0pper L-Isoleucine. ix Page 107 110 113 118 123 125 128 133 135 138 140 144 146 148 FIGURE 44. 45. 46. ESR rotation about the c axis in c0pper L-Isoleucine. Stereographic projection of the point group 4mm (upper) and the location of the four mirror planes and the four— fold rotation axis in the square pyramid (lower). Reduced inverse susceptibility plotted as a function of reduced inverse temperature. Curve a is a Curie-Weiss law; curve b is a high temperature ex- pansion for a two dimensional ferromag- netic Heisenberg exchange. Page 150 156 167 INTRODUCTION Magnetism which results from exchange or super- exchange is in general a very complex three-dimensional many-body problem. However, in some magnetic insulators, the crystal contains low dimensional structures which may be described as "sheets" or chains." These structures, which arise from the physical arrangement of chemical bonds between atoms, frequently result in electron inter- actions which reflect the low dimensional nature of the bonds. Experimentally, one often finds that at suffi- ciently low temperatures, the low dimensional magnetic systems undergo a transition to a three dimensionally or- dered state due to weak interactions between adjacent "sheets" or "chains.” In the ordered state, the stronger low dimensional character manifests itself in the nature of magnetic-field-induced phase transitions. In this work we use a "Spin polarization model" to describe the beha— vior of a magnetic system which undergoes a phase transi— tion in applied field, and specify one particular class of such transitions as "metamagnetic." The three dimensionally ordered magnetic state frequently reflects the symmetry of the crystal lattice. The concept of the magnetic space group and symmetry ope- rations which pertain to axial vectors in a lattice are discussed. These theoretical concepts are then applied to two experimental investigations. The first system, the c0pper (II) complex of the amino acid, L—Isoleucine, was interesting for several rea- sons. The crystal structure of this system is character- ized by chemical bonds which extend only in one plane. In addition, the local coordination of the c0pper ion is five- fold square pyramidal. Low temperature and ultralow tem— perature measurements of the magnetic susceptibility in zero and applied fields are reported and discussed in terms of a model Hamiltonian. The magnetism of the 3d9 copper in the square pyramidal crystal field is investigated using electron spin resonance and discussed using the Van Vleck point charge model for crystal field interactions. The second system, manganese bromide trimethyl amine dihydrate, has the disadvantage that the exact crys- tal structure is at present unknown. X-ray diffractometry and optical goniometryare used to derive the crystallogra- phic point and space groups. Nuclear magnetic resonance measurements in the three dimensionally ordered state are combined with other magnetic studies to suggest a crystal structure and a magnetic space group. Finally a transi- tion which is observed in applied field is discussed and asserted to be metamagnetic. CHAPTER 1 THEORY I. Low Dimensions: An Historic OVerview The existence of long range order as a result of superexchange, depends not only on the nature of the ex- change but on the dimensionality of the interaction as well. Long range order is defined as the existence of a nonzero value for the magnitude of the spin-spin correlation function: W§ (1) 1+A in the limit "A", the separation between spins goes to in- finity. While no rigorous proof exists, it is generally accepted that this implies: £im|< Si A+oo _ 2 §1+A>I- || (2) (The brackets denote the thermal average.) A nonzero value for the spin-spin correlation implies a 2 and therefore for <§i>. Since the nonzero value for <§i> magnetic moment is proportional to , a system possessing a nonzero spin-spin correlation will also possess a sponta- neous sublattice magnetization. Exchange arising from the overlap of electronic wave functions in low dimensions may result in a magnetic spin system which is characterized by the absence of long range order at any finite temperature. 4 For example, a magnetic Spin system which inter- acts through an Ising exchange: Hrs-1): J..S. s. (3) 2.175;] 1] 1z )2 in one dimension has been-shown1 not to exhibit long range order at any temperature above absolute zero. The two di- mensional Ising model was first shown to possess a sponta- neous magnetization by Onsager who presented his solution as a remark during the discussion following the presenta- tion of a paper by Tisza2 in 1942. Several years later, Onsager presented his formula for the spontaneous magneti— zation of the two dimensional Ising model. Onsager pub- lished only the results of his calculations, leaving the derivation to remain a mystery. Four years later, in 1952, C. N. Yang4 finally deciphered the puzzle and presented his rather long and complex derivation of Onsager's results. The three dimensional Ising model was investiga- ted by R. B. Griffiths in 19675. His solution established the existence of long range order for this model. In 1966, Mermin and Wagner proved6 that a system which interacts through a one or two dimensional Heisenberg exchange: H = -_1_.Z. J.. g. .g. (4) cannot possess a spontaneous magnetization at a finite temperature. Stanley and Kaplan showed7 that the absence of Spontaneous magnetization as in the two dimensional Heisenberg model does not preclude existence of a magnetic phase transition characterized by a singularity in the‘ susceptibility. Mermin and Wagner8 also showed that the x-y model: J.. (S. S. + S 2 . S. ) (5) i#j 1] 1x 3X 1y Jy in one or two dimensions does not produce long range order. II. Low Dimension and Metamagnetism In real crystals, the effects of low dimensional correlations are usually seen ina limited temperature range. As the temperature is decreased, the thermal fluc- tuations become small enough so that some weaker interaction produces three dimensional ordering. The low dimensional character can still manifest itself in the behavior of the magnetically ordered spin system in applied field. For example a system which has a strong two dimensional Heisenberg ferromagnetic exchange may be considered as con- sisting of layers of ferromagnetic spins. If in addition, a three dimensional ordered state is produced by an anti- ferromagnetic coupling between the layers, a magnetic field applied in the appropriate direction may produce a magnetic phase transition which consists of reversing alternate sheets to produce a ferromagnet. Such a tran- sition is said to be "metamagnetic." We shall take as our definition of metamagnetism the following: In the pre- sence of an applied field, a ferrimagnetic or antiferromag- netic system may undergo a phase transition characterized by an abrupt increase in magnetization. If the spin sys— tem consists of M sublattices and the transition involves a reorientation of N (N < M) of them, the transition is metamagnetic. The requirement that the magnetic spin sys- tem possess more than one sublattice precludes the exis- tence of a metamagnetic transition in a ferromagnet. In the next section we consider the detailed conditions which result in metamagnetism. III. Spin Polarization Energy A ratherelegant pedagogical technique for exam- ining the behavior of a spin system has been communicated to me by Prof. R. D. Spence.9 At zero temperature, a spin sublattice may be decomposed into basis states which re- present the components of sublattices as polarization "states." For example a two sublattice system will have the spin polarization basis states: I 2 2 In these diagrams, each arrow or "spin" actually repre- sents an entire sublattice. By using linear combinations of these basis states, we can form the sublattice configurations: (13': : Oll>+bl4> 2 I 2 c132=\/= c |2> + dl3> The coefficients depend on the details of the interac- tions (exchange, anisotropy, etc.) and are subject to the normalization: a + b = l (7) The behavior of the magnetic system can be found by calculating the relative energy of each polarization. As this is a zero temperatUre model,the ground state completely characterizes the magnetic behavior. Let us consider, as an example, a simple two sublattice spin sys- tem in the presence of an isotropic antiferromagnetic exchange: H = -J§ g, (8) 1 where the S's represent sublattices and J represents the antiferromagnetic exchange between these sublattices. Then referring to the definitions of the polarization states (Fig. l) we find the energies for the various states are: °=+J E °=+J E °=-J E4°=-J (9) where E1° denotes the energy of the spin polarization state |l> in the absence of an applied magnetic field. The resulting "polarization energy diagram" or PED is shown: 10 E |133>,|‘1>'==F= II>,|2> T The presence of a field applied along the 2 axis of the state I1> will produce a state of mixed polarization: <12 = a(H)I2> + b(H)I3> = V (10) where, as we have indicated, the coefficients are functions of the applied field. Because of the applied field, the sublattice Hamiltonian now has an additional term: _). H = -J§1 §2 - fi - M =.J§1 . g, - fi-r(§1+§z) (11) where F E ngB N = number of spins in sublattice spectroscopic splitting factor 00 ll Bohr magneton 11 The energy of the mixed state ¢1 in the presence of a field applied along the 2 axis is given by: 2 2 0° E = a J2° + b J3° + ab(J23°+J32 ) - 2PHb (12) where J2° is the exchange energy between sublattice 1 and O 2 when both sublattices are in the state I2>. J23 is the exchange calculated with sublattice l polarized as in the state I2> and sublattice 2 polarized as in the state |3>. Since the polarization states I2> and I3> are orthogonal, J23° and J32° are both zero. Substituting from the normal- ization equation we find: E(H) = (1-b2) 12° + b2J3° - ZFHB (13) The local minimum for this state is given by: __ = O = -2bJ o + 2b.] 0 ' 2TH (l4) 2 3 3H TH PH 2 1/2 b = ——5—-5' a = 1" __6"__5' The energy in a field applied along 2 is: 2 E (H) = J2° + J31— J3° -2I‘H #:3- ¢2 J3 JSO‘JZO J3 —J2 Mth °-J °) - 2(J3°-J2°)] O II C.) 2 o_ o (13 J2 ) 12 fl 3) (ll) FIGURE 1. PED for a two sublattice uniaxial Heisenberg antiferromagnet (J does not mix with I3> with H applied along the 2 axis. If we require that each of the sublat- tices in the mixed state: ; ¢2 = cll> + dl3> (16) be normalized to unity: + = c+ + d++ = c+2d = 1 (17a) +=c++0 =c =1 (17b) c = 1 + d = 0 (18) where +(+) represents a sublattice parallel (antiparallel) to the 2 axis.‘ The PED for the two sublattice system in a field applied along 2 is shown in FIGURE 1 for J and ¢1. The sublattices "flap" to a configuration which is perpen- dicular to the field. As the field increases, the sub- sat’ latticasrotate smoothly toward the field. At H = H the magnetization is saturated: H= 0 Hzo H=H$AT 2J1 2J1 ‘ vi 14 No sharp phase transition is produced. Next, consider an additional anisotropic interaction of the form _ 2 .2 H — -k(Slz +32z ) (19) where k is a sublattice anisotropy constant. (The ori— gins of interactions of this type are discussed in the next section.) The total Hamiltonian with a field applied A along z is now: - k(S 2+8 2 I E H — -JS . 