- Email: [email protected]
Role of two-phonon excitations in the inelastic scattering of neutrons from He II
Volume
93A, number
4
PHYSICS
ROLE OF TWO-PHONON EXCITATIONS SCATTERING
10 January
LETTERS
1983
IN THE INELASTIC
OF NEUTRONS FROM He II
R. SRIDHAR and A. SHANTHI Matscience.
The Institute
of Mathematical
Received 30 August 1982 Revised manuscript received
3 November
Sciences, Madras-600
113, India
1982
The suggestion that the multi-excitation component of the dynamic mode is analysed on the basis of a model for the two-particle excitations data. A better agreement is anticipated if a physically relevant potential are also considered.
Recent theoretical attempts [ 1,2] to explain the second peak around the energy transfer ~25 K and the high-energy tail observed in the neutron inelastic scattering from liquid He II [3] have led to the conclusion that the two-particle excitations, which are distinct from the usual phonon-roton excitations, play a dominant role. In this context the following points are worth noting: Wong’s argument [ I], though independent of particle statistics and long-range order, depends on two adjustable parameters - one to take care of the approximations introduced and the other to account for three- or more-particle excitations. Further, Wong’s theory does not explain the second peak. Tripathy et al’s attempt [2] to improve this situation also needs further refinements for the following reasons: the role of single-particle energies wt(k) in determining the two-particle matrix element has been neglected. A closure approximation IO)<0 1 + 11X 11 + 12X2 1= unit operator
(1)
has been used. In this (0) denotes the ground state and In> denotes the n-particle state. This approximation actually amounts to ignoring the structure of the two-particle state in the evaluation of the twoparticle matrix element. Further, the angles between pairs of momenta in the energy-conservation condition (delta function) have been omitted on the ground that the terms in the argument of the delta
structure function is dominated by the two-phonon resulting in a qualitative fit to the experimental is employed and higher (threeor more) correlations
function can be replaced by their averages. On the other hand, it is suspected that if all the angles are included, the second peak itself may vanish! To evaluate the two-particle intensity explicitly, Tripathy et al. [2] introduce limiting forms of S(k) for large and small momentum transfers. It is evident that these limiting forms have not been evaluated taking into consideration the existence of two-particle excitations. Also direct substitution of the limiting forms within the integral is possible because the neglect of the angles within the argument of the delta function directly links small (large) 4 to small (large) w values for a fixed momentum transfer k. The present contribution is a reformulation of the existing approach incorporating a suitable structure factor in the framework of a plausible model for twoparticle excitations. A suitable S(k) which exhibits the necessary limiting behaviour is provided by the arguments put forth by Sridhar and Vasudevan [4] who have obtained an integral equation for S(k) by assuming that there exists a second branch of excitations in liquid He II with a sharply defined frequency and that the first few sum rules are saturated by single- and two-particle excitations. This procedure can be easily generalised to the case when the excitation corresponds to a diffuse peak [S]. In the present contribution S(k) as derived in this reference is used. It has been shown [5] that if the two-particle ex0 031-9163/83/0000-0000/S
03.00 0 1983 North-Holland
Volume 93A, number 4
PHYSICS LETTERS
10 January 1983
citation is described by Ik), = c
f(k,p)
(k,p
lk,p) IO) >
=~~l/~(li,p)pt(pl)pT(P2) P
(2)
wherepI =ik +p,p2 =$k -p,p’f(k)is the density fluctuation operator. If the zeroth, first and third moment sum rules are saturated by single- and two-particle states, the structure factor is given by the integral equation S(k) =
fi2k2 2mwl(k)
R4k4 M -E) - G(k) m2 N
f’(q)@ - d21IS(q)- 1I
(3)
1
with
G(k) =
c w,(k) w2(k,4)f(k,q) [q @>+ q@, q)l ’
(4)
4
The energy 02(k,q) associated with the state Pt(PJ) X pt(p2)]0) is given by c+(k,P)
= wJ(PJ)
+ c+(p2)
(5)
2
with w,(k)
= [h4k4/4m2
t (h2k2/m)p
V(k)] ‘1’ .
