Download Diamagnetic Screening of Transverse Current

406
Supplement of the Progress of Theoretical Physics, No. 69, 1980
Diamagnetic Screening of Transverse Current Correlations
and Neutron Scattering in Metals
Ken SASAKI and Y ukio 0BAT A
Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki 319-11
(Received December 8, 1980)
A general expression for the differential scattering cross section of neutrons
by an electron system is considered in terms of the transverse current-current
response function. Effect of the transverse fluctuating field on the transverse
current-current response function· is found to be important. Such a calculation
is made by using the transverse effective field approach formulated by Pines
and Nozieres. The effect of the fluctuating transverse field appears in the
final expression in the form of dynamic screening. This suppresses the otherwise divergent differential cross section appreciably at smallest q of order
hundredth of inverse Angstrom or less.
§ 1.
Introduction
Since Hebborn and March's review article was published in 1970,n several
important investigations have been made on the dynamic 2>-B> and staticn-9>
orbital magnetism. Nevertheless, we are still left with several important problems to be settled, especially on the dynamic orbital magnetism. We have
among them the problem of neutron scattering by orbital current carried by
electrons in a metal.
Neutron with nuclear spin 1/2 probes the magnetic field in a solid, which
is fluctuating, generally, in space and time. This fluctuating field induces
fluctuating current, and vice versa: the fluctuating magnetic field arises from
the spontaneous motion of charged particles. As is well known, the total orbital
current consists of two parts, paramagnetic and diamagnetic, and the latter
depends linearly on the vector potential acting on the electrons that carry
it. This dualism of orbital current should be considered carefully: the
gauge invariance has to be maintained in any theoretical calculations.
In this paper we try to construct the theory of neutron scattering by
current fluctuation, taking account of the above point. We do it by following the steps outlined below.
In § 2 we represent the neutron scattering cross section in terms of the
magnetic field-magnetic field correlation function, then express it in terms of
the transverse current-current correlation function by making use of the exact
microscopic Maxwell equation incorporating the electromagnetic fluctuations.
The transverse current-current correlation function is related to the corres-
407
Current Correlations and Neutron Scattering
ponding response function with the use of fluctuation dissipation theorem. Thus
we obtain the exact neutron scattering cross section expressed in terms of the
transverse current-current response function.
In § 3 we consider the exact formula for the neutron scattering due to
orbital motion of electrons. Then we adopt the transverse effective field approach formulated by Pines and Nozieres 10> in order to take into consideration
the transverse fluctuating field in calculating the neutron scattering cross section.
In § 4 our theory is applied to the neutron scattering by free electrons
and some numerical calculations are performed to see more specifically how
the transverse fluctuating field affects the cross section.
Section 5 is devoted to the concluding remarks.
§ 2.
2.1.
General scattering formula
Scattering formula in terms of magnetic field correlation function
The magnetic interaction between a single neutron and electrons in a
crystal is described by the interaction Hamiltonianw
(2·1)
where fi a are the Pauli rna trices
A
(J:c=
(01 1)0 '
A
(Jy=
(0i -i)
0
A
'
(J,=
(10
(2·2)
r=
-1.91, the gyromagnetic ratio of neutron, and fLN=ehj2Mc is the nuclear
magneton, M being the proton mass. The magnetic field jj (r, {ri, pJ) at
the position of neutron r arises from electrons in the crystal with coordinates
ri and momenta Pi· We stress that it is an operator and should be subjected
to quantum and thermodynamical averaging before it is compared with any
measurement.*>
The differential cross section cJ2(J/ dQdE' for the scattering of neutrons
per unit solid angle !2 and unit out-going neutron energy E', resulting from
the magnetic interaction in Eq. (2 ·1) is
d 26
( M )2
k'
d!JdE' = 2nh2 Cr ILN) 2k
x
:E
:E P.P"i(A.'k'6'ifi·B(r, {ri, pi}) IA.ko-)\
U.'
2
&(-hw+E. -E.,),
(f(f'
(2·3)
where lko-) and \k' o-') are the initial and final states of the neutron specified
*> We shall always denote quantum mechanical operators by letters with circumflexes.
