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Free-induction decay in a magnetically dilute solid
Solid State Communications, Vol. 5, pp. 935-937, 1967. Pergamon Press Ltd. Printed in Great Britain
FREE-INDUCTION DECAY IN A MAGNETICALLY DILUTE SOLID* T.M. Wu Department of Physics and Condensed State Center, Case Western Reserve University, Cleveland, Ohio, U. S. A. (Received 2 October 1967)
The free-induction decay of a magnetically dilute system with spin 1/2 is evaluated using the Abragam Ansatz, LoweNorberg method and Evans-Powles method separately. We find that the results obtained by Lowe-Norberg theory disagree with those of Abragam Ansatz or Evan-Powles theory. This disagreement is believed to be due to an inadequate expansion for F(t).
NUCLEAR magnetic resonance line shape provides information about the nuclear spin correlations. In liquids and gases these correlations
distance between spin sites with gyromagnetic ratio v, and Gjj is the angle nj makes with the large static magnetic field. H0 and H1 corres-
ultimately arise from the nuclear motion and reflect the space-time correlation of nuclei. 1 However, in solids at temperatures below the activation temperature for nuclear diffusion, where the spatial order is maintained, the line shape provides arising from nuclear a probe spin-spin of the spin interaction. correlations In the field temperature region of interest, the thermal energy is much greater than the Zeeman energy. Under this condition, the Fourier transform of the line shape is equal2 to
pond to the first and second terms in H respectively.
~.
r
~ .i~~
,~ / ~ , /
r
If the line shape were Gaussian, the free induction decay would be Gaussian too. How4 shows ever, a beat free structure, induction which decay is often measurement referred to as Lowe-Norberg beats. Here we investigate theoretically the free-induction decay in a magnetically dilute system. For simplicity we consider an isotropically disordered cubic lattice having a fraction p of the lattice sites occupied by spin 1/2 nuclei and the remaining sites occupied by spin zero nuclei. The basic question is, how does the amplitude of the beats change with p ?
~2 1C~L ,
.
which describes the relaxation of the component of magnetization perpendicular to the applied static magnetic field. This relaxation is called the free Induction decay.
=
E
.
Knowing from the experiment that essentially equally spaced nulls occur in the free induction decay, Abragam suggested2 that the free induction decay (envelope of the Fourier transform of the line shape) could be approximately represented by the form
For the dipolar interaction we use Van Vieck’s truncated Hamiltonian,3 H
.
B 1J(S~ S~ - 1/3
=
~
+ ~1
(1)
F(t)
2 (1-3 cos2 eij)/r~j;nj is the where Blj = 3/2 y’ ~Work supported by the U. S. Air Force Office of Scientific Research through Grant No. AFOSR 565-66.
=
e~2t2~I4sjn(bt)
E”Abragam Ansatz”]
(2)
where a and b are constants to be determined. Expanding this about t = 0, this becomes 935
FREE-INDUCTION
936
Vol. 5, No. 12
DECAY
2 P(t)
= 1 - &
(a2
+ $1
(3) +t
4
(3a4
+ 2rI2b2 +$+
7.
. ..
Using the known second and fourth moments of the line shape’, M, , M1 and the relations 2 M2 = P2 + $
-.4 0
I
I .a
I
(4) M4 =
3a4+
2a2b2
I 1.6
I
I 24
I
32
t
+ g b4
to fit a and b, the extrapolation is obtained. The remarkable fit of this for the case CaF, to the experimental data* 6 with p = 1 and the static magnetic field along the [lo03 axis is shown in Fig. 1 (a). We notice that the initial beat amplitude decreases as p decreases. Extending the original Lowe-Norberg theory4 to the case of a magnetically dilute system, the free induction decay can be expressed as
F(P,t)
=
U(p,t)
=
U(p,t)
l-l
(5)
V(p,t)
10-p)
+ P CO6 (6)
JZi
V(p,t)
{[ C
8,jt
=
2 1 -
5
tan (
jfi
yj12-
T:
[Bijt
tan
Bijt 2I
( -+I
-.40'1
Ifi
(7)
E: j+i
(6ijt --T-l
3
tan
CU)
2
FIG.
1
Free induction decay shapes for F1’ in CaF, for static magnetic field along [IO0 J axis. (a) Abragam’s Ansatz, (b) Lowe-Norberg method, (c) Evans-Powles method. The unit of time is 3a3 /2y” h where a = 2.725 i is the lattice spacing. The circles are the experimental results of Barnaal and Lowe.
Vol. 5, No. 12
FREE-INDUCTION DECAY
The decay curve is numerically evaluated for the first 32 near neighbor interactions. We find that the amplitude of the initial beat is enhanced as p decreases in the region 1 > p > 0. 5, as shown in Fig. 1(b).
