ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Report No. 6 MAGNETICALLY SENSITIVE ELECTRICAL RESISTOR MATERIAL March 1, 1955 to December 31, 1955 E. Katz L. P. Kao Project 2136 DEPARTMENT OF THE ARMY LABORATORY PROCUREMENT OFFICE U. S. SIGNAL CORPS SUPPLY AGENCY CONTRACT DA-36-039-SC-52601 FORT MONMOUTH, NEW JERSEY February 1956

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv OBJECTIVE v A. INTRODUCTION vi B. THEORETICAL PART 1 1. Introduction 1 2. Definitions and General Relations 5 3. The Effects of Crystal Symmetry 10 4. The Number of Nonvanishing Independent Brackets 29 5. Explicit Forms of the Galvanomagnetic Tensor up to B2 33 6. Discussion 41 APPENDIX I Proof of the Theorem Concerning an N-Fold Rotation Axis Along k3 A-1 APPENDIX II Completeness of the Equations (13) B-l APPENDIX III Proof of Table VIII - Zero Brackets for a Threefold Axis of Symmetry C-l ii

LIST OF ILLUSTRATIONS Figure Page 1. Orientation of vectors J, F, B and probes ab in an anisotropic single crystal. 1 2. Counting diagram. 31 Table I. Generating the Eleven Point Groups 11 II. Table of s and ~ Values 12 III. The Values of e(s,G,z) 14 IV. Procedure for Tabulating Bracket Relations 20 V. Bracket Relations for C2h and D2h 21 VI. Bracket Relations for C4h and D4h 22 VII. Bracket Relations for Th and Oh 24 VIII. Zero Brackets. for C3ij Dsi, Ceh, D6h. 25 IX. Zero Brackets for D3i, C6h, D6h 25 X. Relations for Nonvanishing Brackets for C3i, D3i, C6h, D6h 26 XI. The Number of Nonvanishing Independent Brackets an2 + bn + c 34 XII. The Explicit Form of Mo 36 a6 XIII. The Explicit Form of P 36 XIV. The Explicit Form of Q1 37 XV. The Explicit Form of R2 for C2h, D2h 38 XVI. The Explicit Form of R2 for C3i, C4h, C6h, Dsi, D4hy Dh 39 XVII. The Explicit Form of R2 for Th and Oh 40 XVIII. Notations of Various Authors 41 iii

ABSTRACT The isothermal galvanomagnetic tensor components associated with the Hall -effect and the magneto resistance effect are analyzed for arbitrary orientation of the crystal axes in the sample, arbitrary orientation of the magnetic field B, and arbitrary crystal symmetry. The conductivity components in suitable coordinates are expanded in powers of the components of B. The coefficients are the galvanomagnetic material constants, called "brackets"; they are defined in Equation 10. By means of Onsager's relations it is shown that the magneto resistance effect is always even in B, whereas the Hall effect is odd in B only with special geometry (Section 2). Further, the Hall effect is even or zero for some geometric and crystallographic conditions. The effects of the crystal symmetry on the brackets are covered by the theorem of Section 3. The resultant dependencies between the brackets are tabulated completely for all powers of B for the crystal classes other than the trigonal and hexagonal ones. For the latter the bracket relations are given up to the sixth power of B. Formulas for the number of independent brackets up to any power of B are given for all crystal symmetries in Section 4. Explicit forms of the galvanomagnetic tensor components in terms of the brackets for all arbitrary conditions, up to B2, are tabulated in Section 5. The significance of the results obtained is pointed out in Section 6. iv

OBJECTIVE This project aims at developing the understanding of the magneto resistance effect (change of electrical resistivity in a magnetic field) by theoretical and experimental research, with the ultimate aim of developing materials with more favorable magneto resistance properties than are available at present. v

A. INTRODUCTION Owing to administrative circumstances, the present report follows the previous one at an interval of approximately ten months. The work done during this period requires a relatively large amount of reporting. In order to distribute the burden of writing over a reasonable amount of time, it was found to be expedient to report presently on the progress made in this period with the theoretical work. The next report will consist of a similar comprehensive report on the experimental progress without a theoretical part. Regarding the theoretical work reported here, the following remarks are appropriate. 1. Many ofthe problems which remained to be brought to conclusion in the phenomenological theory as sketched in previous reports have since been finished and are reported here. 2. Several additional viewpoints, simplifications, and shorter proofs have been found and are presented. 3. In order to make the report a self-consistent unit, a number of items which have been previously reported are included again. This is true of the main theorem. The bracket relations, which were formerly given up to n < 4, are now available for n < 6 and for many crystals for any value of n. 4. The authors expect to publish the same material in practically the same form very soon. 5. The theoretical developments presented here are, we feel, a good step forward of whatever theoretical material is available in this area of research at present and will be of considerable assistance in the experimental work of our own group and others. 6 Plans for theoretical work in the immediate future are primarily directed towards further development of the electron theoretical part of the theory (see the diagram, Figure 1, of the previous report). vi

B. THEORETICAL PART 1. INTRODUCTION * Consider a volume element of an anisotropic isothermal homogeneous single crystal of arbitrary shape, placed in a homogeneous magnetic field B. In the crystal is maintained a constant current density J by means of a suitable electric field F (see Figure 1). /^~- ^~. O at Figure 1. Orientation of vectors J, F, B and probes ab in an anisotropic single crystal. Evidently F will be a vector function of J and B, F F (JB)** (1) The dependence of F on B represents the galvanomagnetic effect. The crystal symmetry and other physical laws will, in general, restrict the possible forms of (1). It is the purpose of the present paper to find the proper description of F for all possible crystal symmetries, all orientations and magnitudes of J and B with respect to the symmetry axes of the crystal, under the restriction that Ohm's law is valid for fixed B. *Isothermal conditions are assumed throughout this paper without further explicit statement. Throughout this paper we restrict ourselves to nonferromagnetic substances. 1

The vector F can be determined experimentally for any given J and B by measuring the components F1 213 of F in three independent directions. In a direction d one can measure Fd by measuring the potential difference Vab between two probes a and b, without drawing current, and dividing by the distance ab. If d is taken along J, the resulting dependence of F1[ (J,B) is called the magneto resistance effect; if d is normal to J, then Fi (J,B) is called a Hall effect. Both are special cases of the galvanomagnetic effect. Other special cases, such as the Corbino effect, imply a geometry and boundary conditions which are again different. Historically, Lord Kelvin first discovered the magneto resistance effect for Fe in 1856 and also predicted the Hall effect in 1851. After many tries by various workers Hall1 discovered it in 1879. The first empirical for1 mula connecting the two effects was proposed by Beattie in 1896. The name "galvanomagnetic effect" appeared in the literature, meaning the Hall, the magneto resistance effect, as well as some other effects such as the Corbino effect. The dependence of the Hall effect on the magnetic field and on the temperature was studied by many investigators. In 1883 Righil studied the influence of the crystal orientation of the Hall effect. Extensive summaries were 12 5 given by Campbell in 1923 and by Meissner2 in 1935. Briefly, the findings of all this work are that the Hall effect depends on the crystal orientation, and that it is not adequately described by a constant Hall coefficient. The Hall effect is not always an odd function of the magnetic field, contrary to a, suggestion by Casimir in 1945. 1. L. L. Campbell, Galvanomagnetic and Thermomagnetic Effects, Longmans, Green and Co., Inc., 1923. 2. W. Meissner, Handbuch der Experimentalphysik, vol. XI, pt. 2, Leipzig, 1935. 3. See, for example, J. K. Logan and Jo A. Marcus, Phys. Rev., 88:1234 (1952); R. K. Willardson, T. C. Harman, and A. Co Beer, ibid.,: 1512 (1954). 4. H. B. G. Casimir, Rev. Mod. Physo, 17:345 (1945); H. B. G. Casimir and A. N. Gerritsen, Physica~-':1107 (1941)o 2

Numerous authors 'extended Lord Kelvin's work on magneto resistance to nonferromagnetic materials. Grunmach and Wiedert published in 1906-07 the first extensive study for various elements at room temperature. In 1928-30 Kapitza5 went to lower temperatures and higher magnetic fields. In 1897 Van Everdingen1 had discovered the influence of the crystal orientation on the magneto resistance. The effect was studied further by Schubnikow and de Haas, Stierstadt,7 Justi8 and co-workers, Blom, and others. Schubnikow and de Haas, Stierstadt, and, more systematically, Blom, tried to analyze the angular dependence of the magneto resistance on the orientation of the magnetic field relative to the sample by a Fourier analysis. Briefly, findings11 of all this work are that the magneto resistance depends markedly on the crystal orientation, especially at low temperatures; that for low magnetic fields (say less than 1 kilogauss) the magneto resistance is proportional to the square of the field, whereas at high fields the relation is a more complicated function of the field. According to most results, this function is even. In 1905 Voigti2 laid the foundation for an appropriate description of the anisotropy of the Hall and magneto resistance 5. P. Kapitza, Proc. Roy. Soc. London(A), 123:292 (1929). 6. L. Schubnikow and Wo J. de Haas, Comm. Leiden Nr. 207, a,cd (1930); Comm. Leiden Nr. 210, ab (1930). 7. 0. Stierstadt, Z. f. Phys., 80:636 (1933); 8:5310, 697 (1933)8. E. Justi, Leitfahigkeit und Leitungsmechanismus fester Stoffe, Chapter I, Vandenhoeck and Ruprecht, Gottingen, 1948. 9. J. W. Blom, Magnetoresistance for Crystals of Gallium, The Hague,- Martinus Nijhoff, 1950. 10. R. Schulze, Phys. Z., 42:297 (1941); Y. Tanabe, Tohoku Univ. Res. Inst. Sci. Rep. 1:275 (19497; E. Grueneisen and H. Adenstedt, Ann. Phys., 31:714 (1937)7 B. G. Lazarev, N. M. Nakhimovich, and E. A. Parfenova, C. R. Moskau (N.S.), 24:855 (1939). 11. For recent works see, for example, G. L. Pearson and H. Suhl, Phys. Rev. 83:768 (1951); G. L. Pearson and C. Herring, Physica, 20:975 (1954). 12. W. Voigt, Lehrbuch der Kristallphysik, Teubner, Leipzig and Berlin, 1928.

