EGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR MAGNETICALLY SENSITIVE ELECTRICAL RESISTOR MATERIAL June 1, 1954 to September 30, 1954 ERNST. 'KATZ Associate Professor, of PRhysics DEPARTMENT OF THE ARMY LABORATORY PROCUREMENT OFFICE U.S, SIGNAL CORPS SUPPLY AGENCY CONTRACT NO. DA-56-039-ac-52601 November, 1954

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ABSTRACT THEORY Further progress is reported on a general method being developed for determining the symmetry relations of the bracket quantities for the various crystal symmetries. Results of this method are presented in tables. EXPERIMENT Measurements of Bi wires held in a Cardan-suspension are reported as a function of the orientation of the wire defined by the angles ' and i. Their interpretation is discussed qualitatively, but needs further quantitative elaboration. Efforts to coat thin films of bismuth on glass by vacuum evapo:ration are mentioned, ___________________________ ii _____..

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE OF CONTENTS Page ABSTRACT ii A. INTRODUCTION 1 B. THEORETICAL PART 1 1. Purpose 1 2. Definitions of the Bracket Symbols and Related Quantities 2 3. Outer Symmetry Properties of Brackets 3 4. Transformation Under the Symmetry Operations of Reciprocal Lattices 3 5. A Theorem About the Transformation Properties of Brackets Under Rotation About K3 3 6. The Dependence Equations Among the Brackets 7 7. The Tables of Brackets for the Eleven Classes 14 C. EXPERIMENTAL PART 26 1. Equipment 26 2. Materials 26 3. Measurements 28 a. Dependence of the MR Effect on Orientation and on H 28 b. Singleness of Crystals 29 c, Bismuth Coating on Glass Tubes 34 4. Conclusions 34 D. PROGRAM FOR NEXT INTERVAL 35 1. Theoretical 35 2. Experimental 35 E. PERSONNEL 35 iii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN MAGNETICALLY SENSITIVE ELECTRICAL RESISTOR MATERIAL June 1, 1954,to September 30, 1954 A. INTRODUCTION During the summer months progress has been relatively slow due to vacations, changes in personnel, the moving into larger quarters as a result of the expansion of the contract, and finally because of the design and partial construction of equipment for higher H. The theoretical efforts have been directed mainly at developing the insight into the symmetry properties of the conductivity tensor components which was begun in the previous reports. This insight is the necessary foundation for enabling us in the near future to interpret the experimental data described in the second part of this report. The experiments have mostly aimed at measuring for low H the magnetoresistance of bismuth for a sufficiently large number of directions of H with respect to the crystal axes. For low H these data may be described by a magnetoresistance ellipsoid. A few other miscellaneous experiments with Bi are mentioned, B. THEORETICAL PART 1. Purpose In Table V of Report No. 3 some symmetry properties were listed of the bracket quantities describing the dependence of the conductivity components on H. It was stated on page 13 that we wish to derive all the relations among the brackets for all eleven symmetry groups of the reciprocal lattice. This work has been carried essentially to completion and the resulting tables will be useful in all subsequent work for interpreting experimental data. _______________________________________ 1 ____________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2. Definitions of the Bracket Symbols and Related Quantities In order to make the present report more or less self-contained the definition of the bracket symbols is repeated using the same notation as in the previous report. o n-D P1 P n-p-p' aij = A o E [ptPpn-p-pp'j [ H, H2H3 (3.5.8) t~21, p operators t2, and n-p-p operators t'3 in different order. This P[ (tf~)P'(t22) (t92s)n-p-p ] is the sum of all permutations of p' operators tft,. p operators ta2, and n-p-p' operators t13 in different order. This sum consists of n'/p'p'1(n-p-p'): terms. The definition of the vector operator 2 is = -(vExv), (4.2*1) which has the transformation properties of an axial vector, i.e., an antisymmetric tensor of the second rank. In order to set this in evidence we shall denote -ttl = (13) +tt2 (23) +t3 -= (12) The numbers on the right are referred to as "inner indices". From the antisymmetry of the second rank tensor, it follows that interchange of any pair of inner indices within parentheses multiplies the bracket by -1. Only the pairs, (13)(23)(12), will always be used. Furthermore, the outer indices, i, j, shall be denoted as follows: 11 7 a2 12 - oa 21 -- a 22 2 (s = 2) 13 Of a- 23 - t7 31 - 2 32 i 3Y (s = 1 ) 33 -.7 (s = 0). The number of nonthree outer indices will be denoted by s. ____________I______________ 2 ___________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3. Outer Symmetry Properties of Brackets Onsager's relation subjects brackets to some basic symmetry relations, since [p',pn-p-p'lij = (-) Fp',pn-p-p' ji, (4.4.1) or Rule I for n = even outer indices may be transposed Rule II for n = odd transposition of outer indices introduces a - sign Rule IIIfor n = odd and i = j all brackets vanish. The relation 4.4=1 will always be used to rewrite brackets with outer indices 32, 13, 21, in terms of brackets with outer indices 23, 13, 12,respectively. Thus, for n = odd only the outer indices 23, 13, 12, occur with s = 1, 2, while for n = even there are in addition 11, 22, 33, with s = 2, 0O The above rules and relations incorporate the effect of the symmetry due to the inversion center of reciprocal space on the brackets (see also page 12 of the previous report). i. Transformations Under the Symmetry Operations of Reciprocal Lattices The eleven crystallographic point groups of reciprocal space can be generated from the following symmetry elements in addition to the inversion center which generates Sp (a) 2,,34, or 6 - fold axis along k3 generating C2hS6e C4h, C6h, (b) 2-fold axis along k, generating with (a)D2h D3i, D4h, D6h (c) 3-fold axis in the (1,1,1) direction generating Th with D2h and Oh with D4h. These shall be taken up in order.. Theorem About the Transformation Properties of Brackets Under Rotation About Consider a transformation consisting Of a rotation of the coordinate system about k3 through an angle, setting cos = C sin 0 = S ----------------------- 5 --- —-------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The angle 0 must be + 2t/[i, where A, the multiplicity of the axis must. have one of the following values p. = 2,3,4,6. The transformation matrix connecting new and old coordinates is then C SO -S C o (4.6.1) OOI In general, any particular bracket in the new coordinate system will be a linear combination of brackets in the old coordinate system* The coefficients of the old, untransformed, brackets in these linear c6mbinations will depend on 0. Since for any symmetry operation the result before and after transformation must be indistinguishable, there are in principle as many equations as brackets. However, not all are necessarily independent. Actually each of the transformed brackets does not involve linear combinations of all untransformed brackets, but only of relatively few. The following rules refer to this. Let p. p' = m (4.6.2) be the number of inner threes and let 2-s be the number of outer threes. Rule IV All brackets in a single linear transformation equation have the same n, m, s. This rule is immediately evident from the form of the transformation matrix and the rules of tensor transformation. Thus, for given n, m, s, the number of brackets transforming among themselves is relatively small. It is a matter of simple counting to verify that the number of brackets with the same values of n- m, s is, for n = odd s = 1 2(m+l) brackets s = 2 (m+l) brackets n = even s = 0 (m+l) brackets s = 1 2(m+l) brackets s = 2 3(m+l) brackets The various brackets having the same n, m, s may be considered as the components of a multidimensional Vector. The rotation 4.6.1 in this multidimensoibrs Space is a linear Transformation T of this Vector. By means of a co inate trnsfore an oration in this Space, T may be diagonalized and the eigenvectors and. eigenvalues found _______________________________________ 4 ______________________________________4_