12 22 1 2 (20) ) - FH(SlZ+SZZ) This additional term splits the states I1> and I2> in zero field. The energy of each polarization state in zero field is: E °=-2k-Jl° E ° = +J ° E o__ o O=_ O 1 2 2 3 — 2k+J3 E 2k+J (21) 4 3 The energy of the mixed state ¢2 in an applied field is: 2 2 E¢ (H) = -2k-J ° + ‘2F H (22) 2 2 w._ o -2k+J3 J2 The PED for a field applied along 2 with aniso- tr0py is shown in FIGURE 2 for J<0,‘k>0 and |J|>2|k|. In this case, for 0> J. The energies in zero field are as before: 51° = —2k+J E2 = +J E3 = -2k-J (23) Because of the anisotropy, the state |2> is now highest in energy. The PED for this large anisotrOpy case is shown in FIGURE 3. In this situation, a direct transi- J . . . = —— he ure antiferroma netic crit 2F from t p g state Il> to the pure ferromagnetic state I3>. The sublat- tion occurs at H=H tice configurations for fields above and below Hcrit are: 17 H FIGURE 3. PED for a uniaxial antiferromagnet with strong anisotropy (k>>J). CRIT H 18 H< HCR IT HCRIT 5- H —<- HSAT ‘ ll Since this transition has an abrupt increase in magnetiza— tion along the field and involves reversing only one of the two sublattices, the transition is metamagnetic We shall now turn to a more complicated example: the two sublattice canted antiferromagnet. This sublattice arrangement is such that the spins are no longer simply antiparallel along a particular axis but now are tipped or "canted" so they make some angle 9 with the z axis. 9 A 19 This canted system has a net moment in the § direction whose magnitude depends on e, the canting angle. This angle in turn, is a result of competition between the various sublattice interactions. Spin interactions which produce canted systems include competition between (1) first and second neighbor exchanges, (2) symmetric isotrOpic exchange and the antisymmetric Dzialoshinski- Moriya exchange(-D.81x§z) and (3) anisotropic single ion interactions and exchange. Closed form calculations for the two sublattice canted antiferromagnet are given in Appendix B for the case where the competing interactions are the Dzialoshinski- Moriya antisymmetric exchange and an isotropic antiferro— magnetic exchange: —> —> —> H = -J§l-§2 - D - 81x82 (24) In the presence of an applied field, closed form calculations become exceedingly complex. The analysis is often done10 by using numerical methods on a computer. We will instead qualitatively discuss this example of the canted antiferromagnet using the PED technique. In zero field, the antisymmetric Dzialoshinski-Moriya term and the isotropic exchange term interact to produce states of mixed 20 polarization. Recalling the definition of the two sublat- tice basis states, we define two new mixed states: (bl = x (I) = X (25) Assuming D = Dy, the canting angle 9 is found by minimizing the energy and is given by: tan 26 = g (26) where D is the Dzialoshinski—antisymmetric exchange con- 1 stant. In the absence of an applied field, these two states are degenerate. A small field applied along the 2 axis will split the states leaving ¢2, which has the largest moment parallel to the field, lowest. The PED for the canted antiferromagnet is shown in FIGURE 4. 21 FIGURE 4. PED for the two sublattice canted antiferromagnet with no anisotropy. 22 For H 30, the sublattices "flop" into the mixed state ¢2. This state has a net moment which increases with increasing field. At H=Hsat’ the spins are completely polarized along the field. No sharp transition is ob- served. This is consistent with the lack of anisotropy. H=O Hzo H=HSAT 11 Z,H If we now apply an additional interaction of the form: 2 2 27 H = ‘kcslz + S22 ) ( ) we will remove the degeneracy between the mixed polariza— tion states 81 and @2. The canting angle 1n zero field will now also depend on the magnitude of k. If we parame- terize the sublattice orientation energy in terms of an angle 9, we may calculate a by minimizing the polarization 23 energy with respect to 6. The Hamiltonian, =_—>.+_ 2 2 +++ H JS1 S2 k(SlZ +SZZ ) - D-Sle2 (28) with k >> J > 0 . _ 2 2 . 2 2 gives E(e) — —S J cose — DS Sine - kS cos a (29) = —SZJ c056 ~ DSZSine - kS2 sinzg Minimizing E with respect to: 2 2 ks2 a _ = . _ _ . e e 5— — 0 S J Sine DS cose —7— 2 Sin? c057 _ - k - - J Slne - D cose - 7 Sine _ 2D tans ‘ m (30) The mixed state ¢1 will now have the lower energy in zero field. In an applied field, ¢2 will begin to descend in energy due to the Zeeman energy of the large net moment of this state. The resulting PED is shown in FIGURE 5. The relative spin configurations in applied field are shown below: 24 I3) / / FIGURE 5. HCRIT HSAT H PED for the two sublattice canted antiferromagnet with anisotropy. 25 H: O O J produced a metamagnetic transition while k6 FIGURE 7. The mirror operation for an axial vector. 30 Now let us consider the effect of multiplying this operation by anti-inversion: _v m' = m x l (31) We first perform the mirroring operation on the axial vector as shown in FIGURE 7. We then reverse the direc- tion of the axial vector producing the configuration: / m Several of the more common elements of the mag- netic space group are shown in FIGURE 8. We note that a nearest neighbor anti—translation reverses the spins along a given direction. This then doubles the separation between spins which are parallel. If an anti-translation is present in the magnetic space group it (1) produces antiferromagnetism and (2) doubles the size of the magnetic unit cell relative to the crys— tallographic cell. ’f"'.-‘—""“ . -.I - . _.,_ _ __ ‘ . -=—- a - - _——.h_.h _ ._.. 31 <4~ / \ / «L \ ‘Q’uv _/ I 6) FIGURE 8. Symmetry operations for axial vectors. The effects of point group operations which con- sist of rotations,ref1ections and inversions, shown in FIGURE 8 apply to spins which occupy general positions in the magnetic unit cell. If a spin occupies a special po- sition such as on a two-fold rotation axis, in a mirror plane or on an inversion point, the operations produce stringent restrictions on the orientations of the spin. For example, if a spin is located on a two-fold axis,for every component of the spin which is perpendicular to the axis, the two-fold operation produces an equal but Oppo- sitely directed component which will cancel it. The com- ponent of the spin parallel to the two-fold axis will be unaffected by the rotation. Therefore, a spin which is 32 located on a two-fold axis can only have a nonzero moment parallel to the axis. A summary of the requirements of special positions for other operations is given in TABLE 1, Detailed discussions of the spin configurations produced by specific groups will be left until considera- tion of experimental results in later sections. 33 TABLE I Restrictions of special spin positions Spin In or On Position Of n fold rotation axis n fold screw axis mirror plane anti two-fold axis anti n (n>2)-fold axis n-fold anti screw axis anti mirror plane inversion point anti inversion point Spin Orientation Requirements spin must be parallel to n-fold axis no requirement spin must be perpendicular to the mirror plane spin must be perpendicular to the two-fold axis forbidden! spin may not occupy an anti n-fold axis. no requirement spin must be parallel to the anti mirror plane no requirement forbidden! spin cannot occupy an anti inversion point 34 V. Chemical Bonding We shall be discussing the following kinds of chemical bonds in the context of transition metal ions in organic complexes: (1) ionic bonding and (2) covalent bonding which bond the transition metal to its local ato- mic neighbors and (3) "weak ionic" bonding and (4) hydrogen bonds which bond large molecular subunits together to form a crystal. We shall begin our discussion with ionic bonding. Ionic bonding is the result of electrostatic Coulomb inter- actions which causes a positively charged cation to be attracted to a negatively charged anion. The interaction may result between two ionized atoms such as sodium and chlorine in NaCl, or between a polarized system like H20 and an ion such as a transition metal. In this last case, the polar water molecule orients itself so that the oxygen which is slightly negatively charged is pointing towards the positively charged transition metal ion. Covalent bonding is said to occur when there is significant overlap of atomic wavefunctions so as to allow an electron to be transferred from a central ion to the overlap region. In the case of transition metal ions, the orbitals that overlap are not simply the occupied 3d orbitals but usually are "hybridized" orbitals. These orbitals consist of linear combinations of the 3d, 45, and 4p orbitals. Linus Pauling has-indicated13 that the correct 35 combination of orbitals for covalent bonding is the one which is most directed toward the ligands. Having established two methods for "gluing" ions together, let us examine ways of taking these molecular subunits and "sticking" them together to make a crystal. One of the simplest ways is simply the extension of ionic bonding to the larger subunits. As we have seen, ionic bonding holds the sodium to the chlorine in NaCl. This bonding of Na+ to C1- extends throughout the crystal form- ing the lattice. This concept may be extended to larger subunits than two interacting ions. For example, a tran- sition metal ion can coordinate with several negatively charged ligands as to yield a molecular subunit which has a slightly negative charge. An ionic interaction of a somewhat weaker nature may take place between this slightly negative subunit and a correspondingly positiVely charged subunit consisting of organic molecules such as methyl- anime groups. Since the charge is not localized the in- teraction which is frequently called "weak ionic" bonding,is usually weaker than ionic bonding between atoms. (Finally, there is a localized electrostatic interaction called "hydrogen bonding" which is frequently responsible for coordinating large molecular subunits in crystals. Hydrogen bonding is thought to occur when an electron rich atom such as an oxygen in one molecule approaches a slightly acidic (electron depleted) hydrogen 36 atom14. The hydrogen atom may be bonded to nitrogen for example, and carries a partial positive charge. A localized electrostatic interaction,as shown takes place: 2.2 1.95 N l.99 Cu Cu FIGURE 9. The hydrogen bond The actual length of this bond is approximately 2A which is thought to be too long to allow much covalent overlap. Typical energies for this interaction are 6 kcal which is between the covalent bonding (IO—25 kcal) and the Van Der Waal's (l-S kcal ) bonding. The distance separating the atoms and the lack of real electronic wave function overlap tend to imply that a hydrogen bond is 37 an extremely poor path for superexchange. VI. Crystal Field Theory The five d electronic orbitals of an isolated transition metal ion are degenerate. In a crystal, these orbital energies are Split by an electrostatic interac- tion which occurs between the d electrons of the ion and the electronson the surrounding atoms or "ligands." In general, exact calculations of the electrostatic "crystal field" energy are a difficult and complex undertaking. The difficulty comes about because in a crystal, neither the transition metal electrons nor the ligand electrons are completely localized. Due to the rather complex types of bonding which can occur in a crystal, the electronic wavefunctions are no longer adequately described by wave functions of the isolated atom. More complex "hybridized" orbitals which consist of linear combinations of atomic wavefunctions are employed to describe the electrons. However, under certain conditions, the bonding of the transition metal ion to its ligands is mostly ionic. The covalent nature of the bonding is not necessarily non— existent but it is small enough so that to a very good approximation, the electrons are localized on the transi- tion metal ion and can be adequately described by the d 38 atomic orbitals. In this situation, we may use the so- called "crystal field theory." The first investigation of the interactions of the ionic electrons and the crys— talline electric field was by Becquerel15 in 1929. At the same time Bethe16 used group theory in conjunction with the relative strengths of the interactions to show ~how the energies of the atomic orbitals are modified by the ligands. In 1932, Van Vleck17 succeeded, on the basis of a "point charge" approximation, in explaining the "quenching" of the orbital angular momentum and the re- Sulting spin-only magnetic moment of paramagnetic transi— tion metal ions. Van Vleck later went on to show18 that Bethe's original crystal electric field ideas could be generalized and included in-a more modern molecular orbi- tal approaCh. In 1952, however, Kleiner19 pointed out that if the classic electrostatic model is used to calcu- late the splitting of the orbital energies the sign of the splitting is actually reversed if an extended electron cloud model is used in place of the point charge model. A more recent quantum mechanical approach by Tanabe and Sugano20 showed that even in certain "ionic" crystals such as KNiF321, the itinerant or nonlocalized behavior of the transition metal electrons makes the dominant contri- bution to the crystal field Splitting. This strongly suggests that although the point charge model of Van Vleck qualitatively, and often quantitatively yields the correct 39 result, more of the details of the nonlocalized quantum 'mechanical interactions are needed for a comprehensive theory. Having outlined some of the deficiencies of the point charge model, we will use it to qualitatively dis- cuss the effects of the crystal field on the orbital and spin states of transition metal ions. In this theory, we replace the atoms which surround the transition metal ion with point charges. An electrostatic repulsion will occur between the electrons in the d orbitals and these point charges. This interaction will be strongest for those or- bitals which have the greatest spatial extent or "point" towards the ligand point charges. The energy of these orbitals will then be split from orbitals which do not extend toward the ligands. For example, let us consider an octahedral ligand coordination as shown in FIGURE 10a. The spatial extent of the d orbitals is shown in FIGURE 10b. By examining the relative orientations of the d orbitals and the six ligands, we See that the dxz_y2 and dz: orbi- tals point directly towards the ligands, while the dxy, dxz and dyz orbitals all extend between the ligands. Cal— culations show the"eg"22 orbitals (d2 ,2, d'z) are degen- . X 'Y Z O o o " " ‘ erate and split from the remaining t2g (dxy dxz dyz) orbitals as shown in FIGURE 11. FIGURE 10b. Spatial extent of the d orbitals. 