stochastic differential equation study of nuclear magnetic relaxation by spin-rotational...

Physica 1llA (1982) 85-113 North-Holland Publishing Co.

STOCHASTIC DIFFERENTIAL EQUATION STUDY OF

NUCLEAR MAGNETIC RELAXATION BY

SPIN-ROTATIONAL INTERACTIONS

James MCCONNELL

Dublin Institute for Advanced Studies, Dublin 4, Ireland

Received 9 July 1981

The mathematical methods based on stochastic differential equations and the rotation operator, which were developed for the study of rotational Brownian motion and its implications for dielectric dispersion and absorption, are extended so as to yield ensemble averages of certain products of orientational and angular velocity functions. As a consequence, a procedure for calculating nuclear magnetic relaxation times arising from spin-rotational interactions, when inertial effects are included, is presented for molecules of any shape.

1. Introduction

The theory of nuclear magnetic resonance relaxation caused by rotational thermal motion’.‘) was applied by Hubbard in a sequence of papers>‘) to quadrupole interactions, intramolecular dipole-dipole interactions and spin- rotational interactions. The treatment of the last type of interactions is particularly difficult in that it involves the calculation of the ensemble average of the product of functions of orientational angle variables and angular velocity variables. In his earlier investigation Hubbard’) regarded the angles and angular velocities as independent sets of variables, so that the ensemble average of the product was assumed to be the product of the ensemble averages of the function of the orientational variables and of the function of the angular velocity variables. However the orientational and angular velocity variables are not independent and Hubbard4”) later proposed a method based on a Fokker-Planck equation which enabled him to write down a general expression for the Laplace transform of the ensemble average of the product of orientational and angular velocity functions which occur in spin-rotational relaxation studies. For the case of a rotating spherical molecule Hubbard deduced expressions for the spin-rotational correlation time and for the spin-rotational contributions to the reciprocals of the longitudinal and transverse relaxation times.

0378-437 1/82/OOOOAMM /$02.75 @ 1982 North-Holland

86 JAMES MCCONNELL

A method based on Euler-Langevin stochastic differential equations, the ensemble average of the stochastic rotation operator and the Krylov-Bogol- iubov solution of nonlinear differential equations has been found very power- ful for the investigation of dielectric relaxation processes when inertial effects are included’). Indeed the method is generally applicable to processes whose investigation is based on the correlation functions of spherical harmonics. Confining our attention to nuclear magnetic resonance phenomena we have already applied the method to the calculation of spin-lattice relaxation times?. It may be applied without difficulty to the contributions of intramolecular dipole-dipole interactions and of quadrupole interactions to the nuclear relaxation rate of identical nuclei, but not to the contributions of spin-rotational interactions.

It is the purpose of the present paper to extend the above mathematical method so that it will provide the ensemble average of the product of the orientational and angular velocity functions encountered in the study of spin-rotational interactions. In section 2 the formalism for these interactions will be sum- marized, definitions given and the extended mathematical method will be presented in a manner applicable to molecules of any shape. In section 3 a detailed study will be made for the spherical model of the molecules, and the results will be compared with those derived by other methods. Finally, in section 4 the problem for asymmetric molecules will be considered.

2. Spin-rotational interactions

2.1. Definitions and basic equations

We consider the contribution to nuclear magnetic relaxation of identical nuclei in identical molecules. The spin-rotational interaction is the sum over all molecules in a system of the sum of the interactions of the magnetic moments of the nuclei in a molecule with the magnetic field produced by the rotation of that molecule. For later comparison with the results of Hubbard we follow fairly closely the notation of ref. 5. Let us denote by 1; the spin operator of the ith nucleus and by hJi the angular momentum of the molecule that contains this nucleus. The spin-rotational Hamiltonian of the ith nucleus,

hGf =hIi *C’ *Ji, (2.1)

where C’ is a dyadic with the dimensions of a frequency. Hubbard expresses (2.1) as

G( = i V:U:, k=-I

where VT are the spherical components of Ii in the laboratory system and

SPIN-ROTATIONAL RELAXATION TIMES 87

(2.2)

In this equation

b;, = C:,, b:,,, = T Ci, 7 i C:,

V/z ’ (2.3)

where CL, are the constant Cartesian components of the dyadic referred to

axes fixed in the molecule. In (2.2) D L,,, is the rotation matrix for the

transformation of a spherical tensor’@) and ai, pi, yi are the Euler angles specifying the molecular system with respect to the laboratory coordinate system. We see from (2.3) that

b i * = ( - )“b I,,,,. ml” (2.4)

The contributions (l/T,)1 and (l/T*), from the spin-rotational interactions to the reciprocals l/T, and l/T2 of the longitudinal and transverse relaxation times T,, Tz, respectively, are given by

(&), = 2Jl(OO), ($), = Jle-0 + Jdoo), (2.5)

where wo is the angular velocity of the Larmor precession, m

J,(w) = i 1 [C?(t) eio’ + C?(t) e-‘“‘I dt 0

(2.6)

and C!:(t), not to be confused with the dyadic components, is defined by

c:;(t) = (uf(t)U:(O)), (2.7)

where the angular brackets denote ensemble average for thermal equilibrium. We see from (2.2) that

C!(t) = p$, ,$_, b&bL(Dh(ai(t)v Pi(t), n(t)) I 9

x ~kn(W(O)9 Pi(O), yi(“))Jifi(t)Jiv(o))* (2.8)

We take for the molecular frame of reference the principal axes of inertia through the centre of mass and write the components of angular momentum as Ilo,, Izwz, 1303, where I,, I?, I3 are the principal moments of inertia and WI, 02, o3 the corresponding Cartesian components of angular velocity. Then replac- ing Mi,(t) by &o,(t) we express (2.8) as

Cff(t) = he2 p$E, ,$_, bI,b',,r,r"(D:"(ai(t), Pi(t), n(t))

X DL(ai(b), Pi(O), ~iKOh+(~)%(O)), (2.9)

88 JAMES MCCONNELL

a sum of ensemble averages of products of a function of angle variables and

a function of angular velocity variables.