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Another way of stating the same idea is: Among the brackets with the same n, m, s certain linear combinations can be formed which transform into themselves, multiplied by a constant only. These linear combinations are the eigenvectors; the factors are the eigenvalues. Under a transformation T corresponding to a rotation (4, 6, 1) which belongs to the group of the reciprocal lattice, any eigenvector must transform identically into itself. This can be achieved in two ways only: (a) either the eigenvalue = 1, (b) or the eigenvector = 0O In the first case no restriction is derived from the rotation for the eigenvector belonging to the eigenvalue 1. The restrictions, imposed by the crystal symmetry on the brackets, find their expression in vanishing eigenvectors belonging to eigenvalues that differ from unity, It is the purpose at present to lead up to the formulation of the theorem which governs the eigenvalues and eigenvectors. To this end some notations are first introduced. A bracket with given numbers p', p, n will be written with outer indices in the form (a-iB)Z(~+i~)s-z 2-s (4.6.3) 7'^^ip^V^, (4.6.3) where z is an integer and 0 < z < s. For the various possible values of s and z the meaning of this notation is given in Table I. TABLE I THE MEANING OF COMPLEX OUTER INDICES (4.6.3) s z n = odd n = even l-;.,",... "., L... i 0 - [ ]33 21 -i[ 2 [ [ J2 2 0 1 2 2 2 1 2i[ i12 [ ]z+[ ]22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN All brackets have the same p',pn. Further, a function F(m,p,w) is defined by the generating equation m (l+x)P(l-x)m-p =: F(m,p,w)xW, (4.6.4) w=O where w = 0,1,2 --— m. It is easily shown that F(m,p,w) = S(c) (wPq) (-qP) (4.6.5) q with q taking integral values from the largest of 0 and w-p up to the smallest of w and m-p. Strictly speaking it is not necessary to limit q in this manner since terms with integral q values outside this range vanish. The following theorem can now be stated. Theorem The eigenvectors corresponding to a transformation of the type (4.6.1) are ip F(mpw) [m-p, p, n-m](c.i)z(+i)s2- (4.6.6) p=o (one eigenvector for each pair of w and z). The eigenvalues are ei(m+s-2(w+z)) (467) e (4,6.7) The proof of.this theorem is quite lengthy and will not be given here. The manner in which the theorem is applied should be clear from the foregoing. It enables all the symmetry properties of all the brackets to be written down. This will be done in the next section. We add here one remark in connection with the fact that (4.6.6) is a complex eigenvector. If such an eigenvector is to be zero, this implies two equations, namely,, both its real part and its imaginary part must vanish. The conjugate complex eigenvector is then automatically zero and need not be considered separately. Now the complex conjugate of an eigenvector designated by w, z, is the one designated by m-w, s-z. Thus, for the purpose of finding the relations _______________________________ 6 ___________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN between dependent brackets our attention can be restricted to eigenvectors with either w m/2; z < s or (4.6.8) w _ m; z < s/2 On the other hand, for counting the number of free or independent eigenvectors, which is equal to the number of independent brackets, it is easier to let both w and z run through their entire ranges. It is of interest in this connection to discuss the quantity h m + s - 2(w+z), (4.6.9) which ranges from m + s-2z downward in steps of 2; its lowest value being m + s-2z-2w. Every time that h = 0 or an intergral multiple of p, (the multiplicity Of the k3 axis), an eigenvector is free. It is not a simple matter to write down in closed form a formula for the number of independent brackets resulting from this prescription for given n and Ai So far we have obtained a formula for the case p = 2 In that case, which has the symmetry C2h, the number of independent eigenvectors or brackets is forn = odd 3n + lOn + 7 (4.6.10) for n = even n2 + lOn + 8 2 6. The Dependence Equations Among the Brackets In this section all the dependence equations between brackets up to m = 4 will be derived, and since the method is explained in section 5 the equations for m > 4 could be given following;'the same system. In order to make the procedure as systematic as possible another rule which is a consequence of the theorem of Section 5 is first stated. Rule V If h = m + s - 2(w+z) is a multiple of 2 3 4 6 12 none of these but not of 3,4 2 3 12 the vector is restricted for = 3,4,6 2,4,6 3,6 4 none 2,3,4,6..__________________, 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Thus,. for the first few h-values, the following axes are restricted: Restrictions for L = occur for h = 2,3,4,6 1,5,7,11,13,17,19,23, -- 3,4,6 2,10,14,22, ---2,4,6 359,15,21, --- 3,6 4,8,16,20, ---4 6,18 no restrictions 0,12,24, --- In order to apply the theorem, all that is needed is a table of the fuction F(m,p,w), This caa easily be constructed if use is made of the following relations that follow-directly from the definition (4*6-,4) F(m,p,w) = F(m,pw) (7.1) ( p)F(m,p,wl = (- )mP-w)F(mm.nwm p) (4,7.2) w F(m,p,w) = (-) F(m,m-p,w) (4.7.3) F(m,lp,) =. 1 (4.7.4) F(m,m,w) = wm) (4.7.5) F(m=even,.p=odd, w=m/2) = 0 (4.7.6) F(m=even, p=m/2, w=odd) = 0 (4.7.7) F(m,p,w)-F(m,p-l,w)-F(m,p-l,w-l) = F(m,p,w-l) (4.7.8) Table II lists the values of iPF(m,p,w). The dotted lines bound the region of p and w from which the remainder of the tables could be obtained by the symmetry formulas (4.7.1) and(4.7.3). These imply that columns are symmetric for even m-p and antisymmetric for odd m-p, and rows are symmetric for even w and antisymmetric for odd w after dividing the numbers first by ip. In using this table for finding eigenvectors, it is of course practical to divide each row by any common factor. With the help of Tables I and II, the eigenvector for any set of values ma,w,s,z can now be written. For convenience the inner part of those eigenvectors used in Tables IV and V, i.e., the part which follows from m and w without reference to the outer part which follows from s and z according to Table I, has been explicitly listed in Table III. __________________ 8_______________________8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE II THE FUNCTION i'F(m,pw) 0 m = 0 p = p'= w = 0 iF(O,O,O) = 1 O 1 i m =2 \p 0 12 w\ o 1 i -1 1 -2 -0 -2 2 1 -i -1 _m = \p 012 2 w\ O 1 1,-1 -i 1 -3 -i -1 -3i 2 5 -4 1 -5i -1 1 -1 m=l 4 \p 50 1 2 w 0 1 i -1 -i 1 1 -. -2i 0 -2i 4 2 56 0 21 6 5 -4 2i 0 2i 4 4 1 i -I i 1 etc. 1 - -2 0 ' -19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE III INNER PARTS OF EIGENVECTORS FOR VARIOUS m, w m w Inner Part of Eigenvector, -.,.. i -.-...-. f 0 0 [OO,n] 1 0 ([l,0,n-1] + i[Ol,n-l]) 2 0 ([2,0,nn-22] + i[l,1,n-2] - [0,2,n-2]) 2 1 ([2,0,n-2] + [0,2,n-2]) 3 0 ([3,0,n-3] + i[2,l,n-3] - [1,2,n-3] i[O,3,n-3]) 3 1 (313,0,n-31 + i[2,l,n-3J + [1,2,n-3] + 3i[0,,n-3]) 4 0 ([4,0,n-4] + i[3,ln-3' - [2,2,n-4] - i[l>5,n-4])+[o,4,n-4]) 4 1 (2[4,0,n-4] + i[3,1,n-41 + i[l,3,n-4] - 2[0,4,n-4]) 4 2 (3[4,0,n-4] + [2,2,n-4] + 3[0,4,n-4]) All the bracket relations can now be listed. This is done in Tables IV and V. The tables are divided into n = odd (Table IV) and n = even (Table V). In the first four columns the sets of values of m, s, w, and z are listed. Now a pair of such sets, m, s, w, z and m,s, m-ws-z, give conjugate complex eigenvectors. Therefore, only one set of such a pair has been listed; namely the one corresponding to nonnegative h. The two sets of each pair correspond to equal and opposite h values. The next column lists h. The sets of mi s, w, z are arranged so as to have together those h values which correspond to equations for the same axis orders,., listed in the last column. The order is further determined by Rule V, First, sets with h = 0, or a multiple of 12, yielding no equations are listed; then the sets with restrictions for p. = 2,3,4,6; then for - = 35,4,6,etc.r as indicated in the example given under Rule V. The next column lists the equations that follow from setting the eigenvectors corresponding to the set m, w, s, z equal to zero; the last column indicates for which values of. these equations are valid.. Whenever justified^ later equations-use the results of earlier ones in order to obtain their simplest form. Equations of earlier lines, which may be disregarded because they appear later in simpler form, can be recognized by parentheses around the corresponding p. values. For example, in Table IV the case (ms,.lz) = (2,1,1,0) results in two equations which are of interest for. = 3; whereas, for p = 2, 4, 6 the results of this line may be disregarded since they have been combined with the results of (mwstz) = (2,0,1,0). On the line of the latter, the combined results of both sets are listed in their simplest form. l_ 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As before, an arbitrary odd number is denoted by w and an arbitrary even number by e. We have discussed in detail the way in which these tables are constructed in order to permit easy checking of the relations presented. TABLE IV BRACKET RELATIONS FOR n = ODD, m = 0, 1, 2, 3 m w - z h Equations j. O 0 2 1 0 - 0 0 1 0 1 [00CJ13 = [00(cj23 = 0 2346 1 0 1 1 0 1 0 2 1 1 [10eJ2 = [0leJ12 = 0 2346 1 0 1 0 2 [10E].3 = [01E]23 346 [10e]23 = -[01]j13 346 2 1 2 1 0 - - 2 1 1 o0 [20(]1]3 = -[02.]13 = -.[l011]23 (2)3(46) 2 0 1 1 [20C]23 = -[002W]3 = - [1 1. 3 (2)3(46) 2': 0 2 1 2 [11]C12 = 0 346 [20c1 2 = [02W] 12 346 2 0 1 0 5 [200]13 = [020] 13 = [11].13 = 0 246 [20]23 = 0]3 23 3= O12 113 ] 2 = 0 246 3 1 2 1 1 5[30-]12 = '[12e]2 (2)3(46) 3036]12 = -[21c],2 (2)3(46) 3 1 1 0 2) [30e]13 = [03e]23 (3)4(6) 3 0 1 1 2J [0OE]23 = -[035e]3 (3)4(6) [21]G23 = [12e] 3 (3)4(6) [21e],3 = -[12e]23 (3)4(6) 3 0 2 1 5 [30e]12 = [03E]12 = [21e]12 = [12e]l = 0 246 3 0 1 0 4' [0e]113 = [035]23 = [21e123 = [.12e]13 6 [30c]23 = -[03e]13 = -[21.J13 = [12e]23 36 ___________________________ 11 ____________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE V BRACKET RELATIONS FOR n = EVEN, m = 0, 1, 2, 3 m w s z h Equations I O O O O O - - 0 0 2 1 0 0 0 1 0 1 [00e]13 = [00e]23 = 0 2346 0 0 2 0 2 [00e]11 = [OOe]22 346 []OO]12 = 0 346 1 0 1 1 1 0 0 0 1 [1O ]3s = [Ol01]33 = 0 2346 1 0 2 1 I1 [lOw]3 = 2-[lO]22 = -[OlWol2 (2)3(46) 1 1 2 0 1 [01d)]l = -[01o]22 = [lOw]i2 (2)3(46) 1 0 1 0 2 [lowoJ13 = [01w]23 346 i. [lO1233 = -[01]13 346 I o 2 0 3 [o10.1, - [10o122 = [low]o12 = [01 ] ll= [01- 2 [01]2 1 [l12 =0 246 2 1 2 -0 2 1 O O O2 0 2 2 0 2 1 1 0 1i 2[20e]13 = -2[02e]13 = -[lle]23 (2)3(46) 2 0 1 1.1 2[20e]23 = -2[02e]23 = [11e]13 (2)5(46) 2.: 0 2 [20e], = [02eJ22 346 2 1 2 0 2j [02e11 = [20eJ22 346 [20eJ12 = - [02: J 2 346 [llc]11 - -[lleJ22 346 2 0 0 0 2 [20eJ33 = [02e 33 346 [lle]33 = 0 346 2 0 1 0 3 [20e]13 = [20e]23 = [02e]13 = [02]23 = 0 246 [lle]13 = [lle 23 = 0 246 2 0 2 0 4 [l ]12 = [20e11 - [20e]22 36 2[20e] 12 = -[11e]11 [1le]22 36 3 1 l o - - 3 1 0 0 1 330o]33 = -[12w]33 (2)3(46) 3[035]33 = -[21C] 33 (2)3(46) 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE V (continued) BRACKET RELATIONS FOR n = EVEN, m = 3, 4 m w s z h Equations 3. 1 2 1 1 2[21]12 2 = -[530c] 1 + [30)122 = =- -[12a)]11 +[12a]2 2 = 2[03o12 (2)3(46) 3 2 2 0 1 2[120)]:2 = o053]11 [o00322 = = [.21:DJ1 -. [21(] 22 = 2[305]12 (2)3(46) 3 0 2 2 1' 2[30a]ll + [530w]22 + [12W0]22 = 0 (2)3(46) 3 0 2 0 5 2(03w]ll + [05]22 + [21B]22 = 0 (2)3(46) 3 1 1 0 2 [30ao]3 = [035]23 (3)4(6) 3 0 1 1 2J [350c]23 =.-tb3513 (3)4(6) [12(W].3 = [21C]23 (3)4(6) [12J] 23 = -[21c]13 (3)4(6) 3 1 2 0 3) [530>]11 = [3C0]22 = [30]j12 0 246 i * [03c]J1 [035u]22 = [0]2 = 05] 2 = 246 3 0 2 1. 35 [12.u]ll. = [ 12012 = 2 = [21]Jl = = [21~] 22 = [21w]12 = 0 246 3 0 0 0 3= [3120]33 = [2 33 = [1233 = [0 33 = 0 246 5 0 1 0 4 [530wi3 = [21ow]23 = [12w01 3 = [053C023 36 [30w]23 = -[21w]13 = [12U)]23 = -[05] 13 36 4 2 2 1 0 42210 0 42000 4 1 2..2 0 4 2 1 0 1 3[40e6]3 + [22e]13 + 3[04e]13 =.0 (2)3(46):3[40e]23-+ [22E. 2 +.-3o[04]23 = (2)3(46) 4 1 1 1 1 2[40e]13+ [31E]23+ [[e]23 - 2[04e]13 = 0 (2)5(46) 4 0 1 0 5J 2[40123 -[31e]13 - [135]13 - 2[04E]23 = 0 (2)3(46) 4[40e13 -[31SE]23 + [13e]23 + 4[04e]13 = 0 (2)3(46) 4[40e]23 + [316]13 - [13e]l3 + 4[04e]23 = 0 (2)3(46) 4 2 2 0 2 Simpler forms below (346) 4 1 2 1 2 Simpler forms below (346) 4 1 0 0 2 [40c]33 = [04c]33; [31E]33 = -[13e]33 (3)4(6) 4 0 2 2 2 Simpler forms below (346) 4 1 1 0 3 [40e]13 = [40e]23 = [E31"]3 = [31e]23 = 0 246 [22e]33= [226e23 = 0 246 4 0 1 1 [04EJ13 = [04e] = [ ]3 = [23 1 13 = l 23 0 246 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE V (continued) BRACKET RELATIONS FOR n = EVEN, m = 4 m w s z h Equations p 4 1 2 o 4 [31e]1 = -[31e22 36 4 o 2 1 4 [13ejll = -[13e]22 56 2[40e]11 - 2[O04etj = [31e]12 + [13e] 2 36 2[40oe]1 + 2[40<1'22 = [22e]l1 + [22]22 = - 2[04e] + 2[04e]22 36 -3[13]12 + 3[531]2 = 2 [22e]1 - 2 22 = 12[40e]2 - 12[04e]z1 36 -8[40e]12 = 3[11e]1 + [31e]1 36 8[04e]12 = [l13E]l + 3[31eJ]1 36 -[22e]z2 = 3[40e]12 + 3[04eJ12 36 4 0 0 0 4 [40e]33 + [04e]33 = [22e]33 3 [31e]33 = [13e]33 = o 6 4 o 2 o 6 [4oe111 = [o4e]22 4 [40e]22 = [o4e]1 4 [22e]1 = [22e]22 4 [31e]12 = [13e]12 4 [4oe]12 =-[04e]12 4 [22e]2 = 0 4 [31e]Jl = -[13e]22 4 [31e]22 = -[13e]1 4._ The Tables of Brackets for the Eleven Classes Using the results of Tables IV and V, the brackets up to n = 4, can be tabulated for.S2, C2h, S6e C4h, C6h. In order to add the effect of a twofold axis along k1 the results for C2h are subjected to a pernitation 3+1, 1+2, 2-+3. Combining the results of this permutation, which only indicates an additional number of zeros, with the results of the above classes the table of brackets is obtained for D2h, D3i, D4h, Dh:Finally, in order to obtain the results for Th and 0Oh the results of D2h and D4h were (rnntaliy) subjected to the same permutation, this time equating all the results before and after permutation (not only the zeros). It is convenient to use here the easily proved rule that if in any bracket written in the form [(23j'i..12).PP, aat least one of the numbers 1, 2, 3 occurs an odd number of times (inside and outside counted together1)i the bracket vanishes under this permutation. 14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In this way Table VI has been constructed. On each page the bracket symbols for one class are listed. S2 is not listed since all brackets are arbitrary except those which vanish due to (4.4.1). The number of independent brackets for each value of n is listed in the second column; the inside of the brackets is listed in the third column; the outer indices are listed horizontally in the remaining six columns. In cases where one bracket is expressible in terms of one or more others, a choice of independent ones has been made that leads to the simplest expressions of interdependence, In order to see all the dependences, it is important to observe where a symbol is found in the tables. For example, for C4h we find the symbol [200 11 at the place [200]1l and at the place [200]22, signifying that [200]22 = [200]11. Thus, if a symbol occupies its own place it is chosen independently if it occupies another place it is dependent. The tables will be used in the next report for interpreting the measurements on Bismuth (class D3i). They represent, for the field of magnetoresistance and Hall effect, the equivalent of Voigt's well-known tables for the elastic moduli. 15