(6)
Eq. (3) is possible provided
Cf(k,d= P
1.
(7)
In the case when the potential V(k) is a constant, say VO, the integral equation (3) can be solved exactly and the solution denoted by So(k) is used in this present contribution. Further a model for the two-particle excitations is assumed in which the two single-particle components have oppositely directed momenta p J and p2 with the net momentum adding to k. For k = 0 this will be analogous to the Cooper pair. It should, however, be noted that the two-particle excitations are expected to be well defined only for small values of the total momentum k. Following the procedure used in ref. [6] and without using the closure approximation (l), the two-particle matrix element is evaluated to be
k2N2
mS2 [a2 - of(k)]
so that the two-particle will be
X2&, w) =
contribution
-4)
(8)
to the intensity
h4N4 .
m2f12 [02 - wf(k)12
cP f(ktP){(kvQ’o~~@~)S~(P~)~~
x &Co-
)
=
XqGk Vo(k.4)S,(s)So(k-4)~@2
X
t~~(V(,yt’l)[k.(u+k)12 -
Id(k)
q(Pl)
- 9(P2)).
(9)
V. is chosen so as to reproduce the observed velocity of sound C= (Vop/m)1/2 = 237 m/s. There is no procedure by which f(k, p) could be determined uniquely in the present formulation. However, on physical grounds it is expected that the single-particle components in (k), are uncorrelated for large values of the momenta. Hence the coefficient of admixture of the states Ik,p) in Ik), should become negligibly small for large values of the momenta. Consequently f(k,p) is expected to be peaked for small values of p and become vanishingly small for p % 1. As has already been pointed out Ik), itself is well defined for values of k smaller than 2.3 8-l [4]. Also f(k, p) is associated with the spread around the two-particle peak and consequently with the life time of the two-particle excitations. Thus the most probable choice is a gaussian when these excitations are fairly stable so that it can possibly be generalised to include finite-life-time effects. Also such a choice is suggested by the form of S(k, o) for the thermal scattering of neutrons from an isotropic harmonic potential for small values of the wave vector transfer [8]. Consequently the following form is chosen:
(27rc11)3/2 I. f(k,p)=
s1
k3 exp(-
i ~YJk2p2),
(10)
so that condition (7) is also satisfied. Following the analogy of scattering from an isotropic harmonic potential [8] one can choose another form: f(k,p)aev-
:a2(k2
However the normalisation form
+p2)1
.
condition
(7) yields the 199
Volume
93A, number
PHYSICS
4
(27rG# 11. f(k,p)=
a
(11)
f a2p2).
dk k” [S(k) - l] = -1
(12)
0
at least approximately, since only single- and twoparticle excitations have been included in the calculation. Within the limited variation of the parameters oi attempted in the present contribution the following are found to be the best possible choices a!1 = 0.5 84
“2 =9.082.
k= 0.8/i-’
cl
10
20
Neutron
30
40
50
energy
loss
-I
60
70
(K)
z.k!
r
2
L
I 10
Neutron
1
I
20
30
energy
loss
(K)
Fig. 1. Intensity of inelastically scattered neutrons from liquid He II for fixed momentum transfer k versus neutron energy loss in thermal units. Fig. la gives the intensity fork = 0.8 A-’ and fig. lb for k = 0.6 A-r. The dots represent the experimental data [3], the dashed line the results of ref. [2] and the continuous curve the present theory, corresponding to the choice I for f(k) p).
