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On the derivation of the relaxation function for spin systems in the high-temperature approximation
Physica
46 (1970)
ON THE SPIN
391-394
0 North-Holland
DERIVATION
SYSTEMS
OF
IN THE
Publishing
THE
Co., Amstevdaw~
RELAXATION
FUNCTION
HIGH-TEMPERATURE
FOR
APPROXIMATION
G. CASATI CCR Euratom, Received
Ispva,
3 July
Italia 1969
Synopsis The expression is directly derivation
without
it appears
the response linear.
for the relaxation
derived
that,
of the relaxation
to a sudden
that contrary function
in the high-temperature
the formalism
as a consequence
of a spin system
It follows
function
introducing
to what
describes
variation
is commonly
the response
M(t)
of the external believed
of the system
approximation
and Tomita.
of the high-temperature
external field which is not small. As an application ization is calculated.
1. The from t = along the switched
of Kubo
From
magnetic
the temporal
field
is
behaviour
also to a variation
the asymptotic
this
approximation,
of the
value of the magnet-
relaxation function. Let us consider I spin system and assume that --oo to t = 0 it is placed in an external magnetic field applied z axis and equal to H + h. If the small external field h is suddenly off at t = 0 the linear response of the system is given by = q(t) * h + MV’),
(1.1)
where M(t) is the average total magnetic moment at time t, M(o) the static magnetization in the presence of the magnetic field H and q(t) the relaxation tensor. Limiting ourselves to the study of the longitudinal magnetization eq. (1.1) reduces to M,(t)
= Tr(d&))
= h&&)
+ Tr(d,po),
where -4ZZ is the z component of the total magnetic po is the equilibrium density matrix
(I.4 moment
operator,
exp(-P*) PO = x
Tr(exp{-&e/e))
is the total Hamiltonian p(t) = exp(-
+
and
(1.3)
’ of the spin system for t > 0. Furthermore
Zt)plexp
(+Zt), 391
(1.4)
where pi is the initial density matrix
exp(-P{X
Pi= -Tr[exp(
- Izdz,))
(1.5)
-/3{3FZm
and Z - k+4, is the total Hamiltonian for t 5 0. Kubo and Tomitar) have derived a general formula function which is given by the expression &z(t) = /dB Tr{paAz&(f
+ ihi)} -
for the relaxation
@[Tr(paAZ)]2.
(1.6)
In the high-temperature approximation one can expand eq. (1.6) up to the first-order terms in /3and one arrives at the simple result
(1.7) In the same approximation,
expression
Tr(dzd&))
ill,(t)
/Z/3-___-__-
=
-
(1.2) reduces to
Tr (J$?)
/9 ______-
.
Tr(1)
Tr(1)
(1.8)
Expression (1.8) is the starting point of many investigations on spin-spin relaxation ‘2-5). We want to show now that the expression (1.8) may be directly derived in a much simpler manner. Linearity in h will then result as a consequence of the high-temperature approximation and will not have to be assumed. It follows that, contrary to what is commonly believed, expression (1.8) is valid even though the variation Iz of the field is not small. Indeed, in the high-temperature approximation, we have Tr{A,p(t)}
2
s Tr
AZ exp-
,~,j Tr(~z~z(t))
_,~ Tr(d&? Tr(1) ___~~~
Tr(1)
which is just expression
Pi_ Rt . [ ’
(1.9)
( 1.8).
