- Email: [email protected]
Nuclear spin-diffusion constant of a noble gas
Physica
42 (1969) 633-637
NUCLEAR
0 North-Holland
SPIN-DIFFUSION
Publishing
CONSTANT
Co., Amsterdam
OF A NOBLE
GAS
S. HESS* Kamerlingh
Onnes Laboratorium,
Leiden,
Nederland
(Communication Suppl. No. 127~) Received 11 October 1968
Synopsis The nuclear spin-diffusion constant of a dilute noble gas (e.g. sHe) is inversely proportional to a collision bracket obtained from the generalized Boltzmann equation for particles with spin. Although for a pure gas the collision term contains a symmetrized scattering amplitude, that particular collision bracket can be reduced to an O-integral involving a nonsymmetrized scattering cross section.
Introductiom Nuclear spin diffusion in a dilute noble gas (e.g. 3He) can be treated by the kinetic theory of gases much as the classical transport properties. Dealing with particles with spin one has to start from a generalized Boltzmann equation first derived by Waldmannr) rather than the classical Boltzmann equation. The collision term of the Waldmann equation contains the binary scattering amplitude (operator) a and its adjoint ut in a binary way which is not, however, a scattering cross section. Spin diffusion can be characterized D spin if the small nonspherical part lecteds).
Disregarding
to one relaxation
by a single spin diffusion constant of the scattering amplitude is neghigher Sonine polynomials, Dspin in turn, is related
coefficient
2, 0&d by
(1) Here k is Boltzmann’s constant, T is the temperature of the gas and m is the mass of a particle. The relaxation coefficient 0&d is given by a collision bracket integral (to be stated later) obtained from the linearized Waldmann collision term. For the collision of identical particles one has to use a symmetrized scattering amplitude. Nevertheless, the relaxation coefficient Wsd turns out * On leave of absence from the University of Erlangen-Ntirnberg; address : Institut fiir Theoretische Physik, D-8520 Erlangen, Germany. 633
permanent
634
S. HESS
to be given by (n is the number
density)
where the Chapman Q-integrals)
has to be evaluated
with a nonsymmetrized
scattering cross section. Thus the spin-diffusion constant equals the (particle) self-diffusion constant (cf. ref. 3, p. 423) where the pertinent relaxation coefficient has to be calculated as if two colliding particles in a pure gas were distinguishable. This result has already been conjectured by Emery 4). In fact he arrived at eq. (2) with the aid of a plausibility argument. Thus a rederivation of ccl. (2) based on the proper kinetic equation will not be superfluous. Notice, however, that the spin-diffusion constant is equal to the selfdiffusion constant only if no external magnetic field is applied or if the density of the gas is high enough such that the influence of the external field on spin diffusion is negligible. The magnetic field dependence of spin diffusion
has been discussed in ref. 2.
Collision bracket integral. In order to derive eq. (2) we first express in terms of a collision bracket integral defined earlier5) * : Osd = i[S(s’
+
1)1-l <~v$Lo(~&L)>o.
c+,r
(3)
Here V is a dimension velocity variable and s is the spin vector operator. The magnitude of the spin (in units of ti) is S (i.e. S-S = .S(S + 1)). As in ref. 5 the linearized collision term of the Waldmann equation is denoted by c~(. . .). The bracket ( )a refers to a mean value evaluated with an equilibrium distribution function. After an integration over the center of mass velocity of two colliding particles (which shall be labelled by 1 and 2) eq. (3) reduces to
(4) Here (5) is a thermal velocity, y being the magnitude of a dimensionless relative velocity has its usual meanings). The angle of deflection in the centre of * Cartesian convention
components
is applied
of vectors
throughout.
are denoted
by Greek indices and the summation
NUCLEAR
SPIN-DIFFUSION
CONSTANT
mass system is denoted by x. The curly bracket
{...> = j dy eeya ys2x
s 0
0
dxsinx
OF A NOBLE
GAS
is an abbreviation
c{yZru(l -
factors
for
tr1 trz -~~ .. .. (2s + 1)2
where trl and trz are the traces over the spin indices of particles The numerical
635
1 and 2.
in eqs. (5) and (6) have been chosen such that
co+ x)} = LWr),
(7)
where u = a(ys, x) is a scattering
cross section
and LNJ)
is a Chapman
integral. It is understood that the scattering amplitude a occurring in eq. (4) is a “spherical” one, i.e. a shall commute with the total spin si + ss of the two colliding particles : [a, @I + s2)l = 0.