At this stage we introduce the stochastic rotation operator R(t). We see

from (2.9) that

XDkn(~i(O)v pi(O), yi(O))wp(t)wv(O)). Now

(2.10)

DAn(ai(t), Pi(t), n(t)) = DAX(-ri(t), -Pi(t), -ai( 457 C-1

II2

= 3 yln(-Pi(t)3 -ai(t

so

C?(t) = $ bf,bX,~,~“(Yl,(-Pi(O), -ai( &,“‘I m,n=-I

x Yln(-pi(t)7 -Qi(t))Wp(t)wu(o))

I =sp$, ,z_, (-)“bIn,bm,~~~~,(YT~,(-Pi(O),-~i(O)) . ,

X R+(t)Yln(-Pi(O), -ai(0))~p(t)wv("))7

where R(t) is the rotation operator that brings the molecular frame at time

zero to the molecular frame at time t and R+(t) is its adjoint”). Since R(t)

involves the angular velocity through the relation

dR(t) - = -i(J, * o(t))R(t), dt

(2.1 I)

it follows that

However the angle and angular velocity variables though not independent are

separable. This allows us to make the ensemble average firstly over the angular

velocity variables, denoting it by (. . .)w, and then over the angle variables at time

zero. Thus

C?(t) = &T i i ( - )“b&,bLvlprv 1 d[-ai(O)l 1 d[-Pi(O)l M-Pi(o)1 p,v=I m,n=-I

0 0

X YT,-m(-Pi(O), -(Yi(0))(Rt(t)w,(t)o,(O)),YI”(-Pi(O), -ai(

= $g i i (- )“b~,b,yr,I,((R+(t)w,(t)w,(O)),)”,.., ji’.v=I m,n=-I


where -m, n denotes the -m, n-element with respect to the basis Yl,_@(O),

cu(O)), YI,o(P(O), a(O)), YdPKV, 40)). We conclude that

C?(t) = $5 $I ,$_, (- )“b~,b~,I,I,((R(t)w,(t)w,(O)),),,~,. (2.12) . ,

If we sue :ed in calculating (R(t)~~(t)~~(o)),,,, we may be able to find (l/T,), and (l/T,), from (2.5), (2.6) and (2.12).

To perform these calculations it is helpful to define the Laplace transform c;:(s) of c::(t):

Y

c{:(s) = e-“C!:(t) dt, I

(2.13)

0

with C:!(t) given by (2.9). Then, from (2.12), _

c?(s) = $ i i (- )mbhpbbJJ, (,f e~“‘(R(t)w,(t)w”(O)), dt)n_m &Y=I m,n=-I

0

and, from (2.6),

(2.14)

J,(w) = :(cpP(-iw) + c?(k)). (2.15)

As will be explained below in subsection 3.3, we may replace J,(wo) by J,(O) in the extreme narrowing case. Then (2.5) and (2.15) yield

(-3, = k), = 25,(O) = 2cF(O). (2.16)

We write l/T,, for the common value in the extreme narrowing case of (~/TI)I and (1/T2),, and so

$ = 2cF(O). (2.17) ST

The spin-rotational correlation time 7Sr is defined as the integral from zero to infinity of the normalized autocorrelation function of U!(t), so that

Tsr = dt

(U:*(o)U’:(o)) *

From (2.2) and hJi, = lywy we deduce that

(Uf*(O)Uf(t)) = h-* 2 2 b;l:b:,I,I, &,“=I m.n=-I

x (DktdW(O), Pi(O), ~i(o))~:n(~i(~), Pi(t)7 Yi(f))~fi(t)Wu(O))

= 9 $, $_, b’m,vbt,J,Jv

X (DLnC~iCOl, Pi(O), n(O))Dkn(~i(t), Pi(t), yi(t))wp(t)wv(O)),

(2.18)

90 JAMES MCCONNELL

where we have used (2.4) and the property of Wigner functions’2)

&‘k% p, y) = (-)k+mDJ&,,(a, & y).

From a result of Hubbard13) we find that

(2.19)

(D’-k,-m(ai(O), pi(O), yi(O))DL(ai(t), Pi(t), Yi(f))WF(f)mu(O))

= (-)k(D~.-m(ai(0), pi(O), n(O))DL(ai(t), Pi(t), Yi(f))Wp(t)Wv(O)).

Hence

(undue = h-l 2 i h’,,,,h’,,I,!,.

~.Y=I m.n=mI

X (DA,~rn(ai(O), pi(O), ri(o))DL(ai(t), Pi(t), Yi(f))Wv(f)Wv(O))

(2.20)

= c?(t),

by (2.10), since m and n are summation indices. It follows from (2.13) that

I e-“‘( LJ!*(O)U!(t)) dt = c?(s), (2.21)

I (u~*(o)u~<t>> dt = c:(o). (2.22)

0

To obtain the denominator in (2.18) we note that R(0) is the identity

operator, that the Wigner functions in (2.20) for t = 0 are consequently

independent of the angular velocity, and therefore that

(DA,-m(~i(O), Pi(O), yi(O))D&(c-Ui(O), Pi(O), ~i(O))~p(o)~v(o)) = (DA,-m(ai(O), Pi(O), yi(O))DAn(~i(O), Pi(O), yi(O)))(~~(Oh(O))~ (2.23)

For a rotator of any shape14)

(O,(O)%(O)) = 6,” y. II

Then employing (2.19) and the orthogonality relation

jdy idaj d/3 sin PDG(a, P, r)@&(~, P, r) = 8T26ii,6,,,6+

2j+l ’

(2.24)

0 0 0

we see that


(DA.-m(ai(O), pi(O), yi(O))Db(ai(O>, Pi(O), ri(O))>

dfli(O) sin @i(O)DCn(~i(O), pi(O), ri(O)) (2.25) 0 0 0

X DAn(ai(O), Pi(O), ri(O))

= (- 1)“hn” 3 .

Hence from (2.20) and (2.23)-(2.25)

(U = !$ 5 2 G,,S,,(-)mb !,,,b :,I, p.v=l m.n=-I

We conclude from (2.18), (2.22) and (2.26) that

the value of c?(O) to be obtained from (2.14). In the extreme narrowing case we deduce from (2.17) and (2.27) that

(2.26)

(2.27)

(2.28)

Thus the spin-rotational contributions to the reciprocals of the longitudinal and transverse relaxation times are proportional to the spin-rotational correlation time.

2.2. General method of calculating (R(f)~+(f)~~(0))

We have seen in the previous subsection that the evaluation of (l/T,),, (l/Tz),, 7Sr is deduced from the value of the ensemble average over angular velocity space of R(f)~~(t)&0), where R(t) is the rotation operator that brings the molecular coordinate system at time zero to its orientation at time t. In studies on dielectric phenomena we were concerned only with the average over the angular velocity space of R(t), and we shall now show how in principle the previous method may be adapted so as to meet the require- ments of the spin-rotational problem.