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN cmN c N N w NcNm N N N 0 N N N N I u 1 1 (o o o 1 8o I o (U o I o1 c. o o o, (U o ~ ~ cu I I rl O Z 0 g 0 0 o o o o r 0i Moo oo o'0 cu o - H O H H H 0 H H H HH H HN H H rr _ 1 c_ t O f_. NNNN NCD N) N N N) N N N N O H o 000 O OOHr-HO 00 0 0000 0 0 H H O 0 H H O - OH Hi 0 0 C0 0 I-' ' - IOQ HO OH OO H HOJ OOHM' OJH U C CO C COCO cCO CON N N N N N NCOCO NNA N N N N N N NN Nc N N N N |cu- |o --- o o o — o o o o o o o -o o o o o o | o oJ..o o o o _ cuo o o N —~ s ( ~c o ~~~ ~~ ~ ~ ~ ~ o o o ocuo ~o c O \ 00 O CU O 300 00 0 0 0 0 H 0 H 0H 0H 0 0 0 CM 1 oH 0 0 n 00 H 0 0 H OOOO O No o C O H 0 00 H 0 08 ^ 1 Nu NNN CU (U EU N N NU NNNNN N (U A CM 0 000 0004000 0000000000 00-toooo0oc uoooo H-i0 l 0 000000000000 OOOOO O 8 N 0 r88I OO2 _ rHl 0i 04 \ O 0400 Hu - 0 o 0 H H 0 cu Hu o 0O