408
K. Sasaki and Y. Obata
by wave vector k and spin rJ and by k' and rJ', lA.) and lA.') the initial and
final states of the target.w In Eq. (2 · 3) an average is taken over the initial
states of the target lA.) with Boltzmann's factor P, and also over the initial
spin states of the neutron with weight Pd. In this paper we limit ourselves to
scattering of unpolarized neutrons and have Pt = PJ. = 1/2. Then we have
(2·4)
where q = k- k' is the scattering vector and B (q, {ri, pt}) is the matrix
element of B (r, {ri, pi}) between the orbital states of neutron lk) and lk'),
B(q, {ri, pi}) =<kiB(r, {r£, pi}) lk')
(2·5)
We can write the delta function in Eq. (2 · 4) in the form
iJ(hw+E,-E,,) =12-rrh
s= dt exp[ -zt(hw+E,
.
-E.,)/h],
-=
and thus
d 2rJ _ ( M ) 2< )2 ( 1 ) k'
d!JdE'- 2rch2 rtlN
2rch k
where
B ( ± q, t)
s= dt et.,t<iiC q, t)B(- q, O))tot'
-=
(2·6)
is the Heisenberg operator
B(±q, t) =exp(it&/h)B(±q, {ri, pi} )exp( -it&/h)
(2·7)
=B(±q, {ri(t), pt(t)} ),
and <Ji)tot stands for the canonical average of
A,
<A)tot = tr ( exp {- {33{} A) /tr exp {- {33{},
(2·8)
where j{ is the unperturbed total Hamiltonian including the fluctuating electromagnetic field B. Its explicit form is given in § 3.
2.2. Scattering formula in terms of transverse current correlation function
Equation (2 · 6) shows that the differential cross section is essentially the
Fourier transform over t of the time-correlation function of q-Fourier component of the magnetic field which is fluctuating even in the absence of
external field. The fluctuating field is produced by the fluctuating current
carried by electrons and determined by the microscopic Maxwell equation
Current Correlations and Neutron Scattering
409
(2·9)
written in the Coulomb gauge, where
(2 ·10)
holds.
In Eq. (2 · 9) JJ.. (q, ())) is the transverse current operator and
AJ.. (q, ())) the transverse vector potential operator giving the magnetic and
electric fields through
(2·11)
We emphasize that Eq. (2 · 9) is microscopic in nature and entirely free from
the material constants such as magnetic and electric permeabilities. On combining Eqs. (2 · 9) and (2 ·11) we find
B(±q,t)=
=Fi4ncqJ_~_d(J)
_
eqX}J._(±q,(J))
(2 ·12a)
(J)2_ c2q2
e'"'.
(2·12b)
On going from Eq. (2·12a) to Eq. (2·12b) we used the relation
eqX}J..(±q, ())) =eqX (iJ..(±q, ())) ±}0 (±q, ()))) =eqX}(±q, ())),
(2·13)
where ~1 ( ± q, ())) and J( ± q, ())) are the longitudinal and total current operators, respectively, and eq the unit vector in the direction of q. If we use
Eq. (2 ·12b) to calculate the correlation function of fluctuating magnetic field,
we could write
(B (q, t) B (- q, t') )tot = s_~~ d(J)(B (q) B (- q) )~otei .. (t-t')
= (4n)2c2lf~ d(J)
-~
,
([eqX}(q)][eqX}(-q)])~otei"<t-t')
(())2- c2q~2
•
(2·14a)
(2·14b)
where we used the well-known relation for the spectral density
([eq x }(q, ()))] [eq x }( -q, ())') J>tot
=([eq x }(q)] [eq x }( -q) J>~ot~ (())+ ())').
(2·15)
Comparing Eqs. (2·14a) and (2·14b), we find
2 2
(B(q)B(-q))~ot=(47r)2 ( 2 cq2 ~2([eqX}(q)][eqXJ(-q)J>~ot,
(J)
-cq
(2·16a)
410
K. Sasaki and Y. Obata
where cu in the denominator has no infinitesimal imaginary part, smce the
spectral densities on both sides must be real. Equation (2 ·16a) may be written in another form
(2·16b)
by making use of the vector identity 12>
[eq x )(q)] [eq x
JC- q) J
=e/cJJ.(q)}J.(-q))- (eq)J.(q)) (eq)J.(-q))
=JJ. (q) JJ. (-q).
