Multiple scattering and inelasticity corrections in thermal neutron scattering experiments on molecular systems

ELSEVIER Physica B 203 (1994) 116 128

Multiple scattering and inelasticity corrections in thermal neutron scattering experiments on molecular systems t

j. Dawidowsk i a, t, J.R. G r a n a d a a'*' x, R.E. Maye r a, G.J. Cuello"' 2, V.H. Gil let te a, M.-C. Bel l issent-Funel b

aComisibn Nacional de Energia Atbmica, Centro Atbmico Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina b Laboratoire Lbon Brillouin (CEA-CNRS), CE-Saclay, 91191 Gi[2sur-Yvette cedex, France

Received 26 October 1993; revised 24 April 1994

Abstract

In this work we address the problem of multiple scattering and inelasticity corrections in neutron diffraction measurements for structural studies of molecular systems. A variety of pulsed neutron and reactor experiments was performed on hydrogenous samples under different experimental conditions. Monte Carlo simulations were carried out on the basis of a Synthetic Model to describe the neutron-molecule interaction, and they allowed a simultaneous evaluation of multiple, inelastic and beam attenuation processes into the samples. A very good agreement between measurements and simulations was obtained in all cases, for our demanding choice of samples and experimental conditions.

1. Introduction

Neutron scattering techniques are well established as a powerful tool for the investigation of condensed matter, providing in many cases unique information about its structural and dynamical properties. In particular, a great deal of experimental work, using those methods, has been done in recent years to elucidate some fundamental ques- tions about the structure of molecular liquids and amorphous systems.

* Corresponding author. Also at Consejo Nacional de lnvestigaciones Cientificas y T~c-

nicas (CONICET), Argentina. 2 Also at CRUB, Universidad Nacional del Comahue. *The material contained in this paper is part ofJ. Dawidowski's PhD thesis (Instituto Balseiro, May 1993).

The high statistical accuracy achievable today in neutron scattering measurements performed with the use of powerful sources and sophisticated instruments, does not guarantee by itself the attain- ment of reliable information about the system under study, if we consider that some unavoidable corrections must be applied to the observed spectra. However, the development of adequate procedures to treat them has not paralleled the evolution of source intensity and instrumental quality.

In neutron scattering experiments where we seek structural parameters of molecular systems, the inelastic contribution to the observed spectra con- forms to a pedestal over which the characteristic oscillations of the coherent component are superimposed, and that pedestal will be, in general, quite

0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI O 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 1 7 1 - Q

J. Dawidowski et al./ Physica B 203 (1994) 116-128 117

different for measurements performed on stationary or pulsed neutron sources.

A few years ago, Soper [1] considered that specific problem, showing that structural parameters obtained with both techniques present some discrepancies due to uncertainties in the evaluation of inelasticity. Besides the fact that the experimental techniques based on pulsed and stationary sources explore the scattering law S(Q, to) along different paths in the momentum (hQ) and energy (hog) transfer space, and that other dissimilarities there exist in their respective instruments' characteristics, Bermejo et al. [2] emphasized that additional and more subtle problems associated to multiple scattering effects could also be present.

Leaving aside the nontrivial matter of data normalization, the extraction of a cross section from an observed spectrum then requires the application of the pertinent corrections which, although formally well understood from a theoretical point of view as being due to the same basic interaction process, have been traditionally treated in a dissociated manner.

The inelasticity corrections are usually worked out in terms of expansions involving energy-transfer moments of the scattering law, as proposed by Placzek [3], whereas multiple scattering corrections are handled through a variety of schemes based on the method proposed by Blech and Aver- bach [4], originally devised to account for corrections in scattering experiments in which elastic and isotropic processes can be assumed. Approximated analytical methods which consider both inelastic and multiple interactions have been presented, no- tably that due to Sears [5], but in general their limits of validity are not clearly established [6] although the applicability to a particular case was recently assessed [7].

In view of those alternatives, the numerical simulation of a given experiment arises as the most reliable way for the evaluation of inelasticity and multiple scattering effects as a whole, an obvious idea which was already put in practice many years ago [8]. Nevertheless, those simulations require a suitable model for a proper description of inelastic processes, while keeping an acceptable com- puter-time economy at the same time.

