- Email: [email protected]
Magnetic resonance saturation in solids at finite temperatures
Physwa I03A (1980) 295-315 (~ North-Holland Pubhshmg Co
M A G N E T I C R E S O N A N C E S A T U R A T I O N IN SOLIDS AT F I N I T E T E M P E R A T U R E S L J de HAAS, W Th WENCKEBACH and N J POULIS
Kamerhngh Onnes Laboratortum der Rtlksuntversttelt, Leiden, The Netherlandv
Received 23 April 1980
The present paper gives an extension of the Provotorov theory of magnetic resonance saturation to fimte temperatures Such a¢~ extension ~s not straightforward because at finite temperatures when hne-shape and resonance frequency become temperature dependent the weak coupling expansion m a small h f field is no longer convergent In this paper it is assumed that this expansion can be renormahzed by means of spin temperature concepts The results are formulated using a version of the Mort formalism without recourse to a hnear response approximation
1. I n t r o d u c t i o n
The object of this paper ts to give a description of the dynamic behaviour of a system of spins under conditions of magnetic resonance saturation at all temperatures. For the special case that the so-called high temperature approxImation is vahd such a description was given previously by Provotorov~). The density matrix can in this case be expanded to first order m /3 = I/kT. Redfield2), Abragam et alP) and Goldman et al. 45) derived some of the properties of a spm system under magnetic resonance saturation conditions in the more general case that the high temperature approximation is not vahd any more H o w e v e r , as far as we know, an extension to all temperatures of P r o v o t o r o v s treatment was never given We will consider a system of localized spins L subjected to a strong static magnetic field H0. Perpendicular to this field a small magnetic h / field is applied. The spins interact with each other through magnetic dipolar forces When the field H0 is very large only the part of the dipolar interaction that is invariant under rotation around the field direction will contribute to the properties of the system 6) We will therefore only consider this part of the dipolar interaction, the so-called secular part, and will not take into account further interactions such as s p i n - p h o n o n interaction etc., either. Experimentally the system is studied by means of the response of the magnetization M to the h.f field" one must calculate the expectation values associated with M = y ] (] being the sum of the spin operators of the in295
296
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
dwidual particles and y their gyromagnetlc ratio) We will keep our treatment general, collecting the operators of interest m a set of operators called A~(/~ = 0, 1, ), which m our case will represent
0:i,
a2::,,
/i4-- soc
(~,ec being the interaction H a m l l t o m a n ) When the period of irradiation with the h f field is short and this field is small there will only be small variations in the expectation values of the A , Under these so-called non-saturating conditions the expectation values of A , can be deNved by linear r e s p o n s e theory and the r e s o n a n c e f r e q u e n c y and the hneshape can adequately be described by the Morl f o r m a h s m 7) If the s y s t e m gets saturated through prolonged irradiation, a more complicated description Is needed" e g if one irradiates the s y s t e m with a rotating field with the rotation f r e q u e n c y equal to the larmor frequency, co = w~xt(= yHo) even small h.f fields will, through absorbtlon, cause the t e m p e r a t u r e of the s y s t e m to rise in a tlmescale inversely proportional to the h.f field amplitude This t e m p e r a t u r e change in turn m a y affect the r e s o n a n c e f r e q u e n c y and the lineshape. When the rotation f r e q u e n c y w differs f r o m w~xt the changes in the s y s t e m are more comphcated But it m a y be a s s u m e d that in the t h e r m o d y n a m i c limit the properties of our system, if It gets in "equilibrium" after switching the h f. field off, can be described by a canonical density o p e r a t o r s)
~=21 e x p [ - / 3 ( K 3 L
+ K4~,ec)]
(1)
( Z being determined by the normalization condition Tr t~ = 1). The real p a r a m e t e r s K 3 and K 4 in thermal equilibrium take the values K 3= w~xt and K 4= 1, but after prolonged irradiation they can have quite different values We will now consider the possibility of describing these saturation effects more systematically First of all we must avoid to use the linear response approximation. Fortunately one can still use the Mort f o r m a h s m if It is slightly extended This extended Mori formalism will account for the heating of the s y s t e m H o w e v e r , since any heating will affect the s y s t e m for all the subsequent times the m e m o r y function of the Morl formalism must have an infinitely long time tail This m a k e s the m e m o r y function useless as a description of the hneshape and the c o n v e r g e n c e of weak coupling expansions of this function, such as the expansion in the amplitude of the h.f. field is not ensured Also one would like to account for the heating effects in terms of a change in K 3 and K 4. But one m a y not expect the density operator to take at any time the simple canonical f o r m of (1) To be able to use the t e m p e r a t u r e notion we will introduce an auxiliary density o p e r a t o r of the canonical f o r m (1), with
MAGNETIC RESONANCE SATURATION IN SOLIDS
297
time dependent K3(t) and K4(t). This density operator will replace the usual density operator m the definition of the scalar product used in the Morl theory. The auxiliary density operator can be chosen arbitrarily without affecting the v a h & t y of the Mori formalism The m e m o r y function will depend on the choice of the time dependence for K3(t) and K4(t) and for almost all Ka(t) and K4(t) it will exhibit a tail. H o w e v e r , we will a s s u m e - t h e only essential assumption to be made in this paper - that for an appropriate choice of K3(t) and K4(t) the tall in the m e m o r y function can be made to vanish. In section 2 we will give the extension of the Mori formalism which will make it useful in the absence of a linear response approximation with a scalar product defined by means of the canonically shaped auxiliary density operator. In section 3 this formalism will be applied to the system of localized spins with a rotationally symmetric interaction. In section 4 the results will be compared with the aforementioned studies and qualitatively with experimental hneshapes.
2. General formalism
The o b j e c t w e of this section is to write an expression for the expectation values of a set of physical variables {A0, At . . . . } = {A~} as a function of time, t We will consider a system defined by a Hilbert space of states and a Hamfltonian operating on it. With the set of variables {A,} is associated a set of hermitic operators {fi~}. Will demand of this set that Jg(t) is a linear combination of the fi~,,'s and that ,3,o is the identity operator Furthermore the set {fi.~} shall be a linearly independent subset of the linear space of operators The formal expression for the expectation values which will be denoted by A,(t) is
A~(t) = Tr #(t),4,.
(1)
The trace is taken over a base in the Hllbert space states. The time dependence of p is determined by the Liouville equation
~---~(t) = -i L(t)~(t),
Ot
(2)
where by definition L(t)B = [ ~ ( t ) , / ~ ] for an arbitrary operator /3 It will be assumed that ~ ( t ) is independent of time for t ~< 0 and that [t~(t)],=0 = e-a~°J/Tr e-~'~°).
(2a)
298
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
From the Liouville equation and the expression for the expectation values (1) and (2) we will derive an integro-differentlal equation for the set of expectation values A , ( t ) . The derwation of this equation will be closely similar to that of the usual Mort equations In particular there wdl be a matrix g2~(t) which corresponds to the first moment frequency matrix and a function ~"~(t, 7) which takes the place of the memory functionT). The derwatlon will not be entirely the same as that of the Mori equations, though. The primary difference occurs in the choice of a scalar product m the linear space of operators, necessary for the Mori formalism: /3
1 -~ 1 Tr f dA e_Xe(t ) A , e_(~_A)e.)/~, (f~, B)t = Zo(t) 0
Zo(t) = Tr e -oe")
(3)
The unorthodoxy here is the operator /~(t). In the usual Mort formalism the initial Hamiltonian Yg(0)-the same as m the initial density operator (2a)takes its place. O u r / ~ ( t ) can be chosen arbitrarily but we will hmlt ourselves to /~(t)'s which are a linear combination of the A~ which will gwe exp(-/3/~(t)) the shape indicated in [eq. (1) of section 1]"
K ( t ) = K~(t)A~. (Here and m the following a summation convention Is lmphed over repeated mdices/x, u etc ). ~ ( t ) can be denoted
~(t) = n~(t)A,. /~(t) will be further restricted by the assumption that it is equal to ~ ( t ) at t = 0- H~(0) = K"(0) The bdmear form (A,/3) has the usual properties of a scalar product (fi~,/~)t = (/~, fi,)*t (fi~, A/3)t = A(A,/~)t
(a* denotes the conjugate of the complex number a), (3b) (A an arbitrary number),
(A, B + 6~), = (A, t~), + (A, ~),, (A, A), >i o,
(3c) (3d) (3e)
with equality if and only If fi~ = 0 Furthermore this scalar product has the following useful properties: (1, A)t = Z~(t) Tr e-OK")fi,,
(3f)
MAGNETIC RESONANCE SATURATION IN SOLIDS
299
which will be denoted as
(3)0, - (l, 3). ( 3 +,/~+), = (3,/3)*. /3(3, LK(t)13) = ([3 +,/3])0t
(3g) by definltlon
L K ( t ) B =- [/~(t),/~].
(3b)
In the derivation of the integro-differential equation for A . ( t ) use will be made of the (super) operator Pt which projects any operator/~ from the linear space of operators on the linear subspace of which the set {3.} is a basis Pt can be defined by means of the scalar product as P,A = A.g""(t)(3v, 3 )
(4)
The matrix g""(t) can be considered as a metric It is the inverse of g.v(t),
g.~(t) = (A.,
"~v)t,
(5)
whlch is real. symmetric and posihve definite. (Th~s can be proven, the first (real) because 3+. = 3 . with (3g), the second (symmetric) wlth (3b) and the third (positive definite) because the set {3.} Is hnearly independent wlth (3e)) For the scalar product (3,/~), P, with the same argument t ~s a Hermitian super operator
(A. P,i~), = (A. 3 . ) , g " " ( t ) ( 3 . , 1~), = (A.. 3 ) * g " ~ * ( t ) ( A . /3), = (P,A,/3), The projection operator Pt is independent of the choice of the basis for the subspace on which it projects. We will choose m the following the time dependent basis {A,~(t)}"
3 ; = i,
(6a)
A~'. -- A 3 . ( t ) .
p.~ 0,
(6b)
with AA.(t) = 3 . - (3.)0t With the operators A 3 . ( t ) correspond expectation values d A . ( t ) = A.-(A.)ot ( p . ~ 0 ) - w h l c h vanish for t = 0 . With this new base there corresponds a new metric g..(t) which is given by
goo = 1.
go, = O. g,o = O.
and. for/x, v ¢ 0.
