Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering
We have measured, using the TOSCA spectrometer at ISIS, the neutron scattering spectrum of liquid para-hydrogen ( T = 19.3 K and n = 25.4 nm⁻³ ) and liquid ortho-deuterium ( T = 21.3 K and n = 25.2 nm⁻³ ). From the high-energy region of the spectrum, namely 40 meV < ~ω < 1 eV, where the...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering / G. Corradi, D. Colognesi, M. Celli, M. Zoppi // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 499-521. — Бібліогр.: 76 назв. — англ. |
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Corradi, G. Colognesi, D. Celli, M. Zoppi, M. 2017-06-12T18:48:14Z 2017-06-12T18:48:14Z 2003 Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering / G. Corradi, D. Colognesi, M. Celli, M. Zoppi // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 499-521. — Бібліогр.: 76 назв. — англ. 1607-324X PACS: 67.90.+z, 61.12.Ex DOI:10.5488/CMP.6.3.499 https://nasplib.isofts.kiev.ua/handle/123456789/120749 We have measured, using the TOSCA spectrometer at ISIS, the neutron scattering spectrum of liquid para-hydrogen ( T = 19.3 K and n = 25.4 nm⁻³ ) and liquid ortho-deuterium ( T = 21.3 K and n = 25.2 nm⁻³ ). From the high-energy region of the spectrum, namely 40 meV < ~ω < 1 eV, where the incoherent approximation for the center-of-mass motion applies, we have been able to extract the translational mean kinetic energy, which, as expected, turns out to be rather different from the classical values. However, significant deviations from the impulse approximation have been detected and the inclusion of some correction terms (accounting for the so-called final state effects) had to be accomplished in order to accurately describe the experimental spectra. The comparison of the present mean kinetic energy values with the available experimental and simulated data in the literature is quite good and confirms the excellent performances of TOSCA in the spectroscopic analysis of the condensed phases of liquid hydrogens. Використовуючи спектрометр TOSCA в ISIS нами було проведено вимірювання спектрів нейтронного розсіяння рідкого пара-водню (T = 19.3 K і n = 25.4 nm⁻³ ) і рідкого орто-дейтерію (T = 21.3 K і n = 25.2 nm⁻³ ). З високоенергетичної області спектру 40 meV < ~ω < 1 eV, де застосовується некогерентне наближення для руху центра мас, маємо можливість отримати трансляційну середню кінетичну енергію, значення якої, як ми і сподівалися, відрізняється від класичних значень. Проте, було відзначено сильні відхилення від імпульсного наближення і тому певні поправочні доданки (врахування т.зв. ефектів кінцевого стану) повинні бути взяті до розгляду для точного опису експериментальних спектрів. Порівняння наявних даних по середній кінетичній енергії з значеннями експериментальними чи з моделювань, доступними в літературі, є зовсім добрим і підтверджує високу якість процедури спектрального аналізу на спектрометрі TOSCA для конденсованих фаз рідких воднів. This work has been financially supported by C.N.R. (Italy). The authors acknowledge the skillful technical support of the ISIS Sample Environment Section. One of the authors (G.C.) is also indebted to Accademia Nazionale dei Lincei (Italy) for the research grant Borsa di Perfezionamento negli Studi di Spettroscopia Neutronica “F.P.Ricci”. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering Вивчення середньої кінетичної енергії та залишкових ефектів в рідкому водні за допомогою непружного нейтронного розсіяння Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering |
| spellingShingle |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering Corradi, G. Colognesi, D. Celli, M. Zoppi, M. |
| title_short |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering |
| title_full |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering |
| title_fullStr |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering |
| title_full_unstemmed |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering |
| title_sort |
mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering |
| author |
Corradi, G. Colognesi, D. Celli, M. Zoppi, M. |
| author_facet |
Corradi, G. Colognesi, D. Celli, M. Zoppi, M. |
| publishDate |
2003 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Вивчення середньої кінетичної енергії та залишкових ефектів в рідкому водні за допомогою непружного нейтронного розсіяння |
| description |
We have measured, using the TOSCA spectrometer at ISIS, the neutron
scattering spectrum of liquid para-hydrogen ( T = 19.3 K and n =
25.4 nm⁻³
) and liquid ortho-deuterium ( T = 21.3 K and n = 25.2 nm⁻³
).
From the high-energy region of the spectrum, namely 40 meV < ~ω <
1 eV, where the incoherent approximation for the center-of-mass motion
applies, we have been able to extract the translational mean kinetic energy,
which, as expected, turns out to be rather different from the classical values.
However, significant deviations from the impulse approximation have
been detected and the inclusion of some correction terms (accounting for
the so-called final state effects) had to be accomplished in order to accurately
describe the experimental spectra. The comparison of the present
mean kinetic energy values with the available experimental and simulated
data in the literature is quite good and confirms the excellent performances
of TOSCA in the spectroscopic analysis of the condensed phases of liquid
hydrogens.
Використовуючи спектрометр TOSCA в ISIS нами було проведено
вимірювання спектрів нейтронного розсіяння рідкого пара-водню
(T = 19.3 K і n = 25.4 nm⁻³
) і рідкого орто-дейтерію (T = 21.3 K
і n = 25.2 nm⁻³
). З високоенергетичної області спектру 40 meV <
~ω < 1 eV, де застосовується некогерентне наближення для руху
центра мас, маємо можливість отримати трансляційну середню кінетичну енергію, значення якої, як ми і сподівалися, відрізняється
від класичних значень. Проте, було відзначено сильні відхилення від
імпульсного наближення і тому певні поправочні доданки (врахування т.зв. ефектів кінцевого стану) повинні бути взяті до розгляду для
точного опису експериментальних спектрів. Порівняння наявних даних по середній кінетичній енергії з значеннями експериментальними чи з моделювань, доступними в літературі, є зовсім добрим і підтверджує високу якість процедури спектрального аналізу на спектрометрі TOSCA для конденсованих фаз рідких воднів.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120749 |
| citation_txt |
Mean kinetic energy and final state effects in liquid hydrogens from inelastic neutron scattering / G. Corradi, D. Colognesi, M. Celli, M. Zoppi // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 499-521. — Бібліогр.: 76 назв. — англ. |
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2025-11-27T00:36:19Z |
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2025-11-27T00:36:19Z |
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Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 499–521
Mean kinetic energy and final state
effects in liquid hydrogens from
inelastic neutron scattering
G.Corradi 1,2 , D.Colognesi 1 , M.Celli 1 , M.Zoppi 1
1 Consiglio Nazionale delle Ricerche,
Istituto di Fisica Applicata ’Nello Carrara’,
Via Panciatichi 64, 50127, Florence, Italy
2 Università degli Studi di Firenze, Dipartimento di Fisica,
Via G. Sansone 1, 50019, Sesto Fiorentino (FI), Italy
Received April 29, 2003, in final form July 21, 2003
We have measured, using the TOSCA spectrometer at ISIS, the neu-
tron scattering spectrum of liquid para-hydrogen ( T = 19.3 K and n =
25.4 nm −3 ) and liquid ortho-deuterium ( T = 21.3 K and n = 25.2 nm −3 ).
From the high-energy region of the spectrum, namely 40 meV < ~ω <
1 eV, where the incoherent approximation for the center-of-mass motion
applies, we have been able to extract the translational mean kinetic energy,
which, as expected, turns out to be rather different from the classical val-
ues. However, significant deviations from the impulse approximation have
been detected and the inclusion of some correction terms (accounting for
the so-called final state effects) had to be accomplished in order to accu-
rately describe the experimental spectra. The comparison of the present
mean kinetic energy values with the available experimental and simulated
data in the literature is quite good and confirms the excellent performances
of TOSCA in the spectroscopic analysis of the condensed phases of liquid
hydrogens.
Key words: liquid para-hydrogen, liquid ortho-deuterium, mean kinetic
energy, final state effects
PACS: 67.90.+z, 61.12.Ex
1. Introduction
One of the important features of the liquid hydrogens (i.e. H2 and D2), among
all the quantum liquids, is their intermediate character between a deeply quantum
system like helium, where exchange effects can be relevant and characterize the
system features, and a weakly quantum system like neon or methane, where quantum
effects are relatively small and can be calculated using a perturbative approach
c© G.Corradi, D.Colognesi, M.Celli, M.Zoppi 499
G.Corradi et al.
applied to a classical reference model [1]. As a matter of fact, the quantum features
of the hydrogens are known to be relevant [2], even though the presence of effects that
could be ascribed to the particle exchange has never been experimentally detected in
the liquid state. Therefore, H2 and D2 represent an important case where quantum
exchange does not play an appreciable role, but the system structure and dynamics
are still largely influenced by the quantum delocalization properties [3]. In other
words, hydrogens are an effective practical example of a Boltzmann quantum liquid.
