Intermolecular spin-spin interactions in liquids

:Physica 80A :(i9751:t28-14'8 ~} Nonh4totland /',,~,rhing Co.

INTERMOLECULAR S P I N - S P I N INTERACTIONS IN LIQUIDS V.P. SAKUN Institute of tligh Press~re Physics of the L~S R Academy of Sch'~we% 3Ioscow, USSR Received 14 August 1974 Revised 15 January 1975

Kinetic equations for the one-particle spin density malrix for a system of magncti< me[ecuies in a liquid interacting with one another through intermolecular spin ..spin fo~'cesare deri~ ed ~:nder the ff~llowingassumptions: a) "internaF' relaxation prt~zesses are ignored: b) at g = 0 the entire density matrix is factorized, i.e., ~(0) = [-I.:b,(0), 8,(()) being the spin density matri-\ of the ith molecule; c)random motions of particles in the liquid are considered as a classical markovian process. The finite set of integrodifferential equations, with rank equal to the number of different kinds of the molecules, is Obtained for the matrices "&,(t)) averaged ever all [he possi~e traiectories of the molecules in {he liquid. The equation of motion for the magnetic moment, nonlinear in magnetization, is ,.'~riz~:en for a system of identical molecules with S = {- i~: the low-coneentratio,a limit. This equatiot~ is used to investiN~te the form of the equilibrium magnetization-recovery curve.

1. Introduction It is weit k n o w n that tile b e h a v i o u r of" the macroscopic magnetizati,vn ~! a system o f magnetic particles dissolved in a diamagnetic liquid is govcc-aed to a great extent b y the spin-spin imeractions. Stochastic motk,~s ~f molecules in ~he liquid modulate these interactions in a r a n d o m mann~ ~dins gi,,i~-~gr;,~e to re!axation in the spin system. One distinguishes bet~vo:n the two kinds o f such intern:finns, the intra- and t he. i~rtermolecular ones. The N M R relaxation-time meast~rements in liquids have revea}ed that the contributions o f b o t h kinds to the nuclear reiaxation rates often prove ~o be of the same o r d e r o f magnitude~L As foc s3.qem of motecuIes with S = .~, the [!~el°mofe¢~!a~" spin...~spin interactions usuali 3 give the main contribution to 1he relaxation of ~he mag~etic m o m e n t o f the systen'~° Some other p h e n o m e n a in magnetic solutions which arise f r o m the intermoiecnlar spin .... spin interactions, such as the intermolecular O v e r h a u s e r effect, th.e high-resoh~tion N M R - s p e c t r u m line shifts, and so on, have recently been smweyed in rei" 2. T h e derivation o f kinetic eq~aatio'ns for the magnetic m o m e n t o f the system, ;

128

:INTERMOLECULAR SPIN-SPIN |NTERACTIONS IN LIQUIDS

129

taking into account the intermolecu!ar spin--spin interactions, was carried out by a number of authors, /n ref. 3. si,nple balance considerations were used; the approach based on the density-mat fix description was presented in re f. 4; a method emp]o:;ing the nonequitibrium statistical operator has been developed in refl 5. All tb.ese theories use the t~llowing physical assumptions as basic ones, Motions of the magnetic particles are treated in terms of random classical trajectories, the general restriction being (~ ~ r~ -~ 1 ~

(I)

Here ~,~ is the transla{iona{ correlation time, namely the time characterizing the rate o~" deca:~ of cor~'elafio~ belween successive i~xtermolecular configurations, and ~,q d'cl~es the magnitude of the spin-spin interaction. Actually, the value ,,~r.¢ w:.ts usually viewed as a small parameter, the relaxation part of the kinelic eqm:!ions in it being only of second order° The assumption It) enables one, in ~qmip]e, to consider the case of an arbitra~'y concentration of magnetic particles i,: tl:e solution. Furthermore. fi'om (1) it fo{lows that T > r~,

(2)

whe;e T~ is the spin relaxation1 time. The important consequence of eq. (2) [and hey,co ofeq, (1)1 is that one may decouple the infinite chain of equations coupling spin-density matrices of differertt dimensionalities (the so-called gogolyubovBo~n-Grem>Kirkwood-ivo=~ hiex'archy of equations) through the factorization of some of these matrices, usually of the binary ones. Such a decoupling brings :'~bout markovian kinetic equations for the one-particle density operator, hence it is permissible only for systems with im'entzian absovbtion lithe shapes. The restriction (2), however, may be invalid in viscous liquids, whe~e r.¢ becomes large. If T~ ~ r~ the spin subsystem will be appreciably disturbed during the time interval r~, th~s the spin states of the interacting particles will be highly correlated during the rela:
130

V.P. SAKUN

}!towever, as a result of the essential two-particle natme of the theory, the rate c f the "external" relaxation in tiffs approach turns out to be of order V - V being the macroscopic volume of the medium, and hence has an infinitely small wdue, To circumvent this inconvenience one supposes that the actions of the processes taking place in all pairs o f molecules, whicl~ a given molecule makes up with all :the magnetic molecules in the medium, are additive. Thus a somewhat phenomenoiogical point is introduced in the theory. ?, nother approach in deriving kinetic equations, the projection-operator method, has been reported in ref. 9. The authors of that paper have obtained the generalized kinetic equation lbr the whole (i.e. many-particle) spin-system density matrix with:oat any restriction on the magnitude of ~'¢. l~i the presem paper we shall consider the problem of derivation of the kinetic equation !br the one-particle density matrix for the system of magnetic particles dissolved in an inert liquid, provided that the predominant interactions yielding the relaxation of s p i r l s .... wards the thermal equilibrium are the mtermolecular spin-spin ones. We shall deal with the general case of arbitrary T~/r~. This problem is a typical one of manybody nature, and in order to treat it we shall exploit the diagram technique used extensively in statistical dynarnics~°). An essential feature of the diagramm~qc method is th~~t the expansion parameter now will no longer be given by (1) but by the concentration of the magnetic particles in the solution. in t!ae following section we shall present the derivation of the kinetic equa[ions. In doing so we shall discard the restrictions of the previous theories discussed above. Instead of this the lbllowing assumptions are made: I) At the initial moment (t = 0)the whole many-body density matrix is completely factorized, i;(O) = I ] a,(O) o

