Library, t-ui rm»"- >'"*• JUL 14 1969

Mathematical Techniques for EPR

Analysis of S=5/2 Ions in C2 Symmetry.

Application to FE 3+ in Quartz.

Q

z

,*

l#**T OF \

V

%

*°*eA\i o«

"

U.S. DEPARTMENT OF COMMERCE

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NBS TECHNICAL NOTE 372

ISSUED JUNE 1969

Nat. Bur. Stand. (U.S.), Tech. Note 372, 23 Pages (June 1969)

CODEN: NBTNA

MATHEMATICAL TECHNIQUES FOR EPR

ANALYSIS OF S = 5/2 IONS IN C2 SYMMETRY.

APPLICATION TO FE3+ IN QUARTZ.

R. L. PETERSON, L. M. MATARRESE, AND J. S. WELLS

Radio Standards Physics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402

Price 35 cents.

!

Mathematical Techniques for EPR Analysis of S - 5/2 Ions

3+

in C Symmetry. Application to Fe in Quartz

Robert L. Peterson, L. M. Matarrese, and J. S. Wells

ABSTRACT

Various formulas and mathematical techniques useful

for the analyses of the EPR spectra of ions of angular momentum 5/2 in sites of C? symmetry are presented. Special

emphasis is given to the spectrum of Fe in synthetic brown

quartz. Included are: matrix elements of the Racah operators

for arbitrary direction of the axis of quantization relative to

the crystalline electric field axes, spectral line-position

formulas based upon a usage of second-order perturbation

theory which is somewhat different from the usual, and lineintensity formulas.

Key Words: electron paramagnetic resonance; ferric ion;

synthetic quartz

The purpose of this paper is to present various mathematical

formulas and techniques which are useful for the analysis of the EPR

spectra of S = 5/2 ions in fields of Q symmetry. Special application

3+ 12

is made to the spectrum of Fe in synthetic brown quartz.

3+

Fe is an S-state ion with spin S = 5/2. In quartz it apparently

finds itself in a crystalline electric field environment of two-fold symmetry, C . The crystalline potential may be developed in spherical harmonies Y. . Onlv i = 2 and 4 terms need be considered since the im 3,4 matrix elements of Y . for other & values vanish. The C_ symmetry im ^ eliminates odd m values. Since the crystal field potential must be real,

it finally can be written as
v + v + V An + V + V , (1) 20 22 40 42 44' l '

where

Vimn =*« Y „ + a „ * Y, . (2) im im im i,-m

We have here used spherical harmonics with the property

Y, = (-l)

m

Y. *. (3)

x , - m £ m

In constructing the Hamiltonian it is convenient and conventional

to use spin operators which have the same transformation properties as

3

the Y . . Bleaney and Stevens have discussed one such scheme, which

4 5

has been reviewed by Hutchings. Kikuchi and Matarrese have used a

6 (I)

more systematic prescription, based upon the Racah operators, T

3+ i 7

1 These were used in the analysis of Fe in quartz, and will be used

(i) here. The T are generated by the process m

T U \ = [1(1+ 1) -m(m - 1)P [S , T

(i)

]. (4) m

- m

5 (i) With the normalization of Kikuchi and Matarrese, the T necessary m

for the present purposes are

T^2)

= S

2

-X/3,

U r T

±z = S

' ZJ^>>

T^4)

= [35S

4

- 30XS 2 + 25S

2

- 6X + 3X

2

] (5) U r r r

,(4) Ti, = -S'

2 [X-9T14S -7S 2

]'Vr0/8, ±2 ± r r T

±4

= S ± '^0/l6 '
where

X = S(S+1) . (6)

