A STUDY OF THE ANTIFERROMAGNEHC STRUCTURES 5N CUCI ' 2H Q: C4351 ”SH 9; emcf NECL‘éH O 2 2 2 2 4 2 Thesis for the Degree of DH. D. MICHIGAN STATE UNIVERSITY Robert Kleinberg 1963 THE!!! 0-169 IIIIIIIIIIIIIIII'IIIIIIIIIII IIIIIIIIIIIII 31293 01764 0388 LIBRARY Michigan Stave University ' This is to certify that the thesis entitled A STUDY OF THE mmmmmmmmc STRUCTURES 011012“ 20, 00012 ~6HN0 d,bema 111012061120 presen Robert Kleinberg has been accepted towards fulfillment of the requirements for W‘ 6/ degree in 7)%[ 976! QlL/p L.)%W Major (firofessor 7 Date fly ’26) /;6 )’ 0 . . —ur—---\o v‘ - .H mm. .—._y-— _«-. ABSTRACT A STUDY OF THE ANTIFERROMAGNETIC STRUCTURES IN 20, and NiCl '2H 0 CuCl 2 2 ~2H 0, CoCl'ZH 2 2 by Robert Kleinberg An attempt is made to determine the magnetic ordering in cobalt chloride hexahydrate and nickel chloride hexahydrate in the antiferromagnetic state. By applying the Schubnikov group symmetries to the cobalt and nickel chloride X-ray and proton—resonance data, it is found that the possible space groups which describe the magnetic ordering in both salts are P C El and CC %" Magnetic fields are then calculated in all a the possible structures and compared to the experimental fields in an attempt to determine the correct structures. In the first attempt, the fields at the proton positions are calcu- lated after first assuming that the magnetization density about each ion is equivalent to a point dipole placed on the ion site, and that only a magnetic interaction is present. The calculated fields in all the structures are found to be in large disagreement with the experimental fields in both the cobalt and nickel salts. A similar calculation for c0pper chloride dihydrate shows that the structure proposed by Rundle gives fields at the proton positions which are in good agree- ‘I Robert Kleinberg ment with the experimental fields, but the structure proposed by Poulis does not. The results of calculating the Fourier coefficients of the magnetic fields in the assumed cobalt and nickel chlo- ride structures are also not useful in differentiating between the PC and CC structures. Finally, a calculation is made in which the magnetization density is approximated by a distribu- tion of point dipoles. The moments of the dipoles are deter- mined by fitting the fields calculated at the proton positions to the experimental fields. The resulting distributions are all quite similar to one another, but it is concluded that the magnetic ordering in the cobalt chloride is described by the Space group PC 3; and the ordering in nickel chloride is given a by CC % The large disagreement between the simple dipole model fields and the experimental fields is taken to imply that the magnetization density deviates strongly from Spher- ical symmetry and that there may also be'a substantial ex- change interaction between the electron and proton spins. A STUDY OF THE ANTIFERROMAGNETIC STRUCTURES IN Cuc12-2H20, CoC12-6H20, and N1C12-6H20 By Robert Kleinberg A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1963 some? 7/3/54’7’ ACKNOWLEDGMENT The author wishes to express his gratitude to Professor R. D. Spence for suggesting this problem and for his guidance and many helpful suggestions throughout the course of this work. ii "II TABLE OF CONTENTS Page I. INTRODUCTION . . . . . . . . . . . . . . . . . . 1 II. EXPERIMENTAL RESULTS . . . . . . . . . . . . . . 11 X- -ray Analysis . . . . . . . . . . . . . 11 Magnetic Susceptibility . . . . . . . . . . . 12 Local Magnetic Fields . . . . . . . . . . . . 13 Proton Positions . . . . . . . . . . . . . 19 Dipole- Dipole Splitting . . . . . . . . . . . 24 III. THE POSSIBLE ANTIFERROMAGNETIC SYMMETRY GROUPS OF COBALT AND NICKEL CHLORIDE . . . . . . . . 28 IV. CALCULATED LOCAL FIELDS . . . . . . . . . . . . 36 Introduction . . . . . . . . . . . . . . . . 36 Poisson' 5 Analysis . . . 37 Spherically Symmetric Magnetization Density . 4O Unsymmetrical Magnetization . . . . . . . . . 44 Programs . . . . . . . . . . . . . . . . . . . 47 V. COPPER CHLORIDE . . . . . . . . . . . . . . . . 51 Experimental Results . . . . . . . . . 51 Calculation of Local Field Vectors . . . . . . 52 Discussion and Conclusion . . . . . . . . . . 57 VI. SIMPLE DIPOLE ANALYSIS OF COBALT AND NICKEL CHLORIDE . . . . . . . . . . . . . . . . . . . 59 Introduction . . . . . 59 Mirror Plane Maps of the Magnetic Field . . . 59 Local Fields at the Proton Positions . . . . . 66 VII. SECOND ORDER CALCULATIONS . . . . . . . . . . . 85 Introduction . . . . . . . . . . . . . . . . . 85 Fourier Analysis . . . . . . . . . . . . . 86 Dipole Array Calculation . . . . . . . . . . . 93 VIII. DISCUSSION AND CONCLUSIONS . . . . . . . . . . . 110 REFERENCES . . . . . . . . . . . . . . . . . . . . . . 113 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . 115 iii Table N GUI-bot) 10. 11. 12. 13. 14. 15. 16. 17. LIST OF TABLES Atomic Parameters for Cobalt and Nickel Chloride Measured Local Magnetic Field Vectors at Cobalt and Nickel Chloride Proton Positions . Cartesian Components of Local Field Vectors Covariant Components of Local Field Vectors Proton Parameters for Cobalt Chloride . . . Dipole- Dipole Splittings in Cobalt and Nickel Chloride . . . . . . . . . . . . . Symmetry Relations Between Field Vectors in 21/a and CC 2/c Atomic Parameters for Copper Chloride . . Calculated and Experimental Local Field Vectors in Copper Chloride Calculated Local Magnetic Field Vectors at Proton Positions in Cobalt and Nickel Chloride . . . Cartesian Components of Calculated and Measured Local Field Vectors at Proton Positions in Cobalt and Nickel Chloride Comparison of Experimental Field Magnitudes with Calculated Field Magnitudes in Cobalt Chloride Comparison of Experimental Field.Magnitudes with Calculated Field Magnitudes in Nickel Chloride Associations of Calculated to Experimental Local Field Vectors . . . . Ratios of Experimental and Calculated Fields Fourier Coefficients of Magnetic Field in Cobalt Chloride . . . . . . . . . . . . . . . . . . . Fourier Coefficients of Magnetic Field in Nickel Chloride iv Page 12 16 17 17 21 27 34 52 56 72 74 78 79 83 84 92 93 Figure 1. 10. 11. 12. LIST OF FIGURES Page Rotation Antiferromagnetic Resonance Diagram with Rotation about the a' Axis . . . . . . . 15 Stereographic Projection of the Local Field Vectors in Cobalt and Nickel Chloride . . . . 18 Proton Positions and Proton-Proton Vectors in Cobalt Chloride . . . . . . . . . . . . . . . 22 Diagrams of Allowed Schubnikov Space Groups PC 21/a and cc 2/c . . . . . . . . . . . . . 32 Possible Magnetic Structures for Cobalt and Nickel Chloride . . . . . . . . . . . . . . . 33 Stereographic Projections of Calculated Local Field Vectors in PC 21/a and CC 2/a . . . . . 35 Coordinate System and Vectors used to Calculate §i_(§_)....................39 Possible Magnetic Structures for Copper Chloride . . . . . . . . . . . . . . . . . . 53. Stereographic Projections of Poulis, Rundle, and Experimental Local Fields in Copper Chloride at Proton Positions and on Surface of ;1X Sphere . . . . . . . . . . . . . . . . 54 Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of P-Cobalt Chloride . . . . . . . . . . . . . . . . . . 61 Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of C- Cobalt Chloride . . . . . . . . . . . . . . . . 62 Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of P-Nickel Chloride . . . . . . . . . . . . . . . . . 63 Figure Page 13. Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of C- Nickel Chloride . . . . . . . . . . . . . . . . 64 14. Stereographic Projection of the Local Field Vectors in P-Cobalt Chloride at Proton Positions and on Surface of .12 Sphere . . . 68 15. Stereographic Projection of the Local Field Vectors in C-Cobalt Chloride at Proton Positions and on Surface of .13 Sphere . . . 69 16. Stereographic Projection of the Local Field Vectors in P-Nickel Chloride at Proton Positions and on Surface of 01R Sphere . . . 70 17. Stereographic Projection of the Local Field Vectors in C-Nickel Chloride at Proton Positions and on Surface of 12 Sphere . . . 71 18. Labeling of Dipole Positions Along Chlorine and Oxygen Bonds . . . . . . . . . . . . . . 102 19. Dipole Fractions Along Chlorine Bond in P—Cobalt Chloride . . . . . . . . . . . . . . 104 20. Dipole Fractions Along Chlorine Bond in C-Cobalt Chloride . . . . . . . . . . . . . . 105 21. Dipole Fractions Along Chlorine Bond in P-Nickel Chloride . . . . . . . . . . . . . . 106 22. Dipole Fractions Along Chlorine Bond in C-Nickel Chloride . . . . . . . . . . . . . . 107 vi Appendix I-a I-b II III-a III-b IV LIST OF APPENDICES Stereographic Projections of Axial Vector under Monoclinic Heesch Point Groups Classification of Monoclinic Heesch Point Groups According to Stereographic Projection of Axial Vector Equivalence Between Spherically Symmetric Magnetization and Point Dipole . . . Computer Program for Calculating Magnetic Field in Cobalt Chloride Computer Program for Calculating Magnetic Field in Nickel Chloride Computer Program for Calculating Dipole Fractions vii Page 115 115 116 117 120 123 m; o‘!‘ F? R» O I}. or, I. INTRODUCTION The Hamiltonian of a hydrated, antiferromagnetic crystal at low temperature contains three terms which are of interest in this thesis. 9I=II+QI+9<., e p in where 398 is the Hamiltonian of the atomic spin system,g€ the Hamiltonian of the proton system, andE}( in represents the interaction between the tWo systems. In general, ’X e>>9€ p’ and the influence of the interaction 9K in on the motions of the atomic Spins is negligibly small. There- fore the internal magnetic fields in an antiferromagnetic crystal are almost completely due to the atomic spin system, so that only the 98 term is required to calculate the e magnetic susceptibility of the crystal. The Hamiltonian of the proton system is =- H ‘ I - I -T 1 9f}, gnjlr'o ;_J. ggn/unzé ,‘I. , ( > where [1n is the nuclear magneton efi/2mc, gn, the nuclear Splitting factor and approximately 5-58,.15’ the Spin vector th of the j proton, and TI is the operator corresponding go th the magnetic field produced at the j proton by all the other protons. The expression for_2i is TI:2:;§%—2[Ii"'_(11°r1)ri]’ (2) 1.ij rij where rij is the distance from proton i to proton j. Generally, in hydrated crystals only the protons on the same water molecule are capable of giving non-negligible contributions to (2), Since other protons are much further away. The interaction term in the proton system -g’-‘/"n 1.1.21 ’ A} consists of time independent and time dependent parts. The time independent part gives rise to the dipole-dipole split- ting of the Zeeman levels, while the time dependent part gives rise to proton relaxation. The interaction Hamiltonian. .in’ for a proton is 96 1n='gn/“ £s°li+li°Zinj§Jw (3) and the expression for._’I_‘S is S=Z-r_ #10 . 7(8 l°ri)_ri]., (4) where 51 is the vector from the ith atomic spin to the pro- ton. AS in the proton system,_'I_‘S is the operator correSpond- iing'to the magnetic field produced at the proton, but by the atomic Spins. g and P0 are the atomic Splitting factor and 30hr magneton. The secular part of (4) is just the time averaged field at the proton position due to the atomic sys— tem. The time depéhdent part ofl‘s induces transitions be- tween the energy levels of the proton system, by scattering of spin waves. The second term in the interaction Hamiltonian is the hyperfine interaction and has been used by Shulman and Jaccarino1 to describe the field in MnF2., and by Van der Lugt and Poulis2 for Vivianite. This term arises from an overlap of the atomic spin orbits with the protons. The time averaged part of the exchange term between a proton and its nearest ionic neighbor is Akj(lk°<§j>). O. In the nmr experiment a r.f. field is used to induce transitions between levels in the proton system. The fre- quency of the applied field at resonance is related to the internal field at the proton positions by hv = anHt ’ where H1: is the net field at the proton positions, and is the sum of the applied field and the field due to the spin system. Therefore, Since 9? in is of negligible influence on the Spin system, a nmr study of the proton resonances yields information about the Spin system as well as the pro- ton system. At ordinary temperatures the proton resonances give information about the susceptibility of the Spin system, at low temperatures they indicate quite precisely the transi- tion from the paramagnetic to antiferromagnetic states, and finally, below the transition they act as probes to measure the time averaged internal fields. The time averaged Hamil- tonian of the proton system is 9? @131 Eti“—L— [31> ' “(€13 519512] ’(5) 21:32 r312 where Etiigofggi’ and + Aik s > A 6 "li =ZI k TP<- -k ( ) The second term in (5) is quite small so that the local in- ternal field at the proton is given by (6). The effect of the proton-proton interaction is to Split the resonance line into several components. If the exchange contribution to the local field is present in the paramagnetic state too, it is detected by a shift of the resonance frequencies (at constant applied field) from the free proton resonance fre- quency. Shulman and Jaccarinol have observed Shifts in the nmr lines of Man in the paramagnetic state. The shifts are about thirty times greater than those expected from the dipolar interaction and are explained in terms of the ex- change interaction. Bleaney3 has been able to estimate the shift in the fluorine resonance from the Spin resonance hyperfine structure data of Tinkham.4 Other transition group fluorides have also been shown to exhibit hyperfine shifts. On the other hand, many paramagnetic salts do not give hyperfine shifts, even just above the Neel temperature. In c0pper chloride the paramagnetic state nmr lines show no evidence of an exchange shift. Van Kranendonk5 and Moriya6 have calculated relaxation time for the protons in copper chloride using only the dipolar interaction, and their results are in order of magnitude agreement with the observed relaxation time. Sawatzky and Bloom7 report that the protons in cobalt chloride hexahydrate have only a weak dipolar inter- action with the magnetic cobalt ions. In the antiferromagnetic state, the local fields at the proton positions are measured by means of the applied and zero field techniques. In these methods, both the mag- nitude and direction of the field vectors are measured. Riedel,8 and Riedel and Spence,9 show how the symmetry of the local fields are used to enumerate the various possible arrangements of the magnetic moments in the antiferromag- netic state. They argue that the point group symmetry shown by the measured local magnetic fields must transform all axial vectors the same way. Therefore the point symmetry of the dipole moments is the same as for the local fields, since the dipole moments are axial vectors. Thus from the complete set of Space groups which may describe the magnetic symmetry of the crystal, those groups whose Heesch point groups do not correspond to a possible point group of the measured local fields, must be deleted. The space groups that describe the allowed axial vector symmetries in a crystal are known as Schubnikov groups. The complete set of 1651 groups was first derived by Zamorzaev,10 and then by Belov et al.11 This set con- tains the ordinary crystallographic space groups. Belov's derivation is quite straightforward and consists of two basic steps. First, the set of fourteen uncolored Bravais lattices