g..(t) = ( a 3 . ( t ) . A3~(t)),
(7)
300
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
Consequently one can evaluate g~. by differentiating the " f r e e e n e r g y " function 1 F(~, K~'(t)) = - ~ log Zo(t) (8) twice w~th respect to the "field" K 0
0
- ~ g ~ , ~ ( t ) = c~K" O ~K
(9)
F
-/3g~,. can therefore be called a susceptlbdlty. With the projection operator and the scalar product it becomes possible to write the matrix differential equation for the expectation values. This equation will necessarily contain the energy balance equation, which can also be written directly
d H ~ ( t ) A . ( t ) = H~(t)A~,(t ) + Yr t~(t) ~[~(t), H ~ ( t ) a . ] = H~(t)Au(t) dt
(10)
For the details of the derivation of the matrix lntegro-differential equation which proceeds to more or less standard quantum mechanical procedure we refer to appendix A. The equation reads t
Au(t)
AA~(t)g~".(t) + l S20.(t) - / d~'(AA~(r)qFu(t' ~') + l qt0.(t, ~')). (l l) 0
(Here and henceforth the summation convention is taken to extend only over the values of the indices/.t, v different from zero as the coritributions of v = 0 have been written down exphcltly in (l l) as l S20.(t) and f~ d r 1 qz0u(t, r)). It should be stressed that eq. (l l) is a direct consequence of the Llouvdle equation (1) and (2): in the derivation no use has been made of a linear response approximation The expression for J2~. is
~(t)
= g~(t)( AA~(t), iL(t)A~)t
(1 la)
and with (3f) and (3h) one can deduce from this that ^
A
g20~(t) = (1,1L(t).4~,)t = [3(H"(t)A., iLK(t)A~,)t
(1 lb)
qt"u(t, r) is defined by q-~(t, r) = d
[g~'(r)(AA~(r), U~(r)Uv(t)(l - P,) IL(t).4.L],
q'o~(t, r) = - ~ ( H " ( r ) A . , lLr(~')(1 - P.)U~,~(~ ") Uv(t)(1 - Pt) ~L(t)AD
(1 lc) (1 ld)
where the propagator Up(t) is a hnear but not necessarily unitary super-
MAGNETIC RESONANCE SATURATION IN SOLIDS
301
operator which maps the space of operators onto itself Up(t) is defined by Up(0) = 1,
d d-t U p ( t ) =
U,(t)iL(t)(l-
Pt)
3. Application to magnetic systems
In this section we will consider a lattice of spins I with Zeeman and pair interactions subjected to a magnetic field. This system is defined by its Hilbert s p a c e - t h e direct product of the single particle spaces, the magnetic moment operators M~ = 3,I~' ^
i
^
(I'~(a = x, y, z) is the spin operator which acts on the space of the ith lattice site), and the Hamlltonian ~(t) = ~c
+ eo ~ ' ' i,
=
J,,(Ixlx + lyly
(1)
2Izlz)
(la)
I1
The summations over i and j are over the lattice, i is the sum of the individual if' over the lattice. In this section it will be assumed that .I,i describes the dipolar interaction: J"
[247r-~o~ ~ r ' l
Lr,,J /
(lb)
where r,s is the distance between the lattice positions i and j and r,~ the c o m p o n e n t of the vector connecting the lattice i and j in the direction of the external static field We will suppose this static field to be very large compared to the dipolar fields which will impose axial s y m m e t r y around the direction of the static field upon the external properties of the system. T h e r e f o r e the part of the dipolar interaction that is not mvariant under rotation around the field direction has been left out. The remaining invariant part, the secular part, commutes by definition with/~. Because of the long range effect of the dipolar interaction (which shows itself through the demagnetizing effect) it is necessary if this rotational mvarlance assumption is to be experimentally valid that the external shape of the system is also rotationally symmetric around the field direction ¢oext represents the externally imposed field: oJ~t(t) = ~ l ( t ) + oJ[~' e~,
~l~xt= 3/H0
302
L J DE HAAS, WTH WENCKEBACH AND N J POULIS
(ex, e , e~, are the unit vectors in the x, y, and z directions ) tol(t) c o r r e s p o n d s to the h.f field, which is perpendicular to the z direction H0 Is the static magnetic field (which Is m the z dlrectmn) In this section the f o r m a h s m of sectmn 2 will be applied to the spin system The first step m this apphcatlon is the choice of the set {fi,¢} which will be
t
I
t
"' ^' - 2 L"'L"') -= ~ . An = ~ J"(l'xPx + 1,I,
(2a-d)
tJ
The expectation value corresponding to/14 will be denoted by ED(t), while those corresponding to ,4~ to ,3.3 will be written as It to /3. The Hamlltonmn can be written as a linear combination of the operators A s • Y((t) = HS(t)As, w~th
HI(t) = ,o~(t),
H2(t)=
o~¢(t),
H3(t) = to[xt(--- yH0),
H4(t) = 1
(3a-d)
A crucial step m the p r o c e d u r e lmphed by section 2 is the choice of K s ( t ) As has been pointed out, our intention is to select K s ( t ) m such a way that there are no long time tads in the " m e m o r y function" g'%(t, z) This is important for two reasons first we want to mterprete the Mort equations (2.11) in the usual w a y with ~2 describing " r e s o n a n c e f r e q u e n c i e s " and 1/,% determining line shapes This interpretation cannot be maintained If K~'(t) describes an inappropriate density operator and qt% must c o n v e y the m e m o r y of the initial state as it applies for example to the r e s o n a n c e frequencies The second reason is that the expansion of ~ % ( t , 7) in oJt(t) for large difference b e t w e e n t and r - l a r g e in the sense that the system gets saturated in the time interval b e t w e e n r and t - can only be e x p e c t e d to converge ff ~ % ( t , ~') has no long tlme tail. T h e r e f o r e , if one wants to obtain meaningful results f r o m a small to~ approximation, the tail of ~%(t,~-) must be suppressed. In the high t e m p e r a t u r e s a p p r o x i m a t i o n this p r o b l e m does not exist because the long time tails vanish m general at high temperatures, as is clearly d e m o n s t r a t e d by the success of the magnetic resonance saturation d e s c r l p h o n of P r o v o t o r o C ) But at finite temperatures, when a t e m p e r a t u r e d e p e n d e n c e of such quantities as O%(t) and ~ % ( t , z) occurs, there will in general be time tails m g'%(t, ~-). We will now m a k e the a s s u m p t i o n that these tails can be suppressed by choosing KS(t) appropriately. The p r o o f of the validity of this a s s u m p t m n will not be a t t e m p t e d here The assumption, concretely, is if K " ( t ) is chosen to satisfy the equations
M A G N E T I C R E S O N A N C E S A T U R A T I O N IN S O L I D S
K t=0,
K 2=0,
o d ~Au(t)=-~g,..(t)K"(t),
for /z = 3 and 4.
303
(4)
then m the thermodynamic limit ~"~(t. r) vanishes for large I t - r l , l.e m a time scale independent of tot. This choice implies that the auxiliary canonical density operator (1 1) describes the expectation values of A3(I~) and A4(Eo): A3(t) =
(A3)ot, A4(t) = (.3,4)0,
or
AA3(t) = O, AA4(t) = 0.
(5)
The terms proportional to AA3(t) and AA4(t) in eq (2 11) (those with u = 3 and 4) wdl therefore not contribute any time tails occurring in them are irrelevant The remaining components of ~ ' , (i e. with /z = ! and 2) are assumed to have no time tails w a h this choice of /~. This is the key assumption of this section which may be considered to renormalize the expansion in wl. We assume that sufficient c o n d a i o n s for the validity of this assumption are that o~1 be small and that w[Xtlz and Y(sec, the major parts of the Hamdtonian, commute. We will analyse the various terms in eq. (2.11). First of all there are the terms
(,4~)0t, which
are
"averages"
calculated
with
the
auxiliary
density
operator
exp(-C~l((t))lZo(t). (A~)o, and (A2)0, vanish because / ( ( t ) (with K z = 0 and K ~= 0) is lnvariant under rotation around the z-axis (A3) and (A4) can be calculated by the methods of equilibrium stattstical mechanics. Because we are dealing with a system with dipolar interaction, which decays only with r -3 for large distance r between two lattice positions, one must account for the demagnetizing effect. This means that the thermodynamic quantities can only be calculated for a defined sample shape The averages for two different shapes" in sample shape (1) (A,,)~) and in second sample shape (2) (fi,,)~z) are related through [(.4.)gl, qKr,,
(6a)
where •o c
=
EOC2
D.>[(L)o, Ix,.,., Dcz>[( z)~,] ,2,,,,
(6b)
304
L J DE HAAS, W T H
W E N C K E B A C H AND N J POULIS
(D is the demagnetizing factor.) This expresses merely that the two averages should be evaluated at the same "internal" field The matric gu~(t) takes, due to the rotation s y m m e t r y around the z-axis, the following form"
gi
0
0
0
g~
g34
0
g~4 g44
The 3, 4-components can be obtained by differentiating the " a v e r a g e s " (,43)0t and (/L)0t with respect to K 3 and K 4. Here again one must perform the calculation for specific shape and relate It to different shapes by means of eqs (6a, b) Before analysing the terms .O%(t) and q z ~ ( t , r ) m eq. (2.11), we will consider the time derivative of Eo(t) 0.e v = 4) separately One can write L ( / ) ~ , e c = [~asec + 60L~ + t-~Xl " I, ~,ec] =
-
[~,ec,
~',"
i]
= ( - L ( t ) + ~o~xt Lz)a~t • ].
(7)
(Lz denotes the c o m m u t a t o r with I z ' L z B = [fz,/3] for arbitrary /3) This, inserted in the Liouville equation, yields /~/~(t) = - t o ,
• I + to~X' to, • ( ] x e.)
(8)
Consequently, ~t will be sufficient to consider only the equations for I of the functions g2"~(t), ~ ' . ( t , ~-) only the c o m p o n e n t s with v = 1,2, 3 are required. Furthermore, because AA3(t) and AA4(t) vanish, /x can be restricted to the values 1 and 2 Ftrst of all we will calculate g2".(t) The trick for th]s calculation is to avoid commutators with Y(,~ in favor of LK(t) and commutators with I~ (which will be denoted by L . L~B = [f~,/3]). This is possible because one can write L(t) = ~
1
LK +
(to~xt K 3 ( t ) ] 1 K4(t)}.~ + o~7(t)L~
(9)
Inserted m Ou.(t) = gU~(t) (AA#(t), iL(t)AA.(t))t, with (2 3h) for the first term. this yields f2%(t)
([A~, A.])0, + (co~t
K3(t)) K4(t)} (AA~, 1 LzA~), + wl(t)(AA~, l L~A~)t].
(10)
MAGNETIC R E S O N A N C E SATURATION IN SOLIDS
305
The commutators for uS 4 can be evaluated with
e~v is the Levi-Civita symbol. Hence s2%(t) = - g ~ ( t ) [f ~ - - 1- - ~ 6o-va(i~)0/ + ( ea[x' K---~/K3(t)](aA~'e~AA,,) + OJ?(t)(aA~, ~o~aA~)].
(1 1)
Because of the rotational symmetry only (Is) with a = z contributes in the first term and In the second only o-= a. g2"~(t) appears in eq (2.11) m the expression AAJP',(t). AA~ and AA2 are /~ and Iy and AA3 and AA4 are zero (therefore one may take here g ~ = (l/g±)O~'O. The expression becomes
3A.~
= _ [I±.