In Eqs. (5), the subscript r refers to the two-fold axis of the crystalline

electric field. The remaining two orthogonal axes will be denoted by

p and q. The raising and lowering operators, S' , of Eqs. (5) are

defined by

S! = S ±iS . (7) ± p q

The complete spin Hamiltonian for an S-state ion (excluding nuclear

spin interactions) in an environment of C syr

magnetic field H in an arbitrary direction, is

spin interactions) in an environment of C symmetry, and an external

*= o S-fS+ C

20

T<

2 > + C

22

(T fe-^+ T^"KVW

+ C (T<

4)e2iX42 + T

( 4

)e

2iX

42, + c <T<

4 > e"4

^+ T

(4) e 4^). 42 2 -2 44 4 -4

Here (3 is the Bohr magneton (used here as positive) and g is the

° U) g-tensor. The coefficients of the T have been written as a real m

number, C„ , multiplying a phase factor. The C. clearly can be im r y & r ^ m y taken as positive without loss of generality, for |m|>0. The C _ and

C _ must be real but may be positive or negative.

To use perturbation theory at high fields, one diagonalizes the

Zeeman term. If the g-tensor is isotropic, as is the usual case for

S-state ions, and has been shown to be true for iron-doped quartz, the

axis of quantization is just the direction of H. Let this axis be the z-axis

of any x, y, z orthogonal coordinate system. The direction of the z-axis

relative to the p, q, r crystalline electric field axes is given by the Euler

7

angles a, p\ in the convention of Rose. The axes and angles are
illustrated in Figure 1. The transformation equations from the S , S' operators to the

r ±

S , S operators are

z ±

Si = i e±la

fe (cos 6 + 1 ) + S (cos (3 - 1 )] + S e

±1(

* sin 6,

(9)

S = -(S, + S ) i sin (3 + S cos 6, r + - z

where

S = S ± iS . (10) ± x y

(i) The matrix elements of the T in the representation of the eigenstates m v B

| M), -S ^ M ^ S, of S , are given in Tables I - V. Table VI lists, for

z the reader's convenience, the associated Legendre 6 polvnominals r y P.im

for ^ i <4 and m ^ S. , which occur in Tables I and III. The diagonal

elements of the Zeeman term are of course just g(3 HM.

For reasons which will be discussed, we have used a perturba-

, . 8

tion analysis somewhat different from the usual one. In the conventional perturbation theory, one develops the eigenstates and eigenvalues

in perturbation series as

I E ) = | M> + | M

(1

') + | M

U)

) + . . . , (11)

E =e' ' + E

(1) +E(2) + ..., (12) M M M M

where

3C ~- K

Q + V, (13)

3fC |M>= E^|M), (14)

^

e

m>=

emI e

m>>

(15)
and V is the perturbation. Substituting Eqs. (11) - (14) into (15), and

9 performing the standard manipulations, one obtains in the nondegenerate

case,

em= <m|v|m>|, (16)

(2)_ r

(M) 1(mMm')1 Z M £, E

(0)_ E

(0)

' ("J

M M'

(3) = ^(M) y^(M) (M[ V\M') (M'Mm") (M"|v|M) M " M' M" (e[°J - E

( °!) (E

(0)

- E

( M °l) M' 7 M M"'

rp (M) 1(M| VJM^l 2

( M[ V|M)

" M' (E

(0)

- E<°[, 2

^ M M' '

sr (M) The notation 2_^ > f° r example, means that M' = M is excluded from

M' 3+

the summation on M'. Now for Fe in quartz, it turns out that C n is

20

numerically quite large: its correction to the line-position values is

sometimes more than 30% of the free-ion value. Thus (M| v|M) is

(2) "large. " It does not occur in the second-order term E , but does

occur in the third order terras, with the result that for certain lines

and certain orientations, the third-order terms should not be neglected.

(2) However, for (3 = 0° (H parallel to the two-fold axis), the T term

lies completely on the diagonal. Therefore by including it in 3C„, a vast improvement in accuracy in the second-order calculations is

achieved. We systematically have put all diagonal elements of V in

(0) with 3C^. This renormalizes E, , , and gives the following perturbation M

corrections'.