effects. Further contributions to the phenomenological theory were made by 15 4 1 4 15 Kohler, Casimir, Seitz, Juretschke, and other workers. The present paper attempts to give an explicit development and broadening of the general phenomenological theory, which is required for the interpretation of isothermal galvanomagnetic measurements in terms of true isothermal galvanomagnetic material constants. In Section 2 the proper galvanomagnetic constants are defined, the dependence of F on the orientation of J and B is discussed, and some general relations are established. In Section 3 a general method is developed by means of which the effects of crystal symmetry are properly taken into account. In Section 4 formulas are given for the number of independent galvanomagnetic constants for the various crystal classes. In Section 5 explicit forms are given for F up to terms quadratic in the components of B for the various crystal symmetries, while the corresponding forms for higher powers can be elaborated from there on without essential difficulties. The significance of the results obtained is pointed out in Section 6. The objective and plan of attack of the present work is as follows. The ultimate aim of the theory of the galvanomagnetic effects is to describe the function (1) completely in terms of the electronic properties of the material concerned. This task can be divided into two parts. In the first or phenomenological part the function (1) is described in terms of a number-finite or infinite-of appropriate constants that are characteristic for the material. In the second or electron-theoretical part these constants are interpreted in terms of electronic properties. The present paper is only concerned with the phenomenological part. 13. M. Kohler, Ann. Physik, 20:878,891 (1934); 95:365 (1935). 14. F. Seitz, Phys. Rev., 79:572 (1950). 15. H. J. Juretschke, Acta Cryst, 8:716 (1955). 16. The essentials of the present paper were first given in a contract report, with the UoS. S Signal Corps Engineering Laboratories, of November, 1954, by E. Katz, entitled: Magnetically Sensitive Electrical Resistor Material. 4

Here again the work can be divided into several steps. First, proper coordinates will be defined. Second, the galvanomagnetic function (1) must be put in the simplest form consistent with general principles and with arbitrary orientation of B, arbitrary crystal symmetry, etc. This form must then be expressed in terms of functions that are characteristic for the material. Third, these functions must be described by a number-finite or infinite-of constants which are amenable to discussion by the electron theory. Fourth, the restrictions imposed on these constants by the crystal symmetry are to be established, and the number of independent nonvanishing constants should be found. Fifth, the information so obtained must be synthesized; the galvanomagnetic effect, measured under the most general conditions, is then expressed in terms of material constants ready for interpretation by the electron theory. 2. DEFINITIONS AND GENERAL RELATIONS a. Coordinate Systems.-Two sets of orthogonal coordinate systems will be used. (1) The symmetry coordinates ki (i = 1,2,3). These are adapted to the crystal symmetry* as follows: For the groups C1, Si the directions of the coordinate axes are arbitrary. For the groups S2, S4.4 S6, C2, C3, C4, C6, C2h, C3h, C4hy C6h,: the k3 axis is taken along the axis of rotation; * the other axes have one degree of freedom. For the groups T, Th, Td, the coordinate axes are taken along the twofold rotation axes. For all other classes, k3 is taken along the rotation axis of highest order, kl along a rotation axis* normal to k3, and k2 accordOnly the macroscopic symmetry of the crystal, i.e., to which of the 32 crystallographic point groups it belongs, need here be considered. The axis of an improper rotation is understood here as the normal to the corresponding reflection plane. 5

ingly. Vector or tensor components with respect to these symmetry coordinates will carry Latin subscripts. (2) -The laboratory coordinates fX (a = 1,2,3) with x1 along the current density J, x2 in the plane of J and d, and x3 accordingly. In the case of magneto resistance, d is along J, allowing one degree of freedom for x2 and x3 in the plane normal to x1. Vector and tensor components with respect to the laboratory coordinates will carry Greek superscripts. No confusion between superscripts and exponents should arise in practice. The definition of the laboratory system implies =J -_ J 0 _ 0. (2) Denoting the direction cosines between the two coordinate systems by 2i and using the summation convention for repeated indices, we have;f = Jfli l iCT = ~() FP = FJQ b. Assumption I. Ohm's Law. -We assume Ohm's law to be valid for any constant applied magnetic field B, i.e. the current density J is a homogeneous linear vector function of F. Thus, in symmetry coordinates: Ji = aij (B) Fj Fj = Pji (B) Ji, where the conductivity tensor components aij and the resistivity components pji are both functions of B and are related by Pji = Aij/A ~ (5) Here A is the determinant of the vij and Aui is the cofactor of raij in A. The 6

functions aij(B) and pji(B) are characteristic of the material at any given temperature and independent of the geometry of galvanomagnetic measurements. The effects of crystal symmetry are to place restrictions on these functions. However, the direction of current flow J has, in general, no particularly simple relation to the symmetry coordinates, and the results of measurements are most directly expressed in terms of laboratory coordinates. Ohm's law, restated in laboratory coordinates, is Fa = p (B)J' where(6) p (B) = Pji(B)2li = A (B),'/(B) For a = 1 these equations describe the magneto resistance effect; for a = 2 or 3 they represent the Hall effect. The latter form expresses these galvanomagnetic effects in terms of the conductivity components in symmetry coordinates..c. Assumption II. Onsager1s Relations. -The validity of Onsager's relations is assumed: Pji(B) = Pij(-B) and, consequently, (7) j..(B) =.() -ui(-) a~ij(-B) as well as p~P(B) = pa(-B). (8) d. The Parity of the Magneto Resistance Effect.-Equation (8) states that 11 / v i1 p (B) = p(-B), (9) which proves the theorem that the magneto resistance effect is even in B. In the literature 17 there has been some controversy about the evenness of pl", but the above argument shows that under the very broad assumptions stated, pll 17. References 11, 4; D. Shoenberg, Proc. Camb. Phil. Soc., 31:271 (1935); J. Meixner, Ann. Phys., 36:105 (1939); ibid., 40:-105 (194T); B. Donovan and G. K. T. Conn, Phil. Mag., 41:770 (1950). - 7

must be an even function in B without exception. e. Parity of the Hall Effect. —The Hall effect as defined by (6) with a f 1 implies that the Hall electrodes are normal to the current. We shall adhere to this definition, though some experimenters prefer to define the Hall effect as measured with the Hall electrodes on an equipotential when B = 0. In general the Hall effect is neither an odd nor an even function of B. This is true for either definition. However, in a number of special configurations the crystal symmetry may impose a special parity on the Hall effect. The complete list of such configurations is as follows. Consider the crystallographic point group, obtained from that of the crystal by augmenting it with an inversion center. The physical significance of this augmented group is explained in Section 3. Then one can easily prove with respect to this augmented group: (1) If B lies along a 3-, 4-, or 6-fold axis and either J or d is normal to B, then the Hall effect is odd. (2) If B is normal to a 2-, 4-, or 6-fold axis and either J or d is along that axis, then the Hall effect is odd. (3) If B lies along any 2-, 3-, 4-, or 6-fold rotation axis and is coplanar with J and d, then the Hall effect is even. (4) If B, J, and d are normal to the same 2-, 4-, or 6-fold axis, then the Hall effect is even. (5) If B and either J or d lie along any 2-, 3-, 4-, or 6-fold rotation axis, then the Hall effect vanishes. There are no other cases in which the Hall effect is purely even, odd, or zero as a function of B. The "new" galvanomagnetic effect reported by Goldberg and Davis,l8 for example, is a case illustrating points (4) and (5). In their Fig18. C. Goldberg and R. E. Davis, Phys. Rev., 94:1121 (1955). 8

ure 1 the slight discrepancy between the axis direction and the direction of zero Hall effect must be due to an experimental error of imperfect alignment. f. Assumption III. Power Series Expansion of ij(B).-Most galvanomagnetic measurements suggest that aij(B) can be expanded as a series in powers of the components Bk. One of many typical examples is reproduced by A. H. 19 Wilson9 from work by Justi and Scheffers on gold. If a Fourier analysis of polar diagrams of this sort involves significant terms with arguments of the sines or cosines up to n0, then, it is easily shown, significant contributions to the conductivity components aij arise from terms proportional to the nth power of B and vice versa. There is an observed limitation to the appropriateness of a powerseries expansion for galvanomagnetic effects. Experiments20 have shown that the Hall voltage and the magneto resistance at low temperatures contain oscillating terms which presumably are connected with the van Alphen - de Haas effect and are proportional to B sin Bo/B. Such terms do not possess a derivative with respect to B at B = 0 and hence cannot be expanded in powers of B. Consequently, the development presented here does not apply to that part of the galvanomagnetic effects which arisesfrom terms of such a nature. As the third assumption, we write mo n m aij(B) = [m-ppn-m]ij B B2 B3 (10) i Z~ m-p p n-m n=O m=O p=O or, introducing the direction cosines yi of B with respect to the symmetry coordinates, 19. A. H. Wilson, The Theory of Metals, Univ. Press, Cambridge, 1953, p 318. 20. See, for example, P. B. Alers and R. T. Webber, Phys. Rev., 91:1060 (1953); T. G. Berlincourt and J. K. Logan, Phys. Rev., 93:348 (1954); T. G. Berlincourt and M. C. Steele, Phys. Rev., 98:9567(1955-; M. Co Steele, Phys. Rev., 99:1751 (1955). 9