From a purely experimental point of view, dealing with a Boltzmann liquid is
not very different from dealing with a classical or even a superfluid one. However,
the difference emerges in its full glory at the time of interpreting the experimental
results, when a comparison with the theoretical predictions is carried out. As far as
the static properties are concerned, the state of the art of computer simulation has
evolved to such an extent that almost all microscopic static properties of quantum
liquids (including superfluid 4He, with the only possible exception of liquid 3He at
temperature T > 0 [4]) can be precisely evaluated using the Path Integral Monte
Carlo (PIMC) technique [5]. However, even though some interesting and promising
results have begun appearing in the literature [6–12], no simulation approach has
been proved so far to give a complete and precise description of the dynamics of a
quantum liquid. The microscopic dynamics in a Boltzmann liquid is also interesting
if one takes into account that, from a theoretical point of view, this is still an open
problem. Therefore, the possibility of carrying out experiments on real liquids that
map almost exactly this model is of paramount importance, and the liquid hydrogens
represent one of the few examples where the Boltzmann quantum liquid theory could
be applied to real systems.
Quantum effects in Boltzmann liquids originate from the delocalization of par-
ticles whose center-of-mass (COM) positions cannot be associated with individual
points in space, but are characterized by a probability distribution [13]. This, in
turn, is somehow related to the de Broglie thermal wavelength, λDB, which depends
on the particle mass and the system temperature, according to the definition [14]:
λDB = h/(2πMkBT )1/2, (1)
where M is the particle mass, T is the temperature, h is the Planck constant, and kB
is the Boltzmann constant. Strictly speaking, the above definition of the de Broglie
thermal wavelength is applicable to a massive free particle by identifying its momen-
tum, p = Mv, with the momentum of the associated plane wave, p = ~k, where k
is the wave-vector. However, for a dense system of interacting particles a different
definition should be used. Let us consider, for example, a single molecule in a dense
liquid of identical (but still distinguishable, according to the Boltzmann statistics)
particles. If we could take a snapshot of this particle, surrounded by the first shells
of its neighbors, we would see that the space available to the tagged molecule is lim-
ited by the cage-effect produced by the nearest neighbors. This implies the existence
of an upper limit for the fraction of the space, ∆r, available to the tagged-particle
wave-function. However, due to the Heisenberg uncertainty principle, this has an
implication on the minimum size of the momentum variance, ∆p. As a consequence,
500
Mean kinetic energy and final state effects in liquid hydrogens
the mean kinetic energy associated to a single particle (which is determined by the
squared momentum: 〈Ek〉 = p2/2M) assumes a value that is larger than what is
expected on the basis of the classical equipartition theorem (i.e. 〈Ek〉cl = (3/2)kBT ).
Furthermore, the intermolecular interaction details do not contribute very much to
the mean kinetic energy value. This effect is generally observed at very low temper-
atures, below the liquid Debye temperature [15], and is somehow equivalent to the
zero-point motion effect of the molecules composing a solid.
In order to be more rigorous, we introduce the momentum distribution of the
single particle n(p). Classically, this is given by the well-known Maxwell-Boltzmann
distribution that is:
n(p) =
(
β
2πM
)3/2
exp
(
−β
p2
2M
)
, (2)
where β = 1/kBT . The space-Fourier transform of the momentum distribution,
namely:
ñ(r) =
∫
dp exp
(
− i
~
p · r
)
n(p), (3)
represents the one-body density matrix, i.e. the statistical average of the overlap
between the system wave-function and itself, after shifting the coordinates of one
particle by the quantity r (keeping the coordinates of all the other particles fixed). It
is easy to show [16] that the one-body density matrix describes in a rigorous way the
delocalization of a single particle in an interacting system, both caused by quantum
and by thermal effects. We note that the momentum distribution is normalized in
such a way that:
∫
dp n(p) = 1, (4)
which implies that the normalization of ñ(r) is simply: ñ(0) = 1. Starting from the
Maxwell-Boltzmann distribution, one writes:
ñ(r) = exp
(
−Mr2
2β~2
)
(5)
which means that the standard deviation of the single-particle delocalization is:
σ = ~/(MkBT )1/2 , (6)
a result which is very similar to the definition of the de Broglie wavelength (see
equation (1)). It is worthwhile to observe that, by this procedure, we have obtained
a sort of a thermally averaged wave-packet, even for a classical particle obeying to
the Maxwell-Boltzmann momentum distribution.
We have seen that in a classical system, n(p) has a Gaussian shape, given by
the Maxwell-Boltzmann distribution. However, it is known that the Gaussian shape
is lost for a strongly quantum mechanical system, such as 4He in the vicinity or
below the λ-transition [17–22], where the emergence of a long-range tail in ñ(r) is
interpreted as a measure of the Bose-condensate fraction. However, looking at a less
501
G.Corradi et al.
quantum system, like neon, no significant deviation from a Gaussian distribution has
been clearly experimentally detected [23], while recent PIMC simulations indicate
that the momentum distribution of neon maintains its Gaussian shape for several
orders of magnitude, both in the liquid and the solid phase [24]. These results are also
corroborated by the general theoretical finding [25] that the quantum corrections to a
classical Maxwell-Boltzmann momentum distribution retain a Gaussian functional
form up to the ~
2 perturbative order. This suggests that the validity range of a
Gaussian momentum distribution could be extended to include Boltzmann quantum
liquids too.
One might argue that quantum effects in neon momentum distribution are not
so large to evidence deviations from the Gaussian shape. However, this hypothesis
has been also successfully used to interpret deep inelastic neutron scattering data
in liquid helium far from the λ-transition [26], and in dense supercritical 4He [27].
By assuming that a Gaussian shape for the momentum distribution also applies to
the Boltzmann quantum liquids, one can extrapolate the validity of equation (2) by
simply replacing the classical value of the single-particle mean kinetic energy, i.e.
〈Ek〉cl = (3/2)kBT , with its actual quantum mechanical value, 〈Ek〉. Thus, equa-
tion (2) becomes:
n(p) =
(
3
4πM〈Ek〉
)3/2
exp
(
− 3p2
4M〈Ek〉
)
. (7)
Consequently, the one-body density matrix (see equation (5)) writes:
ñ(r) = exp
(
−M〈Ek〉r2
3~2
)
. (8)
Thus, in this framework, the knowledge of the real value of the quantum mechani-
cal single-particle mean kinetic energy is sufficient to determine the single-particle
momentum distribution and its Fourier transform, i.e. the one-body density matrix.
We point out, once more, that this is only a working hypothesis which is mostly
based on the properties of neon. It would be important to know if the same scenario
applies to other Boltzmann liquids characterized by quantum effects of a larger size.
More experimental data and simulations, carried out on liquid hydrogens, could be
quite helpful in solving this problem.
2. The experimental point of view
Measuring the single-particle momentum distribution in a dense interacting sys-
tem is not a trivial task by any means. However, it has been shown that this physical
quantity can be accessed, in an almost direct way, by using high-energy neutrons
[28,29] in conjunction with a technique known as Deep Inelastic Neutron Scattering
or Neutron Compton Scattering (NCS) [30].
As a matter of fact, if the energy of the neutron is much larger than any ener-
gy scale involved in the atomic system interactions (we remind that 5 eV roughly
502
Mean kinetic energy and final state effects in liquid hydrogens
correspond to 58.000 K) the scattering process is not influenced by the local en-
vironment of the target atom after the scattering event. This is generally called
short-time approximation or impulse approximation (IA) [30]. In addition, due to
the large wave-vector transfer, Q, the interference effects caused by the spatial cor-
relation between neighbors become totally negligible and the so-called incoherent
approximation applies [31]: the scattering process simply becomes a sum of single-
atom collision events where the neutron transfers, by effect of the recoil, a large
portion of its kinetic energy to the target atom. This energy, on average, is related
to Q by the simple relation: ~ωr = (~Q)2/2M , where M is the mass of the recoil-
ing particle [31]. In the experimental conditions where NCS applies, the momentum
transfer (i.e. ~Q) becomes so large that the IA holds and the neutron scattering
cross section is uniquely determined by the momentum distribution of the target
atom [30].
The NCS experimental technique has widely been used, in recent times, with
the aim of obtaining information on the momentum distribution function and the
mean kinetic energy of dense quantum monatomic liquids [23,32–38]. However, when
the same technique is extended to molecular liquids (e.g. H2 and D2) the situation
appears more involved and the presence of an intra-molecular structure makes the
interpretation of the experimental data much more difficult [39–44]. In practice,
using NCS on the hydrogens, the energy and momentum transfers are so large that
the scattering cross section is determined by the proton and deuteron (for hydrogen
and deuterium, respectively) nuclear momentum distribution. In this condition, it
becomes rather difficult (if not at all impossible) to obtain from the experimental
data the corresponding function for the molecular COM.
