a? A .5 “W9“ . .wr ¢ 7 _. . .. .Yu .3; v .I V K A~t 0. hi; ‘ 9. fa . a :3 am .me 5” ' .0014 U a; t‘ y d ’3 x.» w; 3 '3‘ ’3' a - . a. J 33... E .- 1” “a. “a ”K“ C . r . JV C . :1 .5 5““ S. r: .. 4P 3.. . .. .1 a...“ 3.. w: n... r... «a.» 3.1. a. a & out. .9. . .fi « . V. .ml .9 ”HM 5.0“ "nth“ H...».. a; u .%J a . . .w tn" .2. O§ no... "‘2‘ t... ,b‘? .1.“ rm 3 .3... . A. .4... i. a 7 &IV ”‘0. a... a“: . . III... ‘ .WO. 0 3. . ”‘t oak! "0 )~ A .h J {I we at .‘a-\ II .. a O. n at ”and“, 5.0. ‘54.. L w? a .o-‘ “’ “in—Imumflmmlflrnluimnwmuuumul 3 . ' ' LIBRARY I 312.919.11219175. J ‘ - ' Michigan State : ' University , i. 3 f, , . ' a: P I Thisistoeertlfgthatthe l, V ‘1 thesis entitled - 4 i' . . .} Anisotropic Zeeman Effect ! ' .| in Crystalline Solids . . i presented by p . ; Donald Ora. Van Ostenburg i , i a: B has been accepted towards fulfillment ‘ J: E of the requirements for l R M S degree mlhxaiu. 1. w _ '. W m (1. Major professor Date May 28, 1953 PLACE IN REI'URN Box to remove this chedtout ftom your record. To AVOID FINES room on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE * - fl =- ANISTROPIC ZEEMAN EFFECT IN CRYSTALLINE SOLIDS BY Donald Ora van Ostenburg A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1953 1:” /' ‘5'/:”3 $53 Acknowledgment I would like to express my sincere gratitude to Dr. Chihiro Kikuchi for the suggestion, patience and invaluable guidance in carrying out this problem. {303.1931 I. II. III. IV. V. VI. VII. Table of Contents Page Introduction 1 IsotrOpic and Anisotropic Zeeman Effect 3 Energy Levels Due to the Crystalline Electric Field in the Ground State of Double Ionized Vanadium A. Quantum Mechanical Methods for Obtaining Energy Levels 8 B. Calculation of Energy Levels 9 Angular Dependence of Energy Level Transitions 21 Conclusion 28 Appendix 30 Bibliography 35 (1) Introduction Recently much information has been obtained concern- ing the radio-frequency study of atoms and molecules. In particular, attention has been given to the study of paramagnetic substances by means of microwave absorption. The great advantage in using these experimental techniques lies in the fact that transitions between energy states differing slightly in energy may be observed. The knowledge obtained from experimental data together with theoretical considerations from quantum mechanics has led to a clearer insight as to the structure of crystalline paramagnetic substances than previously attained by the use of X-rays, specific heat, magnetic susceptibilities, and infrared measurements. In experiments of this character the general procedure is to place the sample in a section of a wave guide or some form of an absorption cell and apply a steady magnetic field at right angles to the microwave magnetic field produced in the cavity. As the steady-magnetic field is allowed to vary, the amount of microwave energy absorbed is seen to pass through various maxima which correspond to transitions among the energy levels of the substance. ‘We should like in the following pages to give as an example a discussion of the quantum mechanical method in obtaining the energy levels of the doubly ionized vanadium ions comprising the Tutton salt, vanadous ammonium sulphate. (2) The Hamiltonian used to calculate the levels contains a term involving theinteraction of the steady magnetic field with the electron spin, and two terms representing the crystalline electric field due to the permanent electric dipole moments of the water of hydration. This latter effect is small and is treated as a perturbation on the former. we employed the perturbation method, and carried out the approximation to the third order. The energy levels are found to be angular dependent with respect to the external magnetic field, and as a result, so are the transitions. The crystalline electric field is shown to give rise to the so called ”fine structure“. A method is given for the evaluation of the constants associated with the electrostatic field. Finally, using the experimental values for these constants as determined by Bleaney et al. we made theoretical plots showing the variation of the fine struc- ture with angle. (3) .II. The Isotropic and Anisotropic zeeman Effect