Exact analytic solution of the spin 32 combined Zeeman-quadrupole Hamiltonian

Journal of MoIecular Structure, 111 (1983) 85-90 Elsevier Science Publishers B.V., Amsterdam -Printed

EXACT

ANALYTIC

R. B.

CREEL

Physics

SOLUTION

and

OF THE

D. A.

Dept.,

SPIN

3/2

85 in The Netherlands

COHBINED

ZEEflAN-OUADRUPOLE

HAMILTONIAN

DRABOLD

University

of Akron,

Akron,

Ohio

44325

(USA)

ABSTRACT The secular equation arising from the combined spin 3/2 Zeeman-quadrupole Hamiltonian has recently been solved in analytic form (refs. 1,2). Energy level valid for all magnetic field strengths, are derived and frequency expressions, in a form highly suitable for analysis of resonance patterns arising from rotation of a single crystal about an axis perpendicular to the magnetic field direction. Theoretical rotation patterns are presented for the six possible transitions. A simple method for measurement of the quadrupole parameters is discussed.

INTRODUCTION Exact

expressions

interactions magnetic

of

the

(I = 3/2)

field

with

3-5).

Approximate

(refs.

6-8).

Recently,

using

relative are

spherical

to the

derived

specify

the

efg

axes

useful

crystal

about

SECULAR

EQUATION

Using

E”

- ;

Euler

x = Zu$(l

i = -2~

Q

$

E

been

obtained

by Goldstein

system,

with

the

frame.

field

l$

=

(ref.

field

+ 2rpkesin2$sin2+

-

0;

?-I(1+ cos26)cos2~]cos2~

+ 3cos%

This

general

the

exact Euler

field

equations angles

representation

by rotation

9),

to specify

H in the

i lOV2,

II; {[3sir?e

using

orientations the

of

paper,

axes

to is

of a single

direction.

is: +

for

direction

frequencies

laboratory

to the

the

of the

principal

arbitrary

derived

In this

1,2).

(efg)

for

were

to specify

(refs.

Zeeman-quadrupole orientations

gradient

derived

solutions

of patterns

as defined

laboratory

+ $/3)

0022-2860/83/SO3.00

analysis

combined

particular

field

transition

to the

perpendicular

angles,

equation E2 _

for

an axis

to the

secular

relative

axes and

for

have

analytic

coordinates

levels

of the

electric

solutions exact

principal

energy

levels obtained

to the

polar

for

particularly

relative

efg

been

respect

(refs.

case,

energy

have

- 1 - nsin2acos2$),

0 1983 Elsevier Science Publishers B.V.

the

x-direction,

efqaxes the

.. 86

::

-' : + 2n(l

z'= -,;(I +' +t/3)2 + ?v$${[3sin2@

- 3sin*e

t4ncosesin2$kin2$ Tine constant

+ 7 + 2nsinzecos2$

h has beenomitted

in the

be in frequency units; vB and n are If the field is chosen ve = yH/Z-. axis

perpendicular

EtiERGY LEVELS

to the

field

AND TRANSITION

The secular

equation

+ cos28)Cos2~

above

T +sfn2s3cos24

- $cos291

equations,

+ 9uk!

the

energy

(4) levels

will

the conventional quadrupole parameters, and to .be in the x-direction, rotation about an

changes

a single

angle,

@.

FREQUENCIES

yields

energy

levels

of the

form

(ref.

7):

Eq = jR.+ Dj/2 E3 = (R - 0)/Z E2 = -(R - S)/2 EI = -(R + 5),‘2,

(5)

where, if co53 = -[X3

- 36XZ - 53Y2]/(X2

R = 6-‘/‘[X

i- (X2

u =

[ax

+ f)

5 =

c$x _ f) _ We i7lustrate

-

+

+ 122)‘/*

12z)‘/2COS(5/3)j’/’

(6)

R2]:/2

(7)

R2]1/2

(8)

the

nature

of the

solutions by considering the simple case of Although this is a situation for which

H parallel. to the z’-axis of the efg. the secular equation is readily solved behavior of the elements R, D, and S. function

of v,

for

two values

(ref. Fig.