A way of circumventing this difficulty was found by Langel et al. by using neu-
trons of lower energy [45]. Actually the intermolecular and intra-molecular degrees
of freedom of the hydrogens, provided the system pressure is kept within reasonable
limits, are decoupled to such an extent that they can be considered almost inde-
pendent. In this situation, we can select an energy-interval of the probe such that
the momentum transfer is large enough that the IA applies to the COM dynamics
of the molecules but, at the same time, the energy transfer is low enough that the
intra-molecular modes are not excessively excited. In this scenario it becomes pos-
sible to obtain the required information for the COM dynamics, provided a reliable
model to describe the intra-molecular modes is available.
We have practically shown that this procedure is possible. In a recent series of
experiments, the neutron spectrometer TOSCA, at the pulsed neutron source ISIS
(Rutherford Appleton Laboratory, UK), was used to obtain a direct information on
the translational mean kinetic energy (i.e. the COM’s one) in liquid and solid para-
hydrogen [46,47]. In addition, we have shown that the lower energy portion (less
than 50 meV) of the hydrogen spectrum derived from TOSCA can be efficiently
used to gain information on the COM self-dynamics in the liquid phase [48,49]. In
these experiments, a key role is played by the possibility of implementing the IA at
the level of the molecular COM dynamics. Actually, this is rigorous only in the ideal
limit: Q → ∞, while some corrections should be applied for any finite value of Q.
503
G.Corradi et al.
3. The theoretical point of view
As already mentioned in the previous section, in the case of low-pressure liquid
H2 and D2 the decoupling between molecular COM dynamics and roto-vibrational
dynamics holds almost exactly [50]. This fact implies that the double-differential
neutron scattering cross-section, (dσ2/dΩdE ′), is separable in the sum of two con-
volution products between COM and roto-vibrational parts, and, moreover, that
the latter are substantially identical to the equivalent quantities observed in the gas
phase [51,52]:
(
dσ2
dΩ dE ′
)
=
(
dσ2
dΩ dE ′
)
r−v
⊗Sself, C(Q, ω)+
(
dσ2
dΩ dE ′
)
r−v, el.
⊗Sdist, C(Q, ω), (9)
where (dσ2/dΩdE ′)r−v and (dσ2/dΩdE ′)r−v, el. are the roto-vibrational double-diffe-
rential cross-sections (total and elastic, respectively) reported in detail in [51–53].
It is important to recall that these two functions are relatively simple (they can be
even expressed analytically), since they are totally independent of the many-body
quantum dynamics of liquid state. They are effected only by some isolated-molecule
parameters, the H (or D) neutron scattering cross-section, and the composition of
the rotationally-excited molecular population in the system. On the other hand,
Sself, C(Q, ω) and Sdist, C(Q, ω) represent the well-known inelastic structure factors
(self and distinct, respectively [31]), which describe the COM dynamics in the liquid
state. It is worth noting that both of them exhibit the same properties (e.g. sum rules
[54] etc.) as in a monatomic system, since the hydrogen intra-molecular structure
has been confined within the other factors of the two convolution products.
In the limit of large transferred wave-vectors, where the COM static structure
factor, SC(Q) approaches the unity:
SC(Q) =
∫
∞
−∞
dω [Sself, C(Q, ω) + Sdist, C(Q, ω)] ' 1, (10)
equation (9) reduces to:
(
dσ2
dΩ dE ′
)
'
(
dσ2
dΩ dE ′
)
r−v
⊗ Sself, C(Q, ω) , (11)
which is precisely the aforementioned incoherent approximation. Its physical mean-
ing is clear and interesting: if Q grows large enough, the collective excitations in a
liquid system, described by Sdist, C(Q, ω), become more and more damped, to such
an extent that, once 2π/Q is much smaller than the typical intermolecular distance,
they completely disappear. What is left is just the single-particle dynamics, which
is actually contained in Sself, C(Q, ω). In liquid hydrogen [55] and deuterium [56],
the incoherent limit appears as already reached at Q = (5 − 6) Å−1. Thus, in what
follows in this section, we will always consider the response of liquid H2 and D2 at
wave-vector transfers larger than this figure.
We have already pointed out that by further increasing the energy transfer the
liquid responds like a collection of weakly interacting particles, as far as the mere
504
Mean kinetic energy and final state effects in liquid hydrogens
COM dynamics is concerned. This is a general achievement, provided the inter-
molecular forces are not infinite anywhere, and implies that, in the asymptotic limit
(ω, Q) → ∞, the self part of the COM structure factor, Sself, C(Q, ω), simply de-
pends on the COM momentum distribution, nC(P ), through the well-known IA
relationship:
Sself, C(Q, ω) =
∫
dP nC(P ) δ
(
ω − ~Q2
2M
− P ·Q
M
)
, (12)
where now M represents the molecular mass. We will show in the rest of this sec-
tion how to derive this formula from the short-time expansion of the intermediate
scattering function and how to correct the IA for the so-called final states effects
(FSE) at any finite value of Q.
Let us introduce the self part of the COM intermediate scattering function,
Iself, C(Q, t), as the Fourier transform of Sself, C(Q, ω):
Iself, C(Q, t) =
∫
∞
−∞
dω exp (−iωt)Sself, C(Q, ω). (13)
It is straightforward to prove [31] that the following relation holds:
Iself, C(Q, t) =
1
N
〈
N
∑
j=1
exp (−iQ · Rj(t)) exp (iQ · Rj(0))
〉
, (14)
where N is the number of molecules in the system, Rj(t) is the position operator
of the jth COM at the time t, and the symbol 〈...〉 represents the usual quantum
statistical average. Using some standard algebra of the Heisenberg representation
[57], one can rewrite the exponential operator at time t, exp (−iQ · Rj(t)), as:
exp (−iQ · Rj(t)) = exp (iHt) exp (−iH ′t) exp (−iQ · Rj(0)) , (15)
making use of the system Hamiltonian, H, and of the same operator modified by
the scattering event concerning the jth COM, H ′:
H ′ = H +
~
2Q2
2M
+
~Q · Pj
M
. (16)
Inserting equations (15) and (16) into equation (14), and using again some Hei-
senberg-representation algebra [58], one finally obtains a complete expression for
Iself, C(Q, t):
Iself, C(Q, t) = exp
(
−i
~Q2
2M
t
)
1
N
〈
N
∑
j=1
Tt exp
(
−i
∫ t
0
dτ
Q · Pj(τ)
M
)
〉
, (17)
where Tt is the time-ordering operator, and Pj(τ) = exp (iHτ)Pj(0) exp (−iHτ) is
the COM momentum at time τ in the Heisenberg representation.
505
G.Corradi et al.
The IA is now obtained from the previous equation if the scattering time t is
assumed to be short, so that only times close to t = 0 are retained. Practically it
means that Pj(t) is expanded in a power series in t: Pj(t) = Pj(0)+Fj(0)t + · · ·,
where Fj is the force acting on the jth COM due to the neighbor molecules, via the
intermolecular potential. Keeping only the first term of the series, one writes:
I
(IA)
self, C(Q, t) = exp
(
−i
~Q2
2M
t
)
1
N
〈
N
∑
j=1
exp
(
−i
Q · Pj(0)
M
t
)
〉
, (18)
whose Fourier transform yields:
S
(IA)
self, C(Q, ω) =
1
N
〈
N
∑
j=1
δ
(
ω − ~Q2
2M
− Q ·Pj(0)
M
)
〉
, (19)
which is exactly identical to equation (12), since the only variable depending on the
system is the momentum of the struck COM, Pj.
We have obtained the IA just by neglecting the Fj(0) term in the expansion of
Pj(t). So, for the IA to hold, one must have: ~Q+Pj � Fj τs, where τs is the typical
scattering time. By increasing Q, τs is of course reduced, but it is also crucial that
Fj = −∂V /∂Rj is not infinite anywhere (with V being the intermolecular potential
energy). In liquid hydrogens, V has a steep repulsive, but not infinitely repulsive,
core. Thus Fj is large, but still finite, and the IA can be attained. However, we
expect that corrections to the IA could be relevant and it is important to determine
the nature and the size of these deviations, following the procedure used in Ref.
[59]. The expectation value in equation (17) may be expanded in cumulants, and,
in addition, Pj(t) in each cumulant term may be expanded in a power series of t:
Iself, C(Q, t) = exp
(
−i
~Q2
2M
t
)
exp
[
∞
∑
n=1
µn(−it)n
n!
]
. (20)
However, it is more useful to separate the IA and non-IA (i.e. FSE) contributions
to the various cumulant coefficients. After having operated such a rearrangement,
one obtains:
Iself, C(Q, t) = I
(IA)
self, C(Q, t) exp
[
∞
∑
n=1
βn(−it)n
n!