A few years ago, a Synthetic Model was introduced [-9] to describe, in an approximate manner,

the incoherent inelastic interactions of neutrons with molecular systems. With a minimum set of input data, it allows the evaluation of fundamental magnitudes like total cross sections, energy transfer kernels, etc., the complete analyticity of the resultant expressions being its main advantage. This fea- ture makes the Synthetic Model a highly conve- nient one to be used as a kernel for calculations in numerical simulations.

In this work we present the results of our studies, both on the experimental and calculational sides, related to the multiple scattering and inelasticity corrections in neutron scattering experiments designed to explore the structure of molecular liquid and amorphous systems. For this purpose we performed a wide set of neutron measurements on water and polyethylene samples of different sizes, some of them specifically intended to exacerbate the effects of those corrections, employing pulsed and stationary neutron sources. The experiments were followed by numerical simulations, based on the Synthetic Model description of the actual scattering law, and under the premise that all neutron processes inside the sample are considered without any further assumption or approximation, except those already borne into the model.

2. Neutron sources and instruments

The scattering experiments were performed using a steady source (reactor Orph6e of the Laboratoire L6on Brillouin - LLB, Saclay, France) and a pulsed source (at the Bariloche LINAC, Centro At6mico Bariloche - CAB, Argentina).

2.1. Reactor experiments

The reactor measurements were performed at the reactor Orph~e of LLB, on the 7C2 spectrometer which is equipped with a BF3 position sensitive detector with 640 cells [10]. The angular step between two adjacent cells is equal to 0.2 ° which leads to a maximum diffraction angle 20 = 128 °. The incident wavelength used in these experiments (2o = 0.712 A) allowed us to cover a scattering vector (Q = 4~(sin 0)/20) range from 0.3 ~,- 1 to 16 A - 1 The neutron wavelength was determined from the

118 J. Dawidowski et al./ Physica B 203 (1994) 116 128

spectrum of a powdered nickel sample. Following a standard method which uses the spectra of vanadium and plexiglass samples, a correction ar- ray was generated to normalize the different detector cells to the same detection efficiency.

2.2. Time-of-flight experiments

The time-of-flight measurements were carried out using the 25 MeV electron LINAC at CAB, in all cases operating at a frequency of 100Hz. A cooled lead target was employed as a fast neutron emitter, and several types of moderator configurations were used to produce different slow- neutron spectra. All moderators were placed at the same position beside the target, in such a way that fast neutrons reached the assemblies from the face opposite to the emitting surface. A series of conver- gent collimators define a circular beam (50 mm in diameter) at the sample position. The detector bank [11] consists of 32 3He detectors (10atm. filling pressure, 1" diameter, 6" active length) placed on a conical surface in backscattering position (156 ° mean scattering angle) designed to satisfy a time- focussing condition:

Qe t = constant (1)

where hQo is the momentum exchanged by elasti- cally scattered neutrons which are detected after a total time-of-flight t. The distance from source to sample is 4.94 m, and the mean distance from the sample to the detector bank is 0.45 m.

The calibration of the Qe scale was performed using the Bragg reflections of the observed scattering spectrum from a polycrystalline copper sample.

3. Numerical simulations

The simulations that we have performed consist in tracking each individual neutron history until the particle leaves the sample, as it was described in a recent paper [7]. This is not an optimized method as that proposed by Copley [12] to simulate double differential cross sections experiments. However, the method employed here is adequate to reveal and characterize the effects of multiple scattering processes under the different experimental situ-

ations analyzed in this work. The use or design of an optimized calculation code will be imperative when a customary correction tool is desired.

At the core of our Monte Carlo calculations we used the scattering function derived from the Syn- thetic Model. Its basic hypothesis, supporting argu- ments, and approximations involved have been previously discussed in a detailed manner [9,13], so that only a brief account of its main characteristics is given in the Appendix. However, we wish to emphasize here that this model gave most satisfac- tory results in neutron and reactor physics applica- tions [14], as well as in the evaluation of inelasticity effects in neutron diffraction work on molecular liquids [15]. The input parameters for the Synthetic Model corresponding to the molecular systems considered in this work, are given in Table 1.

The procedure adopted in our simulations can be summarized in the following steps:

(1) The incident neutron energy is determined from the incident spectrum, taken as a distribution function. In the case of a reactor experiment this energy is fixed.