200
10 January
1983
Table 1
exp(-
However, this form becomes unphysical as it does not depend on the total momentum. The parameters CQ(i = 1, 2) are determined by requiring that the corresponding S(k) should satisfy the requirement [9] (2r~*p)-~ 7
LETTERS
Momentum transfer (A-’ )
Peak position
for case I (K)
calculated
experimental
0.6
21
%20.07
0.8
21
e24.5
The calculated two-particle intensities for two values of the momentum transfer are given in fig. 1. The peak positions for case 1 [eq. (lo)] are given in table 1. The theoretically obtained tail part of the intensity is slightly higher than the experimental observation for k = 0.6 8-l and slightly lower for k = 0.8 8-l _ However, for smaller energy transfers the fit at k = 0.6 8-l is good The fit with experiment for the choice II [eq. (1 l)] not only gives the peak position wrongly but also does not reproduce the width and the high-energy tail. This is essentially due to the lack of dependence on the total momentum. Since the present attempt uses a structure factor consistent with the existence of two- article excitations, has the correct factor [w* _ w11 (k)] -I in the two-particle matrix element and takes into account the angles properly in the framework of the model proposed, it is concluded that the two-phonon excitations can only lead to a qualitative fit to the data. A better quantitative fit is anticipated if a realistic potential is used instead of V. and higherorder (more than two) correlations are properly included. A detailed analysis along these lines is in progress and will be reported elsewhere.
[l] V.K. Wong, Phys. Lett. 61A (1977) 454. [2] D.N. Tripathy and M. Bhuyan, Phys. Lett. 86A (1981) 303. [3] A.D.B. Woods and R.A. Cowley, Rep. Prog. Phys. 36 (1973) 1135;Can.J.Phys.49(1971) 177. [4] R. Sridhar and R. Vasudevan, Acta. Phys. Pol. A61 (1982). 545. [S] R. Sridhar, to be published (1982). [6] D. Pines, Elementary excitations in solids (Benjamin, New York, 1964). [7] R.A. Cowlcy, in: Correlation functions and quasiparticle interactions in condensed matter, ed. J. Woods Halley (Plenum, New York, 1977) p. 596. (81 W. Marshall and S.W. Lovesey, Theory of thermal neutron scattering (Clarendon. Oxford, 1971) p. 59. [9] E. Feenberg, Quantum theory of fluids (Academic Press, New York, 1969).
93A, number
4
PHYSICS
ROLE OF TWO-PHONON EXCITATIONS SCATTERING
10 January
LETTERS
1983
IN THE INELASTIC
OF NEUTRONS FROM He II
R. SRIDHAR and A. SHANTHI Matscience.
The Institute
of Mathematical
Received 30 August 1982 Revised manuscript received
3 November
Sciences, Madras-600
113, India
1982
The suggestion that the multi-excitation component of the dynamic mode is analysed on the basis of a model for the two-particle excitations data. A better agreement is anticipated if a physically relevant potential are also considered.
Recent theoretical attempts [ 1,2] to explain the second peak around the energy transfer ~25 K and the high-energy tail observed in the neutron inelastic scattering from liquid He II [3] have led to the conclusion that the two-particle excitations, which are distinct from the usual phonon-roton excitations, play a dominant role. In this context the following points are worth noting: Wong’s argument [ I], though independent of particle statistics and long-range order, depends on two adjustable parameters - one to take care of the approximations introduced and the other to account for three- or more-particle excitations. Further, Wong’s theory does not explain the second peak. Tripathy et al’s attempt [2] to improve this situation also needs further refinements for the following reasons: the role of single-particle energies wt(k) in determining the two-particle matrix element has been neglected. A closure approximation IO)<0 1 + 11X 11 + 12X2 1= unit operator
(1)
has been used. In this (0) denotes the ground state and In> denotes the n-particle state. This approximation actually amounts to ignoring the structure of the two-particle state in the evaluation of the twoparticle matrix element. Further, the angles between pairs of momenta in the energy-conservation condition (delta function) have been omitted on the ground that the terms in the argument of the delta
structure function is dominated by the two-phonon resulting in a qualitative fit to the experimental is employed and higher (threeor more) correlations
