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Atomic and molecular momentum distributions in quantum fluids by Neutron Compton Scattering
IB ELSEVIER
Physica B 234-236 (1997) 329-330
Atomic and molecular momentum distributions in quantum fluids by Neutron Compton Scattering C. Andreani a, D. Colognesi a, A. Filabozzi a, M. N a r d o n e b, E. Pace c aDipartimento di Fisica, Istituto Nazionale di Fisica della Materia, Universit?t di Roma "Tor Vergata ", Via della Ricerca Scientifica 1, 00133 Roma, Italy bDipartimento di Fisica e Istituto Nazionale di Fisica della Materia, Universitlt di Roma 11I, Via della Vasca Navale 37, 00154 Roma, Italy Dipartimento di Fisica, Istituto Nazionale di Fisica Nucleare, Universit?~ di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy
Abstract
Neutron Compton Scattering (NCS) measurements can provide direct information about mean kinetic energy of molecules in quantum fluids. The principal tool in interpreting NCS experiments is the Impulse Approximation (IA). A different approach to the scattering process, which takes into account the exact total energy of the initial state and the average potential energy of the final one, is investigated. This approach describes experimental data for a quantum molecular fluid (D2) more precisely than IA.
Keywords: Hydrogen; Momentum distribution; Quantum
NCS, that is achieved at high energy (hco > 1 eV) and momentum (q > 20 A - 1) transfers, is a sensitive probe of the single-particle properties of the matter, like mean kinetic energy and momentum distribution n(k). The theoretical framework for the interpretation of NCS makes use of IA, valid in the limit for ({o,q) ~ oo, which assumes that the scattering proceeds incoherently and that particle struck by the incident neutron recoils as a free particle [1]. NCS has been so far mostly applied to quantum monatomic systems and also to some molecular systems like liquid D2 [2] and H2 I-3, 4]. In fact in a real experiment the infinite limits of q and co are never reached. Deviations of the response function S(q, co) from IA, known as final state interactions (FSI), can then occur: these are due to deviations of the final states of the particle *Corresponding author. Fax: + 39-6-2023507; [email protected].
e-mail:
effects
from the plane-wave form [1]. Indeed there is another source of deviations which reflects properties of the initial state of the particle [5]. In Ref. [6], in order to improve IA for 4He, the following expression for the response function was suggested:
S(q, co) =
fn(k)f(h~o
h2(k +
(V)
+ Ei)dk
(1)
where El is the total energy of the initial state and ( V ) is the mean value of the potential energy of final state, which takes approximately into account the effects of the neighbour atoms on the recoiling one. Using a well known sum rule for incoherent structure factor [6], it is possible to get rid of ( V ) , replacing ( E l - ( V ) ) with (Eke): the mean kinetic energy of the initial state. It is very useful in what follows to introduce the scaling variable
0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 9 7 5 1
330
C, Andreani et al. / Physica B 234 236 (1997) 329-330
y = Mco/(hq)-q/2 and the response function F(y, q) = h2q/(M) S(q, co). For a diatomic molecule the scattering can occur in two typical situations: if co is much larger than the typical energy of collective excitations, but lower than the typical vibrational energy of the molecule, one gets scattering from the whole molecule and can extract information on nm(k), the centre of mass momentum distribution of the molecule, if co largely exceeds both the typical energies, scattering happens from the single atom belonging to the molecule, and then the momentum distribution nv(k), describing the relative motion of the atoms in the molecule, is also involved. In this paper we analyse experimental data on quantum liquid D2 at T = 20.7 K and p = 24.5 n m - 3, where the latter framework applies. In [2] IA has been used for this quantum fluid and an overall agreement with the experimental data has been found, but some small discrepancies at the lowest q values [q ~ (30-40) A -1] are evident. In [2] F(y,q), the response function for a single atom, is described as the convolution of two distinct contributions, namely Fro(y,q) and Fv (y, q), coming from the momentum distributions nm(k) and nv (k), respectively:
F(y,q) = 2 f~_o Fm(2y',q) Fv(y - y',q) dy'.