(2·17)
We may use either Eq. (2 ·16a) or (2 ·16b) as the need arises.
2.3.
Scattering formula in terms of transverse current-current response
function
On the other hand we know that the transverse current-current correlation function in Eq. (2 ·16b) may be written, according to the :fluctuationdissipation theorem, 12),13) in the form
(2·18)
where
(2 ·19)
Is the Fourier transform of the retarded Green's function
«JJ. (q, t) JJ. (- q, O)))~ot = - ~ 0 (t) ( [JJ. (q, t), JJ. (- q, 0)] /ot.
(2· 20)
Now, on combining Eqs. (2 · 6), (2 ·16b), and (2 ·18), we arrive at a general
formula for the differential cross section in terms of the transverse currentcurrent response function
d 2cJ
2 ( 1 ) k'
__ ( M ) 2
2h
2nh2 (r !-lN) 2nh k 1- e-f1hm
dJJdE'-
X (
§ 3.
3.1.
(4n) 2c 2l
2 2)2
cu 2 -cq
I
11
m\\
j J. (q ) j J. ( -q ) 1/r,w.
\\tot
(2·21)
Calculation of transverse current-current response function
Transverse effective field approach
Now let us consider in more detail the total current
)(q) in Eq.
Current Correlations and Neutron Scattering
411
(2 · 20). In this paper we confine ourselves to study the contributions from
the orbital current. Then, from the primary meaning of the current, it may
be written as
2
}(q) =JP(q) _ _!!___ ~ A(k)[J(q-k),
me k
where the first term on the right-hand side of Eq.
transform of the paramagnetic current
(3·1)
(3 ·1) 1s the Fourier
(3·2)
and the second 1s the Fourier transform of the diamagnetic current in which
{J(q)
=~
exp(-iqr,)
(3·3)
i
is the Fourier transform of the density.
If we are faithful to the :fluctuation-dissipation theorem, we should take
A (r) for the :fluctuating transverse polarization field in the absence of external
field.12l.w This means that we have to use the total Hamiltonian which includes the :fluctuating transverse field A1. (r) as well as the instantaneous
Coulomb interaction between electrons to obtain the total response function
«11. (q) jl. (- q) »;~!. Such a total Hamiltonian may be written in the Coulomb
gauge as follows :w
(3·4)
where the :fluctuating field A1. (r) emerges in the kinetic energy of the
electron system, the second term is the periodic potential produced by the
array of ions in a crystal, the third is the instantaneous Coulomb interaction
between electrons, and the fourth is the energy of the :fluctuating field.
The current-current response function appearing in Eq. (2 · 21) represents
the coefficient of the linear response of the paramagnetic current to the
external field Aext,l.(r) =Aext,l.(q,w)expi(qr-wt). Thus we suppose that
a :fictitious transverse field Aext,l. (r) is applied to the system. Then the kinetic
energy in Hamiltonian (3 · 4) has to be replaced by
(3·5)
And developing the brackets we find the interaction of the electron system
with the transverse external field
412
K. Sasaki and Y. Obata
~
1
SCint
= ---;;
s
~
d 3r JJ..
(r) Aext,J.. (r).
(3·6)
In this interaction Hamiltonian just appears the operator JJ.. (r), Eq. (3 ·1),
of which the correlation function is being sought. So it would be most
orthodox to apply the linear response theory to calculate the transverse
current-current response function ({..}J.. (q) JJ.. (- q) ))~~!, by using the unperturbed total Hamiltonian j{ as given in Eq. (3 · 4) and the perturbation Jtint in
Eq. (3 · 6). The rigorous approach is very difficult to be carried out, however.