When a substance emitting a bright line spectrum is placed between the poles of a powerful electromagnet, one is able to observe by means of a spectroscope that the original line is split into three or more closely spaced components. The group of lines into which the original is split is called a Zeeman pattern. It is known that when an atom is placed in a weak magnetic field,the magnetic moment associated with the angular momentum of the atom causes it to precess about the direction of the field at a rate determined by the Larmor frequency. Due to the quantum conditions imposed upon such a system, the angular momentum Jh is allowed only those orientations such that its projection on the direction of the field is given by mfl, where m: 1 l/Z‘t 3/2 °°"*J. Thus, we see that the system has 2J+l energy states. As a result of the increas- ed number ofenergy levels, which differ slightly in energy, the atom is able to emit radiation corresponding to the various Zeeman levels. In the preceding paragraph we treated the zeeman effect as being produced by a single or group of atoms. No mention was made of the fact as to how the magnetic field should be oriented with respect to‘Uuesubstance . Indeed, there is no need for such a restriction since the direction is perfectly arbitrary. Hence, we may say that the substance is isotrOpic or that it produces anisotropic Zeeman effect. (1+) New let us turn to the case where the particular atom or ion whose spectrum is to be studied is located in a crystal. It is reasonable to expect that the Zeeman effect produced by such an ion will vary as the magnetic field is applied at different orientations with respect to the substance since the other constituents of the lattice structure produce a magnetic and electric field at the ion depending on their symmetrical arrangement. In the Tutton salt of vanadous ammonium sulphate with which we shall be concerned, the paramagnetic ions are surrounded by an octahedron of water molecules in their immediate vicinity which may be illustrated by Figure l. The four molecules lying in the horizontal plane are separated from the vanadium ion by about 1.9A whereas the other two by a distance of 2.15A. The line connecting the water molecules with the greatest separation is called the tetragonal axis, and if the four water molecules lying in the plane formed a square arragement the figure would be said to have tetragonal symmetry. EXperiments performed to determine the orientation of the tetragonal axis with respect to the magnetic axis of the crystal have shown that there is considerable rhombic symmetry. This effect arises from the fact that the four water molecules lying in the plane do not form a square by a rhombus. The water molecules contain permanent electric dipoles which aline themselves with their negative end toward the positive vanadium ion, which (5) is thus surrounded by a crystalline electric field of tetragonal and rhombic symmetry. Abragam and Pryce. have shown that the Hamiltonion 56-? geH-S t D{S; -l/3(S)(S+19 e E{S; - 3;} may be used to interpret the Spectrum obtained in paramagnetic resonance. The term involving D represents the field of axial symmetry whereas that of E the field due to the rhombic component. The value of g is assumed isotrOpic. we may fix a coordinate system in the crystal by assuming the z axis to lie along the tetragonal axis and the x and y axes to lie along lines of symmetry in the plane of the four water molecules. To obtain an idea as to the structure of a unit cell of such a crystal let us look at Figure 2. The three crystallographic axes (a,b,c) are approximately in the ratio (3,h,2) and two of the crystals magnetic axes Kgand K,lie in the ac-plane whereas K5 is perpendicular to them being along the b-axis. The two ions of the unit cell, of which one is a translation plus a reflection in the ac- plane of the other, have their tetragonal axes inclined at an angle of 22° with the ac-plane. As a result of the two ions per unit cell the spectrum is rather complicated. Hewever, the two spectra can be seperated. As the next step in our discussion let us turn to the calculation of the energy levels. For this purpose we shall resort to the use of