]
, (21)
where the first four βn coefficients can be evaluated making use of the quantum sum
rules valid for isotropic Sself, C(Q, ω) [54]:
β1 = 0;
β2 = 0;
β3 =
~Q2
6M2N
〈
N
∑
i=1
∇2
i V
〉
;
β4 =
Q2
3M2N
〈
N
∑
i=1
(
~∇iV
)2
〉
. (22)
506
Mean kinetic energy and final state effects in liquid hydrogens
Unfortunately a cumulant expansion cannot be terminated at the level of β4, since
β4 > 0 and an unphysical divergence for t → ∞ would occur in equation (21). For
this reason another (coarser) approximation is usually applied, the so-called additive
approach (AA) [57,59]. The AA is simply obtained by expanding the FSE part of
equation (21) up to the t5 term:
I
(AA)
self, C(Q, t) ' I
(IA)
self, C(Q, t)
(
1 +
iβ3
3!
t3 +
β4
4!
t4 − iβ5
5!
t5
)
. (23)
The convergence of I
(AA)
self, C(Q, t) for t → ∞ is now fully guaranteed since I
(IA)
self, C(Q, t)
exhibits at least a Gaussian dependence on t. By taking the Fourier transform of
equation (23), one finally writes:
S
(AA)
self, C(Q, ω) '
(
1 − β3
3!
∂3
∂ ω3
+
β4
4!
∂4
∂ ω4
− β5
5!
∂5
∂ ω5
)
S
(IA)
self, C(Q, ω). (24)
which was first derived by Sears in 1969 [60]. Assuming a purely Gaussian COM
momentum distribution (see section 1 for a discussion of this point), it is straight-
forward to write equation (24) analytically [59,61]:
S
(AA)
self, C(Q, ω) '
{
1 − β3
2µ2
2
(ω − ωr)
[
1 − ω2
d/3
]
+
β4
8µ2
2
[
1 − 2ω2
d + ω4
d/3
]
+
β5
8µ3
2
(ω − ωr)
[
1 − 2ω2
d/3 + ω4
d/15
]
}
S
(IA)
self, C(Q, ω), (25)
where S
(IA)
self, C(Q, ω) retains a Gaussian functional form too:
S
(IA)
self, C(Q, ω) =
1√
2πµ2
exp
[
−(ω − ωr)
2
2µ2
]
, (26)
with µ2 = 〈Ek〉2Q2/3M , ωr = ~Q2/2M , and ω2
d = (ω − ωr)
2/µ2. It is important to
point out that now 〈Ek〉 represents the translational mean kinetic energy, i.e. the
one of the molecular center-of-mass.
4. Relevance of the FSE corrections in the TOSCA spectra
4.1. The TOSCA spectrometer
The neutron scattering experiments were performed on the first version of the
spectrometer (i.e. TOSCA-I), a crystal-analyzer inverse-geometry instrument op-
erating at ISIS, the pulsed neutron source of Rutherford Appleton Laboratory
(Chilton, Didcot, UK) [62]. The incident neutron beam spanned a broad energy
(E) range and the energy selection was carried out on the secondary neutron flight-
path using the (002) Bragg reflection of 10 graphite single crystals placed in back-
scattering around 136.0◦. This fixed the nominal scattered neutron energy, E ′, to
507
G.Corradi et al.
0 200 400 600 800 1000
0
5
10
15
20
25
!ω�(meV)
Q
(
Å
-1
)
Figure 1. Average (Q, ~ω) trajectory in the kinematic plane spanned by the
TOSCA spectrometer (back-scattering section).
' 3.51 meV. Higher order Bragg reflections were filtered out by 15-cm thick berylli-
um blocks cooled down to 30 K. This geometry allows to cover an extended energy
transfer range, even though the fixed position of the crystal analyzers and the small
value of the neutron final energy imply a variation in Q which is a monotonic func-
tion of ω. The resolving power of TOSCA-I was rather good (2% < ∆~ω/E < 5%)
in the energy transfer region relevant to the present experiment (40 meV < ~ω <
1 eV). The extended spectral range of TOSCA makes this instrument a sort of
neutron equivalent of a Raman optical spectrometer, the only difference being the
momentum transfer monotonically growing along with the energy shift (see fig-
ure 1): Qω→∞ → (2mnω/~)1/2, where mn is the neutron mass. Thus, intra-molecular
transitions could be in principle easily observed even beyond the first vibrational
transitions of molecular hydrogen and deuterium (we remind that these are placed at
515.6 meV and at 370.9 meV, respectively [63]). However, it should be also pointed
out that, due to the molecular recoil, the observed shifts are generally much greater
than those observed on a conventional Raman spectrometer.
4.2. The spectrum of liquid hydrogen
In the high-energy region (40 meV < ~ω < 1 eV) of a TOSCA spectrum, the
wave-vector transfer grows to such an extent (5.6 Å−1 < Q < 23.0 Å−1) that
the scattering process is expected to approach the IA regime, at least as far as
liquid samples are concerned. Therefore, as we have already seen in section 2, this
spectrum would mainly contain information relative to the momentum distribution
of a single particle in the system. On the other hand, in the TOSCA low-energy
spectral region (3 meV < ~ω < 40 meV), the size of the wave-vector transfer
(2.9 Å−1 < Q < 5.6 Å−1) is of the order of the reciprocal of the inter-molecular dis-
tances between neighbors (namely, around 3.5 Å for liquid H2 and D2). In this case,
the self contribution to the scattering function is expected to provide information on
508
Mean kinetic energy and final state effects in liquid hydrogens
the molecular dynamics driven by the pair interactions. Even though the transition
between these two regimes is not clearly defined, it is reasonable to assume that in
a liquid, it lies beyond the first peak of the static structure factor S(Q) (which in
liquid H2 and D2 is located around Q = 2 Å−1), in the region where S(Q) becomes
' 1 (i.e., as we have pointed out in section 2, Q ' 5−6 Å−1). Should the momentum
transfer decrease to much lower values, to such an extent that the hydrodynamic
regime would drive the microscopic dynamics, it would be possible to gain direct
information on the long-time behavior of the velocity auto-correlation function and
the self-diffusion coefficient making use of the self part of the scattering function.
But this is not exactly the case of TOSCA, and only an approximate estimate of the
diffusive dynamics is expected to come out from the present hydrogen and deuterium
spectra [48,49].
Let us focus on liquid H2: if pure liquid para-hydrogen is considered, the inelastic
neutron spectrum becomes rather simple. At low temperature (T < 25 K) only the
fundamental rotational state (J = 0) is populated. In addition, since the transitions
to the even states (J = 0 → J ′ = 2, 4, 6, · · ·) are weighted by the small hydrogen
coherent cross-section (σc(H) = 1.8 barn), their intensity is almost two orders of
magnitude smaller than the transitions to the odd states (driven by the incoherent
one, σi(H) = 80.3 barn) [64]. Thus, the observed spectrum reduces, for any practical
purpose, to a set of odd rotational transitions (J = 0 → J ′ = 1, 3, 5, · · ·). Due to the
small moment of inertia of hydrogen, rotational transitions are all well separated
(J = 0 → J ′ = 1 implies an energy jump of 14.7 meV, while J = 0 → J ′ = 3
corresponds to a jump of 87.4 meV). As a consequence, taking into account the
extra shift induced by the molecular recoil, the overlap of different bands is small
and each band can be analyzed almost individually. Applying equation (9) to low-
temperature para-hydrogen in the spectral region (40 meV < ~ω < 1 eV), the
neutron scattering double-differential cross-section simply becomes:
(
d2σ
dΩdE ′
)
=
k′
k
σi(H)
4π
∑
J ′=1,3,5,···
|f(Q)|20→J ′Sself, C(Q, ω) ⊗ δ(ω − ω0→J ′), (27)
where k and k′ represent the incoming and out-coming neutron wave-vector, re-
spectively; f(Q) is the intra-molecular form factor [53], and ~ω0→J ′ stands for the
energy gap between the J = 0 and the various J ′ rotational states. Actually a por-
tion of the high-energy region of the spectra (namely, 100 meV < ~ω < 1 eV),
which involves the rotational transitions beyond the first excited rotational level,
(J = 0 → J ′ = 3, 5, 7) has been already analyzed in the framework of the pure
IA [46,47]. In this case, due to the high value of the wave-vector (Q > 8.0 Å−1)
and energy transfers, the inter-molecular interactions were supposed only to give
rise to a simple renormalization of the COM mean kinetic energy. This, in turn,
affects the width of the molecular recoil peak which describes the free motion of the
molecule. A simple modification of the Young-Koppel theory [53], originally devised
for gaseous H2 and D2, was applied and we could determine the COM mean kinetic
energy in liquid hydrogen and deuterium as a function of the thermodynamic con-
ditions [46,47]. As we see at the end of this subsection, a careful investigation of
509
G.Corradi et al.
the TOSCA spectra will show important deviations from IA, even in this high-ω
scattering region.
The measurement was carried out in the dense liquid phase at T = 19.34 K.