(8)
The nonspherical part of the scattering amplitude which determines the spin-relaxation constant, is negligibly small compared to the spherical part. For convenience we rewrite eq. (4) by using the identity (si -
s2) a+ = a+@1 -
s2) +
[(s1 -
s2), a+]
and eq. (8):
- cosx)}+ +
~n~{y~[s1~, alLa+, s1.d cosx}.
(9)
From the classical Boltzmann equation one would obtain only the first term containing the “cross section” aat but not the second term. Before one can evaluate the traces occurring in eq. (9), one first has to know the spin dependence of the symmetrized scattering amplitude. Symmetrized scattering amfilitzcde. The symmetrized (spherical) amplitude a is related to the unsymmetrized spin-independent amplitude
a&)
scattering scattering
by6)
4x1 s1.s~) = LJ2
(m(x) +
@Q(sI.sz) ao(x- x)).
(10)
The factor 0 equals + 1 or - 1 for Bosons or Fermions respectively. The dependence of a and a0 on the relative kinetic energy (i.e. on ys) is not indicated in eq. (10). The scattering amplitude depends on ~1.~2 only via Q = Q(si*ss). This operator, when applied the spin-exchange operator** ** For S = 4, e.g. one has Q = + + 2~1.
~2.
S. HESS
636
to a wave function (spinor) y(Mi, Ms) depending numbers 441 and Ms of two particles, interchanges
Qy(M1,J42)
=
on the magnetic quantum these quantum numbers6)
y(J42, Ml).
Obviously one has Qs = 1. A reprcscntation invariant
(11)
definition
of Q (eclui\alent
to “‘1. (1 1)) is g$\rtn
:
by
1
QaQ =
Introducing P*i
Qs2Q = sl.
~2,
the projection
= k(l i
(12)
operators
Q)
(13)
into states which are symmetric (+ 1) or anti-symmetric (- 1) with respect to the interchange of the magnetic quantum numbers, one may rcwritc eq. (10) : a(x, Sl.S2) =
$(G&)
p+l(sl*s2)
+
a-&)
(14)
~-1(~1*S2)).
Hcrc the functions Ckto(x) = !&o(x)
+ @ao(x -
2))
(15)
have a definite symmetry under the exchange x + TC- x. Now we are ready to evaluate the spin traces occurring in eq. (9) For convenience
Spin-traces.
Tr = tr:-!! (2.C +
1)’
let us introduce
the abbreviation
.
(16)
Then one has Tr 1 =
1
and
Tr Q = (2s +
1)-i.
(17)
Since Trsl -s2 = 0 and Trsl-szQ
= Trsl-Q
Qs2Q = Trsl.slQ
= S(.S +
1)(2S +
1)-i,
(18)
one obtains Tr P+, = $(l + (2s +
1)-l)
(19)
and P+,=
Furthermore Tr[si,,
;.
(20)
one needs
QILQ,a/11= 2.W + 1).
(21)
NUCLEAR
Red&ion
of
a&d
SPIN-DIFFUSION to
aia_,)
CONSTANT
a Chafwzan integral.
OF A NOBEL
Observing
7dy e+y5
(22) x + rr -
x) and using
X
0
x 2~ ;dX sin X[la,j2
+
la_,j2
-
cos
X(a,a:o
+
a&_,)].
0
Due to eq. (22) this result is equivalent to eq. (2) if the Go,‘) contains the nonsymmetrized scattering cross section 4~)
637
that
sin x dX = 0,
(since the integrands are odd under the interchange the traces (20, 21) one finally obtains from eq. (9) : C,,sd= $zc
GAS
= lao(
=
la,(x)
+ a-o(x)12.
(23)
integral
(24)
Thus it has been shown that the proper collision term of the generalized Boltzmann equation for particles with spin leads to a spin-diffusion relaxation coefficient which is proportional to the L?(l,l’-integral involving a nonsymmetrized scattering cross section although a symmetrized scattering amplitude has been used to characterize the binary collision of equal particles. REFERENCES
1) Waldmann, L., Z. Naturforsch. 12a (1957) 660, 13a (1958) 609; Snider, R. F., J. them. Phys. 32 (1960) 1051.
4 Hess, S., Z. Naturforsch. 23 a (1968) 898.
3) Waldmann, L., in Handbuch der Physik, vol. 12, ed. S. Flilgge (Springer, Berlin, 1958). 4) Emery, V. J., Phys. Rev. 133A (1964) 661. 5) Hess, S. and Waldmann, L., Z. Naturforsch. 21 a (1966) 1529, 23 a (1968) 1893. 6) Waldmann, L., Physica 30 (1964) 17.