(2) The ratio between absorption and total cross sections, at the energy established in step (1), is used to determine whether a neutron is absorbed or not in a given interaction. If the neutron is absorbed, the history finishes.

(3) The free path of the neutron is determined according to the characteristic exponential probability controlled by the macroscopic total cross section. If the path is such that the neutron goes out of

Table 1 Input values used by the synthetic model. The bound cross sections taken were ~r(H) = 81.66b, a(C) = 5.551b, and a(O) = 4.232 b, and the effective masses are given in neutron mass units. The mass of the 'molecular unit' is 18 for H20 and 42

for (CH2)n

Molecule Mode hoJ (eV) M [H] M [O,C]

H20

(CH2),

1 0.070 2.38 342.0 2 0.205 4.768 746.2 3 0.481 3.18 373.1

1 0.022 21.015 21.015 2 0.090 12.0 199.5 3 0.150 1.96 435.27 4 0.360 2.985 217.65

J. Dawidowski et al. / Physica B 203 (1994) 116-128 119

the sample, the history finishes and a new neutron is considered.

(4) Once the collision point is decided, for an incoming neutron with energy E, we obtain a new energy E' from the distribution given by the energy-transfer kernel ao(E ~ E'), according to the Synthetic Model (Eq. (A9)).

(5) The scattering angles 0 and q~ (taken in spherical coordinates) are then determined. Angle ~b is obtained from a uniform distribution, in correspondence with the azimuthal symmetry of the systems that we are treating here, whereas 0 is obtained from the distribution given by the double differential cross section ~ 2 6 / ~ 2 ~ E according to the Synthetic Model (Eq. (A8)).

(6) We return to step (2). Each neutron is tracked until it leaves the sample

or it is absorbed. The output of this simulation is a double-entry table which contains the number of recorded neutrons as a function of the elastic momentum transfer Qe, and the number of collisions experienced by each of them. Emerging neutrons are finally registered according to their detection probability (efficiency of the detection system).

4. Results

4.1. Reactor measurements

We will consider now the results of scattering experiments on water performed at the Orph6e reactor. Two cylindrical water samples placed with their axis perpendicular to the incident beam were employed. The first (big) one was contained in a vanadium tube, 6 mm inner diameter and 70 mm in height, while the second (small) sample was in a quartz tube, 1.1 mm inner diameter and 35 mm in height. The big sample was bathed by a 50mm height incident beam covering entirely its width, and the small one was completely bathed by the incident beam.

Figs. 1 and 2 show the results of those measurements as a function of the scattering angle (2o = 0.712A), with background and empty-can contributions subtracted, for the big and small samples, respectively. The subtraction process was performed along the traditional procedure due to

o H o o S • s o l t b

0.3 ***** Total IlOIIttt.

. . . . . . . . . . . . . . .

° ° * a ° o a ° o , o , o o o

° ' ° 0 . . . . . 310 . . . . . ~) . . . . . 9G . . . . . . . . . . . . . . . . . 120 160 180 Angle (degrees)

?

~ 0.2

e~

0.1

Fig. 1. Angular distribution observed in a reactor measurement (2 = 0.712 ,~) on a cylindrical (6 mm in diameter) water sample, The experimental data are compared with the single, multiple and total intensities from the numerical simulation.

f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

° . °81 . . . . I z ~ , ~ : ~ k ,

- - O : o . . -~ ° . .

080I ° ' ° " : : : : : : : : : : • 0 . . . . . 8'o . . . . . ~, . . . . . ~ . . . . . 1~* . . . . . , ~ . . . . ,a* Angle (degrees)

Fig. 2. Same as in Fig. 1 for the water sample of 1.1 mm in diameter.

Paalman and Pings [16], where the can contribution to the total spectrum is evaluated as

IE( O ) Ac, s c ( 0 )

Ac.c(0)

in which 1~(0) is the measured empty can spectrum, Ac. sc(0) is the cylindrical absorption factor for scattering in the can and absorption both in the sample and can, and Ac, c(0) is the cylindrical absorption factor for scattering and self-absorption in the can. Bearing in mind the relatively small effect of the containers on the observed intensities in these experiments, we felt it appropriate for the present purposes to employ this still widely used can

120 J. Dawidowski et al./ Physica B 203 (1994) 116-128

correction procedure, although in the spirit of a consistent approach as we are pursuing here all contributions to the measured spectrum should be simultaneously considered.