00 n m aij(B) = ' Bn r [m-p,p,n-m]ij yim- p Y2. (lOa) n=O m=O p=O The coefficients in this expansion are designated by the bracket symbols and are independent of B. They are the true phenomenological material constants characterizing the galvanomagnetic behavior of any particular material. They are sums of components of tensors of rank 2n + 2 since the axial vector B is an antisymmetric tensor of rank two. Onsagervs relations imply [m-p,p,n-m]i = (_)n [m-p,p,n-m]ji. (11) Consequently, it is sufficient to consider ij values of 11, 22, 33, 23, 31, 12, only. This will always be done unless stated otherwise. Another consequence of (11) is that all brackets with n = odd and i = j vanish. Denoting by C an arbitrary odd number: [m-p,p,W-m]ii = 0. (lla) The restrictions imposed on the brackets by the crystal symmetry will be described in the next section. The fact that (6) is simpler in terms of Pij then in terms of aJi would suggest a power-series expansion of the former. The latter was chosen since the brackets so obtained permit a simpler electron theoretical interpretation. However, the contents of all that follows are applicable without any modification to p-brackets and a-brackets alike. 3. THE EFFECTS OF CRYSTAL SYMMETRY a. Only Eleven Point Groups Need Analysis.-Tensor components that are material constants must be invariant under the operations of the crystallo*In order to set the tensor character in evidence, the brackets will sometimes be denoted by [(23)m-P,(31)P,(l2)n-m]iJ. The quantities (23), (31), (12) will be referred to as the pairs of inner indices and ij as the outer indices. 10

graphic point group of the crystal considered. If the tensor components are of even rank, they transform identically into themselves under inversion. Consequently,. all tensor components of even rank that are material constants must be invariant under the operations of the point group that is obtained by augmenting the point group of the crystal considered by an inversion center. This is obviously also true for the brackets. Any point group augmented by an inversion center becomes one of the eleven well-known crystallographic point groups which possess such a center. Thus, it suffices to analyze these eleven point groups. They can all be generated by at most two rotations in addition to the inversion center. We shall generate the eleven point groups by means of the elements shown in Table I. In the second row are listed the twenty-one point groups without inversion center which go over into those of the first row by the addition of an inversion center. TABLE I GENERATING TEE ELEVEN POINT GROUPS Point Group I S2 C2h Csi C4h Ceh D2h D3i D4h D6h Th Oh Equivalent CI C2 C3 C4 C3h CaV C3v D2d Dsh T Td Point Cs S4 C6 D] D3 C4v C6v 0 Groups D4 D6 Generating* Elements axis.along k3 2 3 4 2 3 4 6 2 4 axis along kj. -.- - 2 2 2 2 - - axis along [ 111] -.- '.. 3 3 If an axis is taken as a generating element, its multiplicity N is listed at the appropriate place. The inversion center which is a common generating element of all groups is not listed. 11

Under a general rotation each bracket is transformed into a linear combination of other brackets. If the rotation be a covering operation, and thus requires invariance of the bracket, then certain relations must hold between the brackets. Under inversion each bracket is transformed identically into itself. Hence, no relations between brackets can be derived from the requirement of invariance under inversion. b. A Theorem Concerning the Effect of an N-Fold Rotation Axis Along k3.-The effects of an N-fold rotation axis along k3 are covered by a theorem which states that certain linear equations must hold between the brackets. In order to express these combinations concisely, some notations are introduced. Let s be the number of non-threes among a given ij and let G be the number of twos minus the number of ones in ij. The numbers s,G define uniquely one of the six independent pairs of indices ij, and vice versa. Table II gives the relation explicitly for further reference. We write [[m-p,p,n-m] [m-p,p,n-m(s) (12) TABLE II TABLE OF s AND G VALUES ij s G 33 0 0 31 1 -1 23 1 +1 11 2 -2 12 2 0 22 2 +2 Let z denote a nonnegative integer < s and w a nonnegative integer < m. Each of the linear equations referred to by the theorem will be labeled by five integers n, m, s, z, w. All brackets occurring in one equation have the same nm,s, but may differ in p7,. The parameters z,w serve to label the various equations with the same n,m,s, involving the same brackets with different coefficients 12 12

Theorem For an N-fold rotation axis along k3, the brackets satisfy the equations m j Z g(m,pw) 4(sGz) [m-pJpn-m](s, ) = (13) G p=O provided the inequality h = m+s - 2(w+z) - kN (k=0,l,+2...) (14) holds; in other words, h is not a multiple of N. The summation over g is meant to include all ij combimations with constant s. The coefficient g(m,p,w) is defined in terms of binomial coefficients by g(mp,w) = ip ' (-) (m-) ) (15) q=0 The factor e(s~G,z) is given in Table III for all values for which it is defined. The proof of this basic theorem is given in Appendix I. In Appendix II it is shown that the only solution for the complete set (13) for given n,m,s, is that all brackets involved vanish. A consequence is that the equations (13) with the condition (14) represent a.complete description of the symmetry properties of the brackets. For finding the relations between brackets of given nm,s, one will first list all brackets of this set according to their p and G values, next establish their coefficients g~ for all possible values of the parameters wz, and finally write down one equation of the type (15) for each set of w,z values compatible with (14). Shortcuts to this procedure will be explained after some corollaries of the theorem have been proved. c. Some Consequences of the Theorem -In formulating the fundamental theorem a rotation axis was taken along k3. It is simple to apply the theorem 15

TABLE III THE VALUES OF e(s,G,z) n = even n =odd s G ij z 0 1 2 0 1 2 0 0 33 1 - 1 -1 31 1 1 - 1 1 - 1 1 23 i -i - -i i 2 -2 11 1 1 1 2 0 12 21 0 -2i 0 2i 0 2 2 22 -1 1 -1 - to a rotation axis along ki or k2 by permutation of both inner and outer indices. The effect of a threefold axis along the [ill] direction can be taken into account by requiring invariance for the brackets under cyclic permutation of the indices 1,2,5 both in and outside any bracket. Thus the effect of symmetry for the eleven point groups is completely described by the theorem with these generalizations. However, in a number of cases the application of the theorem is greatly simplified by means of some corollaries. Corollary I For N = even (2,4,6) about k3, all brackets for which the index 3 occurs an odd number of times (inside plus outside) are zero. Proof In any particular bracket, the index 3 occurs 2-s+m times. Thus it must be shown that all brackets with (m+s) odd vanish. If (m+s) is odd then h cannot be an integral multiple of the even number N. Thus all equations provided by the general theorem are valid. According to the theorem, given in 14

Appendix II, all brackets concerned must now vanish. q.e.d. In preparation of Corollary II let two brackets be called "adjoint" with respect to k3, if they can be obtained from one another by interchanging the indices 1 and 2*), both inside and outside, and writing the resulting pairs of indices in the conventional order. For example, the brackets [m-p,p,n-m]23 and [pm-pn-m]31 are adjoint. Indeed the first bracket can be written as [(23)mP,(3l)P (12)nm]2. Upon interchanging 1 and 2 this becomes [ (32)P,(l5)m-P, (21)n-m]l. All pairs must now be interchanged in order to appear in the conventional order. The n inner pairs each give a minus sign. The outer pair gives (_)n according to Onsager's relation (11). The result is always a plus sign. The resulting bracket is [(235),(3l)mP, (12)n m]31, which is the same as [p,m-p,n-m]31. Thus the adjoining operation with respect to k3 transforms one bracket into another one, by interchanging p with m-p and G with -G. Corollary II For N -= 4 about k3, nonvanishing adjoint brackets are either equal or opposite. They are equal if the number of occurrences of the index 2 is even, opposite if this number is odd. Proof Under a fourfold rotation about k3, ki transforms into kS and k2 into -k:. Thus the result of this rotation differs from the operation "adjoining" only by a factor (-) to the power of the number of occurrences of the index 2. If the index 2 occurs an even number of times the factor is +1, otherwise -1. Since the fourfold rotation is a covering operation the corollary is proved. *Similarly we define adjoint with respect to ki (or k2) by interchanging the indices 2 and 3 (or 3 and 1). 15

The indices 1 and 2 occur both even or both odd. Indeed the total number of indices 1, 2, and 3 is even, and the occurrence of the index 5 is even according to Corollary I. Thus the corollary is symmetric with respect to the indices 1 and 2. q.e.d. Corollary III If in an equation of the type (15) each-bracket [m-p,p,n-m]ij is replaced by [m-ppn'-m]ij, where n' has the same parity as n, the resulting.equation also belongs to the set (13) and has the same h. Proof The corollary is essentially due to the fact that k3 is the rotation axis. In equation (13) n occurs only in two places: in the brackets, all brackets in one equation having the same n, and in e (see Table III). In the latter the influence of n enters only through its parity, hence different values of n with the same parity lead to similar equations. Since the definition (14) of h does not contain n, the h-values of such equations are equal. q.e.d. A consequence of this corollary is that a change from n to n' with the same parity in any bracket relation leads to another valid relation, i.e., bracket relations for given values of m,s need to be tabulated only for n = even and for n = odd, a fact which has permitted great simplification in the tables of bracket relations that follow. Corollary IV Two equations of the type (13) with equal n,ms having parameter values w,z and w'=m-w, z'=s-z, i.e., h'=-h, are conjugate complex. Proof One must prove that the coefficient ge is transformed into its conjugate complex by changing from w,z to wSz '. According to (15) it is easily shown 16

that g(mp,w) = (.)m+P g(m,p,m-w). Since the definition of g contains a factor i, g(m,p,w) = (-)m g*(mp, w), (16). the asterisk denoting the comrplex conjugate. Likewise, Table III shows that c(sG,z) = e (sB,,z). (16a) Thus by changing from wz to w' z' the equation is multiplied through by the constant factor (-) and each coefficient changes to its complex conjugate. q.e.d. The occurrence of the equations (13) in complex conjugate pairs permits a simplification in listing or surveying all such equations. If the real and imaginary part of each equation is taken separately, one need only consider one equation of each conjugate complex pair and maintain all self-conjugate equations. This can be done in two ways. In the first way one restricts the range of w to values satisfying the selection rule: m - 2w >O, and leaving 0 < z < s free. In this way only equations with h > 0 are selected, and this procedure is most practical for the making of tables of bracket relations. In the second way one restricts the range of z to values satisfying the selection rule: s - 2z > and leaving 0 < w < m free. This way is useful for proving some general relations, for example, those of Appendix III. It is evident that for self-conjugate equations m -2w = s - 2z = 0 is a necessary and sufficient condition. The number of real equations so obtained is equal to the number of original complex equations. 17