quantum mechanics. (6) Tetragonal Axis 0 water of hydration @ Vanadium ion Figure l. (7) Lou Sou-J 'llhp-ol Ills €430 Figure 2. (8) III. Energy Levels Due to the Crystalline Electric Field in the Ground State of Doubly Ionized Vanadium. A. QUantum Mechanical Methods for Obtaining Energy Levels. It is our desire in this section to put forth the necessary ideas from quantum mechanics for the purpose [of calculating the energy levels or the possible values of the energy associated with a dynamical system. To this end one computes the eigenvalues of the Hamiltonion, considered as an Operator, associated with the energy of the system. In the Schroedinger scheme this means that one must compute those values of E for which the equation k V1 = E; ‘f’x possessess well- behaved solutions. The symbolfi‘represents the Hamiltonian, and (ithe electronic wave function asso- ciated with the state of the system of energy E;. The wave function (‘15 called the eigenfuction of# and it is said that E; andifioflébelong to each other. The quantum mechanics deveIOped by Heisenberg connects the Operators associated with dynamical variables with ‘ matrices. The problem is to calculate these matrices be- cause from them the eigenvalues of the system may be obtained by solving fiuasecular equations of their deter- minants. If the matrices are diagonal, the exact eigen- values are just those elements. The matrix elements are no defined by the relation "- - or = m‘fiérgfl A.“ ~00 (9) where% represents the Hamiltonion andtfthe normalized wave function with? its complex conjugate. The Hamiltonion may be represented as a sum of two expressions as fl? :?5£ +l/ where}; is the unperturbed potential anthhe pertur- bation . In our expression for the Hamiltonion 3f : gem and V: D (S;- _ 1/3S(‘S-019+E{S,: -S;} Since the first term is predominant it is necessary that its eigenvalues be solved exactly. This may be accomplished by diagonalizing its matrix. The remaining terms may be treated as a perturbation and their eigenvalues obtained approximately by the method of perturbation theory. With these ideas in mind let us proceed to the calculation of the energy levels B. Calculation of Energy Levels In this section we shall give the procedure and results for calculating the expressions representing the energy levels. The ground state of a free doubly ionized vanadium ion is ,F% where the electron configuration is 3d3 . The orbital momentum is quenched strongly by the crystalline electric field and as a consequence J = 8=%. The wave functions consist of electron spin functions com- binded in such a way so as to make them orthonormal. For the case 88% these expressions are given by (10) £03,: 0((1)o( (2)4(3) V1; =fi[o(<1)o«2)@<3> wammwe) woman/(3)] 9{=—,%5-[s(1)@<2)a<(3) +(3<1)ar(2) @ <3) +4<1>e<2>gmj W1: @(1) (3 (2) (3(3), The terms dandérepresent electron Spin functions where d;(é)refers to an electron with it‘s spin parallel to the external me gnetic field andbi(?)to an electron with antiparallel spin. The number 1, 2, and 3 designate the three electrons in question. Now let us turn to the Hamiltonion to see how one may represent its terms. From the quantum mechanical point Of view each term involving 8 is a spin Operator which is defined in terms Of the auxiliary Pauli spin Operators as, Va=hd s=as a=ac The Pauli Operators are 2 x 2 matrices which have the form f : 0 1 0’ : 0 '1 0' : 1 0 X 1 O 7 1 0 i 0 -1 and where§d=dandge = -9 Likewise, it can be shown that the expressions (6.x tail» and {0’ x wify) have the prOperty Of transforming @ into 0/ and 4 into@ respectively. For this reason they are called electron flip Operators. Lets us now see what effect the various Operators have upon the wave functions. Before doing it is advantageous to introduce a notation due to Dirac.3 namely that Of the ket vector defined as 'S, m>, where S represents the total (ll) electron spin and msthe magnetic quantum number. In this new notation the former wave functions appear as, (p; 3/2, 3/2>«p- | 3/2, 1/2) .5 -|3/2, -1/2> r. “’33- l3/2,-3/2> For purposes Of illustration we shall calculate in detail the effect Of Operators S: and Stupon.