which describe all ordinary three dimensional translational symmetry groups, is expanded to include twenty- two new black-white lattices. These are derived by coloring each translational element in turn in each particular Bravais lattice, considering all possible combinations of colored and uncolored translational elements, and eliminating those re- sulting combinations of elements which lead to identities or do not lead to groups. The second step in the derivation consists of adding all combinations of colored and uncolored symmetry elements to each of the thirty-six lattices, and sifting out those sets which are not groups and those groups which are identities, leaving the 1651 Schubnikov groups. Ten theorems given by Belov and used in his derivation have been translated and are given by Riedel in his thesis. In {1) {3. (n (‘7‘ T1 ’(7 It is clear that the Schubnikov groups which describe a particular crystal must satisfy not only the magnetic point symmetry but also the results of all physical experiments on the crystal. This includes x-ray and susceptibility data as well as the nmr data. With the aid of this experimental data and the use of certain principles, Riedel is able to limit the ordering in Azurite to four possible antiferromag- netic structures. Some of the principles used by Riedel are given here in section 111, where the same process is used to limit the possible orderings in cobalt and nickel chloride. The first extensive analysis of an antiferromagnetic crystal was done by Poulis and Hardeman,12 on CuClZ-ZHZO. They measured the local fields in both the paramagnetic and antiferromagnetic states. On the basis of their experimental data, and using proton positions which have since been shown to be wrong, they postulated what is here called the Poulis structure for antiferromagnetic copper chloride. Later, Peterson and Levy13 made a neutron diffraction study of this salt and determined proton positions which agree with the positions reported by Itoh et a1}:1 who used the nmr technique. In 1957 Rundle15 suggested another antiferromagnetic struc- ture for copper chloride. His structure was made on the basis of Poulis's data and the Peterson and Levy proton positions. Subsequently, Hardeman,16 using the new proton positions, again analyzed the problem and concluded that the Poulis structure was more likely the correct one. Both Rundle's and Hardeman's analysis are based on the simple dipole model, but neither one of them gives the field which is calculated at the proton position due to dipoles on the c0pper ions. Recently, a canted arrangement has been pro- posed by Moriya,17 but as yet there is no definite determina- tion of the antiferromagnetic ordering in copper chloride, and the reason is not too hard to understand. In copper chloride there are eight protons in the unit cell, but they are all related by symmetry operations. Thus, one has only the three components of the local field to work with. But the magnetization density which gives rise to the measured fields is a continuous, unknown func- tion of position in the unit cell, and the attempts to use only the three field components to determine an approximate magnetization density and to determine the ordering has quite understandably met with not too much success. Since the time that Poulis performed his experiments on copper chloride, several other antiferromagnetic crystals have been studied by means of nmr. Some of these are, Azurite,18 LiCuC13o2HZO,19 MnF2% and CuP2~2H20.20 The analysis of the copper fluoride is of consider- able interest, since a neutron diffraction study of this salt has also been made at 4.20K by Abrahams.21 He reports that the Schubnikov group which describes the ordering is PI 21/n, and that the best choice of the three spin direc- tions he assumes, has the spins along the crystal c axis. ‘.;J CC This result is supported by the antiferromagnetic experiments of Peter and Moriya22 who report that the spins are displaced 3.50 from the crystal c axis. If one could determine the translational symmetry of copper chloride from a neutron diffraction study, as Abrahams has done for the fluoride, then of course one could easily distinguish between the Poulis and Rundle structures. At the present time no such study has been made, and in fact, copper fluoride is the only crystal for which a neutron diffraction study has been made at 4.20K. Thus, for the present at least, the nmr technique is the main one for determining the magnetic structure of antiferromagnetic crystals at liquid Helium temperatures. Recently, Spence and Middents‘z3 have completed a nmr analysis of antiferromagnetic cobalt and nickel chloride hexahydrate. Unlike the copper chloride, both these salts have four unrelated protons in the unit cell. Therefore a nmr analysis yields twelve different field components to be used in a determination of the magnetic ordering. It was thought that with this information one could determine the magnetic ordering in these salts. - This thesis is a collection of the results of cal- culations made in an attempt to determine the magnetic order- ings in the cobalt and nickel salts. Calculations have also been made on the c0pper chloride, and they are also reported here. 10 In section II the experimental data on cobalt and nickel chloride are given, and in the following section the experimental data is applied to the seventy-five Schubnikov groups which describe the monoclinic antiferromagnetic sym- metries. In section IV the fields at the proton positions are determined in the magnetic structures proposed in the preceding section. The programs made from the derived equa- tions are also discussed. At this point the analysis of the cobalt and nickel salts is interrupted to present the results of the simple dipole calculation for the fields in the Poulis and Rundle structures for copper chloride. These results are given in section V. The simple dipole analysis of cobalt and nickel chloride is then given in section VI, followed by the results of a Fourier and dipole array analysis given in sec- tion VII. Finally, in the last section, results are reviewed and final conclusions are stated. II. EXPERIMENTAL RESULTS X-ray Analysis The crystal structure of cobalt and nickel chloride has been determined by Mizuno.24 Both crystals belong to the monoclinic prismatic class, having formula weight of two, and Space group classification C‘%. The unit cell dimensions in angstroms for the cobalt and nickel salts are: Cobalt Nickel a 10.34 10.23 b 7.06 7.05 c 6.67 6.57 P 122°20' 122°10' Since there are two metal ions in the unit cell, and since the general position is eightfold in the space group C‘%, the metal ions must lie on‘% inversion centers. The metal ions are located on the corners of the unit cell and the center of the a-b faces. They are octahedrally sur- rounded by four water molecules and two chlorine atoms. The water molecules form a distorted square with the cobalt or nickel ion at the center, while the chlorine atoms are sit- uated on positions along the two normals to the oxygen plane. The remaining two water molecules of the formula unit are situated in the mirror plane and are relatively free, but do 11 t .' 12 take part in the hydrogen bonding scheme. The chlorine atoms are located in the mirror plane, and the oxygen "square" is inclined to the c axis by about 10°. The oxygen atoms in the "square" are referred to as oxygen one atoms, and the free oxygens in the mirror plane are referred to as oxygen two atoms. Atomic parameters for atoms in the unit cell are given in Table 1. Table 1. Atomic Parameters for Cobalt and Nickel Chlorides Cobalt Nickel X Y z x y 2 M .000 .000 .000 .000 ' .000 .000 C1 .274 .000 .171 .271 .000 .167 OI .0312 .208 .251 .0312 .208 .251 OII .288 .000 .702 .288 .000 .702 Magnetic Susceptibility Magnetic susceptibility measurements have been per- 25'27 He has used the a.c. bridge method formed by Haseda. to measure the molar susceptibility of Single crystals of cobalt and nickel chloride. His results show that an anti- ferromagnetic transition occurs for the cobalt salt at a temperature of about 30K, and that the ordering of spins is along the monoclinic c axis, or the cll axis. For nickel chloride the transition temperature is at about 6.20K, with the axis meas betw and 13 the ordering of Spins along the axis perpendicular to the c axis in the a-c plane, or the ci_axis. Local Magnetic Fields The magnetic fields at proton positions in the unit cell have been measured at 1.1OK, using the applied and zero field methods. The applied field method consists of rotating the crystal about two axes in an applied magnetic field and measuring, at constant applied radio frequency, the difference between the resonance fields in the antiferromagnetic crystal and the resonance field at that frequency for the free proton. The result of plotting line separation as a function of the rotation angle gives curves similar to the one shown in Figure 1. From these curves the magnitude and direction of the local fields are determined. Since the local fields in cobalt and nickel chloride are quite strong the zero field method has also been used to confirm the applied field meas- urements. In this method the crystal sample is placed Slightly off center of a modulation coil which is free to rotate about two orthogonal axes of rotation. When the separation of the resonance lines from two protons which have opposite local fields is a minimum, then the modulation field is parallel to the internal magnetic field at the proton positions. The magnitude of the field is then determined by setting the oscillator frequency such that the two signals coalesce at all orientations of the modulation field and ,~... whe p05 giv mea 14 using the relation hv =gPH£ , where H2 is the magnitude of the local field at the proton position. More detailed descriptions of these methods are given by Kim26 and Middents,27 and only the results of the measurements are given here. The local field vectors determined by the applied and zero field methods are given in Table 2. The polar angle 9 is measured from the positive c axis and the azi- muthal angle 49 is measured from the positive a' axis in the a'-c plane. The stereographic projections of the vec- tors in both salts are given in Figure 2. Fields from protons in general positions are labeled with integers l to 4 and 9 to 12. It is clear that the lines within each of these sets are related by mirror plane and two-fold symmetry, and come from protons which are also related by symmetry con- ditions. The mirror plane proton fields are labeled with the integers 5 to 8. The three field components of each set of lines in the orthogonal crystal coordinates and in the co- varient crystal coordinates are given in Tables 3 and 4. The zero field lines in nickel chloride for the protons in general positions have a dipole-dipole Splitting of 46 kc. Dipolar splitting in the cobalt salt is observed in the applied field but not in the zero field measurements. It is therefore presumed that the zero field Splitting iS less that 20 kc. mfix< .m on“ “Dona :ofipmuom npfiz Emummfio mocmcommm oflpocwmeouuowfiwc< coepmpom .H ouswfim 1|“ I . ~.o . u j opa can and ow. and .n o- can ca . on on oe on on o. .r a l _ ... n m n. n . .P .. l l .p o 0., . D CD N o' 1 f Q C O Table Cobal U1 bu (\J H OWN 10 ll 12 Nicke mbwm xooowcr 10 Ill 12 16 Table 2. Measured Local Magnetic Field Vectors at Cobalt and Nickel Chloride Proton Positions —— 3:; Cobalt H (gauss) 9 cp 1 1785 74° 128° 2 1785 74° -128° 3 1785 106° -52° 4 1785 106° 52° 5 1592 77° 0° 6 1592 103° 180° 7 1179 58° 180° 8 1179 122° 0° 9 1397 84° 97° 10 1397 84° —97° 11 1397 96° -83° 12 1397 96° 83° Nickel H (gauss) 9 CP 1 1729 55° 166° 2 1729 55° -166° 3 1729 125° -14° 4 1729 125° 14° 5 1.397 39° 180° 6 1.397 141° 0° 7 1.285 39° 180° 8 1.285 141° 0° 9 1.029 75° 49° 10 1.029 75° —49° 11 1.029 105° -131° 12 1 029 1050 131° Ta Cobal. 17 Table 3. Cartesian Components of Local Field Vectors Hx Hz Hx Hz 1 ~1057 1352 492 -1374 992 5 1551 0000 358 -879 1086 7 1000 0000 625 -809 999 9 -169 1379 146 652 266 Table 4. Covariant Components of Local Field Vectors E E Cobalt h‘ h‘ h' h' x y z z 1 —6 505 9 545 3 282 6 563 5 11 590 O 000 2 388 4 776 7 5 281 0 000 4 169 8 338 9 - 670 9 736 974 l 948 . E 9 Nickel h'x h'y h'z h'z l - 650 2 418 6 517 13 035 5 -1 698 O 000 7 135 14 270 7 -1 565 O 000 6 563 13 127 9 4 198 5 288 l 748 3 495 mUMHoHQU meofiz Una “Hmnoo ca mucuoo> oaofim Hmooq any Mo maofiwoonoum ofinmwuwoouopm .N ousmfim of th ofMi the u its b formu where tion, with the m to de Pr0t0 all f make This min.) fOn.p 19 Proton Positions El Saffar28 has made a room temperature nmr analysis of the cobalt salt, and in conjunction with the x-ray data of Mizuno has been able to determine the proton positions in the unit cell. By rotating a crystal of CoC12-6H20 about its b-axiS in an external magnetic field, and using the formula _ 2 AH=3)1r 3(3 cos O-l) , where (SH is the line pair separation in a known orienta; tion, 9 is the angle that the proton-proton vector makes with the direction of the applied magnetic field, and p is the magnetic moment of the proton, El Saffar has been able to determine the direction and magnitude of the proton- proton vectors in the unit cell. The proton-proton vectors in the mirror planes are all found to be equivalent. They lie in the a-c plane, and make an angle of 260 from the c axis towards the a axis. This set of vectors is called M. Since there are eight mirror plane protons in the unit cell there are four pro- ton-proton vectors in this Set. The eight proton-proton vectors of the general pro- ton positions are inclined towards the mirror planes, so that this set consists of two subsets of four vectors. The vectors within a subset are all identical and are related to the vectors of the other subset by a reflection operation. devis ton p proto ton v vecto are g inves Posit have 1 Spence in the tors 5 to be OnanCe depex. that Ill'Cke 20 The set of eight vectors is called G, and the subsets of four vectors are called G' and G". The subset of proton-proton vectors designated by G' is parallel to 9=74O andCP =-l65°. The subset designated by G" is found to be parallel to G= 74° and 90 =165°. In order to find the proton positions, E1 Saffar devised several hydrogen bonding schemes, and from the pro- ton positions in each of the schemes calculated the proton- proton vectors. The proton positions which give proton-pro- ton vectors which best fit the experimentally determined vectors and the requirements of the water molecule geometry are given by El Saffar. The proton positions used in this investigation are given in Table 5. They differ from the positions originally determined by El Saffar in that they have been corrected to better fit additional data taken by Spence. In Figure 3, the positions of the protons are drawn in the unit cell. The stereogram of the proton-proton vec- tors is also given in Figure 3, and labeled with notation to be described below. It is found that the room temperature magnetic res- onance lines of nickel chloride have the same orientation dependence as the cobalt chloride. It is therefore assumed that the proton-proton vectors and proton positions in nickel chloride are the same as in cobalt chloride. H I 3 4 1 5 € 6 l 7 l 2 8 l C ‘- x 9 10 1 11 I 2 12 l 2 \ posit iIth DESI-e andn 21 Table 5. Proton Parameters for Cobalt Chloride 8 g' 1 x, v, z .097 .324 .274 .073 .286 .199 2 x, l-y, z .097 .676 .274 .073 .714 .199 3 l-x, y, l-z .903 .324 .726 .073 .286 .801 4 l-x, 1-y, l-z .903 .676 .726 .073 .714 .801 1 1 . 