After performing the background measurements of the empty cryostat, we cooled
the empty container at the desired temperature and we measured its time-of-flight
(TOF) neutron spectrum. Then, hydrogen was allowed to condense in the scattering
cell. This was made of SS-316 steel and consisted of an array of 5 tubes (O.D. 3.2 mm,
I.D. 1.6 mm), 64.0 mm high and 5.5 mm apart from one another, so as to minimize
the multiple scattering contributions. After filling up the cell, the pressure of the
gas handling system was set to p = 191 bar (much larger than the corresponding
saturated vapor pressure: pSVP ' 0.8 bar [65]) in order to obtain a high-density liquid
sample: n = 25.43 nm−3 [65]. The reason for this requirement will be made clear
hereinafter. At the bottom of the scattering container, out of the neutron beam, we
had inserted some powder of a paramagnetic catalyst made of Cr2O3 on an Al2O3
substrate in order to speed up the conversion from ortho- to para-hydrogen. The
relative concentration of the two species was monitored observing the scattering
spectrum. In particular, we could observe the progressive disappearance of the J =
1 → J ′ = 1 (quasi-elastic line) transition, which is weighted by the incoherent cross-
section of the proton [31], from the low energy portion of the spectrum. When this
spectral feature was below the limit of detectability (in practice, masked by the J =
0 → J ′ = 0 transition, which is weighted by the coherent cross-section of the proton)
we assumed that the equilibrium had been reached. The equilibration process lasted,
in our case, about 20–25 hours. The estimated concentration of para-hydrogen, based
on the theoretical calculation [65], was assumed to be 99.82%. Then, we started
recording the scattering spectrum up to an integrated proton current of 3065.8 µAh
(roughly, 18 hours of beam time). The stability of the thermodynamic conditions
during the experiment was quite good: temperature fluctuations never exceeded
0.2 K and the pressure stability (2 bar) was strictly related to this value due to the
good realization of the gas handling system. The temperature uncertainty (standard
deviation) estimated for our measurement was 0.02 K, and the density of our sample
(and its uncertainty) was later derived according to [65]: n = 25.43(3) nm−3. In
figure 2(a) we show the raw spectrum of liquid para-hydrogen at T = 19.34 K and
n = 25.43 nm−3 in all the TOSCA-accessible ω-range.
The experimental TOF spectra were transformed into energy transfer data, de-
tector by detector, making use of the standard TOSCA-I routines available on the
spectrometer, and then added together exploiting the narrow angular range spanned
by the whole set of detectors (∆θ = 9.5◦) [62]. In this way, we produced a single
double-differential cross-section measurement along the TOSCA-I kinematic path
(Q, ω). Then, data were corrected for the k′/k factor and subtracted of the tiny
empty can contribution (separately recorded at T = 34.99(5) K, with an integrated
proton current of 982.2 µAh). Two fundamental corrections were carefully per-
formed: multiple scattering evaluation and self-absorption attenuation. As far as
the former is concerned, we simulated both single-scattering and multiple-scattering
neutron spectra measured by TOSCA-I for each sample through the analytical ap-
510
Mean kinetic energy and final state effects in liquid hydrogens
0 200 400 600 800 1000
0.1
1
10
J=0−>J'=7
J=0−>J'=5
J=0−>J'=3
J=0−>J'=1
!ω�(meV)
k/
k'
(
d2 σ/
d Ω
/d
E
')
(a
rb
. u
ni
ts
)
(a)
0 200 400 600 800 1000
0.1
1
10
(b)
k/
k'
(
d2 σ/
d Ω
/d
E
'
)
(a
rb
. u
ni
ts
)
!ω�(meV)
Figure 2. (a) Raw experimental neutron spectrum measured on liquid para-
hydrogen at T = 19.34 K (after the can scattering subtraction). The rotational
transitions from the J = 0 ground state to the odd-J ′ excited states are clearly
identified. (b) Raw experimental neutron spectrum measured on liquid ortho-
deuterium at T = 21.3 K (after the can scattering subtraction).
511
G.Corradi et al.
proach suggested by Agrawal in the case of an infinite flat slab-like sample [66]. All
the details of this procedure have been reported in [49]. Following this stage, self-
absorption correction was applied to the normalized single-scattering experimental
data, still assuming the infinite slab approximation and making use of the afore-
mentioned analytical approach [66]. However, in this case, no model was employed
for the para-hydrogen total scattering cross-section, which, on the contrary, was ob-
tained from the experimental results of direct measurements on the SVP liquid p-H2
at T = 16.0(2) K [67]. At the end of the correction procedure, the processed data,
limited to the 3 meV < ~ω < 1 eV interval, had actually become proportional to the
double-differential neutron scattering cross-section, as in equation (11), plus a small
(and rather flat) sample-dependent background. Making use of the aforementioned
Young-Koppel model [52,53], we were able to calculate the intra-molecular term of
equation (11), (dσ2/dΩ dE ′)r−v, along the (Q, ~ω) trajectory in the kinematic plane
spanned by TOSCA-I. Assuming, as a first working hypothesis, a simple IA form for
Sself, C(Q, ω) in the 40 meV < ~ω < 1 eV range, we set up a simple fitting procedure
of the processed experimental data, Σexp(Q, ω), through the following function:
Σexp(Q, ω) =
[
A
k
k′
(
dσ2
dΩ dE ′
)
r−v
⊗ S
(IA)
self, C(Q, ω) + BII(ω)
]
⊗ RTosca(ω), (28)
where A represents an overall normalization constant, BII(ω) accounts for a poly-
nomial (second order) sample-dependent background, and RTosca(ω) stands for the
instrumental energy resolution [62]. We recall here that this fitting function con-
tained only five independent parameters: A, three polynomial coefficients in BII(ω),
and 〈Ek〉, which is implied by S
(IA)
self, C(Q, ω) as shown in equation (26). The fit, per-
formed through a FORTRAN code coupled with the MINUIT minimization library
[68], showed that equation (28) does not properly describe the liquid p-H2 scatter-
ing law in the 40 meV < ~ω < 1 eV range, since the reduced χ2 could not decrease
below 6.83 (see also figure 3(a)). Thus, in order to improve the quality of our fit,
we replaced S
(IA)
self, C(Q, ω) with S
(AA)
self, C(Q, ω) in equation (28), the latter function be-
ing defined as in equation (25). As far as the AA coefficients are concerned, β3
and β4 were evaluated following equation (22) and assuming a pair-wise additive
inter-molecular potential:
1
N
〈
N
∑
j=1
∇2
jV
〉
= n
∫
dR g(R)∇2v(R) = 169.8 meV Å
−2
,
1
N
〈
N
∑
j=1
(
~∇jV
)2
〉
= n
∫
dR g(R)
(
~∇v(R)
)2
= 1800 meV2Å
−2
, (29)
where n is the molecular density of the system, v(R) the Silvera-Goldman pair-wise
inter-molecular potential [69], and g(R) is the COM pair correlation function, derived
from previous PIMC simulations [47]. On the contrary, β5 was supposed to have only
a Q2 dependence (in the framework of the well-known Gaussian approximation for
Iself, C(Q, t) [54]), and was left as an additional fitting parameter (the sixth). After
512
Mean kinetic energy and final state effects in liquid hydrogens
repeating the fitting procedure including the FSE terms of the AA, we obtained a
much better description of the experimental data, since reduced χ2 stabilized this
time around 1.48. The improvement of the fitting quality induced by the FSE can be
clearly observed in figure 3(a). The experimental estimate of the COM mean kinetic
energy was 〈Ek〉 = 78(4) K.
In order to verify that the insufficiency of the IA to properly describe the exper-
imental data from liquid para-hydrogen was caused by the inter-molecular FSE, we
decided to study a low-density gaseous sample (T = 28.94(2) K, n = 2.06(9) nm−3,
c[p − H2] = 97.7(2)%). As a matter of fact, by increasing the mean free path by
a factor of 2.3, we expected a much lower effect of the FSE on the measured neu-
tron spectrum. The gaseous sample (3.5-mm thick at p = 6.0(2) bar) was mea-
sured on TOSCA-I until an integrated proton current of 745 µA h was reached.
Then, its spectrum was processed similar to the aforementioned liquid p-H2 data
set. A purely IA fitting procedure was performed in a wide energy-transfer range:
4 meV < ~ω < 1 eV. The final results can be seen in figure 3(c), where a very
good agreement between Σexp(Q, ω) and the IA model is evident, the reduced χ2
turning out to be 1.35. The experimental translational mean kinetic energy, in this
case, was estimated to be: 〈Ek〉 = 44(3) K, which is compatible with the classical
value: 〈Ek〉cl = 43.41(3) K. It is worth noting that, as far as gaseous para-hydrogen
is concerned, our findings totally confirm the conclusions drawn by Simmons et al.
[70] about the validity of the Young-Koppel model.
4.3. The spectrum of liquid deuterium
Let us now describe the liquid-D2 inelastic scattering: even though pure liq-
uid ortho-deuterium is considered, its neutron spectrum is not as simple as in the
para-hydrogen case. We still have that at low temperature (T < 25 K) only the fun-
damental rotational state (J = 0) is populated; but in o-D2 the transitions to the
even states (J = 0 → J ′ = 0, 2, 4, 6, · · ·) are weighted by a linear combination of the
coherent and incoherent atomic cross-sections [64]: σc(D) + 5σi(D)/8 = 6.87 barn.
