On the microscopic structure of liquid hydrogens
Among the quantum liquids, the hydrogens cover an interesting position between liquid helium, where quantum exchange gives rise to the macroscopic phenomenon of superfluidity, and neon, where quantum effects are relatively small, so that its features can be evaluated by perturbation methods with...
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| Zitieren: | On the microscopic structure of liquid hydrogens / M. Zoppi, M. Celli, U. Bafile, E. Guarini, M. Neumann // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 283-297. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-1204342017-06-13T03:04:40Z On the microscopic structure of liquid hydrogens Zoppi, M. Celli, M. Bafile, U. Guarini, E. Neumann, M. Among the quantum liquids, the hydrogens cover an interesting position between liquid helium, where quantum exchange gives rise to the macroscopic phenomenon of superfluidity, and neon, where quantum effects are relatively small, so that its features can be evaluated by perturbation methods with reference to a classical system. Nonetheless, the experimental access to the microscopic structure of the hydrogens is not an easy task both because of their intra-molecular structure and the small molecular mass that is comparable with that of the neutron probe. In this paper we discuss the state of the art and summarise the available experimental information on the microscopic structure of the hydrogens. The experimental data for the two systems are compared among them and with the results of quantum Path Integral Monte Carlo simulations. It is found that similar quantities, measured in corresponding thermodynamic points, are rather different for the two systems due to the different weight of quantum effects. Moreover, the comparison with the simulation results shows that, for deuterium, there is a substantial agreement, both at the level of the structure factor and its thermodynamic derivatives. The agreement is less satisfactory for liquid hydrogen. Серед квантових рідин водень займає цікаве місце між рідким гелієм, для якого квантовий обмін приводить до макроскопічного явища надплинності, і неоном, для якого квантові ефекти відносно малі, так що його властивості можуть бути оцінені методами збурень відносно класичної системи. Тим не менше, експериментальний підхід до мікроскопічної структури водню не є легким завданням як через його внутрішню молекулярну структуру, так і через малу молекулярну масу, що співмірна з масою нейтронів. У цій статті ми обговорюємо стан справ і підсумовуємо наявну експериментальну інформацію про мікроскопічну структуру водню. Експериментальні дані для двох систем порівнюються між собою і з результатами розрахунку інтегралів за траєкторіями методом Монте Карло. Знайдено, що подібні величини, виміряні у відповідних термодинамічних точках, є швидше відмінні для двох систем завдяки різним вкладам квантових ефектів. Більше того, порівняння з результатами моделювання показують, що для дейтерію спостерігаються суттєві узгодження як на рівні структурного фактора, так і його термодинамічних похідних. Узгодження є менш задовільним для рідкого водню. 2001 Article On the microscopic structure of liquid hydrogens / M. Zoppi, M. Celli, U. Bafile, E. Guarini, M. Neumann // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 283-297. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 61.25.Em, 61.12.Gz DOI:10.5488/CMP.4.2.283 http://dspace.nbuv.gov.ua/handle/123456789/120434 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Among the quantum liquids, the hydrogens cover an interesting position
between liquid helium, where quantum exchange gives rise to the macroscopic
phenomenon of superfluidity, and neon, where quantum effects are
relatively small, so that its features can be evaluated by perturbation methods
with reference to a classical system. Nonetheless, the experimental
access to the microscopic structure of the hydrogens is not an easy task
both because of their intra-molecular structure and the small molecular
mass that is comparable with that of the neutron probe. In this paper we
discuss the state of the art and summarise the available experimental information
on the microscopic structure of the hydrogens. The experimental
data for the two systems are compared among them and with the results
of quantum Path Integral Monte Carlo simulations. It is found that similar
quantities, measured in corresponding thermodynamic points, are rather
different for the two systems due to the different weight of quantum effects.