(3~

i

In homogeneous solutions (and ! under homogeneous initial condition~), the matrices 5~(0), with i denoting particles of the salve species, me equal to one anotlher. 2) The spin-spin interaction :is pairwise, and apart 1"rma the spin variables, its potential depends only on the difference between the radius vectors of the particles, i.e. it takes the form f ~ 0"~ - r,,),

(4)

with r,, rk being the radius vectors of *i~e ith and the kth particles, respectively. 3) Magnetic particles are considered as point-like. Their random motions in the medium are treated as a classical stationary homogeneous markovian process. Hence, the reaction of the spin system to the medium is ignored. No restriction on the value of r~ will be imposed. The initial condition (3) is rather a strong one. For eq. (3) to be fulfilled, it is necessary that the states of all

INTERMOLECULAR SPIN--SP~N INTERACTIONS IN LIQUIDS

]31

the particles .in the spin subspace should be completely uncorre~ated at the initia! moment. Yet it should be emphasized that the effect of the initial condition vanishes in a sttfficientl3, tong time. This f~.ct enables our kinetic equations to be applied to the systems in a steady state, regardless of whether eq. (3) had been fu!fi]ted at t = 0 or not. For instance, one may tzse them in investigating the absorption line shapes, the steady Overhauser effect, etc. In section 3, equations are written l~or the case of small concentrations of the molecules in the liquid, namely when cr ~ (c is the concentration of the magnetic particles, and r is the "effective" radius of the intermolecular interaction potential) is small. This means that the mean qua~tity of' particles, with which a certain particle interacts "effectively" at a given insta,~t, is a small parameter. In ~;,zction 4 we consider the special case of identical molecules wit!~ S = ½. The markovian limit (2) is considered for the to,w-concentration case. No restrictior~, however, is imposed on the streng:h of tb.e spin--spin interactions. In this case the equation of motion for the macl'oscopic magnetic moment is in geneva/ non.linear in the magnetization, although it ~'educes to the welt-known Bloch linear equation ~*) in commonly adopled approximations, e.g., it" inequality (1) is satistied. ;u section 5. use is made of the equation derived in the preceding section to :.m estigate the form of the equilibrium magnetization-recovery curve in the system of alike molecules with S = J: in the preseilce of an external static magnetic field. Conclusions are presented in sectio~. 6.

2. Derivation of kinetic equations Vve consider the system of N,~ molecules of the kind A.. N,~ molecules of the kind. B . . . . . dissolved in a homogeneous liquid ot:" the macroscopic volume V. The hamiltonian of the total system may be writter~ a;

i-}(~) = ~ *2;(0 + ~ v£ P~k (,',(:) -~.,,te~). i

(5)

i.k

where I'-t~(t)is the dynamical part of the ,~amiltonian of't,~e ith particle ( ~ motecu!e). /7~(:) describes the interaction of the mo!ecuie with both the constant and the time,dependent external fields, r~(t), r~(t) represent random trajectories of the Ah arid ~he kth particles in the medium. We do ~:~ot i~zc!ude itq /:7/~{t)the it~teractio,~ of t~c molecule with its own random local field. As it seen fi-om (4), those terms m hamiltonian (5) which depend simultaneously on r~, r,~ and the internal spatial varietbles of the molecules are also omitted. However, the generalization of oar derivation to the case including the interaction oF the molecules with the random local iields presents no special problem. We neglect these because of the assumed predominant role of the pure intermolecutar interactions in the processes coi> sidered in this paper.

132 "

:VIP,:SAKUN

The density matri:x of the ath particte is defined as ~r.(O =

Sp, q(t),

where the primed symbol Spa denotes the trace operatmn over all spin variables of the system except thOseof the ath molecule. We are interested in the oneparticle density matrix ~#,(t)) averaged over all the possible trajectories of molecules in t:he solution. The density matrix averaged in this way enables us to determine all the additive observables of the system. The density operator of the whole system ~(t) obeys the Liouville equation d ~(t) dt

i h7 [~(t), H(t)], .

It is convenient to introduce the unitary operator

O(t) = H O,(t), i

where / , )) O,(t) = Texp I--tT~ i f dr'/~(t', , O

which is the T exponent, ff)r the translbrmation to lhe interaction representation. For ~, ( 0 = O + ( t ) ~ ( t )

O(t)

we then obtain

d :2;*(t)= i . , . %7 h ~" (o, :*(t)],

(6)

where Lk

,~11)" = ¢:*

Of(t) 07(0

I: '~ ~ (v~(:) -- r>O) O,U)

O,A:).

We start the derivation of kinetic equations fbr /.4~(:), by representing the solutior~ ,,f eq, (6) formally as


(7)

tNTERMOLECULAR SPIN-SPIN INTERACTIONS IN LIQUIDS

133

where

/

,_~(,) = T e x p l - ~ j

i ;

d,'(?*(t'),)

O

Next, tot us expand eq. 17) as a series in the intermolecular interaction,

G,,(t}

:=

Spa~ ()

V Sp,,

dt~ .. 5 dr,,, 0

*I~, n = O

r "

x Z

...

f~, k I

l • ~,,,o,(t,,)] v(O) i m, k m

x 5 dr~ .-5 dr.,, E [(i/2h) l~*,.(r`.)] ... E [(i/2h) ~,i,~i~:,)] /. 0 rap S~ 0

(8)