Thus, their intensity is larger (but just by one order of magnitude) than the transi-
tions to the odd states (driven by a fraction of incoherent atomic cross-sections [64]:
3σi(D)/8 = 0.77 barn), which are, actually, small but not totally negligible. In addi-
tion, the spectral framework is complicated by the moment of inertia of deuterium,
so that the rotational transitions are not well separated as in p-H2 (J = 0 → J ′ = 0
implies no energy jump, while J = 0 → J ′ = 2 corresponds to a jump of 22.2 meV
only). This situation is clearly seen in figure 2(b), where the individual transition
peaks are hardly recognizable. Finally, a further complication is caused by the so-
called elastic transition (i.e. J = 0 → J ′ = 0), which, according to equation (9), also
implies an additional term related to the collective excitations of the liquid and ex-
pressed by Sdist, C(Q, ω). However, limiting our data analysis to the 50 meV < ~ω < 1
eV range, we make sure that Sdist, C(Q, ω) could be totally neglected, as explained in
section 2. Summarizing the discussion above, we conclude that equation (9) applied
to low-temperature ortho-deuterium in the spectral region 50 meV < ~ω < 1 eV
513
G.Corradi et al.
32 64 128 256 512 1024
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40 (a)
k/
k'
(
d2 σ/
d Ω
/d
E
'
)
(b
ar
n/
sr
/m
eV
)
!ω�(meV)
32 64 128 256 512 1024
0.000
0.003
0.006
0.009
0.012
0.015
!ω�(meV)
(b)
k/
k'
(
d2 σ/
d Ω
/d
E
'
)
(b
ar
n/
sr
/m
eV
)
4 8 16 32 64 128 256 512 1024
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
(c)
k/
k'
(
d2 σ/
d Ω
/d
E
'
)
(b
ar
n/
sr
/m
eV
)
!ω�(meV)
Figure 3. (a) Experimental double-differential neutron scattering cross-section
from liquid para-hydrogen at T = 19.34 K (circles with error bars) together with
its impulse-approximation (dashed line) and additive-approach (full line) best
fits. (b) Experimental double-differential neutron scattering cross-section from
liquid ortho-deuterium at T = 21.3 K (circles with error bars) together with
its impulse-approximation (dashed line) and additive-approach (full line) best
fits. (c) Experimental double-differential neutron scattering cross-section from
gaseous para-hydrogen at T = 28.94 K (circles with error bars) together with its
impulse-approximation (full line) best fit.
514
Mean kinetic energy and final state effects in liquid hydrogens
reads as:
(
d2σ
dΩdE ′
)
=
k′
k
∑
J ′=0,1,2,3,···
σJ ′
4π
|f(Q)|20→J ′Sself, C(Q, ω) ⊗ δ(ω − ω0→J ′), (30)
where σJ ′ = 6.87 barn if J ′ is even, and σJ ′ = 0.77 barn if J ′ is odd.
The ortho-deuterium measurement was carried out in the liquid phase at T =
21.3 K using a flat scattering cell. This was made of aluminum, 1.2 mm thick walls,
with a cylindrical-slab geometry. The sample thickness was 2.5 mm and the cell
diameter (55.0 mm) was a little larger than the beam cross-section (roughly 50
mm × 20 mm). After filling up the cell, the pressure of the gas handling line was
set to p = 0.76 bar (slightly larger than the corresponding saturated vapor pressure:
pSVP ' 0.47 bar [71]) in order to make sure that the scattering cell was actually
totally filled with liquid. As in the case of para-hydrogen, at the bottom of the sam-
ple container (out of the neutron beam) we had placed some paramagnetic catalyst
to accelerate the conversion from para- to ortho-deuterium. The relative concentra-
tion of the two deuterium species was checked observing the scattering spectrum in
the spectral region 3 meV < ~ω < 200 meV. When the change in the aforemen-
tioned ~ω range was below the statistical limit of detectability, we assumed that
the equilibrium had been reached. This process took about 23 hours and the esti-
mated final concentration [71] of o-D2 in the sample was 97.4%. After the end of
the equilibration, we started measuring the scattering spectrum up to an integrated
proton current of 2062.5 µA h (equivalent to roughly 12 hours of beam time). The
temperature and pressure uncertainties (standard deviations) were estimated to be
0.3 K and 0.02 bar, respectively. The density of our sample (and its uncertainty)
was later derived according to [71]: n = 25.2(1) nm−3. In figure 2(b) we show the
raw spectrum of liquid ortho-deuterium at T = 21.3 K and n = 25.2 nm−3 in all the
TOSCA-accessible ω-range.
Experimental TOF data from the liquid o-D2 were processed following a similar,
but somewhat simpler guide-line like the one already employed for p-H2 (see the
previous subsection for details): the multiple-scattering contribution was estimat-
ed to be roughly 6.1% of the single-scattering one [72], and then it was neglected
and included in the background (see below). As far as the self-shielding is con-
cerned, since liquid ortho-deuterium exhibits an almost constant total scattering
cross-section [73] in the range of interest of the incoming neutron energy (53 meV
< E < 303 meV, see later), we decided not to perform any corrections. Once ob-
tained the experimental Σexp(Q, ω), we applied to the processed data a IA fitting
procedure similar to the one expressed in equation (28), just replacing the parabolic
polynomial sample-dependent background by a more general one of the third or-
der. The fit was accomplished in the energy transfer range 50 meV < ~ω < 1 eV,
even though the main data features extended up to about ~ω = 300 meV. The re-
sults, reported in figure 3(b), were not totally satisfactory since the reduced χ2 did
not decrease below 2.70. Following the same strategy as for liquid para-hydrogen,
the leading terms of the FSE were introduced in the fitting procedure through the
AA, just replacing S
(IA)
self, C(Q, ω) with S
(AA)
self, C(Q, ω). Its FSE coefficients were either
515
G.Corradi et al.
estimated (β3 and β4), or left as a fitting parameter (β5). The former two were cal-
culated following equations (22) and (29) using a PIMC pair correlation function
(see [74] for the details of the simulation technique), and the potential derivative
terms turned out to be:
1
N
〈
N
∑
i=1
∇2
i V
〉
= 145.5 meV Å
−2
,
1
N
〈
N
∑
i=1
(
~∇iV
)2
〉
= 981.1 meV2Å
−2
. (31)
The new fitting procedure achieved a much better agreement between experimental
data and model function (reduced χ2 = 1.08), as seen in figure 3(b), and the COM
mean kinetic energy was evaluated to be 〈Ek〉 = 65(3) K.
5. Discussion and conclusions
The result for the center-of-mass mean kinetic energy of liquid para-hydrogen
(namely 〈Ek〉 = 78(4) K) was found in good agreement with the corresponding
Path Integral Monte Carlo simulation value: 〈Ek〉(PIMC) = 74.5(1) K [47]. The same
quality of agreement is observed if the present experimental result is compared to the
〈Ek〉 value obtained from the same experimental data through a simpler IA analysis
in a narrower energy transfer range (100 meV < ~ω < 1 eV): 〈Ek〉 = 74(5) K [47].
Unfortunately, as for liquid ortho-deuterium, it is not possible to establish such a
comparison: the most relevant spectral features lie in the 50 meV < ~ω < 100 meV
range, where, on the other hand, the FSE are not negligible. However, making use of
other neutron spectrometers exploring different (Q, ω) trajectories, it was feasible
to extract information on the translational mean kinetic energy from a liquid normal
(i.e. 66.6̄% of ortho) D2 sample at T = 20.0 K and n = 25.7 nm−3. After a pure IA fit
over an extended energy and momentum transfer range (15 meV < ~ω < 120 meV
and 6 Å−1 < Q < 15 Å−1), Mompeán et al. [75] managed to estimate the COM
mean kinetic energy to be 〈Ek〉 = 60(9) K. This figure can be easily reconciled
with our present finding: 〈Ek〉 = 65(3) K, confirming that there is no appreciable
effect of the roto-vibrational molecular population on the COM dynamics in liquid
deuterium.