I
~
B/f4g~ (/~)0e~ +
i±,~(oa[xt K3(t)'l K4(t)Je~
+ to~(t)I~e~].
(12)
I~ Is the component of the vector I perpendicular to H0. With (a × b), -~ e~,a~a~ this can be written
--~]e~
× I± + ~o~(t) x I~
(13) v
and if one defines oJ~fr by the relation
o~[.=
(f~) ~o,~, K~(t)
[3K4g~
- K4(t),
(14)
or
flK4g~ =
(I)~ot
w[~_ to[~t+ K3(t)/K4(t),
(14a)
(13) becomes
(3A~,Ou~) = (w[n e~ × I~ + (oJ, × I±)L
(15)
g20~(t) can be calculated inserting (9) for L(t). The first two terms of (9) do not contribute because of rotahon symmetry and therefore J20.(t) = (1, iL(t)AA~(t)) = l ( O J ? ( t ) L . a A ~ ( t ) )
=
-oJ~(t)~.z(I~)o
= (tol(t) × (I~)0)~Finally we must deal with ~ , ( t , ~-). Thls is the most difficult problem.
q'%(t, r) = d [g'~(r)(3A~(r), U~'(r)Up(t)(1 -- P,) IL(t),4~),].
(16)
306
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
Since we have a s s u m e d that if K ~' is given by eq (4) this function has no indefimtely long time tail, it can be e x p a n d e d m oJ) for any combination of t and r. F o r our p u r p o s e the lowest nontrlvial order m oJ1 is sufficient. In first order (hnear response) in o~1,Iz and/~0 vanish. C o n s e q u e n t l y according to (4) K3 and K4 also vanish in first order in oJl. In lowest order the r derivative therefore does not need to be evaluated for the scalar product and the projection operator, but only for U~l: d U~l(z) = - l L ( r ) ( 1 - P~) U~l(r) dt In lowest order In oJ1, L(z) and L ( t ) can be replaced b y to~XtLz + LD (Lo being defined b y LD/~ = [~,ec,/~]) omitting oJTL~ f r o m L(t). Lz and LD are hermitic operators with our definition of the scalar p r o d u c t and this together with the hermiticlty of P, enables us to write ~(t,
r) = - - g ~ ( r ) ( - l ( 1 - P~)(to~tLz + LD)AA,~ , U~l(,r)Up(t) x (1 - P), i(o~XtLz + LD)AA.(t)L.
(17)
L~AA~ and Lo AA~ v a m s h w h e n / x = 3 or 4" only the values 1 and 2 of tr and v contribute. We a s s u m e d that qt".(t, r) vanishes for large I t - r [ Let us define a characteristic time Tcha~ by the fact that ~ ( t , r) vanishes when It - r[ >> T~na~ Then, ff to~ is asymptotically small, K " ( t ) which varies only in higher than first order in to~ will be constant o v e r times of the order T~har- One m a y therefore take Pt and the scalar product in the expression for ~ " . ( t , r) to be constant m time This m a k e s it possible to solve the equation for U. formally ex
U;~(r) = e-,(,oL L=+ Lo)~ - v,>,~
(18a)
Uv (t) = e'(~ x'L, +Ltg(1- V,)t
(18b)
F u r t h e r m o r e , since the s u b s p a c e of operators spanned by the set {fi,~,} is i n v a n a n t under rotation around the z-axis:
PLz = L~P Applying this to (17), with the c o m m u t a t i o n relation of Lz and LD:L~LD = LDL~ it follows that Up(t)(1 - Pt) = e ''o?'L~ e'LD(1-v')t(1 -- Pt)-
(19)
and Ugh(t) analogously, with t replaced by - t . Finally one m a y insert a P, in front of AA~, = PtAAu to show that (1 - Pt)Lz / I , ~ = 0 Altogether qz~'~(t, z) b e c o m e s -
1
~x,
~ ' ~ ( t , r) = g - - ~ (zaAu, iLD(1 -- Pt) e ''~L L~,-,)e,LDo-v,×t-,)(1 _ Pt) ILo zlA,), (2O)
MAGNETIC RESONANCE SATURATION IN SOLIDS
307
g ' ~ has b e e n r e p l a c e d b y 1/gL since only (r = 1 or 2 gives a n o n v a n l s h i n g result in (17) T h e m a t r i x f u n c t i o n qtu.(t, r) is real and due to the r o t a h o n s y m m e t r y a r o u n d the z-axis the following relationships exist b e t w e e n Its c o m p o n e n t s
~',(t, r) = ~(t,
~-),
~/'~(t, ~-) = -~(t,
7)
(as can be checked w i t h a 7r/2 rotation around the z-axis in w h i c h Ix ~ I~ and Iy--,-I,) This s y m m e t r y c a n be used to e x p r e s s g t ~ in t e r m s of a scalar c o m p l e x f u n c t i o n Gt(O) which is defined b y Gt(O)
= g -~1
(L, IL0(1 - Pt) etLDtI-P')° (1 - Pt) iLD(ix
-}- Iiy))t,
w i t h the relahon ,Lz(ix + i L ) = iiy + ix wh,ch implies that e'°'[X'L,'(ix + , i y ) = e ''°Ix'' (i, + ' L ) one can express ~ m G ext
~ l ( t , r) = Re G t ( t - r) e '~L "-'~,
(22a)
ext
~ ( t , r) = I m Gt(t - ~') e '~" ~t .~
(22b)
Finally gt0.(t, ~-) c a n be calculated a n a l o g o u s l y to g'".(t, ~-) ~0~(t, ~-) = -(1,0to~XtLz + t L D + Itol(t)L.)(l -- Pt)Upl('c)Up(t) ×(1 - P t ) I L ( t ) A A . ( t ) ) . = - ~(to~(t). L iLK.~(I - P.) UT~(r) U~ (t)(l - P t ) I L ( t ) A A . ( t ) ) . In the first e q u a t i o n Lz and L o do not c o n t r i b u t e b e c a u s e t h e y are hermltic. T h e s e c o n d e q u a t i o n is o b t a i n e d with the aid of (2 3f) and (2.3h). This e x p r e s s i o n can be p r o c e s s e d in the s a m e w a y as ~P~'.(t, ~') which winds up as ~0.(t, ~') = - f l t o ~j(r)(I~, K4(t) ILD(1 -- Pt) e'L°(l-P')(t-~) × e"°~"L:"- ~)(1 -- Pt) ILD A.4.(t)),
(23)
This c o n c l u d e s the f o r m a l d i s c u s s i o n of ~ ' . ( t , r) R e s u m i n g the results of this section for I~ (the v e c t o r f o r m e d b y the x and y c o m p o n e n t s of I): l l ( t ) = to~(t)e~ × l±(t) + tol(t) × I~,,~e~ t
- J d~-(l~(~')+ flK4(T)g~(-c)tol(~))qtzz(t, ~-), 0
(24)
308
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
and in addlhon
It(t) = tot(t) x. l+(t), ED(t) = --to~(t).. IAt) + tot(t)" (IAt) × e~to~~)
(24b)
( q ¢ ~ IS the 2 × 2 tensor f o r m e d by the x, y ( = 1,2) c o m p o n e n t s of 1/:% to~f~(t) and q t , , ( t ) must be calculated with a KU(t) given by K~ = 0, Kz = 0 and Iz = -/3(g33K 3 + g34K4),
(25a)
/~D = --/3 (o°43/~3 + g44/~4),
(25b)
and at t = 0 K 3 = to~xt, K 4 = 1 T h e s e equations are exact in their general form. H o w e v e r , only to lowest order in tot does the expression (20) hold for ~±±, but it is assumed that this is an accurate a p p r o x i m a t i o n for small tot owing to the special choice of K " One should note the close similarity of eqs. (24a-b) to the Bloch equations with ~±± and a convolution instead of the usual 1/T2 term In addition there is (24c) for the dipolar energy, which can be obtained f r o m the Bloch equations with the aid of the energy balance equation (2 12) In the next section we will see why an equation for ED lS necessary, when we c o m p a r e the results of the present section with earlier results
4. The lineshape In the previous section we have written equations for I,(t), It(t) and ED(t) (the expectation value of the secular part of the dipolar energy) F r o m these equations a formal expression for the r e s o n a n c e absorbtion lineshape will be obtained in this section This formal expression connects with the existing theory for r e s o n a n c e lineshapes in t e r m s of the Morl m e m o r y funchon8'9). We will consider the p r o b l e m of solving eq. (3 24) in the limit of small hf field to~(t) This limit was already adopted in section 3 It implies that I i will be hnear in tot, whereas the magnitude of Iz and ED IS not coupled to the amplitude of tot The time derivative o f / 1 is also linear in to~ and that of Iz and ED is proportional to the square of tot. This gives occasion to a two time scale b e h a v i o u r for hmltingly small to~. a rapid time scale in which/~ and ED (and also K 3 and K 4) are constant whereas I± and to~ vary In the slow time scale variations of It and E o and of K 3 and K 4 a r e seen, the averaged rate of change of these four quantities being proportional to the averaged square of to~. We can therefore adopt a solution p r o c e d u r e for these equations in the small tot limit in which first one solves (3 24a) for I.(t) with fixed K 3, K 4, It and ED. Subsequently this solution I±(t) is inserted in the equations for Iz and ED (3.24b-c).
MAGNETIC RESONANCE SATURATION IN SOLIDS
309
First, then, one must solve Ii from I i ( t ) = ~o[n(t)ez × l i ( t ) + t o l ( t ) × I t ( t ) t
- f d~'(ll(r) + ~ K 4 ( r ) g ± ( r ) t o l ( t ) ) r l t ± ± ( t ,
(1)
r).
0
The hme dependence of to[n, lz, K 4, g± and q¢~ due to the time dependence o f K 3 and K 4 c a n be ignored here, because it occurs only in the slow time scale (1) is a matrix equation which can be dlagonahzed with the introduction of L = Ix + ily and L = I x - 1Ir. The latter being the complex conjugate of L we will only consider L : t
I+(t) = I o ~ ( t ) I + ( t )
- IoJl+(t)Iz(t) - [ d r ( L ( r ) 0 ext
+ BK4(r)g±(r)~ol+(r))Gt(t
(2)
- r ) e ' ~ " "~
with ~o~+= 6o~ + iwy This equation, because we ~gnore slow time scale time dependences of K 4 etc., is linear in L , oJ~ and time mvariant and can therefore be solved by means of Laplace transforms:
f dt e-S"-to~L(t)
Lt:s)--
(Re(s) > 0)
(3a)
to
and analogously o31+to(S) =
/ dt e-'"-~)o~l+(t)
(3b)
to
and
=f
(3c)
d t e-StGto(t),
0
which yields a Laplace transformed version of (2): - L ( O ) + s L , ( s ) = Io~ff( t ) L ( s ) - ioJ,+A s ) lz( t ) - (Lt(s)
+ flK4(t)gl(t)tbt+r(S))dt(s
- ito~x').