42 (1969) 633-637
NUCLEAR
0 North-Holland
SPIN-DIFFUSION
Publishing
CONSTANT
Co., Amsterdam
OF A NOBLE
GAS
S. HESS* Kamerlingh
Onnes Laboratorium,
Leiden,
Nederland
(Communication Suppl. No. 127~) Received 11 October 1968
Synopsis The nuclear spin-diffusion constant of a dilute noble gas (e.g. sHe) is inversely proportional to a collision bracket obtained from the generalized Boltzmann equation for particles with spin. Although for a pure gas the collision term contains a symmetrized scattering amplitude, that particular collision bracket can be reduced to an O-integral involving a nonsymmetrized scattering cross section.
Introductiom Nuclear spin diffusion in a dilute noble gas (e.g. 3He) can be treated by the kinetic theory of gases much as the classical transport properties. Dealing with particles with spin one has to start from a generalized Boltzmann equation first derived by Waldmannr) rather than the classical Boltzmann equation. The collision term of the Waldmann equation contains the binary scattering amplitude (operator) a and its adjoint ut in a binary way which is not, however, a scattering cross section. Spin diffusion can be characterized D spin if the small nonspherical part lecteds).
Disregarding
to one relaxation
by a single spin diffusion constant of the scattering amplitude is neghigher Sonine polynomials, Dspin in turn, is related
coefficient
2, 0&d by
(1) Here k is Boltzmann’s constant, T is the temperature of the gas and m is the mass of a particle. The relaxation coefficient 0&d is given by a collision bracket integral (to be stated later) obtained from the linearized Waldmann collision term. For the collision of identical particles one has to use a symmetrized scattering amplitude. Nevertheless, the relaxation coefficient Wsd turns out * On leave of absence from the University of Erlangen-Ntirnberg; address : Institut fiir Theoretische Physik, D-8520 Erlangen, Germany. 633
permanent
634
S. HESS
to be given by (n is the number
density)
where the Chapman Q-integrals)
has to be evaluated
with a nonsymmetrized
scattering cross section. Thus the spin-diffusion constant equals the (particle) self-diffusion constant (cf. ref. 3, p. 423) where the pertinent relaxation coefficient has to be calculated as if two colliding particles in a pure gas were distinguishable. This result has already been conjectured by Emery 4). In fact he arrived at eq. (2) with the aid of a plausibility argument. Thus a rederivation of ccl. (2) based on the proper kinetic equation will not be superfluous. Notice, however, that the spin-diffusion constant is equal to the selfdiffusion constant only if no external magnetic field is applied or if the density of the gas is high enough such that the influence of the external field on spin diffusion is negligible. The magnetic field dependence of spin diffusion
has been discussed in ref. 2.
Collision bracket integral. In order to derive eq. (2) we first express in terms of a collision bracket integral defined earlier5) * : Osd = i[S(s’
+
1)1-l <~v$Lo(~&L)>o.
c+,r
(3)
Here V is a dimension velocity variable and s is the spin vector operator. The magnitude of the spin (in units of ti) is S (i.e. S-S = .S(S + 1)). As in ref. 5 the linearized collision term of the Waldmann equation is denoted by c~(. . .). The bracket ( )a refers to a mean value evaluated with an equilibrium distribution function. After an integration over the center of mass velocity of two colliding particles (which shall be labelled by 1 and 2) eq. (3) reduces to
(4) Here (5) is a thermal velocity, y being the magnitude of a dimensionless relative velocity has its usual meanings). The angle of deflection in the centre of * Cartesian convention
components
is applied
of vectors
throughout.
are denoted
by Greek indices and the summation
NUCLEAR
SPIN-DIFFUSION
CONSTANT
mass system is denoted by x. The curly bracket
{...> = j dy eeya ys2x
s 0
0
dxsinx
OF A NOBLE
GAS
is an abbreviation
c{yZru(l -
factors
for
tr1 trz -~~ .. .. (2s + 1)2
where trl and trz are the traces over the spin indices of particles The numerical
635
1 and 2.
in eqs. (5) and (6) have been chosen such that
co+ x)} = LWr),
(7)
where u = a(ys, x) is a scattering
cross section
and LNJ)
is a Chapman
integral. It is understood that the scattering amplitude a occurring in eq. (4) is a “spherical” one, i.e. a shall commute with the total spin si + ss of the two colliding particles : [a, @I + s2)l = 0.