41 %\\\. d (I d ‘x. XY YZ x2 129 FIGURE 11. Splitting of the d orbitals in an octahedral environment. Let us examine the effect of this interaction on the or— bital and spin angular momenta in the case of five elec- trons (Sds). When we examine the energies produced by occupying these orbitals, we must consider the relative strengths of two competing interactions. The first inter- action, the energy which results from OCCUpying the higher eg orbitals can in theory be minimized by occupying only the t2g orbitals. However, there is an electron-electron 42 repulsion interaction which occurs strongly between elec- trons in the same orbital. It is instructive to consider the two limiting cases. In the first, the so-called "weak ligand field" case, the splitting A between the tzg's and the eg's is small compared to the electron-electron repulsion energy. In this case the orbitals will be singly occupied. Hund's rule for coupling spin indicates the five electrons should all be "spin up” to give the maximum total spin of S=S/2. However, the Pauli exclusion principle states that no elec- trons can have the same quantum numbers. Since all five spins have the same spin projection, they must each have a different £2 quantum number. One can see that the sum of all five projections (2, l, 0, -I, -Z) is zero. The result is that the total orbital angular momentum is L=0 (S-state). The other limiting case, the "strong ligand field" occurs when A is large compared to the electron- electron repulsion. In this case the minimum energy occurs with all five electrons occupying the t2g orbitals. The Pauli exclusion principle forces the electrons in doubly occupied orbitals to have their spins "paired" antiparallel. The four "paired" spins haVe zero net spin angular momentum. The only contribution to the spin comes from the single unpaired spin. This results in an S=l/Z spin state for the atom. 43 The strong ligand field results in low spin sys- tems, while the weak ligand field results in high spin. VII- The Internal Field in a Magnetic Sample The magnetization for a paramagnet in an external field is given by:23 M(H,T) = NguBSBS(X)' (32) where Bs(x) is the Brillouin function = ZS+l . (ZS+l)X 1 X Bs(x) ZS coth ZS §§ coth 7— (33) . _ gPBHS Wlth X - ‘E-T— and N = Avogadro's number g = spectroscopic splitting factor “B = Bohr magneton S = spin quantum k = Boltzmann's constant H = internal magnetic field T = absolute temperature As we shall see,the field inside a magnetic crystal is generally not equal to the applied field. The Brillouin function is plotted as a function of the argument "X" u HS (£i£%r0 in FIGURE 12. Let us examine the behavior of the magnetization for small values of X(high temperature and/or low field). FIGURE 12. 44 The Brillouin function. 45 LOP— — 3X For y< p HLorentz _ 3* M M.W. (40) We may calculate the field at P due to the mag- netic material contained inside the sphere by numerically summing the field at point P dueto each dipole. -;+. +3’X.+. A. + “J (I) “3)ni HDIP — g (41) r. J where fij is the dipole located a distance rj from the point P. 3 is a unit vector which points from the dipole to P. The total internal field due to all these effects is: + -'fi 'E I I 42 Hint ’ app + LOR ' HDEM + HDIP ( ) + A _). A + Z‘JJ- +3(n:u.)n fi. = H + (33-- D)¥Lp+ J l l. (43) int app 3 j r.3 J The magnitude of the dipole field is usually small compared to the Lorentz and demagnetizing fields. For this reason it will be neglected in the discussion which follows. We have shown that for small fields or high temperature: 0 _ M. x - H (emu/cc) (44) int 49 where x° is the magnetic susceptibility that would be observed in the absence of demagnetizing and Lorentz fields. Then: M X° = 4n M Happ + ('3— " D) W M _ HapJL 4n _ M p D 1 + (3— D) Happ M.W. Now if we define: : __M_ Xmeas ‘ Happ we can write 0 _ Xmeas - 4n 45) (1 + (3—’-D) Xmeas ——£—- ( M.W. Knowing the physical dimensions of a magnetic sample, its density and its molecular weight, we can correct the expe- rimentally measured susceptibility as long as the measure- ment is made at small applied fields or high temperatures so that the magnetization is a linear function of the field. CHAPTER 2 EXPERIMENTAL APPARATUS: MAGNETOMETER I. Introduction In order to measure the bulk magnetization of magnetic insulators, we have constructed a magnetometer capable of measuring magnetization at liquid He4 tempera- ture in applied fields up to 16 kOe. The design was sug- 26 and is based on the gested by an article by Mcquire original design of an induction magnetometer by Cioffi27. The magnetization of the sample is detected by measuring the EMF induced in a pickup coil when the sample is moved through the coil. The EMF is then electronically integra- ted to give the net change in flux caused by the moving sample. This change in flux is related to the bulk mag- netization of the sample. II. Theory The magnetic fields of a uniformly magnetized sphere are givenz8 by: 3 8w M R A . A + + o Sine = = O O B0111: HOUt 7 (C03 1‘ + 2 ) (46a) + an M 2 (46b) B. =-—— 0 1n 3 if = '—-4" M 2 (46c) in 3 o 50 51 where Bin and fiin are the magnetic fields inside the sphere and M0 = magnetization in emu/unit volume R = radius of sphere r = distance from center of the sphere and r and O are defined: We note that the fields outside the sphere are those of a magnetic dipole of moment 5 = 1%33 MOE. Let us consider the flux through a loop of radius a with the sphere located a distance 2 away from the plane of the loop along the axis of the loop as shown in Fig. 27. The flux through the 100p is given by: '6' ll f Bnorm x d x d¢ (47) j a dK = S a ano Bnorm xdx 52 3 O—— n i . z E? 1 a 2 = Znu f 3 cosEO-l xdx o r = Znu [a X +2 - x dx 0 (x2+z2)3/2 (xz+zz)3/z = 2““ fa 3 zzx - X dx 0 (xz+zz)5/2 (x2+zz)3/2 3 4 (z>R) = Effigfg , '22 ' 1 (48) (32+zz)3/2 (z2+zz)1/2 If the sphere intersects the plane of the coil, we must modify the calculation due to the altered geometry: We may derive p from the equation of the circle which results from the intersection of the plane of the loop and the sphere: p = x + y (49) The equation for the sphere is: R2 = x2 + y2 + Z2 ' (50) Substituting from this last equation we find: 2 _ Zz 1/2 p = (R (51) The flux through the loop from the fields outside the sphere is given by: a ¢out = 2W“ {p (Bout)norm de 54 2 3 8'" R M0 _22 1 - l3- ‘ 3 (a2 +2 z)3/2 R + l ' l. (52) (a2+zz )1/2 R _ p gin — Znufo (Bin) norm xdx 2 8n M = ° (R2 - 22) (53) 3 < 8nZR3MO _Zz ¢(Z-R) = @0111: + Qin = 3 2 3/2 (a2 +2 ) + 1 (54) (32 +2 2)1/2 Thus the flux through the loop has the same functional dependence for all values of 2. As the sphere is moved along the z axis the flux through the loop will change thereby inducing an EMF in the coil: m . 6' E = - (55) Q) ('1‘ Integrating this EMF as a function of time: 3¢ _ f Edt = - f3? dt = -A¢ — 4(2') - 4(2) (56) Since the ¢ involves the magnetization of the sample, we 55 can measure the magnetization by integrating the EMF induced in the coil. We note that this result is independent Of the path of the sample, but involves only the flux indicated at the two end points. In a practical experiment, the sensitivity and noise rejection of the apparatus can be improved by using an astatic pair of pickup coils. The experimental sample coils, shown in FIGURE 13, consist of two coaxial solenoids of N turns counterwound on the same cylindrical form. I There are two advantages to such a system. The first is that any fluctuations in applied magnetic field will affect both coils equally. Since the coils are wound in Opposite directions the EMFs induced by the fluctuating fields will have the Opposite sign and cancel. The second advantage is that in addition to increasing the sensitivity by a factor Of N over that of the single loop, the EMF's induced in the two coils by the moving sample will have the ggmg sign and therefore will add. To calculate this, we must evaluate the flux through both coils as we move the sample from the center of one coil to the center of the other. However, by symmetry, the flux at the two end points should be the same. Therefore, it will suffice to evaluate the total flux with the sample in the center Of one coil and then double the result. The detailed cal- culation involves summing the flux through each of the N FIGURE 13. S6 The magnetization apparatus. SUPER- CONDUCTING \ SOLENOID SAMPLE’// 57 L\\\\\I Li\\\\) SAMPLE / HOLDER // F E L -. \ V \ ‘N \ \\\~ \ ASTATIC PICKUP '//’cous : // \ /// \r N 2 S8 lOOps of the two coils: 2N (DO = .2 Qiai) i=1 2 3 2 = M 21; ~21 . 1 (57) 3 i=1 (32+Ziz)3/2 (a2+ziz)l/2 A computer calculation for the sample coils in our appa- ratus was carried out. The total flux change in moving the sample from One coil to another is: SHZRSM _ = O 3 I AQtot - ZOO ———g——— (2.62 x 10 ) (58) III. The Cryostat The cryostat is shown in FIGURE 13. The sample holder is supported by a piece of 1/4" thin wall cuprO- nickel tubing. This tubing is fed through an 0-ring "Veeco" vacuum seal in the lid. The O-ring makes a sliding contact with the tubing and allows movement of the sample without significantly affecting the vacuum over the helium bath. The static applied field is produced by a super- conducting solenoid consisting Of 14,712 turns of 0.0178 cm diameter copper jacketed Niobrum Zirconium wire wound on a form Of "Synthane" type G-ll fiberglass epoxy.29 The sole- noid produces a field of 1680 oersteds per ampere. The maximum current at 1.1k is approximately 12 amperes which 59 l JI I 0K /"1 I IT , \ IM ‘ - 233K ‘ J IOOK ' 233K .. + I r 905 I M O (XNLS I I I I i I FIGURE 14. Circuit diagram for the amplifier and electronic integrator. {'5 - l i W A 1 445 [ I | _ ‘El fi 2 -RS IM I IOOK) I—Ii-I-IS _, I 55 gfiPmP ' 6 . .49 , 71>— 4334 COM '\—48 pmmwcmm 200K 9 B IO t I II offset ' 2K = M'Z )_L_I‘) '4 chassis 200 200K ? I5 A FIGURE 15. Schematic showing electrical connections Of components. 60 gives a maximum field of 20 kilo-oersteds. The current is determined using a digital voltmeter to measure the voltage across a resistor connected in series with the solenoid. The value of the resistor is 1.31 i .05 milliohms. The field in the solenoid as a function of the voltage across the series resistor is: H(Oersteds) = 1282 x Vre (millivolts) (59) S The sample holder is machined from epoxied linen. The two pickup coils are wound astatically on a synthane type G-ll epoxy form. Each coil consists Of approximately 2500 turns Of number 36 enameled c0pper wire. The output of the pickup coils is connected to a differential ampli- fier and then to an integrator. Both the amplifier and the integrator utilize an Analog Devices Corporation model 233k chopper stabilized Operational amplifier powered by an Analog Devices model 904 dual 15 volt power supply. The circuit is shown in FIGURE 14, and the schematic is shown in FIGURE 15. The gain may be switched from 1000 to 10,000 and the integration time constant can be set to either one or five seconds. The voltage output of the integrator is measured using a digital voltmeter. Typi- cal integrated output voltages ranged from 15 millivolts tO 9.5 volts with a signal to noise ratio Of one at the lowest voltages. The zero Offset Of the Operational 61 amplifiers must be periodically readjusted since the Offset voltage will appear as an integrated signal. Typical drift rates are: l millivolt/second with a gain of 10,000 and a time constant of 5 seconds; 0.5 millovolts/sec with a gain Of 10,000and a time constant Of 1 second; and 5 microvolts/second with a gain of 1000 and a 5 second time constant. IV. Calibration The overall calibration of the system was per— formed using a sample Of ferric ammonium sulfate, Fe2(SO4)3 (NH4)2 (804)-24H20. This material is an S=5/2 paramagnet at liquid H84 temperatures. The magnetization is given by: H 311B kT M(H,T) = (NguB gnaw ( ) (60) The experimental data is shown in FIGURE 16. The solid line is a fit to the expression: M.W. T M(volts) = iK ( emu ) M(H,T) (61) volt-mole where M = is the integrated EMF in volts M.W. = the molecular weight m = the sample mass K = the system calibration constant T = the integrator time constant 62 FIGURE 16. Magnetization Of ferric ammonium sulfate. Solid curve is the theory. /ng 50F— 40— 01 O M (emu N o l IO— 63 L I I 4 5 8 WT (kOe/Kelvin) 64 A = the amplifier gain The derived calibration constant is: K = (2.93 i .05) x 10‘5 em“ (62) mole-volt We have previously derived (eq. 58) a theoretical expres- sion for the EMF for our coils: 3 8W2R3MO EMF = (2.62 x 10 ) ———————— 3 with M0 in emu/cc. TO compare this expression with our experimental value, we must convert it to the same units: 3 4NR M0 = magnetic moment in emu 3 TO express this in emu/mole, we multiply by m/M.W.: 3 4BR M 0 m = magnetization (%%%€) 3 M.W. 3 (2.62x10 )2 M.W. l -5 emu K = = 6.07x10 _ (63) theor (2.62x103)2 mole volts Thus the constant derived from a theoretical expression differs fromthe experimental number by only a factor of two. CHAPTER 3 MANGANESE BROMIDE TRIMETHYL AMINE I. Introduction Manganese bromide trimethyl amine dihydrate (MBTMA), MnBr3[(CH3)3NH]-2HZO is a member of the general class of metallo-halide organic compounds: MX3[(CH3)3NH]~2H20 (where M = C0, Cu, Ni, Mn and X = Cl, Br), which have been the subject of extensive structural and magnetic measurements.30—33 In these com- pounds, the metal ion is generally octahedrally coordinated by two waters and four halides. The octahedra are edge shared along the halides so that the metal ion-ligand structure takes the form of chains or sheets giving rise to experimentally observed low dimensional magnetic behavior. Although the crystal structure of MBTMA has not been determined, measurements of the magnetic specific 35 heat34 and zero field susceptibility have demonstrated low dimensional behavior and a transition to a three dimen- sionally ordered state at 1.56K.36 NMR, magnetization and susceptibility (in applied field) experiments have led to a zero temperature model for the spin system and a tentative crystal structure which is 65 66 consistent with the space group derived from x ray analysis and is similar to the crystal structures for other members of this group of compounds. 