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 0 0 cm| CN J HHC1 (\I I( Cm CNC (N\1 N N N + I; N N 1 1 '""''"T* '"T' 1- 1 N T, o 1 -1 ' 1 ' 1 l-l 'i<^ 1^ + + 1 ' 1 oi 0U 1 T 1 ri N i I I I ++ I o II 4 I - r- - -H, H H 0 H r I C 0 00 0 0 C J 0 \, 0 0o a~ oo o (++ (o o oNI 00 CCCO HOr- 00 HCONOCO HOCOH C OQ CHlN\HON OCN uOHHC\N HO 00 HHO (\4CO HCuOQCuI0MCHC HC\|(0| ( 0 9\ N\V0 C i 0 _ - O - r- - O 0- - - - -0 1 O H O 00 00C 0~ 0 0 OSi 0- 0 0 ~Q0 ~ 0 g0 g0 C n 00o 00 0 11 1 1 I C O Oa coC D g0g N cmNj C) ' ) CMJ)) J N N0 N N 0 CmCIN IN ON C cm NcoC) f)) 00 O O c 0 0 0 O- 0 ~, o 0o 0 - ~.. H r H - 0 0O N N.H..0 H N ~oo 'o o ~ oo 2oN OO0 O1 00 HH 0 H H _ _ _ t _QO l 0 H H 0 N \ | O t + C r8 8 CQ ~ ~~)~~ ei G~o co I II 2280 '~0'" Q00 0 0 i I O O 1 O O OO O O Ol O i~ X~ O — ~) O9 g7 Or) Of) O) Or) O ) 0 0t o O 8 000 000ooo oo0 oo0 0 — 0 C'JRuOHH EH H H o H 0 0 oo ogoo n CM N HCH N H H HH \ - C) r-)(l - lC H 0) H H H H ) 0)0 H H H H1 H H H111 0 ) H H H 1 H O H )ooo H H H ooooooo ooo H ooo o u 0 00 00 - HO _ -_ 0 H _ -0 |- + C!, C HN I I + I I I + I I c~a I I p0,-~ 0 r-H HH 0 0 01 O O O O ~O frl rl1 0 O O O O Xar l O O C rl C r l K OJ *cJ I ICU 0- HO r~0 C 0 | O | | (U N O O aN a | 4 o o O, N g 8- 1 --- 0 H1 H 1 ) H H H))) H H H H H H H H H 0) C)) C)) H.H HH H -i Hq H H H i cH o H H-i H 0 O 00O 00 HH O OI-H O 2O H0 0 OOL'HOO 0000000000 om —OH K'\ HO 00N0 C' 0-A?M r?~~~~O H CuM~~~~~~~~~0 0 0 -O 0i 0 0 O.... 17

