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On free induction decay in a spin system with three-spin interactions in solids
Physics Letters A 184 ( 1994) 290-296 North-Holland
PHYSICS
LETTERS
A
On free induction decay in a spin system with three-spin interactions in solids E.B. Fel’dman Institute of Chemical Physics in Chernogolovka, Academy of Sciences ofRussia, Chernogolovka 142432, Moscow Region, Russian Federation
Received 13 July 1993; revised manuscript received 17 November 1993; accepted for publication 18 November 1993 Communicated by J. Rouquet
A simple model for the free induction decay (FID) in a spin system with three-spin interactions is proposed. It is shown that the NMR line shape is asymmetric. The initial part of the FJD obeys a Gaussian law and its “tail” is exponential. A good agreement with the experimental data of Mefyod [ Pis’ma Zh. Eksp. Teor. Fiz. 55 ( 1992) 4121 is obtained.
1. Introduction The development of high resolution NMR in solids [ 1,2] has attracted attention to the problems of the dynamics of spins with rapidly oscillating interactions. Such conditions are realized, for example, in experiments with magic angle spinning (MAS) of a sample [ 11, in the Lee-Goldburg experiment [ 31, multipulse experiments [ I,2 1, etc. The essential feature of the behaviour of spins in these experiments is the possibility to describe spin dynamics under such conditions with an effective time-independent Hamiltonian. Methods to obtain an effective Hamiltonian Pff have been found earlier [ 4-6 1. Although dipole-dipole interactions (DDI ) of nuclear spins are of two-spin character, Pff is a three-spin Hamiltonian. It is well known [ 71 that the problem of a theoretical explanation of the free induction decay (FID) after a x/2-pulse of a r.f. field, perpendicular to a constant field (applied along the z-axis), is very hard to solve. However, there are several approaches which make it possible to calculate the FID of nuclear spins of 19F in CaF, monocrystal [ 8,9]. The theoretical results [ 8,9 ] are in good agreement with the experimental data [ lo]. The problem considered is essentially a more difficult one, because the Hamiltonian .Pff has a more complex structure than the DDI Hamiltonian. It was shown experimentally that the central line of the MAS NMR spectrum is a Lorentzian line. So, if one neglects the oscillations in FID, which are responsible for satellites, one can conclude that FID obeys an exponential law. Recently [ 111 the FID in the Lee-Goldburg experiment was investigated in detail. It turned out that the initial part of the FID (approximately until the signal decays to one half of its initial amplitude) is a Gaussian function and “the tail” of the FID is an exponential curve. There are no theoretical approaches for the explanation of the experimental results except the formal attempt of ref. [ 111 which was based on the method of memory functions [ 7 1. A simple model is suggested in this report to explain the above-mentioned experimental results. The model is based on considering the main elementary spin processes which occur in the system considered. A good agreement with the experimental data of ref. [ 111 was obtained.
290
03759601/94/S
07.00 Q 1994 Elsevier Science B.V. AlJ rights reserved. SSDI 0375-9601(93)E0921-V
Volume 184, number 3
PHYSICS LETTERS A
2. Effective Hamiltonian
IO January 1994
and a general expression for the FID
We will consider a system of nuclear spins (s= 4) with dipole-dipole lattice. The DDI Hamiltonian is given by
1 bij(3S,Szj_Si*Sj)
S&=
interactions
(DDI)
in some regular
(1)
)
icj
where b, is the DDI constant between the nuclei spins i and j, Sai ((Y=x, y, z) are the Pauli matrices of the nucleus i and y is the gyromagnetic ratio. In the Lee-Goldburg experiment the system is driven by a superposition of a constant magnetic field Ho and a non-resonant linear polarized magnetic field Hi cos( ot). It is necessary for an averaging of the DDI that a magic angle condition holds [ 1,2], i.e. Wl
Iwo-WI
=Jz,
where w. = yHo, w1 = yH,. The effective Hamiltonian form,
for this case is known
[ 12,13 1. For our aims we shall write it in the following
where SZ$ = Six f iSi,,, w,=
w:+(wg-w)2
(4)
and for the Lee-Goldburg experiment the z-axis is directed along w,. Hamiltonian (3) is a three-spin operator except the first term which is not essential in our consideration. The same is valid for the Hamiltonian of the MAS NMR pffMAS [ 1,2 1. However, there is an important difference between Hamiltonian ( 3 ) and p&s. While the Hamiltonian *As is invariant with respect to a 180” rotation of all spins around the x, y-axes (xlz, ylz) of the frame rotating with frequency w, Hamiltonian (3) changes its sign when a similar operation is performed. The function G(t) describing the FID can be represented as [ 1,2 ] G(t)=
Sp{S-
exp( -i.%‘4)S+
exp(iXDefft)} 7
sp{s+s-}
(5)
where S’ = Er Sg . Since for the MAS NMR experiment Heff= pEffAs is an invariant rotation of all spins around the x-axis at 180’) it is easy to obtain from ( 5 ) that G(t)=G(
-t)
with respect to a 180”
.