In the same Figs. 1,2 we present the results corresponding to our Monte Carlo simulations based on the Synthetic Model. They provide a satisfac- tory description of the observed angular distributions, even in the case of the big sample where the multiple scattering effects are severe. On the other hand, when the latter effects are diminished (small sample case, Fig. 2), the close agreement between measured and calculated curves emphasizes the ca- pacity of our scattering function to account for the dynamic effects which control the cross section behaviour. Finally, the departures observed at low scattering angles, are essentially due to the interference component superimposed on the smooth 'self' pedestal [17]. Those matters are further discussed over the next sections.

4.2. Time-of-flight measurements

TOF experiments were performed using three moderator configurations, which produced the spectra shown in Fig. 3. The epithermal spectrum was obtained with a moderator consistent of a 20mm thick polyethylene slab, covered in its emitting surface with a 1 mm thick cadmium plate; the under-thermalized spectrum was obtained from a sandwich type configuration [183, composed of

0,6

~ 0 , 4

~ 0 . 2

0.0

O I • * , , - The rms l • o * p ~ Under -Lhermsl~ed . . . . . Epithermol

•

" "--..-.F"* "*~ • ~ ,

°'0~ ° '~ .~r~ (~) '°

Fig. 3. Incident spectra employed in the time-of-flight experiments, plotted at constant lethargy intervals.

a 20 mm thick polyethylene slab as a pre-moderator, a l m m thick cadmium plate, and a 6mm thick polypropylene post-moderator; and finally, the thermal spectrum was produced by means of a 40 mm thick polyethylene slab.

The incident spectra were determined with a 3He detector directly exposed to the incoming neutron beam, after correcting the observed intensities for the detector efficiency [19], The energy scale for these spectra was calibrated with the known reson- ance energies of In, Cd and Mo filters.

In the following paragraphs we will present the results obtained under each moderator configuration.

4.2.1. Results with epithermal neutron spectrum We employed two disk-shaped polyethylene

samples, 50mm in diameter, with thicknesses of 1 mm and 5 mm, respectively. In Figs. 4 and 5 we present the experimental neutron spectra corresponding to these samples compared with the Monte Carlo results for single, multiple and total scattering. The ordinate scale represents the number of registered neutrons divided by the total number of incident neutrons considered in the simulation. The experimental data have been multiplied by a constant factor, to make them correspond to the scale unit thus defined. Agreement between calculation and experimental result is good in each

15

2

PI~ 1 tuna

2o°°°Oo::22 : It o

. . ~ . . . . i . . . . i . . . . 5o ! ' ' ' ' 60 20 30 40

Q. (A-')

Fig. 4. Scattering on polyethylene using the epithermal incident spectrum. The indicated intensity is the ratio between the number of recorded neutrons and the number of incident neutrons, corresponding to the numerical simulation.


1 4 0

120

T 1 o o

2

~ so

p l . ~ . 5 m a r e - - I l Z l ~ C a ~ m t L L l

~ X J , ~ L

w a t t . o o o o

, , , . , i i , * , , , i , , , , i , , • , i . . . .

- 2 O 30 40 5 0

Q, (A -1) 8 0

Fig. 5. Same as in Fig. 4 for the 5 mm thick polyethylene sample. Numerical results from a free-gas model are also indicated.

case, although small discrepancies at high Qe values can be observed. This region, corresponding to neutrons of energies higher than 1 eV, is especially critical with regard to the determination of the incident spectrum, bearing in mind that the efficiency of the detector employed for that purpose falls down rapidly over that region; as a consequence, an increasing degree of uncertainty may be expected to affect the input spectra used in these simulations at higher energies.

In those figures the large contribution of multiple scattering processes must be noticed, even for the 1 mm thick sample which is a typical one for this kind of measurements (83% transmission over the free-atom energy region). This is not an unexpected result, if one considers the large scattering angle at which our detector bank is placed, consistent with the relative enhancement of the multiple scattering contribution observed in the reactor case at back- ward angles (Figs. 1, 2). The different angular behaviour of single and multiple scattering intensities stresses the fact that the latter is essentially a sample volume effect [5], as opposed to the (equally normalized) single scattering probability which displays the anisotropy of the process as dictated by the scattering function. This matter is further discussed in a subsequent section.