So we adopt here the mean field approximation initiated by Bohm and Pines 15l
and developed by Pines and Nozieres. 10l In this approach we retain only the
part of the fluctuating polarization field that oscillates in space and time
coherently with the external field. Thus the prescription is to replace
Aext, J.. by Aeff, J.. and then calculate the linear response to Aeff, J.. under the
approximation .l.J.. (ri) =0 (except the part absorbed in Aeff,J..). According to
Bohm and Pines 15l and also Pines and Nozieres, tol this transverse effective field
approach is accurate enough to deal with a non-relativistic electron system.
We allow for the Umklapp processes associated with the fluctuating effective
field but confine ourselves to a cubic crystal where the mixing of transverse
and longitudinal phenomena hardly occurs. In this case we can make a formulation which is quite analogous to Pines and Nozieres' original one for a
homogeneous system.
The perturbation due to the transverse effective field Aeff,J.. is given by
(3·7)
where K' are the reciprocal lattice vectors. Equation (3 · 7) is an essential
assumption in the transverse effective field approach. On applying the linear
response theory,12l,tsl we find then
<JJ..(q+K, a>))
= _
_!_
c
.E{x.J..
(q+K, q+K', a>) +~<{J(K-K', a>=O))}Aeff,J.. (q+ K', a>).
K'
m
(3·8)
Here XJ.. (q, q', a>) is the transverse paramagnetic current-current response function as defined by
XJ..(q, q', a>) =({..JJ..P(q)}J..P( -q')))r,.,
= I~~ dt e'"'({..JJ..P(q, t)JJ..P( -q', O)))r
in terms of the retarded Green's function
(3·9)
Current Correlations and Neutron Scattering
413
In Eqs. (3 · 9) and (3 ·10) Jj_ P (q) is the transverse part of the paramagnetic
current as given by Eq. (3 · 2). Notice that the superfix tot is missing on
the double brackets in Eq. (3 · 9). This means that the Hamiltonian working
there is the unperturbed one where A1_ (ri) =0 (i=1, ···, N). The quantity
({5 (K- K', w = 0)) in Eq. (3 · 8) is also defined by
({5 (K- K', w = 0)) =
J_ dt ({5 (K- K', t) )e-iqt.
00
(3 ·11)
00
The mixing of different frequencies, being higher order effect, 1s neglected.
On combining Eq. (3 · 8) and the Maxwell equation for the effective field,
we find
1 :E
(Jj_ (q +K, w) )= - C
A
K',K'
{ xj_ (q+K, q +K', w) +-({5
e 2 (K -K', w=O)) }
m
(3·12)
where Xj_ and p 4K are the matrices of which (K, K') elements are Xj_ (q+K,
q+K',w) and ({J(K-K',w=O)), respectively. Equation (3·12) shows the
linear response of current to external field. Therefore we obtain
((Jj_ (q + K) jj_ (- q- K) ));~!,
=
:E {xj_ (q +K, q+K', w) +!t_({J(K-K', w= 0) >}
K'
m
x {(w2 - c 2 iq + K' /2) 1- 4n (xj_ + !t_P4K)} -l
m
q+K',q+K
(w 2 - c 2 iq + K/ 2).
(3 ·13)
3.2.
Neutron scattering cross section
On substituting Eq. (3 ·13) into Eq. (2 · 21), we obtain
-
2
( M )
2nh2
2
(rf.l.N)
k'
k
1
16nc 2 (q+K) 2
1-e-.eh., [w 2-c2(q+K) 2]
x Im :E{xj_ (q+K, q+K', w) + e 2 (p(K-K', w=O))}
K'
m
(3 ·14)
If we discard the Umklapp processes associated with the transverse screening,
414
K. Sasaki and Y. Obata
we have
Ne 2
X.L(q,q,(J))+--
m
1-
2
(!)
4n 2 2 [ X.L(q, q,
-cq
(3 ·15)
2
Ne ]
(!))+-m
and
-
2
( M )
2nh}
xi {
m
2
(rP.N)
k'
k
1
16nc2l
1-e-.B""' [(J) 2 -c2q 2]
X.L(q,q,(J))+Ne 2/m
}
(J) 2-c 2l-4n[x.L(q, q, (J)) +Ne 2/m] ·
(3 ·16)
In Eqs. (3 ·15) and (3 ·16) q may take on any value outside as well as
inside the first Brillouin zone. Hereafter we preferably use Eqs. (3 ·15)
and (3 ·16) for the sake of simplicity, expecting that effect of the Umklapp
processes on the transverse dynamic screening is much weaker than on X.L
itself.