With the insertion [ 3 K 4 ( t ) g ± ( t ) ( t o [ n ( t ) - w[xt+ ( K 3 ( t ) / K 4 ( t ) ) = - I s ( t ) solution of (4) is
(4) for Iz the
310
L J DE HAAS, WTH WENCKEBACH ANDNJ POULIS
L t ( s ) = ~K4(t)g~(t) ((+ffw~fr(t) - ¢°~xt+ K3(t)/K4(t)) - Ot(s - 1w~t))o51+(s) s - l o ~ " ( t ) + O , ( s - I(D[if)
+
I+(0) S -- lm~tr(t) + Gt(s - 1o)~xt)
(5)
This can be mdentified with the n o t a h o n of the hnear response theory (w~th I+(0) = 0)"
L ( s ) = x++(s)cbl+(s)
(6)
One can apply the reverse Laplace t r a n s f o r m to the solution (6) and c o m p a r e it with the expressions for the time d e n v a t w e s of L as they can be obtained f r o m (2) for e g. a stepped field ¢ol+(w~+(t) = 0 for t < 0, ¢ol+(t) = oJj+ for t > 0) This yields expressions for the m o m e n t s of X++(5)
f d¢o X++(Iw)
ff~(to),
(7a)
oc
f dw
~ox++(lOJ)
= ioJ~ff(to)I~(to) - i ~K4( to) g ±( to)G~(O),
(7b)
zc
to which consequently only the imaginary part of A'++ contributes F r o m here one can go b a c k to the x and y c o m p o n e n t s of I i and introduce the f r e q u e n c y d e p e n d e n t susceptibility tensor through L(s) = ~(s)~,(s) This susceptibility can be e x p r e s s e d m X++" Xxx(S) = xy~(s), Xx,(S) = - x y x ( s ) ,
Xxx(S) = ~(x++(s) + x++(g)),
(8)
xx~(s) = ½(x++(s) - x + + ( ~ ) )
(a denoting the c o m p l e x conjugate of the n u m b e r a). When ¢o~xt is sufficiently large only positive ¢o will contribute significantly to the integrals in eqs. (7) and then it is possible to a p p r o x i m a t e these integrals by
2 1 do) X " O o ) = Iz(to),
(9a)
0
2 f dto X"(ito) = toL~(to)I~(to) - [3K4(to)g~(to)Gto(O), 0
(9b)
MAGNETIC RESONANCE SATURATION IN SOLIDS where X'x'x is minus the imaginary part of With the expression for Gt0(O),
/3K4glOt(O)
= ½(L, ILo(! -
Xxx(X'xx - IX'x'x, X'xx being
311 the real part)
Pt)ILoL),,
(9b) can be elaborated, w h i c h yields m agreement w i t h ref. 4)
f dto(to
totXt)x'(Ito)
3ED.
(10)
0
The explicit e x p r e s s i o n for X"x, i e. the imaginary part of the factor of w~+(s) m (5), is, again for large to~xt which allows us to neglect the conjugate term )? in (8)
,, • XxxOto)=-K4gi/3½[to
Re ¢~(ito - ItoLext) ( t o ext K / K 4) COL -~--toL - - I m ~ ( i t o -- I"t o L~x~.2 )l -t- [Re t_~(ito - ttoL ~xt)] 2 ext
(11)
This Is of the f o r m )(fxtd :
/1(~1' ( t o
- -
to~xt .+. to
KS~K4),
(12)
with
.¥'d = -~/3K4g±
Re a ( l t o
- - I t o Lext) t o
(12a)
[to - w~xt + I m t3(ito - ito~x')] 2 + [Re t3(iw - ito[xt)] 2 x~ ~s the ~maginary part of the susceptibility that one o b s e r v e s m the same s y s t e m when it is in thermal equilibrmm at a t e m p e r a t u r e / 3 ' = / 3 K 4 with an applied static field to~xt = K 3 / K 4 (still without non-secular dipolar interaction), when the f r e q u e n c y of the h.f field is shifted in c o r r e s p o n d e n c e with the static field The equations for the slow time scale b e h a w o u r of I~ and E0 (3.24) can be e x p r e s s e d m terms of this X~:
I~ = to~X'() (to - totxt + KS~K4),
(13a)
to
Eo
2 ,, (to[xt __ to)(to __ to[xt + K a / K 4) = toiX0 to
(13b)
This is consistent with the results of P r o v o t o r o v ~) for the high t e m p e r a t u r e a p p r o x i m a t i o n in which case neither toter nor q' depends on K 3 and K 4. A final r e m a r k can be m a d e concerning the usefulness of the f o r m a l i s m in calculating the lineshape. In view of the lineshapes that have been found experimentally, as depicted In fig. 1, one might fear that G takes a very complicated shape as a function of frequency. A priori this could of course be
312
L J DE HAAS, W T H WENCKEBACH A N D N J
POULIS
Fig I Resonance absorbtlon m CaF2 as a function of frequency (artlbrary units) at 57% polarization from ref 3
t r u e , b u t t h i s Is n o t n e c e s s a r y
one can explain a similar hneshape
with a
s t r u c t u r e f o r G as d r a w n in fig 2a a n d b T h e e s s e n c e o f s u c h a n e x p l a n a t i o n is t h a t t h e c e n t e r f r e q u e n c y
of
G(s-
I o ~ xt) d o e s n o t c o i n c i d e w i t h oJ~n, b u t
d i f f e r s f r o m it b y a n a m o u n t o f t h e o r d e r o f t h e w i d t h as a f u n c t i o n o f ~o o f G, o r m o r e . T h i s l e a d s to a l i n e s h a p e Lorentzian amphtude
in s h a p e
if G d o e s
as s h o w n
In fig
not very appreciably
2c
T h e l a r g e p e a k is
over
its w i d t h .
The
is
flK4g±/ReG(ItO~~ - ioj~xt)
a
b
f
Fig 2 Demonstration of the occurrence of a shouldered absorbtlOn hne using (4 12) when a Lorentzlan shaped G Is assumed, with its center shifted half of its width from ~o~n (a) and (b) show the imaginary and real parts of G as a function frequency in arbitrary umts and (c) shows the resulting absorbtlon hneshape m the same frequency scale
MAGNETIC RESONANCE SATURATION IN SOLIDS
313
and the w~dth Re GOto[~ -
l(~[xt).
5. Conclusion The f o r m a h s m presented in section 2 and elaborated m section 3 offers a d e s c n p h o n of m a g n e h c r e s o n a n c e saturation It is perhaps useful to e m p h a s i z e once more that the validity of this description is quite trivial but that is usefulness is only ensured if our a s s u m p h o n that ~ ' , ( t , ~-) vanishes for large ( t - T ) is valid. Only then can one expand ~ in toj, which gives significance to g2 and ~ independently, and then too the expression (3.20) as a lowest order a p p r o x i m a t i o n of ~ ' ~ ( t , z) can be used. A second r e m a r k can be made about the choice of the projection operator It is u n n e c e s s a r y to define this o p e r a t o r in terms of the same operators of which e x p e c t a h o n values must be calculated In fact in (3.20) a projection o p e r a t o r defined only with {Ix, Iy} as the set of A~'s would gwe the same results. This r e d u n d a n c y of P reflects the fact that our original interest in the f o r m a h s m of section was partly motivated by the need to have a description for more general non linearities in spin systems. With an elaborate p r o j e c h o n o p e r a t o r it is also possible to describe more complicated systems such as for e x a m p l e the one obtained when non secular terms and spin-lattice interactions are retained
Acknowledgements We wish to express our grahtude to Prof. H . W Capel en Dr. J.C. Verstelle for their interest and cooperation in the d e v e l o p m e n t of this p a p e r
Aphendix A
Derivation of equation
(2.9)
The L~ouville equation (2 2) is solved formally by
fi(t)=
O(t)~(t =O)O-l(t), d I)=-i~'(t)O,
U(t =0)= i.
By means of cyclic permutation in the trace expression for the expectation
314
L J DE H A A S , W T H
WENCKEBACH AND N J POULIS
value this enables us to write the Hetsenberg representation of (2 1)
A(t) = Tr fi(t = 0) U-~(t)fi, O ( T ) . The las'~ ~ t ~ , ~ p e ~ working on A.
z~av, 'ere ~ g ~ d ~
~ tt",~ ~t~'a~1 ~" a ~'cqye~T_¢'~'~
O-"tr)AO(t] =- u ( t ) A with
d U ( t ) = U(t) tL(t) dt
U(t--O)=l
and
With this notation the expression for the time-derivative of the expectation values of the set of variables {A.} can be written
d-d-A(t) = Tr p(t = O)U(t)[P, + (l - Pt)] IL(t)A dt The part of this formula with (1 - P t ) can be rephrased with the identity t
U(t)(1 - Pt) = [(1 -/9,=0) -
f
d r U(r)P, -d-i P,U;'(r)]Up(t),
0
which can be verified by differentiating left and right with respect to t (and applying (dMt) P,. = P,.(dMt)P,) When one writes the prolectmn operators explicitly with (2 4) and inserts the previous mdentlty this leads to A. = (Tr fi(t = 0)U(t)A~)g~(t)(A~. iL(t)fi~.) + T r t~(t = 0)(1 - P~=o)Uv(t)(l - Pt) IL(t)A. t
-f
d~'(Tr ~6(t = O)U(r)fi~v) d g~(~') 0
(A~, U~'('r)Up(t)(l - Pt) 1L(t)A.) (The factor ( 1 - P t) in eq. (1) has been squared here, keeping one factor together with iL(t)A, for notational symmetry). The second term vanishes, since [Tr t~(t)/}],=0 = [(B)ot],=0 for an arbitrary o p e r a t o r / } and consequently, i f / } = (1 - P,)/}', [Tr iS(t)(1 - P,)/3'],=o = [Tr t~(t)/3' - (/3')or -
AA.(t)g"~(t)(Afi~('r),/~')tlt=o = 0
For 13"= b~t}' ~ ? - P,) cL~t?ff~, t?tbs s ~ ' s
e~at ~
sc~"orrd ee,"m v~t',.s~'s.