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN cm C( (V (V CM ( c( C C CM cm(cCM CM 0 rl Nl rl H Nr l H H H r H C Q 0 00 K) O U Q - N^ CM U N 0 0 0000000000Jc'Joooo o ~ ~~~~O ~C rO HON O N0 0 H C~t~j H H CD CD H 0 000 OO r-0r- 0004004000 00olol oo0K rHN\ Hr-l 00 HO O rO-JO r J rl t;O OH ~ ~ H *-H-1 ( H ~H ~ ~~ ~ C VOH ~ ~ ~ H CO A^ cm VC O CO cm E()VcmV(V (V V (M C\1(V( VCD (Vc( 0* V' o0 000 OOOH-iH-lO 0000 OC O oOO 000^ H00000n^n0HHo ISu ~ I I I I I I || 0 rl 0 n E-o n 000 O 0Ejoo o o000 0000000 o o oooooooo +' 0 0 0 0 0 C O^ H OON O1NHo H N -1 NH- 1 _ O r1 (O C\1 M 1 H 8o ooQoO ooo o'oooo1o 0 CVJ CUO 1- HHC\JCV 000 88o000 0000000000 oo 0 009 Oor) 0 00 80 0 000 0 0 0 0 c 00 0 40 -t 0 c \j -0 CV JO OJOOd ~0l H 04 0 r- rl I U n rl I\ ( 0 ( r CIIC U c IF IL I r4K~ -jCI I 0 o 0 ooo ----------- oo oooooooo 1 -----------------------

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 8 O CM J 0" cuCM CM cq 0 6I+ cq H|~~~~ r-i{~~~~~ N N N N N N N N N CMH NJ ( H H 0 - r- r-.. H,- H, -4 A M 0 O0- '-ooo oo00-'0o' —000 0000ooo0 — Ht~ Hr-4 00 0 tl r-1 r-1 0H O O, 0 0 0 00 0 0 OU C ~ 0 00O o c cu cu 08 0 0o b <M oH H,1 H H H H H 0 0 0 00 0 0 0 cH 0 H\ 00 H 0 ~Ocy ~301 ~o ~(o o I | {r ~|OO ~ OO |HC) H ) O | xH ) HHC 0) C l H 0 8-x-O 000'-r-'0 rO0oo 0o _>0 _ oo0 000o o- r- 8 ~ 0~0000-'- N 0 08 ri0 0 oo 00 88u H |N|8 | O Or O O O rl O r8 I r O O U - N N O I I 0U COH CO CH iOO rO O CO PO, K\ I I OOOv O O O O CY O O O Ocm C ) O I c O CO CO O N CO I I O IC\;O n HI N8 00 r-IH QQ H H Q 00 H C A\\ n 0~HH 00 KSr~'0 0 0000 00 h I I I I I N I rI g H h 4 0m C') CO)C)tCv) C') CO') Cf)COC )CO., oF on 8 2 oo8o < noo o o..2. - ~.o0 o o 0 o o o-" —'o C-i N 0 00 8 -ot 21)) OCU 0 wqcli~~~~~~~~~~~~~~~~~~~~~~~~c 0o - H HCCH-i <-l (\j Hm r-i-i H (MCM (U H HCCH 0000000000 MHHH HHrCCMOCj HJ *-* 000.-I'-"r-''000 HHoHOO O OO HH0 0HoCOO OH H 00 0 ~ 0 08.. — Oo - 0 0 oU m o\o.o c 8 C 0 Cu H 0 Hc - 0 ( 0 01 K 0 0 <A 0 0 OO 0 r O, 0 -4.oN N M I~9 0 r-l 0 01 0 ro -o 0 0 0 0 CI h"\ C 0i C-i m r a o cu Oi r......... --- — - 19.....