function can be replaced by their averages. On the other hand, it is suspected that if all the angles are included, the second peak itself may vanish! To evaluate the two-particle intensity explicitly, Tripathy et al. [2] introduce limiting forms of S(k) for large and small momentum transfers. It is evident that these limiting forms have not been evaluated taking into consideration the existence of two-particle excitations. Also direct substitution of the limiting forms within the integral is possible because the neglect of the angles within the argument of the delta function directly links small (large) 4 to small (large) w values for a fixed momentum transfer k. The present contribution is a reformulation of the existing approach incorporating a suitable structure factor in the framework of a plausible model for twoparticle excitations. A suitable S(k) which exhibits the necessary limiting behaviour is provided by the arguments put forth by Sridhar and Vasudevan [4] who have obtained an integral equation for S(k) by assuming that there exists a second branch of excitations in liquid He II with a sharply defined frequency and that the first few sum rules are saturated by single- and two-particle excitations. This procedure can be easily generalised to the case when the excitation corresponds to a diffuse peak [S]. In the present contribution S(k) as derived in this reference is used. It has been shown [5] that if the two-particle ex0 031-9163/83/0000-0000/S
03.00 0 1983 North-Holland
Volume 93A, number 4
PHYSICS LETTERS
10 January 1983
citation is described by Ik), = c
f(k,p)
(k,p
lk,p) IO) >
=~~l/~(li,p)pt(pl)pT(P2) P
(2)
wherepI =ik +p,p2 =$k -p,p’f(k)is the density fluctuation operator. If the zeroth, first and third moment sum rules are saturated by single- and two-particle states, the structure factor is given by the integral equation S(k) =
fi2k2 2mwl(k)
R4k4 M -E) - G(k) m2 N
f’(q)@ - d21IS(q)- 1I
(3)
1
with
G(k) =
c w,(k) w2(k,4)f(k,q) [q @>+ q@, q)l ’
(4)
4
The energy 02(k,q) associated with the state Pt(PJ) X pt(p2)]0) is given by c+(k,P)
= wJ(PJ)
+ c+(p2)
(5)
2
with w,(k)
= [h4k4/4m2
t (h2k2/m)p
V(k)] ‘1’ .
(6)
Eq. (3) is possible provided
Cf(k,d= P
1.
(7)
In the case when the potential V(k) is a constant, say VO, the integral equation (3) can be solved exactly and the solution denoted by So(k) is used in this present contribution. Further a model for the two-particle excitations is assumed in which the two single-particle components have oppositely directed momenta p J and p2 with the net momentum adding to k. For k = 0 this will be analogous to the Cooper pair. It should, however, be noted that the two-particle excitations are expected to be well defined only for small values of the total momentum k. Following the procedure used in ref. [6] and without using the closure approximation (l), the two-particle matrix element is evaluated to be
k2N2
mS2 [a2 - of(k)]
so that the two-particle will be
X2&, w) =
contribution
-4)
(8)
to the intensity
h4N4 .
m2f12 [02 - wf(k)12
cP f(ktP){(kvQ’o~~@~)S~(P~)~~
x &Co-
)
=
XqGk Vo(k.4)S,(s)So(k-4)~@2
X
t~~(V(,yt’l)[k.(u+k)12 -
Id(k)
q(Pl)
- 9(P2)).
(9)
V. is chosen so as to reproduce the observed velocity of sound C= (Vop/m)1/2 = 237 m/s. There is no procedure by which f(k, p) could be determined uniquely in the present formulation. However, on physical grounds it is expected that the single-particle components in (k), are uncorrelated for large values of the momenta. Hence the coefficient of admixture of the states Ik,p) in Ik), should become negligibly small for large values of the momenta. Consequently f(k,p) is expected to be peaked for small values of p and become vanishingly small for p % 1. As has already been pointed out Ik), itself is well defined for values of k smaller than 2.3 8-l [4]. Also f(k, p) is associated with the spread around the two-particle peak and consequently with the life time of the two-particle excitations. Thus the most probable choice is a gaussian when these excitations are fairly stable so that it can possibly be generalised to include finite-life-time effects. Also such a choice is suggested by the form of S(k, o) for the thermal scattering of neutrons from an isotropic harmonic potential for small values of the wave vector transfer [8]. Consequently the following form is chosen:
(27rc11)3/2 I. f(k,p)=
s1
k3 exp(-
i ~YJk2p2),
(10)
so that condition (7) is also satisfied. Following the analogy of scattering from an isotropic harmonic potential [8] one can choose another form: f(k,p)aev-
:a2(k2
However the normalisation form
+p2)1
.