The rotation operator obeys the stochastic differential equation (2.1 l), which we now write

dR(t) - = -i(J * w(t))R(t), dt

(2.29)

92 JAMES McCONNELI

omitting the subscript of J because we shall focus attention on one molecule

only. In (2.29) J has components J,, JZ, .Jx in the rotating molecular coordinate

frame that obey the commutation relations

[JZ, J3] = -iJ,, [JJ, J,] = -iJ?, [J,, JZ] = -i13,

which we express briefly as

[J,, J,] = -i(J * eP X e,), (2.30)

e, and e, being unit vectors in the p- and v-directions. We suppose that the

rotation of the molecule is due to thermal motion in a steady state and that the

components o,(t), w:(t), w3(t) of o(t) in (2.29) obey the Euler-Langevin

equations

dW,(t) I,;, - (I>- Z3)w2w3 = -Z,B,w, + I, ~

dt ’

dWz(f) (2.3 I)

zjcjj - (I, - z&o,wz = -Z3B_cw3 + I3 yp,

where B,, BZ, Bj are frictional constants and WI, W?, W, are Wiener

processes. Eqs. (2.31) are nonlinear and w,(t), o>(t), q(t) will be centred but

in general non-Gaussian’5). If the molecule is spherical or linear, o(t) obeys a

Langevin equation and is a centred Gaussian random variable.

Ford16) has given a general method based on earlier studies of Krylov and

Bogoliubov”) of solving a nonlinear stochastic differential equation of the

type

dx(t) - = EO(f)X(f), dt

(2.32)

where x(t) is a random variable that may be an operator, E a small parameter

and O(t) a stochastic operator. Since accounts of Ford’s method have been

published8*‘8), we shall just quote results that are relevant for our purposes.

Eq. (2.29) is obtained from (2.32) by the substitutions

x * R(t), EO(t)++ -i(J - w(t)).

Then we write

R(t) = (I + &‘)(f) + c’F(‘)(t) + c3Fc3’(t) + l 4Fc4’(t) + . . a)@(t)), (2.33)

where I is the identity operator and F”‘(t) are stochastic operators with zero

ensemble averages. The non-stochastic (R(t)) obeys an equation


y = (dP(t)+ ,2fJ(2)(t) + •~fi(~)(t) + •~fl(‘)(t) + . . .)(R(t)). (2.34)

In our previous investigations we were concerned only with the solution of

(2.34) but now we must find R(t) in order to calculate the average value of

R(t)~+(tMo). Since

(eO(t))r, -i((J . o(t))) = 0

because (w(t)) = 0, it is found that

l F(‘)(t) = -i (J - o(t,)) dt,,

0

f ‘1

l *F(*)(t) = - I I

dt, dtz[(J . o(t,))(J . w(t2)) - ((J . o(t,))(J . o(tz)))],

c3F(3)(t)=iidt i [ I dtz dtd(J * o(tl))(J . m(tz))(J . o(t3)) (2.35)

0 0 0

-(J * dtdX(J * m(tz))(J . m(tdN -(J . o(tz))((J . o(t,MJ * o(h)))

-(J . 4tMJ . o(td)(J . @(tz)N -((J * o(td)(J * m(t3)(J * dtd))l,

and that in general

n-1

c”@“‘(t) = -e”.flc”)(t) - i(J - o(t))e”-‘F’“-“(t) - E” F, F”‘(t)fi’“-“(t). (2.36)

We also have

d”(t) = 0, l 2f2(*)(t) = - J

((J - o(t))(J . o(t,))) dtl,

0

f 11

l 3flc3'(t)= i I I

dt, dtz((J . dt))(J . w(td)(J . m(tdNr

l 4O”(t) = idt , jdtz jdt,{((J . o(t))(J . o(t,))(J . W(tZ))(J . o(t3))) (2.37)

0 0 0

- NJ * o(t))(J * w(tdMJ . dtz))(J * dt3N - NJ * dt))(J * dtdM(J * dtl))(J * dtd)) -NJ * o(t))(J * dtdM(J * dtd)(J * dtd))l.

94 JAMES MCCONNELL

On substituting the values of flci’(t) into (2.34) we may be able to obtain (R(t))

in a form suitable for further computation. Then finding F”‘(t) from (2.35), or

from (2.36) and (2.37), and substituting into (2.33) we have R(t) and may

proceed to calculate (R(f)~&(t)ti~(O)) required in (2.12).

In these investigations the operators are independent of the angle variables

and so we may denote ensemble averages either by (. . .). or by (. ’ .). For

convenience we shall adopt the latter notation.

All the above considerations are applicable to molecules of any shape. We

shall now apply the general theory of this section to a spherical molecular

model.

3. Spherical molecules

3.1. Calculation of (R(t)w,(t)w,(O))

When the rotating molecule is spherical in shape, eq. (2.31) reduce to

z&J(t) = -zBw(t) + z F (3.1)

and o(t) is a Gaussian random variable with zero mean. Then, since the mean

value of the product of an odd number of such w’s vanishes, R”‘(t) given in

(2.37) vanishes, as indeed do n”‘(t), n(‘)(t), etc.. It has been shown that?

(R(t)) = [I + rJ2(1 - em”) + r2{J2[:- (Bt + 1) emB* - 4 em”*]

+ (J2)2[$ _ e-B’ + f e-2B’]}

+ r3{J2[& (:B2t2+ 2Bt + 1) emBr - (:Bt + 1) ee2” -3Bf -$e ] (3.2)

+ (J2)2[f - (Bt + $) e-‘* + (Bt + i) e-“’ + d e-3B’]

+ (~2)3[:_; e-Bt + 4 e-2Bl -4 e-3BI]}+. . .,emBGjt,

where

j(j + 1) being the eigenvalue of J2 and

kT Y=m’

(3.3)

(3.4)

where k is the Boltzmann constant and T the absolute temperature. It is

found in dielectric absorption experiments that the value of y does not exceed

a few per cent”‘zo). We see from (3.2) that (R(t)) is a multiple of the identity

and so commutes with J,, J2, J3.


We wish to calculate R(t) from (2.33), (2.35) and (3.2), and use the value so

calculated to obtain (R(t)~+(f)0~(0)). Let us suppose that n is an odd integer. The contribution to R (t)w,( t)0”(0) corresponding to E “F(“)( t) contains only terms with an odd number of w’s and so the ensemble average of the contribution vanishes. We may therefore deduce from (2.33) that

(R(f)uJf)uy(O)) = ((I + ,*F’*‘(t) + l 4Fc4’(t)

+ d’F@‘(t) + . + ~)~p(fh(W(R(W.

For a steady state solution of (3.1) we have*,)

(oi(t,)w,(t,)) = 6@ F e-Bltt-rmt (i, p = 1,2,3).