(8)
The nonspherical part of the scattering amplitude which determines the spin-relaxation constant, is negligibly small compared to the spherical part. For convenience we rewrite eq. (4) by using the identity (si -
s2) a+ = a+@1 -
s2) +
[(s1 -
s2), a+]
and eq. (8):
- cosx)}+ +
~n~{y~[s1~, alLa+, s1.d cosx}.
(9)
From the classical Boltzmann equation one would obtain only the first term containing the “cross section” aat but not the second term. Before one can evaluate the traces occurring in eq. (9), one first has to know the spin dependence of the symmetrized scattering amplitude. Symmetrized scattering amfilitzcde. The symmetrized (spherical) amplitude a is related to the unsymmetrized spin-independent amplitude
a&)
scattering scattering
by6)
4x1 s1.s~) = LJ2
(m(x) +
@Q(sI.sz) ao(x- x)).
(10)
The factor 0 equals + 1 or - 1 for Bosons or Fermions respectively. The dependence of a and a0 on the relative kinetic energy (i.e. on ys) is not indicated in eq. (10). The scattering amplitude depends on ~1.~2 only via Q = Q(si*ss). This operator, when applied the spin-exchange operator** ** For S = 4, e.g. one has Q = + + 2~1.
~2.
S. HESS
636
to a wave function (spinor) y(Mi, Ms) depending numbers 441 and Ms of two particles, interchanges
Qy(M1,J42)
=
on the magnetic quantum these quantum numbers6)
y(J42, Ml).
Obviously one has Qs = 1. A reprcscntation invariant
(11)
definition
of Q (eclui\alent
to “‘1. (1 1)) is g$\rtn
:
by
1
QaQ =
Introducing P*i
Qs2Q = sl.
~2,
the projection
= k(l i
(12)
operators
Q)
(13)
into states which are symmetric (+ 1) or anti-symmetric (- 1) with respect to the interchange of the magnetic quantum numbers, one may rcwritc eq. (10) : a(x, Sl.S2) =
$(G&)
p+l(sl*s2)
+
a-&)
(14)
~-1(~1*S2)).
Hcrc the functions Ckto(x) = !&o(x)
+ @ao(x -
2))
(15)
have a definite symmetry under the exchange x + TC- x. Now we are ready to evaluate the spin traces occurring in eq. (9) For convenience
Spin-traces.
Tr = tr:-!! (2.C +
1)’
let us introduce
the abbreviation
.
(16)
Then one has Tr 1 =
1
and
Tr Q = (2s +
1)-i.
(17)
Since Trsl -s2 = 0 and Trsl-szQ
= Trsl-Q
Qs2Q = Trsl.slQ
= S(.S +
1)(2S +
1)-i,
(18)
one obtains Tr P+, = $(l + (2s +
1)-l)
(19)
and P+,=
Furthermore Tr[si,,
;.
(20)
one needs
QILQ,a/11= 2.W + 1).
(21)
NUCLEAR
Red&ion
of
a&d
SPIN-DIFFUSION to
aia_,)
CONSTANT
a Chafwzan integral.
OF A NOBEL
Observing
7dy e+y5
(22) x + rr -
x) and using
X
0
x 2~ ;dX sin X[la,j2
+
la_,j2
-
cos
X(a,a:o
+
a&_,)].
0
Due to eq. (22) this result is equivalent to eq. (2) if the Go,‘) contains the nonsymmetrized scattering cross section 4~)
637
that
sin x dX = 0,
(since the integrands are odd under the interchange the traces (20, 21) one finally obtains from eq. (9) : C,,sd= $zc
GAS
= lao(
=
la,(x)
+ a-o(x)12.
(23)
integral
(24)
Thus it has been shown that the proper collision term of the generalized Boltzmann equation for particles with spin leads to a spin-diffusion relaxation coefficient which is proportional to the L?(l,l’-integral involving a nonsymmetrized scattering cross section although a symmetrized scattering amplitude has been used to characterize the binary collision of equal particles. REFERENCES
1) Waldmann, L., Z. Naturforsch. 12a (1957) 660, 13a (1958) 609; Snider, R. F., J. them. Phys. 32 (1960) 1051.
4 Hess, S., Z. Naturforsch. 23 a (1968) 898.
3) Waldmann, L., in Handbuch der Physik, vol. 12, ed. S. Flilgge (Springer, Berlin, 1958). 4) Emery, V. J., Phys. Rev. 133A (1964) 661. 5) Hess, S. and Waldmann, L., Z. Naturforsch. 21 a (1966) 1529, 23 a (1968) 1893. 6) Waldmann, L., Physica 30 (1964) 17.