Moreover, the comparison with the simulation results shows that, for deuterium,
there is a substantial agreement, both at the level of the structure
factor and its thermodynamic derivatives. The agreement is less satisfactory
for liquid hydrogen. |
| format |
Article |
| author |
Zoppi, M. Celli, M. Bafile, U. Guarini, E. Neumann, M. |
| spellingShingle |
Zoppi, M. Celli, M. Bafile, U. Guarini, E. Neumann, M. On the microscopic structure of liquid hydrogens Condensed Matter Physics |
| author_facet |
Zoppi, M. Celli, M. Bafile, U. Guarini, E. Neumann, M. |
| author_sort |
Zoppi, M. |
| title |
On the microscopic structure of liquid hydrogens |
| title_short |
On the microscopic structure of liquid hydrogens |
| title_full |
On the microscopic structure of liquid hydrogens |
| title_fullStr |
On the microscopic structure of liquid hydrogens |
| title_full_unstemmed |
On the microscopic structure of liquid hydrogens |
| title_sort |
on the microscopic structure of liquid hydrogens |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120434 |
| citation_txt |
On the microscopic structure of liquid
hydrogens
/ M. Zoppi, M. Celli, U. Bafile, E. Guarini, M. Neumann // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 283-297. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT zoppim onthemicroscopicstructureofliquidhydrogens AT cellim onthemicroscopicstructureofliquidhydrogens AT bafileu onthemicroscopicstructureofliquidhydrogens AT guarinie onthemicroscopicstructureofliquidhydrogens AT neumannm onthemicroscopicstructureofliquidhydrogens |
| first_indexed |
2025-07-08T17:52:38Z |
| last_indexed |
2025-07-08T17:52:38Z |
| _version_ |
1837102173778870272 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 283–297
On the microscopic structure of liquid
hydrogens
M.Zoppi 1 , M.Celli 1 , U.Bafile 1 , E.Guarini 2 , M.Neumann 3
1 Consiglio Nazionale delle Ricerche, Istituto di Elettronica Quantistica,
Via Panciatichi 56/30, I-50127 Firenze, Italy
2 Istituto Nazionale di Fisica della Materia, Unitá di Genova,
Dipartimento di Fisica, Genova, Italy
3 Institut für Experimentalphysik der Universität Wien,
Strudlhofgasse 4, A-1090 Wien, Austria
Received August 23, 2000
Among the quantum liquids, the hydrogens cover an interesting position
between liquid helium, where quantum exchange gives rise to the macro-
scopic phenomenon of superfluidity, and neon, where quantum effects are
relatively small, so that its features can be evaluated by perturbation meth-
ods with reference to a classical system. Nonetheless, the experimental
access to the microscopic structure of the hydrogens is not an easy task
both because of their intra-molecular structure and the small molecular
mass that is comparable with that of the neutron probe. In this paper we
discuss the state of the art and summarise the available experimental in-
formation on the microscopic structure of the hydrogens. The experimental
data for the two systems are compared among them and with the results
of quantum Path Integral Monte Carlo simulations. It is found that similar
quantities, measured in corresponding thermodynamic points, are rather
different for the two systems due to the different weight of quantum effects.
Moreover, the comparison with the simulation results shows that, for deu-
terium, there is a substantial agreement, both at the level of the structure
factor and its thermodynamic derivatives. The agreement is less satisfac-
tory for liquid hydrogen.
Key words: quantum liquids, liquid hydrogen, quantum effects
PACS: 61.25.Em, 61.12.Gz
1. Introduction
Quantum effects, in condensed matter, originate from the de-localization of the
particles which cannot be associated with individual points in the space but are
characterised by a probability distribution [1]. This, in turn, is somehow bound to
the de Broglie thermal wavelength, λDB, which depends on the mass of the molecule
c© M.Zoppi, M.Celli, U.Bafile, E.Guarini, M.Neumann 283
M.Zoppi et al.
Table 1. Evaluation of quantum effects for common quantum liquids. The sub-
script CP and TP refer to the Critical Point and to the Triple Point, respectively.
For helium, TP indicates the λ-point. λDB is the de Broglie wavelength defined
in equation 1. The parameters σ and ℓ represent the hard core diameter of the
particle and the average interparticle distance, respectively.
System
TCP
(K)
TTP
(K)
nCP
(nm−3)
nTP
(nm−3)
(
λDB
σ
)
CP
(
λDB
σ
)
TP
(
λDB
ℓ
)
CP
(
λDB
ℓ
)
TP
He 5.20 2.18 10.47 21.99 1.50 2.31 0.84 1.66
H2 33.19 13.96 9.00 23.06 0.72 1.11 0.44 0.94
D2 38.34 18.71 10.44 25.99 0.47 0.68 0.31 0.60
Ne 44.4 24.55 14.31 37.2 0.21 0.29 0.14 0.26
and on the temperature, according to the definition [2] :
λDB = h/(2πMkBT )
1/2, (1)
where M is the mass, T is the temperature, and kB is the Boltzmann constant.
Strictly speaking, the wavelength associated with a free massive particle can only
be defined assuming that the momentum of the quantum particle, p = Mv, is
identified with the momentum of the associated plane wave, p = ~k = h/λ. This
would imply a slightly different definition for the thermal wavelength.