11. Experimental Large single crystals of MnBr3[(CH3)3NH]-ZH20 were prepared by reacting a 25% aqueous solution Of (CH3)3NH with MnBr2'4H20. A solution of 48% hydrobromic acid, HBr, was then slowly added to the mixture. The resulting pinkish- Orange solution was then filtered and slowly evaporated near room temperature. Within several days, well formed (FIG. 17) orange—pink crystals appeared. The solution was filtered and several small ”seeds" were suspended on threads in the solution and allowed to continue growing. Upon reaching a mass of several hundred milligrams, the crystals were removed from the solution and used in the following experi- ments. A. Crystallography The density Of the crystal was determined by weighing the crystal in air and in metha— nol. The lattice dimensions and crystallographic space group were determined by x ray diffractometry of aligned single crystals using MOKa radiation. The lattice of MBTMA has dimensions: a = 8.45:.02A b = 7.651.02A and c = 8.4Si02A and is monoclinic with an angle of B = 91°56' i 4'. The derived cell volume used in conjunc— tion with the experimentally determined density indicates 67 FIGURE 17. The morphology of manganese bromide trimethyl amine dihydrate. J"- 69 there are two molecular units in the crystallographic unit cell. The extinctions of every other reflection along the crystallographic b axis as well as the presence of the monoclinic angle imply the crystallographic Space group P21/m. The point group Z/m was confirmed as was the mono— clinic angle by measuring the crystal face normals using an optical goniometer. B. Electron Spin Resonance Electron Spin reso- nance was performed on an aligned Single crystal at 9.2 GHz (x-Band) in applied fields up to 20 kilo-oersteds. The mi~ crowave cavity was immersed in liquid He4 and measurements were made from 4.2 to 1.1K. Only one broad resonance (:ZkOe. wide) was Observed. Within experimental uncertainty the g-value and the line width are isotropic. The persis- tence of the single line through the Neel temperature (1.56K) with only a Slight decrease in line width is thought to be due to regions of paramagnetic material. The crystal is very hydroscopic and some Of the surface material may have been dissolved. This dissolved material would then behave as a paramagnetic glass at low temperatures. No evidence Of a R=O Spin wave resonance was Ob- served at the lowest experimental temperature of 1.1K. 70 C. Nuclear Magnetic Resonance Nuclear magnetic resonance experiments were performed on aligned Single crystals immersed in liquid He4 using a marginal oscillator. The spectrum in zero applied field was recorded at T=l.lk and is shown in TABLE II. The line which occurred at 18.74 MHZ (at 1.1K) was recorded as a function of tempera— ture and is Shown in FIGURE 18. Deuterated crystals of MBTMA were prepared by first heating MnBr2°4H20 to drive Off the waters of hydra- tion. The resulting powder was then reacted with (CH3)3NH in D20 in the presence Of hydrobromic acid HBr. Crystals were again produced by slow evaporation. This time, how- ever, the evaporation took place in a desiccator and great care was taken to prevent H20 from contaminating the crys- tals. The spectrum for the resulting deuterated crystals was recorded in zero applied field below TN. The lines at 18.17 and 18.74 MHZ decreased by almost two orders Of mag- nitude. Since it is expected that there is no Significant exchange Of hydrogen atoms for deuterons in the methyl or amine groups, it is probable that only the waters of hydra- tion were replaced by D20. We infer that the two NMR lines which decreased in magnitude in1jugdeuterated crystal are due to the protons in the residual waters Of hydration. NO other lines were affected by the deuteration. An applied field of-300 oersteds was used to probe the rotational symmetry of the magnetically ordered 71 TABLE II Zero field NMR frequencies and line widths in MBTMA (T=l.lK) v(MHz) v(MHz) Av(KHz) Av(KHz) v(MHz) Av(KHz) 1.53 250 8.15 250 26.91 125 1.86 200 9.76 200 28.56 125 1.94 250 18.17 250 3.195 250 18.74 250 7.68 250 23.50 125 72 F IGURE 18. Temperature dependence of a proton nuclear magnetic resonance frequency in MBTMA. ear 73 IB— I6— l4— I2— IO— ”(MHz) CD I 1 I 1 1 OLD H 1.2 L3 L4 T (KELVIN) 74 FIGURE 19. Rotation diagram for proton nuclear magnetic resonance in MBTMA with an applied field of 300 De. (T=l.lK). 75 I30 l50 I70 IIO 3O 50 70 IO 76 state. Rotations were made about each of the three crystal- lographic axes. A typical rotation diagram is shown in FIGURE 19. Although the frequenciesare different for the other lines, the rotational behavior for all lines in applied field is identical. A detailed examination of these rotation diagrams shows that for each local field magnitude, there are four distinct local field directions. D. Magnetic Susceptibility The near zero field susceptibility for MBTMA has been measured by the group 37 The at the University Of Illinois at Chicago Circle. susceptibility and magnetic Specific heat data are shown in FIGURES 20 and 21. The data indicate a TN=1.56K. The ac susceptibility in applied fields up to 16 kilo-oersted has been measured at liquid He4 temperatures using standard mutual inductance coils inside a supercon- ducting solenoid. The data with the external field applied along the crystallographic b axis is shown in FIGURE 22 for sev— eral different temperatures. At temperatures below TN, an anomalous peak at H z 1200 oersteds is Observed. NO such anomaly is Observed with the field applied along the other two axes. The values Of the critical field and the temperature at the peak in susceptibility are Shown in FIGURE 23. This diagram indicates the presence Of at least three distinct magnetic phases. Saturation Of the 77 FIGURE 20. Zero field magnetic susceptibility of MBTMA (after J. N. McElearney gt 21;)- 78 129 _L0 . ... o -E o 0 ..N . ._ 0 A 0 OZ ' -—> o .J : UJ : X o -CI)|-— O o o 3 0. -I£O o a" Do . .. “s o ‘0 .' :-, o .0. . . .. ~— .....0,°’--.‘ o .00 0 o. '0. (\I >1_mv_ :. o .m ON 0.. _ _ < _ Md ®.O ad N._ H (904) 8S spin system was not Observed at the highest experimental field. It is possible that another magnetic phase may exist at fields above 16 k0e. E. Bulk Magnetization The bulk magnetization has been measured as a function of applied field at liquid He4 temperatures in a "snatch coil" magnetometer (see expe- rimental apparatus: Magnetometer). The magnetization with the field applied along the crystallographic b axis is shown for various temperatures in FIGURE 24. As the temperature is lowered from 4.2k to TN one Observes a linear increase in magnetization for small increasing applied fields. The slope dM/dH is consistent with the measured susceptibility. At temperatures close to TN, however, an abrupt change in the lepe occurs at a field of several hundred oersteds. If one extrapolates the high field portion of the magneti- zation curve back to zero applied field, one finds a nonzero magnetization whose magnitude increases as temperature is lowered toward TN' There is no discontinuity in magnetiza- tion in the other directions, At temperatures below TN, the magnetization in the b-direction shows (FIGURE 25) a sudden increase which occurs at H z 1200 De. At higher applied external fields the magnetization continues to rise smoothly, although not linearly with increasing applied field. Saturation of the magnetization was not observed for fields up to 16 k0e. 86 FIGURE 24. Magnetization of MBTMA with the external magnetic field parallel to the b axis. I2r— oo 5 I I M(emu/grom) 0" I 87 88 FIGURE 25. Magnetization of MBTMA below TN (T=l.lK) with the external field along the crystal- lographic axes. 89 O_ .V_ N. O. m m N — _ — — q — O O O MCICQILNSOI o 0 o 0 o 0 o O C C O . . . C V . . . E o o 00 O . O. O L ON 90 III. Discussion A. Evidence for a Proposed Crystal Structure The exact crystal structure for MBTMA is at present unknown. It appears likely, however, that the structure is closely related to that of C0C13[(CH3)3NH]-2H20 (CCTMA). The struc- ture of the cobalt complex,38 shown in FIGURE 26 is charac- terized by chains of edge sharing octahedra which extend parallel to the b axis. The local coordination of the me- tal ion is octahedral consisting of four chlorines and two water molecules. The water molecules lie mostly above and below the cobalts inthe c direction. The unit cell of CCTMA is orthorhombic with four molecular units in the crystal- lographic cell. We assume that in MBTMA, the atomic configuration of the CoClg and.(CH3)3NH+ molecular subunits are still preserved, but that a structural modification occurs which affects only the molecular packing Of these units. The pro- posed unit cell modification is exhibited in FIGURE 27. This change moves the chain running through the center of the CCTMA unit cell down until it coincides with the chains in the ab plane. The cell becomes Slightly dis- torted and descends in symmetry from orthorhombic in CCTMA to monoclinic in MBTMA. The local coordination of the man- ganese ion is presumed to be octahedral with four bromines and two waters of hydration. The edge sharing character of 91 FIGURE 26. The crystal structure of cobalt chloride trimethyl amine dihydrate (after D. B. Lossee 23 a1.). 92 Inlv FIGURE 27. 93 Comparison of the proposed unit cell of MBTMA (lower) with the unit cell of cobalt chloride trimethyl amine dihydrate (upper). 