As we mentioned in section 4, in the case of liquid para-hydrogen, where there
are virtually no difficulties caused by the elastic (J = 0 → J ′ = 0) scattering,
a careful data analysis in the low-energy transfer range 3 meV < ~ω < 40 meV
has been attempted, and successfully carried out [49], making use of the Gaussian
approximation in conjunction with a model for power spectrum of the velocity auto-
correlation function of the COM, f(ω) [54]. It is worth noting that the link between
the present AA description of the FSE and the velocity auto-correlation function ap-
proach exists and can be shown in a straightforward way. According to the Gaussian
516
Mean kinetic energy and final state effects in liquid hydrogens
approximation [54], Iself, C(Q, t) generally writes as:
Iself, C(Q, t) = exp
(
−i
~Q2
2M
t
)
exp
[
−Q2w(t)
]
, (32)
where w(t) is the mean squared travelled path of the COM at time t, and is simply
related to f(ω) by [49,54]:
w(t) =
~
2M
∫
∞
0
dω
f(ω)
ω
{[1 − cos (ωt)] coth(~ω/2kBT ) + i [sin (ωt) − ωt]} . (33)
Since, as we have seen in section 3, we are interested only in the short-time behavior
of Iself, C(Q, t), let us expand w(t) in a power series in t:
w(t) ' w(0) + w(0)(1)t +
w(0)(2)
2!
t2 +
w(0)(3)
3!
t3 +
w(0)(4)
4!
t4 . (34)
Comparing equation (20) and equation (34), and then plugging the latter into equa-
tion (33), one obtains:
Q2w(0) = 0;
Q2w(0)(1) = iµ1 = 0;
Q2w(0)(2) = µ2 =
~Q2
2M
∫
∞
0
dω f(ω)ω coth(~ω/2kBT );
Q2w(0)(3) = −iµ3 = −iβ3 =
−i~Q2
2M
∫
∞
0
dω f(ω)ω2;
Q2w(0)(4) = −µ4 = −β4 =
−~Q2
2M
∫
∞
0
dω f(ω)ω3 coth(~ω/2kBT ). (35)
Making use of the f(ω) sum-rules reported in [76], it is possible to verify that the
previous relationships on the one hand, and the formula µ2 = 2Q2/3M〈Ek〉 together
with those in equation (22), on the other, coincide, provided the COM momentum
distribution has a zero kurtosis [57], as in a purely Gaussian functional form.
In conclusion, in the present study we have measured, making use of the TOSCA
crystal-analyzer inverse-geometry spectrometer, the neutron scattering spectrum of
liquid para-hydrogen (at T = 19.3 K and n = 25.4 nm−3) and ortho-deuterium
(at T = 21.3 K and n = 25.2 nm−3) in the energy and momentum transfer ranges
3 meV < ~ω < 1 eV and 2.9 Å−1 < Q < 23.0 Å−1, respectively. From the high-
energy and high-momentum region of the spectrum (namely 40 meV < ~ω < 1 eV,
and 5.6 Å−1 < Q < 23.0 Å−1), where the incoherent approximation for the center-
of-mass scattering law surely holds, we have been able to extract the COM (i.e.
translational) mean kinetic energy through a fit of the experimental spectra by
applying the well-known Young-Koppel model to describe the intra-molecular roto-
vibrational dynamics of H2 and D2. As expected, the mean kinetic energy turned out
to be greatly larger than the corresponding classical value (i.e. 1.5 kBT ), owing to the
517
G.Corradi et al.
presence of relevant quantum effects in both liquids. However, given the almost iden-
tical thermodynamic conditions, liquid para-hydrogen appeared rather more quantal
than ortho-deuterium: 〈Ek〉 = 78(4) K (para-hydrogen), and 〈Ek〉 = 65(3) K (ortho-
deuterium). Moreover, significant deviations from the impulse approximation have
been detected during the fitting procedure of both sample data (especially in the
40–50 meV < ~ω < 100 meV interval). Thus, the inclusion of some correction terms
(accounting for the so-called final state effects following Glyde’s additive approach)
was necessary in order to accurately describe the experimental spectra. We also
proved that this approach is a simple time-expansion of the well-known Gaussian
approximation, provided the COM momentum distribution exhibits a purely Gaus-
sian functional form. The first two coefficients controlling the intensity of the final
state effects were evaluated making use of the Silvera-Goldman inter-molecular po-
tential and the center-of-mass pair correlation functions of liquid H2 and D2. The
comparison of the present translational mean kinetic energy values with the avail-
able experimental (on o-D2) and simulated data (on p-H2) in the literature is quite
satisfactory and confirms the excellent performances of TOSCA in the spectroscopic
analysis of the condensed phases of liquid hydrogens.
Acknowledgements
This work has been financially supported by C.N.R. (Italy). The authors ac-
knowledge the skillful technical support of the ISIS Sample Environment Section.
One of the authors (G.C.) is also indebted to Accademia Nazionale dei Lincei (Italy)
for the research grant Borsa di Perfezionamento negli Studi di Spettroscopia Neu-
tronica “F.P.Ricci”.
References
1. Barocchi F., Neumann M., Zoppi M. // Phys. Rev. A, 1985, vol. 31, p. 4015; Baroc-
chi F., Neumann M., Zoppi M. // Phys. Rev. A, 1987, vol. 36, p. 2440.
2. Marinin V.S., Pashkov V.V. // Sov. J. Low Temp. Phys., 1977, vol. 3, p. 397;
Marinin V.S., Pashkov V.V. // Sov. J. Low Temp. Phys., 1977, vol. 3, p. 472.
3. Dyugaev A.M. // J. Low Temp. Phys., 1990, vol. 78, p. 79.
4. Ceperley D.M. // Phys. Rev. Lett., 1992, vol. 69, p. 331.
5. Ceperley D.M. // Rev. Mod. Phys., 1995, vol. 67, p. 279.
6. Voth G., Chandler D., Miller V.H. // J. Chem. Phys., 1989, vol. 91, p. 7749.
7. Cao J., Voth G. // J. Chem. Phys., 1993, vol. 99, p. 10070.
8. Cao J., Voth G. // J. Chem. Phys., 1994, vol. 100, p. 5093; 1994, vol. 100, p. 5106;
1994, vol. 101, p. 6157; 1994, vol. 101, p. 6168.
9. Martyna G.J. // J. Chem. Phys., 1996, vol. 104, p. 2018.
10. Cao J., Martyna G.J. // J. Chem. Phys., 1996, vol. 104, p. 2028.
11. Pavese M., Voth G. // Chem. Phys. Lett., 1996, vol. 249, p. 231; Calhoun A.,
Pavese M., Voth G. // ibid., 1996, vol. 262, p. 415.
12. Kinugawa K. // Chem. Phys. Lett., 1998, vol. 292, p. 454.
518
Mean kinetic energy and final state effects in liquid hydrogens
13. Balucani U., Zoppi M. Dynamics of the Liquid State. Oxford, Oxford University Press,
1994.
14. Hansen J.P., Mc Donald I. Theory of Simple Liquids. London, Academic Press, 1986.
15. Andreev A.F. // JETP Lett., 1978, vol. 28, p. 556; Andreev A.F., Kosevich Yu.A. //
Sov. Phys. JETP, 1979, vol. 50, p. 1218.
16. Feenberg E. Theory of Quantum Fluids. New York and London, Academic Press, 1969.
17. Ceperley D.M., Pollock E.L. // Phys. Rev. Lett., 1986, vol. 56, p. 351; Can J. //
Phys., 1987, vol. 65, p. 1416.
18. Sosnick T.R., Snow W.M., Sokol P.E. // Phys. Rev. B, 1990, vol. 41, p. 11185.
19. Azuah R.T., Stirling W.G., Glyde H.R., Sokol P.E., Bennington S.M. // Phys. Rev. B,
1995, vol. 51, p. 605.
20. Azuah R.T., Stirling W.G., Glyde H.R., Sokol P.E., Bennington S.M., Boninsegni M.
// Phys. Rev. B, 1997, vol. 56, p. 14620.
21. Andersen K.H., Stirling W.G., Glyde H.R. // Phys. Rev. B, 1997, vol. 56, p. 8978.
22. Glyde H.R., Azuah R.T., Stirling W.G. // Phys. Rev. B, 2000, vol. 62, p. 14337.
23. Timms D.N., Evans A.C., Boninsegni M., Ceperley D. M., Mayers J., Simmons R.O.
// J. Phys.: Condens. Matter, 1996, vol. 8, p. 6665; Azuah R.T., Stirling W.G.,
Glyde H.R., Boninsegni M. // J. Low Temp. Phys., 1997, vol. 109, p. 287.
24. Neumann M., Zoppi M. // Phys. Rev. E, 2002, vol. 65, p. 31203.
25. Landau L.D., Lifshits E.M. Statistical Physics. Vol. I. Moskow, Nauka, 1978.
26. Woods A.D.B., Sears V.F. // Phys. Rev. Lett., 1977, vol. 39, p. 415.
27. Celli M., Zoppi M., Mayers J. // Phys. Rev. B, 1998, vol. 58, p. 242.
28. Hilleke R.O., Chaddah P., Simmons R.O., Price D.L., Sinha S.K. // Phys. Rev. Lett.,
1984, vol. 52, p. 847.
29. Sears V.F. // Phys. Rev. B, 1984, vol. 30, p. 44.
30. Watson G.I. // J. Phys.: Condens. Matter, 1996, vol. 8, p. 5955.
31. Lovesey S.W. Theory of Neutron Scattering from Condensed Matter. Oxford, Oxford
University Press, 1987.