,

r i, s I

Ti~e t e r m s in (8) c o r r e s p o n d i n g to m = 0 or n = 0 are d e t e r m ; n e d as follows: if m = 0 (n = 0), the expression occurririg in the b r a c k e t s at the left (right) o f / ; ( 0 ) should be r e p l a c e d b y unity. Such a rule wilt f u r t h e r be a d o p t e d t h r o u g h o u t this paper. In o r d e r to as a s u m o f (i = 1 . . . . . m, i n t r o d u c e the eq° (8) can be

p e r f o r m statistical a v e r a g i n g o f eq~ (8) it is conven:iem to express it t e r m s such t h a t in all o f t h e m the integration variables ti, z~:, k = 1 . . . . . n), are positively o r d e r e d in time. F o r this p u r p o s e we n o t a t i o n t~, ( p = 1 . . . . . m + n), c o m m o n to t~ a n d r`k w h e r e u p o n written as

q(0=

5 d4,+,,

Z 5at;" m, n:=O

De

0

0

x ~P" ( e S ' ,~ [-. (i/2h) V*,,,k,(t ~'". ~ ",.}~k.o[-(i/2h)I>;*~.,,..{t,,)],.~(0) x ~. [(it2h)f'~,~('r~)].... £

[(i/2h)V*q,,,~r ~)1].

(9)

~,o~ d e n o t e s the s u m m a t i o n o v e r all the w a y s o f t i m e o r d e r i n g c,9" t~ a n d zk except those which violate the internal .positive ~,tm~e ordering,_ o f t~. (t~ > tz > ... > t,,,), a n d t h a t ofT~., (r~ > z2 > ." > %). tl

t2

{m

,.,

i-7:=:~.-:7T7:.: ................ T~,7-..... q L. . . . . . . . . . .

t

Fig.

I.

a/____

g,_L~.:-..=.~

11i

T2

Diagram corresponding

.0.

to

I: n 0

a term in the

sum

(9),

t 3 ~ ....

: :Vle. SAKUN

]it proves to be convenient to represent the terms of tlhe sum (9) as t~e tbllowir~g d!agrams~,°). We call the points with coordinates t~ ,: !.., tin, -r,, .... r . the vertices of the diagram t The vertex at the point t'. generates the factor +_(i/2h)t~*%,(t'~,) withagiven q~,/%. "* ' The sign ÷ corresponds to the verti~'es of the bottom semiaxis of the diagram: while the sign - to those of the upper one. Particles entering the diagram are represented in it by the asterisks. Each particle (i.e. asterisk} should be Connected b y dashed ("potential") iines with those vertices which correspond to ~;he:interaction of the indicated particle with the other ones. In order to reestablish the analytical expression corresponding to a particular diagram i{: is necessary 1) to construct the expression Sp;~ [ 2 , . ( t , ) 2 , , ( t a ) . - .

2~(t,,,)~(0)2,.(r,,)..-2,.(r,2)2,0:01, r

where 2~(t~)is the factor corresponding to the vertex at: tp; 2) to integrate this expression over t~ . . . . , t .... T~ . . . . , %, fi'om 0 to t, keeping fixed the way of time ordering as indicated in the diagram. In order to obtain 4,,*(t) all kinds of diagrams must be summed up. i.e. the summation

r(r.

t(z zi

must be performed. " Let us consider a diagram in which one may reach ar~y asterisk when moving fiom the ath one along the dashed lines only. We shall call such structures the "a-connected" diagrams. It can be proved (see appendix) that, after the trace operation Sp~ is carried out, only those terms remain in eq. ;,9) which are associated with the "a-com~ected'" diagrams. It can also be shown 'Lhat in '°a-connected ~ diagrams one of the indices i~, k't coincides with a. After statistical averaging it becomes immaterial m uhich particle (among particles of the same kind) a given asteri k correspon& (~:xcept for the asterisk denoting the ath particle). In other words, the result ~oes not depend on the permutation of the particles belonNng to the same kind. The number of the permutations l~ot affecting the ath particle is e,Nat to ,,,A

-

i)!

.,~, ..~

I ] x ! ?" ~! c ':": ; v ' ' = .~J"

" ~,:.. ;~, (N4 "- - - ,¥~,*,~)~ , , ~*:¢,..I ~ , ~ X

Ax;

. . . . . . . . . . . . . . . . . . . .

.

'~

~;'i'~;.' (10) );

!VX

where the ath particle is attributed to the/t ~., --'~s In (10), X runs over the species ._pe,cIe,~. of the particles, and .A~ m: denotes the nmnber o~" particles of species X involved in the diagram of interest. As N a -+ m, eq. (10) tends to N 2 ~ I ] )'~:.~ X

.

(l l)

INTERMOLECULAR SPtN-Sth

tNTERACrlONS 1N 1,1QU1DS

I.~

Hence, after statistical averaging oae can omit the summation over identical pa,ticles :in each sum V~, \-'.. (p = 1 m + n). and multiply instead each term in (9). by the factor (1 I). x:~**~\ % t , , / can now be written as / :,_@

=

+ N;i*

E

2

~n,n=O

O~

j" F .a

- - .

O

O

,

,

n l + ~1 D 1

× Sp,' <[-(i/2,~) I,c,* q~,(t~)]

...

[-(i/210

p',i,,<,,(t,,,)] .g~(O)

-< {(i/2h) );:~*~,,,,(~,,)] ... [(i 2h) 1:,. ~,(r,~]'> . , , r ,~

(12).

the symbol .l~, stands for the "a-connected" diagram while the summation ~ ; , ~,o~,s over all the possible "a-connected*' diagrams except identical ones, i.e. t ,~c~se" ot'tained by the permutations (I0). In (12), (-.-3 means ibe statistical averaging procedure, The statistical!y averaged series (12) will further be represented with the he{p o ~ new diagrams which can be obtained t"rom the previous ones as follows. Fh'st, the tatter should be statistkally averaged. Second, in new diagrams each asterisk, except the ath one,ge~:erates a factor that is equal to the full number oF particles of the ki~td to which the particle corresponding to this asterisk belongs. The he', ,.iiagrams will hereafter ~:,e re!'brred to as averaged diagrams° The fbli~ai:.v.,, summa!ion oF such diagrams

y_ x z , yields
.....

-i .....