5 2+x,*§+y, z .597 .824 .274 .427 .786 .199 6 éix,‘%-y, z .597 .176 .274 .427 .214 .199 7 %_x, %+y, 1-2 .403 .824 .726 .573 .786 .801 8 .é-x,.%-y, 1-2 .403 .176 .726 .573 .214 .801 m m' 9 x, 0, z .191 .000 .547 .274 .000 .838 10 1-x 0, 1-2 .809 .000 .453 .726 .000 .162 11 %¢x,.% , z .691 .500 .547 .774 .500 .838 12 1%- ,-% , l-z .309 .500 .453 .226 .500 .162 The notation which is used to describe the proton positions, proton-proton vectors, and associations of exper- imental magnetic fields to proton positions is now described. There are four Sets of protons in the unit cell of cobalt and nickel chloride. The protons in each set are related by symmetry to the other protons in that set, but are not ‘EL‘Ie 3 . 22 /_-\ [1,1'] [2,2' 0 0 [10,10'] " [12,12'] [99] {[11,11'] [3,3']° : °[4,4'] 10 o 11'0 10? C51 Proton-Proton Vectors 9' C) C) 12 0 12: 9 o 1 y = O y = .— 2 r I) plane plane 3 6 3' c J v 4 / 1 v" / 1: General Proton ‘ Positions 4. “f2 C 2'0/ L Figure 3. Proton Positions and Proton-Proton Vectors in Cobalt Chloride relate sets a the pr to the symmet are la 1‘ t0 positi 9' to (.191, to a 5 this p is ref ring t the of ezence refer: Which withOL 23 related by symmetry to the protons in any other set. The sets are labeled g, g', m, m'. The sets g and g' belong to the proton-proton vectors in G, and the sets m and m' belong to the vectors in M. The positions in g, are generated by symmetry operations on the position (.097, .324, .274), and are labeled from 1 to 8. Positions in g' are labeled from 1' to 8', and are generated from (-.073, .286, .199). The positions in the sets m and m' are labeled from 9 to 12, and 9’ to 12' respectively and are generated from the positions (.191, .000, .547) and (.274, .000, .838). When referring to a Specific proton position the integer which represents this position is written in a bracket, e.g., the position 2' is referred to as g'=[2‘], or simply as [2']. When refer- ring to two or more protons, the integers are written in the order g, g', m, m'. This notation, to bracket all ref- erences to proton positions is used to avoid confusion when referring to the experimentally measured magnetic fields, which are also labeled with the integers l to 12, with or without parenthesis. The representation of the association of a Set of experimental magnetic fields to proton positions is done as follows. Because of the symmetry conditions which relate the fields at the positions in a set of protons, it is necessary to give the field at only one of the proton positions in a set. Thus to define the fields at all the proton positions in the unit cell one need give only the fields at the proton positions [1, 6', 9, 9']. The fields 24 associated with these positions are written in parenthesis in the same order as the positions. For example, [1, 6', 9, 9'] = (9, 2, 5, 7) has the experimental field 9 associ- ated with the proton position [1], the experimental field 2 with the position [6'] and the fields 5 and 7 with the respective positions [9] and [9']. Since all the calcula- tions reported here are done on the [1, 6‘, 9, 9'] posi- tions, it is understood that when only the fields are given in parenthesis, these positions are to be associated with the fields. For example, for (9, 2, 5, 7), (5, 7) and (9,'2), the proton positions are understood to be [1, 6', 9, 9'], [9, 9‘] and [1, 6'] respectively. Dipole-Dipole Splitting It has been mentioned previously that the effect of the dipole-dipole interaction between protons on the same water molecule is to Split the proton resonance lines into a maximum of four components. If, as in the case of c0pper chloride, the protons on a water molecule are related by a two-fold axis of symmetry, the fields have the same magni- tude but different directions. The Splittings for this case have been worked out by Van Kranendonk and Poll.31 For cobalt and nickel chloride the fields are different in both magnitude and direction. This is also true for the applied field experiment because the local fields are large com- pared to the applied fields. The energy levels of the 25 second term in the Hamiltonian given in (5) has been derived by Spence for the more general case. He finds that the frequencies of the four resonance lines are vl=P+2A+s, V2=f-26-s, 793=P+2A -s, v4=p-2a +S. In these equations gee. . P '2'1? T 1 2 [Pl - A +((T) E3) {4(1-COS 912)-3(COS Gl-COS 62)}, where r12 is the interproton distance, 81 and 62 are the angles between 5 and the fields 5 and_H2 at the protons 12 1 l and 2, and 912 is the angle between the fields. When the magnitudes of the field vectors are equal, and only their directions are different these expressions describe the Spectrum in copper chloride. If‘Egfi (Hl'Hz) is large com- pared to IPI , one obtains a single pair of lines whose separation is dipole is obs N0 spl It is detect chlori used t ways t with p Ciatic ured s associ the f‘ Proton than a are st the g the g 26 2 A9 = LEED). (cos 91 cos 82 - cos 912). (8) 3 hr12 In the zero field experiments on nickel chloride a dipole Splitting into two lines with a separation of 46 kc. is observed for the fields which are not in the mirror plane. No Splitting is observed in the cobalt chloride field lines. It is estimated that a splitting of at least 20 kc. can be detected on the scope, so that the Splittings in cobalt chloride must be less than this amount. Equation (8) is used to calculate dipole-dipole Splittings for all possible ways that pairs of experimental fields may be associated' with proton-proton vectors in the unit cell. All the asso— ciations giving results that are in conflict with the meas- ured splittings are not possible. It is clear that in each association, one of the experimental lines must come from the fields 1-4, and the other from 9-12. Also, Since the protons in the group g' are closer to the nearest metal ion than are the protons in group g, and Since the fields 1-4 are stronger than 9-12 in both the cobalt and nickel salts, the g protons must be associated with the fields 9-12, and the g' protons with the lines 1-4. Consider the proton-proton vector [1, 1']. There are sixteen ways of associating fields with this vector. But since the fields in each group of lines are related by inversion symmetry, the sixteen ways are easily reduced to four sets of four associatiOns, where the magnitude of the 27 Splitting is the same for each member of a set. Since cos O=-cos(e+180°), the associations which transform into each other under a substitution of one or both fields by their opposite fields must all have the same dipole-dipole Splitting. Splittings have been calculated for each set of associations for both salts, and the results of these cal- culations are given in Table 6. By comparing these results with the experimental splittings, it is seen that, for each salt, there are eight possible ways of associating line pairs with the vector [1, 1'], since the calculated split- tings of 5 and 16 are possible for cobalt, and 45 and 43 for nickel. Table 6. Dipole-Dipole Splittings in Cobalt and Nickel Chloride Associations Splitting Magnitude Cobalt Nickel observed 20< 46 (9,1), (11,3), (11,1), (9,3) 30 45 (10,2), (12,4), (12,2), (10,4) 5 21 (10,1), (12,3), (12,1), (10,3) 36 10 (9,2), (11,4), (11,2), (9,4) 16 43 tal whic nick proc repr appl cm: 3m III. THE POSSIBLE ANTIFERROMAGNETIC SYMMETRY GROUPS OF COBALT AND NICKEL CHLORIDE In this section, the process by which the experimen- tal data is used to eliminate all the Schubnikov Space groups which cannot describe the magnetic structure in cobalt and nickel chloride is described. The result of this sifting procedure is to leave only two groups which are possible representations of the magnetic symmetry. The x—ray data shows that there are two metal ions in the unit cell, and they each occupy positions of«% sym- metry. The susceptibility measurements indicate that the moments on the ion Sites are in the a-c crystal plane. The stereograms of the local fields measured in the nmr exper- iments show that the magnetic field point group symmetry is- described by each of the Heesch point groups 21', ml', 2‘ , 2', iv. And finally, the experiments of E1 Saffar prove m . m4 that two protons are in the crystal mirror plane, and two are in general positions. If the principles stated by Riedel and Spence are applied to the experimental data on cobalt and nickel chloride then six possible groups are easily determined. The principles are: 1. The Schubnikov Space group must have as its 28 poir the this are each oper whic imen the mono and ster afp Ere 29 point group one of the possible point groups predicted by the experimental nmr data. The point groups which satisfy this condition have been given above. These point groups are obtained by drawing the stereograms which result when each of the 11 monoclinic Heesch point groups is allowed to operate on an axial vector, and then collecting the groups which give stereograms that are equivalent to the exper- imentally determined stereograms of the axial vectors in the cobalt and nickel salts. The symmetry diagrams of the monoclinic Heesch point groups are given in Appendix Isa, and the classification of these groups according to the stereographic patterns is given in Appendix I-b. 2. A magnetic moment may not lie on an anticenter, in a mirror plane, or across a two fold Symmetry axis. This condition quickly eliminates the point groups and 2! m. at 3. Schubnikov symmetry elements must coincide with 3 m. corresponding uncolored elements of the Federov group C This condition insures that the colored group elements do not introduce atoms to new positions in the unit cell. The magnetic Space group may be of the same or lower symmetry as the crystal Space group, but never higher. If the principles 2 and 3 are applied to the 52 monoclinic Schubnikov groups remaining on Belov's list, after the use of principles one and the discarding of the grey Space groups, then there remain Six groups which are acce and alon disc is e pres. eigh' givil fielc antit fore1 alEn sets the 1 but 1 the J 5133. D051 [0!- of L 30 acceptable descriptions of the magnetic ordering in cobalt 2 c-f' PC a, P 2 ch, Ccc, and nickel chloride. They are P C 1, CC2. The difference between the PC set of groups and the CC set is that in the former set the centering translation is an antitranslation, and in the latter the translation along the c axis is the antitranslation. In each set, the groups of lower symmetry may be discarded for the following reason. The general position is eightfold, and if the 2' and m' symmetry elements are present in the colored space group, then the fields at all eight proton positions are related by Symmetry Operations, giving one set of four lines on the stereogram of the fields. On the other hand, the positions related by an antitranslation and either 2' or m' are fourfold. There- fore, the eight proton fields associated with eight equiv- alent protons in general positions are divided into two sets of fields. The fields within each set are related by the translational, and either 2' or m' symmetry elements, but the fields in one set are not related by symmetry to the fields in the other set. Thus in the case of the lower symmetry groups, the fields at the eight general proton positions should give two sets of stereographic patterns for a total of eight different fields. The stereogram of the measured local fields consists of only one set of lines from each set of equivalent general pro met has won gro are is ti0 hat and yie the in- axi. the DOS. at Sch al of th: to: Dr it. 31 proton positions, and it is concluded that the point sym- metry of the magnetization density about each metal ion has the full symmetry %;. Otherwise the two sets of lines would not coincide. Thus the only acceptable Schubnikov groups are PC'%% and CC %. The diagrams of these groups are given in Figure 4. The notation of Riedel and Spence is used to indicate antisymmetry operations. In this nota- tion the antioperations are represented by diagonal cross hatching. It is noted that in CC % the planes m' and c, and also a and n are superimposed. The two allowed groups yield two possible ordering schemes, and they are labeled the P and C structures. The diagrams of the two orderings in the cobalt and nickel chlorides are given in Figure 5. Since the possible Space group symmetries of the ‘ axial vectors are now known, they can be used to determine the symmetry relations between the fields at the proton positions in the unit cell. If it is assumed that the field at g=[l] is in the positive quadrant, then by applying the Schubnikov symmetry elements to this field, the fields at all other g positions are generated. Since the magnitudes of the fields are the same, it is necessary to give only the signs of the field components. The results of these considerations are given in Table 7, and the stereographic projections of the fields at the proton positions are given in Figure 6. In this section, it has been shown that on the basis of e poss of C the used expe is r 32 of experimental data and symmetry arguments, there are two possible ways of ordering the electron Spins in the crystals of cobalt and nickel chloride. In coming to this conclusion, the magnitudes of the local magnetic fields have not been used. It does not seem unreasonable, at this point, to expect that a calculation of the local fields is all that is required to distinguish between the two structures. P3; CCE Ca C .3 L- g .111- ++1 5+ 1 ’7 1 *1 Er [WW/2+ 4.