The plot in Fig. 1(c) (using 32 nearest neighbors) shows that the amplitude of the beat decreases as p decreases, in good agreement with the Abragam Ansatz. Finally, we should like to mention that the results obtained by using the Lowe-Norberg theory do not agree with those obtained by Abragam’s Ansatz or by the Evans-Powles method, which we believe is a better method. The reason is, as pointed out by many authors, 2 ~ that the expansion of F(t) in Lowe-Norberg’s theory is arbitrary and diverges for times not much longer than those for which it was computed.
Recently Evans and Powles” derived the free induction decay formula by using a Dyson type expansion in power of H1. Their result gave good agreement with experiment. Adapting this method to a magnetically dilute system, we get the free induction decay as F(p,t)
=
F(p,t)
+ F1(t) +
.
B F (p,t) 0
=
U
Acknowledgment Dr. D. J. Scalapino - The for author helpfulwishes discussions to thank and
t
(—q—)I
+ p cos
[(1-p)
(8)
.
With
937
(9)
suggestions.
(—!f_) St
F1(p,t)
—f
=
t
B1
j
sin
B [t
U
((1-p) + p cos
t
(J~L..)) 2 (10)
t dt’
U
[(1-p)
+ p cos
[Bki(tt) 2
+
Bkj 2
tj1
References 1.
BLOEMBERGEN N.,
PURCELL E.M. and POUND R.V.,
Phys. Rev. 73, 679 (1948).
2.
ABRAGAM A.,
3.
VAN VLECK J.H.,
4.
LOWE I. J. and NORBERG R. E.,
5.
KITTELL C. and ABRAHAMS E., Phys. Rev. 90, 238 (1953). Their result of fourth moment for magnetic 2Mfield oriented along the [100] crystaYaxis has a small numerical error. It should be M4(p) = 3p 2(1) (0.63 + .2.2.L ). p
6.
BARNAAL D. and LOWE I.J.,
7.
EVANS W. A. B.
The Principles of Nuclear Magnetism, Clarendon Press, Oxford (1961). Phys. Rev.
74, 1168 (1948). Phys. Rev. 107, 46 (1957).
Phys. Rev. 148, 328 (1966).
and POWLES J. G.,
Phys. Letters 24A, 218 (1967).
Das Blochabklingen den freien magnetischen Induktion eines verdtlnnten magnetischen Spin 1/2 Gitters wird auf drei Arten berechnet: Die Resultate der (1) Lowe-Norbergtheorie stimmen mit denen des (2) Abragamansatzes und denen der (3) Evan-Powlestheorie nich fiberein. Dies Nichtübereinstimmung scheint auf eine unvollst~ndigeEntwicklung der F(t)-F’unktjon in (1) zuri~ckführbarzu sein.
FREE-INDUCTION DECAY IN A MAGNETICALLY DILUTE SOLID* T.M. Wu Department of Physics and Condensed State Center, Case Western Reserve University, Cleveland, Ohio, U. S. A. (Received 2 October 1967)
The free-induction decay of a magnetically dilute system with spin 1/2 is evaluated using the Abragam Ansatz, LoweNorberg method and Evans-Powles method separately. We find that the results obtained by Lowe-Norberg theory disagree with those of Abragam Ansatz or Evan-Powles theory. This disagreement is believed to be due to an inadequate expansion for F(t).
NUCLEAR magnetic resonance line shape provides information about the nuclear spin correlations. In liquids and gases these correlations
distance between spin sites with gyromagnetic ratio v, and Gjj is the angle nj makes with the large static magnetic field. H0 and H1 corres-
ultimately arise from the nuclear motion and reflect the space-time correlation of nuclei. 1 However, in solids at temperatures below the activation temperature for nuclear diffusion, where the spatial order is maintained, the line shape provides arising from nuclear a probe spin-spin of the spin interaction. correlations In the field temperature region of interest, the thermal energy is much greater than the Zeeman energy. Under this condition, the Fourier transform of the line shape is equal2 to
pond to the first and second terms in H respectively.
~.
r
~ .i~~
,~ / ~ , /
r
If the line shape were Gaussian, the free induction decay would be Gaussian too. How4 shows ever, a beat free structure, induction which decay is often measurement referred to as Lowe-Norberg beats. Here we investigate theoretically the free-induction decay in a magnetically dilute system. For simplicity we consider an isotropically disordered cubic lattice having a fraction p of the lattice sites occupied by spin 1/2 nuclei and the remaining sites occupied by spin zero nuclei. The basic question is, how does the amplitude of the beats change with p ?
~2 1C~L ,
.
which describes the relaxation of the component of magnetization perpendicular to the applied static magnetic field. This relaxation is called the free Induction decay.
=
E
.
Knowing from the experiment that essentially equally spaced nulls occur in the free induction decay, Abragam suggested2 that the free induction decay (envelope of the Fourier transform of the line shape) could be approximately represented by the form
For the dipolar interaction we use Van Vieck’s truncated Hamiltonian,3 H
.