For a dense system of interacting particles, the relation connecting λDB (or any
similarly defined thermal wavelength) with the space probability distribution is not
trivial. In addition, the actual size of the space probability distribution gives rise
to different effects, depending on the value of its ratio with the two other length
scales, namely the molecular size and the interparticle distance. If we define σCM
as the full width of the space probability distribution of the molecular centres of
mass, this quantity should be compared with σ, the usual hard-core parameter of
the pair potential, and with ℓ, the average interparticle distance. As a 0-th order
approximation, we can assume that σCM ≈ λDB. Thus, the relevant quantities can
be calculated and are reported in table 1.
Depending on the different ratios between λDB and σ, or between λDB and ℓ,
substantially different quantum effects are observed. For liquid helium, λDB is larger
than ℓ. This allows a quantum exchange between neighbouring particles and pro-
duces the macroscopic effect of superfluidity. However, as the density decreases (e.g.
around the critical point), the ratio becomes < 1 and the quantum exchange is
quenched. Nonetheless, the quantum size of the particle is sensibly larger than σ
and therefore large quantum effects are still expected due to the de-localization of
the particles (quantum diffraction). On going from helium to hydrogen, and deu-
terium, and neon, the expected exchange effects are even smaller. We see that for
deuterium and neon, the value of λDB never exceeds the size of σ and, to reproduce
the experimental data, it is sufficient to take into account the quantum diffraction
effect only. For neon, this ratio is so small that the expected quantum effects can
284
On the microscopic structure of liquid hydrogens
be effectively taken into account using perturbation theories on a classical reference
system [3]. For the hydrogens, instead, the quantum diffraction features are expected
to be rather large (λDB exceeds the value of σ) but no quantum exchange effect is
expected for those systems. In fact, even for liquid hydrogen at the lowest temper-
ature, i.e. close to the triple point, the ratio between λDB and ℓ is < 1. This ratio
could increase by decreasing the temperature, but in this case the liquid freezes and,
due to the zero-point motion of the molecular centres of mass, the kinetic energy
increases and so does the effective temperature.
To sum up, we observe that the hydrogens, among the quantum liquids, play a
very special role. On the one hand, the exchange effects appear to be negligible, and
therefore the Boltzmann statistics can be applied. On the other hand, the quantum
diffraction effects, i.e. those due to the spatial broadening of the centre-of-mass
wavefunction, are quite relevant and cannot be evaluated by a simple perturbation
theory. In addition, due to the same electronic structure, the hydrogens are expected
to experience, at least to a first approximation, the same intermolecular potential.
Therefore, the different behaviour that is observed in the two systems can only be
attributed to a different size of the quantum effects that, in turn, are determined by
the different molecular mass.
From the experimental point of view, to determine the microscopic structure
of liquid hydrogens is not a trivial task. On the one hand, the small number of
electrons make the hydrogens almost transparent to X-rays. The alternative experi-
mental technique, namely the neutron diffraction, turns out to be difficult too. The
small molecular mass, comparable with that of the probe, makes the inelasticity
effects quite relevant and this implies that the neutron diffraction data are heavily
affected by inelasticity effects. In addition, among the hydrogens, there is a sub-
stantial difference between the two isotopes. For deuterium, in fact, the coherent
neutron cross section, i.e. the parameter that is in front of the microscopic struc-
ture factor in the scattering cross section, is of the same order of magnitude of the
incoherent one. This implies that, qualitatively, the interesting signal lies on top of
a background that is of a similar order of magnitude [4]. For hydrogen, instead, the
ratio between the incoherent and the coherent cross sections is almost two orders of
magnitude. This makes the required signal extremely small, with respect to the inco-
herent background, and explains why no generally accepted experimental structure
factor exists, to date, for liquid hydrogen.
We are confident that a great part of the experimental problems could be dealt
with in a not too far future. Many experimental groups are working on it (see, for
example, [5]) and small partial achievements are obtained, month after month. How-
ever, the interpretation of the experimental results needs also a reliable theoretical
framework that is necessary to extract the relevant information from the experimen-
tal spectra. Dealing with quantum liquids is certainly more complex than with the
classical ones. As a matter of fact, theories to evaluate the microscopic structure
of simple classical liquids have reached a satisfactory level [2]. For quantum liquids,
instead, the theoretical approach is still in its infancy, even though the Path Integral
Monte Carlo (PIMC) simulation technique can be used to compute, given a suitable
285
M.Zoppi et al.
intermolecular potential, the microscopic structure of monatomic quantum liquids
[6,7]. The technique was initially implemented to simulate Boltzmann particles, and
has been recently improved to take into account the quantum statistics as well. For
example, using a PIMC simulation technique that is able to take into account the
Bose-Einstein statistics, Ceperley has shown that beautiful results can be obtained
for the microscopic structure of superfluid liquid helium [8]. Of course, dealing with
a Boltzmann quantum liquid is much less demanding, from a computational point
of view.
