Neutron total cross-section model for liquids and its application to light water

Annals of Nuclear Energy 38 (2011) 1687–1692

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Neutron total cross-section model for liquids and its application to light water A.D. Viñales a, J. Dawidowski a,b,⇑, J.I. Márquez Damián a,b a b

Centro Atómico Bariloche and Instituto Balseiro Comisión Nacional de Energía Atómica, Universidad Nacional de Cuyo, 8400 Bariloche, Argentina Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina

a r t i c l e

i n f o

Article history: Received 14 December 2010 Accepted 14 April 2011 Available online 10 May 2011 Keywords: Neutron cross-sections Liquids Diffusion Light water

a b s t r a c t In this article we present a model that describes the total cross-section of molecular liquids, aiming to provide a good description in the subthermal energy range with a minimum number of parameters, amenable to perform calculations of interest in Nuclear Engineering. We follow the spirit of Granada’s Synthetic model that was successful in the thermal and epithermal ranges. The model describes the incoherent cross-section and is suited for hydrogenated liquids. We apply the model to light water at room temperature. The results compare favorably with experimental data. We compare with other models that describe the subthermal cross-sections. In the description we stress the need to have a good knowledge of the vibrational intermolecular energy spectrum, and the diffusion constant. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Liquid light and heavy water are the most widely used moderator and reflector materials in nuclear reactors. The knowledge of their dynamic and structural features, plays an essential role in the calculation codes employed in Nuclear Engineering, that make use of libraries of neutron cross-sections. Although much progress has been made in the experimental and theoretical aspects of water dynamics, surprisingly, the libraries employed by those codes are still based on models and experimental data that were obtained and processed during 1960s (Mattes and Keinert, 2005). Of particular interest in Nuclear Engineering is the knowledge of the neutron total cross-section rT. Neutron transmission measurements that allow the determination of rT can easily be performed in very short times. Also, as recently reviewed (Granada et al., submitted for publication), the analysis of rT as a function of the energy by means of suitable models, can be employed to obtain valuable information about the structure and dynamics of a given system. These considerations, emphasize the need for updated models and experimental of neutron total cross-sections. Particularly, the knowledge of total cross-sections for subthermal neutrons (energies below 103 eV) is required for reactor calculations in the design of cold sources for research reactors. Experimentally, the total cross-section for light and heavy water was measured by Zaitsev et al. (1991) at cold and ultra-cold neutron energies (reaching 106 eV). On the theoretical side different ⇑ Corresponding author at: Centro Atómico Bariloche and Instituto Balseiro Comisión Nacional de Energía Atómica, Universidad Nacional de Cuyo, 8400 Bariloche, Argentina. Tel.: +54 2944 445165; fax: +54 2944 445299. E-mail address: [email protected] (J. Dawidowski). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.04.015

approaches have been followed. Although successful descriptions have been attained, a simple model is still lacking. The problem of formulating a simple model amenable to perform calculations to describe the dynamic behavior of the neutron interaction with molecular liquids, was successfully tackled by Granada (1985) with the Synthetic Model. The model (essentially incoherent) was applied to a wide variety of hydrogenated liquids. Total cross-sections are described with a minimum input dataset related with the frequency spectra of the molecules in the thermal and epithermal energy ranges. However, the diffusive regime (energies below 1 meV) is not described by the Synthetic Model, because it is based on the free-gas model, and is out of its range of applicability. The Synthetic Model describes the neutron–molecule interaction with a free-gas model with a minimum input dataset, based on the description of the frequency spectrum with Einstein oscillators. Effective parameters (mass, temperature and vibrational factor) are employed, that vary with the incident neutron energy that represent the molecular internal degrees of freedom. A simple approach to the problem of formulating a scattering law that describe the diffusion and free-gas limits for low and high energy-transfers, respectively, was provided by Egelstaff and Schofield (1962). The model was employed to analyze quasi-elastic neutron scattering data, but the total cross-section calculated by integrating the scattering law and its comparison with experimental data was not explored at that time. The most successful model formulated hitherto to describe the total cross-section was introduced by Morishima et al. and particularized for light water. Employing the Egelstaff—Schofield model for the translational motion, and vibrational densities of states from experimental data (Morishima and Aoki, 1995; Edura and