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN N 0 0 0 o 0 0 0 0 0 00o oNOo 000 O rOO O 400O O 0 0 c - oo g o R o o o co o | H0o o0 00 OOOHO OQOO0O00O0 OO000rOHOOOOOHO 0 0H C o r n 0- 1 0 "I 0 Q OO 8 r' 00 000 O0OO'0'H OOOOOOHOO OM o 0 OOO_1HO O CU Q HKO O H H II O 0 H PI 0 N 0 > 8 00 0 Oor-ooo ooooooo o Hoooooo~ooooO _ I _ __ _ _ (J __J 00002 000000 0000000000 0o00OOOOOOOOO H-00C Q — 'r --- OO*-H-lr-tO O -- 0 —r'- -- - -l — -- -- r — - 0 0 0 0 — 0 --- — 8 0 O CU 0 rlCV l (UCU 20 r-4C)n u.Ri2 Rr- A2 U j(UIirC: E: d E:~~~~~-44nj4m 4 Z 0A

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 00 0 -Q Cl OJ cm C WC CM Cu cvC ( cm I,H HI ( | o o | ~ ~ (, - H H ( ~C oo0 0 H o ' oooO OOi 0 O00 10 i r01 0 0 0 00CM H1 1+ H 1 1 w I I 0 00 0 0 sd l l l l ld. 0 0 0 O 0 0 0 0 O. OJ CU I CO T f) CD CO CD Y)l) C') C CO CaCCC CO C) r<0 0 _:t I n r ^'~lT CU S -,o, C |O C| o ~oP8 OICI l | 80 ~ OO o |NNO O I O O O O O O O O O I OC) O O ( O O O Cr)ONO O^1 O CC"M |||NI I V)I I 21 -000 0 -00 - 00 -~pgI..o o03 3 mo...~oo.C <CI I I CQ [U CO CO CO CO CO CO CO CO CO CO CO CO(U (U ( C C CC C CQ Q0 1 000 O O QJ0 0 00 K 0-0000 0H00 O j CUM CO Q So C\U H::) H,-t 000 %000r-lOO 0000000000 88O0'00O0CUACU8^0 H CM 00 0 0 000 Q i0oOO 0000000000 O0'0000C000 i 8l 0 Or010 OCMOH~r-l Q^Or-CUOOOt-lr- O -^Oi-tV H K' 00 0 0N 0 K' 0 1 000 H 0 ~ ~ ~~~ H r H-1~~~~~ CU H^ ~ ~ ~ ~ ~ I HHH H " 0 0 CU C CO 21..

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN N N cm N N( N( H 0 0 O0 0 0 -0 rH S - r —0 - C N O O JO i lr H N COCOCO( c CO l _ N C) N NM N NC H|I 0 OOO OOO O 000Jo000 0 000 OK\Hoo00000ooo 0 H- OH M H n r< H r-l 0 o H 00 C I ~S( 11 I A cu c m c OO O 00 OO OHOO O 00o 0 OO O OO O000000OrO00 Oj 0 H 0 QHC Hn H *0 H0 0 H 0 0 CUj I O 0 0yci,~~~~0 0 H o \ 0 000 oo00o 0 0000000000 oo0-tooooooj0oo0000 i 0 o8 000o 000 8ooooooo0 H CN H H HH HC\O 0 CUH rH (lrrl N H H ( AH HN H cuo 0 3 oo0 330ooo 0000000000 00-00000ooooo0000 0 02280 0 00 cuj H 0 000 00000000 '00 0 000000 00 CJ0000 0 0 Od' 0. OH-IO OCJOHOHr- Q! HcdOOC HH 0-Or-H n H 0 0 NHCtiJ 0H JH 0o o 0-l oo oo M 0\ 0 o o _...... — -22,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 00 00 cu N ~~~~~~~~~~~~~~~~~~~~ (U~ ~ ~ ~ ~ ~c ~ 04 I j -r cc cm H cm H4 H -4 HH 0 (M C\J 0j 0 OOH 000000 OONOHOHOOO OO0000000 o 0 0 0 CM 04 0 00000 ccl t 4* + H OjH ~ ~ r-H HH H\ 0 0 0 00 0 0 ** 04 040 cc io _:c _\ () C( (4) H (CO (CO CO CO (44 (44 (44 (4)(I W (44 (44 H-1 0 0o0 0 OOOOH0 0004000000 OOOOOK'\HOOOOOOHqO 0 H O H H4C K4C\H-\ 00 4 0 00 0 0) (i ) () H 4 (4) (m (cm (CM CO C\ (cm4( c( 4 C o O o Ooo'0o0 o 0o o0o00o oHoo o0 0400 OOOOOOOH i OO ^ ~ 1-1 I-. * — 1~~~~~~ ~ ~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~-1 ^-.4^^,\ 0 0 ~ H 0 HO H K 04 +p II ~ 04 (4) 0440 0)40 0)0)0) Q( - COoc CO ( 0) ) cQ 04) Wi~~~~~~~~~~~rc ' 00 ' 08 0000000000 8 8. 000O00048O00O Q co c'S O 0c o E4~~ ~~~~~~~~~~ N 0 CM' CM oQyo oo o o o o3 oo o oo^o~ 0~~~~~~~~~0 0 H 0 H-4H (I H 'I HCM H 04HH 04H HC W IN ~r( -=t ~ c~ r. -111 C0J 0 000 088OJO00 0000000000 00-=0000040(JO0O c~~ 0400 0 004* cli cli 0 -t. 0 0 (\j 04 -t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0~~~~ tt~ ~~IIt CM r-I H W 14 Iw H-1 H04H HHH WH0 H H 04 H HH H (|H (44 0 000 OO0O OOOOOOOOOO 0000000000 -si8OO H 0 OOO 0 0OQ0O0 C\COS~cu c\j 0 0J 04040 *00 0 4C4 H O H J W K\ * II It II IQ ON I IL j r' 04 a4 04 a4 a4 --------—. ---- _ - 25 --- —-------