It follows from (2.37) that**)

&p’(t,) = _ kT 52 I’ I I

,-BWd df2 = g 57 1 _ e-4),

0

0 0 0

We see from (2.36) that

t t

l 4Fc4’(t) = - I c40”(t,) dt, - 1 E*F(*)(f,)E*@*)(t,) dt,

0 0

-i (J * o(t,))E3FC3’(t,) dt,.

0

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

Let us calculate (IuP(t)Wy(0)), (E*F’*‘(t)o,(t)W,(O)), and (e4F’4’(t)w,(t)oy(0)). From (3.6)

(Io,(t)wV(0)> = 6 J kT ems1 “I ’

since t 5 0 in (2.6) and all subsequent equations of section 2. From (2.35) and

(3.6)

r2F~2~(f)=-~df,~dt2[,~,J~~o.(f,)o,(~2)-~J2~d~,~dt2e~B”1~1$l, 0 0 0 0

(3.10)

96 JAMES MCCONNELL

and therefore f 'I

(~'F'~'(t)~,(t)oy(o)) = - i: JJ, j- dt, j dtz(w,(t)w,(t,)o,(tz)o,(O)) r.v=l

0 0

+ y yJ2S,,.(Bt - 1 +eeBt)emBf, (3.11)

on introducing y from (3.4). From the properties of Gaussian variables23)

(w,(t)o,(t,)w,(tz)w"(o)) = (o,(t)wr(tl)>(w,(tz)w,(O)>

+(o,(t)Ws(t2))(o,(tl)ov(O))+(Wp(t)W"(O))(Wr(t,)o,(f2))

= F 2{6,,6,,e- c )

B('mtL+*2) + (6,,6,, + tipy8,s) e-B(f+*~-'z).

Thus t ‘I

I I dtl dtZ(~~(t)~,(t1)~s(t2)~,(0)) = Y F emBr

0 0

X {(eB' - 1 - Bt)S,,S,, + (em” - 1+ Bt)(6,.& + ~Jis)l

and on employing (2.30) we deduce that

(3.12)

(3.13)

(E2FC2)(t)W,(t)oy(0)) = -y F ((1 - 2 e-” + e-2Bt)JpJv

+ (- e-” + Bt eeBt + eezBt) i(J . e, X e,)}.

Since flC2) and flC4) are non-stochastic, we see from (3.9) that

(3.14)

f

(E4FC4)(t)WIL(t)W,(0)) = -y 6,” emBr 1 E4fln4)(tl) dt,

0

- I

~2~n(2’(t,)(E2F’2’(t,)Wlr(t)W,(0)) dt,

0

Now, by (3.8),

- i (w,(t)(J - W(tl))e3F”‘(tI)Wy(0)) dt,. (3.15)

0

f kT 3

-- I S,, eeBt l 4fF4)(t,) dt, = 6, J2 ems’

(3.16)


which is a convolution24). Next we have from (3.7) and (3.10)

* f

- I

,2L!(2)(t,)(,2F(2)(t,)Op(t)~V(0)) dt, = -gJ* 1 dt,(l - e-Bf1)

0 0

X(%(t) [i’d2 f ,.:_, dt, c JJSw,(t2)wS(tl) - yJ’(Bt, - 1 + emstl) w,(O) I >

.

0 0

(3.17)

Using (3.12) we deduce that

‘I f2

I I dt, dt, i J~.~(,(o,(t)w,(t,)o,(t,)w,(O))

r.s=l 0 0

= y $bet{&J2 (e-"l- 1 + Bt ,) + J,J,(e”’ - 2 + e-Br1)

+ (e-Bf1 - 1 + Bt,) i(J * e, X e,)}. (3.18)

Cancelling terms in (3.17) and (3.18) and integrating with respect to t, we obtain

- I l Z~n(2~(t,)(~2F~2~(t,)~p(t)~,(0)) dt,

0

= -y*~J*{J,J,(l ++e-*’ I -381 - 3Bt e-*’ - 3 e-**’ + I e ) (3.19)

+ i(J . e, x e,)(i em*’ - Bt em*’ + :B*t* e-*’ - e-**’ + Bt em**’ +: ee3*‘)}.

The equation in (2.35) for E3Ff3’(t) gives

-F J* $, Jbwb(t2) e-B(*3mfr)+ [

2, Jcw,(t3) emB(f2-‘4)

so that

3

+ 2 Jdcod(t4) emB(t2-f3’ , d=I

f

-i I

(o,(t)(J - o(t,))e3F’3’(t,)o,(0)) dt,

0

=jdt,jdt2j2dt3jdt4(A-B-C-D),

0 0 0 0

(3.20)

(3.21)

98 JAMES MCCONNELL

where

A = Oj_, J,JbJ,Jd(W,(t)W"(t,)~b(~Z)~~(t~)~d(t4)W"(o)), . . 1

B = FJz “$=, J,Jh(~,(t)o,(t,)ob(tz)w,(0)) e~B(fq~*4)

C = FJ’ ai, J,Jc(W,(f)W,(f,)w,(t,)w,(0)) emB”2-‘4’, (3.22)

D = y j2 i J,Jd(W,(t)WU(t,)Wd(t4)W,(0)) emB(*‘+‘). a.d= I

To evaluate A we extend (3.12) to the product of six w’s and employ (2.30) to derive the relations:

JJJJJJJ, = P(.J2 - Z)(.P - 21), o.b.c= I

(3.23)

= ~ A=, J,JJ,J,JJ, = J2(J2 - Zj2. . .

After a long but elementary calculation we deduce from (3.21)-(3.23) that

+ [2J2 _ z]J,J, 1 &, . . . i df4 e-B(1+t~-t~-t3+r~

0 0

+ [2J2 - 3Z].JJ, + [J2- II i(J . eP X e,)

SPIN-ROTATIONAL RELAXATION TIMES

+ J2s,,) j dt, . . .i’ dt4 e-B(f+Il-f2+f3--t4)

0 0

+ {[2_J2 - SI]J,J, + (2J2 - 41) i(J * ep X e,)}

xjdfl... jl dt4 e-B(t+t,+t,mt,-r,) 1,

0 0

99

(3.24)

and we see that the multiple integrals are convolutions. The value of (e4F’4’(t)~C(f)~,(0)) is now obtained from (3.15), (3.16), (3.19)

and (3.24). If we wished to find (~~F”)(t)o,(t)o”(O)), the calculation would be extremely long; for example, corresponding to A in (3.22) we would have a summation which involves the ensemble average of the continued product of 8~‘s and this consists of 105 terms. We shall not therefore take the calculations further. For our purposes we do not require an explicit expression for the rotation operator R(t) but such an expression may be written down from (2.33), (2.35), (3.6)-(3.10), (3.20), (3.2) and (3.3). The value of (R(f)uP(f)uy(0)) is obtainable from (3.2), (3.3), (3.5) and our calculated values

of (IWw(t)Wy(0)), (~2F’2’(t)W,(t)~,(0)), (E4F’4’(t)o,(t)o,(0)). Explicit values of the integrals occurring, which in fact are not required for the investigation of spin-rotational interactions, may be derived by inverting the Laplace transforms of the convolutions25).