Ell )= , (19) M
(2)

= y

(M) I(M|V|M')1

M M

(3) v- (M) yi (M) (M|V|M') (M'|v|m") (M"lvlM)

M

= » - <' - *$ <> - -£i)

'

At (3=0 the third-order terms in this reformulation are

entirely negligible at K-band frequencies in iron-doped quartz. For

(2)

(3 = 90 , the T term has off-diagonal elements, and the two perturbation treatments tend to be equivalent in their results. (These conclusions could be drawn as soon as rough estimates of the C„ were

known).

The line-position formulas, through second order in the modified perturbation treatment, are given in Table VII. The definitions of

the terms appearing in this Table are given in Table VIII. H corresponds to the (E / - E . ) transition, H to the (E . - E . ) transition

5/2 3/2 2 j/2 1/2

and so on. One other point may be discussed here. C symmetry, by itself, does not define the orientations of the p and q crystalline field axes,

For example, from a crystallographic analysis, one knows that the optic

axis of quartz (Z) lies in the p, q plane, and that the X-axis of the quartz

3+

coincides with the two-fold axis of one of the sites at which the Fe

resides. (See Fig. 1). The angle a, defined as measured from the paxis, is not immediately measurable. The measured angle is a - 0,

where 9 is the (arbitrary) angle from p to Z. For example, when H is

parallel to the optic axis, parameters such as C cos 2(0 - ^ ,) ^ nd

C „ , sin 4(0 - X ) appear in the line position formulas. Thus the

44 44

^

angles 5 X and X „ , which locate the V_ and V . . lobes in the p, q 22 44 22 44

plane, can be measured only relative to 0, that is, relative to the optic
axis. In reference 1, we arbitrarily set p to coincide with Z (i. e. ,

= 0); this was purely for convenience.

We turn finally to the line -intensity formulas. We consider

only the case for which H is along the r-axis, and the microwave field

H lies in the p, q plane (Fig. 2). It was for this situation that an

interesting effect was uncovered in iron-doped quartz: for H nearly

perpendicular to the optic axis, and for small values of H, several

lines are readily observable. However, when H is rotated into orientations near the Z-axis, all the lines disappear into the noise. Substitution of the parameter values, determined from the high-field data,

into the line -intensity formulas shows that this effect is to be expected,

and further confirms our choice of parameter values, including the

orientations of the crystalline field lobes. However, line intensities are

difficult to measure accurately, and so the determination of the parameters from line-position analyses is still to be preferred.

The interaction between the ion and the microwave field is

(see Fig. 2)

5C, = l^oH

l

2 o I

S cos 3? + S sin 3? p q

-i$ i<3? S e + S e

(22)

Here we have used Eqs. (7) and (9), setting both a and (3 equal to zero. (The value of a here is actually arbitrary, as it is in the spinHamiltonian matrix for .H parallel to the r-axis (Table IX). However,

the same value of a must be used in both places for consistency; we use

a = 0. ) For H parallel to the r-axis, the 6X6 matrix of the spin

Hamiltonian separates into two independent 3X3 matrices. This case
is given in Table IX. The | 5/2), | 1/2), and | -3/2) states couple

together to form three exact eigenstates of the Hamiltonian, which we

call |E ), k = 0, 1, 2. Similarly, |-5/2), |-l/2), and | 3/ 2) couple to

form | E ). The secular equations for determining the energies E and

eigenstates | E ) are thus cubic and can be solved analytically.

Writing

|EJ= a* |± 5/2) +b* |± 1/2)+ c* |T 3/ 2), (23)

one finds the nonvanishing matrix elements of 3C, (Eq. (22)) in the |E ) basis to be

<E; Ik, | E;> = i gp^, j^5 e

l»(a;* c- + cf a+) + 3e

iS

b;* b*

+ ^8 e V

(\> 1 c k

+ c

£ b

fc

) \. (24)

The line intensities are proportional to the absolute-value-squared of

Eq. (24).