94 1f ate PIP—1r , 95 the octahedra in CCTMA which results in a chain of metallo halide bridges is also thought to be present in MBTMA. With the proposed modification of the relative arrangement of these ”chains," the length of the a axis in MBTMA should be half that of CCTMA and there will be only two molecular units inthe cell. TABLE III is a summary of the crystallo- graphic parameters Of both systems. The chemical unit MnBr3[(CH3)3NH]-2H20 contains fourteen protons. As there are two molecules in the chemical unit cell, the cell con- tains a total of 28 protons. Atoms may be located at ”general” symmetry positions in the unit cell as follows: place the first atom in the unit cell at some position (X,Y,Z) which is £33 on a rotation axis, in a mirror plane or at an inversion point. We then perform each of the symmetry operations in the group and generate the remaining general positions. The group P21/m has four symmetry Ope- rations:"2",mirror, inversion and identity in its point group 2/m. Thus there are four general positions in the crystallographic unit cell which are related by symmetry operations. We could place all twenty eight protons in the MBTMA unit cell in seven groups of four general posi- tions. The four protons in each group are said to be "equivalent." Physically this means that aside from rela- tive differences in orientations produced by the symmetry operations, the local atomic surroundings of each proton would be the same. This last statement implies that 96 TABLE III Comparison of unit cell parameters of MBTMA and CCTMA MnBr3[(CH3)3NH]-2H20 C0C13[(CH3)3NH]-ZHZO Number of molecules in cell 2 4 Lattice dimensions a 8.45 i .02 X 16.671 K b 7.65 1 .02 X 7.273 X c 8.54 1 .02 3 8.113 X B= 91°56' 97 physical prOperties such as the magnitudes of the local magnetic fields of all four protons in this symmetry rela- ted group, would be identical. The fields, of course, will, in general, have four different directions. A dif- ferent group of four protons which were generated from another "general" starting position (X', Y', 2') need not have the same atomic surroundings or physical properties. Thus, for the seven groups of four protons we would expect seven distinct local field magnitudes. However, the Spa- tial locations of the protons are not all independent. One of the protons, for example, belongs to the amine (NH) unit. Since there are only two such units in the cell, we cannot place these protons (and nitrogens) in general posi- tions which would result in there being four of them in the unit cell. We must then, place them in "special" posi- tions. A special position is a particular location in the unit cell, such as in a mirror plane or at an inversion point. An atom which is placed at one of these positions is unaffected by one or more of the symmetry Operations. For example, an atom located in a mirror plane is unaffected by the mirroring Operation. There are a total of eighteen methyl (CH3) protons. Since eighteen is not a multiple of four, a least two and perhaps more of these protons are in special positions. The deuteration experiment shows that there are two distinct local field magnitudes for the water protons. Since there are eight H20 protons we may place a] tv 01 OI SI 98 all of them in general positions producing the requisite two groups of four. TABLE IV Shows a possible assignment of special and general positions and the resulting number of local field magnitudes. This assignment produces nine local field magnitudes which is consistent with the number of zero field proton resonances. Finally, because the cell only contains two manganese ions, these also must be in special positions. Let us elaborate on the reasoning for the asser— tion that the local coordination of the metal ion is simi- lar in both MBTMA and CCTMA. The broad peak in the magne- tic specific heat data above T for both systems indicates N low dimensional magnetic behavior. This behavior, which is thought to be due to short range correlations, is also evident in the nonzero (extrapolated) moment observed in the bulk magnetization measurements above TN. This sup- ports the assertion that chains of metal halide bridges are present in both MBTMA and CCTMA. The water molecule which completes the octahedral local coordination of the metal ion in the cobalt salt appears to have the same location in MBTMA. Evidence for the location of the waters of hydration is provided by the zero field susceptibility and NMR below TN. The zero field susceptibility data along the c axis has the smallest value at the lowest ex- perimental temperature. We infer this axis behaves much 99 TABLE IV Assignment of proton special positions in MBTMA Chemical Number in Gen- Number in Spe- -Number of unit eral positions cial Positions Distinct (4 operations) (2 operations) Proton Local Field Magnitudes CH3 12 6 3+3 = 6 NH 0 2 0+1 = 1 H20 8 0 2+1 = 2 100 like the parallel axis in a uniaxial antiferromagnet. The spins in MBTMA then are mostly along the c axis. If we assume the waters of hydration which complete the octahe— dra are located above and below the manganese ion along the c direction, we can calculate the local field at the protons due to the manganese moment. A typical configura- tion for a water molecule bonded to a manganese ion is Shown in FIGURE 28. O N 5 Z 3+ + FIGURE 28. Bonding of a water molecule to a manganese ion. 101 The magnetic dipole field at a distance ”r" from the manganese is given by: g: 311(U'IJ)‘IJ (64) A ->- where n is a unit vector which points along r and u is the manganese magnetic moment: {I = 115 :3 (65) If we assume the magnanese moment points along the line joining the manganese and the oxygen, we calculate that the magnetic field at the water proton is 4425 oersteds. The nuclear magnetic resonance frequency for a proton in a field of this magnitude is 18.84 MHz. The apparent excel- lent agreement of the calculated frequency with our experi- mental values of 18.17 and 18.74 MHz is somewhat fortuitous since this calculation used only one manganese ion, while the real field is due to all spins in the lattice. A dipole sum was carried out using our proposed crystallographic lattice with the spins coordinated antiferromagnetically along the chains with the moments in the + and - c direc- tions as shown in FIGURE 29. The sums were calculated by computer over the volume of a sphere which contained 6800 manganese spins. The result gave a field of 4354 oersteds or a nuclear magnetic resonance frequency of 18.54 MHz. The magnetic moment used in this calculation is based on a 102 FIGURE 29. Magnetic dipole arrangement used in computer calculation. spin of S=5/2 and the g-value of 2.12 which was derived from the ESR data. Let us next consider, at least quali— tatively, the details of the interactions which determine the spin state of Mn++ in MBTMA. 103 B. Magnetism of 3d5 S=S/2 Doubly ionized manganese is a transition metal ion with a 3d5 orbital configuration. As previously indi- cated it is likely that the local coordination in MBTMA is octahedral. In octahedral symmetry the five d orbitals are split by the crystalline electric field interaCtions as shown in FIGURE 12. If the energy separation "A" be- tween the t2g and eg states is not large (the "weak ligand fieldélimit), the minimum energy configuration is achieved by singly occupying each orbital. Each of the five d electrons are coupled "spin up" resulting in the maximum spin angular momentum, S=5/2. The fact that all the spins are parallel requires that the orbital angular momentum be i=0; an "S"-state configuration. This zero net angular momentum will cause all matrix elements involving L (in particular H°L and L°S) to vanish in first order. Since as we have shown earlier these matrix elements are respon- sible for the deviation of the g-value from 2, manganese with S=S/2 should have a g equal to 2. In addition, one would not expect any crystalline electric field splitting of the iS/Z, 13/2 and :1/2 Kramer's doublets. There are however, excited configurations of the d orbitals which are produced by doubly occupying one of the orbitals. The resulting S=3/2 system will have a nonzero orbital angular momentum. However, because of the resulting large 104 electron-electron repulsion produced by the double occu- pancy of one of the orbitals, matrix elements calculated in second order between the ground (i=0) state and this excited state are reduced by the large energy difference which appears in the denominator. Experimentally, one normally finds the g value very nearly 2 for S=5/2 manga- nese in octahedral coordination. MBTMA, for example, has a g value of 2.12:.02 which is isotropic within experimental (E.S.R.) resolution. Only one resonance line with a width of approximately 2 kilo-oersteds was Observed. This width is quite usual for manganese which in addition to having an electron spin of S=S/2, has a nuclear Spin of I=5/2. Each of the six (28+l) spin states is split by the hyperfine interaction into Six (21+l) components giving a total of 36 levels! In a con- centrated magnetic system such as MBTMA, spin-spin inter— actions broaden the individual levels so that the resonance line would appear as one broad signal. The fact that only one such broad resonance is observed, indicates the three Kramer's doublets (S= 5/2, 13/2, :1/2) are only slightly split by the crystal field. This again is consistent with ...). a small spin orbit coupling due to the L=0 (S-state) orbi- tal ground state. 105 C. The Ordered State: Metamagnetism As previously discussed, the zero field suscep- tibility data indicates the sublattice magnetization in MBTMA is mostly along the c axis. The large peak in the b axis susceptibility data and the nonzero extrapolated moment above TN, are both consistent with the formation of a net moment produced by canting the Spins. The magnetic phase transition which is observed below TN occurs with H applied along the b axis. This implies the transition which is observed is not a spin flOp, which occurs with H parallel to the sublattice magnetization but is more likely a meta- magnetic transition. If the ordered state consisted of a uniaxial antiferromagnetic system, one would observe a spin flOp only with the field applied along the direction of sublattice magnetization. If the sublattices are canted, the zero field susceptibility data along each of the three axes behaves differently whereas a uniaxial system would be characterized by a parallel and a perpendicular suscepti- bility. The facts that the magnetization in small applied fields vanishes smoothly for decreasing fields and that there were no discontinuities in the NMR rotation patterns in applied field imply that the total spin configuration in the ordered state has no net moment. The observations that MBTMA: (1) has a magnetic transition in applied field which appears to be metamag- netic; (2) exhibits four distinct local field directions 106 for all proton lines; (3) has an antiferromagnetic unit cell, suggest the following model. We postulate that the system behaves as a four sublattice canted antiferromagnet, with the following zero temperature sublattice Hamiltonian: + + + + (S -S + 52-83) + + k 2 o H - -I‘H-2Si - zsig. - J 1 4 1 AF 1 + + ' 1117(51 §3 + S2 @743 (66) 2V . ° 1 0 With k IJAF|>IJAFI and J AF’ JAE both negative and where gi represents an anisotropy axis whose direction is different for each sublattice but are related by symmetry. Here again, the S's represent sublattices. The model is shown with the interactions indicated in FIGURE 30a. The sublattices are mostly along the + and - c directions and are canted in the + and - b directions. The magnetic tran— sition is presumed to be metamagnetic and results in the spin configuration shown in FIGURE 30b. The canting angle 9 may be calculated from _ ' 67 AM — Ng “B S $1n0 ( ) where AM is the field induced change in magnetization at the phase transition. Using the observed value of 1.91 x 103 emu/mole we deduce a canting angle of approxi- mately 4°. We shall qualitatively discuss the behavior of the system in an applied field by using the previously derived sublattice polarization energy diagrams (PED). 6.4 3 FIGURE 30a. Zero temperature sublattice model for MBTMA with H=0. x,H . I 3 FIGURE 30b. Zero temperature sublattice model for . , > MBTMA With H - HCRIT 108 We shall take as our polarization basis states the following: I 2 | 2 4 “>3 H I2)= 2:; 13): 1 3 1 2 1234 l4>= .__."—_’ 15>= :‘2. 16>= II" 3 ‘ 4' -——.- Q4 In zero applied field, these polarization states are mixed by the competition betweentfluaanisotropy and the antiferromagnetic exchange, to form the following states: a|l> + b|2> = >X: (68a) 1 4 3 24 cl3> + d|4> << (68b) 3! '9' II 452 The polarization state 02 is higher in energy in zero field because of the exchange energy JAF' Let uS consider first the behavior of this system with a field applied along the x(b) axis. I». 109 FIGURE 31. PED for manganese MBTMA with H along the b axis. It 110 HSAT HCRIT 111 The applied field couples some of the state |4> into ¢1. This produces a net moment so that the energy of the state decreases because of the Zeeman interaction. The state 02 has a larger moment than 01, so that it des— cends in energy faster than 01 as shown in FIGURE 31. For H < HC the system is polarized mostly along + and - rit’ 2 but is canted in the + and - x directions. As the field is increased, the sublattices rotate towards the field. At H = H sublattices l and 4 reverse their direction crit so that they coincide with 3 and 2 respectively. This re- sults in an abrupt increase in the magnetic moment. The moment then continues to increase until H = HSAT' With the magnetic field applied along the z(c) axis we must consider a new mixed polarization State. (113 = e|4> + f|6> = \/ (69) Again the state ¢1 is lowest in zero applied field. The state ¢3 is Significantly higher in energy than 02 because of the assumption JAE > JAF' The applied field mixes some of the ferromagnetic polarization |6> with 01. This causes the energy of ¢1 to decrease because of the Zeeman inter— action. The state 03 has a large moment and descends rapidly in energy with increasing applied field. The PED for this situation is Shown in FIGURE 32. FIGURE 32. 