............

i .......... :.... r

,

Fi.-...~,~. ~ Schematic .mcture'o~"W-set".

i) Apart from the intemat potential bonds, some of the vertices of such a set are connected by th: dashed lines with mHy one (common to aH these vertices) particle which is sited otlt of the set and differs from a. This particle is plotted in tig. 2 as the asterisk i, (i ¢- a), while the indicated set is pictured there as a square. All potential bonds connecting its vertices with the ith particle are designated in fig. 2 by the single heavy line segment.

-2) The remaining vertices of the averaged diagram (that is, the ones not entering the Set)with which the ith particle is connected are disposed tef~ of the set. The set with such features will be hereafter referred to as the "/-set" (i.e., the set connected with the ith particle). We also include in the/-set the part

=

<(o) i1 <,(0),

of t h e entire initial density matrix (3) (I~v denotes the product over all particles Of the/-set). We shall write below for brevity i-set for all such sets, even though the particle giving rise to the set differs from the ith one. The other group of averaged diagrams is composed of the ones without such sets. We call them the diagrams of type I~ whereas ~he averaged diagrams involving /-sets will be related to type II diagrams. It is evident that any given diagram of type II can be unambiguously produced from some diagram of type ~ by adding an appropriate amount of suitable/-sets. N:)w, we consider random motions of the magnetic particles to be independent of one another and describe them with the help of a distribution function of the markovian type, pt~.~ ~ ' ( .fq, tq; { r } ~

H p,,,~r~

(q)

"

.-,_~ ' • ...." :r~ t j O , to) ~.(q-l) . t._,)

, tq l',

P; (rl :~), t, i (o) to) e~ ('I °) to)

(13)

i

where P ({r}q, t,~;...," vjo, ¢°: to) is the multidimensional distribution function for the vector {r} formed of the spatial coordinates of all the mag~etic particles in the solution. P~ ,r~('(v>,tp I r~(P- ~), t,,_~) is the conditional probability of finding, rdtp) = r~v~ if r~(q,_ ~) = r~XX'-~. We assume the random process to be stationar:, and homogeneous. Then P~(r"~P',GI ri("-~), t,_~)

=

Pf¢y/ (,,) -- /x,, i: , t,,

-

iv... 1 I, O. O' .

Let us take a particular type-I diagram and label its particles ( - asterisks) by the index L Let t:~,di) be the coordinate of the right most vertex among those vertices with which the itb particle is connected through the potential lines. The addition of the i-set to the ith particle implies the following procedure. One adds a number of asterisks and vertices to that part of the diagram which lies to the right of t,'~,~di) whereupon one connects them in a proper manner by the dashed lines only with one another and, besides, wil~h the particle i. After this has beer+. done, the diagram of type 1I will arise. An/-set can be added, of course, not only to one partic?e but also to a few ones. I n averaging a type-II diagram with 7~heaid of the distribution function (t3), one can at first average all its /-sets, that is, perform the partial averaging of the diagram keeping fixed the spatial coordinates

INTERMOLECULAR SPIN-SPIN INTERACTIONS IN LIQUIDS

137

of the particles not entering any'/-set. We note that if property (4) of the inter° molecular interaction is the case, one can average different #setsindependently of one another. Let us now take a particular averaged diagram of type iI containing a given /-set (apart from the indicated/-set, the diagram may include the other ones as well). Denote the time arguments of the vertices entering this /-set by /, (~i ), [p(i) = 1, ...,p,~=~(i)], and the arguments of the remaining vertices of the diagranl by ((R) (i) .r(R) p~m, [p(R) = l . . . . . p~.~(R)]. Both /,p(~ and t~(R~ are ordered in time, (i) t~,,.(i) > t'(" > ... > t~~,,,,,,,, / rl(R)

.....

..

p > tm ~(i)

>

(R) "" > j~p.:°,(m,

while .iheir mutual disposition may be arbitrary and is determined by the averaged diagram involved, We can, first of all, perform the summation over ali those averaged diagrams which differ fi'om one another only ii arrangement of t;ll ~ relative to ~p(R). /(R) Such averaged diagrams correspond to identical integrands but difli~z~nt limits of integration. The summation, clearly, amounts to the single term wi~h the same integrand but independent integration over lhe time variables of the/.set; namely, the integration in this term is performed as follows "

'pmax(R) ~ 1

.[ d'](e'''"

[.

0

0

.oma x ( i ) t

dt;2SR, I

dt?"~.,

0

j"

dt;i]:;~ ,.

(t4)

0

An analogous procedure can be performed on a!l i-sets entering the diagram, and on all the averaged, diagrams of type [I. Such a partial summation of the series. (t2) results in the renormalizafion of ,;-sets in the sense of rearrangement of the limits on integrals, as given b7 (14). We suppose such a renormalization to be performed throughout in the following, thus we shall only deal with the renormalized i-sets hereafter. We shall designate them re-#sets. ~t can easily be seen that a re-i-set results in the following a~alyticat expression (the quantities belonging to the re-i-set are labelled by i) tmi.(i

)

t r (|1 ~m(~) ¢ n U ) -

t

Ill 51) where N[.° is the number of particles of kind X involved in the re-i-set ut3der consideratiom The remaining notation in. (15) is obvious. Let us now add, by turns, the re-/-sets of all kinds to the ith particle of some diagram of type I and sum up all the diagrams obtained in this way (fig. 3), The

v:P,

:

s:, ummat~on~' :' ~ of the series sho~vn in fig.• 3, dearly, amoim,~s to t l ~ ' of the re-i-sets (which are p~c~ure~'d as squares with the dra-wn diagonals), i,e,. to Z s~'~(~') n f t } = O

g" E

(16)

D;I(i) I'~

Combining (t 5) and (16) one can readily yetiS3' that the series composed of all the kinds of re-i-~ts added to the ith ~article is idemical with the series on the righthand side of:eq. (12), if one puts there a =: i and t = t~.(i). Hence the total sum of the series pictured in fig. 3 is equal Io the "initiative" diagram of type I (i,e,. the leftmost dla~ram in fig. 3). in which, however, (~* ~ ' ~,

Fig. 3. Squares with d,'awn diagonals pictt~re"re.i-sets".