MAGNETIC RESONANCE SATURATION IN SOLIDS
315
References I) B N Provotorov, Soviet Physics Jetp 14 (1962) 1126, JETP 41 (1961) 1582 2) A G Redfield, Phys Rev 98 (1955) 1787 3) A Abragam, M Chapelher, J F Jacqumot and M Goldman, J Magn Resonance 10 (1973) 322 4) M Goldman, J Magn Resonance 17 (1975) 393 5) M Goldman, Spin Temperature and Nuclear Magnetic Resonance m Sohds (Clarendon, Oxford, 1970) 6) A Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961) 7) See e g B J Berne and G D Harp, Advances m Chemical Physics Vol XVII (IntersclenceJohn Wdey, New York) p 63 8) F C Barreto and G F Retter, Phys Rev B6 (1972) 2555 9) D Klvelson and K Ogan, Adv Magn Resonance 7 (1974) 72
M A G N E T I C R E S O N A N C E S A T U R A T I O N IN SOLIDS AT F I N I T E T E M P E R A T U R E S L J de HAAS, W Th WENCKEBACH and N J POULIS
Kamerhngh Onnes Laboratortum der Rtlksuntversttelt, Leiden, The Netherlandv
Received 23 April 1980
The present paper gives an extension of the Provotorov theory of magnetic resonance saturation to fimte temperatures Such a¢~ extension ~s not straightforward because at finite temperatures when hne-shape and resonance frequency become temperature dependent the weak coupling expansion m a small h f field is no longer convergent In this paper it is assumed that this expansion can be renormahzed by means of spin temperature concepts The results are formulated using a version of the Mort formalism without recourse to a hnear response approximation
1. I n t r o d u c t i o n
The object of this paper ts to give a description of the dynamic behaviour of a system of spins under conditions of magnetic resonance saturation at all temperatures. For the special case that the so-called high temperature approxImation is vahd such a description was given previously by Provotorov~). The density matrix can in this case be expanded to first order m /3 = I/kT. Redfield2), Abragam et alP) and Goldman et al. 45) derived some of the properties of a spm system under magnetic resonance saturation conditions in the more general case that the high temperature approximation is not vahd any more H o w e v e r , as far as we know, an extension to all temperatures of P r o v o t o r o v s treatment was never given We will consider a system of localized spins L subjected to a strong static magnetic field H0. Perpendicular to this field a small magnetic h / field is applied. The spins interact with each other through magnetic dipolar forces When the field H0 is very large only the part of the dipolar interaction that is invariant under rotation around the field direction will contribute to the properties of the system 6) We will therefore only consider this part of the dipolar interaction, the so-called secular part, and will not take into account further interactions such as s p i n - p h o n o n interaction etc., either. Experimentally the system is studied by means of the response of the magnetization M to the h.f field" one must calculate the expectation values associated with M = y ] (] being the sum of the spin operators of the in295
296
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
dwidual particles and y their gyromagnetlc ratio) We will keep our treatment general, collecting the operators of interest m a set of operators called A~(/~ = 0, 1, ), which m our case will represent
0:i,
a2::,,
/i4-- soc
(~,ec being the interaction H a m l l t o m a n ) When the period of irradiation with the h f field is short and this field is small there will only be small variations in the expectation values of the A , Under these so-called non-saturating conditions the expectation values of A , can be deNved by linear r e s p o n s e theory and the r e s o n a n c e f r e q u e n c y and the hneshape can adequately be described by the Morl f o r m a h s m 7) If the s y s t e m gets saturated through prolonged irradiation, a more complicated description Is needed" e g if one irradiates the s y s t e m with a rotating field with the rotation f r e q u e n c y equal to the larmor frequency, co = w~xt(= yHo) even small h.f fields will, through absorbtlon, cause the t e m p e r a t u r e of the s y s t e m to rise in a tlmescale inversely proportional to the h.f field amplitude This t e m p e r a t u r e change in turn m a y affect the r e s o n a n c e f r e q u e n c y and the lineshape. When the rotation f r e q u e n c y w differs f r o m w~xt the changes in the s y s t e m are more comphcated But it m a y be a s s u m e d that in the t h e r m o d y n a m i c limit the properties of our system, if It gets in "equilibrium" after switching the h f. field off, can be described by a canonical density o p e r a t o r s)
~=21 e x p [ - / 3 ( K 3 L
+ K4~,ec)]
(1)
( Z being determined by the normalization condition Tr t~ = 1). The real p a r a m e t e r s K 3 and K 4 in thermal equilibrium take the values K 3= w~xt and K 4= 1, but after prolonged irradiation they can have quite different values We will now consider the possibility of describing these saturation effects more systematically First of all we must avoid to use the linear response approximation. Fortunately one can still use the Mort f o r m a h s m if It is slightly extended This extended Mori formalism will account for the heating of the s y s t e m H o w e v e r , since any heating will affect the s y s t e m for all the subsequent times the m e m o r y function of the Morl formalism must have an infinitely long time tail This m a k e s the m e m o r y function useless as a description of the hneshape and the c o n v e r g e n c e of weak coupling expansions of this function, such as the expansion in the amplitude of the h.f. field is not ensured Also one would like to account for the heating effects in terms of a change in K 3 and K 4. But one m a y not expect the density operator to take at any time the simple canonical f o r m of (1) To be able to use the t e m p e r a t u r e notion we will introduce an auxiliary density o p e r a t o r of the canonical f o r m (1), with
MAGNETIC RESONANCE SATURATION IN SOLIDS
297
time dependent K3(t) and K4(t). This density operator will replace the usual density operator m the definition of the scalar product used in the Morl theory. The auxiliary density operator can be chosen arbitrarily without affecting the v a h & t y of the Mori formalism The m e m o r y function will depend on the choice of the time dependence for K3(t) and K4(t) and for almost all Ka(t) and K4(t) it will exhibit a tail. H o w e v e r , we will a s s u m e - t h e only essential assumption to be made in this paper - that for an appropriate choice of K3(t) and K4(t) the tall in the m e m o r y function can be made to vanish. In section 2 we will give the extension of the Mori formalism which will make it useful in the absence of a linear response approximation with a scalar product defined by means of the canonically shaped auxiliary density operator. In section 3 this formalism will be applied to the system of localized spins with a rotationally symmetric interaction. In section 4 the results will be compared with the aforementioned studies and qualitatively with experimental hneshapes.
2. General formalism
The o b j e c t w e of this section is to write an expression for the expectation values of a set of physical variables {A0, At . . . . } = {A~} as a function of time, t We will consider a system defined by a Hilbert space of states and a Hamfltonian operating on it. With the set of variables {A,} is associated a set of hermitic operators {fi~}. Will demand of this set that Jg(t) is a linear combination of the fi~,,'s and that ,3,o is the identity operator Furthermore the set {fi.~} shall be a linearly independent subset of the linear space of operators The formal expression for the expectation values which will be denoted by A,(t) is
A~(t) = Tr #(t),4,.
(1)
The trace is taken over a base in the Hllbert space states. The time dependence of p is determined by the Liouville equation
~---~(t) = -i L(t)~(t),
Ot
(2)
where by definition L(t)B = [ ~ ( t ) , / ~ ] for an arbitrary operator /3 It will be assumed that ~ ( t ) is independent of time for t ~< 0 and that [t~(t)],=0 = e-a~°J/Tr e-~'~°).
(2a)
298
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
From the Liouville equation and the expression for the expectation values (1) and (2) we will derive an integro-differentlal equation for the set of expectation values A , ( t ) . The derwation of this equation will be closely similar to that of the usual Mort equations In particular there wdl be a matrix g2~(t) which corresponds to the first moment frequency matrix and a function ~"~(t, 7) which takes the place of the memory functionT). The derwatlon will not be entirely the same as that of the Mori equations, though. The primary difference occurs in the choice of a scalar product m the linear space of operators, necessary for the Mori formalism: /3
1 -~ 1 Tr f dA e_Xe(t ) A , e_(~_A)e.)/~, (f~, B)t = Zo(t) 0
Zo(t) = Tr e -oe")
(3)
The unorthodoxy here is the operator /~(t). In the usual Mort formalism the initial Hamiltonian Yg(0)-the same as m the initial density operator (2a)takes its place. O u r / ~ ( t ) can be chosen arbitrarily but we will hmlt ourselves to /~(t)'s which are a linear combination of the A~ which will gwe exp(-/3/~(t)) the shape indicated in [eq. (1) of section 1]"
K ( t ) = K~(t)A~. (Here and m the following a summation convention Is lmphed over repeated mdices/x, u etc ). ~ ( t ) can be denoted
~(t) = n~(t)A,. /~(t) will be further restricted by the assumption that it is equal to ~ ( t ) at t = 0- H~(0) = K"(0) The bdmear form (A,/3) has the usual properties of a scalar product (fi~,/~)t = (/~, fi,)*t (fi~, A/3)t = A(A,/~)t
(a* denotes the conjugate of the complex number a), (3b) (A an arbitrary number),
(A, B + 6~), = (A, t~), + (A, ~),, (A, A), >i o,
(3c) (3d) (3e)
with equality if and only If fi~ = 0 Furthermore this scalar product has the following useful properties: (1, A)t = Z~(t) Tr e-OK")fi,,
(3f)
MAGNETIC RESONANCE SATURATION IN SOLIDS
299
which will be denoted as
(3)0, - (l, 3). ( 3 +,/~+), = (3,/3)*. /3(3, LK(t)13) = ([3 +,/3])0t
(3g) by definltlon
L K ( t ) B =- [/~(t),/~].
(3b)
In the derivation of the integro-differential equation for A . ( t ) use will be made of the (super) operator Pt which projects any operator/~ from the linear space of operators on the linear subspace of which the set {3.} is a basis Pt can be defined by means of the scalar product as P,A = A.g""(t)(3v, 3 )
(4)
The matrix g""(t) can be considered as a metric It is the inverse of g.v(t),
g.~(t) = (A.,
"~v)t,
(5)
whlch is real. symmetric and posihve definite. (Th~s can be proven, the first (real) because 3+. = 3 . with (3g), the second (symmetric) wlth (3b) and the third (positive definite) because the set {3.} Is hnearly independent wlth (3e)) For the scalar product (3,/~), P, with the same argument t ~s a Hermitian super operator
(A. P,i~), = (A. 3 . ) , g " " ( t ) ( 3 . , 1~), = (A.. 3 ) * g " ~ * ( t ) ( A . /3), = (P,A,/3), The projection operator Pt is independent of the choice of the basis for the subspace on which it projects. We will choose m the following the time dependent basis {A,~(t)}"
3 ; = i,
(6a)
A~'. -- A 3 . ( t ) .
p.~ 0,
(6b)
with AA.(t) = 3 . - (3.)0t With the operators A 3 . ( t ) correspond expectation values d A . ( t ) = A.-(A.)ot ( p . ~ 0 ) - w h l c h vanish for t = 0 . With this new base there corresponds a new metric g..(t) which is given by
goo = 1.
go, = O. g,o = O.
and. for/x, v ¢ 0.