On calculating the imaginary part of the expression in the braces in Eq.
(3 ·16), we finally obtain the differential cross section of neutrons by orbital
current :fluctuation,
d 2<J
d!JdE' =
(
-
M )2
2ntl
k'
1
( r/J.N) 2 k 1- e-.811.,
16nc2q 2 Im X.L (q, q, (J))
(J)) )2+ [4n
x~~--~--~~----~~~-=~~~~~----~--~~
2 - c 2q 2-4n ReX.L (q, q,
2 •
[(1) 2 ImX.L (q, q,
(J)p
(J)) ]
(3·17)
This expression may be rewritten in the form
(3 ·18)
where
((I) ) + c2q2ReX.L(q,q,(J))
4n
J+
(J) ) 2 + c;
te(q,(J))= [ 1- ( cq
2
2
[
4n
c2limX.L(q,q,(J))
J
2
(3·19)
and (J)p= (4nNe 2/m) 112 is the plasma frequency.
We first notice that the offending denominator (1)2 - c 2 q 2 in Eq. (3 ·16) has
disappeared in Eqs. (3 ·17) and (3 ·18). This suggests the internal consistency
of our approach.
Second, we see that the effect of two kinds of interaction between electrons on the transverse current-current response function is well separated
Current Correlations and Neutron Scattering
415
in Eq. (3 ·18). That is, the effect of instantaneous Coulomb interaction is
exclusively confined to X1_ (q, q, (J)), while the effect of interaction mediated
by the fluctuating electromagnetic field appears as a diamagnetic screening
factor IC ( q, (J)).
Third, Eq. (3 ·17) reduces to
d 26
(
M )2
d!JdE'=- 2nh2
k'
16n
1
(rfJ.N)2k 1-e~pfl., c2limX_j_(q,q,(J)), (3·20)
if all terms except c2 q 2 are ignored in the denominator of Eq. (3 ·17).
Equation (3 · 20) is essentially the scattering formula derived in conventional
ways by Szabo,16l Schneider,m and Lovesey and Windsor. 8> In these formulae
the retardation effect as well as diamagnetic screening effect accompanied by
the fluctuating transverse field are completely ignored. This truncation is
not always permissible, especially in the region of smallest q, as is shown
in the subsequent sections.
§ 4.
Scattering cross section on a free electron model
In 1956 Elliotes> calculated the inelastic cross section of slow neutron
by electrons in a metal on a free electron model with special reference to
sodium and found that it is very small except within a few minutes of arc
of the forward direction, where the experiments were virtually impossible
at that time because of the nearness of the unscattered beam. However, in
view of the remarkable progress in the small angle neutron scattering technique which has been made recently,t 9> it would be of practical significance
to examine new features introduced in the cross section by the diamagnetic
screening effect of which no account was taken by Elliott and explore the
feasibility to detect the effect by small angle neutron scattering experiments.
We apply general formula (3 ·17) or (3 ·18) to a free electron model
with parameters pertaining to copper; 3.6147 A for the fcc lattice constant
and one electron per atom for the density of electrons. The transverse paramagnetic current-current response function (3 · 9) was first calculated by
Szabon' 16> for free electron gas at T = 0,
_ __!_[1- (A- w)2]2lni1+A-w/AI+__!_[1- (A+
4A 8
A
1-A+w/A
4A 8
w)2]\n~1-A-w/AI}
A
1+A+w/A '
(4·1a)
=;
2 t{[1-(A+
~)Te(1-(A+ ~Y)-[1-(A- ~YTe(1-(A- ~Y)}.
(4·1b)
416
K. Sasaki and Y. Obata
On substituting Eqs. ( 4 ·1) into Eq. (3 ·19) we have
2E w
1
3w
1---F
-+-XP 1+--+-2mc A
2
).