The microscopic structure of the quantum liquids listed in table 1 is affected by
quantum diffraction effects depending both on the temperature and the molecular
mass. For monatomic liquids, like helium or neon, the calculation of the microscopic
structure is more direct than for the molecular liquids. However, as the hydrogens
are characterised by a simple intramolecular structure, it is sufficient to take into
account the many-body quantum effects on the centre-of-mass distribution and to
deal with the intramolecular structure using suitable models. As a consequence, in
order to compare the experimental results with the theory, one has to combine the
results of the PIMC simulations (centres of mass structure) with those obtained
from a suitable theoretical calculation of the intramolecular structure.
We observe that, for a homonuclear diatomic molecule, the diffraction cross sec-
tion can be expressed as [4]:
dσ
dΩ
= u(Q) [S(Q)− 1] + v(Q) + P (Q), (2)
where S(Q) is the static structure factor of the molecular centres of mass and the
functions u(Q) and v(Q) are molecular form factors which are interpreted as the
intermolecular and intramolecular neutron cross sections respectively. The unknown
function P (Q) accounts for the inelastic scattering corrections. We point out that
in writing equation 2 we totally neglect the orientational correlations. This is a
reasonable assumption for liquid hydrogens, as it is well known that this is well
verified even in the solid phase [9].
If we apply a free rigid-rotor model to the molecule, then the expressions for the
molecular form factors assume a very simple analytical form [10]. However, it is well
known that this simple model is insufficient to take into account, quantitatively,
the microscopic structure of deuterium and should be generalised [4]. Young and
Koppel [11] have shown that, in the gas phase, a free rotating harmonic oscillator
model gives a substantially correct description of the neutron cross section of the
hydrogens. Therefore, to a first approximation, the molecule can be modelled as
a freely rotating harmonic oscillator. In this case, the molecular form factors can
be calculated and are expressed as a sequence of functions approximating the true
behaviour [12].
To the lowest order, the functions u(Q) and v(Q) become the familiar rigid
rotor functions that are now modulated by a Debye-Waller factor generated by the
harmonic oscillator [12]. These are:
u(0)(Q) = 4a2coh
[
exp
(
−
λ2
DWQ2
2
)
sin(QDe/2)
(QDe/2)
]2
(3)
286
On the microscopic structure of liquid hydrogens
and
v(0)(Q) = 2(a2coh + a2inc) + 2(a2coh − ba2inc) exp(−2λ2
DWQ2)
sin(QDe)
(QDe)
, (4)
where λDW = (~/2Mωv)
1/2 is the Debye-Waller wavelength and ωv is the circu-
lar frequency of the molecular vibration. The parameter De represents the average
equilibrium distance of the two nuclei while a coh and ainc represent the coherent and
incoherent scattering length [13]. For orthodeuterium b = 0, while for parahydrogen
b = 1 [12]. The sequences converge rapidly and it is found that their limiting be-
haviour can be well represented by equations 3 and 4 if the parameters λDW and De
are replaced by slightly different effective parameters [12].
2. An overview of the experimental results on the S(Q)
The experimental determination of the structure factor of helium, over a wide
range of thermodynamic state points, is rather old [14] and that of liquid neon is
even older [15]. Instead, no reliable information was available on liquid hydrogens till
recently [16]. In fact, in order to correct the neutron diffraction data for inelasticity
effects, researchers have mostly used the so-called Placzek correction procedure [17].
This is a perturbative treatment whose critical parameter is the ratio between the
mass of the neutron and that of the target particle. For helium, this value is 1/4 and
the theory is still manageable. However, the theory holds for a monatomic system
and could only be applied to a molecular system by introducing the concept of
effective mass (atomic mass 6 mE 6 molecular mass) which in turn depends on
the energy of the incident neutrons. For hydrogen, the mass ratio would become, at
best, 1/2 which makes the Placzek correction procedure unmanageable.