Corollary V If in an equation of the type (13) each bracket is replaced by its adjoint, the resulting equation also belongs to the set (13) and has the same |h. Proof Each equation (13) contains adjoint pairs of brackets, characterized by p,G and p "=m-pt'=-, since s:ummations over p and G occur. The coefficients of adjoint brackets in one equation are g(m,p,w) E(sQz) and g(mp',w) c(s,G',z). Thus, replacing all brackets by their adjoints is equivalent to interchanging the above coefficients without changing the brackets. We shall prove that, apart from a constant factor, this change in coefficients transforms the equation into its conjugate complex, which according to Corollary IV also belongs to the set and has opposite h. It follows directly from equation (15) that mew -m * g(mp,w) = (_) i g (m,p':,w), (17) where p'=m-p. It is also easy to verify in Table III that ~(s,G,z) = (-)Zi's (sz), (l7a) where 1' = -G. Thus by changing from p,G to piQg, the coefficients are multiplied by the constant factor (-)nih and change to their conjugate complex. q.e.d. A consequence of this corollary is: For any real relation between brackets its adjoint relation is also valid with the same coefficients. This fact is extensively used in constructing the tables of bracket relations which follow. A number of other corollaries follow from the symmetry of g and e. The principal ones are listed below, proofs being left to the reader. They are useful for checking relations among brackets. 18

Corollary VI Any relation between brackets with s = 1 (9 = + 1) is invariant for the substitution G= -9 followed by reversal of the sign of the coefficients of all terms with 91 = -1. Corollary VII Any relation between brackets with s = 2 is invariant for the substitution 91 = -9 followed by reversal of the sign of the coefficients of all terms with = O. Corollary VIII Any relation between brackets for s = 1, n = even, is transformed to a valid relation for n = odd by changing the sign of all brackets with 9 = + 1, and vice versa. Corollary IX For m = odd any relation between brackets is invariant for the substitution ml = m-p followed by reversal of the sign of the coefficients of all terms with outer indices 51 and 12. d. The Procedure for Tabulating the Bracket Relations. —It was stated that the brackets are linear combinations of components of tensors of even rank. Thus the analysis can be restricted to eleven out of thirty-two point groups. Moreover, Onsager's relations permit one to use only six pairs of indices ij, with equations (11) and (lla) valid for all point groups. The theorem, given by equations (13) and (14), allows complete tabulation of all bracket relations for all these groups. Often the application of the corollaries, especially I and II, is helpful in obtaining the tables for certain groups from those of other groups. The brackets of the group S2 can be tabulated completely by using equations (11) and (lla) only. It is to be remembered that all the tables that follow are constructed according to the convention that k3 is taken along the ro19

tation axis of highest order and k1 along a rotation axis normal to k3, if there be one. TABLE IV PROCEDURE FOR TABULATING BRACKET RELATIONS Table of + Corollary I + Corollary II + Invariance Yields Under Cyclic Table of Group bout k About k About k Permutation Group _ I~UVKV 1,. ~ ~Permutation Group S2 + C2h C2h + D2h C2h + C4h D2h + D4h D2h + Th D4h + Oh Csi + D3i Csi + C6h C6h + D6h The first six groups of the last column of Table IV are seen to be completely derivable from the Corollaries I and II and the invariance under cyclic permutation, as indicated by the + signs. The groups D3i, C6h, D6h are based on C3i, which required the direct application of the general theorem, assisted by the various corollaries. We have not found any simple rule yielding the complete tabulation for C3i. e. Bracket Relations for Czh and D2h —Table V for C2h and D2h is constructed on the basis of Corollary I. The effect of symmetry is manifest entirely in the vanishing of certain bracketsj all nonvanishing brackets are independent. Thus, we have used three symbols to indicate the state of a bracket whose inner part is given by the second column and whose outer indices 20

appear in the first row. The inner parts of brackets contain the symbols e for an arbitrary even number and C for an arbitrary odd one. + means the bracket is independent both for C2h and D2h. $ means the bracket is independent for C2h but vanishes for D2ho 0 means the bracket vanishes both for C2h and for D2h. Examples: [203]23 is found to be zero as shown by [eew]23 in the eighth row, [204]12 is found to be independent for C2h and zero for D2h as shown by [eee] in the first row. The outer indices 11, 22, 33 cannot occur with n = odd, according to equation (12), Table V is complete for all n. TABLE V BRACKET RELATIONS FOR C2h AND D2h ij 23 31 12 11 22 33 [eee] 0 0 ' + + + c> [ecmx] + 0 ' 0 0 0 1) I a Io [wecn] q + 0 c 0 0 [cxe] 0 0 + j [cixm] 0 0 G ed [wee] + O 0 0 [eowe] + 0 [eew] 0 0 + + means bracket independent for C2h and D2h. G means bracket independent for C2h, zero for D2h. 0 means bracket zero for C2h and D2h. f. Bracket Relations for C4h and D4h.-The effect of symmetry is manifested in two ways: either a bracket is zero or it is equal to plus or minus its 21

adjoint (as defined in Corollary II of the theorem). Of each such pair of adjoint brackets, one bracket can be chosen as independent. In the first, fourth, fifth, and eighth or last row self-adjoint brackets may occur. A self-adjoint bracket may be forced to vanish if it must be minus its adjoint. Thus it turns out that four symbols are needed, whose meaning is explained under the Table VI, which gives the complete bracket relations for all n. Examples: [202]12 = -[022]12 for C4h, zero for D4h (first row). [220]12 = self-adjoint and -[220]12, hence zero for C4h and D4ho [202]11 = +[022]22; one of the pair is independent, both for C4h and D4h. TABLE VI BRACKET RELATIONS FOR C4h AND D4h ij 23 31 12 11 22 33 [eee] 0 0 - 4 --- t > [eclm] 4 -, - 0 0 0 0 " [wew] -G X1 \ 0 0 0 0 [uxue] 0 0 4 4 ---- 4- - [aYIY] 0 \ / a [wee]. k. -, 0 [ece ] ' "' 0 [eew)] 0 0 means bracket is one of an independent pair and equal to its adjoint for C4h and D4h. G-means bracket is one of an independent pair and equal to minus its adjoint for C4h, zero for D4h. 4-means -Gand in addition zero if self-adjoint. 0 means zero for C4h and for D4h. Dotted lines connect adjoint places. 22

g. Bracket Relations for Th and Oh.-Table VII for these groups is derived from Table V for D2h and Table VI for D4h by requiring that brackets remain invariant under cyclic permutation of all indices. The six permutations of the indices 1,2,5 fall into two groups of three cyclic permutations. Brackets belonging to two such cyclic groups are pairwise adjoint with respect to kl, k2, and k3. If adjoint brackets are to be equal, such as happens in Oh, or if brackets are pairwise self-adjoint, the two cyclic groups coincide. For example, we have for Th a cyclic group of three equal brackets, obtained by cyclic interchange of inner and outer indices from the first one: [202]11 = [220]22 = [022]33 The other cyclic group of three equal brackets can be obtained from these by a transposition of two indices. Thus by interchanging 1 and 2, i.e., adjoining with respect to k3, one obtains: [022]22 = [220]11 = [202]33 For Oh all six are equal. On the other hand, the bracket [220]33 is selfadjoint with respect to k3, hence there is only one cyclic group of three brackets derived from it and they are equal both for Th and Oh, namely: [220]3 = [022]11 = [202]22 h. The Bracket Relations for C3i, D3i, C6h, D6h —No simple rules for the complete tabulation are available and the theorem plus corollaries will be used. In order to obtain the most compact form for the results, the following scheme has been adopted. For any particular bracket we must first decide whether or not it is zero. For Csi all zero brackets are listed in Table VIII. The proof that these brackets are zeros for C3i is given in Appendix III. The groups D3i, C6h, and D6h have the same zeros and the additional zeros listed in Table IXo The latter was obtained from Tables V, VI, and VII according to the procedure shown in Table IV. 23

TABLE VII BRACKET RELATIONS FOR Th AND Oh ij 23 I 31 12 11 22 33 [eee 0 0 0 + —+- -+ [eon] + 0 0 0 0 0,I [wea)] 0 '+ 0 0 0 0 [acxe] 0 0 + 0 0 0 [x] 0..0 0 [ece] 0 O+ 0 [eew] 0 0 + + means nonvanishing bracket for Th and for Oh, 0 means zero bracket forTh and Oh. In Th + is one of an independent cyclic set of three equal brackets. In Oh + is one of an independent permuted set of three or six equal brackets (three, if brackets are pairwise self-adjoint). Dotted lines indicate places of brackets of the same set. If a bracket does not vanish according to Tables VIII and IX, then its relations to other brackets are shown in Table X, up to m = 6 inclusive. Table X is arranged in three parts, according to s = 0, 1, or 2. Each part consists of seven sub-tables for m = 0, 1,... 6. The sub-tables, except the simplest ones, have the form of a core array of coefficients bordered by brackets. This arrangement represents a double-entry table, similar to the familiar trigonometric tables: brackets on the left are equal to the linear combinations of those at the top, with the listed coefficients, whereas brackets on the right use these same coefficients with those at the bottom. That this arrangement, in which adjoint brackets stand at opposite ends of 24