— / i / [t / If}; V I I I. ' 5 «4f— 4 : -' ' : 1: fl 1; fiL“ : I + : '5‘ [I . I : l ' ‘—" / T K T K / fi :/ u . I. - - I 3 l JLL- !l JE:+_ +90 +1? *1? ° i' at‘l 4 Figure 4. Diagrams of Allowed Schubnikov Space Groups PC 31 and Cc 2'. a c 33 //// / i/ ,g/ / c Co C12°6H20 / 1 /, / / / / 4/ / / / / ‘— Ni C12 6H20 Figure 5. Possible Magnetic Structures for Cobalt and Nickel Chloride 34 Table 7. Symmetry Relations Between Field Vectors in P 21 C 7f' and Cc % Proton Hx Hy Hz Hx Hy Hz 1 + + + + + + 2 + — + + _ + 3 + _ + _ + - 4 + + + _ _ _ 5 - — - + + + 6 - + - + - + 7 - + - - + - 8 - _ - - - _ 9 + 0 + + O + 10 + 0 + - o - ll - O — + 0 + 12 - 0 - _ o - [11: Figu [5][8]_ O [11][12] 35 [6][7] A ‘ [9][10] [2][3] Figure 6. [1][4] [4][8] [2][6] [9][11] [inst Stereographic Projections of Calculated Local Field Vectors in PC %% 111 Ila-'81 mag: reSI two of 1 one cry: this cula deve the C0111 IV. CALCULATED LOCAL FIELDS Introduction In the previous section the angular relations be— tween the local fields are used to determine the possible magnetic orderings in the cobalt and nickel salts. The results are that there are two possible orderings in the two salts. It seems that a calculation of the magnitudes of the local fields at the proton positions would enable one to distinguish between the two structures for each crystal. Such a calculation is described in the rest of this thesis. In this section the equations used to cal- culate the fields in the both structures of the salts are developed. Since there is no indication of exchange between the electron and proton Spins, it is assumed that the main contribution to the local field arises from the dipole- dipole interaction between the metal ions and the protons. It is also assumed that the field at the proton positions is due to the Space and time averaged moment of the unshared metal ion electrons, so that the problem is reduced to de- termining the fields at a point outside of a static distri- bution of magnetization. This problem is considered here and expressions are derived for the magnetic field at the 36 pr01 and fiel mag: plac cry: and sati ing symh a di and fIOm prOg the [1ij no ma: 37 proton positions in the P and C structures of the cobalt and nickel salts. In the first attempt to derive equations for the fields at the proton positions, it is assumed that the magnetization density in the crystal is approximated by placing a point dipole on each ion position. For each crystal the ordering schemes given in Figure 5, are used and the fields calculated from these Simple dipole models satisfy the symmetry conditions discussed in the preced- ing section. In the second attempt, a non-Spherically Symmetric magnetization density is approximated by placing a distribution of point dipoles about each ion position and writing the resulting field as a sum of contributions from each member of a set of dipole arrays. Finally the programs made from the derived equations are discussed. Poisson's Analysis The problem of determining the magnetic field H in terms of the magnetization M” is now considered. The dia- gram in Figure 7 is used to describe the derivation. It is assumed that the exchange interaction between the pro- ton and electron Spins is negligible, so that the field at the proton positions is due to Space and time averaged moment of the unshared electrons. Further, Since there is no electric current density at the proton positions, one may assume that H_is equal to the negative gradient of a mag Z€I whe and 15 1 tiot the 38 magnetic scaler potential: L1 :2 "‘:7 4)) (9) The divergence of the magnetic flux density §,is zero in all Space: VLB : O) (10) “ where §.is assumed to be equal to a linear function of H and Mn §=5+WM. (11> From the above three equations, Poisson's equation v14) =11TT V'M) (12) is easily obtained by substituting equation (9) into equa- tion (11), and then (11) into (10). Poisson's equation has the well known solution32 I 49 = SMCI') - V(-§)6Lt, (13> where R is the distance between the volume elementzxt at .E" and the observation point defined by the vector 5, The magnetic field vector is then obtained by taking the nega- tive gradient of the potential, according to equation (9). The integral which results for H is (14) \-_:1 = SMC:‘1~Q—__— oL'C. wh whe 311C 39 where B I G .2 VV(JR-) RS“ TR:— (15) When M(£f) is Spherically symmetric, the simple If... dipole case results, and the potential is given by '1 q; .. 74.4 v“), (16) where the dipole moment 2 is defined by and the magnetic field is given by [j =/g§._. <17) P(X1,X2,X3) I” 161 In x1 Figure 7. Coordinate System and Vectors used to Calculate H(r). ‘(D '3. m0< de1 in the dip git In In to Chl an; 4O SphericalIy Symmetric Magnetization In Appendix 2, it is Shown that the magnetic field outside of a spherically symmetric magnetization density is equivalent to the field due to a point dipole, so that the model considered here is one in which the magnetization density about each ion in the crystal is approximated by a point dipole. To obtain the field at a point P(X1,X2,X3) in the crystal, the contributions from all the dipoles in the crystal must be summed. The contribution from a Single dipole is given by equation (17), so that the field at P is given by 131(7) e Z 444.115“). In matrix notation the field is written as H1 G11(Ri) 6120251) G13(Ri pi i. H3 G31(Ri) G32(Ri) G33(Ri p3 In the antiferromagnetic crystal, all dipoles are assumed to be pointing in the same direction, so that in cobalt chloride, where only y3%0 H1, G13 H2 = ’13 Zn; G23 ) (18) ‘5 G H 3 3 and in nickel chloride, where only P1 # 0 wh an: whe poi pri and whe ord for fflj 41 H1 G11 1 H3 G31 where“; 311, since a two sublattice model has been assumed, and the dipoles of the two lattices have Opposite sense. Now the components of the tensor §(Ri) are given by I—Rsf CF where 5i is the displacement vector from the iig ion to the ,7 u . . ~ GAlm (Bi) = 3(X1-x4i)(Xm-xm_;) ' 8m ' (20) point P(X1,X2,X3), and the XiI are the coordinates of the 132 ion. These coordinates may be written in terms of the crystal basis vectors. The ions on the corners of the primitive lattice have coordinates given by (ha, kb, 1c), and the centered ions are located at ((ht%)a,(k+%)b,Ic), where h,k,1 are integers. In the orthogonal crystal co- ordinate system these positions are x1=ha cos 9 x2=kb (21) x3=Ic-ha sin 9 for the primitive lattice, and x1=(h+%) a cos 9 x2=(k+-:-) b (22) x3=Ic-(h+%) a sin 9 for the centered ion positions, where G=P’-90°. 31" co' and Whe: If abo C0: 42 If the values for the Gim given in equation (20) are now substituted into equations (18) for H, then the cobalt field components are given by (Xl'xli)(X3-X3i) .= 03.1 '. _ .. 11 P3: Di 3(X2..x2i)0.(.3 x31) , (23) 1 ‘ ‘ 2(x )2 (x )2 (x ) 3‘x3i ‘ 2‘X2i ' 1-X11 and the fields for nickel chloride are 2(x 2 (x )2 (x 2 1‘x1i) ' 2'X21 ’ 3'x31) 13 = ,1 Z g1 3(X1-x1i)(X2-x2i) (24) 1 i i 3(X1-Xli)(X3-x3i) where 5/2 _ 2 2 2 Di - [(Xl-Xli) +(X2-X2i) +(X3-X3i) J . If finally, the values of the xii are substituted into the above formulas, one obtains for cobalt chloride, the field components and f :1: ( 43 . (leha cos 8)(X3-Ic+ha Sin 9) Hi — P3:E::3 G. D + h,k,‘l l “ 1 . - +— _ + +— (x1 (h 2)a cos 6)(X3 lc (h 2)a Sin 8):], D! (X2-kb)(X3-Ic+ha sin 9) H2 = p3ZE; 3~E7 D + h,k,l 1 ~ 1 . (Kg-(k7)b>(X3-1°*(h*'z')a 9:1,. 65> D! 2(X3 -lc+ha Sin 9) 2-(X2-kb) 2-(X1-ha cos O)2 3=P3th3 TEY . '+ D 3 2(X -Ic+(h+%)a Sin 9)2'(X2-(kt%)b)2-(X1-(h+%)a c059}%] ‘ *‘ ’ a D! and for the field components in nickel chloride 2(Xl -ha cos 9) W—(X —kb) u-(X -lc+ha sine)2 “1M12[ D. h,k,l 2 1 2 “ . 1 . 2 2(x -(h+l)a cos 9) -(x -(k+—)b) -(x -lc+(h+—)a sm 9) 6 ‘1 2 2 2L 3 '4? J (26) u k D! _ 3 (X1 -ha cos 6)(X2 -kb)+ (X1 -(h+-) a cos O)(X2-(k+— —)b) ‘y123[a'1ggl_} D . D' h,k,‘i (X -h 05 e)(x -Ic+ha sin e) :21 3 H 3 D N h,k,l S 1 ~ 1 . (X1'(h+2)a cos 9)(X3-(l+-2—)c+ha Sin 9)] D! whe 44 where 5/2 D =[3X1-ha cos 6)2+(X2—kb)2+(X3-lc+ha sin 6)%] D'=(X (h+l)a cos e)2+(x -(k+l)b)2+(X -lc+(h+l)a sin 6)2 5/2 1‘ 2 2 2 3 2 and where, referring to Figure 5,0" =1, S =-1 for the P- , structure fields, and 0' = 5 =(-1) for the C-structure. In copper chloride the magnetization is also along the X1 direction, and the copper ions also occupy positions in a C centered unit cell. Thus, the structure of copper chlo- ride is similar to the structure of nickel chloride, and by using copper chloride parameters, the copper chloride fields are easily obtained from the above equations which have been derived for nickel chloride. Unsymmetrical Magnetization Density In the general case, Efir) is not known, and to cal- culate it from crystal wave functions is quite difficult, since this involves, first the determination of a set of approximate crystal wave functions, and then evaluation of the integrals which result when a linear combination of the functions is put into the integral for 3. After this is accomplished there are still at least five constants to be determined. Considering the labor involved in an approx- imate calculation of this nature, it was decided to use a different method to solve for the integral in equation (14). The method of approximating the integral by a distribution of point dipoles placed about the metal ions is the method (16 to wh It of On is to pm of to dis 37. 45 deve10ped here. The integral for §,is first approximated by a series to obtain .15. =Zi 31:21, Eigi : (27) where fli is the field at r due to the i-F—h- dipole, and p=2mi. c It is also noted, that if the strength of the i'E-ll dipole, m.=f,gfl then Eéfi=1. H(r) is thus approximated as the sum of the contributions from a distribution of point dipoles. On considering the crystal lattice translational symmetry it is seen that if 31(3) is the field at E due to a dipole m.=f.2 at r.', then it is equal to the field at r - r.‘ due -1 1 -1 - -1 to the dipole 9i located at the origin. Thus the computer program which is used to calculate the field from a lattice of point dipoles placed at the ion sites may also be used to calculate the fields in a lattice that has equivalent distributions of dipoles about each of the metal ions. Iffli = fig” and 1f fliflgi) represents the f1e1d at r, due to dipole‘mi at £i" and if 50' = 0, then §i(51) = Ei'gq’ti) and EOE-351') = 11.0 fits-£5) = 910' g Thus EO(B ) = £0. g(3i)9 wt NC An is '(J 46 and me = me. 1f 21.. = 2.. so that 510:9 = £12 9 : f‘ - 1 " - 21. 9(3) f0 0 - 1 = fi = C H (R ) ?;-—o 1-—o 1 ’ where Ci = $1 . 0 Now, from (27) one has that on = = . .. ' 5(3) Z111 201 go (5 £1). (28) L. '=o And if the fact that an f0 = 1 - 2: fi, (29) :1 is used in equation (28), then 3(5) _ _ ‘1’ ‘71—- $3.0) - Zfi [9:51) - 9(303- <30) ('31 Equations (28) and (30) may now be used to deter- mine the dipole fractions for each of the dipoles in an assumed distribution of dipoles about the metal ions. First the simple dipole fields due to each dipole of the distribu- tion are calculated at the proton positions. In cobalt and nickel chloride there are four protons and thus twelve field components. There may be up to twelve dipoles in the dis- tribution, since there are twelve experimental field com- ponents. Using the 144 calculated field components in 47 equations (28) and (30) gives twelve equations for the twelve experimental field components in terms of the twelve unknown fi's. ‘The Gijggi) are then the 144 components of a twelve by twelve matrix which must be inverted to solve for the fi' In this way the dipole fractions may be calculated for any dipole distribution of twelve or less dipoles. The equations (28) and (30) have been derived for the general case where none of the fi have been assumed to be equal. It is clear that two dipoles which are related by mirror plane or two fold symmetry must have the same dipole fractions. If ff and fm are two equal dipole frac- tions for the "32 and mill dipoles which are related by a symmetry operation, then f$[§(§1)-§(Bo)] + fm[§(§m)-§(§o)] = 2 ff[§(_fii)*§(§ul - gay]. (31) 2 Therefore if the average of the fields calculated at two positions related by the symmetry Operation is placed into equation (28) or (30), the fraction solved for is tWice the value ofithe two equal fractions at each point. Programs To make the numerical calculations for the magnetic fields and the dipole fractions, some of the above equations have been programmed in 160 Fortran-A, for use with the Control Data 160 computer. The programs to calculate the 48 spherical and cartesian magnetic field components have been made from equations (25) and (26), and are given in Appendix 3. These programs are used to compute the field quantities at an arbitrary point in both the P- and C-structure simple dipole crystal models. The field programs have been extensively code checked. First, copper chloride parameters were used in the nickel program and the field was calculated at the proton position used by Poulis for his calculation. The only difference between the two calculations is that in the present calcula- tion the sum is taken over eight unit cells, whereas Poulis summed over nine atoms. The agreement between the two cal- culations is quite good. The programs were tested for two- fold and mirror plane symmetry by calculating the magnetic field at differen proton positions. The results of these calculations are shown in the stereogram in Figure 9. In equations (18) and (19) it is noticed that Hx of cobalt is equal to Hz of nickel chloride, if the same parameters are used in the two equations. A calculation has been made which used the nickel parameters in the cobalt program. The resulting value of Hx for cobalt was found to be equal to the value of Hz calculated from the nickel program. It has also been verified that field values calcu- lated at nearby points satisfy the condition thatcurl E be equal to zero. For some cases, fields had to be cal- culated for a set of points located on the surface of a th SL1 IE. ha} We 49 sphere of small radius, and at the center of the Sphere. When the average field on the surface of the sphere was determined, it was found to be equal to the field which had been computed at the center of the Sphere, as it should be, according to the mean value theorem for harmonic functions. And lastly, several numerical hand calculations were made, which agreed with the computer results. The programs for solving equations (28) and (30) for the dipole fractions fi are given in Appendix 4. Most of these programs consist of the matrix inversion, which is carried out by means of the Gaussian elimination method. These programs were quite easily code checked by computing the inverse of a three by three matrix. The computer re- sults were equal to the results of a hand calculation. The rest of these programs were also checked out by means of hand calculations. The convergence properties of the field programs were also investigated. It was found that the convergence is much better in the copper chloride program, than in the cobalt and nickel programs. Most of the cobalt and nickel field calculations were made with sums over 216 unit cells. There is some difference between the results of summing over this number of cells, and the results from summing over a greater number of cells. But there is also a large enough error in the coordinates of the protons to give an error 50 in the calculated local fields of approximately the same magnitude as the change in the fields due to summing over a greater number of cells. The slowness of the computer and the round-off error also discouraged one from making an extensive number of computations of sums over more than 216 unit cells. It is noted here that there is slight disagreement in the fields calculated at positions related by twofold or mirror symmetry, because-of the lack of con- vergence and the round-off error. V. COPPER CHLORIDE Experimental Results In this section, the results of the investigations into the magnetic structure of copper chloride are dis- cussed- Since there is only one set of equivalent proton positions in the crystal, the computational analysis of the copper chloride is more limited and straightforward than for the cobalt and nickel chlorides. Also, the re- sults of the calculations are much less ambiguous than for the other two salts. The crystal structure has been determined by Harker,3 but recently Peterson and Levy13 have made a neutron dif- fraction analysis and are able to determine the proton positions. Copper chloride belongs to the rhombic bi- pyramidal class, having formula weight of two, and space_ group classification.thn. The unit cell dimensions, in 8 angstroms, are a = 7.38, b = 8.04, and c = 3.72. Because of twofold and mirror plane symmetry, the general position is eightfold. Since there are eight hydrogen atoms in the unit cell, they must all be related by symmetry operations. From the neutron diffraction studies, it is known that the two hydrogen atoms in a water molecule are related by a twofold symmetry axis of rotation. The atomic parameters 51 52 for atoms in the unit cell are given in Table 8. Table 8. Atomic Parameters for C0pper Chloride At om X y 2 Cu .000 .000 .000 C1 .250 .000 .370 O .000 .250 .000 H .082 .307 .130 The magnetic fields at the proton positions have been measured in the antiferromagnetic state by Poulis and Hardeman.12 Their results, corrected to T = 00K, are Hx?:539oe Hy?:3780e H2513800e H=7600e The stereographic projection of one of these fields is given in Figure 9, and labeled ‘Ex.’ Calculation of Local Field Vectors Both Poulis and Rundle have proposed ordering schemes for this crystal, and the two different structures are given in Figure 8. In the structure proposed by Poulis, the moments in the C-centered planes are ferromagnetically coupled to each other, and neighboring planes are antifer- romagnetically coupled. The Rundle structure differs, only in that the C- centered moments are antiferromagnetically coupled to their 53 neighbors at the corners of the cell. If these ordering schemes are compared with the ordering of the C-nickel chloride structure, it is seen that the Poulis ordering is identical with the C-nickel chloride, while the Rundle ordering requires only the reversal of sign on the cen- tered moments. Poulis Structure Rundle Structure Figure 8. Possible Magnetic Structures for Copper Chloride. The fields which are calculated at the proton posi- tions in both the Poulis and Rundle structures are used to distinguish which of the two structures gives the best agree- ment with the experimentally measured fields. The calcu1a=. tion is made in the simple dipole model approximation. 54 Figure 9. Stereographic Projections of Poulis, Rundle, and Experimental Local Fields in Copper Chloride at Proton Positions and on Surface of .12 Sphere 55 Equation (26) gives the internal magnetic field at the pro- ton position in a crystal with p1 f O, and only slight changes are required to convert the nickel program to a pro- gram which is used to compute the local fields in the two copper chloride structures. Since there is uncertainty in the proton positions, fields are also calculated at positions on the surface of a .12 sphere which is centered on the proton position. This is done to see if there is better agreement between the experimental and calculated fields under small translations of the proton position. The angular components of the local fields are given in the stereographic projection of Figure 9. The results of the field calculations on the small Sphere centered on the proton positions are also displayed on the stereogram. The closed and open circles represent the local magnetic fields which are calculated at the proton positions. The large boundaries enclosing the calculated field circles in the positive quadrant are drawn to just enclose all the points which represent fields at positions on the surface of the sphere. It is seen from these projections of the calculated field vectors, that the variations of the pro- ton positions do not markedly change the field vector at the proton positions. Thus the assumed structures give local field directions which are quite different from each other, and the directions are not brought into agreement 56 with each other, or the measured field directions when the proton position is varied by small increments. The results of the Poulis and Rundle structures are labeled 'P' and 'R' respectively. The field components and magnitudes corresponding to the stereographic projections, are given in Table 9. The Poulis and Rundle field components and magnitudes which are computed from the program, are given in the first two rows of the table. In the last columns of these rows are the dipole moments calculated by dividing the experimental field magnitudes by the corresponding calculated field magnitudes. In the next two rows are the Poulis and Rundle Table 9. Calculated and Experimental Local Field Vectors in Copper Chloride Structure Hx Hy Hz H )1 Poulis 101 484 14.7 495 1.54 Rundle 680 187 180 728 1.04 Poulis 155 743 22.6 760 Rundle 710 195 188 760 Experiment 539 378 380 g 760 1.02 field components and magnitudes, in oersted units, obtained by multiplying the first two rows by their reSpective dipole moments. In the last row is given the experimental field 57 components and magnitudes. The dipole moment given here is obtained from the equation P’= 8a :9 , where g is the Lande g factor for the free electron, and is equal to 2, P0 is the Bohr magneton and is equal to .927x10-20 erg/gauss, and ga is the measured antiferromag- 3, 34 netic g factor? and equals 2.19. Discussion and Conclusion The stereogram in Figure 9 shows that the agreement between the angle of the calculated and experimental fields is much better for the fields calculated from the Rundle structure. And from Table 9 it is seen that there is also very good agreement between the magnitude of the field for this structure and the magnitude of the experimental field. The dipole moment for the Rundle structure compares quite favorably with the measured dipole moment. On the other hand, the local field calculated from the Poulis structure does not agree with the experimental data as well. The difference in the angles is large and is not decreased by altering the proton positions as iS sug- gested by Hardeman. The large dipole moment illustrates the poor agreement between the field magnitudes. From these results, one must conclude that the field calculated from the Rundle structure is in very good agreement with the experimental field, whereas the field 58 calculated from the Poulis structure is, on the contrary, in poor agreement with the experimental data. It iS there- fore concluded that the Rundle structure is a much better representation of the ordering in copper chloride than is the Poulis structure. VI. SIMPLE DIPOLE ANALYSIS OF COBALT AND NICKEL CHLORIDE Introduction The results of a simple dipole analysis of the cobalt and nickel chlorides is described in this Section. As in the preceding calculation for copper chloride, the main approach here is to calculate the local fields for each ordering in the two salts. Then the results from the two structures are compared to the experimental fields in an attempt to find the best fit to the experimental data. In the case of nickel and cobalt chloride, there are two protons in the antimirror plane, so that at these protons the y-component of the magnetic field must be zero. It was thought that the magnetic structures could be deter- mined by comparison of the fields in the neighborhood of these protons. Thus for the cobalt and nickel salts, two types of calculations have been made. First, the magnetic field is calculated in the mirror planes of the salts. Then the magnetic fields are calculated at the proton posi- tions. Mirror Plane Maps of the Magnetic Field The results of the first set of calculations are used to map the magnetic fields in the mirror planes. To 59 60 make the calculations, the cobalt and nickel Simple dipole programs are used. Since the twofold symmetry axis is perpendicular to the mirror plane, it is necessary to cal- culate fields in only one-half of the unit cell mirror plane. To make the maps, the fields are calculated at fifty points in one-half of the mirror plane, and then the field lines are drawn to satisfy the angles calculated for these points. The results of the calculations are Shown in the maps of Figures 10 to 13. The first conclusion about the maps is that they are quite similar to the maps which are sketched by using only the dipoles on the corners of the mirror plane in the unit cell. This indicates that only contributions from moments in the unit cell are of significance in determining the direction and magnitude of the local fields at the pro- ton positions. This fact is also indicated by the close agreement between the present calculation of the fields in copper chloride over twenty-seven unit cells with PouliS'S calculation over nine atoms. The result of these two com- parisons is that the main contribution to the fields at the proton positions comes from the magnetization density in the unit cell. From the map of the magnetic field in the P-cobalt chloride, it is seen that the direction and sense of the field lines in the vicinity of the proton positions is in qualitative agreement with the experimental mirror plane 61 Figure 10. Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of P—Cobalt Chloride 62 k“ 2 Figure 11. Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of C—Cobalt Chloride K /\ a. C! Figure 12. Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of P— Nickel Chloride W m /\//\ a 0 Figure 13. Map of the Magnetic Field in the Mirror Plane of the Simple Dipole Model of C-Nickel Chloride 65 fields in cobalt chloride. Nevertheless, in order for the calculated fields to coincide with the experimental fields, for the association m,m' = (5,7), the calculated field at the m position must be rotated by 53 degrees, and the cal- culated field at the m' position must be rotated by 35 de- grees. Thus, although the calculated fields lie in the proper quadrants, large changes in the angles are required to bring the calculated vectors into quantitative agree- ment with the experimental field vectors. The association (8,6), requires very large changes in the angles, but from the map it appears that a magneti- zation along the oxygen bonds might lead to this type of angular distribution. The associations of m to 6, and m' to 5 or 8, seem to be highly improbable, since they require that the fields at the proton positions be in opposite sense to the fields coming from the vicinity of the nearest metal ion. The association m,m' = (7,6) also seems to be invalid since it requires that the field vectors at the proton positions be pointed in towards each other requiring very large curva- ture in the field lines coming from the ion at the origin. It is seen in the C-structure map, that the fields at the proton positions are approximately in the same direction and sense. It would therefore take considerable distortion of the fields to reverse the sense of one of the proton fields. It is therefore clear that neither m or m' 66 can be associated with the field 6. The same is true for the field 7, since associating this field to either of the proton positions requires that the field at the proton be in opposite direction to the flow of field lines coming from the nearest metal ions. Thus the only remaining asso- ciations are (5,8) and (8,5). Both of these require large angular rotations of one of the calculated vectors, and also, they do not appear to be very probable from study of the map. From the P-nickel chloride map it is seen that the calculated and experimental fields are in very good agree- ment with each other, and a change of only seventeen degrees is required to bring m into coincidence with fields 5 or 7. The associations for this structure are (5,8) and (7,6). The C-nickel chloride map Shows poor agreement with the experimental fields, but it is clear that there are two possible associations, Since a rotation of the both vectors by about forty degrees brings them into agreement with the experimental fields 5 and 7. The associations are (5,7) and (7,5). Local Fields at the Proton Positions The magnetic fields are computed at proton positions g,g',m,m' = [l,6',9,9'] in both the cobalt and nickel salts. Also, since the proton positions are not accurately known, fields are calculated on the surface of a .1A Sphere cen- tered about each of the given proton positions. 67 The angular relations between the directions of the local field vectors are shown in the stereographic projec- tions of Figures 14 to 17. The closed and open circles represent the local field vectors calculated at the proton positions. The closed circles represent vectors with pos- itive z-component, the Open circles, negative z-component. The results of the field calculations on the small Spheres are also displayed on these stereograms, and as before, the boundaries enclosing the lines are drawn so as to just enclose all the points representing fields from the positions on the surface of the corresponding Sphere. It is seen from the results that the variations of the proton positions does not markedly change the field at the proton position, so that there is no agreement, for a given salt, between the sets of fields calculated from the two types of ordering. Nor is there agreement between the calcu- lated and experimental fields. The directions and magnitudes of the calculated fields are given in Table 10. Table 2 of measured local field vectors is again given here for comparison with the calculated fields. In Table 11, the components of the cal- culated and experimental fields along the three orthogonal crystal coordinates are given. 68 Figure 14. Stereographic Projection of the Local Field Vectors in P-Cobalt Chloride at P oton Positions and on Surface of .1 Sphere 69 Figure 15. Stereographic Projection of the Local Field Vectors in C-Cobalt Chloride at roton Positions and on Surface of .l Sphere 7O Figure 16. Stereographic Projection of the Local Field Vectors in P-Nickel Chloride at Proton Positions and on Surface of .IR Sphere 71 Figure 17. Stereographic Projection of the Local Field Vectors in C-Nickel Chloride at Proton Positions and on Surface of .12 Sphere 72 Table 10. Calculated Local Magnetic Field Vectors at Proton Positions in Cobalt and Nickel Chloride Cobalt Chloride A H(3'3) 0 cp P-Structure g = [1] 500 -76° 71° g' = [6'] 657 -64° 75° m = [9] 715 23° 0° m' = [9'] 518 24° 0° C-Structure g = [1] 596 -54° -79° 8' = [6'] 879 -76° 53° m = [9] 581 54° 0° m1 = [9'] 662 60° 0° Nickel Chloride H(X'3) 9 ‘P P-Structure g = [1] 712 78° —25° g' = [6'] 829 79° -11° m = [9] 322 23° 0° m' = [9'] 285 -390 0° C-Structure g = [1] 456 -77° -39° g' = [6'] 673 —40° -550 m = [9] 496 2° 0° m‘ = [9'] 593 4° 0° 73 Table 2. Measured Local Magnetic Field Vectors at Cobalt and Nickel Chloride Proton Positions J Cobalt H (gauss) 9 cp 1 1785 74° 128° 2 1785 74° -128° 3 1785 106° -52° 4 1785 106° 52° 5 1592 77° 0° 6 1592 103° 180° 7 1179 758° 180° 8 1179 122° 0° 9 1397 84° . 