32. Peek D.A., Schmidt M.C., Fujita I., Simmons R.O. // Phys. Rev. B, 1992, vol. 45,
p. 9671.
33. Peek D.A., Schmidt M.C., Fujita I., Simmons R.O. // Phys. Rev. B, 1992, vol. 45,
p. 9680.
34. Herwig K. W., Sokol P.E., Sosnick T.R., Snow W.M., Blasdell R.C. // Phys. Rev. B,
1990, vol. 41, p. 103.
35. Sosnick T.R., Snow W. M., Sokol P.E. // Phys. Rev. B, 1990, vol. 41, p. 11185.
36. Andreani C., Filabozzi A., Nardone M., Ricci F.P., Mayers J. // Phys. Rev. B, 1994,
vol. 50, p. 12744.
37. Bafile U., Zoppi M., Barocchi F., Magli R., Mayers J. // Phys. Rev. Lett., 1995,
vol. 75, p. 1957.
38. Bafile U., Zoppi M., Barocchi F., Magli R., Mayers J. // Phys. Rev. B, 1996, vol. 54,
p. 11969.
39. Mayers J. // Phys. Rev. Lett., 1993, vol. 71, p. 1553.
40. Andreani C., Filabozzi A., Pace E. // Phys. Rev. B, 1995, vol. 51, p. 8854.
41. Bafile U., Zoppi M., Celli M. // Physica B, 1996, vol. 226, p. 304.
42. Bafile U., Zoppi M., Celli M., Magli R., Evans A.C., Mayers J. // Physica B, 1996,
vol. 217, p. 50.
43. Bafile U., Zoppi M., Celli M., Mayers J. // Phys. Rev. B, 1998, vol. 58, p. 791.
519
G.Corradi et al.
44. Andreani C., Filabozzi A., Pace E., Colognesi D., Zoppi M. // J. Phys.: Condens.
Matter, 1998, vol. 10, p. 7091; Andreani C., Pace E., Colognesi D. // Phys. Rev. B,
1999, vol. 60, p. 10008.
45. Langel W., Price D.L., Simmons R.O., Sokol P.E. // Phys. Rev. B, 1988, vol. 38,
p. 11275.
46. Celli M., Colognesi D., Zoppi M. // Eur. Phys. J. B, 2000, vol. 14, p. 239.
47. Zoppi M., Celli M., Colognesi D. // Eur. Phys. J. B, 2001, vol. 23, p. 171.
48. Zoppi M., Celli M., Colognesi D. // Europhys. Lett., 2001, vol. 53, p. 34.
49. Celli M., Colognesi D., Zoppi M. // Phys. Rev. E, 2002, vol. 66, p. 21202.
50. Van Kranendonk J. Solid Hydrogen. New York, Plenum Press, 1983.
51. Sears V.F. // Can. J. Phys., 1966, vol. 44, p. 1279.
52. Zoppi M. // Physica B, 1993, vol. 183, p. 235.
53. Young J.A., Koppel J.U. // Phys. Rev. A, 1964, vol. 33, p. 603.
54. Rahman A., Singwi K.S., Sjölander A. // Phys. Rev., 1962, vol. 126, p. 986.
55. Bermejo F.J., Kinugawa K., Cabrillo C., Bennington S.M., Fak B., Fernández-
Dı́az M.T., Verkerk P., Dawidowski J., Fernández-Perea R. // Phys. Rev. Lett., 2000,
vol. 84, p. 5359; Zoppi M., Neumann M., Celli M. // Phys. Rev. B, 2002, vol. 65,
p. 092204.
56. Zoppi M., Magli R., Howells W.S., Soper A.K. // Phys. Rev. A, 1989, vol. 39, p. 4684;
Zoppi M., Bafile U., Magli R., Soper A.K. // Phys. Rev. E, 1993, vol. 48, p. 1000;
Zoppi M., Bafile U., Guarini E., Barocchi F., Magli R., Neumann M. // Phys. Rev.
Lett., 1995, vol. 75, p. 1779; Zoppi M., Soper A.K., Barocchi F., Magli R., Bafile U.,
Ashcroft N.W. // Phys. Rev. E, 1996, vol. 54, p. 2773.
57. Glyde H.R. Excitations in Liquid and Solid Helium. Oxford, Clarendon Press, 1984.
58. Mahan G.D. Many-particle Physics. New York, Plenum Press, 1990.
59. Glyde H.R. // Phys. Rev. B, 1994, vol. 50, p. 6726.
60. Sears V.F. // Phys. Rev., 1969, vol. 185, p. 200.
61. Andersen K.H., Stirling W.G., Glyde H.R. // Phys. Rev. B, 1997, vol. 56, p. 8978.
62. Bowden Z.A., Celli M., Cilloco F., Colognesi D., Newport R.J., Parker S.F., Ricci F.P.,
Rossi-Albertini V., Sacchetti F., Tomkinson J., Zoppi M. // Physica B, 2000, vol. 98,
p. 276–278.
63. Huber K.P., Herzberg G. Constants of Diatomic Molecules. New York, van Nostrand
Reinhold Company, 1979.
64. Sears V.F. // Neutron News, 1992, vol. 3, p. 29.
65. McCarty R.D., Hord J., Roder H.M. Selected Properties of Hydrogen. NBS Mono-
graph, vol. 168, 1981.
66. Agrawal A.K. // Phys. Rev. A, 1971, vol. 4, p. 1560.
67. Celli M., Rhodes N., Soper A.K., Zoppi M. // J. Phys.: Condens. Matter, 1999, vol. 11,
p. 10229.
68. James F. MINUIT Minimization Package: Reference Manual. Geneva, CERN Program
Library, 1994.
69. Silvera I.F., Goldman V.V. // J. Chem. Phys., 1978, vol. 69, p. 4209.
70. Herwig K.W., Simmons R.O. // Mol. Phys., 1992, vol. 75, p. 1393.
71. Prydz R. NBS Report No. 9276, Boulder, CO, 1967 (unpublished); Roder H.M.,
Childs G.E., McCarty R.D., Angerhofer P.E. NBS Technical Note No. 641, 1973 (un-
published).
72. Corradi G. Determinazione dell’energia cinetica traslazionale media del deuterio liq-
520
Mean kinetic energy and final state effects in liquid hydrogens
uido tramite misure di spettroscopia neutronica. Master degree thesis. University of
Florence (IT), 2001 (unpublished).
73. Morishima N., Mizobuchi D. // Nucl. Instr. and Meth. A, 1994, vol. 350, p. 275;
Morishima N. // Ann. Nucl. Energy, 2000, vol. 27, p. 505.
74. Neumann M., Zoppi M. // Phys. Rev. A, 1991, vol. 44, p. 2474.
75. Mompeàn F.J., Garćıa-Hernández M., Bermejo F.J., Bennington S.M. // Phys.
Rev. B, 1996, vol. 54, p. 970.
76. Singwi K.S., Tosi M.P. // Phys. Rev., 1966, vol. 149, p. 70.
Вивчення середньої кінетичної енергії та
залишкових ефектів в рідкому водні за допомогою
непружного нейтронного розсіяння
Дж.Корраді 1,2 , Д.Колонезі 1 , М.Целлі 1 , М.Зоппі 1
1 Національна Рада природничих досліджень,
Інститут прикладної фізики ’Нелло Каррара’,
вул. Панціатічі 64, 50127 Флоренція, Італія
2 Університет м. Флоренція, фізичний факультет,
вул. Г.Сансоне 1, 50019 Сесто Фіорентіна, Італія
Отримано 29 квітня 2003 р., в остаточному вигляді – 21 липня
2003 р.
Використовуючи спектрометр TOSCA в ISIS нами було проведено
вимірювання спектрів нейтронного розсіяння рідкого пара-водню
(T = 19.3 K і n = 25.4 nm−3) і рідкого орто-дейтерію (T = 21.3 K
і n = 25.2 nm−3). З високоенергетичної області спектру 40 meV <
~ω < 1 eV, де застосовується некогерентне наближення для руху
центра мас, маємо можливість отримати трансляційну середню кі-
нетичну енергію, значення якої, як ми і сподівалися, відрізняється
від класичних значень. Проте, було відзначено сильні відхилення від
імпульсного наближення і тому певні поправочні доданки (врахуван-
ня т.зв. ефектів кінцевого стану) повинні бути взяті до розгляду для
точного опису експериментальних спектрів. Порівняння наявних да-
них по середній кінетичній енергії з значеннями експериментальни-
ми чи з моделювань, доступними в літературі, є зовсім добрим і під-
тверджує високу якість процедури спектрального аналізу на спект-
рометрі TOSCA для конденсованих фаз рідких воднів.
Ключові слова: рідкий пара-водень, рідкий орто-дейтерій,
середня кінетична енергія, ефекти кінцевого стану
PACS: 67.90.+z, 61.12.Ex
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