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Morishima, 2004), they calculated the total cross-section in a wide range of energies. While a good agreement in the subthermal range with Zaitsev data was achieved, the variation of its behavior with temperature did not follow the experiments (Edura and Morishima, 2006). The reason for the disagreement was that while the model employed a considerable number of parameters to describe the vibrational spectrum, this parameterization was not completely reliable (due to lack of experimental information) in the whole temperature range from freezing to boiling points. At present the experimental information about the frequency spectrum as a function of temperature is incomplete and does not follow the present-day quality standards. The best available information was provided by Novikov et al. (1986), that obtained a complete set of information on the dynamics of light water in a series of temperature dependent measurements that were suitable to reconstruct the double differential scattering cross-section of H in light water, but the results lack of details in the corrections applied to the data and show discrepancies when compared to other series of experiments. Furthermore, the data do not compare favorably with Molecular Dynamics simulations. In this paper, we set about the task of formulating a simple model suited to describe the neutron total cross-section in the subthermal range, and also capable to reproduce the results already known in the thermal range, with a minimum number of input parameters, following the spirit of Granada’s Synthetic Model. The model is based on Granada’s previous description of water in the thermal range, with a translational scattering law based on the Egelstaff–Schofield model. The frequency spectrum is described by Einstein oscillators, with a weight based on experimental data. We show that the diffusion constant and the collective part of the frequency spectrum are deeply interrelated parameters in the description of the total cross-section. We show that with our model a good agreement with Zaitsev data is achieved, employing the diffusion constant of the water molecule known by different experimental methods. The multiphonon contributions to the total cross-section is analyzed, showing that in the low-energy limit they follow the ’1/v’ law. In contrast, the main term describing the molecular diffusion differs from that law.

Eq. (2) is the basis to calculate the total cross-section, starting from models formulated on the scattering law S(Q, e). It is easy to generalize Eq. (2) for a molecule. Let m be the index of the atom in the molecule, with Nm the number of equivalent atoms, rmb its boundatom cross-section, and Sm(Q, e) its scattering law, then

rðE0 Þ ¼

X

Nm

m

rmb 2 2k0

Z

1

Q dQ

0

Z

emax

Sm ðQ; eÞ de:

ð4Þ

emin

2.2. Molecular dynamics We will focus our attention on molecular isotropic systems, following the formalism developed by Granada (1985). The formalism will describe the incoherent part of the scattering law, so it will be a good description for hydrogenated systems. We will assume that the translational, vibrational and rotational degrees of freedom are not coupled. This means that the Hamiltonian that governs the evolution of the molecular coordinates of the system can be written as the sum

H ¼ Htr þ Hrot þ Hvib :

ð5Þ

The instantaneous position of the species m in the molecule can be written as

Rm ðtÞ ¼ rc ðtÞ þ dm ðtÞ þ um ðtÞ;

ð6Þ

where rc(t) is the position of the molecular center of mass, dm(t) is from the center of mass to the nucleus equilibrium position and um(t) its displacement from the equilibrium position. As a consequence of the uncoupled motions the intermediate scattering function can be expressed as

vm ðQ; tÞ ¼ vtr ðQ ; tÞvmrot ðQ ; tÞvmvib ðQ; tÞ;

ð7Þ

where the dependence is on the modulus of the vector Q, due to the isotropy of the system. In a liquid there are not truly free rotations. Instead, (as pointed out Nelkin, 1960) rotational oscillations that can be treated with the same formalism as vibrations, take place. Calling generically ‘‘oscillations’’ to rotations and vibrations, Eq. (7) can be written as

2. Theoretical framework

vm ðQ; tÞ ¼ vtr ðQ ; tÞvmosc ðQ ; tÞ:

In this section we will develop the theoretical framework to describe the total cross-section.

Then, the corresponding van Hove scattering function will be the convolution

Sm ðQ ; xÞ ¼ Str ðQ; xÞ  Smosc ðQ ; xÞ:

2.1. The total cross-section The neutron microscopic scattering total cross-section can be written as an integral of the van Hove scattering law as Dawidowski et al. (1999)

rðE0 Þ ¼

rb

Z

dX dE

4p

k SðQ; eÞ; k0

ð1Þ

where rb is the bound-atom cross-section, e = E0  E the difference between the incident and the emergent neutron energy and k0 and k the modules of the incident and the emergent neutron wavevectors, respectively. The integral is performed over all angles and final energies. A change of integration variables to Q, e makes explicit the kinematic range of integration, resulting

rðE0 Þ ¼

rb 2 2k0

Z 0

1

Q dQ

Z

emax

SðQ ; eÞ de;