97° 10 1397 84° -97° 11 1397 96° -83° 12 1397 96° 83° Nickel H (gauss) G (P 1 1729 55° 166° 2 1729 55° -l66° 3 1729 125° —14° 4 1729 125° 14° 5 1.397 39° 180° 6 1.397 141° 0° 7 1.285 39° 180° 8 1.285 141° 0° 9 1.029 75° 49° 10 1.029 75° -49° 11 1.029 105° -131° 1 029 105° 131° 1.. N 74 Table 11. Cartesian Components of Calculated and Measured Local Field Vectors at Proton Positions in Cobalt and Nickel Chloride P-Cobalt P-Nickel Hx Hy Hz Hx Hy Hz 1 154 459 -125 -632 292 153 6' 151 571 -287 799 -157 132 9 281 0 658 -128 0 297 9' -212 0 473 183 0 -220 C-Cobalt C-Nickel Hx Hy ‘ Hz Hx Hy Hz 1 -95 472 -352 -351 278 -96 6' -499 -683 -200 -248 .353 -510 9 471 0 339 ll 0 502 9‘ 585 0 348 -43 O 599 Magnitudes of Experimental Field Components (gauss) Cobalt Nickel Hx Hy Hz Hx Hy Hz 1-4 1057 1352 492 1374 343 992 9-12 169 1379 146 652 750 266 5-6 1551 0 358 879 0 1086 7-8 1000 O 625 809 O 999 75 By comparing the calculated and experimental stereo- grams, it is quickly seen that there is considerable qual- itative agreement between the experimental and P-Structure stereograms. The C-Structure stereograms do not appear to be in such good agreement. In the case of P-cobalt chloride, the mirror plane fields must undergo shifts in e of 53 and 35 degrees in order to coincide with the mirror plane experimental field vectors. The shifts which are required for the general proton fields are much smaller. The most direct associa- tions of experimental fields to calculated fields are [l,6',9,9'] = (4,12,5,7) and (12,4,5,7). The next best asso- ciations are (9,4,5,7), (9,1,5,7) and (12,1,5,7). For C-cobalt a reasonable association is obtained if one of the mirror plane fields is shifted by 60 degrees into the experimental field vector 8. Then the other mirror plane vector need not be shifted by more than a few degrees. The general proton fields are then associated with the experimental fields 12 and 2. The associations are [1,6',9,9‘] = (12,2,5,8) and (12,2,8,5). If the field at g' goes into the experimental field 3, then the associations (12,3,5,8) and (12,3,8,5) are also possible. For P-nickel the obvious associations are (l,10,5,8) and (1,10,7,6), and only comparatively small angular shifts are required to bring the calculated and experimental fields into coincidence. But it is noted that these associations 76 require g' to have the smaller field. The best associa- tions with large field on g' are (ll,4,5,8) and (12,4,5,8). In the case of C-nickel chloride, the associations (12,4,5,7) and (12,4,7,5) are possible, if the mirror plane protons are diSplaced 40 degrees, and the field at g' is displaced onto the field vector 4. Other possibilities are (ll,2,5,7) and (ll,2,7,5). As to be expected, there is much disagreement be; tween the cartesian components of the calculated and experimental fields. This is because a small change in the angle of a vector causes a much larger change in its carte- sian components. Therefore to best compare the calculated to experimental fields, one Should compare the angles and magnitudes of the vectors. A study of the magnitudes of the calculated fields is made from Tables 12 and 13. Since there are two protons in general positions and two in mirror plane positions, there are four possible ways of associating the magnitudes of the experimental and calculated fields. It is clear that each of the associa- tions which have already been discussed must belong to one of the four ways of associating the field magnitudes. For a given association of magnitudes, a least squares dipole moment is calculated from the formula 4 Zn -H . <>= P E Hci2 i=1 77 where pri is the experimental field magnitude to be associated with the calculated field magnitude Hci' The calculated fields are then multiplied by this dipole moment and compared to the experimental fields. The results of this calculation for the cobalt and nickel chlorides are given in Tables 12 and 13. The first column in these tables contains the experimental field magnitudes, and in the succeeding columns are the calcu- lated field magnitudes which are associated with the exper- imental magnitudes in the first column. The associations are listed at the top of each calculated field column. In the fifth row are the dipole moments which have been calcu- lated from the experimental fields and the calculated fields which are directly above them. The products of the dipole moments and the calculated fields are next tabulated in rows 6 to 9. Finally these fields are compared to the experimental fields and the standard deviations are given in row 10. The data in these two tables is diSplayed so that the associations having the smallest deviations are in the center. Therefore the two center columns are the best fits to the experimental fields. one column for the P-, and one for the Castructures. names Heeeeeesesxms 78 Hem eom mom eHH ems Hem mam flee emwwwwwwm coma omma ewes eeme eema meme mesa mesa. seas omms Hoes mews oees mesa mesa meme, mews Nome mews Heed eema eoma omms ooeH HHNH eeeHA some fleas meme eoea mass ewes mama Home coed mesa ooo.m sHH.m emH.m esa.m ese.m eme.m mme.m oem.m Aexw mee11 Hem mee HMm man wee was was 6844 Hem mee Hem mee was eds man man Nona ose ose eon eon ooe see eon. see send eon eon see osw see ooe see ooe mesa As.e.o.Seam.e.o.HeAe.e.H.6efle.e.H.6efis.e.s.oehe.m.6.aefi.e.e.a.oeme.e.o.ae oaowm HHwDOOID pamwm pHmDOOnm owowm.uooxm .1 eeesoaeo “Hence ea mmeseeemez.eaeem pmumaaoamo new: moospflemmz pamwm Hmpaosfluoaxm mo conflumaeoo .NH manmh 79 vemwm Hmpmmswemaxme eee mem ome em eee eee mem mem emwwmwwmm meme eeee; mmee‘ mmme. see eoe ooe see meme smee meme eeme eeee eoe eee eme 66m some emme eeme oeee. emee eeme emme meme oese 6m0e eeoe eeoe mmee emee eome meme ooee eoee omee eme.m mem.m eme.m eee.m eee.m eee.m moe.m moe.m AexV mom eoe mem eoe mem mmm mem mme meme eoe mom eoe mom .mmm mem mmm mem some mme mee eme eme men mee eme eme omoe eme eme eme mee eme eme mes mem omee em.m.6.eeem.s.e.eeee.m.e.6eem.e.e.eves.m.e.oeem.m.e.6eee.m.o.eeem.s.6.ee eeeeee.emexm eeeeoeeo eexuez ee eeeseeeeez eeeee eeeeesueeo nee: mmeseenewz eeeee eeeemEeemnxm me someeeneeo .ee.eeeee 80 In Table 12, for the cobalt chloride, the deviation is least for the P-Structure magnitude association which correSponds to the angular associations. The next best P- cobalt association is discarded Since it requires that the smaller general magnetic field be at the general proton which is closest to the cobalt ion. The remaining associations are rejected too, Since in each of them the mirror plane fields are in reverse order and quite out of line, with the exper- imental fields, and in the last case the general proton fields are also in reverse order from the experimental fields. Thus for P-cobalt chloride, the association (12,4,5,7) gives the best fit for both the angles and magnitudes of the exper- imental local fields. The remaining angular associations also correspond to this magnitude association, but are not in such good agreement as the stated association. In the case of C-cobalt structure, the best magnitude association correSpondS to the angular associations (12,2,8,5), and (12,3,8,5) and the association (9,2,8,5). The next best magnitude association allows for the angular associations (12,2,5,8), (12,3,5,8). On comparison of the associations for the two cobalt structures, the P-cobalt structure appears to be the better representation of the magnetic ordering than the C-structure. Although the magnitudes of the local fields in both the cal- culated structures agrees with the experimental magnitudes, the angular agreement is much better for the P-cobalt 81 association (12,4,5,7). If these results are compared to the results of the dipole-dipole Splitting associations in Table 6, it is seen that the best point dipole association does not agree with the observed splittings. Therefore Since the associations which do satisfy the Splitting requirements are poor ones on the dipole model for both the P- and C-structure the re- sults of this calculation for the case of cobalt chloride must be considered to be inconclusive. For nickel chloride the results are quite similar. From Table 13, it is seen that the agreement between the field magnitudes for the P-nickel is not good. The magni- tudes of the mirror plane fields are just about half of the values of the experimental fields, and the magnitudes of the general proton fields are also out of line with the experi- mental fields. Further the best angular association requires the general proton position nearest the nickel ion to have the smaller field. In view of the inability of the simple dipole P-Structure to satisfy the conditions that the field magnitudes agree with the experimental magnitudes and that the proton nearest to the source should have the larger field magnitude, this association must be considered a poor repre- sentation of the ordering in nickel chloride. The best associations are (ll,4,5,8) and (12,4,5,8). For the C-nickel structure, the correSpondence be- tween the experimental and calculated fields is excellent 82 for one of the associations. The deviation of 73 is the smallest of all the associations considered in both cobalt and nickel. Unfortunately the angular associations (12,4,7,5) and (ll,2,7,5) which correSpond to this magnitude association are not too good. The next best magnitude association also has a relatively small deviation, and it correSponds to the angular associations (ll,2,5,7) and (12,4,5,7). For nickel chloride it seems that the point dipole model C-structure fields give the best fit to the experimental fields, and the best associations are (12,4,7,5) and (ll,2,7,5). The latter association agrees with the observed dipole-dipole Splitting. The calculations for the fields in the crystals have been made on the assumption that there is no exchange field at the proton positions. If it is true that the atomic spins are not canted, as Moriya has proposed for the case of copper chloride, then any exchange field that may exist in the two salts is along the z axis for cobalt chloride and along the x axis in the nickel chloride. If these field components are now disregarded in the calculated stereograms and another effort is made to associate the calculated stereogram fields to the experimental fields, only in the case of C—cobalt does one find an improved fit for the association (9,2,8,5), and this association does not satisfy the dipole-dipole Splitting requirement. 83 The results of this section are tabulated in Table 14. The associations which agree with the results of the dipole- dipole Splittings are marked with an s and the best dipole model associations are marked with a d. Table 14. Associations of Calculated to Experimental Local Field Vectors P-cobalt C-cobalt P-nickel C-nickel (12,4,5,7)d (12,2,8,5) (l,10,5,8)d (12,4,7,5) (9,4,5,7) (12,3,8,5)s . (ll,4,5,8)s (ll,2,7,5)s (12,1,5,7)5 (12,2,5,8) (12,4,5,8) (9,1,5,7)5 (12,3,5,8)S (9,2,8,5) It has been mentioned several times that the general protons are at different distances from the nearest metal ion. It is of interest to calculate the ratios of the fields at the g and g' proton positions for the two structures and compare them to the ratio of the experimental fields. The re- sults of such a calculation are presented in Table 15. The conclu- sions of this calculation are that the P-cobalt and C-nickel struc— tures are the better representations of the magnetic ordering . 84 Table 15. Ratios of Experimental and Calculated Fields A Cobalt Nickel Ex 1.28 1.68 P- 1.31 1.16 C- 1.47 1.48 VII. SECOND ORDER CALCULATIONS Introduction In the preceding section a simple dipole analysis of the cobalt and nickel salts is described. It is found that one can make a choice between the two structures for a given salt, for the structure which gives the better agree- ment with the experimental stereograms. But further con- sideration shows that the associations which give the better fit, do not give dipole-dipole Splittings which are in good agreement with the experimentally observed Splittings. It is therefore desirable to verify the simple dipole analysis with different calculations, preferably ones which allow for the non-Spherically symmetric magnetization which seems to be implied by the previous results.. Two such calculations have been made, and the results of these calculations are described in this section. The most straightforward calculation is to expand the magnetic field in a Fourier series, and then determine the first coefficients in the series for the different structures. There then exists the possibility that only one of the two possible structures will Show nice convergence for the first coefficients. The advantage of this method is that no assumption is made about the magnetization density. 85 86 The disadvantage is that there are at most only four terms in each series from which the coefficients are calculated. The second method is the one used by Poulis and Hardeman for copper chloride. In this calculation the mag- netization density iS approximated by a distribution of point dipoles about the metal ions, and the fractional Share of each dipole is calculated from the experimental fields. There then exists the possibility that the correct structures may be determined from the sets of dipole fractions. This method has already been developed in Section IV. Fourier Analysis To begin this calculation, it is assumed that the Fourier expansions of the scaler magnetic potential 0, and the magnetic field H, are ¢ = ACE) e i£.£ , E. where ACE) = g¢(g)e'§-£ AT, and H(r) =2 fiQQel- _r_ , 33 where 87 The vector 3 is equal to eiqi where ei are the con- travariant basis vectors and are equal in magnitude and direction to the edges a, b, and c of the unit cell. The qi then, are the atomic parameters for an arbitrary position in the crystal. It is also recalled, that the covarient basis vectors ej, defined by the condition ej°ei= Si define a reciprocal lattice in which the vector §_= 2TThiei, and where hlii, h2=m, and h3=n.' Now . ~ .K' H1 = -V¢)= -eJZ 2'“ ith(1,m,n)e1-' _r_ , K but H may also be expressed in terms of its covarient com- ponents, giving H = eJ H. — J so that . e iK-r Hj = -2Tflzth(1,m,n)e —— And it is now clear that §'= erj where bj = -iZTTth(I,m,n). Now, in order to go from exponential to sine and cosine series, relations between the coefficients A(I,m,n) in the expansion for Hj must be found. These are easily determined when H is subjected to the Schubnikov symmetry conditions in both the P and C struc- tures. Both structures contain m',2', and i. If the field at (q1,q2,q3) is (H1,H2,H3), then, under m', the field at (ql:02,q3) is (H1,H2,H3). Applying these conditions to the 88 integral A(I,m,n) = 3W* e-iK jSHJe £)°Lt results in the relation A(l,m,n) = A(l,m,n). In a like manner the relation A(I,m,n) = -A(Tflm,fi) is derived from the 2‘ symmetry condition, since under this element 1 q2 (H1,H2,H3) goes to (H1,H2,H3) when (q, ,q3) goes to (0' ,q2,a'3). And finally A(1,m ,n)= -("f .55) for the sym- metry element i, Since H remains invariant under a change in sign for the three position elements. It is only in the translational symmetry that the two magnetic structures are different. The P-Structure has / the antitranslation Iza+b which causes H to go to -H when .2. _. _ 1 (q1,q2,q3) goes to (q1+%,q2+§,q3). The result of applying these conditions to the above integral is a restriction on the indices I and m. One obtains the result that I+m must be equal to an odd integer. In the case of the C-structures, there is the anti- I translation I; , and the translation [a3]; . Under [.1112 2 2 'H remains invariant when (q1,q?,q3) goes to (q ,q 1+1 21_ Q3) 2 2 and one obtains the condition that for the C-Structure I+m must be equal to an even integer. In a Similar manner, the antitranslation 1:%_gives the condition that n must be an odd integer. 89 If these three relations among the ,A(l,m,n) are now used in the Fourier expansions of the Hj’ the exponential series from plus to minus infinity may be transformed into a Sine and cosine series from zero to infinity. The final re- sults obtained from this substitution are -8TT12: I[A(I,m,n)cos(2TTmy)cos 2TT(Ix+nz) + K :11 ll A(Ifim,fi)cos(2TImy)cos 21T(Ix-nz)] H = STTi 2: m[A(1,m,n)sin(2Tfmy)Sin 2TICIx+nz) + K ,J M A(l,m,fi)sin(2TTmy)sin 21T(lx-nz)] n1 w -81Ti E: n[A(1,m,n)cos(2TTmy)cos 21T(lx+nz)- K :12 (A) ll A(I,m,fi)cos(21Tmy)cos 21T(Ix-nz)] If each of these equations is now multiplied by the appropriate Sine and cosine functions and then integrated over the unit cell, integrals for the A(1,m,n) are obtained. They are ‘4 T A(T ) i 5 chos 2TT my cos 2Tl'(lx+nz) d‘f ,m,n = —' N 7 8T‘ 30052 2Tme cos2 21T(lx+nz) ~ 1 Sstin 2‘“ my Sin 211' (Txi-nz) (it . m A(l,m,n) - - 8—I'F . 2 . 2 ~ d 5111 21T my S1n 2'\T(lx+nz) I: ,5 i 5H3cos 2me cos 2TT(Ix+nz) n A(l,m,n) = 8TT Scos2 ZTTmy cos2 2TT(Ix+nz) 90 From the last three equations it is seen that each of the coefficients is equal to an integral over a component of 5. Further, there may be up to three different integrals giv- ing the same coefficient. Therefore, if the integrals are approximated by sums, there are four terms in the sums over H1 and H3, and only two terms in the sum over H2, since in this last sum the mirror plane components are zero. Now, these approximations are not very good and it is expected that the values of the coefficients determined from them will be different. Therefore when there exists more than one integral for a given coefficient, the average value of the coefficients is used. The sums for the first few coeffi- cients in both structures are given below. For the P-struc- tures the coefficients are: _ .31 ii: A(lOO) - 4TT Hlj cos 2Trxj, A(0 1 0) s 0, 4 4 _ .1. .1 . 