In order to gauge the influence of the system's dynamics details, we have replaced in our calculations the energy-transfer kernel and the double differential cross section given by the Synthetic Model, by the corresponding expressions for

a free-gas model [20]. In doing so, we kept the actual total cross section of polyethylene [21], in order to maintain the relationship between the mean free-paths and the sample's dimensions. Moreover, to make a more fair comparison with the gas model prescriptions, we considered the ( C H 2 ) n polymer as a mixture of carbon and hydrogen atoms with effective temperatures given by the free-atom expression in the Synthetic Model [9] [Eq. (A2)] :

k B T e f f , = k a T + ~ [ ( n a + ½ ) h c o ~ - k a T ] , (2)

where ka is the Boltzmann constant, 2 runs over all oscillation modes used in the Synthetic Model, characterized by an energy h~ox, an occupation number na and a mass Ma. v indicates the atomic species, My is the mass of atom v, and T is the temperature of the sample. Using kaT = 0.0253 eV, we obtain for the hydrogen and carbon atoms,

kaTeffn = 0.105 eV,

kaT~ff c = 0.037 eV.

The results thus obtained are also shown in Fig. 5. Single scattering components as described by the synthetic and gas models coincide, because over the epithermal energy range the synthetic model tends to the free-gas description. However, the multiple scattering component is under- estimated by the gas model, because it does not properly account for the thermalization process. This fact is further emphasized in Fig. 6, where we show the energy distributions for neutrons which underwent 2, 4, 6 and 8 collisions, according to our simulation for the present case; it is evident that a more realistic scattering function which accounts, albeit in an approximated manner, for the molecular excitation modes, yields a more complex and intense slowing-down power than the simpler one (see the cut-off energy of the epithermal incident spectrum, Fig. 1).

4.2.2. Results with thermalized neutron spectrum Using this incident spectrum (Fig. 1), we carried

out scattering experiments on polyethylene (PLE) and light water samples.


The polyethylene sample was a disk 1.75mm thick and 80 mm in diameter. Fig. 8 illustrates the observed scattered spectra from the PLE sample, compared with our Monte Carlo simulations. The water sample was contained in a disk-shaped vanadium can, 2 mm thick and 60 mm in diameter. The empty-can contribution to the scattering spectrum was also measured and subtracted. In Fig. 7 we show the experimental spectrum of

40000

3 0 o o o

~ 20000

10000

0.01 0.1 V 1 ) 10 100 E n e r g y ( e

Fig. 6. Energy distributions of scattered neutrons after a different number of collisions, as described by the Synthetic Model and a free-gas model, in the 5 mm thick polyethylene sample. The symbols represent results obtained with the bound-atom model; squares: spectrum after 2 collisions; triangles: 4 collisions; circles: 6 collisions; stars: 8 collisions. The continuous curves indicate the distributions obtained with the gas model, which coincide with their corresponding bound-atom results at high energies.

0.0

.,~ 0.4

0.2

0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water

• ooooo start .

*.0** TotAl scatt.

o o o o

f o o .

0.0 0 ' 10 20 30 Q, ( A -1 )

Fig. 7. Scattering on water, using the thermalized incident spectrum.

scattering by water along with our Monte Carlo results.

Through those results, we observe again a good agreement between calculation and experimental data, even under conditions of large multiple scattering intensity. As in the case of polyethylene, we also tested a free-gas model for water. The effective temperatures for hydrogen and oxygen are, respectively:

k B TeffH = 0.115 eV,

kBTerfo = 0.037 eV.

1.0 . . . . . . . .

0.8

7 O 0,6

0.4

0.2

0.0 0

Polyethylene

A ~ . , M u l t i p l e s c a t L

10 20 SO

qe ( A - ' )

Fig. 8. Same as in Fig. 7 for the 1.75 mm thick polyethylene sample.

0.008

0.002

L 0.00!

0.000

Water . . . . . . gx-pertmental

,, Polyethylene

"k . J ° ° °

~ . - . ~ • - u - ° . ~

, L I , . . . . I . . . . . . , , , I . . . . . . . , I

t O 2 0 ' 3 0

q. (A -1)

Fig. 9. Experimental ratio between scattered and incident spectra, for the water and I mm thick polyethylene samples, using the thermalized incident spectrum. Numerical results based on the synthetic and free-gas models are also shown. The values corresponding to the water sample have been shifted to clarify the drawing.