3.2. The Laplace transform of (R(f)WP(f)Wy(0))

In the case of a spherical molecule eq. (2.14) becomes

the integral being the Laplace transform of the operator (R(f)WP(f)mY(0)). As an illustration of the method of calculation and approximation we take the first term

_ (~~Jpj, i dt, / dt2 f dt, / df4 e-B(t-tt+r*+h-td

0 0 0 0

(3.26)

on the right-hand side of (3.24) and we approximate (R(t)) in (3.2) by e-B’%. We shall calculate the Laplace transform of the expression (3.26) multiplied

100 JAMES MCCONNELL

by e-BGi’, noting that

0 0 0 0

=- J,J, e-(~+~Gi"~e~BGif,,~(B+BGi)f,,-(2B+BG,)~~e-(B+BGi)f

The Laplace transform of this is

(S + BGj)(S + B + BGj)3(S + 2B + BGj)’ (3.27)

By inverting this we may find the value of (3.26) multiplied by e-BGI’26). Let us write (3.27) as

zkT J,J” -’ IB ((S/B) + Gj)(l + (S/B) + Gj)3(2 + (S/B) + Gj)’

(3.28)

For the small values of j with which we shall be concerned, Gj defined by (3.3) is of order y. The factors 1 + (s/B)+ Gj, 2+ (s/B)+ Gj are at least of order unity. However for values of s/B of order Gj or less, and so for the extreme narrowing case when s will be taken equal to zero, (s/B) + Gj is of order y. This will raise the order of (3.28) to J,J,ykT/(IB). Then in order to obtain an approximation of order J,J,r*kT/(ZB) it will be necessary to include the term yJ* in (3.2) when approximating (R(t)). Similarly, if the denominator of the Laplace transform had contained a factor (s + BGj)‘, we would have had to include terms proportional to y* in the approximation of (R(t)).

On performing the calculations we obtain

co

I e-“‘(R(t)w,(t)wV(0)) dt 0

kT = “’ I I

Z yBJ2 S + B + BGj + (S + B + BGj)(s + 2B + BGj)

+ Y2 [ f.l”+ iJ’ .14+ J* BJ*

S+B+BGj-S+2B+BGj-(S+2B+BGi)2 $J"-dJ* B4J2

+ S + 3B + BGj + (S + B + BGj)‘(s + 2B + BGj)’ 4 2

+ (S + B + BGj)*(S + 2: : BGj)*(S + 3B + BG,) 11


kT +i(J*e~xe,)yI

I BI I s+B+BGi-(s+B+BGj)2-s+2B+BGi

t B2 1 4 s+B+BGj-(s+B+BG,P-s+2B+BGj+s+3B+BGj

4 2-

+(s + B + B%$(s +I:B + BGj)’

B4(2J2 - 41) ’ (S + B + BGj)‘(S + 2B + BGj)2(s + 3B + BGj) II

- I+2yJ2+y7sJ4+~J~]+ 21+5y.J2 s+BGj s+B+BGj

3yBJ2 1 I$’ +(s+B+BGj)2-s+2B+BGj+s+3B+BGj

[

2.J2 - 21+ 2y(J4 - 52)

+ YB4 (S + BGj)(s + B + BGj)3(s + 2B + BGj) 2J2- 31

+(s + B + BGj)3(s + 2B + BGj)’

2J2- 51 + (S + B + BGj)2(S +2B + BGj)2(s + 3B + BGj) II +. . *. (3.29)

We note that the quantities inside the brackets are multiples of the identity.

3.3. Spin-rotational relaxation times

We return to eq. (3.25) for c?(s). Let us suppose that all the nuclei are in equivalent positions, and let us take the third axis of the molecular frame through the nucleus in which we are interested. We assume that the third axis is an n-fold axis of symmetry2’) with n 2 3. This allows us to writez8)

c&= c\\, cl1 = c&= Cl, c;, = 0 (p# 4). (3.30)

We then see from (2.3) that

b;, = bd=O, bb3= C\l. (3.31)

The b’s are independent of i and we shall henceforth omit their superscript. We shall also write c:(s) simply as c(s) because it is independent of i and because the zero superscripts may be discarded, since this is the only c:!(s) function that

102 JAMES MCCONNELL

we shall meet in this subsection. Thus we express (3.25) as

C(S) = gpJ?, I$_, (-Jmbwbmu (j e-“‘(~(t)~,(tMQ) dt)n__m, (3.32) . .

0

where n,-m denotes the n,-m-matrix element in the representation with basis

elements YI.-I@(O), a(O)), YI.o(P(O>. ~W), YdPKV, a@)).

The matrix element may be written down from (3.29) by putting J2 = 21 and

Gi = Gr, where from (3.3)

G,=2y+y2+%~‘+....

In the present representation

-1 0 0

2 .J3= ! 0 0 0 0 0 1

(3.33)

1 Y (3.34)

the rows and columns being numbered in the sequence - 1, 0,l. The matrices

of (3.34) may be obtained from those of Rose*? by making the substitutions

MI*-hJ,, My +, - h.Jz, M3 ++ - fiJ3,

in order to take account of the minus sign in the commutation relation (2.30).

We see from (3.33) that

In the extreme narrowing case of w,, @ kT/(ZB)3@) we may replace s by zero in

(3.29) when calculating J,(oo) from c(s) as given by (2.15). Then (2.17) yields

I T,, = 2c(O). (3.35)

In order to deduce c(0) from (3.32) we must perform the summations over

m, n, J_L, v involving the b’s and the operators outside the curly brackets of

(3.29). A brief calculation gives

where we have employed (3.31). Then we deduce from (3.34) that

&$, ,$_, (-Y’bhmvMJ * e, X eu)]n,_m = -2C: - 4CICll. (3.37) , 3

On evaluating J,J, from (3.34) and substituting we likewise find that


j, m $_, (-)‘“bnrbmvC’J,),,-m = o. (3.38) . , Eqs. (3.36)-(3.38) were already given by Hubbard3’). In (3.29) the terms that require special attention for s = 0 are all proportional to J,JV. On account of (3.38) they give zero contribution to c(O), and for the purpose of calculating l/T,, from (3.35) they may be disregarded. However for the sake of complet- ing a record of this calculation we shall retain them.