(6)
Hence, the shape of the central line of the MAS NMR is a symmetric function of the frequency [ 141. If at the time t =O the polarization was directed along the x-axis then the same is valid at any moment of time. A different situation occurs in the Lee-Goldburg experiment, in which relationship (6) is not valid. At t> 0 there are two components of the polarization, P, and Py, p = Sp{S, exp( - i%effi)SX exp ( iZtit)} X SPE]
>
p = Sp{S,exp( Y
-i%efft)SXexp(iX”fft)} SPE)
3
(7)
although P,,= 0 at t=O. The Fourier transform of (5), which determines the line shape in the Lee-Goldburg experiment [ 3 1, is then asymmetric. This unusual property can be explained with a simple four-spin model. Let us consider a four-spin system, in which spins are connected by the interaction 291
PHYSICS LETTERS A
Volume 184, number 3
10 January 1994
(8)
(*) , due to the spins 2, 3 and 4 at spin 1, is defined as representing the zzz-part of (4). The field h zz hi,L’ = L we
1
(2bkj + bkl )b,SjzSb
.
i#k#l
For the sake of simplicity we shall suppose that the constants of the DDI of all the spins are equal (bkj= b). Then one finds from ( 10) that hi:’ = $
e
.
(S,,S,,+S,,S,,+S,,S,,)
(10)
The field hii) is equal to - 3b2/2w, for six mutual orientations of the spins and to 9b2/2co, only for the two orientations when spins 2, 3, 4 have the same projections along the z-axis. So in this example the line shape function is asymmetric.
3. A model to calculate G(f) Let us introduce [ 91 a polarization density F( h, t) of spins which at time t are in a field h defined by Hamiltonian (3 ). Hamiltonian (3 ) commutes with S,. This is in accordance with the fact that one of the elementary spin processes in the system considered is the precession of spins in a longitudinal local dipole field. However, the last term of (3) is responsible for the change in the field h. It was shown [ 15 ] that the dynamics of the spin system with three-spin interactions is analogous to the dynamics of spins in liquids. This is so because the local dipole field (see, for example, (9)) is determined by z2 independent contributions, where z>> 1 is the number of neighbouring spins. The process of changing the field h can be considered as a spectral diffusion [ 16 ] in analogy with the consideration of such a process in liquids [ 7 1. The simplest equation which takes into account both spin precession and spectral diffusion is
aF(h, t) =ihF(h,
~
at
t) +AF(h,
t)
,
(11)
where the operator A describing the spectral diffusion [ 17 ] has the form
AF(k t)=&-‘(h)
a
&g(h)
Wk ah
t) >
(12)
where D is the coefficient of the spectral diffusion and g(h) is the NMR line shape which can be determined from
g(h)=
7 G(t) emih’dt . -co
(13)
The Gauss approximation of G(t), which is widely used for the case of two-spin DDI, is not good, because G(t) # G( - t) as was mentioned above. So shall assume that for three-spin interactions G(t) can be written as G(t)=exp(-+M2t2)(1-$M,P),
where Mz and MS are the second and the third moments of the NMR line. Now it is easy to find that 292
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Volume 184, number 3
PHYSICS LETTERS A
10 January 1994
Table 1 Orientation
t
[lOOI [1101
0.74 0.39 0.98
11111
g(h)=
&
s(h) exp( -h2/2M2)
(15)
and M’h+$h”. 2Mz
dh)=l-
(16) 2
Using eqs. we obtain
(ll),
WA T) =ixF(x, aT
~
(12),
2) +
formulae
d
-exp(fx’) 4(x)
(13)-(16)
$
and
4(x)
the dimensionless
aF(x
ew(-lx2) -&-
?)