It is instructive first to consider two special cases. Assume

that C = C =0 and that C A i 0. From Table IX one sees at once

42 22 44

that the states | ± — ) are uncoupled from the remaining states, and

hence are eigenstates of the spin Hamiltonian. Thus, for one value of

± ± t

k the coefficients bn of Eq. (23) are unity, and a, and c. are zero, while

k

± ±k

T

k

for the other two values of k, b, is zero, and a, and c, are not. By

k k k

inspection of Eq. (24), one sees that (E |^C | E ) then depends upon $

1$ -1$ only through e or through e , so that all line intensities are independent of <£. That is, although there are lobes of V in the p, q

plane, the line intensities are uniform for H anywhere in that plane.

Next consider C) 4 and C „ „ = C. =0. This case is

22 44 42
more complicated than the preceding, but still quite tractable. By

± T ±

expressing both a and c in terms of b from the secular equations,

one can draw the desired conclusion without further determining the

coefficients or the energies. After a small amount of manipulation, one

can write Eq. (24) as

< e;ik 1 i<>= swrx i'^W ' V

f -5 "V(f -3

- E

k'

(25)

'f 3

- E

i""

£ 5 -V/ V.3-E

k

f 3" E

i"

c c 3 3

where f = (- - |3c| - - ), f 3 = < — 1 5C | ^ >' etc ' Scluaring the absolute

value, one has

i<e;i kX>i 2

= <*W2

K'I

2

'\ |2

{

(Aw

+ B

k/ cos2 ($ - x

22'

+ (Aki " Bk/ Sin2 ( * " VI

(26)

i$ -i(<£-2\ ) where A ,, and B . are the coefficients of e and e 22 , respec

ki ki

tively, within the braces of Eq. (25), and are real. In this case lineintensity extrema do exist, occurring at $= X. and every 90 therefrom.

When C , C and C A are all considered, the equations

22 42 44

for the line -intensity extrema are quite complicated (transcendental)

and we will not give them here. However, since C , C^ <<C

2Z

f °r

iron in quartz, we can expect that the line -intensity extrema will be

shifted but slightly from the V lobes, and this is indeed the case
1 ± ± T

experimentally. The coefficients a , b , c were determined by COmputer from the analytic solutions to the secular equations for many

values of H, and the locations ($) of the maxima and minima in the line

intensities were calculated. Agreement with these results is observed

to within our ability to locate the intensity extrema experimentally.

10
M 5/2 3/2 1/2 -1/2 - 3/2 -5/2

5/2 Up

3 20

2n/5~

3 P

21

n/10

6 22

3/2 2n/5~ 3 21 3 20

2^2"

3 21

1 P

-s/T 22

1/2 n/Io

6 22

2n/2~ „

3 P

21 3 20

J-P

n/2~ 22

-1/2 -i-P

•s/2" 22

-i P

3 20

-2N^

p

3 21 6 22

-3/2

^T P

22

- 2^

3 21

- 2 P

3 20

- 2^5"

3 21

-5/2 ^10

6 22 3 21

il P

3 20

Table I . Matrix of T ' = S - X/3 . The P. are associated

r ' im

Legendre polynomials (see Table VI).

11
M 5/2 3/2 1/2 - 1/2 - 3/2 -5/2

5/2 f-^o

2 ^

Q

+

, 3 21 6 22

3/2 ¥^1 -* Q

3 20

2^ +

3 21 n/2 22

1/2 4^zz 2^ - 3 21

- i Q

3 20 W °22

-1/2

\[Z 22

-i-Q

3

w20

" 2^

Q

+

, 3 21 6

w

22

-3/2

n/2 22

- 2n/2~ Q , 3 21 -f °20-2^ +

3 21

-5/2

6 22

-2^5" - 3 21

H Q

3 20

,(2) /2 (2) ,(2) Table II. Matrix of T^' = s'

+ />&. < M | T* ^ | M' ) = < M' j T^ '

| M >'

:

'

n/6" . 2 2 ia

20 " 4

CU^ = —— sin (3 e Q*

2 c^(co.p*li».2to

Q21

= *2 sin P( cos P* 1)e

2ior

12
M |: 5/2

5/2

3/2

2 40

3^

p

2 41

1/2.