2. The usy@totic value. In this section we shall derive the asymptotic value of the magnetic moment from eq. (1.9) which we have seen to be valid even though lz is not small. To do this, let us divide A, into two orthogonal operators ~
z
=
A? Tr(Z’k,) Tr(X2)
-- t-J&,
where the total Hamiltonian
X’ for t > 0 is supposed to consist
(2.1) of a Zeeman
RELAXATION
term Zz
FUNCTION
IN HIGH-TEMPERATURE
and a dipole-dipole
APPROXIMATION
term %d:
.%=%‘z+xd=--Hd,+s& According
393
(2.2)
to eq. (2.2), (2.1) becomes :
dlz = - &
H H;, + Hz
+ -A
(2.3)
where
Using the decomposition
(2.3), we have
Tr{~zex{-~X,>J?zexp(~X’~)}=
=(
H;J
HJ
Tr(Z2)
+ Tr(iz2&)),
(2.5)
where we have used the fact that Tr(&?JZZ) = 0. In the representation &? is diagonal, we have Tr{2zsZ(t)}
=
C l
m>j2 exp(iwd)
;
Wmn -
-% -
where En
h
. (2.6)
For very large systems the Hamiltonian ~4? shows a quasi-continuous spectrum, so that the sum (2.6) may be transformed into a Fourier integral. From the Riemann-Lebesgue theorem we then obtain lim Tr{2Z2Z(t)} t+m
= lim Y/(W) exp(iwt) dw = 0 t t-+.x?--oo
(2.7)
whenever the integral JY_ /f(o)l dw exists. Eq. (1.9) together with eqs. (2.5) and (2.7) gives: H H;
+ Hz
1
(2.8)
’
where x0 is the static susceptibility. The asymptotic value (2.8) of the magnetic moment is in agreement with that derived through thermodynamic considerations on the equilibrium states. Acknowledgement: The author is greatly indebted to Professor Scotti for his most valuable advice and criticism throughout this work.
A.
t Here we have assumed that the diagonal terms contained in&T in the representation where 2
is diagonal, do not give rise in f(w) to a S-singularity.
this is still an open question.
As far as we know,
394
RELAXATlON
1)
Kubo,
2)
Caspers,
FUNCTION
K. and Tomita, \fT. I., Theory
IN HIGH-TEn!IPERATURE
K., J. l’hys. of Spin
Sec. Japan
Relaxation.
9 (1954) Interscience
APPROXIMATION
888. Publishers
1964). 3)
Tcrwicl,
4)
Tjon,
5)
R. H. and Mazur,
P., I’hysica
32 (1966)
J. A., Physica 30 (1964) 1; 1341. G., I’hysica 32 (1966) 2017. Sauermnnn,
1813; 36 (1967) 289.
(New
York,
46 (1970)
ON THE SPIN
391-394
0 North-Holland
DERIVATION
SYSTEMS
OF
IN THE
Publishing
THE
Co., Amstevdaw~
RELAXATION
FUNCTION
HIGH-TEMPERATURE
FOR
APPROXIMATION
G. CASATI CCR Euratom, Received
Ispva,
3 July
Italia 1969
Synopsis The expression is directly derivation
without
it appears
the response linear.
for the relaxation
derived
that,
of the relaxation
to a sudden
that contrary function
in the high-temperature
the formalism
as a consequence
of a spin system
It follows
function
introducing
to what
describes
variation
is commonly
the response
M(t)
of the external believed
of the system
approximation
and Tomita.
of the high-temperature
external field which is not small. As an application ization is calculated.
1. The from t = along the switched
of Kubo
From
magnetic
the temporal
field
is
behaviour
also to a variation
the asymptotic
this
approximation,
of the
value of the magnet-
relaxation function. Let us consider I spin system and assume that --oo to t = 0 it is placed in an external magnetic field applied z axis and equal to H + h. If the small external field h is suddenly off at t = 0 the linear response of the system is given by = q(t) * h + MV’),
(1.1)
where M(t) is the average total magnetic moment at time t, M(o) the static magnetization in the presence of the magnetic field H and q(t) the relaxation tensor. Limiting ourselves to the study of the longitudinal magnetization eq. (1.1) reduces to M,(t)
= Tr(d&))
= h&&)
+ Tr(d,po),
where -4ZZ is the z component of the total magnetic po is the equilibrium density matrix
(I.4 moment
operator,
exp(-P*) PO = x
Tr(exp{-&e/e))
is the total Hamiltonian p(t) = exp(-
+
and
(1.3)
’ of the spin system for t > 0. Furthermore
Zt)plexp
(+Zt), 391
(1.4)
where pi is the initial density matrix
exp(-P{X
Pi= -Tr[exp(
- Izdz,))
(1.5)
-/3{3FZm
and Z - k+4, is the total Hamiltonian for t 5 0. Kubo and Tomitar) have derived a general formula function which is given by the expression &z(t) = /dB Tr{paAz&(f
+ ihi)} -
for the relaxation
@[Tr(paAZ)]2.