ð2Þ

emin

emax

emin

  2 h k0 Q Q : ¼ 1þ m 2k0

ð9Þ

2.3. Translational part To describe the translational motion in a molecular liquid different approaches can be employed, depending on the aim of the model. Granada (1985) and Nelkin (1960) employed the free translational model of the molecular unit to describe the cross-sections in the thermal and epithermal range. However, in the case of the the diffusive regime that concerns us, the neutron interaction time with the system is long compared with the mean collision time in the molecule translation. Thus, to describe translations, we will adopt the Egelstaff–Schofield model (Egelstaff and Schofield, 1962) that describes correctly the long and short collision times regimes. The intermediate scattering function is written as

vtr ðQ ; tÞ ¼ expðcd ðtÞQ 2 Þ;

ð10Þ

where cd(t) is

where emin and emax are the limits of the kinematic range

  2 h k0 Q Q ; ¼ 1 m 2k0

ð8Þ

ð3Þ

" # 12 i h 2 2 cd ðtÞ ¼ D t  s : tþs kB T

ð11Þ

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In Eq. (11) D is the diffusion constant of the water molecule, T is the absolute temperature, and s is a characteristic relaxation time. For short times (t  s) in Eq. (10) we get the limit



   hD kB T 2 t  it Q 2 vtr ðQ; tÞ ¼ exp 2skB T h

ð12Þ

that corresponds to the expression for a free gas of molecules where

D

s

¼

kB T ; M

ð13Þ

where kB the Boltzmann constant, and M is the effective mass of the molecule, that differs from the bare molecule mass, because several molecules must be moved in the diffusive process (Egelstaff, 1967) due to the molecule interaction with its neighbors. On the other hand, for long times (t  s) Eq. (10) reduces to

p

e2 þ ðhDQ 2 Þ2 

3. The model In this section we will show the model we developed, based on the formalism presented in the last section.

We begin by expanding (18) in phonons, resulting 2

exp

Ds1 Q 2



where h. . .iR means an average over all the molecule orientations b is the unit vector in the direction of Q. and Q For rotational oscillations (evolution of the vector dm(t)) Nelkin (1960) showed that the same formalism is valid and the vibrational masses are relates with the Sachs-Teller tensorial mass (Sachs and Teller, 1941). For our purposes, these oscillations will be treated in the same way as vibrations.

ð14Þ

which is the expression for the simple diffusion. By Fourier transform of the Eq. (10) we get the scattering law

1

ð22Þ

3.1. Scattering law

vtr ðQ; tÞ ¼ expðDtQ 2 Þ

Str ðQ ; eÞ ¼

b :Ck i ; Mkm ¼ 1=h Q m R

12

   12  s1 2  exp e þ ðhDQ 2 Þ2 þ DsQ 2 K 1 2kB T h

e

 2 h Q cm ðtÞ 2

! ¼1þ

1 X



2 h 2

Q2

n ðcm ðtÞÞn

n!

n¼1

ð23Þ

If we write cm(t) of Eq. (19) as

ð15Þ

cm ðtÞ ¼

m X

amk expði-k tÞ

ð24Þ

k¼1

where K1 is the modified Bessel function (Abramowitz and Stegun, 1965), and we have defined the time s1 as



 h 2kB T

s1 ¼ s2 þ

2 !12 ð16Þ

:

Eq. (15), together with (13) is the scattering law that we will employ in this work to model the translational motion of the molecule.

where m = 2mf, and amk and -k are defined according to the following expressions

amk ¼

-k ¼

8 < ðnk þ1Þk hxk M m : nk k hxk M m



2.4. Oscillatory part

1 6 k 6 mf ;

ð25Þ

mf þ 1 6 k 6 m

xk ; 1 6 k 6 mf ; xk ; mf þ 1 6 k 6 m

ð26Þ

From these definitions, Eq. (23) can be written as This part corresponds to the coordinates um(t) and dm(t) (Eq. (6)) that evolve as oscillations as mentioned above. The displacements can be decomposed in normal modes as

um ðtÞ ¼

3N3 X

Ckm qk ðtÞ

2

 2 h exp Q cm ðtÞ 2

! ¼1þ

where N is the number of atoms in the molecule. In the summation of Eq. (17) the three free translations are excluded. The intermediate scattering function can be written as

! ! 2 2  2 h  2 h m vosc ðQ ; tÞ ¼ exp  Q cm ð0Þ exp Q cm ðtÞ 2 2

ð18Þ

where

Q2

n

n!