g..(t) = ( a 3 . ( t ) . A3~(t)),
(7)
300
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
Consequently one can evaluate g~. by differentiating the " f r e e e n e r g y " function 1 F(~, K~'(t)) = - ~ log Zo(t) (8) twice w~th respect to the "field" K 0
0
- ~ g ~ , ~ ( t ) = c~K" O ~K
(9)
F
-/3g~,. can therefore be called a susceptlbdlty. With the projection operator and the scalar product it becomes possible to write the matrix differential equation for the expectation values. This equation will necessarily contain the energy balance equation, which can also be written directly
d H ~ ( t ) A . ( t ) = H~(t)A~,(t ) + Yr t~(t) ~[~(t), H ~ ( t ) a . ] = H~(t)Au(t) dt
(10)
For the details of the derivation of the matrix lntegro-differential equation which proceeds to more or less standard quantum mechanical procedure we refer to appendix A. The equation reads t
Au(t)
AA~(t)g~".(t) + l S20.(t) - / d~'(AA~(r)qFu(t' ~') + l qt0.(t, ~')). (l l) 0
(Here and henceforth the summation convention is taken to extend only over the values of the indices/.t, v different from zero as the coritributions of v = 0 have been written down exphcltly in (l l) as l S20.(t) and f~ d r 1 qz0u(t, r)). It should be stressed that eq. (l l) is a direct consequence of the Llouvdle equation (1) and (2): in the derivation no use has been made of a linear response approximation The expression for J2~. is
~(t)
= g~(t)( AA~(t), iL(t)A~)t
(1 la)
and with (3f) and (3h) one can deduce from this that ^
A
g20~(t) = (1,1L(t).4~,)t = [3(H"(t)A., iLK(t)A~,)t
(1 lb)
qt"u(t, r) is defined by q-~(t, r) = d
[g~'(r)(AA~(r), U~(r)Uv(t)(l - P,) IL(t).4.L],
q'o~(t, r) = - ~ ( H " ( r ) A . , lLr(~')(1 - P.)U~,~(~ ") Uv(t)(1 - Pt) ~L(t)AD
(1 lc) (1 ld)
where the propagator Up(t) is a hnear but not necessarily unitary super-
MAGNETIC RESONANCE SATURATION IN SOLIDS
301
operator which maps the space of operators onto itself Up(t) is defined by Up(0) = 1,
d d-t U p ( t ) =
U,(t)iL(t)(l-
Pt)
3. Application to magnetic systems
In this section we will consider a lattice of spins I with Zeeman and pair interactions subjected to a magnetic field. This system is defined by its Hilbert s p a c e - t h e direct product of the single particle spaces, the magnetic moment operators M~ = 3,I~' ^
i
^
(I'~(a = x, y, z) is the spin operator which acts on the space of the ith lattice site), and the Hamlltonian ~(t) = ~c
+ eo ~ ' ' i,
=
J,,(Ixlx + lyly
(1)
2Izlz)
(la)
I1
The summations over i and j are over the lattice, i is the sum of the individual if' over the lattice. In this section it will be assumed that .I,i describes the dipolar interaction: J"
[247r-~o~ ~ r ' l
Lr,,J /
(lb)
where r,s is the distance between the lattice positions i and j and r,~ the c o m p o n e n t of the vector connecting the lattice i and j in the direction of the external static field We will suppose this static field to be very large compared to the dipolar fields which will impose axial s y m m e t r y around the direction of the static field upon the external properties of the system. T h e r e f o r e the part of the dipolar interaction that is not mvariant under rotation around the field direction has been left out. The remaining invariant part, the secular part, commutes by definition with/~. Because of the long range effect of the dipolar interaction (which shows itself through the demagnetizing effect) it is necessary if this rotational mvarlance assumption is to be experimentally valid that the external shape of the system is also rotationally symmetric around the field direction ¢oext represents the externally imposed field: oJ~t(t) = ~ l ( t ) + oJ[~' e~,
~l~xt= 3/H0
302
L J DE HAAS, WTH WENCKEBACH AND N J POULIS
(ex, e , e~, are the unit vectors in the x, y, and z directions ) tol(t) c o r r e s p o n d s to the h.f field, which is perpendicular to the z direction H0 Is the static magnetic field (which Is m the z dlrectmn) In this section the f o r m a h s m of sectmn 2 will be applied to the spin system The first step m this apphcatlon is the choice of the set {fi,¢} which will be
t
I
t
"' ^' - 2 L"'L"') -= ~ . An = ~ J"(l'xPx + 1,I,
(2a-d)
tJ
The expectation value corresponding to/14 will be denoted by ED(t), while those corresponding to ,4~ to ,3.3 will be written as It to /3. The Hamlltonmn can be written as a linear combination of the operators A s • Y((t) = HS(t)As, w~th
HI(t) = ,o~(t),
H2(t)=
o~¢(t),
H3(t) = to[xt(--- yH0),
H4(t) = 1
(3a-d)
A crucial step m the p r o c e d u r e lmphed by section 2 is the choice of K s ( t ) As has been pointed out, our intention is to select K s ( t ) m such a way that there are no long time tads in the " m e m o r y function" g'%(t, z) This is important for two reasons first we want to mterprete the Mort equations (2.11) in the usual w a y with ~2 describing " r e s o n a n c e f r e q u e n c i e s " and 1/,% determining line shapes This interpretation cannot be maintained If K~'(t) describes an inappropriate density operator and qt% must c o n v e y the m e m o r y of the initial state as it applies for example to the r e s o n a n c e frequencies The second reason is that the expansion of ~ % ( t , 7) in oJt(t) for large difference b e t w e e n t and r - l a r g e in the sense that the system gets saturated in the time interval b e t w e e n r and t - can only be e x p e c t e d to converge ff ~ % ( t , ~') has no long tlme tail. T h e r e f o r e , if one wants to obtain meaningful results f r o m a small to~ approximation, the tail of ~%(t,~-) must be suppressed. In the high t e m p e r a t u r e s a p p r o x i m a t i o n this p r o b l e m does not exist because the long time tails vanish m general at high temperatures, as is clearly d e m o n s t r a t e d by the success of the magnetic resonance saturation d e s c r l p h o n of P r o v o t o r o C ) But at finite temperatures, when a t e m p e r a t u r e d e p e n d e n c e of such quantities as O%(t) and ~ % ( t , z) occurs, there will in general be time tails m g'%(t, ~-). We will now m a k e the a s s u m p t i o n that these tails can be suppressed by choosing KS(t) appropriately. The p r o o f of the validity of this a s s u m p t m n will not be a t t e m p t e d here The assumption, concretely, is if K " ( t ) is chosen to satisfy the equations
M A G N E T I C R E S O N A N C E S A T U R A T I O N IN S O L I D S
K t=0,
K 2=0,
o d ~Au(t)=-~g,..(t)K"(t),
for /z = 3 and 4.
303
(4)
then m the thermodynamic limit ~"~(t. r) vanishes for large I t - r l , l.e m a time scale independent of tot. This choice implies that the auxiliary canonical density operator (1 1) describes the expectation values of A3(I~) and A4(Eo): A3(t) =
(A3)ot, A4(t) = (.3,4)0,
or
AA3(t) = O, AA4(t) = 0.
(5)
The terms proportional to AA3(t) and AA4(t) in eq (2 11) (those with u = 3 and 4) wdl therefore not contribute any time tails occurring in them are irrelevant The remaining components of ~ ' , (i e. with /z = ! and 2) are assumed to have no time tails w a h this choice of /~. This is the key assumption of this section which may be considered to renormalize the expansion in wl. We assume that sufficient c o n d a i o n s for the validity of this assumption are that o~1 be small and that w[Xtlz and Y(sec, the major parts of the Hamdtonian, commute. We will analyse the various terms in eq. (2.11). First of all there are the terms
(,4~)0t, which
are
"averages"
calculated
with
the
auxiliary
density
operator
exp(-C~l((t))lZo(t). (A~)o, and (A2)0, vanish because / ( ( t ) (with K z = 0 and K ~= 0) is lnvariant under rotation around the z-axis (A3) and (A4) can be calculated by the methods of equilibrium stattstical mechanics. Because we are dealing with a system with dipolar interaction, which decays only with r -3 for large distance r between two lattice positions, one must account for the demagnetizing effect. This means that the thermodynamic quantities can only be calculated for a defined sample shape The averages for two different shapes" in sample shape (1) (A,,)~) and in second sample shape (2) (fi,,)~z) are related through [(.4.)gl, qKr,,
(6a)
where •o c
=
EOC2
D.>[(L)o, Ix,.,., Dcz>[( z)~,] ,2,,,,
(6b)
304
L J DE HAAS, W T H
W E N C K E B A C H AND N J POULIS
(D is the demagnetizing factor.) This expresses merely that the two averages should be evaluated at the same "internal" field The matric gu~(t) takes, due to the rotation s y m m e t r y around the z-axis, the following form"
gi
0
0
0
g~
g34
0
g~4 g44
The 3, 4-components can be obtained by differentiating the " a v e r a g e s " (,43)0t and (/L)0t with respect to K 3 and K 4. Here again one must perform the calculation for specific shape and relate It to different shapes by means of eqs (6a, b) Before analysing the terms .O%(t) and q z ~ ( t , r ) m eq. (2.11), we will consider the time derivative of Eo(t) 0.e v = 4) separately One can write L ( / ) ~ , e c = [~asec + 60L~ + t-~Xl " I, ~,ec] =
-
[~,ec,
~',"
i]
= ( - L ( t ) + ~o~xt Lz)a~t • ].
(7)
(Lz denotes the c o m m u t a t o r with I z ' L z B = [fz,/3] for arbitrary /3) This, inserted in the Liouville equation, yields /~/~(t) = - t o ,
• I + to~X' to, • ( ] x e.)
(8)
Consequently, ~t will be sufficient to consider only the equations for I of the functions g2"~(t), ~ ' . ( t , ~-) only the c o m p o n e n t s with v = 1,2, 3 are required. Furthermore, because AA3(t) and AA4(t) vanish, /x can be restricted to the values 1 and 2 Ftrst of all we will calculate g2".(t) The trick for th]s calculation is to avoid commutators with Y(,~ in favor of LK(t) and commutators with I~ (which will be denoted by L . L~B = [f~,/3]). This is possible because one can write L(t) = ~
1
LK +
(to~xt K 3 ( t ) ] 1 K4(t)}.~ + o~7(t)L~
(9)
Inserted m Ou.(t) = gU~(t) (AA#(t), iL(t)AA.(t))t, with (2 3h) for the first term. this yields f2%(t)
([A~, A.])0, + (co~t
K3(t)) K4(t)} (AA~, 1 LzA~), + wl(t)(AA~, l L~A~)t].
(10)
MAGNETIC R E S O N A N C E SATURATION IN SOLIDS
305
The commutators for uS 4 can be evaluated with
e~v is the Levi-Civita symbol. Hence s2%(t) = - g ~ ( t ) [f ~ - - 1- - ~ 6o-va(i~)0/ + ( ea[x' K---~/K3(t)](aA~'e~AA,,) + OJ?(t)(aA~, ~o~aA~)].
(1 1)
Because of the rotational symmetry only (Is) with a = z contributes in the first term and In the second only o-= a. g2"~(t) appears in eq (2.11) m the expression AAJP',(t). AA~ and AA2 are /~ and Iy and AA3 and AA4 are zero (therefore one may take here g ~ = (l/g±)O~'O. The expression becomes
3A.~
= _ [I±.
I
~
B/f4g~ (/~)0e~ +
i±,~(oa[xt K3(t)'l K4(t)Je~
+ to~(t)I~e~].