(1.6)
In the high-temperature approximation one can expand eq. (1.6) up to the first-order terms in /3and one arrives at the simple result
(1.7) In the same approximation,
expression
Tr(dzd&))
ill,(t)
/Z/3-___-__-
=
-
(1.2) reduces to
Tr (J$?)
/9 ______-
.
Tr(1)
Tr(1)
(1.8)
Expression (1.8) is the starting point of many investigations on spin-spin relaxation ‘2-5). We want to show now that the expression (1.8) may be directly derived in a much simpler manner. Linearity in h will then result as a consequence of the high-temperature approximation and will not have to be assumed. It follows that, contrary to what is commonly believed, expression (1.8) is valid even though the variation Iz of the field is not small. Indeed, in the high-temperature approximation, we have Tr{A,p(t)}
2
s Tr
AZ exp-
,~,j Tr(~z~z(t))
_,~ Tr(d&? Tr(1) ___~~~
Tr(1)
which is just expression
Pi_ Rt . [ ’
(1.9)
( 1.8).
2. The usy@totic value. In this section we shall derive the asymptotic value of the magnetic moment from eq. (1.9) which we have seen to be valid even though lz is not small. To do this, let us divide A, into two orthogonal operators ~
z
=
A? Tr(Z’k,) Tr(X2)
-- t-J&,
where the total Hamiltonian
X’ for t > 0 is supposed to consist
(2.1) of a Zeeman
RELAXATION
term Zz
FUNCTION
IN HIGH-TEMPERATURE
and a dipole-dipole
APPROXIMATION
term %d:
.%=%‘z+xd=--Hd,+s& According
393
(2.2)
to eq. (2.2), (2.1) becomes :
dlz = - &
H H;, + Hz
+ -A
(2.3)
where
Using the decomposition
(2.3), we have
Tr{~zex{-~X,>J?zexp(~X’~)}=
=(
H;J
HJ
Tr(Z2)
+ Tr(iz2&)),
(2.5)
where we have used the fact that Tr(&?JZZ) = 0. In the representation &? is diagonal, we have Tr{2zsZ(t)}
=
C l
m>j2 exp(iwd)
;
Wmn -
-% -
where En
h
. (2.6)
For very large systems the Hamiltonian ~4? shows a quasi-continuous spectrum, so that the sum (2.6) may be transformed into a Fourier integral. From the Riemann-Lebesgue theorem we then obtain lim Tr{2Z2Z(t)} t+m
= lim Y/(W) exp(iwt) dw = 0 t t-+.x?--oo
(2.7)
whenever the integral JY_ /f(o)l dw exists. Eq. (1.9) together with eqs. (2.5) and (2.7) gives: H H;
+ Hz
1
(2.8)
’
where x0 is the static susceptibility. The asymptotic value (2.8) of the magnetic moment is in agreement with that derived through thermodynamic considerations on the equilibrium states. Acknowledgement: The author is greatly indebted to Professor Scotti for his most valuable advice and criticism throughout this work.
A.
t Here we have assumed that the diagonal terms contained in&T in the representation where 2
is diagonal, do not give rise in f(w) to a S-singularity.
this is still an open question.
As far as we know,
394
RELAXATlON
1)
Kubo,
2)
Caspers,
FUNCTION
K. and Tomita, \fT. I., Theory
IN HIGH-TEn!IPERATURE
K., J. l’hys. of Spin
Sec. Japan
Relaxation.
9 (1954) Interscience
APPROXIMATION
888. Publishers
1964). 3)
Tcrwicl,
4)
Tjon,
5)
R. H. and Mazur,
P., I’hysica
32 (1966)
J. A., Physica 30 (1964) 1; 1341. G., I’hysica 32 (1966) 2017. Sauermnnn,
1813; 36 (1967) 289.
(New
York,