K * < * \ / V _., /_<1, <: 'J] < <3 a r O a < < a a k X 4 \ n * * C- 44 qD ( 0(0 4 H- < < H;, (0 H1 * * 0 OH,9,/ \ + CH N k'.,<... * <,., o I- _ ___ __ _( ('(' (0ai *H 7 C 7 r7 CO Q C ------- CO C 3 *-H (0 0 I -1 (0 0, *^h < A ppo1 = I..... O Hi (0 HU 0 0 uaAa = u ppo = u I (0 0~ H 0^H co +> I- __S 3 3 ^ r0 aG=u pp0 =u 25

rows and columns, is possible is due to Corollary V, For example, for s = 1 m = 4 we have [3se]23 = [4oe]31 + 3[o4e]31 [i3e] 31 = 3[4oe]23 + [o4e]23 The two equations are adjoint with respect to ks. TABLE X RELATIONS FOR N01WANTSHING BRACKETS FOR C3i, D3i, C6h, D6h s = 0 m 0 [ooe]33 1 _._ 2 [2oe]33 = [o2e]33 5 [1i2o]3 -5 [3o0]33 [2 a]353 = -3[o3)]533 4 2[4oe]33 = [22e]33 = 2[o4e]33 [5 oco]33 [a-] 33 -2 [ 23a]33 [340w-]355 -5 [ 4 35 [o5O]33 6 [6oe]33 [33e]33 [o6e]33 [5ie]33 0 -3/10 0 [15e]33 [42e]33 -6 0 9 [24e]33 [o6e]33 [33e]33 [6oe]33 26

TABLE X (Continued) s = 1 n = even m 0 1 [low]31 = [o3]23 [~]31 = -[100]23 2 2[20[oe]31 = - [lie]23 = - 2o[2e]31 2[2oe]2 = [iie]1 = - 2[o2e]2 5 [3soC]31 = [12)]31 = [21W0]23 = [03CD]2 [30CU] = -[l]2 = [21w]1 = [030]1 - [3cm]23 = I'["- '1]23 = [2]31 [o 31 ~~~4 ^[4oe]31 [o4e]31 [3se ]23 1 3 [13e ]31 [22e ]31 -3 -3 [22e]23 [13e]23 -3 -1 [3le]31 [04e]23 [40e]23 5 [500)]3 [05(~D]2 1 ~ [5~0D]23 [o05)]31 [41w]23 -2 3 [14w]3 I [41w]31 2 3 [14wC]23 [saD]31 -4 6 [230 ]23 [3s)] 23 -4 6 [23]3 31 [05o)]23 [5sW]31 1 [05w)]31 [s5c]23 6 [6oe]31 [o6e]31 [s-e ]23 1 3 [ise ]31 [42e ]31 -2 -3 [24e]23 [33e]23 -2 2 [33e]31 [24e ]31 -3 -2 [42e]23 [lse ]23 3 -1 [51e]31 [ose]23 [6oe]23 The same table can be used for n = odd if each bracket with outer indices 23 receives a minus sign, according to Corollary VII. 27

TABLE X (Continued) s = 2 n = even m 0 [ooe]l = [ooe]22 1 [10I]ow1 = [o01I]12 = [l0]22 [~01]22 = -[1la]12 = [O1w]ll 2 [iie]12 = [2oe]11 - [2oe]22 [o2e]11 = [2oe]22 [o2e]22 = [2oe]ll [lie]ll [= - [e]2 2[2oe]12 = -2o2e]12 3 [30:)]l [30)]22 [21W]12 -1/2 1/2 [12)]12 [1:2] -1 -2 [21w] 22 [12,a]22 -2 -1 [21 ]11 [03w]12 -1/2 1/2 [30W]12 [03)]22 [03W]11 4 [4oe]11 ([o4e]l + [4oe]22) [o4e]22 [4oe]2 [o4e]12 [3ie]l2 -3/2 /2 25/2 [13e]12 [31e]11 1 3 [ise 22 [22e]11 -5/2 1/2 7/2 [ee]22 [31e]22 -1 -3 [13e]l1 [o4e]1 1/2 1/2 -1/2 [4oe]221 22e]12 -3 -3 [22e]12 [o4e]22 ([4oe]22 + [o4ej11) [4oe 11 [o4e]12 [4oe]12 5 [50o311 [os5]12 [50s]22 [41i]12 1 3 -1 [14w]12 [32)11 il - -6 1 [23w]22 [320)]22 1 6 -3 [23] 11 [23w]12 -3 -4 3 [320:12 [14w]11 0 2 -3 [41w]22 [14]U2 -3 -2 0 [41w]11 [o05w]22 [5so]12 [o5o]11 28

TABLE X (Concluded) 6 [6oe]ll [6oe]22 [o6e]ll [oee]22 [6oe]12 [o6e]12 [33se11 [5se]12 -1 1 -3 3 [15e]12 [5ie]11 2/5 18/5 -3/10 [15e]22 [42e]1l -4 -2 3 6 [24e]22 [51e]22 -8/5 -12/5 -3/10 [s5e]ll [42e]22 -2 -4 6 3 [4e] 11 [42e]12 -2 -3 0 [24e]12 [33e]12 2 -2 -2 2 [33e]12 [33e]22 4 -4 1 [33e]11 [o6e]22 [06e]11 [6oe]22 [o6e]ll [o6e]12 [6oe]12 [33e]22 s =2 n = odd m ~~~0 1~[oc0j...)]12 1 2 [oaoz12 = [20o] 2 [21e]12 = -[o03e]12 [i2e]12 = -3[3oe]12 4 240au]12 = [2] 2[04z2 0]12 5 2[4ie]12 = [23e]12 = -[05e]12 2[14e]12 = 3[32e]12 = -[5oe]12 6 [51C]12 = - 3/10[33CD]12 [15)]12 [42]12 = -6[60c)]12 + 9[06] 12 [24)]12 = 9[60W] 12 - 6[061o] 12 4. THE NUMBER OF NOIVANISHING INDEPENDENT BRACKETS Let I(n) be the number of nonvanishing independent brackets for each point group as a function of n; further let P(n) be the number of possible brackets, E*(n) the number of valid equations (13) between them, E(n) the number of equations (13), valid and nonvalid, and K(n) the number of nonvalid 29

equations (13) corresponding to h = kN. Then evidently I(n) = P(n) - E*(n) = P(n) - E(n) + K(n). (18) Case 1. Groups CN (N = 1,2,3,4,6) It will be shown in Appendix II that PC(n) = EC(n) for a rotation axis along k3, reducing (18) to ICN(n) = KCN(n) ~ (19) In order to evaluate KCN(n), a counting diagram as shown in Figure 2 can be used. For any of the six:allowed combinations of s,z when n is even (three when n is odd), each possible equation (13) is represented by a point in the h,m plane. The equations for a given value of n are represented by lattice points m < n. Since by definition 0 < w < m, we have -m + (s-2z) < h < + m + (s -2z); hence, for each s,z pair (subgraph), the points fill a triangular array. For purposes of counting it is convenient to think of each point as the center of a one-by-two rectangle, the rectangles filling the area. The points (equations) satisfying h = kN lie on a series of equidistant horizontal lines spaced by a distance N. For the group C1 all the points must be counted as a function of n, i.e., up to m = n inclusive, leading to IC(n) = Pc(n) = EC(n) = Kc1(n) = aon2 + bon + co, (20) where, for n = even, ao = 3, bo = 9, Co = 6, and, for n = odd, ao = 3/2, bo = 9/2, co = 3. For CN(N > 1) the number of points lying on the horizontal lines h = kN must be: counted- and is easily seen to be of the form an2 + bn + c, where a = ao/N (21) stems from the main triangle area covered by these lines (i.e., the rectangles of the lattice points on these lines) while 50

. m,. o m. ~ m *. = 0 z = I. *. mh h.. o m.. o m... ~* o dS 0 ~ ~ 9 ~ I* *.. o,~ * * * * m 0.0. _.. *.. * _* _. s = Os = s = 2 iz = O 0 = C z = Points represent E-equations (13) in h~m plane for the various possible combi_hl. _h nations of s and z. Arrows indicate K-equations for which h = M. The case illustrated is N = 3- For n = even all six diagrams are valid; for n = odd., only those with parentheses. * O 9 0i O C t d = * - 0 S _ * * *_ 9 9,As = 20 zs = 8 s _2 Points represent E-equations (13) in h,m plane for the various possible combinations of s and z. Arrows indicate K-equations for which h = kN. The case illustrated is N = 3. For n = even all six diagrams are valid; for n = odd, only those with parentheses. Figure 2. Counting diagram. 31