parameters
x=h/&,
z=t&,
,
(17)
where B= D/LV:~. It is necessary to solve ( 17 ) when F(x, 0) = 1. Let us introduce a dimensionless parameter E= M3/Mk5. The values of M2, M3 for CaF, monocrystal were calculated (with Hamiltonian (3) ) in ref. [ 181. Calculated values of e are given in table 1. As one can see from table 1 t can be approximately equal to 1 (for the orientation [ 1111). However we shall assume that ECK 1 to obtain an approximate solution of ( 17 ). It will be shown that this solution is in good agreement with experimental results [ 111.
4. A solution of eq. (17) We shall start with e=O when eq. ( 17) can be written
aox,z) aT
Introducing E(x
,
a new function
7) =exp(
and Fourier
f(k,z)=
exp(-+x2)7
=ixF(x,T)+dexp(ix”)$
- jx’)F(x, transformating
as
aF( x. T)
.
(18)
P(x, T), 7)
(19) E(x, z),
sj &x,T)eikxdx, --co
(20)
one can rewrite eq. ( 18 ) in the form
(21) Equation (20))
(21) can be solved by the standard
procedure,
and the solution
satisfying
the initial condition
(see
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Volume 184, number 3
10 January 1994
PHYSICS LETTERS A
f(k,O)=fiexp(-tk*)
(22)
f(k,r)=@exp(1/02-k/d-
tk2)
exp{ - ( 1/D2) [6r+ (1 -Dk) e-DT]}
(23)
The observed FID [ 111 can be expressed through F(x, t) as co exp( - tx2)F(x,
z) dx=
Finally, one obtains G(r)=exp[-(l//62)(&+e-yj,)+1/62].
(25)
When Dr+z 1, it is easy to find from (25) that G(r)-exp(-fr’),
(26)
and if &>> 1 we have from (25) that G(z)=exp(
G(7)
is given by
1/62--/fi).
(27)
Thus G( 7) obeys a Gaussian law for &-=.+z1 and it obeys an exponential law for d7>> 1. This conclusion is in good agreement with experimental data from ref. [ 111 where it has been shown that the initial part of the FID (up to 50% from its values at r=O) is a Gaussian curve and “the tail” of the FID is exponential. It is also important to point out an analogy between the developed theory and the theory describing the effect of molecular motions on NMR spectra [ 19 1. The developed approach is sufficient for the explanation of the MAS NMR experiment [ 141. The exponential FID [ 141 is a consequence of the condition 6~ 1 for this experiment. But the approach is not sufficient for the Lee-Goldburg experiment when the shape of the central resonance line is asymmetric. Now we take into account the effect of the asymmetry of g(h). We shall try to find the approximate solution of eq. ( 17), F(x, r), in the form (28)
F(x, z) =Fo(x, r) +@I (x, 7) , where F0 (x, 7) is the solution of ( 17 ) when E= 0 and FO(x, r) can be written as F,(x,
z)
=exp $ (
e-2I57 exp[i( - 6 - $ eeD7+ m >
1-e-oT)x/d]
.
Substituting (28 ) into eq. ( 17 ) and equating terms of order E in both parts of ( 17 ) one obtains that F1 (x, T) satisfies the equation
@I (x, r.) =ixF,(x,T)+dexp(fx')&
a7
~FI(x, r) +td(X2-1) ax
>
aFo(x, 7) ax .
(30)
An analysis of eq. ( 30) is implemented by a method absolutely analogous to the case of E= 0. The contribution of (EF,(x, T) to the observed FID [ 111 can be written as
294
Volume 184, number 3
PHYSICS LETTERS A
co
fz
10 January 1994
7’
I
exP(-jx’)Ft(X,r’)
dx=-fe
5
eXP[q%(y(u’,
z’), (u’+r’)/26)
r’))]@o(P(u’,
du’
0
--oo -T’ +te
5 0
ew[nWu’,
7’)l@o(~(d,
7’),
(u’+7’)/26)
du’,
(31)
where g,(x/b)=h
(-x-$x2-ln]l-xl),
y/(z)=iF(O,T)
9,
@0(x, y) Now
y/z-r(k+
G(7)
(32) y(u,
7)=
1-exy-7)l
y)lty(7)exp[ -J(k+
can be calculated
,
(33)
FYI.