J. Dawidowski et al./ Physica B 203 (1994) 116 128 123

In Fig. 9 we present the results of the free-gas model compared with the Synthetic Model and the experimental data, for both water and polyethylene, in the form of a ratio between the scattered and the incident spectra. We can see that the evaluation based on a free-gas scattering law is unsatis- factory, especially at low Qe values as a consequence of its inherent inability to describe the thermalization process, once we request a proper behaviour over the epithermal range, through the use of the effective temperatures defined above.

5. Analysis of results

5.1. Reactor data

The angular distributions obtained from the two water samples in the reactor measurements differ in a significant way, basically due to the large contribution of multiple scattering processes in the big sample. If we subtract from both distributions the corresponding multiple scattering components evaluated in our simulations, we obtain the results shown in Fig. 10. To reduce the effect of the limited statistics of numerical calculations, we used smoothed curves to represent the above-mentioned components. The single scattering components thus derived from both samples are essentially identical. This is not an unexpected outcome,

0 .08 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

m 0 . 0 6 " - - S y n t h e t / e M o d e l

~ 0 . 0 4

0 . 0 2

i i i , , i . . . . . . . . 0 . 0 0 0 . . . . . . . . . 3 0 . . . . ' ' ' ' ' 8 0 . . . . . . . . . 9 0 ' ' ' ' ' ' ' 1 2 0 1 5 0

Angle (degrees)

Fig. 10. Comparison of single scattering contributions, derived from the reactor measurements on water samples of different sizes, and that predicted by the Synthetic Model. See text for details.

because the primary attenuation factors in both situations are almost constant, as shown by Sears [5]. We also present our calculated single-scattering curve in Fig. 10, showing the excellent agreement with the experimental structure factors over the region dominated by inelasticity effects.

Bearing in mind that the Synthetic Model only describes incoherent scattering processes, the discrepancies observed between experimental and calculated results at low Qe are due to the coherence effects associated to intermolecular correlations in liquid H20.

Finally, the calculated single scattering components were subtracted to the 'experimental' ones, and the result is shown in Fig. 11. Ideally, this process should leave the interference component alone, after the removal of the inelastic pedestal. A good agreement between both sets of data is apparent over most of the Q range covered, except in the upper end where the 'big sample' data, most affected by multiple scattering effects, departs from the expected zero value. On the other hand, at lower Q values where the 'liquid' structure factor is revealed, the agreement between both experiments is noticeable, considering that the small sample's dimensions are optimized for this kind of measurements, while the big one would not have been normally used, on account of the severe corrections required.

A last comment concerning Fig. 11 is in order. The data shown there should not be considered as

0 . 0 0 6

0 . 0 0 4

¢3

0 . 0 0 0

- 0 . 0 0 4

I l a l o 1 . 1 z a z a o o o o o 6

o

-°'0°8o ........ ~ ........ '~ ... . . . . . . .~ ........ i~ ....... i6o Angle (degrees)

Fig. 11. Difference between the experimental and calculated intensities shown in Fig. 10. See text for details.

124 J. Dawidowski et al./ Physica B 203 (1994) 116-128

the final S(Q) pattern of light water derived from these experiments. Rather, we have used them as an exercise of full application of Monte Carlo techniques, to tackle in a simultaneous and consistent way inelasticity and multiple scattering effects. A comprehensive structural study based on the measurements reported here, must also take into account additional information about the scattering system (e. g. isothermal compressibility), a care- ful procedure for the subtraction of can contributions, and the use of adequate algorithms to test the behaviour of the resultant distributions in the Fourier transformed (real) space.

5.2. Time-of-flight data

The TOF scattering spectra of water, corresponding to the thermalized and under-thermalized cases and after normalization for those incident spectra, are shown in Fig. 12. In the same way as in the reactor case, we subtracted from them the multiple scattering components from our numerical simulations, and obtained the distributions shown in Fig. 12. Both data sets are in fairly good agreement, although some residual discrepancies are apparent. Of course, in this kind of measurements, the single scattering spectrum is calculated as an inte- .qral which contains, amongst other things, the

0 . I '

2 2 2

2

2 2 0 - o

"t, "::Oo • . o o o • o T o t J t l l O l t t e l ' ~ [

• • • • o n • o • °

. . . . . . . 'z: . ~'::::

• • , . . , S i n f l e s c a t t e r i n g

• " : : ' ' . , . ,

!