2
tc(q W) = {
'
2
7C
2
[
2
2
). 4
__1_[14J.. (A.- w)2]\ni1+A.-wiA.I+__l_[11-;. +w 1;. 4J.. (A.+ w)2]ln~11+;.J..-wiA.I}2
+w I;.
8
8
;.
;.
+{~2 ~;[[1-(J..+ ;rre(1-(A.+ ;r)-[1-(J..- ;rre(1-(J..- ;r)J}2·
(4·2)
In Eqs. (4 ·1) and (4 · 2) Xp = 3Ne 2h218m2 c2 EF is the Pauli paramagnetic susceptibility of a free electron gas, EF = h 2P/I2m the Fermi energy and (x)
the step function as usual. The reduced scattering vector A. and energy transfer w are defined by A.=qi2PF and w=hwi4EF, respectively.
e
108
~
- lm XJ./Xp cf q2
-1m X.L/Xp c2 q2 K
ld
"bN
(.)
a..
106
;.<
.......
....
;.<
105
.§
*
4
-o 10
c
0
N
N
0'
ld~
(.)
a..
~
2
>(' 10
E
I
PF
tO'
EF
= 1.3586 z.-j
=7.0328 eV
0
10 0.0
W
~
~
M
M
W M
W ( =1'iw/4EF l (x103 l
M
IM
Fig. 1. The unscreened and screened transverse paramagnetic response function, - Im 'X.1_(q, q, w)
/c'q' and -Im 'X.1_(q, q, w)/c'q' "(q, w), on the free electron model with parameters appropriate for copper in the region of smallest q where the diamagnetic screening effect is salient.
The asterisk on the ordinate indicates the limiting maximum value of the screened transverse paramagnetic response function as wave vector q tends to zero, (8~<)-'(1 +4n'/3x,p)- 1
= (8~<) _, (1 +4~</i'X.Li) -•.
Current Correlations and Neutron Scattering
417
We have made a numerical calculation of the screened transverse paramagnetic current-current response function as a function of energy transfer,
- Im X.L (q, q, w) I c 2 q 2/C (q, w), for different wave vectors q. Seven sections of it
are depicted with solid lines in Fig. 1 for smallest reduced vectors A= 0.2,
0.5, 1.0, 2.0, 3.0, 4.0 and 5.0 X 10- 8, on which the effect of diamagnetic
screening is salient.
The sections of unscreened response function,
- Im x.L (q, q, w) I c"q", are also drawn there with broken lines for the same
reduced wave vectors to contrast with the screened ones.
It is readily shown that the maximum of -ImX.L(q,q,w)lc"q" as a function of hw diverges as q- 2 as q----'>0, while that of -Im X.L(q, q, w)lc 2 q21C(q, w)
tends to a finite value represented by the point marked with an asterisk
on the ordinate in Fig. 1. In fact, after some elementary but tedious calculus
we can show that the w coordinate of the crest of the imaginary part of the
screened response function tends to zero in cubic power of A as A----'>0, i.e.
(4·3)
The value of -ImX.L(q,q,w)lc2 q21C(q,w) at the crest tends eventually to
the limiting maximum value (8n-)- 1 (1+47rXpl3)- 1 =(87r)- 1 (1+4n-IXLI)- 1 as
A---"0,
(4·4)
We have also calculated the quantity - Im x.L (8, w) I c2 q2/C (8, w) for a given
energy of incident neutron Ei as a function of scattering angle 8 and energy
loss hw. Four different scattering conditions have been chosen: E 1 = 10.0 me V,
8 = 0.0° (forward scattering) ; Ei = 10.0 me V, 8 = 3.34° X 10- 8 ( = 12") ; Ei = 2.0
meV, 8=0.0°; Ei=2.0meV, 8=6.67°X10- 8 (=24"). The calculated screened
response functions, which may be thought as the inelastic neutron down scat"
tering spectra, are shown in Fig. 2 along with the unscreened ones.
We can draw several conclusions from Fig. 2:
First, all screened response functions for different scattering conditions
are bounded by a common lower limit -(8n-)- 1 (1+4IXLI)- 1 in marked contrast to the unscreened ones which diverges as q- 2 to minus infinity.