8 42

-1/2

8 43

- 3/2

16 P44

- 5/2

3/2 —3\/5" „

P

41

-±1 P

2 40 ^LLp

•jr 41

_5_^2~

8 42

1/2 3 ^10

8 42 n/8~ 41

15 P

40

16 44

5 ^2"

8 42

-N/10

8 43

1/2 '10 8 43

5^2"

8 42

15 P

40

1L p

^8 41 8^ P

42

-3/2 —n/5~ — P

16 44 7

^2" P„o 8 42

-11 P

n/8" 41

"-Up

2 40 2 41

•5/2 ^L p

16 44 8 43

-± -s/10 P..

8 42

- 3\/5~

2 41

A! p

2 40

Table IIL Matrix of T

( ^ = | [ 35S^ - 30X3^ + 255^- 6X+ 3X ]. The P

£

are associated Legendre polynominals (see Table VI).

m

13
M j 5/2

5/2 T R

40

3/2

W£ +

2 R

41

l/2_

!^ R

42

.1/2. ~ R

43

- 3/2

^

R

+ 16 44

_5/_2.

3/2 3n/5"

R

45

2 41

R

2 40 n/8~ R

41 8 42

Js R+ 16 44

1/2 3 V10

R

8 42

-1/2 10

R

8 43

•-il R"

<s/s" 41

- 5I

^

R«

15 R

40 I ^<z

15 R

40 s/s" R41

/10 + R

8 43

l^R«

3/2 \/5"

R

- 5 16 44 ^2"R

8 42 ^8"

R

41

il R

2 40

3^ +

2

R

41

5/2

^

R

- 16 44

-\fTo~ R

8 43 8

^° R

42

3n/5~ - ~Z~ R

41

15

R

2 40

(4) - n/To" / 2 2 Table IV. Matrix of T '= —5- S (X - 9 - 14S - 7S ) . 2 8+ r r (M|T'_4

2>|M>(M^2

4) |Mr. R

40 ^s2

(7c

2

-l)e

2iff

; Rt=-^- s(c±l)(l ± 7c- 14c

2

)e

2i

"; R*„ = 3^§"<c ± 1) 2

( 1 T 7c + 7c

2)e

2i

"; R^= ,2, 2, ^ 1V 2 2ia. -2lJ| s(c±l)'(2cTl)e aa ; R|4

= 21 / s%:± 1)V

s = sin (3, c = cos p. ' "*
M 5/2 3/2 1/2 - 1/2 - 3/2 - 5/2

5/2 15 s

2 40 ! - <i !^<2 8

b43

•s/5" +

16

&44

3/2 3^ - 2 b

41

"i^ s

2 b40

- 15

S

+ W8- 41 8 42

^5" + 16 44

1/2

f

^

S

4

_2

- 15

sn/8" 41

15S40 !^4+

2

-<s/lO +

8

b

43

-1/2

8

b43 1^S4"2 15S40

15

S

+ ^

41 f^ s«

-3/2 •s/5" _-

16 ^44 1^2 15

S

"

n/s" 41

"^

S

2 40

-3N/5" _ +

2 b41

-5/2 n/5" -^10 - 8

b43

3 i— - 8 42

-3^5" - 2 b41

11 s

2 b

16 "44 40

Table V. Matrix of T^ = —^ s

+

' 4

• < M | T

(4

? | M* ) = (M 1

| T^4)| M )*

^0 4 4ia ± n/70 3

s e ; S °40 " 16 ° c ' "41 " 4

, (c ± l)e

4iff ;S^4#s2

(c ± l) Ze

4iff

;

s = sinp; c=cosp.

15
csl

-s

2

y(3c

2

-l) -3sc 3s

2

3 |c(5c2

-3) "|s(5c2

-l) 15s

2

c -15s

3

14 2 5 2 15 2 2 3 4

4 ±-(35c -30c +3) ~|sc(7c -3) =f s (7c -1) -105s c 105s

o' C d.