5 ~o 16 ~o ~5 30 .e

q. (A-')

Fig. 12. Comparison between experimental results on water samples, using the time-of-flight technique for under-thermalized (circles) and thermal (triangles) spectra. Empty symbols indicate total scattering intensities, whereas full symbols denote the single scattering components obtained after subtraction of the calculated multiple scattering contribution.

shape of the incident neutron spectrum [22]. The discrepancies we are referring to simply make that fact visible, even enhanced by the large effective scattering angle of our detection bank which implies a wide energy window for each TO F channel.

5.3. On the detection efficiency

Based on other pieces of information obtained in our Monte Carlo simulations, we deemed it appropriate to add here a few comments about the effects introduced by the detection system on the observed spectra.

For the sake of illustration, let us consider the detector bank used in our reactor measurements. Its efficiency is given by the formula

e(2) = 1 - e x p l - 3~56] (3)

where the neutron wavelength 2 is expressed in A. It is quite obvious that, as a consequence of inelasticity, the efficiency will be also a function of the scattering angle and the sample's characteristics. This effect can be quantitatively assessed by means of a numerical simulation, whence 'observed' spectra are recorded assuming either the actual detector, or a black one. Their ratio gives us an effective efficiency, as shown in Fig. 13 for our big sample case. There we have displayed the efficiencies for detection of single scattered and multiple

0 . 3 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ 0 . 3 0

qP

0

~ 0 . 2 5

~ I D . 2 0

0.15

o o o o o SlaSle soatt . . a= .m MURtl)] e Imatt. w w • • l T o t a l s t a r t .

i

o Q

a

o . • w a e

o o

o o

o

o

• • o o o

• • o

0 l ~ u ~ e s c a r f , n

Angle (degrees)

Fig. 13. Effective efficiency as a function of the scattering angle, and for different scattering orders, as calculated for the reactor experiment on the water sample of 6 mm in diameter. See text for details.

J. Dawidowski et al. / Physica B 203 (1994) 116-128 125

scattered neutrons, as well as the corresponding total effective efficiency. In the same figure, the system's efficiency for a wavelength equal to that of incident neutrons (2 = 0.712 ~,) is indicated, which is of course the same for scattered neutrons which underwent elastic interactions. We can see that for single scattering processes, the effective efficiency changes about 33% over the angular range from 0 ° to 128°, for measurements on a light water sample.

The results shown in Fig. 13 clearly demonstrate that the effective detector efficiency is closely related to the sample's nature through its dynamical properties, and also to its geometry through the multiple scattering contribution. These effects lead to an even more complex situation in the T O F case, where there is also a window of incident energies contributing to each time channel. Although an important fraction of those undesirable effects can be minimized by a wise choice of sample dimensions and experimental conditions (shorter neutron wavelengths in reactor measurements, small scattering angle in T O F instruments), we wish to recall again that elastic and inelastic components of an observed spectrum do not scale in the same manner after normalization by the incident spectrum, because the 'ideal scatterer' employed to determine the latter is usually different in size and nature from the actual sample under study.

6. Conclusions

The work reported here was aimed at a quantitative study of the inelasticity and multiple scattering effects in neutron scattering experiments on molecular systems. Although we concentrated on the magnitude and behaviour of those effects in experiments designed for structure factor deter- minations, the adopted approach can be extended to correct the observed spectra in double-differential cross section measurements on those systems.

On the experimental side, a wide range of measurements were performed, using pulsed and stationary neutron sources, on hydrogenous samples of different sizes to enhance the contributions due to the effects under study. In correspondence with each data set, Monte Carlo simulations were

performed on the basis of a Synthetic Model to describe the interaction of slow neutrons with mol- ecules. A good agreement between experimental and calculated results was obtained in all cases, which renders confidence on the model's ability to produce kernels which are representative of the actual ones. Under those conditions, it is possible to simulate neutron histories from source to detector to produce a unified, consistent, and accurate evaluation of these corrections, rather than treating them in a separated, and generally inconsistent way. We may therefore expect that, under the normal conditions of a well-designed experiment, the data correction procedure proposed and tested here against very demanding cases, should yield highly accurate results.