|3/2, 3/é7 SZ - S 12+ S 22+ Saz(for the three electrons) so 52' 3/2, 3/2> - (812+ 312+ st) 4(1)“(2)°<(3) - S,zo((l)0((2)0((3) + szzq<1)a«2)a«3)+sfl «(norczwm = [s 124(1id(2)4(3)+ 0((1) [512“(234(3)l «(1)4(2)[S_,z°<(3)] now using the fact that s’zo((1)=g°((1) likewise for 0((2) and 4(3) 8 Z| 3/2, ya): 3/2n 0((1)°<(2)°((3)=3/2fi | 3/2, 312) #************* For the Operator S we have s| 3/2, 3/2) . (s 4 s + s )ar(1>q’(2)°(<3) = (3,11; s: + 3 +23 5 + 28 s + 2qu )o((1)o<(2)o<(3) The Sf'for example, gives Siam = (3;. s‘y+ s"z )°((1) : [3231/26X may) (SK-15y) + 1/2 (Sp-13y) (sx+1sy)]ar(1) and s;d(l):Sz[Sz 4(1)] =sz[1/2ho((13 =1/2fisg‘(1)=1/l+fio((1) Using the results Of the flip Operators defined above, (sx-isyM sap ' (sx+ 13y =0 so 1/2 (sX 1. 13y) (SK-18y)°((l)=l/2fi (sx+1sy)\3(1)=1/2no((1) (12) Therefore, S 10((1) . 1/43 0((1) 1- l/2fid(l): 3/1+fi°((1) Likewise we have, s: «(2) = yttrium) 8:40) = 3/nnzad3) Now evaluating the portion 2[S'-Sz 9 SS, + §'%O((l)d(2)°((3) we have for the first part 2 [s s at(1)°< (2)0!(3)]' -s[ + s sax - s,y 51y]°‘(1)‘«2)‘«3) :2 812 IX = 2[ 3,23 zz+ +1/2 (smasly) (Sax' iszy)’+l/2(S,x- is”) (s,x + 1sIL y)] o<<1>o<<2>at<3> z 2[S,ZS ,Zloi(l)°‘(2)°i(3)=2/‘+h Oi(1)°‘(2)°((3)= 1/2?! t’((.'l-)'=((2)<>i’(3) The other terms give zero since (Sx+iSy)°( =0 ; so combining all terms s I 3/2, 3/2>= [Mm + 3A1: .3/tm +1/2s.1/2h+1/2h‘]o((1)°1(2>o((3) 1' 15M: |3/2, 3/27 . In a similar manner one can calculate the results Of the same Operators upon.the remaining wave functions. In essence what we have done is to calculate the eigenvalues a of SI and S belonging tol3/2,3/€>. In short we may sum up the above results along with the results Of the other wave functions in the expressions, s2] 3, 1:1,): m,‘hlS,m3> 3- . s '5, 121,): s (s + 1) 1455,1113) In order to Obtain all Of the matrix elements it is necessary to define the following identities which may be proved in a similar fashion; (l3) [SX+iSyIIS m): fl (8 + l)-m‘ (m +1) hlS 111,1» 1) [sx-isyllsmpv-irw + 1)-n,(m;l)fi fi,|s m -1) [Sx+isy].lS, m5) = [Si-183;} W3 ( 8+1 ) -mJ(m!1) 'SJ 111‘ +1) : VS(S+l)-ms(m;-13[SX+iSy]lS,m +1) -\/S (3+1)-ms(m,, +1) VS(S+l)- (m;1)(mp|s m42> [Sx-iSyHS.m Ina-fl (S+l)-m (m-l )\/s(s+1)-(m-1) (m;2)ls 111-2) Up to now we have defined the axis of quantitization along the first tetragonal axis which corresponds to the crystals electric z - axis. When the magnetic field is oriented along this axis, the first two terms Of the Hamiltonian give diagonal elements in the matrix repre- sentation and the third, elements Off the main diagonal. In a like manner when the magnetic field is along the x- electric axis all Of the terms in the Hamiltonian give nan-diagonal elements. ‘We mentioned before that since the first term H-S is the dominant factor in the calculations it must be diagonalized exactly. In other words its eigenvalues must be Obtained exactly. Upon expanding this term H-S a HZ sZ + 1/2 ( Hx + 1 Hy)(SX- iSy) + l/2(Hx-1Hy) (sx+ iSy) it is easily seen that there will always be non-diagonal elements as long as the terms (Sx-iSy) and (Sx+iSy) are present. In order to eliminate these terms it is necessary to define a new axis Of quantitization, such that the only (1%) component Of H-S is Bin-Szowhere z'is the axis Of quantitzation along the external magnetic field vector. The arrangement may be illustrated in the following figure where xe,yé, 2e are the crystal's electric axes and x; y; z: are the axes Of a coordinate system associated with the external magnetic field, which lies along 2'. 6, «NW are the usual Eulerian angles. Ida! Lise Having defined a new axis Of quantitization which may be oriented in any position with respect to the electric axis, it is now necessary to transform the wave functions to a new coordinate system. The wave functions defined in (2) may be thought Of as four basis vectors defining a four dimensional space and so, in order tO carry out a coordinate transformation it is only necessary to construct linear combinations Of the original. These we shall denote by Pfiwhich satisfy SZ'Y‘= mg); The mathe- matical details in calculating the new wave functions are (15) 1,. given by Kramer's. In his notation r' = or; M r .(g) 1 -.- (g) 7‘ 3 '6; to!" 6; isin 38‘?!) ,(chS g e; 2. 