A discussion of the relationship between the second moment of the absorption and the Van Vleck second moment

Physica

41 (1969) 389-392

A DISCUSSION SECOND

0 North-Holland

OF THE

MOMENT

OF THE

Publishing

Co., Amsterdam

RELATIONSHIP

BETWEEN

ABSORPTION

SECOND

AND THE

THE

VAN

VLECK

MOMENT

J. I. KAPLAN Battelle

Columbus

Laboratories,

Columbus,

Ohio,

USA

Received 10 June 1968

Synopsis Previous derivations of the equivalence of the Van Vleck second moment to the second moment of the true line shape have been marred by approximations. In this note a correct derivation is given and the lowest order correction term is shown to be negligible in the limit of high magnetic fields and temperatures. Experimentalists working with electron spin resonance (EPR) or nuclear magnetic resonance (NMR) in solids have relied heavily for many years on the theoretical interpretation of the line widths given first by Van Vleck in a 1948 article in Physical Review 1). In this paper he gives the following expression for the “mean square absorption frequency” about the origin Y = 0,

Tr[H, Sz]s Av =

,52 -3C42

’

where H = Hzeeman+ HDipole_Digole(Trunoated). 1 referthe reader Vleck’s original paperr) or one of the standard workss-4) for the characterization evaluating

to Van

on spin resonance

of what is meant by HDipole_Dipole(Trunoated). Upon

eq. (1) one obtains

Av= Vo”-

Tr[Hm(T) >s212 hs Tr(.Sz)s

’

(2)

where YOis the resonance frequency of a single noninteracting Zeeman spin. Writing v = va + x and substituting this on the left hand side of eq. (2) one obtains Av

=

-

Tr[Hm(T), hs Tr(Sz)s

ss] =

vv,

(3)

which is the so often used expression for the second moment of the absorption taken about the center of the absorption line. To go from eq. (2) to eq. (3) 389

390

J. I. KAPLAN

Van Vlecki)

assumed (I) the center of the experimentally

is at va and (II) the resonance

is symmetric

observed resonance

about YO.

Eq. (1) as written in Van Vleck’s original paper 1) was given a motivational derivation rather than a rigorous derivation. In the standard texts on the subjects+) this lack was recognized and derivations leading to eq. (3) were given starting from exact relationships involving X”(Y), the imaginary part of the complex spin susceptibility. In all of these derivations approximations were introduced between the exact initial result and the final eq. (3) which raise doubts as to their validity. Since eq. (3) is such a commonly used formula it seems of interest to obtain its correct derivation and an accurate estimation of its range of validity. Our derivation starts with the exact expressions for the normalized and second moments of x”(v) about Y = 0 which ares)

first

c-3

J q”(v) dv
ooo

=

fr Tr e--Hfl[SS, (H//z,Ss]]

sx”(v)dv =

Tr e-H@[S-12,

~~2x”b) dv Tr e-H@[[S-/2, o~xrYv) dv

S+/2]

(4

’

H/h], [H/h, S-t/2]]

Tr e-Ha[S-/2,

S+/2]

’

(5)

where B = 1lkT and S+ = S, f iS1/. At high temperatures (also assumed by Van Vleck) eqs. (4, 5) can be simply evaluated and one findse) = vo +
(6)

(vs> = v; + 3,,.

(7)

Kambe and Usuis) after deriving eqs. 4 and 5 comment that “Another important problem related closely to ours is the spin-absorption. The shape factor (i.e., the absorption) in this case is defined by A(v) = v-ix”(v) so the area and first moment of this paper (i.e., of x”(v)) correspond to the first and second moments of A(v).” It then follows that eqs. 4 and 5 in terms of the absorption A(v) are given as

s sc-3

A(v) dv

= :

A(v) V

0

&,

9

(8)

THE

VAN

VLECK

SECOND

MOMENT

OF THE

ABSORPTION

391

co m

1VA(V) dv

=

J

I

.

(9)

A(v) dv ____ V

0

It has been shown6) that, contrary to what Van Vleck assumed, the center of the absorption is not at ve, so we will replace assumption (I) by (I)’ which is that the center of the absorption is at L, as yet undetermined. After making the change of variable v = i + x one obtains, using the properties (I)’ and (II) of the absorption line, that eqs. 8 and 9 become +m

= __

+m

1

I --oo

A(x) dx

1

,

(10)

+ . . . dx


s

:A(4 dx 1 ;

--co

(11)

1

dx

1 +$+... A

Solving eq. (10) for A one has approximately

that

+CO i = +L

x2A(x) dx + . . . .

where the total absorption j A(x) dx has been normalized Eq. (11) on expansion becomes

=

;-i%

-co

Substituting

(12)

vo s --oo

A(x) dx + . . . .

to one.

(13)

eqs. 6, 7 and 12 into eq. (13) one has

~0”+ 3vv = vo”+ 2,, +

+m P

xzA(x) dx + J -m

+o

(( > 2

p,

vo

>;4A(x)

dx

_-oo

4

(14)

392

THE

VAN

VLECK

SECOND

MOMENT

OF

THE

ABSORPTION

or i- c-z S @A(X)

coo vv =

-co

dx

4

s -ca

(15)

>.

Thus we find as expected that at high magnetic fields the Van Vleck second moment equals the second moment of the true line shape expect for a small correction term. The correction term for a magnetic field of 10 kG and a line width of 10 G only modifies the result by 1 part in 106. Substituting eq. (6) into (12) we also find that the center of the absorption line is located at 2 ; N va + --
(16)

The exact form of the correction term indicated in eq. (15) can be readily found by keeping the next highest order terms in eqs. 10 and 11. The part of the dipole-dipole Hamiltonian that was neglected in only using Hnn(~) can be shown to give rise to additional correction terms7).

REFERENCES

1) 2)

Van Vleck,

3)

Pake,

4)

1962) p. 85. Slichter, Charles

5)

1963) p. 50. Kambe, K. and Usui,

6)

McMillan,

7)

Kaplan,

Abragam, (Oxford,

J. H., Phys. A.,

The

Rev.

74

Principles

( 1948) 1168.

of

Nuclear

Magnetism,

Oxford

University

Press

Inc.

York,

1961) p. 108. George

E.,

Paramagnetic P., Principles

of Magnetic

T., Progr.

M. and Opechowski, J. I., J. them.

Resonance,

Phys.

theor. W.,

W.

Benjamin,

Resonance,

Phys.

Canad.

A.

8 (1952)

J. Phys.

44 (1966) 4630.

Harper

& Row

302.

38 (1960)

1168.

(New

(New York,