Table VI. Associated Legendre Polynominals, P .

(s = sin (3, c = cos (3)

16
H = H + A ^- + H 5i + 3'-1 5 ' - 1

. 2 5 >- 3

1 o 2 VA

2 H^ 2H

1 -AL2 ZH^ ^ - ^ ^2 2

V 2V V 2A V V V

H=H + A, + " 1131

V

51

.

1 V

3. -1 + V5, -1 2 o 1 H

2 -A2 H

2 -A1 2H

2 -Ai2 ^2 _ ^2 3H

2 + A12

4H2+ A

z

H = H + 2H

3 V

31

+ 4H

3 V

51

4H

3

V

3,-1

6*3 V

5, -1

° H

2 -A 2 4H 2 -A 2 4H 2 -A 2 9H 2

- A

2

3 1 3 12 3 1 3 12

V^ 2V^i V ri 2A

i V * i V c; i Y* ? H = H - A + 53 31 51 1 3, -1 5, -1 5, -2

4 o 1 H . + A H + A 1 2H . + A, 2 .2 3H. - A, 4H, - A

4 2 4 1 4 12 4H - A. 4 12 4 2

2V V V V V 2A V

H = H -A " + —I1 S

' 2 -l- + h^L . M-.+ H5 + A

2 H

5 +A

l 2H

5 +A

12

2H

5 +A

1 3H

5 + A

12 16H

2 -A 2

5 2

Table VII. Line Position Formulas. H = fiw/gP , H. corresponds to the o o 1 (E - E . ) transition, etc.

17
gPoA

] . (1 - 3c

2

) C

2Q

- ^6 s 2

c 22

C

22

4 If (3 - 30c

2

4 35 c4

, C

40

JK"4

+ ^N/70 s*c_C_ '^-?- -^ n/T0 s 2

(1 - 7c

2

) c C

42 42

PA, = 2(1 - 3c

2

) C, n

- 2\/6 s 2 c C - ^

(3 - 30c

2 + 35c

4

) C o 2 20 22 22

15 . ~ :

'40

^- N/70 s 4

C . .C.. + ~ ^0 s^ (1 - 7c^) C/1 C

4 44 44 2 42 42

2r 2

g P

o V

5, 3 - 2 ° S [^ S22

C

22 - g ^"° " " 7C » S 42

C

42

+

8

^° *

S 44C

44j

3 c-„ 2 n2

S 44C

44

+ 20s

2

c

2 15

[

C

20 " ;§"

C

22

C

22 - IT (3 * 7C

" )C40

+ ^-^0 (4 - 7c

2

) c /l C A ^

- - *JTo s 2 r C 1 4 42 42 8 44 44 J

g

2

p

2

V = 8s

2

T2^ s_C_ + I^n/To (1 - 7c

2

) s_C_ - -^n/70 sV.C o J , 1 [_

o 2 2 + 8s c

is/6 22 22 16

[

C20-

42 42 16 44 44

? 75 2

c__C +ff (3 - 7c ) C

40 sfc 22 22 16

^-n/H) (4 - 7c

2

) c A c A +~*JTo s 2 rc: 1

8 42 42 16 44 44

J

2_ 2. 5 r 2

Po"V

5, 1 " I ['So + 1" (1 + C "> C

22

C

22 " IT

'' " " 7c '' C

40

+2n/T0(1_6c2+7c

4

) c/,C +|n/70s 2

(1 + c

2

) c C I

2

4 42 42 8 44 44 J

h-ioc 2 !"-2— s c - In/To (5 - 7c

2

) s„ C + |n/70 s 2

s /1/( C /1/1 ] L^ 22 22 8 ' 42 42 8 44 44j

Table VIII. Formulas supplementary to Table VII. (Continued on next

page)

18
S VV

3, -1 = 1 ['\o + it <> + ^

C

32

C

22

+ TS ' " " ?c2 ' C

40

- |

-JTO (1 - 6c

2 + 7c

4

) c 42

C

42

- | ^70 s 2

(1 + c 2

) c 44 C

44 ]

2

+ 18c2

["i"

S

22

C

22

+ K° < 5

- 7c2 > S

4 2

C

4 2

" !