m X

amk1 . . . amkn expðiXðk1 ; . . . ; kn ÞtÞ:

ð27Þ

In the summation, the indexes k1, . . ., kn run from 1 to m, and we have defined

Xðk1 ; . . . ; kn Þ ¼ -k1 þ    þ -kn :

ð28Þ

The oscillatory scattering law, derived from Eq. (18) can thus be expressed as

! ! 2 2  h h cm Q 2 dðeÞ þ exp  cm Q 2 2 2  2 n h 1 m Q2 X X 2 amk1 . . . amkn dðe  hXðk1 ; . . . ; kn ÞÞ  n! n¼1 k ;...;k ¼1

Smosc ðQ ; eÞ ¼ exp 

mf X



1 k

k¼1

2 h 2

k1 ;...;kn ¼1

k¼1

cm ðtÞ ¼



n¼1



ð17Þ

1 X

hxk M m

ðnk þ 1Þeixk t þ nk eixk t



ð19Þ

where mf indicates the total number of normal modes with frequencies xk. At t = 0 cm(t) is

cm ¼ cm ð0Þ ¼

mf X

1

k¼1

hxk Mkm

ð2nk þ 1Þ

ð20Þ

In Eqs. (19) and (20) the Bose mean occupation number in the energy level ⁄xk is

nk ¼ ðehxk =kB T  1Þ1 : k

ð21Þ

The coefficients Mm (vibrational masses) result from the normal mode expansion (17)

n

1

ð29Þ The complete scattering law can be written from Eq. (9) as

! ! 2 2  h h cm Q 2 Str ðQ; eÞ þ exp  cm Q 2 2 2  2 n 2 h  1 m Q X X 2 amk1 . . . amkn Str ðQ ; e  hXðk1 ; . . . ; kn ÞÞ  n! n¼1 k ;...;k ¼1

Sm ðQ ; eÞ ¼ exp 

1

n

ð30Þ

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to the well-known tetrahedral network characteristic of liquid water. From Eq. (37), M1 is 24 neutron mass units.

3.2. Total cross-section To calculate the total cross-section, we introduce the expression (30) into Eq. (2). Thus the total cross-section can be decomposed in two terms

rm ðE0 Þ ¼ rm0 ðE0 Þ þ rm1 ðE0 Þ:

ð31Þ

The first term rm0 ðE0 Þ is originated in the zero-phonon term of (30) and is expressed as

r0 ðE0 Þ ¼

rmb

Z

2

2k0

1 0

! Z 2 emax  h 2 Q exp  cm Q dQ Str ðQ ; Þde: 2 emin

ð32Þ

The second term rm1 ðE0 Þ expresses the phonon contribution. After a change of variables in e it is expressed as 1 Z rmb X

!

2 h

2

Q 2  h r1 ðE0 Þ ¼ 2 Q exp  cm Q 2 2 n! 2k0 n¼1 0 Z  m e max X amk1 . . . amkn Str ðQ ; eÞde  m

2

1

n dQ ð33Þ

where emax y emin are defined as

emax ¼ emax  hXðk1 ; . . . ; kn Þ emin ¼ emin  hXðk1 ; . . . ; kn Þ

ð34Þ ð35Þ

The final expression for the total cross-section is calculated by adding the contribution of each atomic species as in Eq. (4) resulting

rðE0 Þ ¼

X

rH2 O ðE0 Þ ¼ 2rH ðE0 Þ þ rO ðE0 Þ H

emin

k1 ;...;kn ¼1

We will show the calculated cross-section of water at 24 °C (kBT = 25.6 meV). In our calculations the phonon expansion converged with five phonons in the energy range from 2.106 to 102 eV. The value of the diffusion constant employed was D = 0.242 Å2/ps, based on the data presented by Pruppacher (1972) from radioactive tracer techniques. The scattering total cross-section was calculated as

Nm rm ðE0 Þ:

ð36Þ

m

4. Application to light water In this section we will apply the model developed in the previous sections to light water at 24 °C. The vibrational frequency spectrum that we will employ is the same as the one presented by Granada in Ref. Granada (1985). However, since in this paper we pursue a precise description of the cross-section in the subthermal energies we need a more precise approach in the vibrational spectrum at low energies. To this end, based in the frequency spectrum reported in Refs. Bellissent-Funel et al. (1995), Morishima and Aoki (1995), we added a vibrational mode at 8.56 meV. This Einstein oscillator represents the collective vibrations following the spirit of the Synthetic Model. Thus our vibrational spectrum is represented by four modes as indicated in Table 1. Regarding the vibrational masses, we also followed the Synthetic Model. The mass M1 corresponding to the intermolecular energy mode ⁄x1, was calculated taking into consideration the normalization condition for the masses (Granada, 1985; Novikov et al., 1986)