11') the new system being denoted by primes. The monomial 5"” 1'” i ,l' ’l is related t0 “P5,.“ by the expression . r“ 14"” J'” 3 ‘ I - (3) W10»). {Jun-)1 For the case J: 3/2 we have (if (-4! Mi (f)1(;i:(°<'i‘*ei)(ewi) )(*i Mi GM?)1 (7‘)3 "M (flier) Now expanding, substituting from (3) and letting 2 Fa}: (p; 1. .3 g! ((31.1 8 Z; W :f__,€ we see that, i I t It 9“ p ‘00 V‘M s ‘\ NV” 8 J- ’3 I *t ' r r" v .lp% 73-! o( 3.’(P,1+3E.°\@ (ft +3fi.( q V“: + (”fl/(’"fJ I *1. é o 9 r‘ " 9.: ‘ 71:1[9‘110‘ e W%+fi.’(ai Of -246 at? +fi!('€€ dick)“: 4e’otfi!‘€_a/] 2 air/«'ewuwr a» a ) "m i 2. L . f e ‘1‘ Arcs—(ex iii-rear 3. 3"; ._L _ . ' ‘ 2% -Y;[[ efillg’i' +371.c>((s(fl:.--375_1,,1€(’”;1 4. 432:3! (Ki-‘2] (16) These expressions are the new wave function which we shall use to compute the matrix. Using equation (1) it can be seen that for the first or unperterbed portion of the Hamiltonian the matrix is diagonal and has the form 3/2ng 0 O O O l/2gQH O O O O '1/2ng 0 O O O -3/2geH Using the perturbed portion Of the Hamiltonian, the D component gives rise to the matrix; 80 a, a1- 0 a, -a. O at a, 0 -a0 -a' O -2. -82: 210 where 1 «4" a: D§3cost 9 -1) a= iDfi sin 26e a1: - ED sin 6e Y . 2 ‘ 2 2 a a a* an a* and the E component gives, b; b, ta 0 b“ -b. O b,’ 121. 0 ”be -bl O h; -h, bo where b0: 1321351; 820524! b, :E fiisin 6e“V (005250005 6-isin2v’) (17) ~15? . .15? q 2;? b1- E 3e (cosge +sin§e ) ‘9; b,* 13,- bf. Finally, combining the D and E matrix we have for the total Hamiltonian in matrix form, 3/2qu 0 O O 7% _._. O 1/2g‘H o O O O l/2gQH O O O O 3/2gQH a. + b. a, +b, a‘ 4b.. 0 \ 4 a“ + h, ~(a‘ +b) 0 83b; 8.4+ b" O -(a.+b) -(a'+b.) 0 a4 4b“ -(a+b) a-tb. The next step in computing the energy levels is to Obtain the eigenvalues Of the above matrices. For this we resort to the perturbation theory Of quantum mechanics. If one has a matrix «which can be expanded. as a power series in terms Of a parameters then we man write, 9‘ g 0“ our“ “’41.... the term al.,being a diagonal matrix. The parameteré must be a small numerical parameter and each matrix arz-m at” must be independent Of e. The symboldrepresents a perturbed andofan unperturbed matrix respectively. Our next step is a diagonalizedso that we may Obtain the eigenvalues from the diagonal elements. Let us proceed to o .1 do this by introducing the unitary matrix S where S = 8+ e S + 1 £38 + ----- such that the similarity transformation, (18) A = sots" (1+) gives the matrix A which is diagonal. Since A is a transform Ofo(its diagonal elements are eigenvalues Of clor A‘,‘ a A, (5) wherefihis the n'th eigenvalue. Again the S matrix with superscripts are independent Of £ . We finally assume that each eigenvalue isga power series ineiso that A'- A.+£A1 + t" A1. + ------ the A's denoted by I, II,----being independent of £ . In order to find S we multiply (H) on the right by S Obtaining A S = S ti and substituting, (A° +3111 retAz+ ~--) (So-u 81+ 281+ ~----) I ' (8°45 Sta-5‘ 81+ ---) (orator-aw) Now by equating like power Of la we may, knowing oi from the beginning, find the values or A', A“: ----A'and s', sf ---s” . Having calculated values Of A‘,‘we may substitute into (5) to Obtain the eigenvalues. We may acquire any degree Of accuracy by equating the coefficients Of higher powers ore. In the various books on quantum mechanics which give expressions for the eigenvalues, the approximation is usually not carried out beyond the second order. Since we were in- terested in Obtaining the third order calculations, it was necessary to develop them in general. The complete derivation is given in the appendix. The results appear (l9) o . . . . , A“ = doom. 1- 50““ +5 0:51 a: +£ Z an. AM 045‘;- ) (“I'm-dial ‘: («.31 ~01“) q “I did!- “but“ 0“; ”‘3; 43m. .1. 0‘ Z'M ‘ Z»){°‘fi 4:») I; (0‘0 "f')("‘:£’°‘:.~) The primes appearing on the summation signs denote omission Of those values for which the dummy indices equal n. Substituting from the matrices the four eigenvalues correct to third order appear in Figure (3). These values give the energy levels Of the system. Now that we have the four energy levels let us turn to the problem of seeing what information may be found from the energy transitions. (20) a l .i tDN lg- + «o‘eoQ$txmiU&< 1 aa a. . w . n Sees Jessa- .. 