^° s2s

44 C

44j

'

2 « 2^ 3

2 -5 2

-7 2 j 2 IVTO 2 = 2

S Po V

5,-l

= —2

S C

L T

S C

40

+ 2V? C C 42°42

- < 3 + ^' <=44 C

44]

2 + [* " " 3c2) S

42

C

42

+ (1 + ^'^C»\ 'j

2 2

«VV

5. .3 = ^1—? K ^40 - ^ ^ s 2

(1 + c 2, c 42

C

42

2 4 12

+ 2(1 + 6c + c ) c. „C. . 44 44J

+ 2^4^[^\2C42- (1 + C2 > S

44

C

44]

2

A = A + A

12 1 2

s = sin (3 c = cos (3

s 22

= sin2(* - X^) c^ = cos 2 (a - X^)

s 44 = sin 4 <<* " X

44 ) c 44

= cos 4 ^ " X

44 >

s 42

= sin2 <«" X

4 2

) C 42

= C ° S 2(a " X

42 )

Table VIII. Formulas supplementary to Table VII.

19
M

5/2

5/Z_

5 ^ 10 ^

2 G+ T C

20

I 5 ^ + T C

40 45^ -2iX

+ TC

42

6

111. 1/1. _5/2_ jlU/L

2 44

42

3/2

1/2 2^3C

22

e 22;

45 ^ 21X,

-y c

42

e 4

2

G "3 C

20

+ 15C40

15 2i|X \/5C e

2 42

42

-3/2 2 44

15 r_ 2iX

T

^5C

42

e 4

-5/2

1/2

2^C

22

e

2iX 22

2

G "3 C

20

2 ^40

2 G+ T~

C

20

15 „ + ^ C

40

^C

22

e2iX22

^<s/T4C e4iX44

2 44

+ ^/^

^C

22

e 2iX2 >

45^, -2iX.., + T

C4f

42

1 8

"2 G "3 C

20

+ 15C40

15 rcr, +2TX

2^*22

•12

'VsCeHz^.^

42

3/2 ^4C44e-4iX44

2^3-C

22

e

2iX-22

3 20

45——- C

2

C

40

Table IX. Spin -Hamiltonian matrix for C symmetry,

for H parallel to the two -fold axis. G = g(3 H

20
(Optic Axis)

Figure 1. Relation between the p, q, r crystalline electric field axes

and the x, y, z quantization axes. The optic axis (Z) of the

quartz crystal is also shown.

21
Figure 2. The orientation of the microwave field H in the p, q plane,

used in the line-intensity analysis. The optic axis (Z) is

also shown.

22
REFERENCES

1. L. M. Matarrese, J. S. Wells, and R. L. Peterson, J. Chem. Phys.

50, 2350-2360 (1968).

2. A. R. Cook and L. M. Matarrese, J. Chem. Phys., 50, 2361-2364

(1968).

3. B. Bleaney and K. W. H. Stevens, Repts. Progr. Phys. 16. 108

(1953).

4. M. T. Hutchings, in Solid State Physics (Academic Press, New York,

1964), edited by F. Seitz and D. Turnbull, Vol. 16, p. 227.

5. C. Kikuchi and L. M. Matarrese, J. Chem. Phys., 33_, 601 (1960).

6. G. Racah, Phys. Rev., 62, 438 (1942).

7. M. E. Rose, Elementary Theory of Angular Momentum (John Wiley

& Sons, Inc., London, 1957), p. 50.

8. This procedure is discussed briefly by E. U. Condon and G. H.

Shortley, The Theory of Atomic Spectra, (Cambridge Univ. Press,

Cambridge, 1951), pp. 35-36.

9. E. Merzbacher, Quantum Mechanics (John Wiley &: Sons, New York,

1961), Chap. 16.

GPO 8S1 -765

23


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