(12)
I~ Is the component of the vector I perpendicular to H0. With (a × b), -~ e~,a~a~ this can be written
--~]e~
× I± + ~o~(t) x I~
(13) v
and if one defines oJ~fr by the relation
o~[.=
(f~) ~o,~, K~(t)
[3K4g~
- K4(t),
(14)
or
flK4g~ =
(I)~ot
w[~_ to[~t+ K3(t)/K4(t),
(14a)
(13) becomes
(3A~,Ou~) = (w[n e~ × I~ + (oJ, × I±)L
(15)
g20~(t) can be calculated inserting (9) for L(t). The first two terms of (9) do not contribute because of rotahon symmetry and therefore J20.(t) = (1, iL(t)AA~(t)) = l ( O J ? ( t ) L . a A ~ ( t ) )
=
-oJ~(t)~.z(I~)o
= (tol(t) × (I~)0)~Finally we must deal with ~ , ( t , ~-). Thls is the most difficult problem.
q'%(t, r) = d [g'~(r)(3A~(r), U~'(r)Up(t)(1 -- P,) IL(t),4~),].
(16)
306
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
Since we have a s s u m e d that if K ~' is given by eq (4) this function has no indefimtely long time tail, it can be e x p a n d e d m oJ) for any combination of t and r. F o r our p u r p o s e the lowest nontrlvial order m oJ1 is sufficient. In first order (hnear response) in o~1,Iz and/~0 vanish. C o n s e q u e n t l y according to (4) K3 and K4 also vanish in first order in oJl. In lowest order the r derivative therefore does not need to be evaluated for the scalar product and the projection operator, but only for U~l: d U~l(z) = - l L ( r ) ( 1 - P~) U~l(r) dt In lowest order In oJ1, L(z) and L ( t ) can be replaced b y to~XtLz + LD (Lo being defined b y LD/~ = [~,ec,/~]) omitting oJTL~ f r o m L(t). Lz and LD are hermitic operators with our definition of the scalar p r o d u c t and this together with the hermiticlty of P, enables us to write ~(t,
r) = - - g ~ ( r ) ( - l ( 1 - P~)(to~tLz + LD)AA,~ , U~l(,r)Up(t) x (1 - P), i(o~XtLz + LD)AA.(t)L.
(17)
L~AA~ and Lo AA~ v a m s h w h e n / x = 3 or 4" only the values 1 and 2 of tr and v contribute. We a s s u m e d that qt".(t, r) vanishes for large I t - r [ Let us define a characteristic time Tcha~ by the fact that ~ ( t , r) vanishes when It - r[ >> T~na~ Then, ff to~ is asymptotically small, K " ( t ) which varies only in higher than first order in to~ will be constant o v e r times of the order T~har- One m a y therefore take Pt and the scalar product in the expression for ~ " . ( t , r) to be constant m time This m a k e s it possible to solve the equation for U. formally ex
U;~(r) = e-,(,oL L=+ Lo)~ - v,>,~
(18a)
Uv (t) = e'(~ x'L, +Ltg(1- V,)t
(18b)
F u r t h e r m o r e , since the s u b s p a c e of operators spanned by the set {fi,~,} is i n v a n a n t under rotation around the z-axis:
PLz = L~P Applying this to (17), with the c o m m u t a t i o n relation of Lz and LD:L~LD = LDL~ it follows that Up(t)(1 - Pt) = e ''o?'L~ e'LD(1-v')t(1 -- Pt)-
(19)
and Ugh(t) analogously, with t replaced by - t . Finally one m a y insert a P, in front of AA~, = PtAAu to show that (1 - Pt)Lz / I , ~ = 0 Altogether qz~'~(t, z) b e c o m e s -
1
~x,
~ ' ~ ( t , r) = g - - ~ (zaAu, iLD(1 -- Pt) e ''~L L~,-,)e,LDo-v,×t-,)(1 _ Pt) ILo zlA,), (2O)
MAGNETIC RESONANCE SATURATION IN SOLIDS
307
g ' ~ has b e e n r e p l a c e d b y 1/gL since only (r = 1 or 2 gives a n o n v a n l s h i n g result in (17) T h e m a t r i x f u n c t i o n qtu.(t, r) is real and due to the r o t a h o n s y m m e t r y a r o u n d the z-axis the following relationships exist b e t w e e n Its c o m p o n e n t s
~',(t, r) = ~(t,
~-),
~/'~(t, ~-) = -~(t,
7)
(as can be checked w i t h a 7r/2 rotation around the z-axis in w h i c h Ix ~ I~ and Iy--,-I,) This s y m m e t r y c a n be used to e x p r e s s g t ~ in t e r m s of a scalar c o m p l e x f u n c t i o n Gt(O) which is defined b y Gt(O)
= g -~1
(L, IL0(1 - Pt) etLDtI-P')° (1 - Pt) iLD(ix
-}- Iiy))t,
w i t h the relahon ,Lz(ix + i L ) = iiy + ix wh,ch implies that e'°'[X'L,'(ix + , i y ) = e ''°Ix'' (i, + ' L ) one can express ~ m G ext
~ l ( t , r) = Re G t ( t - r) e '~L "-'~,
(22a)
ext
~ ( t , r) = I m Gt(t - ~') e '~" ~t .~
(22b)
Finally gt0.(t, ~-) c a n be calculated a n a l o g o u s l y to g'".(t, ~-) ~0~(t, ~-) = -(1,0to~XtLz + t L D + Itol(t)L.)(l -- Pt)Upl('c)Up(t) ×(1 - P t ) I L ( t ) A A . ( t ) ) . = - ~(to~(t). L iLK.~(I - P.) UT~(r) U~ (t)(l - P t ) I L ( t ) A A . ( t ) ) . In the first e q u a t i o n Lz and L o do not c o n t r i b u t e b e c a u s e t h e y are hermltic. T h e s e c o n d e q u a t i o n is o b t a i n e d with the aid of (2 3f) and (2.3h). This e x p r e s s i o n can be p r o c e s s e d in the s a m e w a y as ~P~'.(t, ~') which winds up as ~0.(t, ~') = - f l t o ~j(r)(I~, K4(t) ILD(1 -- Pt) e'L°(l-P')(t-~) × e"°~"L:"- ~)(1 -- Pt) ILD A.4.(t)),
(23)
This c o n c l u d e s the f o r m a l d i s c u s s i o n of ~ ' . ( t , r) R e s u m i n g the results of this section for I~ (the v e c t o r f o r m e d b y the x and y c o m p o n e n t s of I): l l ( t ) = to~(t)e~ × l±(t) + tol(t) × I~,,~e~ t
- J d~-(l~(~')+ flK4(T)g~(-c)tol(~))qtzz(t, ~-), 0
(24)
308
L J DE HAAS, W TH WENCKEBACH AND N J POULIS
and in addlhon
It(t) = tot(t) x. l+(t), ED(t) = --to~(t).. IAt) + tot(t)" (IAt) × e~to~~)
(24b)
( q ¢ ~ IS the 2 × 2 tensor f o r m e d by the x, y ( = 1,2) c o m p o n e n t s of 1/:% to~f~(t) and q t , , ( t ) must be calculated with a KU(t) given by K~ = 0, Kz = 0 and Iz = -/3(g33K 3 + g34K4),
(25a)
/~D = --/3 (o°43/~3 + g44/~4),
(25b)
and at t = 0 K 3 = to~xt, K 4 = 1 T h e s e equations are exact in their general form. H o w e v e r , only to lowest order in tot does the expression (20) hold for ~±±, but it is assumed that this is an accurate a p p r o x i m a t i o n for small tot owing to the special choice of K " One should note the close similarity of eqs. (24a-b) to the Bloch equations with ~±± and a convolution instead of the usual 1/T2 term In addition there is (24c) for the dipolar energy, which can be obtained f r o m the Bloch equations with the aid of the energy balance equation (2 12) In the next section we will see why an equation for ED lS necessary, when we c o m p a r e the results of the present section with earlier results
4. The lineshape In the previous section we have written equations for I,(t), It(t) and ED(t) (the expectation value of the secular part of the dipolar energy) F r o m these equations a formal expression for the r e s o n a n c e absorbtion lineshape will be obtained in this section This formal expression connects with the existing theory for r e s o n a n c e lineshapes in t e r m s of the Morl m e m o r y funchon8'9). We will consider the p r o b l e m of solving eq. (3 24) in the limit of small hf field to~(t) This limit was already adopted in section 3 It implies that I i will be hnear in tot, whereas the magnitude of Iz and ED IS not coupled to the amplitude of tot The time derivative o f / 1 is also linear in to~ and that of Iz and ED is proportional to the square of tot. This gives occasion to a two time scale b e h a v i o u r for hmltingly small to~. a rapid time scale in which/~ and ED (and also K 3 and K 4) are constant whereas I± and to~ vary In the slow time scale variations of It and E o and of K 3 and K 4 a r e seen, the averaged rate of change of these four quantities being proportional to the averaged square of to~. We can therefore adopt a solution p r o c e d u r e for these equations in the small tot limit in which first one solves (3 24a) for I.(t) with fixed K 3, K 4, It and ED. Subsequently this solution I±(t) is inserted in the equations for Iz and ED (3.24b-c).
MAGNETIC RESONANCE SATURATION IN SOLIDS
309
First, then, one must solve Ii from I i ( t ) = ~o[n(t)ez × l i ( t ) + t o l ( t ) × I t ( t ) t
- f d~'(ll(r) + ~ K 4 ( r ) g ± ( r ) t o l ( t ) ) r l t ± ± ( t ,
(1)
r).
0
The hme dependence of to[n, lz, K 4, g± and q¢~ due to the time dependence o f K 3 and K 4 c a n be ignored here, because it occurs only in the slow time scale (1) is a matrix equation which can be dlagonahzed with the introduction of L = Ix + ily and L = I x - 1Ir. The latter being the complex conjugate of L we will only consider L : t
I+(t) = I o ~ ( t ) I + ( t )
- IoJl+(t)Iz(t) - [ d r ( L ( r ) 0 ext
+ BK4(r)g±(r)~ol+(r))Gt(t
(2)
- r ) e ' ~ " "~
with ~o~+= 6o~ + iwy This equation, because we ~gnore slow time scale time dependences of K 4 etc., is linear in L , oJ~ and time mvariant and can therefore be solved by means of Laplace transforms:
f dt e-S"-to~L(t)
Lt:s)--
(Re(s) > 0)
(3a)
to
and analogously o31+to(S) =
/ dt e-'"-~)o~l+(t)
(3b)
to
and
=f
(3c)
d t e-StGto(t),
0
which yields a Laplace transformed version of (2): - L ( O ) + s L , ( s ) = Io~ff( t ) L ( s ) - ioJ,+A s ) lz( t ) - (Lt(s)
+ flK4(t)gl(t)tbt+r(S))dt(s
- ito~x').