1 1 1 þ ¼ : M 1 M M mol

ð37Þ

M is the effective mass of the diffusing unit, that was taken as four molecular masses (i.e. 72 neutron mass units). This number is supported by molecular dynamics studies (Martí et al., 1996), that show that at room temperature, the molecules participate in a mean of about 3.5 intermolecular hydrogen bonds, as corresponds

Table 1 Values of the parameters for this model used in the calculations. Energies are given in meV and masses in neutron mass units. atom

⁄x1

⁄x2

⁄x3

⁄x4

M1

M2

M3

M4

H O

8.56

70

205

481

24 24

2.38 342

4.77 746

3.18 373

ð38Þ

O

where r (E0) and r (E0)were calculated from Eqs. (31)–(33) with five phonons, with rHb ¼ 81:67 barns, and rOb ¼ 4:25 barns (Mughabghab et al., 1981). To compare with experimental data, the absorption ‘‘1/v’’ contribution was added employing the constants rHabs ¼ 0:3326 barns and rOabs ¼ 0:00019 barns (Mughabghab et al., 1981). In Fig. 1 we compare our model with experimental data measured at various temperatures. Zaitsev’s data (Zaitsev et al., 1991) were measured at 24 °C, Herdade’s (Heinloth, 1961) at 18 °C, Herdade’s (Herdade et al., 1973) at 23 °C and Russell’s (Neill et al., 1968) at 20 °C. Also compared in Fig. 1 are the results of our model with Granada’s Synthetic Model. As we pointed out Granada’s model was optimized in the thermal range. Our model compares favorably in the subthermal range with Zaitsev data. In Fig. 2 we show the different components of our calculation. We observe that the phononic terms clearly show a ‘‘1/v’’ behavior, while the 0-phonon term, dominated by the diffusive translation does not follow this behavior. It is useful to check the results of our calculation with the standard NJOY code (MacFarlane, 1994), usually employed and benchmarked in Nuclear Engineering calculations. To produce an input for this code to resemble the model proposed in this paper as closely as possible, we implemented a light water model using the LEAPR module with the parameters shown in Table 2. LEAPR computes S(Q, e) convolving the partial results for translation and oscillators with the S(Q, e) for a solid-like frequency spectrum. To run the model in LEAPR the lower energy modes (x1 and x2) were included in this frequency spectrum as Gaussian distributions, with widths indicated in Table 2, while the upper energy modes (x3 and x4) were treated as Einstein oscillators.

Total cross section per H2O molecule (barns)

m

5. Results

Zaitzev 1991 Herdade 1973 Heinloth 1961 Russell 1966 Synthetic Model 2

This model with D=0.242 Å /psec Morishima 2004

1000

100 0.01

0.1

1

10

E0 (meV) Fig. 1. Total cross-section of H2O at room temperature over the subthermal energy range compared with experimental data (see references).

A.D. Viñales et al. / Annals of Nuclear Energy 38 (2011) 1687–1692

Total cross section per H2O molecule [barns]

10000 0 phonon 1 phonon 2 phonons 3 phonons 4 phonons 5 phonons Absorption Total

1000

10

0.01

0.1

1

10

100

E0 (meV) Fig. 2. Detail of the components of the calculated total cross-section of H2O.

Table 2 Parameters of the NJOY/LEAPR model. Mode

Type

Energy (meV)

Weight

Diffusion Intermolecular Rotational Scissoring Stretching

Diffusion (c = 2.76370) Gaussian (r = 2 meV)a Gaussian (r = 2 meV)a Oscillator Oscillator

– 8.56 70 205 481

0.01389 0.04167 0.42024 0.20968 0.31452

Included as part of a continuous frequency spectrum.

Total cross section per H2O molecule (barns)

model and LEAPR/NJOY at subthermal energies and a general agreement with experimental results. Above 30 meV we observe that our model (calculated with five phonons) deviates from the experimental data, due to the lack of more phonon terms.