On putting j = 1 in (3.29), employing (3.33) and expressing the results as power series in y we find that

_

Iii e-“‘(R(t)o,(t)wy(0))i=, dt I 0

=$ [ . ] 6 J + y -S,J - $ (J * eP X e,) + y2[~i3,,I + $i (J * e,, X e,)]

- :J,J, + &&Jy + ;y’J,J, + . . . I . (3.39)

For the reason given in the previous subsection it is to be expected that, if we were to continue our calculations so as to include the l “F@‘(t)-term on the right-hand side of (3.5), the coefficient d of -y’J,J” would be altered. The other terms on the right hand side of (3.39) are in agreement with the result of Hubbard32). Eq. (3.32) combined with (3.36)-(3.39) yield

c(o)=~{(cg+2ci)(l- y ++?y2+. * .) + (2c: + 4c,c,,)(:y - gy2+ . . .)}

= &!&{(Ci + 2c:) - y(C, - C,,)’ + +p(CL - c,,)2 + . . .},

and so, from (3.35),

$ = $$cc;+ 2c:)- y(C,- c,,)2+&2(cL- c,,)2+. . .}. (3.40) ,r

To obtain 7Sr we note that for the sphere (2.28) becomes

Now

$, $, (-)mb-m,fibmp = i i (-)mb-,,,b,,S,,G,, &,“=I m,n=-I

= $ ,$_, (-)‘“b&An.-m&v = Cf + 2C:, 1 .

(3.41)

104 JAMES MCCONNELL

by (3.36), and we may express (3.41) as

1 - = F (Ci + 2Cf)7,,. TV

If we write .$ = CJC,,, (3.40) and (3.42) yield

T,, = - ; 1-y 1 (5- I)'+yy2(6- I)‘+. . . x$57 2t2+ 1 1

(3.42)

(3.43)

in agreement with Hubbard3’). When C, = C~I, (3.43) reduces to

where we have written B _’ as Q, the friction time that occurs in the

discussion of the Debye and Langevin equations. Then T,~ is independent of

the orientation, as it should be according to an earlier result of Hubbard for

the spherical molecule’4).

McClung35) carried out an investigation of spin-rotational interactions for

spherical molecules by employing the eigenfunction expansion procedure of

Fixman and Rider36) to obtain a series expansion for the orientational-angular

velocity conditional probability density from the Fokker-Planck equation.

Applying numerical methods he calculated a correlation time which charac-

terizes the anisotropic spin-rotational interactions. If we denote this cor-

relation time by T&, then in our notation37)

1 25 + 1 ’ 2ZkTCi T,,= h*B

{ (+ + 2 (yrB7:j.

On comparing this equation with (3.42) we find that

3(2t2 + l)~,, = (25 + l)‘B-’ + 2(5 - l)?:,.

When we substitute for T,, from (3.43) into (3.44), we obtain

7,1 ’ = l_$y+yy2+. .,

which agrees with the result of McClung and his collaborators38).

4. Asymmetric molecules

4.1. General equations

(3.44)

We now consider the case of a molecule with no special symmetry properties,

whose rotational Brownian motion is governed by the Euler-Langevin equations

(2.31). With an obvious generalization of Y”~ satisfying (3.4) we choose E in


(2.32) as given by

(kT)“’ E = (I,1&B:B:B:)“6

105

and expand the components of the steady state angular velocity:

wi(t) = E&l!‘)(f) + E*W!*)(t) + E3Wj3’(t) + ’ * *.

Then w{“(t) is a centred Gaussian random variable obeying

l *(Wj’)(t)W(‘)(S)) = 6,, -e kT -B,I~-~I m

4 .

On the other hand

(o,(t)o”(s)) = 6,, {F e-B~‘*-sJ

(4.1)

(4.2)

(4.3)

+ 1,--L *(kT)‘e- ( ) 1, Ido

B,lt-sl[l - (B + B, _ B ) t _ S _ e-(B,+B,-B,)lt-sl~ + . . . 0% + B, - B,J2 3

,

(4.4)

where II, p, u is a cyclic permutation of 1, 2, 334. We immediately make some simplifications. In order to calculate relaxation

times we shall work in the three-dimensional representation given by (3.34).

From these we deduce that

and we see that J:, J:, J: commute with each other. As a consequence of this

(R(t)) = (I + i s (I- ewB+).C + * . .] e",

where

G = - $, CD!" + Di">J:,

(4.6)

(4.7)

(4.8) D\+, D$d-, D$‘)=&

Di"-&,I, 2B2B3(B2 + B3) - B,(B: + &B, + B:)

B&B:(B2 + J33)

+ I Bz(B2 + B3) - 2B: + I BJ(& + B3) - 2B:

* B,B&(& + B3) 3 B,B:B#%+ B3)

(12 - 131*

- I&W2 + B3) ’ 1 etc.?. (4.9)

106 JAMES MCCONNELL

Thus 01” is a correction to D!” of order y. The same is true of the summation

terms in (4.6) with respect to 1. We must therefore restrict our calculations for

the asymmetric rotator model of the molecule to corrections of just one order

in y to the Debye-Perrin limit4’).

4.2. Calculation of (R(t)o,(t)w,(O))

We return to (2.33) with E given by (4.1) and deduce that

(R(t)qL(t)m”(O)) = {(ql(~)WV(O))I + kF(‘)(G$L(0&(0))

+ (E*F’*‘(t)W,(f)W,(O)) + . . . 1

(R(t)). (4.10)

On account of the restricted order of approximation available for (R(t)) in the

case of an asymmetric molecule we shall not proceed beyond the terms

written explicitly in (4.10). In the spherical model (EF”)(t)Wp(t)O,(0)), which is

proportional to the ensemble average of a sum of products of three angular

velocity components, vanished because they were centred Gaussian variables.

In the present case wi(t) is given by (4.2) and4’)

5

e--Bi(7--12) coj’)(t&of)(f*) dfz, (4.1 I)

where i, j, k is a cyclic permutation of I, 2, 3 and

Then from (2.35)

(~F(‘)(f)~p(~)dON = -i i Jr 1 (w,(t)o,(tl)w,(O)) dt,, r=I

(1

(4.12)

(4.13)

On substitution from (4.2)

(o,(t)o,(tl)w”(O)) = •4(0~)(f)W!‘)(tl)OZ1)(0))

+ E4(oZ’(t)o!*)(f,)0~)(0))+ E4(0:l)(f)0!')(tI)W22)(0)) + . . ., (4.14)

and we see from (4.3) and (4.11) that each term on the right-hand side

vanishes unless CL, u, r are 1, 2, 3 in some order. Moreover

J, = (J * e,)

= (J - e, X e,), for p = 2, v = 3 or p = 3, Y = 1 or p = I, v = 2; -(J * ep X e,), for w = 3, v = 2 or p = I, v = 3 or I* = 2, u = 1.