b = bo/N for N = odd b = (bo + 1)/N for N = even, n = even (22) b = (bo + 1/2)/N for N = even, n = odd stems from circumference points with h = kN whose rectangle areas stick out beyond the main triangle area. The values of c are most easily evaluated in practice by solving for c for some low value of n for which ICN has been counted. In this way the formulas of Table XI for the C classes result. Case -2. Groups DN (N = 2,5,4,6) The quantities on the right-hand side of (18) will now be interpreted after the effect of a binary axis along kl has been taken into account. This effect is manifest in two ways: certain brackets vanish, reducing P(n), and certain equations about k3 must be dropped, reducing E(n), as will now be shown. According to Corollary I, all brackets for which the index 1 occurs an odd number of times vanish. In any bracket the index 1 occurs p + n - m times inside and 1/2(s - G) times outside, together p + n - m + 1/2 (s - 9) times. On the other hand, any equation (13) about ks contains brackets with the same nm,s, but different p,G. The coefficient ge of any such bracket is ip+l /2(s) is times a real factor. Thus, if n - m + s is even, then all brackets with imaginary coefficients in the equations (13) about ks vanish due to the binary axis about kl, while if n - m + s is odd, those with real coefficients vanish. Thus, every equation (13) about k3 becomes equivalent to its complex conjugate, and in order to find E N(n) it is only necessary to count points h k kN for which h > 0, i.e., the upper half of the counting diagram of Figure 2. Consequently EDN(n) = /2 E(n). (25) The number of possible brackets after introducing the binary axis ki is clearly PDN(n) = IC2h(n), (23a) 32

or with (18) and (20) IDN(n) = ICh(n) - 1/2 [ICl(n) - ICN(n)]. (24) Substituting the results found under Case 1 for Ic2h and IC,, we have for n = even IDN(n) = 1/2 ICN(n) + 1/2 n + 1 (24a) n = odd IDN(n) = 1/2 ICN(n) + 1/4 n + 1/4 (24b) Case 3. Groups Th,Oh and the Isotropic Case For the class Th, analysis of the effect of cyclic permutations leads immediately to ITh = 1/3 ID2h while for Oh case various simple ways of counting yield the results of Table XI. 5. EXPLICIT FORMS OF THE GALVANOMAGNETIC TENSOR UP TO B2 For the practical purpose of reading off the explicit form of the galvanomagnetic tensor component p = F/J in terms of the brackets, the tables of this section have been compiled. These give pl for arbitrary orientation of the sample and the magnetic field with respect to the crystal axes and for all crystal structures except S2. It was shown in Section 2b that 2 _ Aij CZ 1 rpal=.i Ma = —,^, 1 (25) where Aij is the cofactor of aij in A = det aijo and ~i is the direction cosine of the laboratory coordinate axis a with respect to the symmetry coordinate axis i. For a = 1 the equation represents the magneto resistance, for a 1 the Hall effect, and this is true for all that follows. The expansion of all cij in powers of B leads to 33

TABLE XI THE NUMBER OF NONVANISHING INDEPENDENT BRACKETS an2 + bn + c Group n a b c C1=S2 even 3 9 6 odd 3/2 9/2 3 C2h even 3/2 5 4 odd 3/4 5/2 7/4 C3i even 1 3 2 odd 1/2 3/2 1 C4h even 3/4 5/2 2 4k+l 3/8 5/4 11/8 4k-1 3/8 5/4 7/8 C6h 6k-2 1 4/3 6k 1/2 5/3 2 6k+2 J 8/3 6k-l 17/12 6k+1 1/4 5/6 23/12 6k+3 J 5/4 D2h even 3/4 3 3 odd 3/8 3/2 9/8 Dsi even 1/2 2 2 odd 1/4 1 3/4 D4h even 3/8 7/4 2 4k+l 3/16 7/8 15/16 4k-1 3/16 7/8 1/16 D6h 6k-2 5/ 6k 1/4 4/3 2 6k+2 7/3 6k- 1 13/24 6k+l 1/8 2/ 29/24 6k+3 7/8 Th even 1/4 1 1 odd 1/8 1/2 3/8 Oh even 1/8 3/4 1 4k+l 1/16 3/8 9/16 4k- l1/16 3/8 5/16 34

= pal B21 + B21+) B2T at Pai '"~C aP 2B M B2 a)Q + t 1 + Qa1 B + ( M B -- (26) Table XII gives Mo. Table XIII gives the nonvanishing coefficients of ~i Ij in Po ~ Table XIV gives the coefficients of 7k, the direction cosines of B in the crystal coordinate system, resulting in Q1. For the purpose of tabulating the coefficient of B2 we write (2 M = R2 = Rk Yk Y7 - (27) al Tables XV, XVI, and XVII give the RkQ for the various classes. Coefficients for higher powers than B2 can be obtained similarly if needed. These tables are useful, for example, in determining what measurements must be made in order to determine all the independent brackets (material constants). In principle, a number of measurements at least equal to the number of independent brackets is needed. It is expedient to choose these with care. For example, no matter how many magneto resistance measurements are made, even with different samples, allowing variation of li and 7k, one cannot obtain all brackets individually. Combinations of Hall and magneto resistance measurements work most efficiently. In Table XIV ~P has been used as an abbreviation defined by l 1 a 1 a k -= i lj - aj i, where, if t = 1, kI automatically vanishes in agreement with the evenness of the magneto resistance, while, if a / 1, the subscripts ijk must be a permutation of 123 of the same parity as the superscripts lcB. In that case -k is the direction cosine between the symmetry axis k and the laboratory axis 3. 55

TABLE XII THE EXPLICIT FORM OF Mo C2h [ooo] ([ooo]1 [00ooo01] 2- [ooo]2) D2h [000]33 [5ooo]1 [ooo]22 C3iD3iC4h 2 [000oo]33 [000ooo]11 D4hC6hD6h Th Oh [ooo]11 TABLE XIII THE EXPLICIT FORM OF pC 1 i1 1 1 c 1 c 1 a l,,1 2.2.3 3 1.. 2 + a2 1 -2 C2h |[~~~]22 [o ~]33| [oo L33 [ooo] ] | [oo ]11 [ooo]22- [~~~] 12 [ooo 33 [ooo ]1 C3iD3i C4hD4h 000ooo ] 11 [oo o ]33 [ooo ] 11 [ooo ]233 [ooo ]1 C6h,D6hJ Cehh2eh 2 2 2 ThOh 0 [ooo ]11| [ooo] [~]1 36

o o o o o 0 0 0. I. ~ L_ J. o O 0 O O O O O O O O O CCL C UL CU COXU OC, —I H H H H 0 0 1 g I I* O o o 00 o 0 0, O 0 0 o 0 n H cu + O OJ O O O i-i r-i. 0 0 0 + + <t HC O O O +O N H CU CO H H O1 O O - O - rl 0 0 0 0 0 (x <- r- rr O4 O H 0 0 0- 0- 0 lobe _ H (U ~> a o 0 H H 0 Fa CO. rlr- _ Ct c CUl r1 4H HH CUH CU O O 00 0 0 0 0 o 0 0 -H — H ' OO O- O. O O H + + Hrl C H H H r-O o r. r-4 ol ' 0 0 r-U r- CU 0 0 0 0 0o 0 0 0 -- " — OJ OJ OJ 0 0 0 0 0 H H 0 0 0 c(. H ci. d. rfl Ci2.l ca.H cq+ ~ i Ci --- i- i ';.l: 1 I 0 c 37

TABLE XV THE EXPLICIT FORM OF R2 FOR C2h, D2h 1i1 {([~]0 [200 + [200] +[0031) A/B + 2([10oo]2 [100]31 - [000oo] [200]12) A + ([ooo]33 [200]22 + [1oo]2) AB} + + 1 ([ooo1] [200]2 + [1002) B/A00] - [ooo] [200]2) B + ([]oo11 [2001] + [00oo]1) AB + + l31 ({[~ooo] [cool~] [200] + [ooo]11 [100]23 + [ooo]22 [1002* ++ (2[1oo]23 [loo]l - [ooo]12 [200]33) (1 - AB) [ooo]2 }/[ooo]33+ + f{[oo n [30022 50 [11 2 5 125 51 12 55 ~ ~ ]J3 a1210 [ ( + A2 + [612 + (1121 + 1 ){ ([ooo]33 [200]12 - [1]23 [100]3 ) (1 + AB) - ([0ooo]3 [200],, + [Ioo3) A - ([ooo] [200]22 + [loo]23) B}* 121 {([ooo], [ [o2 o]) /B + 2([o1o] 01 -[00 [o1]2]12) A + [] [ o2o]22 + 0 [o1o]25) AB} + - 02 112 [000]3 022 25 51 2 535 12 11 530]1 1 a + 1212 {([ooo]3 [020]22 + [ooi2) B/A+ 2([olo]1 [01o0]2 - [ooo000]33 [o2o]2) B+ ([ooo]11 [o20] + [010oo]1) AB }+ - R22a + 1 {([ooo]22 [02o] [ooo]11 + [oo]11 [01]2 + [ooo]22 [10]2 + (2[010]2 [olo010] - [00ooo]12 [020o] ) (1 - AB) [ooo][ooo] + + (L?12 + 1,){ ([ooo]o3 [020]12 - [1010 [010]23) (1 + AB) - ([ooo] [020] + [o10o]2) A - ([ooo]3 [020]2 + [010]23) B} a1 1 002+51' 1f_1 {([ooo]2 [0 11+ [ 0190] + [12) / [ooo] - 2[o2]12 A + [002]22 AB } [ooo] 30 + + 2122 {([ooo]ll [0o2]22 + [0o1]12) / [000]22 - 2[o02]12 B + [0021l AB [000]53 + -c2 "33 01 2 1313 (1 - AB)2 [oo213 [000]11 [000]22 / [o000 +; (11 + a1) { (1 + AB) [o02]12 ([0ooo22 [00211 + [0o]11 [002]22 + [ool]12) / [ooo]22 [ooo] (02 0 0l~1 0a1 * 01(Zl+[ 01 * - (R + R) (1213 + 1312) ([000ooo]11 [011]23 - [oo1]12 [o10o]1- [000ooo]12 [011]31) + (si+ 111) ([000]22 [11]51-[ool]12 [010]23 -[ooo]12 [011]23) (1 - AB) - (B31 + K) (1s2 + 13[2) ([~~]11 [11]2 - [001]2 [100]31 - [0oo]12 [101]l) + (1311 + 1113) ([ooo]22 [101]1- [00]12 [1oo]2-[ooo]2 [1o]23)( - AB (121 ([ooo]s3 [11011 + 2[01 0]3 [100]31) A/B + (2[00o123 [010]1 + 2[o0o]02 [o]1 - [ 110) A +.2l{([ooo]j [110o]22 + 2[1o00]2 [o1]2) B/A + (2[100]25 [olo]0 1 + 2010oo]2 [00oo]l -[000o [10] [ 12) B + + ([ooo]1 [10]5 + [ooo]3 [10o11 + 2[1oo]31 [oio3) AB}* + - (00+R~) + 1a13 3 ([00ooo] [110] [0~]11 + 2[0]22 [o10]31 [0~1~]1 + 2[1o0]2 [ 01]23 [ ]11). + + (2[10oo]2 [010]31 +2[olo]25 [00oo] - [~oo]12 [110o],) (1 - AB) [ ooo]12 }*/[0ooo] + +12 11 ) ([ooo] [110] - [100] [010o] [olo] [l00]l) (1 + AB) - +t 2 3[1525 [12 [25 51225 51 - ([0ooo] [ 1o]1 + 2[oo]31 [100l]1) A - ([ooo]3 [1o]22 + 2[100o]2 [oo] ) B } A [ooo012/['ooo]l A/B = 1/(B/A) = [ooo]22/[ooo]11 B [ooo]12/[ooo]22 * means zero for D2h 38