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN |NO|OO|OOO OOO o l O 3 O Or | O O O OOOO. 0 0 000 | c (C Cu (C CC (CuU CU C 0 000 0000 0 H 00000000 OOOOOOOK\r-iOOOOOH 0l o r-r i o roi 0 H 0 0 0 0 cM Cu Cu Cu Cu CM CM Cm -|1 o oo o | ~ ~ o~~'cM~~ ooo ~o o oooo~~~ ~~ ~oooooo ~^~ o HQ 0 H KIH rH ^0 2 0 H 0 0 OJ N l cm ACl ClY C C M C n 0 000 QOOHO 0 000;;00 8000Cu0Cu\oHoo0ooooooo0Hoo H 0 H 0 00 C\1 H-< II w_ O ^ ) N O 0 0000 o o0oo o o0 o o o CCH CU' CCu H 0 <3 Q ^ CuCuH COCMH CuCMH y y0C~ 0 00 ^ CuCu uOHCO -M CU H HCuCO CuM C qHCuO OJ 80' 000 0888QOO 0000000000 gg0000000Cug NO000 0 nCMCM Hu H HO cy NQ 0C oo0 80000 00000 00000 8800 oo o o o ooo 0 0 CM 0 0 ~ ooo a Fs~ ~~~~~~~~ - 0 rlri0 0 C i r400 rinrjRC \ \ - i 5Ii~~~~~~~~~~~- -

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN P \0.0 P~~~- 0 H W 8, \ P L,, o N o H HN ) R 0 r00 H r\ 00H 0 000rr Oo r HroOr 0 0 0 o o ~ o ~ o oooooro oooooo888 0 0oooooooo ooo8oo o00o 8 rIHri, Ii)1\) 1 r)1 H 000 008)H 0 oo0o oo 0 O 00 O O 0000000000 o0 o 00 0 8 r H\) 3 25TI) MH \) P w K) 1 NrN 1 I r~II oo r oroooooo~ ooo oo o r 0 0 0 0 O 0 H H\ 00w O00M 0000 000 0000 00 00o0 0... 2 (1 6 1-3 ( 0 ~ 0o 00 OO0 O 0 0 0 0 3 H(16 HOH (13(13 0 R 00 P 00000P00P14 ^000 0rro o o oo2 G o W 0 0IQ 0I 0I 0I ' Pooooo\!A\!ooooooo ooo oroooo FJ 0 o 800 o \ --- 25 ---------------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN C. EXPERIMENTAL PART 1* Equipment The holder with Cardan suspension mentioned in the previous report was completed. In this holder, the bismuth sample is encased in a lucite capsule with copper gallium well electrodes separated by a thin mica sheet. The connections were made with two small silver wires in spiral form in order that the rotation of the capsule may be almost torque free. The MR effect of silver wire was checked to be negligible in the range of the magnetic fields used. The sample wire can be rotated around its own axis; the angle of this rotation is denoted by Q * It also can be rotated around two mutually perpendicular axes. One of these axes has a fixed position perpendicular to the magnetic field and is called the ~ axis; the other axis r is perpendicular to both the Q and 0 axes. Thus r represents the angle between the sample and the 0 axis. The sample holder permits turning around these axes within a 1-1/2-inch gap between the magnet poles. (See Figure 1). In order to enable us to detect slight deviations from the law Ap/p = BH accurately, it was necessary to calibrate the magnetic field with more precision than " 5.. The new calibration of the same magnet was done with the Cooperation of Dr. W. C. Parkinson of this laboratory by the method of proten spin reasonance. This method is capable of calculating the values of the magnetic field H accurately to the fifth digit, which is more than needed since the current reading was only accurate to about 0.5 Therefore all calibration errors will be due to the readings of the magnet current I. Graphical analysis of the dependence of H on I showed H = 199.5 I (1I <6 amp ) compared to the old calibration H = 196 I -10 (+ 5) 2. Materials The bismuth used was from the same sources described' in Report No. 2. Besides the samples prepared by the drawing method, some samples prepared by coating bismuth on small glass tubing, wire tried for the MR effect* __________________________ 26 _____________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Fig. la. Mounting Diagram of Wire AB in Cardan Suspension with Angles i, O., and MIr. I /_ 0 Fig. lb. The Trigonal Axis is Set, During Measurement, to Point at P. ------------------- 27

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3. Measurement s a. Dependence of the MR Effect on Orientation and on H. Since the theoretical development requires many measurements with various orientations, in order to find the fundamental constants of the bismuth crystal, the MR effect was measured for various '4 in steps of 100 from 0 to 900, and at various 0 in steps of 20~ from 0 to 3600. The angle 9 was to set the trigonal axis, the wire axis and the 0 axis in the same plane. This would make possible to align the trigonal axis along the 0 axis. As reported previously the trigonal axis can be determined by looking at the cleavage plane. under a reflection microscope. However, in this way the angle between the trigonal axis and the wire axis is only determined to within + 2~, consequently the plane containing the two axes cannot be determined with sufficient accuracy as required to set the angle Q. This plane is determined instead by using the criterion that if the @ and 0 axes coiniide (t=0) then ~ is a maximum as a function of 9 or 0 when the trigonal axis is in the plane of H and the current through the sample i.; From the fact proved in the previous report that \P/+H \P -H it was expected that the measurements at 0 = 90~ or 270 at any r would give the same value of. The values were found to be the same within experimental error, provided i is sufficiently anall (in our case < 20 ma). The value I2 p1 90 (X=270 was used to "normalize" -the measurements done on different occasions on the same sample. (See Fig. 2) It was found expedient to use a magnetic field of 594 gauss. This would insure 1 At - f -P p +H \p-H and yet would cause enough deflection in the galvanometer, with i from 2 to 20 ma. In order to calculate the fundamental constants of bismuth crystals one must determine beside the position of the trigonal axis with respect to the directions of H and i, the position of a binary axis and the resistivity of the sample. It has been tried to find the binary axis by an etching 28