32a- in; a . 1 g z . . . eating A? a»: 13w.» 4 aux; m- a: an as s. n . Q... a a a «an... .... a. o . , ? 3x. .a finan gens . .1 am . K Figure 3 (21) V. Angular Dependence Of Energy Level Transitions In this section we shall be concerned primarily in showing the angular dependences Of energy level transitions by means Of curves calculated theoretically. we mentioned previously that the orbital angular momentum was quenched by the crystalline electric field and that the electron spin determined the Observed spectroscopic structure. Now if the crystalline electric field has cubic symmetry the spin quadruplet remains degererate, however, if the field is Of lower symmetry the quadruplet is split into two doublets. Furthermore, if a magnetic field is applied the remaining degeneracy is removed. The following is a schematic diagram of the energy levels: *Fg, 4’ 2- 0". cubic electric zero external magnetic field field but field electric field .Of symmetry lower than cubic If the effect due to the crystalline electric field were absent the only interaction would be that between the electrons and the external magnetic field, the Observed spectrum would (22) then consist Of a single absorption curve the energy levels being equally spaced. Due to the interaction Of the crystalline electric field the levels become un- evenly spaced and the single line is split. This splitting is called the fine structure. The four values Of A‘represent these levels in terms Of the magnetic field, the angle Of orientation Of magnetic field with \ respect to the axis Of the crystal, and various constants. The selection rules for the Observed spectrum are A M: :1 where M is the electronic magnetic quantum number for strong fields. Therefore, we may substract successive levels to Obtain formulas for the energy Of the transitions. The expressions are again a function Of the field, the Eulerian angles, and the constants. TO illustrate the electric and magnetic axes let us assume the crystal to be cut cylindrically about the i‘axis. Then:- 2,. 2'- 'H' Ya \ W’ :JH‘V H X“ The axes with subscript e are the electric axis and those Xe subscripted H are a set related to the direction of the (23) magnetic field which is taken along in. TO get an idea as to the variation Of the fine structure with crystal orientation we shall present two examples. For this purpose it is sufficient to calculate the eigenvalues to second order., For the first example, let us assume the magnetic field to lie in the xe,ye plane and rotate the crystal about the 28 axis. Hence, 6 --2’1,¢=a andil’ranges from O to 2‘fl'. After simplification Of the various tri- gonometric terms, the three expressions representing the energy Of transition‘become, L f; (V) '3 hi’ = ng - D + 3.3.. .. 3E c0525” _. 3E1cos’+" ng gQH . t t p n 1 {,(W) 3', hV 7- geH + .Jé-FH £1.) "' gall - 322.3052Hgéficoshv' f (Y) = by = geH + D + 33‘ - 3E cos 2F- 3E‘co MV’ 3 geH g H These terms show the variation Of the fine structure with the angle 9’. It is evident that they are symmetric about 180. i.e. the pattern repeats itself every half cycle. If the interaction due to the crystalline field were absent, the single resonance linewould correspond to hV 3 gQH. In order to evaluate the constants D and E one must plot the curves from experimental data and then adjust the con- stants in the theoretical curves until they fit. Bleany: Penrose and Plumpton determined these constants to be, 1730 gauss E = 530 gauss D at 20 K D 90°K D 1720 gauss E = 5M0 gauss. (2%) A relative plot of these three functions is shown in Figure (h). The value assumed for H is 10,000 gauss so that the constants are in the ratio E: D: H8 l: 3: 20. The relative plOt shows a maximum separation of about six units which in reality corresponds to 3,000 gauss. From the curves it can be seen that as the crystal is rotated through 360. the maxium separation decreases to a general minimum at 900 and then increases to the same maximum again at 180°. Since the curves are symmetrical about 1800 this pattern repeats it self from 180.to 360.. Now, for the second example, let us study the fine structure as the crystal is rotated about the xe axis. This is the same as changing the orientation of the magnetic field in the 28, ye plane. The Eulerian angles have the values,‘P'-' 0 (arbitratily)?