o0 0 0 0 + cmN cm + o CU Qrl I CU rlOo 0 * - - *H. -i / r - 0 Ot CM <H CM iH 0 ~ 0 0 i H -l - H < rl H 11 o 0 CM r-l r- H I 0-CM 0 0 H 0 H H O-o a H 0 H I + o 0 0 0 0 -lCM * rK\*M I M CMJ C-' N K 0 0 CY CY r 0 o 0 0 0 0 0 H C N CM 0 0 0 0 L~ ~ + + O0 H H 0 H ^< HCO I HH +,- 0 - C0 f0 *H dH1 OH *H H - HH 0 0 0 0+10 0. 0. 0 0 + 0 0 I + 0 + 0 m ~ c 0 N N 0 0 + O -W- N -* - -- 0 HN 0 H N0 o 0 0 0 0 0 0 0 I '- + H _ 0 0 0 0 0 -H + H + >0 c 0 0 0 0 H H + HICO H - H H-J ' —.r- 0 0 HN- V ( r10 rlr CFHr-A r-H NX\ 0 - -1 -i c m C Q r v rQ h kM 0 c- -- fH H- 0 H H -O 0 O 0 0 HI H H ~ ~~< 0 '~~ 0 0 0 ~~0 ~ ~l ~ v ~ 0 (H 0 LH 0 m + + + + HE-CM CCX H C C (CM 0 H CM l C CMi HC CO J CU Ci Ck H C CC10 C X H * CM H H 0.-. * -*- ' * - <-* < i-lIrYl rrf0 0 i o 0-0 00 0 0 0 0 H 0- H 0 0 — *-HI 0 0H oi-1 0-1 01-0i- 0 0-i i-I 0 0H- -.. 0 0 0 0 0 0 O O O I O I " 0 O H H O O Pi CO o, - 0 -, 0 0 0 0 0 0 *H. i~ r, - - - H l H O 'O ' - - 0 0,-I 0 N N N N N + N + + + O+ - - - - - H0 OH 0 1 0 0 C - C lO Hl CM OH O O 0 0 0 0H < d H 0 I H H + + + + + + + + $ H H N ~\ H^ d r-i ~ ~ HCM O O O O O I ~IrH O M cflcd c L-0.......... 0 i 0ro -- P^ Pt^ I Q) 0) Q I I I 0 u) ) U) g0 0 0 0 0 N0 0 N Il CY c I IC. 59 r I- - I.-I I- rI r r-IH H IH r- Ci- I - + +

. ---% H0 ~ O OJ o-..IC\J " ^C\ r —7 'H C\JCJ 0 0 b r-l 0 0 0 H 0' o L-j L-*c t + rH-I r-H I C\ + Hr- q-J 0 r-H 0 H *H H, 0 CMU0 -P H-O 0 - 0 0 0 - o 0 OJ c* 0 0 - 0 H ^-P 0 L r "O 0 0 OO O CO 0 ri CY)-1bO + + _ 0H + H H 0 ~ r — r.-. o= i —,i iI -— P g KC\ 0 0 -H E-1 0. 0 O O-P r-0 0 0 0 OJ H Cr ~ O 0 H,r-4.H H0CH H H 3 *H0,, r- * —rd '0, -H C H CM ia-) (L) I 0 0 0 CH IC P,< pr-. —I. ~"..-I C\*H) 0 d,, o -- 0 O~ ( -' m WI\ \H 0 Hr — 0. O,. CH 'q C010 OJ10 C0101 O0 ii Pt- - I O ' ~1 ~ ~ ~ ~- '~ 'r"! r'"4 O0 0 bDO E- ' OI 01 +O O O: - ~H C 01 +,-4,-4,-I. r —I~+ H H H.- H H 0 O H O 0 H CM H E-i p-0 0 HC'! -d H H-I ( H v O '", ' ""l r —t -l O 00 I aO O,'.-I,',- — 0 O O O 0 rO,- 0" OJ OJ,.,- 0 0 0) - Hl H H C CO r-) 01 01J L0 H-4 \J Id t Co R ^ O" 0 0 H r r-l a + 0 0 0 r-1 r r o I * O 0 0 0 0 0 0 0 0 r-0H 0 JO O:: 0 -4o H -- 0 ' H ~~~O 0 H.- I,-.,-. O "'d,m — H, 01j -, r I 0 a0 H C 01 C HCO HHr 0 0 r-l 0 0 H H.),-i-' N - C r, O H O HH N HH NC 0 C i H HH HHH Ol 0J 11 O — J. ~ ^ ~ C'0 H I+ 4 C rvi 0 0 -C rl 8Cr 3 CM C 0-~ 0~l 01 P,., '- -I HH- iC 0 - r,- P I( r 1 II +6 F.-l ~ d '-1korS0

6. DISCUSSION The anisotropy of galvanomagnetic effects, first studied by Righi in 1883, appeared in many works from time to time. However, the crystallographic effects have never been taken into account comprehensively. An overall investigation of the crystallographic effects of the 32 point groups upon the isothermal galvanomagnetic effects is attempted in the present paper. The results include as special cases the work of Voigt and of Juretschke concerning the tables of brackets, the work of Seitz, Pearson and Suhl, and Goldberg and Davis concerning the formulas for cubic crystals, and some of the work of Kohler, insofar as it is concordant with Onsager's relations. For D3i and m < 4, Juretschke's15 results agree completely with ours presented in Table X, except for a different notation. In order to facilitate the comparison of our bracket notation with the notations used by Seitz and by Pearson and Suhl for the case Oh, Table XVIII has been prepared. TABLE XVIII NOTATIONS OF VARIOUS AUTHORS Seitz's Pearson and Suhl's Bracket Notation Notation Notation Go = l/po '[ooo]ll ~~~a ~[100]23 = [001]12 PBt~~~ ~~[200]22 = [002 11 Y [l]0123 = [110]12 + y + 8 [2oo]11 = [oo2]33 a [loo]23/[oo:]_l b ([2oo]22[ooo]l+[ oo] 23)/[ooo] 2 ([200]11- [00]22-[100])/[000]11 Icl2 1 231

The methods developed here can be extended to all thermo-galvanomagnetic effects. We hope to follow up this paper by one which gives the electron theoretical definition of the brackets and further developments regarding the effects of crystallographic symmetry by means of the electron theory of solids. Many investigators limit themselves to cases where the magnetic field is either parallel or normal to the current, or in the plane of the current and the Hall probes. Such limitations seem unnecessary with the broadened definitions.of the galvanomagnetic effects, which permit the magnetic field to be arbitrarily oriented (Seitz did this for the magneto resistance of cubic crystals including terms up to B2). Therefore all the galvanomagnetic effects bearing various conventional names, such as the transverse and longitudinal magneto resistance, the Hall effect, the "planar Hall field," and the Corbino effect, are included as special cases. They can all be analyzed in terms of an ascending power series of the magnetic field; the only known exception is the oscillatory behavior at very low temperatures. The Corbiho effect, about which, to the best of our knowledge no work has been done on single crystals, will be dealt with in a separate paper. We have also brought to light certain properties regarding the parity of the galvanomagnetic effects, about which a certain lack of consistency is to be found in the literature. In particular, the magneto resistance is necessarily an even function of the magnetic field, while, contrary to the odd-Hall-effect convention, it was proved here that the Hall effect is in general neither an odd nor an even function of B, but can be purely odd or purely even or zero when proper conditions are satisfied. These contentions are firmly supported by experiments, for example by Logan and Marcus, and by Goldberg and Davis. 42