The turning point was determined by the availability of new diffraction instru-
ments that use a pulsed neutron source instead of a reactor. In fact, on a reactor
source, the incident neutrons are of constant energy and the momentum transfer,
~Q, is changed simply by changing the scattering angle. On the contrary, using a
pulsed source, the scattering angle is kept fixed and the spread in Q is obtained
from the energy distribution of the incident neutrons. Thus, there is an intrinsic
advantage in using time-of-flight (TOF) neutron diffraction for measuring the static
structure factor S(Q) of liquids composed of light molecules. This is simply related
to the fact that, here, a complete information can be obtained maintaining the scat-
tering angle to a rather small value. As the inelasticity effects increase by increasing
the scattering angle, it appears that choosing a sufficiently small scattering angle
the inelasticity effects can be kept within an acceptable level, even for a light system
like hydrogen.
The first measure of the microscopic structure of deuterium gas on a standard
powder diffractometer convinced us that the problem of the inelastic scattering cor-
rections could be greatly reduced using only the small angle scattering detectors [18].
Following this achievement, we took advantage of the construction of a dedicated liq-
uid diffractometer (SANDALS, at ISIS, UK) that uses an almost forward scattering
287
M.Zoppi et al.
0
1
2
3
4
5
6
7
0.0
0.5
1.0
1.5
2.0
2.5
EXP.
PIMC NWB
PIMC LJ
S
(Q
)
Q(A
-1
)
Figure 1. Microscopic structure factor for the centres of mass distribution of liq-
uid deuterium close to the Triple Point. The dots with the error bars (almost
invisible in the figure) are the experimental points, while the lines represent the
PIMC simulation results using two different intermolecular potentials. The con-
tinuous line is obtained using the NWB phenomenological potential, while the
dashed line represents the Lennard Jones results.
geometry. Using SANDALS, the structure factor of liquid deuterium was measured
both in the vicinity of the triple point [4] and close to the freezing transition [19].
Once the know-how was established, a diffraction experiment on liquid deuteri-
um was also carried out at a reactor source [20], which allowed us to refine the
experimental results. In fact, for the two experiments, the inelasticity corrections
are located in different spectral regions and therefore, from a critical comparison of
the two experimental determinations, we were able to obtain an accurate determi-
nation of the structure factor of liquid deuterium which, in turn could be positively
compared with the simulation results [21]. This comparison is reported in figure 1.
Here, the experiment is compared with the PIMC simulation results obtained using
either the Norman, Watts, and Buck (NWB) [22] or the Lennard Jones (LJ) [23]
intermolecular potential (cf. the continuous and the dashed lines in figure 1).
Thus the residual difficulty for performing an experiment on hydrogen is the
unfavourable ratio between the coherent and the incoherent cross section. Because
of this ratio, the coherent scattering contribution, which carries the information
on the intermolecular structure, is expected to become almost invisible due to the
overwhelming incoherent background. In fact, a rough calculation [12] assuming that
the intermolecular (centre of mass) structure factor of liquid H2 is similar to that
of D2, would predict a coherent modulation of the order of 4% of the incoherent
288
On the microscopic structure of liquid hydrogens
background. So, the experiment appears to be intrinsically difficult, also because we
expect that the coherent response of hydrogen will be smaller than that of deuterium
due to the larger quantum effects induced by the smaller molecular mass. However,
the situation improves if pure para-hydrogen is considered. In this case, due to the
decrease of the intramolecular scattering term at low Q for para-hydrogen, this ratio
would becomes 10% in the region of the main peak of the structure factor and the
experiment would appear feasible, with a reasonable accuracy.
Based on this premise we have carried out a neutron diffraction experiment
on liquid para-hydrogen on SANDALS. However, as we will show in the following
section, the predictions were too optimistic and it was impossible to extract a reliable
microscopic structure factor from the large incoherent background.
One might argue that, in the end, an experimental determination of the mi-
croscopic structure factor of hydrogen is not strictly necessary. A sufficiently accu-
rate S(Q) has been determined for deuterium and, using a suitable intermolecular
potential, it was shown that PIMC simulations could reproduce the experimental
structure factor. Therefore, the hydrogen structure factor could be determined, to
a good approximation, using the same potential and a PIMC simulation applied to
hydrogen.
On the other hand, there is a number of questions that are waiting for an answer.
For example, to which extent the hydrogen and deuterium intermolecular potentials
can be considered equal? Is it not possible that the quantum mechanical g(r) (or
S(Q)) can be obtained from the classical ones simply by a convolution with a suitable
single-particle (center of mass) distribution function? Of course, the present state
of the art of the PIMC technique can give an answer to these and other questions,
but the final assessment, in our opinion, can only rely on the comparison with
solid experimental data. For this reason only the experimental results can definitely
answer these and similar questions.