(4.15)


In evaluating (R(f)~~(t)0~(0)) from (4.10) we already know (~&(t)ti~(O)) from (4.4) and we now calculate (eF”‘(t)o,(t)o,(O)) from (4.13) and (4.14). Employing (4.3) and (4.11), and remembering that CL, V, r are all different we have

E4(W’2’(t)o”‘(t,)W(‘)(0)) P r Y

t

= hdW* e-B,t

I”& I

e B& e-B,It,-tzl ,-Bvltzl df2

--CD

0

= UW2 e-B,t

tt

I”& II

e-BA e(B,+Bv+B,h df2 +

I

e-B4~ e(B,-B,+B,)*z df2

--cc 0

t

+

I

eBA e(B,-Bv-B,)‘z df2

I

‘I

= A&T)’ e-B,, IA I

e-BJ’ B,+B,+B,+

e(B,-W, _ e-BA

BP-B,+B,

+e

BJ, e(B,-B,-B,N _ e(B,-B,N,

B,-B,-B,

Similarly, we deduce that

E4(W(‘)(t)O(2)(f,)0(‘)(o)) = h,oZ e-B,t e-Wl

CL r Y

qJ” B,+B,+B,+

h,(kT)2 eFBp’ emBgl ,4(,(‘)(,),(‘)(,,),1)(0)) =--- P , IJ, B, + B, + B;

On using the relation

which is a consequence of (4.12), and (4.19, we conclude that

(eF(‘)(f)W (t)w (0)) = - (kT)2 ’ P L’ T;l;ir’(J.efixeV)[(I~-r,)((B,-B.)(eg~-B,+B,) e-(B,+B,)t e -Bwt

+ B,(B, - B, + B,) - B,(B, - IL) + (I’ - I’) -B,t e-(B,+B,N

+ (B, - B.);B, - B, - B,) - B,(B, - B, - B,) (4.16)

108 JAMES MCCONNELL

Lastly we require the value of (E*F”‘(t)o,(t)W,(O)) for (4.10). From (2.35)

(~*F’*‘(t)o,(t)~,(O)) = - ,$, JJJ, j dt, i’ dt*~(W,(t)o,(tl)o,(t*)W,(O)) 0 0

- (or(t,)o,(f*))(WLL(t)WY(O))}. (4.17)

Since it was pointed out at the end of the previous subsection that we must

restrict our calculations to corrections of order y, we replace each mi in (4.17)

by cwi e (‘) Then proceeding as in subsection 3.1 for the sphere we now obtain

(e*F’*‘(t)W,(t)o,(0)) = F (_ (I-es;;); - e-Bvf) J,J”

F v IL y

-B,l -B,f e -(B,+B,)t -

B,(& - B,) - B, (i,, - B,) + B,B, I i(J * e X e,) .

I (4.18)

4.3. The Laplace transform of (R(t)w,(t)o,(O))

In calculating Jo” e~“(R(t)o,(t)w,(O)) from (4.10) and (4.6) it must be

remembered that G defined by (4.7) is not a multiple of the identity, and so

care must be taken in performing integrations. It may be shown that43)

lim eG’ = 0. t-a

(4.19)

The integrals that appear in our calculations are of the types

zc m

I e --(I’ eG’ dt,

I t emb* eGr dt (a 2 0, b > 0).

0 0

We write the first integral as Jo” exp[(G .- aZ)t] dt. Since

$ {(G - aI)-’ exp[(G - aZ)t]} = $ [(G - al)-’ + tZ + g (G - al) + . . .)

= exp[(G - aZ)t],

a

I exp[(G - aZ)t] dt = (G - al)-’ lim exp[(G - aZ)t] - (G - al)-‘.

t- 0

Now

lim exp[(G - aZ)t] = lim [e” em”‘] = 0, f-m ,400

by (4.19), and so


00

I e-“’ eGr dt = (-G + al)-‘. 0

It may likewise be shown that m

I t e-“I eGr dt = (-G + bI)-*.

0

Eq. (4.4) yields

109

(4.20)

(4.21)

+

( ) Zp - Z, * (kT)z eWBF’[l - (B, + B, - B,)t - e-‘Bp+B~-B~)*] + . . . .

1, 4Ju (B, + B, - &I* 1

Using this equation, 14.16) and (4.18), and employing (4.20) and (4.21) we deduce that

P

1 e-“‘(R(t)w,(t)oJO)) dt = S,,J F [(-G + [B, + s]l)-’ P

0

+ i f$J?{(-G + [Bp + ~11)~l-(-G + [Bp + Bi + s~r)L)]

* WI* zJu(B, + B, _ BP)2 [C-G + LB, + ‘I’)-’

- (-G + ,;, + B, + s]Z)-' - (BP + B, - B&)(-G + LB, + slI)-*l

(W* -mi(J ‘ew xe.)[(z~-zr){(B~~~~~~:f~~~~~~

+ (-G + [B, + B, + s]I)-’ _ (-G + [B, + SW B,(B, - B, + Br) B,(B, - Bv)

(-G + [B, + s]I)-’ + (Iv - zr’{ (B, - B,)(B, - B, - B,)

_ (-G + [B, + B, + s]l)-‘+ (-G + LB, + SW’ B,(B, - EL - B,) B,(B, - Bv)

(kT)*. - r i(J * e,, X e,)

(-G + [B, + s]I)-’

PV B,(B,, - Bv)

_ (-G + [B, + s]I)-’ + (--G + LB, + Bv + slI)-’ B,(B, - Bv) B,Bv 1

-~_&(_G+s~)-l- (-G + [B, + s]Z)-‘- (-G + [B, + s]I)-’

+(~;+[;v”+Bv+s]Z)-‘]+.... (4.22)

110 JAMES MCCONNELL

In the above, p and u are the numbers such that p, cr, p is a cyclic

permutation of 1, 2, 3 and r is the number which with distinct values of Z.L and

v constitutes the set 1, 2, 3.