(2)
We have changed the functional form of these two components according to Stringari approximation [Eq. (1)]:
fm(Y'q)=fnm(k)6( y
k'q q
k2 + ~q 2M (E~) ) dk 2q
Fv(y,q) fnv(k)f(y
k.q 2 -q~ kq~-~(E~i)) dk
. . . .
0.1
I
. . . .
I
. . . .
I
....
0.08
:L ~ " 0.04
0.02
0 -20
-I0
0
I0
20
y (A-1) Fig. 1. Experimental scaling function F(y,q) for scattering angle 0 = 36.00 ° (q = 31.9 A -~ at the maximum of the recoil peak) compared with IA result [2] (dashed line) and our result from Eqs. (2) and (3) (full line). All the functions are broadened by instrumental Voigt resolution (a = 0 . 9 1 4 A 1, F = 2.272 ,~ 1) [2].
the maximum of the curve from of y = 0. These experimental shifts have been analysed for a large q range (32 ,A- 1 < q < 79 A- 1) together with our model prediction for the shifts and a linear fit (l/q) has been performed for both of them. The slopes for the two lines are: ~exp = - 32 A-2 and Ctth= -- 39 ~,-2 for the experimental data and the model prediction, respectively. The slight discrepancies of our model with respect to experimental data, still present, can be clearly explained remembering that FSI effects, which changes the shape of the F (y, q) have been completely neglected, because the effect of the potential V has been evaluated only in an approximate way. Future work will be devoted to develop an accurate calculation of FSI in NCS from diatomic molecules.
(3) where (E~i) and (Eli) are the initial kinetic energy of the molecular and vibrational motion, respectively. We have computed F(y, q) of Eq. (2), using = 4 meV and
References [1] G.I. Watson, J. Condens. Matter 8 (1996) 5955. [2] C. Andreani, A. Filabozzi and E. Pace, Phys. Rev. B 51 (1995) 8854. 1-3] W. Langel et al., Phys. Rev. B 38 (1988) 11275. [4] J. Mayers, Phys. Rev. Lett. 71 (1993) 1553. 1.5] J. Mayers, C. Andreani and G. Baciocco, Phys. Rev. B 39 (1989) 2022. [6] S. Stringari, Phys. Rev. B 35 (1987) 2038.
Physica B 234-236 (1997) 329-330
Atomic and molecular momentum distributions in quantum fluids by Neutron Compton Scattering C. Andreani a, D. Colognesi a, A. Filabozzi a, M. N a r d o n e b, E. Pace c aDipartimento di Fisica, Istituto Nazionale di Fisica della Materia, Universit?t di Roma "Tor Vergata ", Via della Ricerca Scientifica 1, 00133 Roma, Italy bDipartimento di Fisica e Istituto Nazionale di Fisica della Materia, Universitlt di Roma 11I, Via della Vasca Navale 37, 00154 Roma, Italy Dipartimento di Fisica, Istituto Nazionale di Fisica Nucleare, Universit?~ di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy
Abstract
Neutron Compton Scattering (NCS) measurements can provide direct information about mean kinetic energy of molecules in quantum fluids. The principal tool in interpreting NCS experiments is the Impulse Approximation (IA). A different approach to the scattering process, which takes into account the exact total energy of the initial state and the average potential energy of the final one, is investigated. This approach describes experimental data for a quantum molecular fluid (D2) more precisely than IA.
Keywords: Hydrogen; Momentum distribution; Quantum
NCS, that is achieved at high energy (hco > 1 eV) and momentum (q > 20 A - 1) transfers, is a sensitive probe of the single-particle properties of the matter, like mean kinetic energy and momentum distribution n(k). The theoretical framework for the interpretation of NCS makes use of IA, valid in the limit for ({o,q) ~ oo, which assumes that the scattering proceeds incoherently and that particle struck by the incident neutron recoils as a free particle [1]. NCS has been so far mostly applied to quantum monatomic systems and also to some molecular systems like liquid D2 [2] and H2 I-3, 4]. In fact in a real experiment the infinite limits of q and co are never reached. Deviations of the response function S(q, co) from IA, known as final state interactions (FSI), can then occur: these are due to deviations of the final states of the particle *Corresponding author. Fax: + 39-6-2023507; [email protected].