6. Summary and conclusions

100

1

a

1691

Zaitsev 1991 Heinloth 1961 Russell 1966 2 This model with D=0.242 Å /psec LEAPR/NJOY

1000

100

0.01

1

100

E0 (meV) Fig. 3. Comparison of our results and LEAPR/NJOY. Above 30 meV, our model (indicated with a dashed line) calculated with five phonons deviates from the experimental data.

LEAPR allows to employ either the Egelstaff or the free-gas models to describe the translational motion of liquids. We run it in the Egelstaff mode for Hydrogen, while Oxygen was modeled as free gas. Two parameters are required for the diffusive description: the translational weight wt and a constant c proportional to the diffusion constant:

MH M M D MH D c¼ ¼ h  wt h

wt ¼

ð39Þ ð40Þ

where MH is the mass of the Hydrogen atom, and D is the diffusion constant. The results (Fig. 3) show a good agreement between this

In this work we presented a model that describes the liquid incoherent total cross-section with a minimum number of input parameters employing a synthetic frequency spectrum. We followed the idea of Granada’s Synthetic Model, and improved its results in the subthermal energy range. The main difference between our model and Granada’s, is that our model describes the translational part of the scattering function with the Egelstaff–Schofield model, to allow the calculation of cross-sections at subthermal energies, while Granada’s model employs a free gas treatment for translations, suitable for thermal calculations. Another difference is that our model develops a scattering law, with a vibrational part described by the phonon expansion, whereas Granada’s model develops an effective function based on the gas model plus a one-phonon term, with effective temperatures, masses and vibrational factors depending on the neutron incident energy as a parameter. As shown in Figs. 1 and 2 after adding five phonon terms, a good description of the total cross-section (coincident with the Synthetic Model) is achieved up to 30 meV. To describe cross-sections at higher energies, more phonon terms must be added. On the other hand, at low energies, we observe in Fig. 1 that below 1 meV both models differ, and our model provides a better description of the experimental data. Since the addition of more phonon terms makes the calculation times longer, a simple prescription to minimize times when calculating the total crosssection is to combine a calculation of the higher energies with the Synthetic Model, and a second of the lower energies with our model. It is also worth to compare our model with Morishima’s. In Fig. 1 we see that both are remarkably coincident starting from 0.03 meV. Below this energy our model provides a better description of the experimental data. It must be stressed that the degree of complexity of Morishima’s model aims to the description of angular and energy distributions, rather than concentrating in the calculation of the total cross-section. Thus, it describes the frequency spectrum in greater detail, with Gaussian functions so it has a considerably larger number of input parameters. To test the numerical soundness of our calculation program we checked our results against the code NJOY, with the input provided by our model. The LEAPR/NJOY results show good agreement with our calculations in the subthermal range, and continue to agree with the experimental values at thermal energies. This agreement motivates the use of our model as a fast prototyping tool to test modifications in LEAPR/NJOY inputs. The model proposed in this work can be applied to a wide variety of hydrogenated molecular liquids where incoherent scattering dominates. Normally, a previous knowledge of intermolecular frequency spectrum (not necessarily very accurate) will be needed to generate the Einstein oscillators and their relative weight. On the other hand, we need to know the detail of the intermolecular part of the spectrum and/or the diffusion constant. Both are interdependent in the calculation of the total cross-section, in the sense that a variation of one of them will affect the other. In the case of water, treated in this paper, we represented the low-energy region of the spectrum by an Einstein oscillator with a frequency based on observed spectra, and a weight based on a simple hypothesis on the molecular coordination number. The diffusion constant was based on slight variations of values known from other techniques. If a precise knowledge of the frequency spectrum can be obtained

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from neutron inelastic scattering experiments, then the technique based on total cross-section measurements could be employed to determine the diffusion constant as an alternative to quasi-elastic neutron scattering. It is worth to add that a study of diffusive effects as a function of temperature should include both the measurement of total crosssections as well as low-energy frequency spectra, since it is wellknown that the diffusion constant and the intermolecular portion of the spectrum are very sensitive to temperature. The possibility to employ this technique as a customary tool to investigate diffusion remains to be developed in the future. Acknowledgments We acknowledge ANPCyT (Projects No. PICT Raíces 2006-574), and Universidad de Cuyo (Project No. 06/C288) for financial support. We are grateful to Prof. J.R. Granada for the useful discussions during the elaboration of this work. References Abramowitz, M., Stegun, I., 1965. Handbook of Mathematical Functions. Dover, New York. Bellissent-Funel, M.-C., Chen, S.H., Zanotti, J.-M., 1995. Phys. Rev. E 51, 4558.

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