A similar procedure can be performed on all part!ties of each diagram of type t ~md on all type-] diagrams. On the other hand, this is nothing other than the summation of all the possible diagrams of type II. As a result, we have instead ot"(.12) (,~-,*(t)~- =/~,,(0) + yg"'

~

SZ Z

sp , C {-,°(i/2h) >:

.i"d~; ...

~"

dr;,,+.

,..

%(0)

, .,)., ,z! [(ii2h) V;],,,(r.)].-- [(i/2h) ;;:;~,,(r,)] ), ,' . (17) ,

,

The symbol I"~'> indicates that only the "a-c( mected'" diagrams of type I give a corer!but!on to eq. (17). Thus. we have transf'ormed the sum (~ 2) ,:)vet the diagrams of both types I and t! to the sum over only type-1 diagrams, in which, however, )he matrices <¢?*(t')~,, i ~ a. occur with the arguments t' = t~i.(t) while the argument c.f the matrix .,!~(t') is s~il! t' =, O,

Fig, 4 Gene'.a! view of a particulzr term in the sum (t7). The shaded sqttares designate the irreducible structures of vertice~s,asterisks and :potential Hnes.

INTERM()LECULAR SPIN-SPIN INTERACTIONS tN' LIQUIDS

t,39

Each ~erm in (17) can be represented graphically as lbllows, The shaded squares pictured in fig. 4 denote the diagrammatic parts without/-sets, att vertices and particles (.~ asterisks) o f the Mth square being disposed toward the left or those of the M - t s~ one (i.e., a potential line does not connects these squares), and so on. The subdivision of the diagram into such parts should be pert~rmed in the most complete way, in that none of these diagrammatic parts may ['urther be divided into more shallow parts with such properties, w e call the diagrammatic structures corresponding to these paris the irreducible structures. The wavy arrow ending at 'Jm vertical line segment at r = 0 shows that the argument o£ the matrix ,~,%~,* )< occurring in the diagram is t' 0, It should be emphasized that the irreducible parts of the diagram may be averaged independently. This is the case due to the same reasons which have already enabled us to average the/-sets in at: indew~Jdent way. Finally, we note that each one of the irreducible parts includes one certex at least. Eq. (17) together with tig. 4 immediately yield

! <.%U)) = d.(O) +

)

:>.;;q

(I ~;)

T(irl w~ere V v . . . denotes the summation over all the pos~ibte kinds of the h'reducible diagrams, tn (18), ~.o-a(J <'* ' )) occurs in each diagram with the argumcnt t' = t.,,, which is equal ~;_othe coordina~e of the rightmost vertex of the corresponding irreducible diagram. This fact is indicated by displacing (and curving) the vertical segment to the left of the point t = 0. Let us rewrite eq. (18) i~ the customary notation, v-,,~t,,:"

=

% "( 0 )

,*,, N ; - ~

Z

m,n=-O

. . V E . ;d,, D~ t"~ ~

.

• ' "

.

(x]~' N: '?~') Sp: { [-(i/h)l)'~,(f~)]

f

("~ f m +

~)

0

0

.°.

[i-(i/h))?i:,,k,~(6,,)]@,,(r,,+,,),,,

~* (t,,~.U))> . . . . . . . ) [(i/t0.t~>* ;< [' [1
I"/ , * ~ , ( r ~ ) ] \

(t9)

In this expression, the subscript .I'] ~r~ shows that the coordinate.~, h, ~'~, and the indices of the particles, if represented as diagrammatic structures, combine in the 41 / irreducible diagrams, qhe prime on the summation in t119) signifies that lp > k,,. (p = I . . . . . m + n), in consequence o( which the numerical factor .~ ahead of each P~ is dropped. Differentiation o f eq. (t9) with respect to t brings about the

140

V:P, SAKUN

i

kmeuc equatmns, which m the Schr6dinger representation take the fb!lowing form

: '

_a u,,(,> > : i ( dt

~

" (%(t)),

..

li.(t) + Z Sp, [{ P,,.) P

@.(t)>]

)

+ C4,(t) i~*(t) u°~+ ..(n, (20)

wl~ere r.t2

R" ,,' ,( ) = N.~'

x

t + Jl- I Ira

t

Z

E

E'

r~,n=O m+nT.~ l

!) t

]'a Or)

I dt] "'" I 0

" " ~z,.~,,>~ s~,' {If-o/h>

, ( XI~ ~,x ~tTNx(D)') d&+,:

0

...

~-<,/,,> n'.,o~,..>) {~r." * (t,.+,,)) '

x (~.I ((~* (t:,n(i))))

× I(~/h>)%,u.))..-iJ, ~(u,,) ~=(,>)] ),.,,,.

(21)

I n (20), Spy denotes the trace operation over the spin variables of the pth particle alone. The set of eqs. (20), (21) is the desired result of our calculations. It should b e emphasized that the matrices ( 4 " ) and the remaining parts of eqs. (20), (21) are averaged independently of one another. This is a rigorous result of the present theory, in contrast with the previous theories discussed in the introduction. A distinctive feature of ~he system of eqs. (20), (2t) is its closed character with respect to the one-particle density matrices, i.e. neither two- nor more-particle density matrices are involved in the system. This fact, of coarse, has stemmed from the assumed "uncorretated" initial state (3) of the spin system, and it is the well-known result of the theory of in~versible processes~°). Since all @~(t)) for the particles of the same kind are equal to one another, as also are 4,(0), the sum over p in eq. (20) in fact amounts to tLat o~er only the species of the particles. This is also true for eq. (21). Thus fi<- rank of the system (20), (2t) is equal to the number of species of magnetic ['arcMes dissolved in the liquid.