1 . A(lOl) - 411’ [2 ZH cos 2Tl'(xj+zJ)+§: H3j cos 21ij+zjfl , i=1 j=l 2 4 A(Oll) =-25_-n_ [% le H23 sin 2'" yjsin 2Tl"zj -—2Z_1H3jcosz\ryjc0521rz] , _ j: 91 and for the C structure, _ __i_ . A(OOl) — 4.T 2:. H3j cos 2Ter, 1:1 4 l>: w A(lll) 2TT 3 ‘ 1 Hljcos ZTT'yj cos 2 (xj+zj)- J: 2 4 %Z szsin 2ITyjsin 27T(xj+zj)+% :- . H3jcos ZITyjcos 27(xj+zj{]. J=1 j-l The results of using these equations to calculate the coefficients for the associations in cobalt and nickel chloride which have been discussed in previous sections are given in Tables 16 and 17. Unfortunately, these results do not give a very good indication of the structures. For the P-cobalt there are two associations which give sets of converging coefficients, but in the case of the C-cobalt chloride there are three associations for which the two coefficients are almost the same. Since the sums have been taken over only four points one cannot say for certain that the P-cobalt structure is preferred over the C-structure. A Similar Sit- uation exists in the case of nickel chloride, where both the P- and C-structures have associations which give sets of converging coefficients. In fact, it is curious that the association (l,10,5,8) which does not satisfy the dipole- dipole Splitting requirement, has the best convergence for 92 the P-nickel chloride. Thus because of ambiguity and use of only four points the results of the Fourier calculations cannot be taken too seriously. Table 16. Fourier Coefficients of Magnetic Field in Cobalt Chloride P-Cobalt 3%: A(lOO) 3%: A(lOl) 3;:A(011) (12,1,5,7) 35 514 2 843 -79 000 (9,1,5,7) 31 122 2 050 -78 664 (9,3,5,7) -15 534 -6 655 -10 288 (12,3,5,7) -11 140 -5 851 -10 624 (12,4,5,7) -11 140 —5 851 -81 040 C-Cobalt gill-Moon 31E A(lll) (12,3,8,5) -15 980 -16 000 (12,3,5,8) -6 514 -24 000 (9,3,8,5) —10 786 -16 150 (9,3,5,8) —1 318 -24 170 (12,1,8,5) 4 966 -11 460 (12,1,5,8) 14 434 —19 480 (9,1,8,5) 10 162 -11 610 (9,1,5,8) 19 630 -19 630 93 Table 17. Fourier Coefficients of Magnetic Field in Nickel Chloride P-Nickel in A(lOO) 'figI.A(lOl) :rA(011) (ll,4,5,8) -17 800 11 560 1 376 (ll,l,5,8) -13 138 —4 415 5 436 (ll,2,5,8) -13 138 -4 415 23 276 (ll,3,5,8) -17 800 11 560 19 220 (l,10,5,8) -18 890 -20 000 - 640 C-Nickel 2;: A(OOl) gig A(lll) (11,2,7,5) 1 700 -5 065 (11,3,7,5) -39 910 3 985 (11,4,7,5) -39 910 5 145 (11,1,7,5) 1 700 -3 905 (12,4,7,5) -39 910 -7 770 Dipole Array Calculation In section IV it is Shown that the integral for the 11(5) = Sago-(=3 alt , magnetic field at a proton position may be approximated by a sum over the fields at the proton positions due to an array of point dipoles Situated about 94 the metal ions. The coefficients of the fields due to each of the dipoles is just the fraction of the total moment given to each dipole of the distribution. These fractions are calculated by using equation (28) or (30) and the auxil- iary equation (31). The assumption of a dipole array and the calculation of the dipole fractions for this array constitutes what is here called the dipole array calculation. The use of equation (30) is quite similar to the use of (28), and Since the latter requires less computation, the former is described in detail here. The essential difference between the two equations is that in (28), no attempt is made to subtract the effect due to the spherically symmetric part of the magnetization from the experimental fields. Instead, an attempt is made to directly fit the dipole fractions to the experimental data. In equation (30), the contribution to the magnetic field from the Spherically symmetric part of the magnetization is subtracted from the experimental mag- netic fields, before the dipole fractions are fitted to them. Before the subtraction is made, the reciprocal of the dipole moment is calculated from the experimental and calcu- lated fields. The derivation of this quantity, K, is made by means of the least squares method and is given as .MZ C) m K: N=l,12 95 where the HeX and G0. are the measured and calculated local 1 1 field components at the proton positions. The situation to be described is Shown in the Sketch below. A metal ion is at the origin of the coordinate system,, and a proton is situated at position P. The ith dipole of a distribution about the origin has a moment‘gi, and is situ- ated at r'i. The field at P due to the dipole of momentjgi at 3'1 is now calculated. The displacement vector between )5i and P, is R, It is noted that the vector rfrfi is equal to 3, so that the field at rgr'. 1 due to a dipole situated at the origin is equal to the field at r due to a dipole situ- ated at r‘.. - 1 IH The fields are now calculated for all £7£'i’ where i = 1,12. From each Hji about this proton under considera- tion, where j = 1,3 and represents one of the three coordi- nate directions, the field H. jO at the proton position due to 96 a dipole at the origin is subtracted. Also the HjO are subtracted from the corresponding experimentally determined field components which have been multiplied by K. Finally each "corrected” experimental field component is written as a linear combination of the "corrected” Hji' This same procedure is applied to the three remaining proton positions and one finally has each of the twelve corrected experimental fields written as a linear combination of twelve correspond- ing calculated field components. Since the coefficients are the dipole fractions, the coefficients of Hji are equal to the coefficients of H , and ki one has a system of twelve linear equations in twelve unknowns, from which the twelve fi are calculated. The dipole fraction for the central dipole can then be calculated from the condi- tion (29). The use of equation (28) is more straightforward than the above procedure, since it is not required to calcu- late a least squares dipole moment, and to subtract the part of the field which arises from the Spherically symmetric part of the magnetization. It is seen from equation (28) that each of the twelve experimental field components is written directly as a linear combination of the correspond- ing calculated field components. There is also another important difference between the use of the two equations. Since the dipole fraction at the origin is calculated di- rectly in this method, it is the first term in the linear 97 combination, and a symmetrical dipole distribution cannot be obtained if all twelve dipoles are used in the distribution. It was decided to use distributions of only eleven dipoles to avoid the possibility of true symmetry being destroyed by an overcorrection in the calculation. The programs used to perform the computations are given in Appendix 4, and have been checked by hand calcula- tions. If the distributions have all the dipoles in the mirror plan, or if the auxiliary equation (31) iS used for those dipoles related by mirror plane symmetry, all the y components of the fields associated with the mirror plane protons are zero, and a Small subroutine is inserted into the given programs to eliminate the two rows of zero in the field matrix, and in the experimental field vectors. The input for the programs is the experimental fields and the set of fields calculated from the dipole model programs at the points about the four proton positions which correSpond to the assumed dipole distribution. The output is the set of dipole fractions for that distribution and association of experimental fields to proton positions. Since it is not possible to measure the proton positions to which the measured fields belong, the associa- tions used in the computations deserve some comment here. Initially it was thought that if in a given distribution of dipoles, a large number of different associations are used, then only for the proper association and structure would a 98 reasonable set of dipoles be obtained. A reasonable set of fractions being one that has a strong fraction on the metal ion, exhibits the twofold and mirror plane symmetry and from which a smooth physically meaningful magnetization density can be deduced. On the contrary, it has been found that for a large number of associations, there are distributions-efor both the structures—-which are acceptable choices for repre- senting the magnetization density. That is, one cannot diS- tinguish between the two structures by means of the dipole array calculation alone. The two problems of determining the translational symmetry and of determining the magnetiza- tion density distribution cannot be Simultaneously solved by this method. Thus, since both the Fourier analysis and the dipole array analysis do not distinguish between the two structures, the results of the dipole model calculation must be used to determine the translational Symmetry, and possible associations of measured fields to proton positions. These associations must be in agreement with the dipole Splittings measured from the zero field lines. Finally, the dipole array calculation is used to determine the magnetization distribution about the metal ion, and possibly as a condition on the few associations considered. If the simple dipole, two sublattice, monoclinic model is a reasonable approxima- tion to the physical Situation in the crystal, then this approach should work, otherwise some of the initial assump- tions must be in error. 99 In preliminary calculations, dipole fractions are calculated for three types of dipole distributions. In the type I distributions the dipoles are equidistantly placed on three orthogonal great circles of a sphere of 1.132 radius. The center of the Sphere is on the metal ion. In the type II distributions the dipoles are placed along the chlorine bonds, and in the type III distributions, they are placed along the chlorine, and oxygen-one bonds. The nota- tion which is used to define the positions along the bonds is illustrated in Figure 18. One set of positions along the chlorine bond is obtained by dividing the bond length into sixths and then labeling out from the metal ion. These positions are labeled with single integer numbers. The posi- tions related to this set of positions by inversion symmetry are labeled with primed numbers. The positions which are labeled with double integer numbers are obtained by dividing the bond length into fifths. Along the oxygen bonds the 'A' positions are one-fifth the bond length away from the metal ion, and the 'B' positions are two-fifths the bond length away. Calculations of dipole fractions are made for the type I distributions for many associations of measured fields to proton positions. In all of these sets of dipole fractions the twofold and mirror plane symmetries are lacking. For many cases of both P- and C-structures, sets of fractions are found which have a large dipole fraction on the metal ion position, 100 but because of the lack of Symmetry in all of the results, one is not able to distinguish between the two structures by means of this type of distribution. A similar experience is encountered with the results of calculations for which the type II distributions are used. In each of the calculations of dipole fractions for distribu- tions in which the dipoles are placed only on the chlorine bonds, there is an absence of twofold symmetry between the dipole fractions. The results obtained from this type of distribution are also inconclusive and further calculations are not pursued. It is of interest to note that very little difference is found if the fractions from a distribution along the chlorine bond at the Single integer positions is compared to the fractions obtained from a distribution over the double integer positions. That is, a slight Shift of the dipole positions does not change the results by very much. Type III distributions are the only ones which give sets of dipole fractions which possess the required symmetry. The sets which show the best consistency among the two cobalt and two nickel salts are obtained by using equation (28), in which the dipole fractions are fitted directly to the exper- imental fields. The distribution used in these calculations has dipoles at the origin, 11,11’,3,3',6,6', and the A posi- tions. Several calculations are made which have dipoles on the B positions, instead of the A positions. For these cal- culations it is found that the relative distributions of 101 moment along the chlorine bonds does not change, but the fraction of moment on the B position is less than the moment on the A position. The dipole fractions at all the A or B positions are found to have mirror plane Symmetry, and the auxiliary equation (31) is used for the calculations pre- sented here, since it decreases the dimension of the calcu- lated field matrix, increasing the accuracy of the inversion process. The results of using this procedure are found to be the same as the results which do not average the dipole fields for the dipoles which are related by the mirror plane. As may be seen in the Figures of dipole fractions the com- bined dipole fractions of the Al-+ A4, and A2 + A3 positions are about the same. It has already been mentioned that the results of this calculation cannot be used to distinguish between the two structures in each of the two salts, Since Similar re- sults are obtained for either structures for the same or different associations. Therefore it is not necessary to give here all of the dipole fractions calculated. In Fig- ures 19 through 22, the graphs of the dipole fractions along the chlorine bonds are given for associations which have been previously considered from the Simple dipole model. The two numbers which are associated with each graph are the sums of the dipole fractions at the Al + A4 positions and the A2 + A3 positions. 