(34)
up to terms of the order e inclusively.
The result is
G(7)=exp[-(l/~2)(b7+e-dT)+1/62]-~it~(7)~3(7)exp[-~~2(7)] & exp[Q,(Y(u’,b7))]@o(y(u’,b7),
-
s
exp[~,,(y(u’,d7))l~O(y(u’,d7),
(u’+d7)/26)
W+W/W
du’
du’ >
-a
,
(35)
where K(T)=
!g,
e-2Lir
X(z)=exp
&
- i
- $e-“+
.
262
(
>
(36)
The main feature of (35 ) is the appearance of imaginary terms which lead to a change in the phase of G( 7). It means that the y-component of the polarization of the system is not zero although it was equal to 0 at 7=0. This circumstance was mentioned in an experimental work [ 111. The FID for the case D7 -SC 1 is Gaussian and for DZB 1 it is exponential. This conclusion is also in accordance with ref. [ 111. Thus, the developed theory explains the spin dynamics in the system with three-spin interactions.
Acknowledgement The work was supported,
in part, by a Sloan Foundation
Grant awarded by the American
Physical Society.
References [ 11 U. Haeberlen, High-resolution NMR in solids (Academic Press, New York, 1976). [2] M. Mehring, High-resolution NMR spectroscopy of solids (Springer, Berlin, 1976). [3] M. Lee and W.I. Goldburg, Phys. Rev. 140 (1965) A1261. [4] U. Haeberlen and J.S. Waugh, Phys. Rev. 175 ( 1968) 453.
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10 January 1994
[ 51 B.N. Provotorov and E.B. Fel’dman, Zh. Eksp. Teor. Fiz. 79 (1980) 2206. [ 61 V.L. Bodneva, A.A. Milyutin and E.B. Fel’dman, Zh. Eksp. Teor. Fiz. 92 ( 1987) 1376. [ 7 ] A. Abragam, The principles of nuclear magnetism (Clarendon, Oxford, 1961) . [ 81 K.W. Becker, T. Plefka and G. Sauermann, J. Phys. C 9 ( 1976) 4041. [ 91 G.E. Kamaukh, A.A. Lundin, B.N. Provotorov and K.T. Summanen, Zh. Eksp. Teor. Fiz. 91 (1986) 2229. [lo] I.J. Lowe and R.E. Norberg, Phys. Rev. 107 (1957) 46. [ 111 A.E. Mefyod, A.V. Jaroslavtsev, V.E. Zobov, A.V. Ponomarenko and M.A. Popov, Pis’ma Zh. Eksp. Teor. Fiz. 55 (1992) 412. [ 121 V.A. Atsarkin, A.E. Mefyod and M.I. Rodak, Fiz. Tverd. Tela 21 (1979) 2672. [ 131 V.E. Zobov and A.V. Ponomarenko, preprint no. 657cP, L.V. Kirensky Institute of Physics, Krasnojarsk ( 1990). [ 14) M.A. Alla, E.T. Lippmaa, B.N. Provotorov and E.P. Rull, in: Abstr. V. Allunion Conf. on New methods of NMR and ESR in solid state chemistry (Chemogolovka, 1990) p. 13. [ 151 B.N. Provotorov and E.B. Fel’dman, Zh. Eksp. Teor. Phys., in press. [ 161 K.M. Salikhov, A.G. Semenov and Yu.D. Tsvetkov, Electron spin echo and its application (Nauka, Novosibirsk, 1976). [ 171 E.B. Fel’dman and B.N. Provotorov, in: Abstracts 26th Congress Ampere (Athens, 1992) p. 558. [ 18 ] V.E. Zobov and M.A. Popov, Zh. Eksp. Teor. Fiz., in press. [ 191 P.W. Anderson and P.R. Weiss, Rev. Mod. Phys. 25 (1954) 316.