3. Low concentr~tioas of magnetic particles If the concentrations of the spin particles are small, one may neglect three- and more-particle "collision" processes, that is. one may restrict oneself to those terms in (21) which are proportional to the first power in tile concentrations. These terms correspond to the diagrams involving only two particles (i.e., asterisks). The time A, arguments with which the matrices (.a!) are to be tak~ a are indicated in fig. 5 using wavy arrows. All the two-particle diagrams are seen to be of the irreducible

INTERMOLECULAR SPIN-SPIN INTERACTIONS IN LtQUIE S

141.

.... L 2 2 q 't

0

t

0

,/'"° !

.!

1:

O

1:

0

+ o,, t

0

Fig. 5. Two-particle diagrams.

type, thus they )ield analytical expressions which occur in eq. (21). it can easily be v,zrified that the sum of all the two-particle diagrams is equal to ¢t3

nl,tl=O n'-+n>~

×

f

t

fm

0

" ~

0

"fn -

¢

t m

tm

1

Sp~<[[-(i/h)

~ ,,,,(t,)] *

* ,h • .. [--(i/h) ~~ W

x [(i/h) ~2*(%)1 ... [(i/h) P'*,(r~)], 1

r;a _ [

l

"

<~,, "*(t,,,L~ '

~* <~(t.)>

(i/h)I2*(t)])

t'm - I

+ I dr., ... j" dr,, .[dt, ... S dr,,, 0

0

rn

× Sp~ <[[-(i/h) vL(t,)l ~* ×

rn

. •

• ,. I - d / h )

P;';(t,,)]


',

[(i/h) ~.~(r,.)] ..-[(i/h) P,*(r.,)]. (i/h) .oh,()1>),

(~2)

the particle b being attributed to the species B. In eq, (22L the first (second) commutator has arisen from the two-particle diagrams in which the rightmost vertex is sited at the upper (bottom) semiaxis. The series in m, n in (22) can easily be summed up, whereupon eq. (22) takes a simple R,:m, namely f

3-" Nu ]" dt' B

O

Spb <[g,b (t, t'.) [/d'2(t'))

"* '

% (~

c)(i/h) ;Lo)]>, (23)

where

g,~(t, t'):=

Texp

dt" k~b(t ) .

!~:42: :

v.e. SAKUN

:l

Substitution of (23) for ~*(0 into eq. (20) yields kinetic equations for the oneparticle density matrix @.(t))i linear in the concentrations o f the particles

d <(~.,(t))= i [~,aA(t)},/fA(t)] dt h "

i [
N,.

(< :;,x> )] +

with /~a,(t) = -h-2[~A (t) I dr' S Px ([ ~'*ax(t), O !

,gAx- (t, t') [P*x(t'), (~*(t')) (ax(t~*")>] SAx~+(t, t')]>

(t).

U.... X

(25)

Both eqs. (20) and (23) are independent of the particular choice of the particles a and b among the particles of the corresponding kinds. For this reason all the operators in eqs. (24), (25) are labelled only by A, X signifying the species of the Particles. T h e first two terms on d~e right-hand side of eq. (24) describe the intraand intermolecular coherent effectsa), while the term involving ~Ax(t) is the relaxational term which is due to the intermolecular spin-spin interactions. The one-particle hamiltonian [lA(t) may be taken as

FL,(C) = f/o: + f?~,,(r), where /4o~ = -g{~Hg~ (A), H being a constant external magnetic field in the z direction; g,(A), (~ = x,y, z), are the spin operators, and I)~A(t) is the term dc~.,o°ibing the interaction of the molecule with a time-d% ~endent external magnetic field H~(t). The kinetic eqs. (24), (25) may be applied ito the cas~ ef arbitrary stre~,~gth of the iJ~termolecular spin-spin interaction, arbitrary spin potarisation a,, well as to the case of arbitrary T~/T~. However, the concentrations of molecules must be sufficiently small. The condition tbr smallness of concentrations of interacting molecules depends on the interaction potential as well as on the properties of the distribution function (13), and hence should be given for each particular case separately. If Ht(t) is small, then one can neglect the operators ~H(t), Itt~x(t) in eq. (25). In this case, it is convenient to represent eq. (24) in the matrix ibrm. Let

t~o~- I-,(x))

--

~,,.,,, (x) im(.r)).

INTERMOLECULAR SPIN-SPIN INTERACTIONS IN LIQUIDS

143

Then a simple calcutation yields

~,n,(R;)i/~Ax(0 Im2(.4)) =-

,~.

,

t m:~( ),m~(A, ) ~,mt(X),m2(X).

IdU(m=,(A)l

" '" (<,dr)/tm,dA).~

0 /

x (ml(X), <~'x(/')',)im,(X))K,,. ( t -

~ 1~1

1' ]

,(A),/,,3(A). '//l(~)~ , (126)

where K.(~.

- t' t

= ~

ml(A), m3(A), m~(X)~ mz(A), m,(A), m , ( X ) l

exp tlt~o,,,,,,, ( A ) + it' [,,),,,:,,,,4(A) + o,,,, ,,,,.( ~ )]).

m( X:

-~

6 2

dt dr'

(m~(A), re(X)! ~,~,: (t, t') Ira..,(,4), m,(.¥))

:< (:m(.4), m2(X)l -'~,,(k-(t. ~') im2(A)./~,(k ))

(127)

a n d ~.,~,,,,,,. -: (,J,,, - o~,,,,, p . , m') = l i ~ ) I n : ' } .