= 0 or7r, and 6 ranges from 0 N1. Again substituting these values into the energy level expressions and substracting successive levels so as to satisfy the selection rules, we have after simplification, 5(6) wen-mm .3W,3E.3Ecoszo 2 hggH 2 1 ._ 313‘4 3E+6DEcosl+o ch '- £(e)= 3_I&_:"+3DE+2_D_‘+m-1DE -Dsin 29 861“ 2 32 32 "16 + 2 gQH geH _ 8121' c039 +£31.45 BE:- 3??) c0529 +(§_§1+%%£-3§-3%§)cos‘+0 q. geH (25) {3(9) = gQH + D + -3D +33; 315+ 3E1+ 6DE_+ 3D 43Ec9529 2 * hgeH 2 ’ a. 3313’ + 3E +6DE cosue tinge H . These three expressions are plotted in Figure (5). The constants D, E and H have the same value and it is seen that the maximum separation is again approximately 3,000 gauss. As the angle 0 is variedfrom 0. to 3600 the fine structure changes from a maximum at 0° to a minimum at about 1+3, . The functions £(6) and 12(6) are symmetric about 180° but f,(e) is not. As 9 goes from 180’ to 360°a similar pattern is observed with the exception of g (6) which is inverted. In each plot of Figure (h) and (5) the constant term gQH was neglected since it shifts the lines by an equal amount and doesrmt effect the Spacings. 150° :éo' 40' ‘3 [(0 160 120' 90' 60' V co a) 6' «n cu '1 c> f. «p en Figure 5‘ (28) Conclusion In our discussions and calculations we have not considered the presence of the other non-equivalent ion in the unit cell. 'We have shown by means of quantum mechanical calculations how one may interpret the fine structure of the microwave absorption spectrum of a single ion of the salt V (NH, )1 (80., {61110. The other ion complicates the spectrum and in essence what one sees is a superposition of the two spectra.’ It is possible to separate the individual spectra in the sense that one may study a single ion, and so no serious difficulty arises. In studying the actual spectrum there are many lines in place of the single fine structure lines mentioned. These are caused by the nuclear and electronic interaction and are called the hyperfine structure. Such an effect may be included in the Hamiltonian and the calculations carried out in the same manner. To be specific vanadium 51 has a Spin of 7/2 which causes each fine structure level to be split into eight levels. The increased energy levels give rise to eight new absorption lines in place of each orihginal fine structure line. If the crystal is rotated in the plane w ich bisects the angle between the two tetragtnal axes, it is reason- able to expect that two spectra will coincide. This stems from the fact that at all timesthe magnetic field will make equal angles with the two ions. In experiments (29) performed to verify this fact it is found that the three sets of hyperfine structure lines do not coincide simultaneously. In other words when the hyperfine spectra corresponding to the electronic transitions -%-0§ coincide the two other sets do not. It is thought that this discrepency may due to the two non-equivalent ions not having identical rhombic fields. It is our desire to investigate this effect in the near future in order to see how the t 0 fields are related. The effects of the sulphate and ammonium ions upon the spectrum were neglected since their dipole moments are much weaker. The water of hydration plays an important role in determining the properties of such crystals and it may be considered as an integral part of the crystal lattice. This view is different from that usually held concerning water of crystallization since we are inclined to think of it as being attached or associated by rather weak electrostatic forces. (30) Appendix This section will be concerned with the derivation of the third order perturbation formula. We begin with the expression, (80 +581 4- tab Sui» ---~ ) Cd°+ 50:14:14.1?” )7- (A0 +5 A: +£zA‘ + ---) (S.+£ 31+ 6‘ 31+ --- ) and equate the coefficients of the first, second, and third powers OfSinorder to evaluate the various terms. In this particular case the perturbation consists of 1 (40nd ), so the following represents the set of matrix equations which are to be used: 0 O. O. s: «838A (1) 3° .1 :0 O z 3 : 0‘8 + ‘4 S = S A + S A (2) a. 1.: 01 to z: ‘ 1 £245 +a