112 PED for MBTMA with H along the c axis. 113 Hi. HCRIT H SAT 114 For H < H the Spins rotate toward the field. The cri- crit’ tical field with Happ along the z axis is higher than H along the x axis again_because of the assumption crit o 1 AF > JAF ‘ with Happ along z(c), beyond the highest experimental field J This presumably raises the critical field, of 16 k0e. and therefore it was unobserved. At the criti— cal field, sublattices 3 and 4 reverse and become colinear with sublattices l and 2 respectively. This again results in an abrupt increase in magnetization. The moment con- tinues to grow until H = H Finally, a field applied sat' along y, rotates all spins toward the field until H = Hsat' No Sharp transition is observed. Let us examine the implications of this model for the magnetic space group. D. The Ordered State; The Magnetic Space Group X-ray studies and optical goniometry Show the chemical space group is le/m with the two fold screw axis parallel to the crystallographic b-axis. The magnetic space group may be formed by taking some or all of the elements of the crystallographic group and replacing these elements with anti elements. All of the magnetic Space 39 from P21/m are listed in TABLE V. groups which result Each of these groups is derived from the point group Z/m. Each group contains four symmetry Operations. Except for 115 TABLE V Magnetic space groups resulting from le/m Group Ferromagnetic Antiinversion Antitranslation P21/m Yes No No 1 P21/m NO Yes No I le/m No Yes No I ! P21 /m Yes No No ' PZC Zl/m No Yes Yes PZS Zl/m No No Yes 116 the last two groups which contain eight. If we assume the magnetic spins are confined to one plane and canted along the two fold axis (the b-c plane in the crystal) the number of distinct spin directions within the magnetic unit cell produced by each point group is shown in FIGURE 33. We recall there are only two manganese ions in the chemical unit cell. Therefore, they must be located in special positions. The special positions for the crystallographic space group P21/m occur with the atom located at inversion centers or on the mirror plane. In Section IV of the theory, SYMMETRY AND THE MAGNETIC SPACE GROUP, we tabulated the restrictions placed on the orien- tation of a Spin if it occupies a special position in the magpetic unit cell. To reiterate: a magnetic ion cannot be located on an antiinversion center and a Spin which is in a mirror plane must be perpendicular to the mirror plane if it is real or must lie in the plane if it is an anti mirror plane. The mirror planes in MBTMA are perpen- dicular to the b (the two fold) axis. Our proposed model for the spin arrangement has the Spins along + and - c and canted in the + and - b directions. This canting is not allowed if the Spins are located in either a mirror plane or antimirror plane. Therefore, the spins mpgp occupy the only remaining Special position, an inversion center . FIGURE 33. 117 The magnetic point groups formed from 2/m. 119 Our proposed spin model also requires a total of four unique directions in the magnetic unit cell. The point groups 2/m and 24/mf, which both contain inversion centers, result in only two distinct directions. As pre- viously indicated, antitranslation will reverse the spin orientations in alternate crystallographic unit cells. Antitranslation will then produce a total of four unique directions if added to the groups Z/m and 2//m/ resulting in the groups Z/mfI and 27m1I. The groups Z/m/ and Zl/m do not contain inversion centers and do result in four unique directions in the unit cell. The requirement of four unique spin orientations in the unit cell is met by a group which contains BEER inversion and antitranslation 40 or neither operation. This requirement is satisfied by the groups 2/m’ 2’/m Z/mll’ and 2/mf The first three groups contain antiinversion centers. If the manganese ion is located on an inversion center in the crystallographic unit cell, the correspond— ing magnetic unit cell must not contain antiinversion centers. Therefore, if the manganese is on an inversion center, the correct point group is 2/m1/thich results in the magnetic space group PZszl/m' The subscript Zs indi- cates the magnetic unit cell is doubled along some direc— tion perpendicular to b, relative to the crystallographic 120 unit cell. Determination of the direction in which the unit cell doubles requires further experimental evidence. CHAPTER 4 COPPER L-ISOLEUCINB MONOHYDRATE I. Introduction Bis L-Isoleucinato c0pper (ll) monohydrate. C12H24CuN204-H20, the c0pper (II) complex of the amino acid L-Isoleucine, (hereafter referred to as copper L-Isoleucine) is an example of a five-coordinated transi- tion element ion in which the local coordination forms a square pyramid. Although there have been extensive Spec- 41-45 troscopic investigations of similar systems and some magnetic measurements,46-47 there have been few reports of . . . 43 the low temperature behaVior of five-coordinated systems. We shall present and discuss the magnetic susceptibility in near zero and applied fields and electron spin resonance (ESR) spectra of powder and Single crystal samples of Cu2+ L-Isoleucine monohydrate. The crystal structure is orthorhombic with four chemical units in the crystallographic unit cell. The unit 0 O 0 cell dimensions are a=9.451A b=21.67A c=7.629A and the reported space group is P212121.49 The c0pper atom lies approximately in the center of the base of a square pyramid. Two oxygens and two nitrogens form the base of the pyramid 121 122 FIGURE 34. The square pyramidal coordination of copper L-Isoleucine showing the derived principal axes of the g tensor. Nol'3l'l 0"""\..'\"" I. . uz.l\l B \ y. 1 o , ., i .3. 0" II. , ..\\. 017--------0 FIGURE 35. 124 The crystal structure of copper L-Isoleucine (after Weeks 35 g1.). "A—nh-b ' 125 26. '6?” V i 7“- . \ \ ’ 1 -\ ‘,' c f ,— ,9‘s 0‘ ’. \ \1 . -..» -/ "V7 ‘ "b 0 Cu - N .0 -—-—- hydrogen bond 126 'and a water oxygen completes the top of the pyramid (FIGURE 34). The symmetry of this configuration is 4mm(C4v). The molecules are "hydrogen-bonded” along the a and c axes, but are well isolated in the b-direction. (See FIGURE 35.) These structures will be referred to as "sheets" in the ac plane but in fact the c0pper L-Isoleucine molecules are indeed three dimensional; it is only the bonding which is two-dimensional. This two- dimensional Character suggests the possibility of two- dimensional magnetic behavior. 11. Experimental Copper L-Isoleucine monohydrate, C12H24CuNZO4'H20 was prepared by reacting basic copper carbonate (CuCO (CuCOs'Cu(OH)2), with L-Isoleucine, (CH3CH2CH(CH3)CH(NH2) (OOH)),in water. The resulting dark blue solution was filtered and slowly evaporated near room temperature. The complex crystallized in thin deep blue diamond shaped platelets (FIGURE 36). The largest single crystals had a mass of 15 milligrams. The crystals were oriented for the various single crystal experiments by using the external morphology after comparing the morphology with the known lattice parameters using X-ray diffractometry. Since, as will be shown, the magnetic suscepti- bility results indicate the possibility of nonstoichiometric (Hum FIGURE 36. 127 The morphology of copper L-Isoleucine. 129 copper, the copper content was determined by neutron acti- vation analysis of a 35.6 milligram powder sample. The sample was irradiated at the M.S.U. "TRIGA" nuclear reactor. There are two naturally occurring isotopes of copper: 63 u C (70%) and Cu65(30%). The analysis was performed on the 1.348 MeV gamma ray of the Cu64 isotope which results from neutron capture by Cu63. The half-life of Cu64 is 12.75 hours, which was sufficiently long to enable accurate counting without a significant correction for half~life. By comparing the activity of the sample of copper L-Isoleucine with a standard containing a known amount of COpper, the content in the c0pper L-Isoleucine could be established. The results indicated there were 6.8 mg of copper in a 35.6 mg sample of copper L-Isoleucine, or a mass concentration of 19128. The theoretical concentra— tion of copper in the Isoleucine complex is 18.6%, so that to within the accuracy of the analysis, there is no un— combined copper. A. Magnetic Susceptibility - Zero Field The magnetic susceptibility of powder and aligned single crystal samples was measured in a field of less than SOe. from 0.01K to 4.2K. Temperatures from 4.2 to 1.1K were achieved in a conventional He4 cryostatso. The ultra- 3 4 low temperatures were produced in a He -He dilution 130 51 . . Both conventional ac mutual inductance refrigerator. coils and a superconducting quantum interference device (SQUID) magnetometer were used.52 The experimental susceptibilities are rather large (X z 16 emu/mole) and significant corrections for the Lorentz and demagnetizing fields were made. The expres- sion for the corrected susceptibility in terms of the measured susceptibility is derived in Chapter I (THE IN- TERNAL FIELD IN A MAGNETIC SAMPLE) and is given here for convenience. Xm o — X .... where as before: x0 is the susceptibility that would be observed in the absence of any demagnetizing or Lorentz fields; xm is the experimentally measured susceptibility; D is the demagnetizing factor; 0 is the density and M.W. is the molecular weight. Numerical values for the cor- rections which were made to the susceptibilities of both single crystal and powder samples are given in Table VI along with figures indicating the sample shapes used in the various experiments. The numbers were obtained by assuming that the Single crystals approximated oblate spheroids of the appropriate axial ratios. The corrected susceptibility data are shown in FIGURES 37 and 38. The powder data and the single crystal data exhibit similar 131 TABLE VI Demagnetizing corrections of c0pper L-Isoleuc SAMPLE NO. 1 (SQUID) 1 .09mm Lfimszé— 1 . I.5nm m 0 X 0 X = ————.a—__ X = a l+0.016x2 b 1-0.032x§ SAMPLE NO. 2 (COILS) 5mm m o Xa O x = X = a 1+.0165x2 b 1-.0477x§ SAMPLE NO. 3 (POWDER) 575mm -—«:%3-—Khnm for various samples 132 FIGURE 37. Zero field susceptibility of copper L-Isoleucine. Curve "a" is a Curie Weiss law and curve ”b" is a high tem- perature expansion for a two dimensional ferromagnetic Heisenberg exchange. 133 20. 1 l I '3 O C C 1— d -— G C C G .. ‘4“) 4 _ C O l J (D N m C C 4 ‘u 3 . 1 “ill (9 N (mow/mam T(KELVIN) 1 1 FIGURE 38. 134 Inverse susceptibility of copper L—Isoleucine. Solid line is a Curie Weiss law. 135 b X (mOIe/emu) (I) 1.0 2.0 3.0 T(KELV|N) 4,0 5.0 "“‘*‘-"-~—‘-" 'i-L-J- :— ;-'._ .- _.'_; _;_ 4__-:___ _. _--___' ‘__-_e_~_-._,___ _,_ .__ 1-. __.. _ __.___,_ _. _ .» ~_, 136 behavior over the entire experimental temperature range. In all cases, the susceptibility is Curie-Weiss like from 4.2K down to 0.05K. The data then passes through a large (X216 emu/mole) well defined peak at TC = 0.117K. Below this temperature the susceptibility falls, levels off, and then begins to rise again near 0.03k. B. Magnetic Susceptibility - Applied Field The ac susceptibility was measured in applied static fields up to 200 Oe. in the temperature range from 0.01 to 0.3K. The experiment was performed using conven- tional mutual inductance coils inside a superconducting solenoid.53 At low fields, the critical temperature is depressed for all orientations of the magnetic field but somewhat more with H applied parallel to a or c than with H along the b axis. At low temperatures (below .lK) a peak (FIGURE 39) in the susceptibility is observed with H parallel to the a and c axes at Hcrit = 140 De. FIGURE 40 is a composite graph of the applied field susceptibility data for all three axes. The diagram reveals the presence of at least three distinct magnetic phases with a triple point at HTP 2 150 0e.and TTP 2 0.050K. The internal field with H along b is Significantly affected by the demagnetizing field. Our previous methods 137 FIGURE 39. Magnetic susceptibility of copper L—Isoleucine with H parallel to the c axis. X(orbi’rrory units) 138 fT=68.8mK . O T=50.lrnK O ' O T=4l.8mK . \ o O O 1 I FIGURE 40. 139 Composite H-T phase diagram for copper L-Isoleucine. 