296
PHYSICS
LETTERS
A
On free induction decay in a spin system with three-spin interactions in solids E.B. Fel’dman Institute of Chemical Physics in Chernogolovka, Academy of Sciences ofRussia, Chernogolovka 142432, Moscow Region, Russian Federation
Received 13 July 1993; revised manuscript received 17 November 1993; accepted for publication 18 November 1993 Communicated by J. Rouquet
A simple model for the free induction decay (FID) in a spin system with three-spin interactions is proposed. It is shown that the NMR line shape is asymmetric. The initial part of the FJD obeys a Gaussian law and its “tail” is exponential. A good agreement with the experimental data of Mefyod [ Pis’ma Zh. Eksp. Teor. Fiz. 55 ( 1992) 4121 is obtained.
1. Introduction The development of high resolution NMR in solids [ 1,2] has attracted attention to the problems of the dynamics of spins with rapidly oscillating interactions. Such conditions are realized, for example, in experiments with magic angle spinning (MAS) of a sample [ 11, in the Lee-Goldburg experiment [ 31, multipulse experiments [ I,2 1, etc. The essential feature of the behaviour of spins in these experiments is the possibility to describe spin dynamics under such conditions with an effective time-independent Hamiltonian. Methods to obtain an effective Hamiltonian Pff have been found earlier [ 4-6 1. Although dipole-dipole interactions (DDI ) of nuclear spins are of two-spin character, Pff is a three-spin Hamiltonian. It is well known [ 71 that the problem of a theoretical explanation of the free induction decay (FID) after a x/2-pulse of a r.f. field, perpendicular to a constant field (applied along the z-axis), is very hard to solve. However, there are several approaches which make it possible to calculate the FID of nuclear spins of 19F in CaF, monocrystal [ 8,9]. The theoretical results [ 8,9 ] are in good agreement with the experimental data [ lo]. The problem considered is essentially a more difficult one, because the Hamiltonian .Pff has a more complex structure than the DDI Hamiltonian. It was shown experimentally that the central line of the MAS NMR spectrum is a Lorentzian line. So, if one neglects the oscillations in FID, which are responsible for satellites, one can conclude that FID obeys an exponential law. Recently [ 111 the FID in the Lee-Goldburg experiment was investigated in detail. It turned out that the initial part of the FID (approximately until the signal decays to one half of its initial amplitude) is a Gaussian function and “the tail” of the FID is an exponential curve. There are no theoretical approaches for the explanation of the experimental results except the formal attempt of ref. [ 111 which was based on the method of memory functions [ 7 1. A simple model is suggested in this report to explain the above-mentioned experimental results. The model is based on considering the main elementary spin processes which occur in the system considered. A good agreement with the experimental data of ref. [ 111 was obtained.
290
03759601/94/S
07.00 Q 1994 Elsevier Science B.V. AlJ rights reserved. SSDI 0375-9601(93)E0921-V
Volume 184, number 3
PHYSICS LETTERS A
2. Effective Hamiltonian
IO January 1994
and a general expression for the FID
We will consider a system of nuclear spins (s= 4) with dipole-dipole lattice. The DDI Hamiltonian is given by
1 bij(3S,Szj_Si*Sj)
S&=
interactions
(DDI)
in some regular
(1)
)
icj
where b, is the DDI constant between the nuclei spins i and j, Sai ((Y=x, y, z) are the Pauli matrices of the nucleus i and y is the gyromagnetic ratio. In the Lee-Goldburg experiment the system is driven by a superposition of a constant magnetic field Ho and a non-resonant linear polarized magnetic field Hi cos( ot). It is necessary for an averaging of the DDI that a magic angle condition holds [ 1,2], i.e. Wl
Iwo-WI
=Jz,
where w. = yHo, w1 = yH,. The effective Hamiltonian form,
for this case is known
[ 12,13 1. For our aims we shall write it in the following
where SZ$ = Six f iSi,,, w,=
w:+(wg-w)2
(4)
and for the Lee-Goldburg experiment the z-axis is directed along w,. Hamiltonian (3) is a three-spin operator except the first term which is not essential in our consideration. The same is valid for the Hamiltonian of the MAS NMR pffMAS [ 1,2 1. However, there is an important difference between Hamiltonian ( 3 ) and p&s. While the Hamiltonian *As is invariant with respect to a 180” rotation of all spins around the x, y-axes (xlz, ylz) of the frame rotating with frequency w, Hamiltonian (3) changes its sign when a similar operation is performed. The function G(t) describing the FID can be represented as [ 1,2 ] G(t)=
Sp{S-
exp( -i.%‘4)S+
exp(iXDefft)} 7
sp{s+s-}
(5)
where S’ = Er Sg . Since for the MAS NMR experiment Heff= pEffAs is an invariant rotation of all spins around the x-axis at 180’) it is easy to obtain from ( 5 ) that G(t)=G(
-t)
with respect to a 180”
.