Second, the lower bound amounts practically to half the perfect diamagnetic susceptibility of a superconductor - (47r) -1, a value large enough
to be detected in a normal condition where XL is of order of -10- 6 •
Third, the energy transfer hwM at which the scattering cross section becomes maximum increases with incident energy E 1 : hwM=0.7, 2.25, 7.5, and
45.0 tJ.e V for Ei = 2.0, 10.0, 50.0 and 500.0 me V in the case of forward scattering, respectively.
Fourth, the diamagnetic screening is most prominent in the forward scattering and rapidly reduces its effect with deflection angle. The maximum
deflection angle within which the effect of diamagnetic screening is detectable
418
K. Sasaki and Y. Obata
I I
rz\- I E1 = 2.0 meV, 9 = 0. 0 deg , unscreened I
I I
1 \--IEi=IO.OmeV, 9=0.0deg, unscreenedl
I
I
I
I
I
I
I
I
I
1
\
\
;f'
\
'
\
'
'
\
/
r /,
,
~,
'
~
E1=2.0meV, 9=6.67 xiO deg, unscreenedl
~
'~')(IE1 =10.0 meV, 9=3.34 x 10 deg, unscreened I
--.lv........... '-.......... ....
f· . ' ·-·-·
"'
!Bxi'u ...437Cxpl1 ~
--- '-..,, :upper bound for-ImXl/Xpc2q 2 K
...........
.......
·- -.. . -~..:-
I
--~---------
--
!E 1=10.0meV, 9=0.0deg, screened)
I E1= 2.0meV, 9 =0.0 deg, screened I
where
---- • - ImX !B,wi!Xpc!lf, unscreened
- - • - ImXIB,wi/Xp c2q2K(8,w), screened
with
E1 =energy of incident neutron
9 =scattering angle
101' - ' - - - - - - - - ' ' - - - - - _ . __ _ _ _..___ _ _.......~..._ _ _ _.J...
1.0
2.0
3.0
4.0
5.0
ENERGY LOSS 'hw lp.eVI
Fig. 2. The inelastic neutron down scattering spectra with and without diamagnetic
screening on four different scattering conditions: energy of incident neutron E, = 10.0
meV, scattering angle 0=0.0"; E,=10.0meV, 0=3.34"X10- 8 (=12,.); E,=2.0meV,
0=0.0"; and E,=2.0meV, 0=6.67"xl0- 8 (=24..), which are calculated on a free electron model with parameters appropriate for copper. All scattering spectra with diamagnetic screening, -Imx1 (q,q,w)/c'q',.;(q,w), are bounded by a common upper
limit (87!') -• (1 +47!'/JCLI) -•.
decreases as the incident energy Ei increases.
Fifth, to summarize, the effect would be detectable by small angle
neutron scattering, if the angular resolution of less than 20 seconds of
arc and the energy resolution of less than 10 /leV were attained simultaneously for neutrons with incident energies less than 10"'20 me V.
Current Correlations and Neutron Scattering
§ 5.
419
Concluding remarks
In the preceding sections we observed that the diamagnetic screening
effect predominates and reduces the otherwise divergent neutron scattering
cross section considerably in the region of smallest q, giving it a common
upper limit independent of the scattering condition. However, as estimated
at the end of § 4, angular and energy resolutions required to detect the
effect is too high to attain even by the most advanced techniques which are
available now. We hope that these difficulties will be overcome and the
diamagnetic screening in a normal metal will be confirmed experimentally in
the not very distant future.
Acknowledgements
This paper is dedicated to Professor Ryogo Kubo on occasion of his
sixtieth birthday. The authors would like to express their hearty thanks to
Professor Kubo for his continual advice and encouragement. The authors
are grateful to Dr. ]. Chihara for the stimulating discussion. They are also
benefited from discussions on the experimental techniques in small angle neutron
scattering with Drs. K. Doi, M. Iizumi, H. Betsuyaku, S. Funahashi, K. Gesi
and H. Tomimitsu. This study was partly supported by Grant-in-Aid from
The Ministry of Education, Science and Culture.
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