3. Thermodynamic derivatives of S(Q)
Even though the microscopic structure factor represents the final aim of our ex-
perimental efforts, the lack of information on the S(Q) of hydrogen does not mean
that additional experimental information could not be obtained from the experi-
ments.
In fact, considering equation 2, we observe that the most relevant dependence
on the thermodynamic conditions of the sample is in S(Q). Therefore, since our
experimental data were taken in selected thermodynamic points (cf. table 2), and
by assuming that P (Q) is constant, we could derive an experimental determination
of the thermodynamic derivatives of S(Q). This procedure was successfully used for
liquid deuterium data close to the triple point [4,20].
However, the same procedure could not be applied to the hydrogen data. In fact,
the size of the inelastic scattering corrections, that is expected to be proportional
to the total cross section, becomes now relevant. In other words, due to the large
intramolecular contribution to the scattering cross section, even small fluctuations
289
M.Zoppi et al.
Table 2. Details of the thermodynamic conditions of the experiments. The esti-
mated error on the temperature is 0.1 K, for the H2 experiment, and 0.5 K, for
the D2 experiment. The uncertainty on the pressure is 0.1 bar. As a consequence,
the error on the density of hydrogen is 0.03 nm−3 and that on the density of
deuterium turns out twice as large.
State T (K) p (bar) n (nm−3)
D2 – 1 20.7 2.2 25.42
D2 – 2 20.7 20.1 25.84
D2 – 3 22.0 20.2 25.45
D2 – 4 20.7 40.2 26.30
D2 – 5 23.5 40.6 25.46
H2 – 1 17.1 2.0 22.22
H2 – 2 17.1 29.9 23.00
H2 – 3 20.3 29.9 22.22
H2 – 4 17.1 16.2 22.61
H2 – 5 18.6 15.9 22.19
in the instrument electronics become now crucial and tend to generate noise that
submerges the coherent signal. For these reasons, a different procedure was used
for hydrogen [24]. Making use of the experimental differences, and of equation 2 to
remove the molecular form factor, we were finally able to obtain the thermodynamic
derivatives of the structure factor S(Q).
In figure 2 we report [∂S(Q)/∂n]T , the experimental density derivative, at con-
stant temperature, of the hydrogens. As it was anticipated, the deuterium deter-
mination appears more precise than that of hydrogen. The two sets of data are
qualitatively and quantitatively very similar, even though small differences can be
observed from a superposition of the two figures. The experimental temperature
derivative at constant density, [∂S(Q)/∂T ]n, is reported in figure 3. Again, the deu-
terium case is more definite and the two figures are qualitatively very similar. How-
ever, some quantitative difference can be observed, especially in the position of the
main minimum.
A quantitative comparison between the two sets of data is reported in the two
following figures. In figure 4 the deuterium data for the density derivative are rep-
resented by the continuous line (a spline fit through the experimental points) while
the hydrogen data are the points with the error bars. The quantitative comparison
is rather good, even though the hydrogen data seem to suggest a slight shift to
smaller Q-values. In addition, some difference appears in the small-Q region where
the hydrogen data seem to rise faster to the zero-level. In figure 5, we report the
temperature derivative (the symbols have the same meaning) but now the shift of
the hydrogen data toward smaller Q-values appears more evident than before.
While for the first observation, namely the shift that is observed in both figures
290
On the microscopic structure of liquid hydrogens
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
D2
[d
S
/d
n]
T
(
nm
3
)
Q(A
-1
)
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
H2
Figure 2. Density derivative, at constant temperature, of the structure factor
of the hydrogens. The two pictures appear quantitatively similar, even though
the deuterium data (lower figure) appear much cleaner that the hydrogen ones
(upper figure).
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
D2
[d
S
/d
T
]
n
(
K
-1
)
Q(A-1)
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
H2
Figure 3. Temperature derivative, at constant density, of the structure factor of
the hydrogens. Also in this case the two pictures appear quantitatively similar.
However, a slight difference can be observed in the depth of the minimum that
appears larger for deuterium.
291
M.Zoppi et al.
0
1
2
3
4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
D2
H2
[d
S
/d
n]
T
(
nm
3
)
Q(A
-1
)
Figure 4. Direct comparison between the density derivatives of S(Q) of the hy-
drogens. The line represents the deuterium behaviour while the hydrogen data
are represented by the black dots with the error bars.
0
1
2
3
4
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
D2
H2
[d
S
/d
T
]
n
(K
-1
)
Q(A
-1
)
Figure 5. Direct comparison between the temperature derivatives of S(Q) of the
hydrogens. The line represents the deuterium behaviour while the hydrogen data
are represented by the black dots with the error bars.