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN, method. The technique has not yet been successful. The determination of the resistivity involves breaking up the sample to average the cross-sectional area viewed Uadne'a microscope, and thus will have to wait until the binary axis is determined. A few test samples were tried and the resistivity was found to be of the order of 100 x 10 Q6 c0. The data of measurements will be kept for later evaluation. A family of curves representing a typical A2 measurement in various directions is shown in Fig. 2. Fig. 3 shows the set of measurements for one particular sample which exhibits no shift of the maximum and minimum points. Th7 curves show accurately sinusoidal shapes, indicating that if (Ap/p) 2 were plotted as a function of direction, an ellipsoid would result, This manner of plotting will be discussed with the theory of the next report. At the closing of the report it was tried to raise the magnetic field intensity enough to show the next higher power term, or AP = BH2-CH4.p This was done with the sample in the position Q 0. From the above relation it is seen that Ap/pH2 should be plotted versus H2. A typical plot is shown in Fig. 4, from the data given in Table 1. Note the expanded ordinate scale. This plot shows a straight line relation over its major part, permitting the determination of B and C. The deviation of the first point is within the limits of error, which are larger on the left side of this plot than on the right. Table 2 gives values of B and C for a few of the samples measured. b. Singleness of Crystals. To be sure of our samples being single crystals, we studied the direction of the cleavage planes after successive breaking while the sample was held fixed under the microscope. It was found that if the sample looked continuously smooth by the naked eyes (i.e., no cracks, no bubbles when reflecting light) then it had con-s stant direction of the cleavage plane. Our samples are now being taken from the center part of a long piece; the end parts are broken and the cleavage planes checked to be the same. Only single crystals are used for measurements as reported under a). 29

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 0 III0 0 II II Ia(D _ _ _ _ _ _ is 111?vno, en ^^ )S o o o o oo a) s @ O O~tO x - 30? X o r - 30 I --- —----------- CDl O~~~~~~~~~~1 t,~ Ix du

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 0 0 0 0 o.. o Ea) 1 --- —----------------- 51 ---------------------- dv 15 -

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN,_______,____ o tO I ro0 0 X "~ 601 x ' ld 32 iD ^ ^ sf0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE VII DATA FOR PLOTTING -P vs H2 IN FIG. 4, TAKEN FROM MEASUREMENTS ON SAMPLE NO. 3 IN TABLE VIII H xP x 104 H2 x 10-5 x 109 A P x1pH2 398 7.53 1.584 4.75 786 28.12 6.199 4.54 1144 57.90 13.11 4.41 1500 96.60 22.50 4.29 1850 140.0 34.25 4.09 2180 186.0 47.52 3.92 TABLE VIII THE CONSTANTS B,C IN THE RELATION -- = B - CH2 pH2 I *; ---- - - -. ', I I Trigonal Axis B x 109 C x 1016 B x 109 C x 109 No. From Wire Axis 0==0 0=0 0==90 ==9 Remarks |1 To be 11.8+0.1 7 4.45+0.1 0.8 99.8 % Bi determined Zone Treated 2 18 + 2~ 12.98+0.05 6.06+0.1 2.8+0.3 - 99.8 f Bi — "'~ — ~~ — Zone Treated 3 0 + 4~ 4.62+0.05 1.4+0.1 - Fitzpatrick Bi -"~ - -~ -- ~ ~not zone treated 4 2 + 2~ /v20 - 8.3 Fitzpatrick Bi 4~2 + 2~ ".J20 — cl. 8.5 not zone treated 5 - 5.0+0.2 3+0.5 - Coated Bi 99,8/,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The samples made by drawing Baker and Adamson Bi were found to be as good as those drawn from Fitzpatrick Bi in being single crystals. The zone melting of a Bi sample in a transverse magnetic field of about 6,000 gausses did not influence the orientation of the trigonal axis, nor did it insure the singleness of crystals. In fact it was frequently found that new cracks were introduced. c. Bismuth Coating on Glass Tubes. Tubes were drawn to have outer diameters of about 0.2 mm. They were placed horizontally at various heights above a porcelain crucible with melting bismuth in high vacuum, The bismuth used was from the 99.8% pure Baker and Adamson lot. It was found that: (1) The coating depends on the height. At some high temperature the coating was grainy and not very conductive up to the height of about 1 cm above the melt, and was smooth, conducting, at higher levels. The boundary wassharply noticeable. However, at lower temperature the boundary was lower and at a temperature close to the melting point of bismuth it disappeared. (2) The coating is better if the evaporation is done at lower temperature during a longer time. 3) The vacuum should be better than 10"5 mm. This:was done with a mercury diffusion pump and a liquid air trap. (4) The glass tube should not be too hot. The temperature was not determined, however. This may well be connected with the results mentioned under (1). (5) Most of the samples obtained this way exhibited very little or no angular variation in the A measurement. This shows that the sample obtained this way is poly-crystalline or perhaps amorphous. The constant B in the equation A~ -= BH2 for a few samples was found to be between 4.5 x 10 -9 and 6.8 x 10-, which is of the same order of magnitude as the average of B- values for pure Bi wires reported in previous reports, possibly somewhat smaller. 4. Conclusions The technique for checking singleness of crystals was improved. The wires ere measured in all orientations with respect to H by means of a Cardan suspension. The results indicate an "ellipsoid of magneto resistance," but detailed interpretation is deferred until the next report when the theory will be further worked out. 4 ________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Accurate measurements at fields up to 2500 gauss show a term in Bi proprotional to H4. Efforts towards coating a thin Bi film on glass were successful in establishing conditions for obtaining a smooth, adherent conducting film. D. PROGRAM FOR NEXT INTERVAL 1. Theoretical In the further development of the theory, an effort will be made toward discussing completely the MR ellipsoid, and toward discussing other properties of the bracket symbols, for example their temperature dependence.. Experimental Certain aging effects of bismuth MR will be studied, Further work will be done on the MR of thin Bi films. The rangeof Hwl be extended to the order of 10,000 gauss. Consideration will be given to work at lower temperatures and to the design of ac measuring equipment using very low measuring currents. -E, PERSONNEL Of the persons mentioned in the previous report, the following do no longer work on this project: Miss K. Hanchon, Mr, B3 Crane (temporary), and Mr. Alsaqqar. At the end of the summer, Mr. S, H. Yeh worked for approximately six weeks on the problem of the lhin films, Dr. T. S. Change looked briefly into some theoretical aspects of the azimuth dependence of the MR effect. These two men had only a limited period of time available since they had to leave Ann Arbor with the beginning of the fall term. Mrs. Suits is still here, helping with some theoretical work. In the period of June, July, August, and September, 1954, the major part of the theoretical work reported was done by Mr. Kao and Dr. Katz, whereas the experimental work other than the thin films was done by Mr. Tantraporn and Mr. Patterson. 35