140 H(Oe) I50 — I00— 50 50 TImK) I00 I50 141 of correcting for the demagnetizing and Lorentz fields are no longer applicable because of the large fields and ano- malous behavior of the susceptibility along the a and c axes, which indicates M is no longer a linear function of H. However, we may crudely estimate the highest internal field which would result from a 200 Oe.field along the b axis. Let us assume that M = xH (71) The internal field is given by: 4n _ o p Hint Happ + (3—' D)X Hint MTWT H = app (72) 4 o 1'(31 ' D)X MIWT Hint Using the data in TABLE V1 for sample number 2: HPaX= 20° = 113 De. Int l-(.0477)l6 C. Electron Spin Resonance Electron Spin resonance experiments were per— formed on aligned single crystals at temperatures near 1°k. The data was taken using a rectangular cavity immersed in liquid He4 Rotation diagrams were obtained at x-band 142 (9.2 GHz) about all thru crystallographic axes. In addi- tion, the a axis data were taken at k-band ( 22 GHz) to improve the resolution of the lines (FIGURE 41). Since these experiments were performed in the paramagnetic state at relatively low fields (the argument for the Brillouin function is 0.2), we may use our previously derived expres- sions to correct the internal field for demagnetizing effects: :. fl _ O p Hint Happ + (3 D) X HintMTWT HaPP Hint = -4 ' p 1'('3_ ' D)XoM——.W. 8° = ~133- = h" (1-(51 - D)x ---p H H, 3 OM.W. uB int “B a _ _ 11'. - p - gmeas (1 (3 D)x0 111—1117 (73) Plank's constant where h microwave frequency C II The corrected rotation diagrams are shown in FIGURES 42-44. Each diagram is symmetric about the crys— tallographic axes. This is consistent with the point group 222 (three orthogonal two-fold axes) which results from the reported space group P212121. 143 FIGURE 41. ESR signals with H in the bc plane of copper L-Isoleucine. Curve a is for H parallel to c. Curve b is with H 50° from c. FIGURE 42. 145 ESR rotation about the a axis in copper L-Isoleucine 140 Om... ON: 81 O OO.N . O_.N CNN MW’ FIGURE 43. 147 ESR rotation about the b axis. 148 cm on. on on o. c... on- 8. 2.- cm. _ _ H _ fi — _ _ _ _ o o T cod I . l o_.m Effie/o 10.10 IOIO I I 8:6. _ e e _ _ _ L _ _ _ o FIGURE 44. ESR 149 rotation about the c axis. 150 cm 9 on on o. 0... 8. on. 2... cm. _ _ _ . q __ _ _ _ _ Q C I oo.m I o_.m 0:; I CNN _ _ _ _ _ _ _ _ _ _ o 151 The ESR signal for the b and c axis rotations consists of a single line of essentially constant width which exhibits only a slight g-factor anisotropy. The a axis Signal consists of two lines (FIGURE 41) with a large g—factor anisotropy and an anisotropic line width. The rotation data for each axis was analyzed54 using the expression g2 = a + 8 cos 20 + Y sin 20 (74) which gives the g value at the relative rotation angle ¢. The coefficients in this expression are given by: 2 2 2 2 2 2 8 + _ 8 ‘8- 8 '8- , a = + g B = + cos 20+ y ='—:7—— sin 20+ (75) 2 2 where the maximum and minimum g values (g+ and g_) occur at the relative angles 0+ and 0_ respectively. These para- . 3 meters are then used to determine a matrix W whose compo- nents are given55 by: Wll = OIa+Ba W22 OLb+8n 33 7 4c W22 = O‘a‘Ba w33 = O‘b'Bb W11 ‘ O‘c’Bc W12 = Ya W23 = Yo W13 = Yc W12 = W W23 ' w32 w13 = W31 21 152 where, for example, a is the parameter from the fit to a the a axis rotation data. The eigenvalues of this matrix are the squares of the three principal g-values. The eigenvectors indicate the orientation of the principal axes relative to a set of fixed axis, which in this case are the crystallographic axes. Two rotationally inequiva- lent g tensors have been derived from the b and c axis data and each of the two rotation patterns of the a axis. The principal g values and direction cosines for each of these tensors are given in TABLE VII. The orientation of the derived principal axes relative to the square pyramidal local coordination of the copper atom is shown in FIGURE 60. The two g tensors have their maximum g values in the direction of the "tip" of the pyramid. The pyramids and the g tensors are rotated i51° from the crystallographic b axis in the ac plane. The g values along the crystallographic axes are derived from the principal axis g values and are given: ga = 2.147i.005 gb = 2.130:.005 gC = 2.130:.005 Finally, if one assumes the anisotropy in the line width is unresolved hyperfine structure due to the c0pper (I=3/2) nucleus, one can estimate a hyperfine inter- 1 action of approximately 0.02 cm.' , which is quite reason- able for copper. 153 TABLE VII Principal values and direction cosines of the principal axes of the two g tensors Site 1 Site 2 g1 2.213 f 0.005 2.213 t 0.005 g2 2.042 f 0.005 2.042 1 0.005 g3 2.142 1 0.005 2.142 1 0.005 x1 (0.00, 0.78, -0.63) (0.00, 0.78, 0.63) x2 (0.00, 0.78, 0.63) (0.00, 0.78, -0.63) (1.00, 0.00, 0.00) (1.00, 0 00, 0.00) 154 III. Discussion A. Magnetism of 3d9 Electrons in a 4mm Crystal Field In c0pper L-Isoleucine, the Cu2+ ion has an elec- tronic orbital configuration of 3d9 which is equivalent to having a Single vacancy or "hole" in one of the d orbitals. This orbital configuration always results in an S=1/2 spin state. The spatial extent of each of the five d orbitals is Shown in FIGURE 8 in Chapter I, CRYSTAL FIELD THEORY. In the isolated ion, the energies of these orbitals are all degenerate. To see how the orbital dege- neracies are liftedjjlthe crystalline electric field of copper L-Isoleucine, we shall first examine the behavior of the d orbitals in the symmetry of the local coordina- tion of the ligands. As previously indicated, the copper atom is coor- dinated with five ligands. The spatial arrangement of the two nitrogen and three oxygen ligands approximates a square pyramid (see FIGURE 34). The point group symmetry of this configuration is 4mm (C4V),which consists of a four fold axis and four mirror planes. The sterographic projection of the point group diagram as well as the Spatial location of these operations in the square pyramid are shown in FIGURE 45. The symmetry of the irreducible representations56 in the point group 4mm are shown in TABLE VIII. FIGURE 45. 155 Stereographic projection of the point group 4mm (upper) and the location of the four mirror planes and the four-fold rotation axis in the square pyramid (lower). 157 TABLE VIII Irreducible representations of the point group 4mm 4mm (C4V) A1 2, X2+Y2, 22 A2 Iz B1 xz-Y2 B2 XY E x, Y, xz, YZ, I I X Y Under the symmetry of this group, dXZ and dyz both belong to the same irreducible representation and as such, cannot have their degeneracy lifted by an interac— tion having the symmetry of 4mm. However, the other three orbitals d 2 X '7 action. To determine the nature of the splitting, we must 2, dzz and dxy may be split by such an inter- consider the details of the interactions of the nine copper electrons with the surrounding atoms in the copper L-Isoleucine crystal. We can simplify our analysis of the copper system in the manifold of the nine d electronic wave functions by first comparing the relative crystalline electric field energies of each of the orbitals. The lowest energy con- figuration occurs with the four lowest orbitals doubly 158 occupied and a "hole" in the highest orbital. Instead of performing the analysis based on the nine electrons, one can invert the order of the energy levels of the orbitals and carry out the analysiSIEdng a single positively charged "hole" in the ground state orbital. We shall therefore begin by qualitatively determining the relative energies of the five d-orbitals in the presence of the 4mm crystal- line electric field of copper L-Isoleucine. The 2 axis is placed along the tip of the pyra— mid and the x and § axes pass through the corners of the base of the pyramid: The separation of each ligand atom from the central copper atom is also indicated. By examining FIGURE 8, which Shows the spatial orientation of the d-orbitals, we can see that the dx2_yz orbital will point towards each of the four ligands in the base of the pyramid. Taking the ligand atoms as point charges this state should have the highest energy due to the repulsion of the copper electrons by the 159 electronic charge on the ligands. The next lowest energy level is the dz2 orbital which points toward the one water oxygen located at the tip of the pyramid. Next is pro- bably dxy which is in the base of the pyramid between the ligands. Finally, the dxz orbitals which fall between ligands and extend above and below the base of the pyramid are related by the four fold rotation axis which passes through the tip and center of the base. These last two levels are degenerate and probably lowest in energy. The resulting energy level diagram and the inverted picture used in the "hole" analysis are: in dxz: Y2 .= dyz I? 1.2 ' 38: 1 1 dXY xy I T dzz dxz )1 (ggyz III—44:73, Y Having deduced the relative position of the energy levels, we may Show how this affects the magnetic moment. .In par- ‘ ticular, we shall show in which direction relative to the square pyramidal coordination, the spectroscopic Splitting factor (g-factor) deviates most from 2. 160 The Zeeman energy of the magnetic moment of an electron including both the spin and orbital contributions is given by H = fi-H = '“Bfio (f + 28) (76) where “B is the Bohr magneton and H0 is the applied field One can measure the energy difference between levels with an electron spin resonance experiment. One writes the Hamiltonian using an effective "g" factor: I II -guBH'S (77a) -guBHOSZ e.g. (77b) The energy difference between adjacent levels is: AB = -guBHO(mi-mj) (78) However, magnetic resonance experiments involve magnetic dipole transitions which obey the selection rule Am = i1 Setting the energy difference equal to the energy of the photon: hv = AB = guBHo (79) This effective g factor differs from the "Lande" g factor which is given by = h gLande ‘ 2 fi— '”8<(f+2§) 2(L+§)> §y_ J “B 1 2 2 2 -1JB<2 h 11B 8 = £ (2+1) + 3 '( +1) + 1 S(S+l 80 Lande 2 7 J J 7 ) ( ) Comparing equations (12) and (13a), with equation (7) we see that the "effective" g factor as well as the Lande gr factor will be different from 2 only when I is nonzero. In the magnetic resonance experiment, this orbital effect manifests itself as an additional contribution to the Zeeman energy: E = (81) The maximum deviation from g=2 will occur when this term 57 is largest. The d-orbitals can be written as linear combinations of the spherical harmonics Y?: 1 ‘ 2 + -2 d z = Y = ...._ Y Z 2 dx2_y2 2 (Y2 2 ) l 2 -2 d =.__ (Y -Y ) = _l l -l 2 2 d (Y +Y ) XY [2' X2 /2_ 2 z 162 The matrix elements of the components of the Operator L are given by: m n _ ‘ H 5mm m n _ l m n — 7 - l _ PI— 1 + 7 i/(IL‘I'II) (2-n+l) 6111,11'1 m n _ 1 m _ n — 2I< YZIL+ L_| Y2> _ 1 _ _ — E /(Q n) (Q, n+1) Om’n+1 1 ‘ 2r “(I*n)(£'n*1) 6m,n-1 Returning to our analysis of 3d9 copper with the single "hole" in the "ground state" dxz-yz, we see the zero order matrix element of H’L is: 2 2> L 2 +0 1 -2 '2 A A 0 ) 163 Indeed there are no nonzero diagonal elements for any of the other d-orbitals. In second order, the energy is given by: 2 Il 5(2) = i XE;Y BE (83) x2_y2 k The first excited orbital is dzz. However, matrix elements of the Hamiltonian between dx2_y2 and dzz are zero. There is a matrix element of the 2 component of the Hamiltonian between dx2_y2 and the next highest orbital dxy’ which is nonzero: :3 1 _ (dxz-YZIHszldxy> -2 1 2 2 2_ = _ 2 2 2HZ which gives a contribution to the energy: 2 2H E(2) _ I 2' (84) E° - E° X2_y2 xy The matrix elements of the x and y components in the Hamiltonian are zero between these states, therefore the largest deviation from g=2 occurs with the external field parallel to the z axis. There are matrix elements for the rump, 164 x and y components between other states, but those terms involve higher energy states and therefore are reduced by the larger energy denominator. Experimentally, the largest g value (2.21) for copper L-Isoleucine was indeed observed with the external magnetic field along the z axis of the pyramid. B. Magnetic Susceptibility The copper L-Isoleucine molecules are linked in two dimensions by hydrogen bonds which extend in the a and c directions (see FIGURE 35). The lack of much covalent character of hydrogen bonds should produce only weak super exchange between copper Spins. Pauling has indicated58 that in certain instances the hydrogen actually resonates between equilibrium positions on the opposite ends of the "bondJ' It may then provide the exchange mechanism for the interactions between the copper Spins. The experimental susceptibility well above the critical temperature fits a Curie Weiss law: X—_—.—— C (85) T-0 with a positive (ferromagnetic) Weiss constant of 0w = 0.240k, and a Curie constant of C = 0.466. Below approximately 1°K, the susceptibility deviates from Curie-Weiss behavior. This is perhaps best shown by 165 FIGURE 46 which is a plot of C/xT versus J/kT. In this type of plot, a Curie Weiss law is a straight line which intercepts the x axis at 0=T(curve a). Curve b is obtained using the first 10 terms of a high temperature expansiOn (see Appendix A, THE METHOD OF HIGH TEMPERATURE EXPANSION) by Baker and Rushbrooke59 for a two dimensional square net lattice of spins that interact through an isotrOpic Heisenberg exchange: 11:21): 8.8. 