With the insertion [ 3 K 4 ( t ) g ± ( t ) ( t o [ n ( t ) - w[xt+ ( K 3 ( t ) / K 4 ( t ) ) = - I s ( t ) solution of (4) is
(4) for Iz the
310
L J DE HAAS, WTH WENCKEBACH ANDNJ POULIS
L t ( s ) = ~K4(t)g~(t) ((+ffw~fr(t) - ¢°~xt+ K3(t)/K4(t)) - Ot(s - 1w~t))o51+(s) s - l o ~ " ( t ) + O , ( s - I(D[if)
+
I+(0) S -- lm~tr(t) + Gt(s - 1o)~xt)
(5)
This can be mdentified with the n o t a h o n of the hnear response theory (w~th I+(0) = 0)"
L ( s ) = x++(s)cbl+(s)
(6)
One can apply the reverse Laplace t r a n s f o r m to the solution (6) and c o m p a r e it with the expressions for the time d e n v a t w e s of L as they can be obtained f r o m (2) for e g. a stepped field ¢ol+(w~+(t) = 0 for t < 0, ¢ol+(t) = oJj+ for t > 0) This yields expressions for the m o m e n t s of X++(5)
f d¢o X++(Iw)
ff~(to),
(7a)
oc
f dw
~ox++(lOJ)
= ioJ~ff(to)I~(to) - i ~K4( to) g ±( to)G~(O),
(7b)
zc
to which consequently only the imaginary part of A'++ contributes F r o m here one can go b a c k to the x and y c o m p o n e n t s of I i and introduce the f r e q u e n c y d e p e n d e n t susceptibility tensor through L(s) = ~(s)~,(s) This susceptibility can be e x p r e s s e d m X++" Xxx(S) = xy~(s), Xx,(S) = - x y x ( s ) ,
Xxx(S) = ~(x++(s) + x++(g)),
(8)
xx~(s) = ½(x++(s) - x + + ( ~ ) )
(a denoting the c o m p l e x conjugate of the n u m b e r a). When ¢o~xt is sufficiently large only positive ¢o will contribute significantly to the integrals in eqs. (7) and then it is possible to a p p r o x i m a t e these integrals by
2 1 do) X " O o ) = Iz(to),
(9a)
0
2 f dto X"(ito) = toL~(to)I~(to) - [3K4(to)g~(to)Gto(O), 0
(9b)
MAGNETIC RESONANCE SATURATION IN SOLIDS where X'x'x is minus the imaginary part of With the expression for Gt0(O),
/3K4glOt(O)
= ½(L, ILo(! -
Xxx(X'xx - IX'x'x, X'xx being
311 the real part)
Pt)ILoL),,
(9b) can be elaborated, w h i c h yields m agreement w i t h ref. 4)
f dto(to
totXt)x'(Ito)
3ED.
(10)
0
The explicit e x p r e s s i o n for X"x, i e. the imaginary part of the factor of w~+(s) m (5), is, again for large to~xt which allows us to neglect the conjugate term )? in (8)
,, • XxxOto)=-K4gi/3½[to
Re ¢~(ito - ItoLext) ( t o ext K / K 4) COL -~--toL - - I m ~ ( i t o -- I"t o L~x~.2 )l -t- [Re t_~(ito - ttoL ~xt)] 2 ext
(11)
This Is of the f o r m )(fxtd :
/1(~1' ( t o
- -
to~xt .+. to
KS~K4),
(12)
with
.¥'d = -~/3K4g±
Re a ( l t o
- - I t o Lext) t o
(12a)
[to - w~xt + I m t3(ito - ito~x')] 2 + [Re t3(iw - ito[xt)] 2 x~ ~s the ~maginary part of the susceptibility that one o b s e r v e s m the same s y s t e m when it is in thermal equilibrmm at a t e m p e r a t u r e / 3 ' = / 3 K 4 with an applied static field to~xt = K 3 / K 4 (still without non-secular dipolar interaction), when the f r e q u e n c y of the h.f field is shifted in c o r r e s p o n d e n c e with the static field The equations for the slow time scale b e h a w o u r of I~ and E0 (3.24) can be e x p r e s s e d m terms of this X~:
I~ = to~X'() (to - totxt + KS~K4),
(13a)
to
Eo
2 ,, (to[xt __ to)(to __ to[xt + K a / K 4) = toiX0 to
(13b)
This is consistent with the results of P r o v o t o r o v ~) for the high t e m p e r a t u r e a p p r o x i m a t i o n in which case neither toter nor q' depends on K 3 and K 4. A final r e m a r k can be m a d e concerning the usefulness of the f o r m a l i s m in calculating the lineshape. In view of the lineshapes that have been found experimentally, as depicted In fig. 1, one might fear that G takes a very complicated shape as a function of frequency. A priori this could of course be
312
L J DE HAAS, W T H WENCKEBACH A N D N J
POULIS
Fig I Resonance absorbtlon m CaF2 as a function of frequency (artlbrary units) at 57% polarization from ref 3
t r u e , b u t t h i s Is n o t n e c e s s a r y
one can explain a similar hneshape
with a
s t r u c t u r e f o r G as d r a w n in fig 2a a n d b T h e e s s e n c e o f s u c h a n e x p l a n a t i o n is t h a t t h e c e n t e r f r e q u e n c y
of
G(s-
I o ~ xt) d o e s n o t c o i n c i d e w i t h oJ~n, b u t
d i f f e r s f r o m it b y a n a m o u n t o f t h e o r d e r o f t h e w i d t h as a f u n c t i o n o f ~o o f G, o r m o r e . T h i s l e a d s to a l i n e s h a p e Lorentzian amphtude
in s h a p e
if G d o e s
as s h o w n
In fig
not very appreciably
2c
T h e l a r g e p e a k is
over
its w i d t h .
The
is
flK4g±/ReG(ItO~~ - ioj~xt)
a
b
f
Fig 2 Demonstration of the occurrence of a shouldered absorbtlOn hne using (4 12) when a Lorentzlan shaped G Is assumed, with its center shifted half of its width from ~o~n (a) and (b) show the imaginary and real parts of G as a function frequency in arbitrary umts and (c) shows the resulting absorbtlon hneshape m the same frequency scale
MAGNETIC RESONANCE SATURATION IN SOLIDS
313
and the w~dth Re GOto[~ -
l(~[xt).
5. Conclusion The f o r m a h s m presented in section 2 and elaborated m section 3 offers a d e s c n p h o n of m a g n e h c r e s o n a n c e saturation It is perhaps useful to e m p h a s i z e once more that the validity of this description is quite trivial but that is usefulness is only ensured if our a s s u m p h o n that ~ ' , ( t , ~-) vanishes for large ( t - T ) is valid. Only then can one expand ~ in toj, which gives significance to g2 and ~ independently, and then too the expression (3.20) as a lowest order a p p r o x i m a t i o n of ~ ' ~ ( t , z) can be used. A second r e m a r k can be made about the choice of the projection operator It is u n n e c e s s a r y to define this o p e r a t o r in terms of the same operators of which e x p e c t a h o n values must be calculated In fact in (3.20) a projection o p e r a t o r defined only with {Ix, Iy} as the set of A~'s would gwe the same results. This r e d u n d a n c y of P reflects the fact that our original interest in the f o r m a h s m of section was partly motivated by the need to have a description for more general non linearities in spin systems. With an elaborate p r o j e c h o n o p e r a t o r it is also possible to describe more complicated systems such as for e x a m p l e the one obtained when non secular terms and spin-lattice interactions are retained
Acknowledgements We wish to express our grahtude to Prof. H . W Capel en Dr. J.C. Verstelle for their interest and cooperation in the d e v e l o p m e n t of this p a p e r
Aphendix A
Derivation of equation
(2.9)
The L~ouville equation (2 2) is solved formally by
fi(t)=
O(t)~(t =O)O-l(t), d I)=-i~'(t)O,
U(t =0)= i.
By means of cyclic permutation in the trace expression for the expectation
314
L J DE H A A S , W T H
WENCKEBACH AND N J POULIS
value this enables us to write the Hetsenberg representation of (2 1)
A(t) = Tr fi(t = 0) U-~(t)fi, O ( T ) . The las'~ ~ t ~ , ~ p e ~ working on A.
z~av, 'ere ~ g ~ d ~
~ tt",~ ~t~'a~1 ~" a ~'cqye~T_¢'~'~
O-"tr)AO(t] =- u ( t ) A with
d U ( t ) = U(t) tL(t) dt
U(t--O)=l
and
With this notation the expression for the time-derivative of the expectation values of the set of variables {A.} can be written
d-d-A(t) = Tr p(t = O)U(t)[P, + (l - Pt)] IL(t)A dt The part of this formula with (1 - P t ) can be rephrased with the identity t
U(t)(1 - Pt) = [(1 -/9,=0) -
f
d r U(r)P, -d-i P,U;'(r)]Up(t),
0
which can be verified by differentiating left and right with respect to t (and applying (dMt) P,. = P,.(dMt)P,) When one writes the prolectmn operators explicitly with (2 4) and inserts the previous mdentlty this leads to A. = (Tr fi(t = 0)U(t)A~)g~(t)(A~. iL(t)fi~.) + T r t~(t = 0)(1 - P~=o)Uv(t)(l - Pt) IL(t)A. t
-f
d~'(Tr ~6(t = O)U(r)fi~v) d g~(~') 0
(A~, U~'('r)Up(t)(l - Pt) 1L(t)A.) (The factor ( 1 - P t) in eq. (1) has been squared here, keeping one factor together with iL(t)A, for notational symmetry). The second term vanishes, since [Tr t~(t)/}],=0 = [(B)ot],=0 for an arbitrary o p e r a t o r / } and consequently, i f / } = (1 - P,)/}', [Tr iS(t)(1 - P,)/3'],=o = [Tr t~(t)/3' - (/3')or -
AA.(t)g"~(t)(Afi~('r),/~')tlt=o = 0
For 13"= b~t}' ~ ? - P,) cL~t?ff~, t?tbs s ~ ' s
e~at ~
sc~"orrd ee,"m v~t',.s~'s.
MAGNETIC RESONANCE SATURATION IN SOLIDS
315
References I) B N Provotorov, Soviet Physics Jetp 14 (1962) 1126, JETP 41 (1961) 1582 2) A G Redfield, Phys Rev 98 (1955) 1787 3) A Abragam, M Chapelher, J F Jacqumot and M Goldman, J Magn Resonance 10 (1973) 322 4) M Goldman, J Magn Resonance 17 (1975) 393 5) M Goldman, Spin Temperature and Nuclear Magnetic Resonance m Sohds (Clarendon, Oxford, 1970) 6) A Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961) 7) See e g B J Berne and G D Harp, Advances m Chemical Physics Vol XVII (IntersclenceJohn Wdey, New York) p 63 8) F C Barreto and G F Retter, Phys Rev B6 (1972) 2555 9) D Klvelson and K Ogan, Adv Magn Resonance 7 (1974) 72