(6)
Hence, the shape of the central line of the MAS NMR is a symmetric function of the frequency [ 141. If at the time t =O the polarization was directed along the x-axis then the same is valid at any moment of time. A different situation occurs in the Lee-Goldburg experiment, in which relationship (6) is not valid. At t> 0 there are two components of the polarization, P, and Py, p = Sp{S, exp( - i%effi)SX exp ( iZtit)} X SPE]
>
p = Sp{S,exp( Y
-i%efft)SXexp(iX”fft)} SPE)
3
(7)
although P,,= 0 at t=O. The Fourier transform of (5), which determines the line shape in the Lee-Goldburg experiment [ 3 1, is then asymmetric. This unusual property can be explained with a simple four-spin model. Let us consider a four-spin system, in which spins are connected by the interaction 291
PHYSICS LETTERS A
Volume 184, number 3
10 January 1994
(8)
(*) , due to the spins 2, 3 and 4 at spin 1, is defined as representing the zzz-part of (4). The field h zz hi,L’ = L we
1
(2bkj + bkl )b,SjzSb
.
i#k#l
For the sake of simplicity we shall suppose that the constants of the DDI of all the spins are equal (bkj= b). Then one finds from ( 10) that hi:’ = $
e
.
(S,,S,,+S,,S,,+S,,S,,)
(10)
The field hii) is equal to - 3b2/2w, for six mutual orientations of the spins and to 9b2/2co, only for the two orientations when spins 2, 3, 4 have the same projections along the z-axis. So in this example the line shape function is asymmetric.
3. A model to calculate G(f) Let us introduce [ 91 a polarization density F( h, t) of spins which at time t are in a field h defined by Hamiltonian (3 ). Hamiltonian (3 ) commutes with S,. This is in accordance with the fact that one of the elementary spin processes in the system considered is the precession of spins in a longitudinal local dipole field. However, the last term of (3) is responsible for the change in the field h. It was shown [ 15 ] that the dynamics of the spin system with three-spin interactions is analogous to the dynamics of spins in liquids. This is so because the local dipole field (see, for example, (9)) is determined by z2 independent contributions, where z>> 1 is the number of neighbouring spins. The process of changing the field h can be considered as a spectral diffusion [ 16 ] in analogy with the consideration of such a process in liquids [ 7 1. The simplest equation which takes into account both spin precession and spectral diffusion is
aF(h, t) =ihF(h,
~
at
t) +AF(h,
t)
,
(11)
where the operator A describing the spectral diffusion [ 17 ] has the form
AF(k t)=&-‘(h)
a
&g(h)
Wk ah
t) >
(12)
where D is the coefficient of the spectral diffusion and g(h) is the NMR line shape which can be determined from
g(h)=
7 G(t) emih’dt . -co
(13)
The Gauss approximation of G(t), which is widely used for the case of two-spin DDI, is not good, because G(t) # G( - t) as was mentioned above. So shall assume that for three-spin interactions G(t) can be written as G(t)=exp(-+M2t2)(1-$M,P),
where Mz and MS are the second and the third moments of the NMR line. Now it is easy to find that 292
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Volume 184, number 3
PHYSICS LETTERS A
10 January 1994
Table 1 Orientation
t
[lOOI [1101
0.74 0.39 0.98
11111
g(h)=
&
s(h) exp( -h2/2M2)
(15)
and M’h+$h”. 2Mz
dh)=l-
(16) 2
Using eqs. we obtain
(ll),
WA T) =ixF(x, aT
~
(12),
2) +
formulae
d
-exp(fx’) 4(x)
(13)-(16)
$
and
4(x)
the dimensionless
aF(x
ew(-lx2) -&-
?)