4. Sher~: ceixelation times In this section we consider the case of arbitrary strength of tke intermotecular interaction. We shall also assume that a sufficiently short correlation time r~ exists. such filat tile inequality Yr >t. v.~ is fulfilled. At this point it should be emphasized that the inequality T~ }> ~v, does not necessitate the inequality m~,~ ~ 1, provided that the relaxation in a spin system is due to the intermoiecula, interaction alone. This follows from the fact that ~ ~,0 c" ~, in which case the inequalit2 T;. > ~r~may always be fulfilled by lowering the :oncentration of the magnetic molecules. We consider here a system of identical particles with S ~ therefore the index of the species of the particles will further be omitted. The assumption that ro exists means that the correlation functions K(t) are equal to zero as t > r~. 1i\ further. 7;. >t">r~, then the matrices @~(l')) prove to be slowly varying functions in the integrand in eq. (26). hence (26) becomes

(tn~] .~(t). lm2) ~ -

~

(m.~t.,@(t)) ira4)(rest ,~,~<;li"\j/Pn,,)

(:~: ",),,,o. (28)

~1::~

V.P, SAKUN

where i

09 \

:

s(.,

m~,n%, m ~ ) f =

dt'K ( ,t

:DI2,~ FI~I~; ill 6

t;ffl ~ 1"/'l3 , i H S ~ !

1112 ~ t~14 , O16/!

0

Substituting (28)in eq. ([24) we obtain the markovian equation of motiov for the one,paruele density m a t r i x (6(t)), nonlinear in (~(t)) and too cumbersome. In order to simplify it we exploitthe fact that the distribution function (13) is homogeneous and isotropic, Foi ~this purpose we express the operator ,q,b (t, t') as follows

~o~(t, r) = sg;' (t, r) + Ec~ o~o~°~(t, t')&(a) + ~ ,,ope("b'(t, t') ,~,(b) + E o~p"(~b:'(t, t') ~,(a). ,fa(b). 1~

(29")

~,/~

a, b indicate two different molecules of the same kind and g~,(x) are the spin operators of the xth particle. S~"~) (t, t'), (8,/~ = 0, x, y, z), are random functions of t and t' and symmetric under permutations of 8, ft. Statistical averages of the type

(30)


are invariant relative to rotations about the z axis. Substituting (29) into (27) and taking account of the invariance properties of the f u n c i o n s (30) we obtain the following equation tbr the averaged magnetic moment of the particles #(t) gfl Sp (@(t)) g) =

dp

,, = ell x [ d ' + W(t)] dt

i#~ + jffy

k (ff= - ff~o))

T~

T~'

W~ ( 2 2 -- k --- ff~ + / z D , ky (3 I}

where i,j, k are the unit vectors i:n the x, y', z directions, respectively. Here we have inlroduced the followir~g notation: 7 ....

h

,

7 W, Hi(t) = H~(t) - ~ (ffz - p.:(O)x1, liv z

tt'r(t) = H,x,,.(t),

1,1~.

1

i";

H ' = n + .~!L,_ ~ _ I ~o~..+ 2"

H ' = kH',

\'/"z°'/iV7"V~'~ t)

(,,/°)

w, + 2w,, \

1) w, ~-. ' T~7

,0).

(o>..

INTERMOLECULAR SPIN-SPIN INTERACTIONS IN LIQUIDS

145

t

W1 = N,lim_,,.~

.f dt' dta,dt' [ - ½ Re Q=, o~, + I m

( -.a-Qo,,~, oo + zQ~,, o~

0

- -}~Qxx,-.z "- ½Q.,~=,ox -" Qoz. oo + ~Q~, oo -" -~:,;Q:~,,,z)]," t 6 2

[ - 2 Re Q<, ~: + l m (Qoo, x.~ + ~Q:.:,,,.~ + Q~, oo)1, Wz = N lira f dr' - - dt dr' 0

IV3 = N lira f dt' I--~ ,50

|

7 ,d~ 7:

oo. . ox + Qoo, oo ..... ¼Q~...,,=

,

- ~ Q . .....

0

+ Re (¼Qo=,= -" Qoo..o~.

z.e.,,.,..~

+ .tO 8 ~ X X , =~) + hn (¼Q,~., yZ -" Qo~,o>. + ~Q~.o.0] t

f

W,, = N lira

d2 [Re (Qo= ~:~ + 2Qoo, <,~ d ( - ~ 7 dr--7-

O

+ hn (2Qo~. o,, + 2Q.v~,.,D], t

Ws

=

N lira f dt' t ~' Ct3

d2 .dt . dt'. . [ . " '~:Q:"'*'"~ . . .

~(

0

+ Re (2Qo:,,,:~ + ~Q:,~,>.~,~ + h n (Qo, ,,~. + Qo~,,~:)],

I¢~, =

Nlim,_fdc ,.

6 2

---

dt dr'

(-2

Re Qox, ~,:~,+ I m Q . , ~..0,

0

t

W7 . N ', . l i.m . f d.t ' . d2 [Re Qo.~ x~ + Irn (Qo> o~ .... ,~) ~ ~,,,:)] ,-, :~:, dt dt'

(32)

0

where Re and Im mean th~ real and imagilmry parts, respectively, #~o> is the thermal-equilibrium value of l-t-; we have introduced it in (3t) to eliminate the "infinite-temperature defecC' resulting from the semiclassical nature of the presem

V.P. SAKUN

146 :

i

ithieorv.The corresponding mathematical procedure consists in rep~:~cingin (24), (~--~) (o'.4) (~x) by (hA) ( ~ ) . -- ~,.4~*(°)z~°~,,xwhere , b~°~ are the equilibrium density ma|rices. Unlike the Bl°ch's equation~l), eq. (3t) is nonlinear in #. Apart from the last term on the right-hand side, the nonlinearity in (3 l) arises also from the dependence of ~he "relaxation times" T~, 77'2 on the instantaneous value of the deviation of f~-fi~om its equilibrium value #~o~. if It - lg(°~ is small, (31) reduces to the usual Bloch f o r m . At the present time it is hardly possible to ebserve the nonlinear intermolecular effects in normal liquids inasmuch as they can arise only in highly polarized spin systems, i;e. provided gflH > kT, with T being the "spin" or heat-bath temperature. The most suitable medium for observing these effects is apparently the liquid 3He with paramagnetic ions dis~;olved in it. We note that the coefficie:nts Wz, I4t4, W~,, W7 along with all the nonlinear terms in eq. (31) vanish in the extreme-narrowing limit h-~g[IHr,~ ~ t. This can readily be verified taking into account the property QZ~.;:?, =Q~,7. :~ ant! the fact that the functions QT~L~; become fully isotropic as H ~ 0. No restriction on the magnitude of o)~, the strength of the intermolecular spinspin interactions, has been imposed in tl'e course of the derivatisn of eq. (31), though it was assumed that a correlation time r¢ exists such that the correlation functions (27) are equal to zero as t - t' > z~, and in addition that ~ >> ~ . It; however, the interaction is sufficiently small so that the condition ¢.,~r,~ ~ 1 is fulfilled, one may expand the integrands in (32) in power series in the spin-spin interaction and restrict oneself thereupon to its lowest powers. It then turns out that the expansions of the coefficients W2, ~ , W6, !4/'7 start with terms ,,-.to3. Therefore, in the commonly used Born approximation (~(,~) kinetic equations for the magnetic moment become again linear and identical in appearance with the Bloch phenomenological equations.