condition
(7) yields the 199
Volume
93A, number
PHYSICS
4
(27rG# 11. f(k,p)=
a
(11)
f a2p2).
dk k” [S(k) - l] = -1
(12)
0
at least approximately, since only single- and twoparticle excitations have been included in the calculation. Within the limited variation of the parameters oi attempted in the present contribution the following are found to be the best possible choices a!1 = 0.5 84
“2 =9.082.
k= 0.8/i-’
cl
10
20
Neutron
30
40
50
energy
loss
-I
60
70
(K)
z.k!
r
2
L
I 10
Neutron
1
I
20
30
energy
loss
(K)
Fig. 1. Intensity of inelastically scattered neutrons from liquid He II for fixed momentum transfer k versus neutron energy loss in thermal units. Fig. la gives the intensity fork = 0.8 A-’ and fig. lb for k = 0.6 A-r. The dots represent the experimental data [3], the dashed line the results of ref. [2] and the continuous curve the present theory, corresponding to the choice I for f(k) p).
200
10 January
1983
Table 1
exp(-
However, this form becomes unphysical as it does not depend on the total momentum. The parameters CQ(i = 1, 2) are determined by requiring that the corresponding S(k) should satisfy the requirement [9] (2r~*p)-~ 7
LETTERS
Momentum transfer (A-’ )
Peak position
for case I (K)
calculated
experimental
0.6
21
%20.07
0.8
21
e24.5
The calculated two-particle intensities for two values of the momentum transfer are given in fig. 1. The peak positions for case 1 [eq. (lo)] are given in table 1. The theoretically obtained tail part of the intensity is slightly higher than the experimental observation for k = 0.6 8-l and slightly lower for k = 0.8 8-l _ However, for smaller energy transfers the fit at k = 0.6 8-l is good The fit with experiment for the choice II [eq. (1 l)] not only gives the peak position wrongly but also does not reproduce the width and the high-energy tail. This is essentially due to the lack of dependence on the total momentum. Since the present attempt uses a structure factor consistent with the existence of two- article excitations, has the correct factor [w* _ w11 (k)] -I in the two-particle matrix element and takes into account the angles properly in the framework of the model proposed, it is concluded that the two-phonon excitations can only lead to a qualitative fit to the data. A better quantitative fit is anticipated if a realistic potential is used instead of V. and higherorder (more than two) correlations are properly included. A detailed analysis along these lines is in progress and will be reported elsewhere.
[l] V.K. Wong, Phys. Lett. 61A (1977) 454. [2] D.N. Tripathy and M. Bhuyan, Phys. Lett. 86A (1981) 303. [3] A.D.B. Woods and R.A. Cowley, Rep. Prog. Phys. 36 (1973) 1135;Can.J.Phys.49(1971) 177. [4] R. Sridhar and R. Vasudevan, Acta. Phys. Pol. A61 (1982). 545. [S] R. Sridhar, to be published (1982). [6] D. Pines, Elementary excitations in solids (Benjamin, New York, 1964). [7] R.A. Cowlcy, in: Correlation functions and quasiparticle interactions in condensed matter, ed. J. Woods Halley (Plenum, New York, 1977) p. 596. (81 W. Marshall and S.W. Lovesey, Theory of thermal neutron scattering (Clarendon. Oxford, 1971) p. 59. [9] E. Feenberg, Quantum theory of fluids (Academic Press, New York, 1969).