292
On the microscopic structure of liquid hydrogens
(4 and 5), we can suggest a simple, intuitive, explanation, the difference in the rising
edge at small Q, that is observed in figure 4, is totally unexpected. In fact, when the
data are plotted in reduced units of Q (i.e. as a function of Q∗ = Qσ, where σ is the
effective molecular diameter), it can be noted that the rising edge and the positive
peak of the density derivative, and the whole temperature derivative, become almost
coincident provided that the effective diameter of hydrogen is increased by ∼ 2%.
As a matter of fact, liquid hydrogen is expected to show larger quantum effects
with respect to deuterium. This is consistent with the data reported in table 1. One
aspect of the increased quantum behaviour is an expected increment in the effective
molecular size of hydrogen with respect to deuterium. If we reasonably assume that
the two isotopes experience the same intermolecular potential, that is determined
by the same electronic structure, we note that the observed increase of the effective
molecular size is a signature of the expanded single particle wavefunction.
4. Comparison with the PIMC simulations
The experimental results have been compared with those of a PIMC simulation
carried out in similar thermodynamic conditions. For both hydrogen isotopes, we
used the same intermolecular potential, namely the NWB potential [22]. The sim-
ulations were carried out using N = 500 classical particles and Trotter numbers P
(= number of bids in the classical isomorphic polymer system) ranging between 1
and 32 [25]. After a suitable number of equilibration moves, starting from a random
initial configuration, the various stochastic configurations produced by the PIMC
sequence were used to evaluate the thermodynamic averages. Typically, a minimum
of 10.000× 5×N moves was used to average the radial distribution function, g(r).
The PIMC results were analysed as a function of P and we found that P = 16 was
sufficiently accurate for deuterium, while P = 32 was sufficiently large to represent
the correct quantum limit of hydrogen.
The limited number of particles (N = 500) was not so large to produce a simula-
tion g(r) sufficiently extended to obtain a reliable S(Q). In order to extend the cutoff
radius from ≈ 14 A (i.e. half the size of the simulation box), to a more suitable value
of ≈ 30 A, N should increase by a factor of 10, and the CPU time would increase,
approximately, by two orders of magnitude. Thus, we had to devise a different way
of extending the range of g(r), so that we could overcome this limit. To this end,
we used an extension procedure, suggested by Verlet [26], which approximates the
function h(r) = g(r)− 1 using a damped oscillating form
h(r) = (A/r) exp(−r/r0) sin(r/r1). (5)
The parameters A, r0, and r1 are obtained by fitting the functional form (5) to the
simulation results, starting from the third zero of h(r).
In figures 6 and 7 we report the comparison between the experimental results and
the PIMC simulations for the thermodynamic derivatives of the structure factor of
the hydrogens. We observe that, for the case of deuterium, there is nice agreement,
also quantitative, between the PIMC calculations and the experimental results. For
293
M.Zoppi et al.
0
1
2
3
4
5
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
D2_exp
NWB
[d
S
/d
n]
T
(n
m
3
)
Q(A
-1
)
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
H2_exp
NWB
Figure 6. Density derivatives, at constant temperature, of the structure factor
of the hydrogens. Comparison between experiment and PIMC simulations using
the NWB intermolecular potential. We observe a rather good quantitative agree-
ment for the deuterium data (lower figure) while for hydrogen (upper figure) the
agreement cannot be considered more than qualitative.
0
1
2
3
4
5
-0.04
-0.03
-0.02
-0.01
0.00
0.01
D2_exp.
NWB
[
dS
/d
T
]
n
(
K
-1
)
Q(A-1)
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
H2_exp.
NWB_ext
Figure 7. Temperature derivative, at constant density, of the structure factor
of the hydrogens. Comparison between experiment and PIMC simulations using
the NWB intermolecular potential. We observe a rather good quantitative agree-
ment for the deuterium data (lower figure) while for hydrogen (upper figure) the
agreement cannot be considered more than qualitative.
294
On the microscopic structure of liquid hydrogens
hydrogen, instead, the agreement is less satisfactory. In the density derivative (fig-
ure 6), the data appear a little out-of-phase with respect to the PIMC results. In
order to increase the agreement with the simulation, the experimental points should
be located to smaller values of Q, thus increasing the difference already observed in
figure 4 with respect to deuterium. In addition, a further mismatch is observed in
the first portion of the experimental function where, a simple shifting of the data
is not sufficient to recover the required agreement. A similar qualitative result is
observed in figure 7, but now a simple shift of the simulation results to higher values
of Q would appear to be sufficient.