e-mail:
effects
from the plane-wave form [1]. Indeed there is another source of deviations which reflects properties of the initial state of the particle [5]. In Ref. [6], in order to improve IA for 4He, the following expression for the response function was suggested:
S(q, co) =
fn(k)f(h~o
h2(k +
(V)
+ Ei)dk
(1)
where El is the total energy of the initial state and ( V ) is the mean value of the potential energy of final state, which takes approximately into account the effects of the neighbour atoms on the recoiling one. Using a well known sum rule for incoherent structure factor [6], it is possible to get rid of ( V ) , replacing ( E l - ( V ) ) with (Eke): the mean kinetic energy of the initial state. It is very useful in what follows to introduce the scaling variable
0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 9 7 5 1
330
C, Andreani et al. / Physica B 234 236 (1997) 329-330
y = Mco/(hq)-q/2 and the response function F(y, q) = h2q/(M) S(q, co). For a diatomic molecule the scattering can occur in two typical situations: if co is much larger than the typical energy of collective excitations, but lower than the typical vibrational energy of the molecule, one gets scattering from the whole molecule and can extract information on nm(k), the centre of mass momentum distribution of the molecule, if co largely exceeds both the typical energies, scattering happens from the single atom belonging to the molecule, and then the momentum distribution nv(k), describing the relative motion of the atoms in the molecule, is also involved. In this paper we analyse experimental data on quantum liquid D2 at T = 20.7 K and p = 24.5 n m - 3, where the latter framework applies. In [2] IA has been used for this quantum fluid and an overall agreement with the experimental data has been found, but some small discrepancies at the lowest q values [q ~ (30-40) A -1] are evident. In [2] F(y,q), the response function for a single atom, is described as the convolution of two distinct contributions, namely Fro(y,q) and Fv (y, q), coming from the momentum distributions nm(k) and nv (k), respectively:
F(y,q) = 2 f~_o Fm(2y',q) Fv(y - y',q) dy'.
(2)
We have changed the functional form of these two components according to Stringari approximation [Eq. (1)]:
fm(Y'q)=fnm(k)6( y
k'q q
k2 + ~q 2M (E~) ) dk 2q
Fv(y,q) fnv(k)f(y
k.q 2 -q~ kq~-~(E~i)) dk
. . . .
0.1
I
. . . .
I
. . . .
I
....
0.08
:L ~ " 0.04
0.02
0 -20
-I0
0
I0
20
y (A-1) Fig. 1. Experimental scaling function F(y,q) for scattering angle 0 = 36.00 ° (q = 31.9 A -~ at the maximum of the recoil peak) compared with IA result [2] (dashed line) and our result from Eqs. (2) and (3) (full line). All the functions are broadened by instrumental Voigt resolution (a = 0 . 9 1 4 A 1, F = 2.272 ,~ 1) [2].
the maximum of the curve from of y = 0. These experimental shifts have been analysed for a large q range (32 ,A- 1 < q < 79 A- 1) together with our model prediction for the shifts and a linear fit (l/q) has been performed for both of them. The slopes for the two lines are: ~exp = - 32 A-2 and Ctth= -- 39 ~,-2 for the experimental data and the model prediction, respectively. The slight discrepancies of our model with respect to experimental data, still present, can be clearly explained remembering that FSI effects, which changes the shape of the F (y, q) have been completely neglected, because the effect of the potential V has been evaluated only in an approximate way. Future work will be devoted to develop an accurate calculation of FSI in NCS from diatomic molecules.
(3) where (E~i) and (Eli) are the initial kinetic energy of the molecular and vibrational motion, respectively. We have computed F(y, q) of Eq. (2), using
References [1] G.I. Watson, J. Condens. Matter 8 (1996) 5955. [2] C. Andreani, A. Filabozzi and E. Pace, Phys. Rev. B 51 (1995) 8854. 1-3] W. Langel et al., Phys. Rev. B 38 (1988) 11275. [4] J. Mayers, Phys. Rev. Lett. 71 (1993) 1553. 1.5] J. Mayers, C. Andreani and G. Baciocco, Phys. Rev. B 39 (1989) 2022. [6] S. Stringari, Phys. Rev. B 35 (1987) 2038.