B 1J(S~ S~ - 1/3
=
~
+ ~1
(1)
F(t)
2 (1-3 cos2 eij)/r~j;nj is the where Blj = 3/2 y’ ~Work supported by the U. S. Air Force Office of Scientific Research through Grant No. AFOSR 565-66.
=
e~2t2~I4sjn(bt)
E”Abragam Ansatz”]
(2)
where a and b are constants to be determined. Expanding this about t = 0, this becomes 935
FREE-INDUCTION
936
Vol. 5, No. 12
DECAY
2 P(t)
= 1 - &
(a2
+ $1
(3) +t
4
(3a4
+ 2rI2b2 +$+
7.
. ..
Using the known second and fourth moments of the line shape’, M, , M1 and the relations 2 M2 = P2 + $
-.4 0
I
I .a
I
(4) M4 =
3a4+
2a2b2
I 1.6
I
I 24
I
32
t
+ g b4
to fit a and b, the extrapolation is obtained. The remarkable fit of this for the case CaF, to the experimental data* 6 with p = 1 and the static magnetic field along the [lo03 axis is shown in Fig. 1 (a). We notice that the initial beat amplitude decreases as p decreases. Extending the original Lowe-Norberg theory4 to the case of a magnetically dilute system, the free induction decay can be expressed as
F(P,t)
=
U(p,t)
=
U(p,t)
l-l
(5)
V(p,t)
10-p)
+ P CO6 (6)
JZi
V(p,t)
{[ C
8,jt
=
2 1 -
5
tan (
jfi
yj12-
T:
[Bijt
tan
Bijt 2I
( -+I
-.40'1
Ifi
(7)
E: j+i
(6ijt --T-l
3
tan
CU)
2
FIG.
1
Free induction decay shapes for F1’ in CaF, for static magnetic field along [IO0 J axis. (a) Abragam’s Ansatz, (b) Lowe-Norberg method, (c) Evans-Powles method. The unit of time is 3a3 /2y” h where a = 2.725 i is the lattice spacing. The circles are the experimental results of Barnaal and Lowe.
Vol. 5, No. 12
FREE-INDUCTION DECAY
The decay curve is numerically evaluated for the first 32 near neighbor interactions. We find that the amplitude of the initial beat is enhanced as p decreases in the region 1 > p > 0. 5, as shown in Fig. 1(b).
The plot in Fig. 1(c) (using 32 nearest neighbors) shows that the amplitude of the beat decreases as p decreases, in good agreement with the Abragam Ansatz. Finally, we should like to mention that the results obtained by using the Lowe-Norberg theory do not agree with those obtained by Abragam’s Ansatz or by the Evans-Powles method, which we believe is a better method. The reason is, as pointed out by many authors, 2 ~ that the expansion of F(t) in Lowe-Norberg’s theory is arbitrary and diverges for times not much longer than those for which it was computed.
Recently Evans and Powles” derived the free induction decay formula by using a Dyson type expansion in power of H1. Their result gave good agreement with experiment. Adapting this method to a magnetically dilute system, we get the free induction decay as F(p,t)
=
F(p,t)
+ F1(t) +
.
B F (p,t) 0
=
U
Acknowledgment Dr. D. J. Scalapino - The for author helpfulwishes discussions to thank and
t
(—q—)I
+ p cos
[(1-p)
(8)
.
With
937
(9)
suggestions.
(—!f_) St
F1(p,t)
—f
=
t
B1
j
sin
B [t
U
((1-p) + p cos
t
(J~L..)) 2 (10)
t dt’
U
[(1-p)
+ p cos
[Bki(tt) 2
+
Bkj 2
tj1
References 1.
BLOEMBERGEN N.,
PURCELL E.M. and POUND R.V.,
Phys. Rev. 73, 679 (1948).
2.
ABRAGAM A.,
3.
VAN VLECK J.H.,
4.
LOWE I. J. and NORBERG R. E.,
5.
KITTELL C. and ABRAHAMS E., Phys. Rev. 90, 238 (1953). Their result of fourth moment for magnetic 2Mfield oriented along the [100] crystaYaxis has a small numerical error. It should be M4(p) = 3p 2(1) (0.63 + .2.2.L ). p
6.
BARNAAL D. and LOWE I.J.,
7.
EVANS W. A. B.
The Principles of Nuclear Magnetism, Clarendon Press, Oxford (1961). Phys. Rev.
74, 1168 (1948). Phys. Rev. 107, 46 (1957).
Phys. Rev. 148, 328 (1966).
and POWLES J. G.,
Phys. Letters 24A, 218 (1967).
Das Blochabklingen den freien magnetischen Induktion eines verdtlnnten magnetischen Spin 1/2 Gitters wird auf drei Arten berechnet: Die Resultate der (1) Lowe-Norbergtheorie stimmen mit denen des (2) Abragamansatzes und denen der (3) Evan-Powlestheorie nich fiberein. Dies Nichtübereinstimmung scheint auf eine unvollst~ndigeEntwicklung der F(t)-F’unktjon in (1) zuri~ckführbarzu sein.