parameters
x=h/&,
z=t&,
,
(17)
where B= D/LV:~. It is necessary to solve ( 17 ) when F(x, 0) = 1. Let us introduce a dimensionless parameter E= M3/Mk5. The values of M2, M3 for CaF, monocrystal were calculated (with Hamiltonian (3) ) in ref. [ 181. Calculated values of e are given in table 1. As one can see from table 1 t can be approximately equal to 1 (for the orientation [ 1111). However we shall assume that ECK 1 to obtain an approximate solution of ( 17 ). It will be shown that this solution is in good agreement with experimental results [ 111.
4. A solution of eq. (17) We shall start with e=O when eq. ( 17) can be written
aox,z) aT
Introducing E(x
,
a new function
7) =exp(
and Fourier
f(k,z)=
exp(-+x2)7
=ixF(x,T)+dexp(ix”)$
- jx’)F(x, transformating
as
aF( x. T)
.
(18)
P(x, T), 7)
(19) E(x, z),
sj &x,T)eikxdx, --co
(20)
one can rewrite eq. ( 18 ) in the form
(21) Equation (20))
(21) can be solved by the standard
procedure,
and the solution
satisfying
the initial condition
(see
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PHYSICS LETTERS A
f(k,O)=fiexp(-tk*)
(22)
f(k,r)=@exp(1/02-k/d-
tk2)
exp{ - ( 1/D2) [6r+ (1 -Dk) e-DT]}
(23)
The observed FID [ 111 can be expressed through F(x, t) as co exp( - tx2)F(x,
z) dx=
Finally, one obtains G(r)=exp[-(l//62)(&+e-yj,)+1/62].
(25)
When Dr+z 1, it is easy to find from (25) that G(r)-exp(-fr’),
(26)
and if &>> 1 we have from (25) that G(z)=exp(
G(7)
is given by
1/62--/fi).
(27)
Thus G( 7) obeys a Gaussian law for &-=.+z1 and it obeys an exponential law for d7>> 1. This conclusion is in good agreement with experimental data from ref. [ 111 where it has been shown that the initial part of the FID (up to 50% from its values at r=O) is a Gaussian curve and “the tail” of the FID is exponential. It is also important to point out an analogy between the developed theory and the theory describing the effect of molecular motions on NMR spectra [ 19 1. The developed approach is sufficient for the explanation of the MAS NMR experiment [ 141. The exponential FID [ 141 is a consequence of the condition 6~ 1 for this experiment. But the approach is not sufficient for the Lee-Goldburg experiment when the shape of the central resonance line is asymmetric. Now we take into account the effect of the asymmetry of g(h). We shall try to find the approximate solution of eq. ( 17), F(x, r), in the form (28)
F(x, z) =Fo(x, r) +@I (x, 7) , where F0 (x, 7) is the solution of ( 17 ) when E= 0 and FO(x, r) can be written as F,(x,
z)
=exp $ (
e-2I57 exp[i( - 6 - $ eeD7+ m >
1-e-oT)x/d]
.
Substituting (28 ) into eq. ( 17 ) and equating terms of order E in both parts of ( 17 ) one obtains that F1 (x, T) satisfies the equation
@I (x, r.) =ixF,(x,T)+dexp(fx')&
a7
~FI(x, r) +td(X2-1) ax
>
aFo(x, 7) ax .
(30)
An analysis of eq. ( 30) is implemented by a method absolutely analogous to the case of E= 0. The contribution of (EF,(x, T) to the observed FID [ 111 can be written as
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Volume 184, number 3
PHYSICS LETTERS A
co
fz
10 January 1994
7’
I
exP(-jx’)Ft(X,r’)
dx=-fe
5
eXP[q%(y(u’,
z’), (u’+r’)/26)
r’))]@o(P(u’,
du’
0
--oo -T’ +te
5 0
ew[nWu’,
7’)l@o(~(d,
7’),
(u’+7’)/26)
du’,
(31)
where g,(x/b)=h
(-x-$x2-ln]l-xl),
y/(z)=iF(O,T)
9,
@0(x, y) Now
y/z-r(k+
G(7)
(32) y(u,
7)=
1-exy-7)l
y)lty(7)exp[ -J(k+
can be calculated
,
(33)
FYI.