5, Equilibrium magnetization-recovery curve Let the spin system consisting of a macroscopic amount of like molecules with spins one-half interacting with one another be prepared at t = 0 in the "uncorrelated" state (3) and let #fl0) ¢ #~o) Which law wiit the magnetization/~: follow in approaching its equilibrium value, provided the time-varying magnetic field is H~(t) = 0? To give the answer, let us .~;o/ve eq. (31) putting for simplicity ~.~(0) = #.,.(0). Define ~ = (l/h),) (#~ - I,~)); then it follows from (31) that

5r

__

¢

Tl

~2~2,

(33)

INTERMOLECULAR SPIN-SPIN INTERACTIONS IN LIQUIDS

147

where

77

=

W,

+

IVy,

"

1 +

t

' :

g

/

.

The solution of eq. (33) is ~(t)

= ~(0)[l

-

~.~~,r,~ (0)(e

-'/~

-

1)1 - ~ e - ' / n

.

As seen, the longitudinal magnetization-recovery curve is nonexponential. As for the trans~,erse magnetization, we do not give here ~he corresponding curve since it has too cumbersome an analytic appearence,

6. Conclnsion

Although the kinetic equatiotts derived in the pref;ent pape: were applied to N M R or ESR systems, the generalization to other cases presents t~o special problem. After taking account of"internat"' relaxation processes these equations become as well applicable to optical systems. For instance, one may use them :o study the problem of energy transfer in a liquid activator-sensitizer system. By assumin,:{ the distribution function (l 3) to be diffusional and putting the diffusion coefficient equal to zero one may extend our kinetic equations into the solid state. One may then apply them ~o examine the ESR ( N M R ) absorbtioi~ line shapes in magnetically diluted solid samples, the cross-retaxatio~ effects as well as the radiationless enerLJ transfer problem, elc, The characteristic of the one-particle kinetic equations (20). (21) is that they are applicable not only to the limiting migrationqike cases (i.e.~ when localized excitations exist duc to the fast "internal" relaxation), and via-impurity-band cases (when the intermolecular intera~ctions prevail over the spin-thermal-bath ones) of propagation of the spip~ excitation, but also to all intermediate situations.

Acknowledgments

The author would like to thank Professor I. V. Ateksandrov for drawing attention to the problem, and Dr. N. N. Koi st and Dr. T. N, Khazanovich lbr helpful discussions.

]48

V.P. SAKUN

Appendix Denote the vertices, to which one may get when moving from a only along the dashed lines, by black dots and the remaining ones by empty circles. Each diagram (:fig. 6a) can be related i n a one-to-one manner to the one (fig. 6b) in which the ieftmost empty circle is sited at the same t' coordinate as in 6a but on the opposite Semiaxis, while the remaining vertices retain their places. As can easily be seen, the diagrams 6a and 6b are mutually cancelled out, and hence the complete sum of the diagrams containing empty circles vanishes as well. Thus only "a-connected" diagrams give a contribution to (9).

i

not

a*:: ~ - - - ~ : 1 i

---'~--ooo

a,:--: ' i :', " ',i/ ,, / i

,o,

a

b

L/

,: "°

Fig. 6. Two diagrams mutually cancelled out. Explanations are in the text.

Now let the leflmost vertex of an "a-connected" diagram not be bound directly with the ath particle. Performing the analogous procedure on this vertex we find that the sum of all these diagrams is also equal to zero.

References 1) G,W.Nederbragt and C.A. Reilty, J, chem. PIL ~ 24 (1956) ~! 10. C. A, Reilly and R. L. Strombotne, J. chem. Phys. 26 (1957) 1;~38. F, A, Bovey, J. chem. Phys. 32 (t960) 1877, C. MacLean and E.L.Mackor, J. chem. Phys. 44 (t966) 2708° 2) V, Sinivee, J. Magn. Resonance 7 (1972) 127. 3) l.Solomon, Phys. Rev. 99 (1955) 559. 4) V.Sinivee, Eesti NSVTA Toim~ F•as. Mat, ;8 (1969) 468. 5) T. N. Khazanovich, Phys, Letters A29 (~ '59) 601, T. N. Khazanovich and V. Yu, Zitsermaa, Mol, Phys. 21 (1971) 65. 6) N.N.Korst and T,N°Khazanovich, Zh, Ek~p~ Teor. Fiz. 45 (1963) 1523. 7) H,Silescu and D, Kivelson, J. chem. Phys. 48 (1968) 3493. 8) J,H, Freed, J. chem. Phys. 49 (1968) 376. J, H. Freed, G.V. Bruno and C.F. Polnaszek, J. phys, Chem, 75 ( 197 l) 3385, 9) P.N.Ar~,res and ILL, Kelley, Phys. Rev. I34 (1964) A98. 10) S, Fujita, Introduction to Non,Equilibrium Quantum Statistical Mechanics (W, B. Saunders Co., Philadelphia, London, t966). i i) ,A~Abragam, The PrinciNes of Nuclear Magnetism (Oxferd Univ. Press, London, 1961).