4), it 1 shows

‘leads to the most bizarre graphs of R, D, and S as a

of n.

2.0

0.0

LARh4OR FREQUENCY
Elements R, II, and ‘S as a function 1. v = 4. (a} n=O (b) n=l. ,a

4.0

6.0

LARMOR FREQUENCY iu,> nf va for

the

case

H parallel

to z’

87

same

The

energy

set

of

levels

constructed

parameters

used

from

in Fig.

Eqns.

obviously be-written as piecewise

linear

separate regions; v3 < v /2, v

2 v0 (V

-

Eqns.

5,

Es 2

El.

-9.0

the

ranking

Thus,

0.0

the

Fig. 2. = 4. % The E4 - E,, remaining

four

\Js = E2 - E1 = S are

shown

LARMOR

Fig. 3. = 4.

3.2 FREQUENCY

of v0 for

transitions,

are

graphed

high

field

directly

4.8 bv.1

Transition frequencies See text for notation.

levels

for

1.6

satellites, in

Fiq.

case

same

notation

of

3.2

4.8

FREQUENCY

H parallel

(v,)

to z' with

~3 = Ej - El,

oarameters

given

EL 2 E3 2

cross.

VL+ = EI+ - El, the

the can

three

the

maintained

never

the

with

With

.

LARMOR

Zeeman-quadrupole v1 = Es - E,,

1.6

but

2, with

the

(2

automatically touch,

in Fig. Za, 4,5)

v

-

k,)

Energy levels as a function (a) n = 0 (b) n = 1.

and

>

0.0

FRECUENCY

the

"Q

may

Fig.

(refs.

v0

4

is always

levels

, and

shown

to

functions

3.2

transitions,

0.0

Qi2

of levels energy

1.6 LARMOR

4

5 are

Referring

1.

in Fig.

1~2=

3.

by vs = EL - Es=

The 0 and

1.

0.0

1.6 LARMOR

as a function of v0 for (a) n = 0 (b) n = 1.

3.2

4.8

FREQUENCY

H parallel

(u,)

to z' with

SINGLE

CRYSTAL ROTATION PATTERNS-

Recently,. crystal

experimental

about-an

patterns

axis

_to extract

patterns

of frequencies

perpendicular

values

to the- field

of the quadrupole

(refs. 10. 11). Either perturbation ties have been used. With the.field

obtained

by rotating

have been fit

parameters.and

a single

to theoretical.

efg directions

theory or numerical ly calculated chosen in the x-direction,

the

frequen-. exact

solutions are in a form highly suitable for the analysis of rotation patterns evolving fromrotation of a crystal about an axis (z) perpendicular to- the field. With this geometry, rotation of the crystal changes a single Euler angle 0 with the other two, 6 and @,, remaining constant. Theoretical

rotation

patterns

of the energy levels

and transition

are Shown in Fig. 4, for Y& = v0 = 10, n = l/2, with the field plane of the crystal. The exact solutions should prove useful routines

where perturbation

theory

is not Sufficiently

frequencies

in the y’z’ in fitting

-

accurate.

18.2

CO'10.4

d

z -I

2.5

z u -g -5.2 ‘3 -13.1

-21.0

!

’

a 40

’

’ 80

ROTATION

Fig. 4. gotation (a) energy levels

’ 120

ANGLE

s

I 160

’

-,.o!

room temperature

I

I

single

I

I

r--r160

120

80

ROTATION

(PHI)

vj between the two lower levels

In the previous diagonalization

,

40

patterns with H in the y’z’ (b) transition frequencies.

We have reanalyzed the transition

’

ANGLE

(PHI)

- plane of the efg system, crystal (2 l/Z)

rotation’pattern

data for

in a-paradichlorobenzene.

analysis (ref. 12), the transition frequency was calculated by of the Hamiltonian matrix and compared to data in a three para-

meter fit (2 Euler. angles and n), taking into account chemical shift effects. In the reanalysis, in which chemical shift effects have been ignored, the exact equation

for

vS was &pared

three Euler angles. locating is shown in Fig_- 5.

to data in a five

parameter fit

the efg axes relative

(w,,

n, and the

to the sample tube).