In order to write down the matrix representatives of the operators occur-

ring in (4.22) in the representation defined by (3.34) we put

Dj” + Di*’ = D I. (4.23)

Then from (4.5) and (4.7)

$D,+:D~+D~+~ 0 :D,- :D2 -G+uZ= 0 D,+Dz+a 0

:D, - fD> 0 !D,++D,+D~+~

and therefore

(-G + al)-’

:D,+fD*+Dj+a 0 -__ f(D, - 02)

(Dz+D3+a)(D3+D,+a) (02 + D3 + aMD3 + DI + a)

:(D, - J&J -(Dz+D3+a)(D3+D,+a)

0

(4.24)

Omitting subscripts for the moment we may say that when s = 0 in (4.22),

a = B or 2B except for -G + sl where u = 0. Since D = yB approximately by

(4.8), (4.9) and (4.23), and since G defined by (4.7) is of order D, the

non-vanishing elements of (-G + ~1))’ are in general of order B-‘, the first

term on the right hand side of (4.22) is of order kT/(ZB) and the others are of

order ykT/(ZB). However (-G)-’ is of order ZBI(kT) and so produces a

contribution of order kT/(ZB), as it did in eq. (3.29) for the sphere.

4.4. Calculation of spin-rotational relaxation times

A prerequisite for the calculation of the different spin-rotational relaxation

times is the value of (so” e-“‘(R(t)o,(t)wV(0)> dt) required for substitution into

(2.14). It is seen from (4.22) that in the integral there occur operators which

are more complicated than the Z, i(J . eP x e,), .Z,.Z, met in the study of the


spherical rotator. A great calculational difficulty arises from the presence in

Jo” (Racy) dt of terms like (-G)-‘JJ,. This difficulty disappeared in the spherical model where the JJ,-terms did not contribute to c?(O). In order to derive a satisfactory expression in the J,J,-terms it would be essential to extend the value of (R(t)) in (4.6) to at least one higher order in y. This would be laborious but the means of doing it is available3?.

It is not difficult to see that, when the results of the present section are applied to a spherical molecule, we obtain agreement with those of section 3. Indeed (4.24) reduces to

(-G + al)-’ = (20 + a)-‘I.

Then the last term in (4.22) is a multiple of J,J, gives no contribution to c(0). To order ykT/(IB) to Jt (R(f)WW(t)Wy(0)) dt the contribution

and so, as in subsectioti-3.3, the other terms in (4.22) give

D may be approximated by yB and thus the last expression becomes

which agrees with (3.39) in the approximation of the present section. At the present state of our knowledge the most that one can do for a totally

asymmetric molecule is to explain how the various relaxational times asso- ciated with spin-rotational interactions are related to c?(s) through the eqs. (2.5), (2.6), (2.13), (2.17), and (2.27), to show that c?(s) is related by (2.14) to the Laplace transform of (R(f)tiF(t)wy(0)) and to express this by (4.22). The investigation is entirely theoretical. Since, as has been pointed out in a recent study of the dielectric relaxation of asymmetric polar molecules”), there is no obvious way of determining B,, Bz, B3, a comparison with experiment is not yet possible.

Special cases of the asymmetric molecule, other than the spherical model, are being currently investigated.

6. Conclusion

It has been found possible to apply the averaging procedure used pre- viously for functions of orientational variables to products of functions of orientational and angular velocity variables encountered in the study of nuclear magnetic spin-rotational relaxation phenomena. An analytical method

112 JAMES MCCONNELL

has been developed and this yields results which are in agreement with those

obtained, by very different methods, by Hubbard and by McClung and his

collaborators for a rotating spherical molecule. It has been shown how the

method could be employed for a molecule of arbitrary shape, and attention

has been drawn to some of the calculational difficulties that would be

encountered. It may be concluded that the mathematical approach based on

the stochastic rotation operator is adequate for the investigation of the

nuclear magnetic relaxation processes arising from spin-lattice, in-

tramolecular dipole-dipole, quadrupole and spin-rotational interactions.

References

1) A. Abragam, Nuclear Magnetism (Clarendon, Oxford, 1961).

2) P.S. Hubbard, Rev. Mod. Phys. 33 (1961) 249.

3) P.S. Hubbard, Phys. Rev. 131 (1963) 1155.

4) P.S. Hubbard, Phys. Rev. A8 (1973) 1429.

5) P.S. Hubbard, Phys. Rev. A9 (1974) 481.

6) P.S. Hubbard, Phys. Rev. A24 (1981) 645, erratum to ref. 4.

7) P.S. Hubbard, Phys. Rev. A24 (1981) 645, erratum to ref. 5. 8) J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, London,

New York, 1980).

9) J. McConnell, Physica 102A (1980) 539.

10) Ref. 8, section 7.3.

11) Ref. 8, sections 7.3 and 7.4.

12) M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

13) Ref. 4, eq. (5.5).

14) Ref. 8, section 11.4.

15) Ref. 8. section 11.1.

16) G.W. Ford, Notes on Stochastic Differential Equations (unpublished, 1975).

17) N.M. Krylov and N.N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton Univ. Press, Princeton, 1947).

18) G.W. Ford, J.T. Lewis and J. McConnell, Proc. R. Ir. Acad. 76A (1976) 117.

19) K.F. Herzfeld, J. Am. Chem. Sot. 86 (1964) 3468.

20) Y. Leroy, E. Constant and P. Desplanques, J. Chim. Phys. 64 (1967) 1499.

21) Ref. 8, eq. (5, 3.8).

22) Ref. 8, eq. (9, 2.6). (9, 2.8). (9, 2.17).

23) Ref. 8, eq. (3, 4.18).

24) M.G. Smith, Laplace Transform Theory (Van Nostrand, Toronto, New York, Princeton, 1966).

25) Ref. 8, Appendix C.

26) Tables of Integral Transforms. Bateman Manuscript Project Vol. I (McGraw-Hill, New York,

Toronto, London, 1954) p. 232. 27) G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of

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29) Ref. 12, pp. 28, 29.


30) Ref. 1, p. 279. 31) Ref. 5, p. 489. 32) Ref. 7, eq. (5.10). 33) Ref. 7, eq. (5.20). 34) Ref. 3, eq. (2.13b). 35) R.E.D. McClung, J. Chem. Phys. 73 (1980) 2435. 36) M. Fixman and K. Rider, J. Chem. Phys. 51 (1%9) 2425. 37) Ref. 35, eq. (66). 38) G. Levi, J. P. Marsault, F. Marsault-H&ail and R.E.D. McClung, J. Chem. Phys. 73 (1980)

2443, eq. (10). 39) G.W. Ford, J.T. Lewis and J. McConnell, Phys. Rev. Al9 (1979) 907. 40) Ref. 8, eq. (12, 5.6), (11, 5.18), (11, 5.23). 41) Ref. 8, pp. 194, 195, 217, 218. 42) Ref. 8, eq. (11, 1.16). 43) Ref. 8, eq. (11, 5.28). 44) J. McConnell, Physica 105A (1981) 593.

stochastic differential equation study of nuclear magnetic relaxation by spin-rotational...

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