APPENDIX I Proof of the Theorem Concerning an N-Fold Rotation Axis Along k3 Let kl'k21k31 be a set of symmetry coordinates of a point and let kj"k2tk31" be the transformed set of the same point after rotation of the coordinate axes through an angle 0 = 2i/N about k3. With respect to these two systems of axes the components of a tensor T of arbitrary rank are related by ij... = a j - TT... a T (A.l) 4Ti...;k~. o.oT where (cos X sin X 0O ait = (-sin 0 cos 0. (A.2) \ 0 0 1 This is true in particular for the components of the position vector k of a point, of the components of the magnetic field vector B, and of the components alj of the second-rank conductivity tensor. Thus by such a rotation of coordinate axes any equation between singly primed tensor components is transformed into one with doubly primed tensor components. If the rotation is a "covering" operation of the crystal, then the equation must be invariant, i.e., its form in terms of singly primed quantities must be identical to that in terms of doubly primed quantities. This principle can be applied to equation (10). Thus, for a covering operation the coefficients (brackets) in the singly primed and doubly primed forms of (10) must be identical, hence can be written without any primes. Now the doubly primed forms of equations (10) can be transformed by applying the equations A.1 and A.2 to the doubly primed components of a and B. The transformed equations so obtained in terms of singly primed components A-1

must be equivalent to equations (10) in their direct singly primed form. This equivalence requirement yields certain identities between the brackets. It will be shown that these identities are just described by equations (13) and (14). In order to simplify the proof we shall not apply the above reasoning to the Cartesian components of the tensors involved but rather refer to a pair of coordinate systems of a different type (complex, nonorthogonal) defined by k" V= kl' + i k2' k2' = k - i k2'g (A.-3) k3' = k3s and likewise for double primes. Quantities referring to the complex coordinates will be marked by a bar throughout. We define the complex components of B by B.' = Bl' + i B2' B2' = B10 - i B21 (A.4) B3' = B3 ' and likewise for double primes. The complex components of the second-rank conductivity tensor are defined by* *For a tensor of arbitrary rank n the complex component Tij... is defined as follows. a. Replace every index in the given order by a symbol according to this scheme: the index one by (11+i2), the index two by (1 -i2), the index three by (13). b. Multiply the n-fold product of symbols so obtained according to the associative and distributive law, but do not use the commutative law. c. Replace each "term" of the symbolic polynomial so obtained by T with the indices of that term in the given order and with a coefficient equal to the product of the coefficient parts (upper parts) of the "factors" of that term. The resulting polynomial in T is the desired expression. For example, one wants to find the appropriate definition of T123. According to (a) he writes the symbol (li+i2)(11-i2)(13). According to (b) he obtains the symbolic polynomial (11)(11)(13) - (11)(i2)(13) + (i2)(11)(13) - (i2)(i2)(13). According to (c) the definition is now T123 = T113 - i T123 + i T213 + T223s A-2

11 - a22 + i(a12 + 21) 11 + a22 - i(12 - 21) a1 + ij -( ll +a22 + i(a12 a21) 11 a2 - i(a + a ) a - ia (A.5) a31 + io32 31 5 2 533 According to Onsager's relation oij(B) = oji(-B), (A.6) hence terms below the diagonal of A.5 are dependent and it suffices to consider those above the diagonal. For convenience we shall now introduce another notation. Let s denote the number of ones and twos together, and z the number of twos, among the indices of a iji. This definition of z is consistent with that given in Section 3b. The numbers s,z define uniquely one of the six independent pairs of indices ij and vice versa. We write (note the indices between parentheses): sz ) - aij * (A.7) Thus, for example, a(11) is another notation for a23. It can be verified by direct substitution of these definitions and comparison with A.1 and A.2 that the complex tensor components as defined transform under a rotation of coordinates about k3 according to i. = Zy aik aj *.*.* "aT T...T (A.8) \j...T with the diagonalized matrix ai ( 0 e o ). (A.9) 0 0 1 In particular, A-3

Bi" = B1' e'-i B2" = B2j ei' B3s = B3' g.)(B) = &(s,z)(B) e-io(s-2Z) We are now ready to apply the invariance principle. The equations to which it will be applied are the expansions of a(sz) and a(s z) in powers of Bk' and Bk", respectively, analogous to equation (10): 00 n m sZ( ) =), Cs z) (n,m,w) W B2-W 3nm - (A.ll) n=0 m=0 w=0 Comparison of A.11 with equations (10) and (11) yields Cs Z)(nm,w) - (1/2) (-)w I g(m,p,w) c(s,e,z) [m-pfp,n-m](s ), (A,12) p=0 G where g and e are the functions defined in the text by equation (15) and Table III. Equations of identical form, but with double primes, hold for the components with respect to the doubly primed coordinate system. From the equality of singly and doubly primed brackets for a covering operation of the crystal, arrived at earlier, it follows now that SC( ) (nm,w) = C ) (nmw). (A.15) (sz) (sliz) According to the plan outlined at the beginning of this section, one must now express the doubly primed components of asz) and Bk" in the doubly primed analog of A.11 in terms of the singly primed ones by means of A,10. We obtain oo n m (s,z) = Cs z) (nmw) eih g-w B -m-w n- (A.l4-) n=0 m=0 w=0 A-4

with h = m + s - 2(w + z). Comparing A.14 with A.11 it is seen in view of A.13 that the two results are only compatible if either Csz) (n," w) = (s,z) (n,m,w): 0 (A.l5) or eih = 1 In connection with A.12, the equations A.15 are identical with (15) and (14) in the text, and the theorem is proved. A-5

APPENIDIX II Completeness of the Equations (13) Consider the complete set of equations (13) for given n,m,s, that is, the equations (13) for all values of 0 < w < m and 0 < z < s, regardless of the value of h. Theorem The only solution of the complete set of equations (13) for given n,m,s is that all the brackets with the given n,m,s vanish. Proof In the power-series expansion (10) the brackets for given n,m,s, with the six independent pairs of outer indices, are independent constants (as long as no symmetry restrictions are introduced). The moduli of the transformations from Bi to Bi and from cij to aij or a(s,z) are nonvanishing, according to A.4 and A.5, Consequently, in the power-series expansion A.11 the constants C(s z) for given n,ms, are independent constants and the equations A.12 for given nms can be inverted, leading to homogeneous expressions of the brackets in terms of the C's. The complete set of equations (13) for given n,m,s, states that all C's vanish; consequently, all brackets with these n,m,,s values vanish. q.e.d. In this proof use has been made of the fact that for given n,ms, the number of brackets [m-p,p,n-m] (s,) is equal to the number of constants C(sZ) (n,m,w), hence equal to the number of equations (13) [without (14)]. This equality is easily verified. Indeed, the number of equations for given n,m,s, is determined by the ranges of the integers w and z, given by 0 < w < m and 0 < z < s, while the number of brackets is determined by the ranges of the inB-l

tegers p and l/2(s - @), given by 0 < p < m and 0 < 1/2(s - 9) < s. The theorem of this appendix is useful in spotting vanishing brackets and in counting nonvanishing ones. It also follows from this theorem and Appendix I that the symmetry properties of the brackets are completely described by the equations (13) with the condition (14). B-2

APPENDIX III Proof of Table VIII - Zero Brackets for a Threefold Axis of Symmetry Consider the set of equations (13) for given values of n,m,s and different w,z. According to Corollary IV and the remarks made immediately thereafter, it is expedient to select from the original complex equations those satisfying the selection rule s - 2z > 0 (C.l) and equating their real and imaginary parts to zero. The number of real equations so obtained is equal to the number of the original complex equations with s - 2z unrestricted by C.l, It is also equal to the number of brackets having the given values n,m,s that occur in these equations (see Appendix II). A first type of zero bracket arises through the application of the theorem of Appendix II. If no equation of the set is invalidated by h = kN, then all the brackets of the set vanish. The h values of the equations, for any particular value of s - 2z, range from m + s - 2z downward in steps of two for the various values of 0 < w < m. In order to avoid h = 0, this type of analysis is restricted to m + s = odd. In order to avoid h = + 3 as well, it is necessary that m + s - 2z = 1 (C.2) for all values of z, compatible with the given nms values and C.1. This condition can only be realized in three ways. First, m = 1, s - 2z = 0, and s = 0; consequently, z = 0 and n must be even. Second, m = 1, s - 2z = 0, and s = 2; consequently, z = 1 must be the only compatible z value, necessitating n = odd (see Table III). Third, m = 0, s - 2z = 1; consequently, s = 1, and the only z value compatible with C.1 is z = 0. The corresponding zero brackets are, C-l

respectively, [lo)]33 [01u]33 [ioe]l2 [ole]12 (C.3) [oon 23 [oon] 3 A second type of zero brackets can arise for m + s = even. According to the definitions of g and e, the coefficients ge of any bracket in the complex equations are ip+l/2(s-).is times a real number. Thus, all the real parts of the equations with given n,ms, contain only brackets of one parity of p', defined by p' = p + 1/2(s - g), (C.4) and all the imaginary parts of these equations contain only brackets of the other parity of p'. The brackets with given n,m,s thus fall into two subgroups according to the parity of p'. In each subgroup there are as many real independent equations as there are brackets. There is one way in which the condition h = kN can affect one of these two subgroups without affecting the other. The brackets in the unaffected subgroup must then vanish on the same grounds, as before the condition h = kN was applied. In order that h = kN shall affect only one subgroup, it is necessary and sufficient that it affect only self-conjugate complex equations. According to Corollary IV, these occur only for h = O; namely, if both (s - 2z) = 0 (C.5) (m - 2w) = 0 hence m,s, and m + s are all even. Since h = 6 is inadmissible, the possible values of m are 0,2,4. These can be realized in three ways, compatible with C.5. First, s = 0; consequently, z = 0 and n = even. The case m = 0 must here be excluded since the remaining subgroup is empty. Second, s = 2, n = odd; consequently, z = 1. On the same grounds as in the first case, m = 0 is excluded. C-2

Third, s = 2, n = even; consequently, z = 1 according to C.5. But the equations with s = 2, z = 0 must now also be admitted, while no equation that is not selfconjugate complex should be ruled out by h = k.3. This configuration allows only m = 0. The corresponding zero brackets are: [lie]33 [31e]33 [l3e]33 [llu]12 [S3c]12 [isco]12 (C.6) [ooe ]12 It can be shown that a threefold axis produces no other zero brackets than those listed in C.3 and C.6. C-3