The fit

a9

Fig. 5. Rotation pattern of transition v5 in a - paradichlorobenzene. Solip line is theory, cross marks are data.

. -_

-0.2

-1.2

ROTATION The

fit

+ 0.005, -

not

to O-8",

angles the

be

same

axis,

efg

axes

relative of

orientation

followed

S, which, (three

in turn,

eqn.

where,

the

to

chemical

shift

rotation

ANQLE (PHI) RADIANS obtained

shift

for

compared

effects

Euler

value were

angles;

32.7".

Since

bein

taken

the

into

-CI= -2.58'

6 is so

rotations

effects

of0.07iZ

small

almost

sign

of

0 + ':,can-

pattern.

ROTATIONS v5, and

lab

axes

n, and can,

Zeeman-quadrupole

by rotation

Y and can

independent)

from

chemical

1.8

the

three

Euler

in principle, transitions

of a single

crystal

be

at an by 4S0

anales

specifying

determined

from

initial

arbitrary

and

about

90°

an

to H.

coefficients

Y, = AcosP+,

for

four

perpendicular The

j,,

the

to the

previously

indistinguishable,

THREE

parameters,

the

obtained

19norin9 a single

FROM

quadrupole

measurement

axis

z.

to

since

was

virtually

from

PAWETERS

The

accurate

Q + $ = -32.3"

9 are

determined

compared

agreement

and

$ and

QUADRUPOLE

the

more

Reasonable

compared about

n = 0.0675

considered

account.

the

gave

0.8

Z (eqns.

be written

using

eqn.

3, 4) can

as functions (5).

For

be written of the

example,

in terms

four

at

the

of R,

transition initial

D, and

frequencies

orientation

(3), + Bsin24,

A, B, and

orientations

+ C,

C are of

(9)

functions

a, + 45"

and

of vO,

ho,

n, e, $ and

eO_

Using

Y1 and

Y,

6, + 90°1

Y1 = -Asin2+,

+ Bcos2+,

•i-C

(10)

Y2 = -Acos2+,

- Bcos20,

+ C.

(11)

Similar are

equations

independent,

frequency cos200.

there The

arise are are

Euler

for

the

obtained. enough

angles

function When

equations are

only

Z.

Thus,

combined

with

to solve

for

determined

six the v~,

within

equations, known

pure

n, cos%, the

of which

usual

four

quadrupole

~0~29,

and

multiplicities.

90

This

method

is

We know it

exact solution. appearance

of

the

fourth

power

highly

accurate,

not

on use of

based

information (eqn.

Relationships

of

the

quickly the

measure

results

points If

to

metry is

parameter.

straight

forward

Exact

if

theory

enter

in more detail

quick,

into

the

is used

coefficients

X, Y, and Z are

frequency

method which

will

at low fields.

to

but

vQ.

theory

form since

the

func-

frequencies

at low fields.

be satisfactory We would

through

at high since of

the efg

fields,

they

not

fitting

the

magnetic

using

secular

and shift

both

shift

may swamp effects equation tensors

tend to discuss this case in a future paper, in which be described

of

dis-

raised

to

expect

many data

and (11).

solution

provided

for

fraction

pure quadrupole

obtained

are

perturbation

in simpler

as those

performed

the

a sizeable

and the quantities

and angles (10)

v0 cc vB due to

be satisfactory

arise

a 3-rotation

as precise are

into

v0 is

but

the

about

as in refs.

measurements

be incorporated

should

in the

arising

error, when frequencies

(9-11)

frequencies,

parameters

be-nearly theory

for

the same form,

to devise the

be satisfactory

The parameters and angles

disposed

It may be possib!e

if

to eqns.

transitions

symmetrically

equations

round-off

measurements

similar

A, B, and C in exactly are

not

intermediate

The method

4).

instead of exact theory. tions

will

into

the

exact

effects

including are

the

should

from a small shift

and perturhation

effects We in-

coincident. rotation

asym-

method will

theory.

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R. 8. Creel, E. von Heerwall, C. F. Griffin, 58 (1973) 4930.

and R. G. Barnes,

J.

Chem. Phys _