(34)
up to terms of the order e inclusively.
The result is
G(7)=exp[-(l/~2)(b7+e-dT)+1/62]-~it~(7)~3(7)exp[-~~2(7)] & exp[Q,(Y(u’,b7))]@o(y(u’,b7),
-
s
exp[~,,(y(u’,d7))l~O(y(u’,d7),
(u’+d7)/26)
W+W/W
du’
du’ >
-a
,
(35)
where K(T)=
!g,
e-2Lir
X(z)=exp
&
- i
- $e-“+
.
262
(
>
(36)
The main feature of (35 ) is the appearance of imaginary terms which lead to a change in the phase of G( 7). It means that the y-component of the polarization of the system is not zero although it was equal to 0 at 7=0. This circumstance was mentioned in an experimental work [ 111. The FID for the case D7 -SC 1 is Gaussian and for DZB 1 it is exponential. This conclusion is also in accordance with ref. [ 111. Thus, the developed theory explains the spin dynamics in the system with three-spin interactions.
Acknowledgement The work was supported,
in part, by a Sloan Foundation
Grant awarded by the American
Physical Society.
References [ 11 U. Haeberlen, High-resolution NMR in solids (Academic Press, New York, 1976). [2] M. Mehring, High-resolution NMR spectroscopy of solids (Springer, Berlin, 1976). [3] M. Lee and W.I. Goldburg, Phys. Rev. 140 (1965) A1261. [4] U. Haeberlen and J.S. Waugh, Phys. Rev. 175 ( 1968) 453.
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10 January 1994
[ 51 B.N. Provotorov and E.B. Fel’dman, Zh. Eksp. Teor. Fiz. 79 (1980) 2206. [ 61 V.L. Bodneva, A.A. Milyutin and E.B. Fel’dman, Zh. Eksp. Teor. Fiz. 92 ( 1987) 1376. [ 7 ] A. Abragam, The principles of nuclear magnetism (Clarendon, Oxford, 1961) . [ 81 K.W. Becker, T. Plefka and G. Sauermann, J. Phys. C 9 ( 1976) 4041. [ 91 G.E. Kamaukh, A.A. Lundin, B.N. Provotorov and K.T. Summanen, Zh. Eksp. Teor. Fiz. 91 (1986) 2229. [lo] I.J. Lowe and R.E. Norberg, Phys. Rev. 107 (1957) 46. [ 111 A.E. Mefyod, A.V. Jaroslavtsev, V.E. Zobov, A.V. Ponomarenko and M.A. Popov, Pis’ma Zh. Eksp. Teor. Fiz. 55 (1992) 412. [ 121 V.A. Atsarkin, A.E. Mefyod and M.I. Rodak, Fiz. Tverd. Tela 21 (1979) 2672. [ 131 V.E. Zobov and A.V. Ponomarenko, preprint no. 657cP, L.V. Kirensky Institute of Physics, Krasnojarsk ( 1990). [ 14) M.A. Alla, E.T. Lippmaa, B.N. Provotorov and E.P. Rull, in: Abstr. V. Allunion Conf. on New methods of NMR and ESR in solid state chemistry (Chemogolovka, 1990) p. 13. [ 151 B.N. Provotorov and E.B. Fel’dman, Zh. Eksp. Teor. Phys., in press. [ 161 K.M. Salikhov, A.G. Semenov and Yu.D. Tsvetkov, Electron spin echo and its application (Nauka, Novosibirsk, 1976). [ 171 E.B. Fel’dman and B.N. Provotorov, in: Abstracts 26th Congress Ampere (Athens, 1992) p. 558. [ 18 ] V.E. Zobov and M.A. Popov, Zh. Eksp. Teor. Fiz., in press. [ 191 P.W. Anderson and P.R. Weiss, Rev. Mod. Phys. 25 (1954) 316.
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