170' 1 3 where J is the exchange constant between a copper atom and its four nearest neighbors on the same two dimensional "sheet." The parameters which give the best fit to the experimental data are g = 2.18 and J/k = +0.120K (ferro- magnetic). This exchange compares favorably with the Curie-Weiss constant which is ZJ/k (.244K) for the same number of nearest neighbors. The system is at best only approximated by a square net. A two dimensional exchange is however, consistent with the nature of the crystal lattice. The experimental susceptibility data was also compared with a two dimensional x-y model60 and a two dimen- sional Ising61 model. The agreement between the experi- mental data and these last two theoretical models was not as good as with the two dimensional Heisenberg model. FIGURE 46. 166 Reduced inverse susceptibility plotted as a function of reduced inverse temperature. Curve a is a Curie-Weiss law; curve b is a high temperature expansion for a two—dimen- sional ferromagnetic Heisenberg exchange. 168 There are three aSpects of the data near and be- low the critical temperature which are interesting and not entirely understood: (1) the peak in the susceptibility measured along all three axes; (2) the anisotropic beha- vior in applied field below TC and finally, (3) the rising susceptibility at the lowest experimental temperature. The peak in the susceptibility at T=TC and the subsequent decrease in susceptibility for slightly lower temperatures is thought to indicate a transition to a three dimensionally ordered antiferromagnetic state. This three dimensional ordering is caused by an antiferromagnetic coupling between sublattices on adjacent "sheets." This coupling may be dipolar or a "real" superexchange. Further evidence for the existence of a three dimensionally ordered state below Tc’ is supplied by the susceptibility data in applied field. The anomalous peak in the susceptibility as a func- tion of applied field indicates a discontinuous change in magnetization. This implies the existence of a sublattice magnetization with the associated longrange ordering. The exact nature of the ordered state and the magnetic transition are not at present understood. The fact that the zero field susceptibility is approximately isotropic below the critical temperature contraindicates a uniaxial antiferromagnet as does the Observation of a mag- netic phase transition with the applied field along two different orthogonal axes. As we have shown in the previous 169 theoretical discussion of the uniaxial antiferromagnet, a sharp transition is observed only with the field applied along the sublattice magnetization. The observed transi- tions indicate then, that the system consists of more than two sublattices. However, more experimental evidence such as N.M.R. or neutron diffraction in the ordered state is needed to illuminate the nature of the Spin system in the ordered state. Finally, the third aspect of the magnetic beha— vior, the rising susceptibility below 0.03K is still not understood. This effect was at first thought to be due to impurities or nonstoichiometric copper. A calculation of the impurity concentration necessary to produce the observed susceptibility can be made by assuming the rising susceptibility is due to a paramagnetic impurity described by: X=% The Slope of the experimental values yields a Curie con- stant C = 0.6. The theoretical expression for the Curie constant is: Ngzu 2 C = B S(S+1) (86) 3k where N = Avogadro's number spectroscopic splitting factor 00 II 170 Bohr magneton LIB: k = Boltzmann's constant 8 = spin quantum number The theoretical Curie constant for copper (assuming g=2.18) is 0.44. For an S=1 system such as nickel, C = 1.17 and for an S=5/2 system such as manganese, C=5.13. Comparison of these calculated constants with the experimental Curie constant derived fromthe ultralow tem- perature data indicates the sample is composed of: (l) 100% S=l/2 impurity; (2) 50% S=l impurity or (3) 12% S=5/2 impurity. Neutron activation analysis eliminates the possibility of uncombined copper and the nominal quality of the reagent grade materials rules out other transition metal impurities in such high concentration. The source of this behavior remains unclear and requires further experimental investigation. CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS The two antiferromagnetic insulators discussed in this thesis have been shown to exhibit interesting mag- netic behavior at low temperatures. The crystallographic space group of MBTMA was determined using X—ray and opti- cal goniometry to be P21/m. A plausible model for the crystal structure has been derived from the Space group which is consistent with (1) the structure of an associated compound CCTMA, (2) the number of molecules in the unit cell (3) the lattice dimensions (4) the number of zero field proton magnetic resonance line (5) the agreement between the calculated and experimental values for the NMR frequencies of the water proton and (6) the low dimensional behavior above TN. The local coordination of the manga— nese (11) ion is apparently octahedral with four bromines and two waters of hydration. Magnetization and nuclear magnetic measurements made in the ordered state indicate that the magnetic state is characterized by a four sublat- tice canted antiferromagnet with a canting angle of approxi— mately 4°. A field indoced magnetic phase transition is observed at HC = 1200 oersteds applied along the b 171 M ‘* 172 axis. This transition is asserted to be metamagnetic and results in a net moment. The crystal structure of c0pper L-Isoleucine is characterized by "sheets" of hydrogen bonded copper L-Isoleucine molecules. The local coordination of the c0pper ion is five-fold and approximately square pyramidal. The two dimensional crystal structure is reflected in the magnetic properties above TN. The zero field susceptibi— lity is isotropic and is described in the temperature range from 4.2 to 0.5k as a two dimensional square net magnetic lattice which interacts through a ferromagnetic Heisenberg exchange. A transition to a three dimensional antiferro— magnetic state occurs at TC = 0.117K. A field induced mag- netic phase transition is observed below TN with the field applied along the a and c axes. The susceptibility at ultralow temperatures exhibits an anomalous increase with decreasing temperature. The spin state of the 3d9 c0pper ion in a crystal field of 4mm symmetry has been experi- mentally investigated using ESR in the paramagnetic state. The largest g value was found to occur with H applied along the "tip" of the square pyramidal local coordination. This is consistent with a qualitative theoretical analysis based onthe Van Vleck point-charge model for crystal field interactions. 173 Further work clearly includes determination of ‘the exact crystal structure of MBTMA. Once this is done the spin arrangement can be determined by "orienting" the spins so that the calculated magnetic dipole fields at the proton Sites agree with the results of the zero field re- sonance. The nature of the transition in applied field may be further specified by observing the water proton lines-above the transition, and again "orienting" the spins so as to produce calculated fields at the proton Sites which agree with the Observed resonances. Neutron scat- tering will probably not yield good results in this mate- rial due to the inelastic scattering from the large number of protons in the unit cell. The specific heat of copper L-Isoleucine should be measured at ultra low temperatures to determine how much entropy is involved in the transition at 0.117K. If the result differs substantially from the expected S=R2n2, further experiments should be done at ultralow temperatures. The NMR at temperatures below TC can be used to derive the orientation of the copper spins in the ordered state. At still lower temperatures, the NMR may be able to indicate whether the anomalous increase in susceptibility is due to a rearrangement of the magnetic sublattices. If the nuclear magnetic resonance of the water protons can be followed 174 through the field induced transition, the nature of the transition and the resulting Spin configuration could be determined. Finally, if a high spin transition metal ion com- plex of L-Isoleucine can be grown, the effect of the 4mm crystal field symmetry on the exchange and the Single ion anisotropy may be investigated for different orbital configurations. 10. 11. 12. 13. 14. 15. 16. REFERENCES E. Ising, Z. Phys., 81, 253 (1925) D. C. Mattis, The Theory of Magnetism, Harper and Row (1965), p. 256. L. Onsager, Phys. Rev., 88 117 (1944). L. Onsager, Nuovo Cimento Suppl., 8, 261 (1949). C. N. Yang, Phys. Rev., 88, 809 (1952). R. B. Griffiths, J. Math Phys., 8, 478 (1967). N. D. Mermin and H. Wagner, Phys. Rev. Lett., 17, 1133 (1966). H. E. Stanley and T. A. Kaplan, Phys. Rev. Lett., 11, 913 (1966); J. Appl. Phys., 88, 975 (1967). Mermin and Wagner, Ibid. R. D. Spence, Private communication (to be published). C. W. 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The quantum— mechanical-thermal average involves evaluating the trace of the Hamiltonian over all possible eigenstates of this 1023 many-bodied system. At sufficiently high temperatures such that kT is large compared to the interaction energies, we may approximate a solution to the thermodynamic quanti- ties associated with the Hamiltonian through the use of a "high temperature expansion." In this analysis, we expand the partition func- tion "2" as follows: 2 E tre'8H B2H2 3 3 tr(1-BH+ 2! 'T+...) 2 ter — gT trH3 + m nS - BtrH + i N where n5 is the number of thermodynamically accessible states. Let us examine this expansion for the Heisenberg Hamiltonian: + + H = -22 J.. S.°S. rem». 180 We shall restrict the exchange to nearest-neighbor interactions. Jij‘ H + —> 2J2 S. S. . . 1 J 1] Then the partition function becomes: _ 2J + + 2J ;_ + .+ 2 Z ‘ ns ’ ET tr¥. 51"Sj + ET 2! tr(¥.si Sj) 13 1) 2J 3 1 + 3 - — 3Ttr(2 §i-s.) 4+ kT ° ij 3 In order to evaluate the sums inside the traces, we must specify the arrangement of the spins. As an example, we shall take a two dimensional square net lattice. Each spin will have four nearest neighbors. The first sum: 2 §.o§. ij 1 J is the sum of nearest neighbor pairs. Since there are N spins in the crystal, each having four nearest neighbors, there are 4N such terms. This means that the linear term in the partition function is ++ - 31 (4Ntr(S.-S.) kT 1 J The quadratic term in the partition function produces three kinds of interactions: 181 . . . + + 2 1) two ion interactlons (Si-Sj) _. ++++ 2) three ion interactions (Si°Sj)(Sj°Sk) + + + + 3) four ion interactlons (Si-Sj)(Sk 58) We may examine these various cases through the use of inter- action diagrams. In the first case we find: 0 o 0 ,0, E F o::‘.‘:o o o ,0, o o . o:__,‘o o ‘o’ o : : o ‘o’ o o —> + 2 Thus there are 4N terms of the form (Si°Sj) . In the second case: 9 o o 1 : z '. o----<5 o o 9 o o—---—o---o : 182 ++++ we find lZN terms of the form (Si-Sj)(SjoSk). Since there are 4N nearest neighbor pairs in the crystal, the quadratic term in the expansion must contain (4N)z or a total of 16N2 terms in the sum: + -> 2 (2 51.3.) ij 3 This means the four ion interactions must produce: 2 16N - 4N - 12N = 16N2 - 16N = l6N(N-l) terms of the form: + + -> -> Thus the quadratic term in the expansion for the partition function is: Z + -> —> -+ + + 1 2J 2 . . ‘2‘? (a) (4N tTCSi-Sj) "' 12N tr(Si Sj)(si 8k) + (16N(N-1)) tr(Si°Sj)(Sk°S£)) We have, at least for these first few terms, reduced the problem to evaluating the traces of pair interactions. However, the counting problem and the associated interac- tion diagrams become very complex for higher order terms. The counting is usually done by simulating the lattice by a mathematical algorithm on a computer. APPENDIX B THE TWO SUBLATTICE CANTED ANTIFERROMAGNET We assume a Hamiltonian of the following forms: H = -D-Sle2 - J 51-82 - k(Slz+SZZ) l 1 + +1 6 = a l> + b 4> = a . + b = <: 1 I l +2 +2 2 _ 2 2 2 E - -D(ab+ba) + J(a -b ) -k(2a ) = -2Dab + J(az-bz) - Zkaz recalling the normalization: — 2Db(l-b2)1/2 + J(l-sz) - 2k(l-b2) m I — -2D(l-b2)1/z - Db(1-b2)-1/Z(-2b) - 4Jb+4kb ,1 -2D(1-b2)1/2 + 2962(1-62)‘1/2 - 4b(J-k) -zn(bz(1-bz)‘1/2 - (l-b2)1/2) -2b(J-k) 2 _ 1+b2 1/2 -2b(J-k) b D (l-bz) 183 184 k-J (l-sz) ‘ 2b(1-b2)1/2 26(1-62)1/2(5fil) = (l-sz) Square both sides: 2 4bzcl-bz) (18132 = (l-2b2)2 _ 2 4(b2-b4) (£51) = 1 + 4134 2 - 2 _ 2 4 1 + (5E!) b4 - 1 + (551) 462 + 1 = 0 let y = b2 _ 2 _ 2 4 1 + (55;) 2 -4 1 + (Efiij y + 1 = 0 k-J 2 k-J 2’ k-J 2 y _ 1 + (—D—) Haw—T) ) (M D) ) 2 1+(ka 2 b = i J?— Now in the limit D goes to zero, the system must reduce to the uniaxial case which means b must be zero. Bim k-J = m D+0 D k-J 2 k-J 4 k-J 2 21m y z (T) i “(T ’ (‘15—) D+0 “35532 185 22 (1 H- 1) NIH Therefore only the - root is correct! Then: b _ + (1 ”3131323 - / (1 + (£51362 - (1 + ($312) 1” 2(1 + 1134).?) a = (1 - b2)1/2 The two roots for b correspond to canting along + or -x directions. P'""' 11‘ “11]”ij[12111157111ijES