102 55 e- 6 44 {5 - 3' 33' 4 1- 4| 55* -L 6' Figure 18. Labeling of Dipole Positions Along Chlorine and Oxygen Bonds 103 Dipole Fraction at Metal Ion Equals l Ass ' t' OCla 1on Al=+ A4 A2 + A3 3 ll 11' 3' 6' 104 (12,1,5,7) (12,3,5,7) - 25 -.41 —.23 -.39 I I' II 1' (9,1,5,7) (12,4,5,7) - 40 -.45 — 38 -.43 III fi ll _. (9,3,5,7) -.45 -.43 __ I . I - Figure 19. Dipole Fractions Along Chlorine Bond in P-Cobalt Chloride 105 (12,3,8,5) (12,3,5,8) -.36 -.32 -.34 -.30 (9,1,8,5) -.35 -.31 (9,3,8,5) . (9,3,5,8) - 36 l -.30 _ 34 -.32 l (9,1,5,8) -.30 -.28 - (12,1,8,5) , _ 1 (12,1,5,8) —.44 ' l l -.34 -.36 -.33 Figure 20. Dipole Fractions along Chlorine Bond in C-Cobalt Chloride 106 (ll,4,5,8) -.25 -.24 1 l . (ll,l,5,8) -.23 -.22 n l , (l,10,5,8) -.24 —.23 Figure 21. (11,3,5,8) -.25 -.26 (ll,2,5,8) -.24 -.22 Dipole Fractions Along Chlorine Bond in P-Nickel Chloride (11,2,7,5) -.41 —.39 I ' ' (11,4,7,5) -.47 —.45 I I | l J (12,4,7,5) -.30 -.33 107 I Figure 22. (11,3,7,5) -.41 -.39 I ' T (11,1,7,5) -.43 -.42 1 I Vi I Dipole Fractions Along Chlorine Bond in C-Nickel Chloride 108 In some reSpects, the sets of dipole fractions are quite Similar to each other. In each distribution the dipole fractions on the oxygen bonds are large and of opposite Sign compared to the fraction at the origin. For the P-cobalt, and C-nickel distributions, the fractions along the chlorine bonds are Smaller, and the fractions along the oxygen bonds are larger, than the correSponding fractions along the C- cobalt and P-nickel bonds. Nevertheless the fractions on the oxygen bonds are large in all cases. Since only eleven dipoles are used in each distribu- tion, one may use the correSponding fractions in each distri- bution to calculate the twelfth field component and compare the result with the experimental values. It is not unreason- able to expect that if the dipole fractions do have meaning, then they Should give a field at the twelfth position that is equal to the experimental field. It is curious that many sets of dipole fractions give a value to this field component that is not too different from the experimental value. In cobalt chloride the value of the field component is .494, and the P-cobalt association (9,1,5,7) gives a value which differs by 1.5 from this value. The next best agreement is given by the association (12,3,5,7), with a difference of 2.9. The C-cobalt associations (12,3,5,8) and (9,3,5,8) are off by 1.1 and 1.5 reSpectively. For nickel the field is equal to .9918, and the P-nickel association (ll,4,5,8) gives a component which is off by 3, while the C-nickel associations (ll,2,7,5) 109 give components that are off by 2 and 0 respectively. If these results are taken seriously, then it is concluded that the C-nickel structure is the best choice for the nickel chloride. The results for the cobalt salt seem to be incon- clusive. It is noted that the fractions for the P-cobalt association (9,1,5,7) and (12,3,5,7) are almost identical with the set of fractions for the C-nickel associations (ll,2,7,5) and (11,3,7,5). This observation and the results in Table 15, are taken to imply that the P—cobalt structure is the best fit for cobalt chloride. VIII. DISCUSSION AND CONCLUSIONS Several calculations have been employed here in attempts to determine the magnetic ordering in cobalt and nickel chloride, and in copper chloride. The results in the case of copper chloride are fairly conclusive, and the Rundle structure appears to be the correct one for copper chloride. For cobalt and nickel chloride the results of the calculations are not satisfactory, in the sense that none of the calculations gives a clear cut distinction between the two possible structures. The first part of solving the problem consists of limiting the number of possible structures to two. This is accomplished by discarding those space groups which do not conform to the experimental data. Since the experimental data indicate that either a 2' or m' symmetry element iS present, the structure cannot be triclinic. Also, the ob- served number of field lines limits the sublattice structures to two. It thus turns out that the salts have two sublattice monoclinic structures. The second part of the problem consists of distinguish- ing between the two possible structures for each salt. This turns out to be quite difficult for Several reasons. The un- known that is to be found here is the translational symmetry 110 111 element. But, in order to determine it, one must know the magnetic interactions which are present in the crystal, and also one must know the magnetization density. In order to make the calculations which are described here it is assumed initially that only the magnetic interaction need be consid- ered for a field calculation, and that the magnetization den- sity is Spherically symmetric. On the basis of these assump- tions the fields at the proton positions are calculated by means of a simple dipole model, and it is found that the calculated fields are in large disagreement with the experi- mental fields in both salts. It therefore becomes necessary to make a calculation which does not require one or both of the assumptions, and it was thought that this could be accom- plished by calculating the Fourier coefficients of the mag- netic field in each of the assumed structures. But here again the results of the calculation are indecisive. Finally a cal- culation is made in which it is assumed that the magnetization density may be approximated by a distribution of point dipoles. It turns out that the results of this calculation are also ambiguous. Distributions of dipoles are obtained by fitting the moments of the dipoles in a chosen distribution to the eXperimental fields. In general the resulting distributions for the two different structures of a given salt are very similar, and it is quite difficult to distinguish which set of distributions has more meaning than the other. The final choice is that the cobalt chloride magnetic structure is given 112 by the Space group PC 3;, and the nickel chloride structure is given by the Space group CC % At the onset of this work it was thought that the interpretation of the dipole array fractions would not be difficult, but it now appears that in order to completely understand the results of the dipole array calculations, further study of the method is required. One must understand the meaning of the negative fractions, not only in terms of the unsymmetric magnetization density but also in terms of a possible exchange field. The possibility of an exchange interaction between the proton and electron Spins is suggested by the considerable differences between the Simple dipole fields and the experimen— tal fields. If the magnetization is unsymmetrical, it cannot be assumed that the electron cloud is confined only within the radius of the nearest atoms. That is, in Spite of the fact that there are no observed exchange interaction shifts in the paramagnetic state, the possibility of exchange interaction in the antiferromagnetic state cannot be dismissed. Before undertaking a detailed study of the magnetiza- tion density and exchange forces, it would be nice to know the translational symmetry of the magnetic ordering. In nickel chloride the transition temperature is 6.20K, so that a neu- tron diffraction study of this salt is certainly possible. A comparison of the results of such a study, to the results given here, would be of interest. mNO‘UIb 10. 11. 12. 13. 14. 15. 16. 17. 18. REFERENCES Shulman, R. G. and Jaccarino, V., Phys. Rev. 108, 1219 (1957). Van der Lugt, W. and Poulis, N. J., Physica 27, 733(1961). Bleaney, B., Phys. Rev; 194, 1190(1956) Tinkham, M., Proc. Roy. Soc.(London) 5236, 535(1956). Van Kranendonk, J. and Bloom, M., Physica 22, 545(1956). Moriya, T., Prog. Theo. Phys.(Japan) 16, 33(1956). 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Rev. 112, 1544 (1958), 113 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 114 Spence, R. D. and Murty, C. R. K., Physica 21, 850(1961). Shulman, R. G. and Wyluda, B. J., J. Chem. Phys. 22) 1498(1961). Abrahams, S. C., J. Chem. Phys. 29” 56(1962). Peter, M. and Moriya, T., J. Appl. Phys. 22, 1304(1962). Spence, R. D. and Middents, P. W., To be published. Mizuno, J., J. Phys. Soc. Japan 22, 1412(1960), 29, 1574 (1961). Haseda, T. and Kanda, E., ibid, 22, 1051(1957). Haseda, T. and Date, M., ibid, 13, 175(1958). Haseda, T., ibid, 22, 483(1960). Kim, D., "Nuclear Resonance Studies of Antiferromagnetic Crystals in Zero-Field.” Dissertation, Michigan State University(l962). Middents, P. W., "A Nuclear Magnetic Resonance Study of CoBr ~6H20 in the Antiferromagnetic State.” Dissertation, Mich1gan State University(l963). E1 Saffar, Z. M., J. Phys. Soc. Japan 21, 1334(1962). Van Kranendonk, J. and P011, J. D., Results of their calculations are given in a paper by Poulis, N. J., Hardeman, G. E. G., Van der Lugt, W. and Hass, W. P. A., Physica 23” 280(1958). Stratton, J. A., Electromagnetic Theory. McGraw-Hill Book Company, Inc. New York and London(l94l). Harker, D., Z. Krist. 22, 136(1936). Itoh, J., Phys. Rev. §;, 852(1951). Date, M. and Nagata, K., To be published. 115 APPENDIX I a. Stereographic Projections of Axial Vector under Monoclinic Heesch Point Groups. 2! ‘ \ ' \ O V O /m 2'/m 2'/m' 2/m 1' b. Classification of Monoclinic Heesch Point Groups according to Stereographic Projection of Axialeector. 2‘ 21' m m' ml' 2/m 2'/m' 2/m' 2'/m 2/m 1' If and then 116 APPENDIX II Equivalence Between Spherically Symmetric Magnetization and Point Dipole M.= 0 outside of some region G =Itf(£) inside G A V :HKD— All +— ——1 Dr + @rDG (P rs'me Dcp VIM : If“? 33%;) = 4;!) case Outside of G, V24): 0, and (1) may be expanded in surface harmonics. But, in order for the potential to be continuous across the boundary of G,4>must vary as the first Legendre polynomial, or as the first moment. 117 APPENDIX III-A COMPUTER PROGRAM FOR CALCULATING MAGNETIC FIELD IN COBALT CHLORIDE 2 FORMAT (512) 3 FORMAT (3E14o8) 4 FORMAT (4E1408) 7 FORMAT (3F1004) 8 FORMAT (6H$J$L$K96Xo2HPXo12Xo2HPYv12Xc2HPZo12X92HCX012Xo2HCYo12X02 2HCZ) 9 FORMAT (31206E14o8) 12 FORMAT (212) I3 FORMAT (/) l4 FORMAT (6F1006) 15 FORMAT (28HCOBOLT$CHLORIDE$GEN$OBJ$NOo9) 16 FORMAT (60X96F1007) 17 FORMAT (3F7o4) PUNCH 15 A310034 8:7006 C36067 COH=084495 SOH=053484 S=A*COH 27 T=A*SOH TWO=200 READ 29 JMAXoKMAXoLMAXoNNgMM JJ=2*JMAX+1 271 KK=2*KMAX+I 28 LL=2*LMAX+1 JSUP=I+JMAX KSUP=I+KMAX LSUP=I+LMAX 29 CMAX=C*LMAX TMAX=T*JMAX 19 READ 120 NoM READ 179 V1. V2. V3 20 READ 179 CXoCYcCZ X=V1+CX Y=V2+CY Z=V3+CZ N=N+I 3O =-loO 32 XS=X-S/TWO 321 YS=Y~BITWO ZS=Z+T/TWO XB=X+JSUP*S 33 ZB=Z-TMAX-CMAx-T XSB=XS+JSUP*S 34 341 35 38 39 40 401 41 42 421 422 423 46 461 462 463 47 118 zsa=zs-TMAx-CMAx-T cx=o cv=o cz=o PX=0 pv=o 92:0 00 1 JPRIME=I¢JJ J=JPRIME-JSUP XB=XB~S xeso=xe*xe ZB=ZB+T+LL*C SGNzloO xse=xsa-s xsaso=xsa*xse ZSB=ZSB+T+LL*C oo 5 LPRIME=19LL L=LPRIME~LSUP SGN=SGN*U ze=ze~c zaso:ze*za zesoe=Two*zeso xu=xa*za xzu=zesoz-xaso BRIC=XBSQ+ZBSQ zse=zsa~c zsaso=zseizsa ZSBSOZ:TWO*ZSBSQ xus=xsa*zsa xzsu=zsesoa-xseso anlcs=xsaso+zsaso YB=Y+KSUP*B YSB=YS+KSUP*B BINX=O BINY=O BINZ=O oo 6 KPRIME=10KK K=KPRIME-KSUP va=va~e veso=ve*va YU=ZB*YB zu=xzu-vaso BRAC=8RIC+YBSQ GL=BRAC*BRAC*SORTF(BRAC) YSB=YSB~B vsaso=vsa*vse vus=zsa*vsa zus=xzsu-Ysaso BRACS=BRICS+YSBSQ GLS=BRACS*BRACS*SQRTF(BRACS) 466 467 468 501 502 503 504 61 67 63 64 10 II 119 BINX=BINX+XU/GL+XUS/GLS BINYzBINY+YU/GL+YU$/GLS BINZ=BINZ+ZU/GL+ZUS/GLS PX=PX+XU/GL-XUS/GLS PY=PY+YU/GL-YUS/GLS PZ=PZ+ZU/GL-ZUS/GLS CONTINUE CX=CX+SGN*BINX CY=CY+SGN*BINY CZ=CZ+SGN*BINZ CONTINUE CONTINUE RCX=300*CX RCY=3 00*CY RPX=3.0*PX RPY=3.0*PY RHOC=SQRTF(RCX*RCX+RCY*RCY) RHOP=SORTF(RPX*RPX+RPY*RPY1 HC=SORTF(RCX*RCX+RCY*RCY+CZ*CZ) HP=SQRTF(RPX*RPX+RPY*RPY+PZ*PZ) C0NV=0017453293 AZMTHCzATANF(RCY/RCX1/CONV AZMTHP=ATANF(RPY/RPXI/CONV ZNTHC=ATANF(RHOC/CZ)/CONV ZNTHP=ATANF(RHOP/PZ)/CONV PUNCH 2 JMAXOKMAXOLMAXONOM PUNCH 7o XoYoZ PUNCH 14c HCoAZMTHCoZNTHCcRCXoRCYoCZ PUNCH 14o HPcAZMTHPcZNTHPoRPXqRPYoPZ PUNCH 16o RCXoRCYoCZoRPXcRPYoPZ 1F (M-N) 10.10.20 CONTINUE NN=NN+1 IF (MM-NN) 11011019 CONTINUE PUNCH 13 STOP END END 120 APPENDIX III-B COMPUTER PROGRAM FOR CALCULATING MAGNETIC FIELD IN NICKEL CHLORID 2 FORMAT (5121 3 FORMAT (3E14081 4 FORMAT (4514081 7 FORMAT (3F10041 8 FORMAT (6H$J$L$KQ6X02HPX012X92HPY912X92HPZQ12X02HCX012X92HCY912X. 2HCZ) 9 FORMAT (31296E14981 12 FORMAT (212) 13 FORMAT (I) 14 FORMAT (6F1006) 15 FORMAT (28HNICKEL$CHLORIDE$GEN$OBJ$NO¢81 16 FORMAT (60X. 6FIOO7) 17 FORMAT (3F704) PUNCH 15 A310023 21 837.05 C=6057 22 COH=084650 23 SOH=053238 S=A*COH 27 T=A*SOH TWO=200 READ 20 JMAXO KMAXQ LMAXo NNQMM JJ=2*JMAX+I 271 KK=2*KMAX+1 28 LL=2*LMAX+I JSUP=1+JMAX KSUP=1+KMAX LSUP=1+LMAX 29 CMAX=C*LMAX TMAXiTidMAX 19 READ 120 NOM READ 170 V19 V29 V3 20 READ 17o CX! CY. CZ X V1+CX Y V2+CY Z V3+CZ N=N+1 3O U:-IOO 32 XS=X-S/TWO 321 Y$=Y-B/TWO Z$=Z+T/TWO XB=X+JSUP*S 33 ZB=Z“TMAX-CMAX~T XSB=XS+JSUP*S 34 341 35 36 37 38 39 40 401 41 42 421 422 423 424 46 461 462 463 121 ZSB=ZS~TMAX~CMAX~T CX=0 CY=O CZ=0 PX=0 PY=O PZ=O DO 1 JPRIME=10JJ J=JPRIME-JSUP XB=XB-S XBSO=XB*XB XBSOZ=TWO*XBSQ ZB=ZB+T+LL*C XSB=XSB-S XSBSO=XSB*XSB XSBSQZ=TWO*XSBSO ZSB=ZSB+T+LL*C SGN=1 00 DO 5 LPRIME=1.LL L=LPRIME-LSUP SGN=SGN*U ZB=ZB-C ZBSO=ZB*ZB ZU=XB*ZB X2U=XBSOZ~ZBSO BRIC=XBSO+ZBSQ ZSB=ZSB-C ZSBSO=ZSB*ZSB ZUS=XSB*ZSB XZSU=XSBSO2-ZSBSO BRICS=XSBSQ+ZSBSO YB=Y+KSUP*B YSB=YS+KSUP*B BINX=0 BINY=O BINZ=0 DO 6 KPRIME=19KK K=KPRIME-KSUP YB=YB-B YBSO=YB*YB YU=XB*Y8 XU=XZU-YBSO BRAC=BRIC+YBSQ GL=BRAC*BRAC*SORTF(BRAC) YSB=YSB-B YSBSO=YSB*YSB YUS=XSB*YSB XUS=XZSU-YSB$O BRACS=BRICS+Y$BSO GL$=BRACS*BRACS*SQRTF(BRACS) 466 467 468 501 502 503 504 61 67 63 64 10 11 122 BINX:BINX+XU/GL+XUS/GLS BINY=BINY+YU/GL+YUS/GLS BINZ=BINZ+ZU/GL+ZUS/GLS PX=PX+XU/GL-XUS/GLS PY=PY+YU/GL-YUS/GLS PZ=PZ+ZU/GL~ZUS/GLS CX=CX+SGN*BINX CY=CY+SGN*BINY CZ=CZ+SGN*BINZ CONTINUE CONTINUE GCY=3 00*CY GCZ=300*CZ GPY=3.0*PY GPZ=3.0*PZ RHOC=SORTF(CX*CX+GCY*GCY1 RHOP=SORTF(PX*PX+GPY*GPY1 HC=SQRTF(CX*CX+GCY*GCY+GCZ*GCZ) HP=SQRTF(PX*PX+GPY*GPY+GPZ*GPZ) CONV=0017453293 AZMTHCIATANF(GCY/CX1/CONV AZMTHP=ATANF(GPY/PX1/CONV ZNTHC=ATANF(RHOC/GCZ)/CONV ZNTHP=ATANF(RHOP/GPZ)/CDNV PUNCH 2 JMAX9KMAX9LMAX9N9M PUNCH 79 X9Y9Z PUNCH 149 HC9AZMTHC9ZNTHC9CX9GCY9GCZ PUNCH 149 HP9AZMTHP9ZNTHP9PX9GPY9GPZ PUNCH 169 CX9GCY9GCZ9PX9GPY9GPZ IF (M-N) 10910920 CONTINUE NN=NN+1 IF (MM-NN) 11911919 CONTINUE PUNCH 13 STOP END END 123 APPENDIX 1V COMPUTER PROGRAM FOR CALCULATING DIPOLE FRACTIONS. FORMAT (I2) FORMAT (6HMATRIX) FORMAT (7HINVERSE) FORMAT (6HVECTOR) FORMAT (I39 2F2094) FORMAT (3F1096) FORMAT (11F10o6) FORMAT (11F1093) \OGQO‘UlbUNH FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT (212) (30X93F10o6) (6F1006) (F1096) (9F1093) (3X92F2004) (I) DIMENSION A(12912)9 L(12)9 M(12)9 V(12)9 SAB(12)9 XPH(12) SUBROUTINE INVER NONLOCAL A9L9M9N SEARCH FOR LARGEST ELEMENT DO 80 K819N L(K)=K M(K)=K BIGA=A(K9K) DO 20 I=K9N DO 20 J=K9N IF (ABSF(BIGA)-ABSF(A(IoJ)))10920920 BIGA=A(I9J) L(K)=I M(K)=J CONTINUE INTERCHANGE ROWS J=L(K) 1F (L(K)-K) 35935925 DO 30 I=19N HOLD=-A(K9I) A(K9I)8A(J9I) A(J9I)8HOLD INTERCHANGE COLUMNS 1=M(K) IF (H(K1-K) 45945937 DO 40 J=19N HOLD=-A(J9K) A(JoK)=A(J9I) A(J9I)8HOLD DO 55 I=I9N IF (I-K) 50955950 A(I9K)=A(I9K)/(-A(K9K)) 124 55 CONTINUE REDUCE MATRIX DO 65 1:19N DO 65 J=19N 56 IF (I-K) 57965957 57 IF (J-K) 60965960 60 A(I9J)=A(I9K)*A(K9J)+A(I9J) 65 CONTINUE DO 75 J=19N 68 IF (J-K) 70975970 70 A(K9J)=A(K9J)/A(K9K) 75 CONTINUE A(K9K)=190/A(K9K) 80 CONTINUE FINAL ROW AND COLUMN INTERCHANGE KBN 100 K=(K-1) IF (K) 15091509103 103 I=L(K) IF (I-K) 12091209105 105 00 110 J=19N HOLD=A(J9K) A(J9K)=-A(J9I) 110 A(J9I)=HOLD 120 J=M(K) IF (J-K) 10091009125 125 DO 130 I=19N HOLD=A(K91) A(K9I)=-A(J9I) 130 A(J9I)=HOLD GO TO 100 150 CONTINUE RETURN END 48 READ 19 N READ 99 NN9MM 22 READ 99 IN9JM LIL=3*NN J1M=1+LIL JOHN =2+LIL JUDY =3+LIL 23 READ 69 XVAR9YVAR9ZVAR IN=IN+1 A(JIM9IN)=XVAR A(JOHN9IN)=YVAR A(JUDY9IN)=ZVAR PUNCH 349 XVAR9YVAR9ZVAR9A(JIM9IN)9A(JOHN9IN)9A(JUDY9IN) 1F (JM-IN) 27927923 27 CONTINUE NN=NN+1 IF (MM-NN) 31931922 31 CONTINUE PUNCH 2 16 PUNCH 79 ( ( A(I9J)9 J=19N)9I=I9N) 46 161 17 14 49 18 I2 11 21 41 125 PUNCH 2 N=N-2 DO 46 1=I9N A(29I) = A(39I) A(39I) = A(49I) DO 46 J=49N JJ=J+2 A(J9I) = A(JJ9I) CONTINUE PUNCH 889 ( ( A(I9J)9 J=19N )9 CALL INVER CONTINUE PUNCH 3 PUNCH 889 ( ( A(I9J)9 J=19N )9 READ 19 KAL KAL=4*KAL DO 41 K=19KAL READ 69 (XPH(J)9 J=I912) XPH(2)=XPH(3) XPH(3)=XPH(4) DO 49 J=49N JJ=J+2 XPH(J)=XPH(JJ) CONTINUE PUNCH 19 K PUNCH 4 PUNCH 889 ( XPH(J)9 J=19N ) PUNCH 4 SUM1=0 SUM2=O DO 11 1=I9N V(I)=O DO 12 J=19N V(I)=V(I) +A(19J)*XPH(J) CONTINUE SAB(I) 3 ABSF( V(I) ) SUMI = SUMI + V(I) SUM2 = SUM2 + SAB(I) PUNCH 59 I9 V(I)9 SAB(I) CONTINUE PUNCH 999 SUMIO SUM2 DO 21 I=19N V(I) = V(I)/SUM1 SAB(1) 3 SAB(I)/SUM2 PUNCH 59 I9V(I)9 SAB(1) CONTINUE CONTINUE PUNCH 19 STOP END END I=19N 1=19N ) ) MICHIGAN STATE UNIV. LIBRARI: IIHIHWIW [III HIIIIIIVIHIHIIHHIIIIHHII WIN IN! I II 31293017640388