5. Discussion
By means of neutron diffraction, we have measured the thermodynamic deriva-
tives of the structure factor of the liquid hydrogens close to the triple point. Due to
the relatively small incoherent background we succeeded in obtaining the S(Q) of
liquid deuterium. However, no such results could be obtained for hydrogen, due to
the combined effect of larger inelasticity effects and an even larger incoherent back-
ground. Nevertheless, the thermodynamic derivatives of the microscopic structure
factor were obtained in both cases. This experimental information was analysed by
itself, by comparing the deuterium and the hydrogen data, and in comparison with
the simulation results.
The direct experimental comparison evidenced a different behaviour between
light and heavy hydrogen that could be qualitatively attributed to a different size
of the hard core diameter of the two molecules. This difference is consistent with a
different quantum behaviour related to the mass difference of the two isotopes.
As far as the comparison with the PIMC simulation is concerned, we observe that
the simulation results would suggest an even further shift of the hydrogen data to
smaller Q values in order to obtain a nice agreement with the experiment. Moreover,
a difference in the small-Q interval, already observed in the comparison between the
two experimental density derivatives (cf. figure 4) remains unexplained.
The observed differences could be attributed either to residual systematic errors,
of unknown origin, that have not been accounted for in the data analysis of the hy-
drogen experiment, or to the calculated results, especially on going from deuterium
to hydrogen. We tend to exclude problems from the simulation technique. In fact,
PIMC simulations have been successfully applied to even more quantum systems
without problems. However, it may be that the problem stems from the intramolec-
ular model that was used in the data analysis of the experiments. For example, it is
possible that the free rotating harmonic oscillator model could be good enough for
deuterium but insufficiently accurate for hydrogen.
All we can do here is to further check the experimental data. To this end, we have
planned a new experiment, now on a reactor source, to validate the experimental
data of liquid hydrogen. In fact, even if this experiment seems to be more difficult, in
principle, the expected systematic errors would affect, in this case, a totally different
region of Q. Therefore, from a critical comparison of the two sets of data, we are
295
M.Zoppi et al.
confident to extract a more precise information.
To sum up, we have shown that the microscopic structure of the hydrogens ac-
tually depends, as expected, on the isotopic mass which, in turn, acts on the size
of the quantum behaviour of the molecule. Some results are qualitatively consistent
with the theoretical predictions, even though a fully satisfactory quantitative agree-
ment hasn’t been obtained yet. We hope that further experimental data that will
be available in the near future may bring the whole picture to an overall agreement.
However, should this not happen, the observed discrepancies would raise a further
interesting question concerning the peculiar behaviour of one of the most studied
substances on earth and the most abundant, in absolute, in the universe.
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До мікроскопічної структури рідкого водню
М.Зоппі 1 , М.Челлі 1 , У.Бафіле 1 Е.Гуаріні 2 , М.Нойманн 3
1 Інститут квантової електроніки,
1-50127 Флоренція, Італія
2 Національний інститут фізики матерії,
Генуя, Італія
3 Інститут експериментальної фізики,
А-1090 Відень, Австрія
Отримано 23 серпня 2000 р.
Серед квантових рідин водень займає цікаве місце між рідким гелі-
єм, для якого квантовий обмін приводить до макроскопічного яви-
ща надплинності, і неоном, для якого квантові ефекти відносно малі,
так що його властивості можуть бути оцінені методами збурень від-
носно класичної системи. Тим не менше, експериментальний підхід
до мікроскопічної структури водню не є легким завданням як через
його внутрішню молекулярну структуру, так і через малу молекуляр-
ну масу, що співмірна з масою нейтронів. У цій статті ми обговорює-
мо стан справ і підсумовуємо наявну експериментальну інформацію
про мікроскопічну структуру водню. Експериментальні дані для двох
систем порівнюються між собою і з результатами розрахунку інтег-
ралів за траєкторіями методом Монте Карло. Знайдено, що подібні
величини, виміряні у відповідних термодинамічних точках, є швидше
відмінні для двох систем завдяки різним вкладам квантових ефектів.
Більше того, порівняння з результатами моделювання показують, що
для дейтерію спостерігаються суттєві узгодження як на рівні струк-
турного фактора, так і його термодинамічних похідних. Узгодження є
менш задовільним для рідкого водню.
Ключові слова: квантові рідини, рідкий водень, квантові ефекти
PACS: 61.25.Em, 61.12.Gz
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