A few specific points are worth to be emphasized. Besides the always present inelasticity effects when scattering by hydrogen is involved, our results show that a nonnegligible amount of multiple scattering may exist even in cases where a (usually considered as) small sample is being used. We have also performed TO F experiments specifically designed to verify that effect [23], using PLE samples of the same thickness and different diameters (20, 50, and 80 mm) placed behind a thick Cd mask with a hole of the same diameter (20 mm) as the incident neutron beam; those measurements clearly showed the increase of multiple scattering intensity with the sample's lateral dimensions, even though the volume of the sample bathed by the beam was the same. Furthermore, by using PLE samples 5mm thick and diameters as indicated above, we found from the difference between the 'zero extra-volume' sample (20 mm diameter) and the 50 mm diameter one, that a considerable amount of multiple scattering at least of order 3 - still exists for that experimental configuration.

Concerning the efficiency of the detection systems, and beyond the usual knowledge of its behaviour as a function of neutron energy, we have discussed on quantitative grounds the effective efficiency that a detector will have depending on the nature and geometry of the sample under study, and the implications of this fact on the absolute normalization of the measured data.

Besides the simulations discussed here involving the simultaneous evaluation of inelasticity, multiple


scattering, and beam attenuation effects, work is in progress to complete a package of optimized rou- tines including the can contributions in the Monte Carlo calculations.

Finally, we wish to emphasize that the results presented here can be extended to cases where the coherent contribution to the total spectrum plays an important role, under the reasonable approximation of treating the interference component as due to elastic processes only. The incorporation of both contributions in a complete numerical simulation, should provide an efficient way to describe the corrections, and it may contribute to the refinement of structural parameters in liquid and amorphous systems, to a degree more appropriate to the high quality of experimental data achievable with present day instruments.

Acknowledgements

Special thanks are due to Mb.ximo Schneebeli and Pablo D'Avanzo for their technical support during LINAC operation, and to Luis Capararo for his assistance in the experimental setup assem- bly. One of us (J.D.) wishes to acknowledge finan- cial support of CONICET in the form of a post- graduate fellowship, while another (V.H.G.) wishes to acknowledge a partial support from Fundaci6n Antorchas (Argentina) and C.E.A. (France).

Appendix: Main features of the Synthetic Model

The scattering system is considered to be an ideal molecular gas at temperature T. Although the motion of the molecular unit may be severely hin- dered in a real system due to the presence of its neighbors, it is only for very slow neutrons that this collisional regime will become dominant. Clearly, the onset of such a situation will depend on the particular system under study, but for most real cases it implies neutron energies (a few meV) which are outside the region of main interest in neutron diffraction work or reactor-physics studies.

The internal degrees of freedom of the molecule are assumed to be not coupled. This is a first approximation to the real situation which is valid as

long as the amplitudes of the atomic oscillations around their equilibrium positions are small compared to the interatomic distances in the molecule. Each of the 2 internal modes is represented by an Einstein oscillator with angular frequency to~ and effective mass Ma. Those effective masses must satisfy normalization conditions imposed by the value of the (spherically averaged) Sachs-Teller tensorial mass, and the requirement that the free-atom cross section - determined by the bound-atom cross section and the atomic mass M - be approached at high neutron energies (large compared with kBT and the largest hto~). Also, neutrons with these energies 'see' the scattering nuclei as possessing a kinetic energy associated with a temperature ~r, determined by the system's temperature T and the mean energies E~ of the oscillators representing the normal modes of the molecule.

The different forms that the actual scattering function S(Q, to) takes according to the magnitude of the energy transferred in the process, motivate the main features which characterize the model, that is, the appearance of different effective transla- tional masses, temperatures, and vibrational factors. The synthetic scattering function T(Q, to; Eo) therefore uses the incident neutron energy Eo as the variation parameter, to determine the values of those effective quantities across the energy range. In this manner, a kind of envelope represents the combined effect of the quantum excitations of the internal modes, thus avoiding the complexities involved in a detailed description of them.

The above considerations are, in fact, equivalent to the argument that an incident neutron is able to transfer at most an energy Eo, so that the incom- plete first moments of the actual scattering function determine an effective mass

1 _ 1 [1 - P~.(Eo)]

// Mmo~ l + ~ M~,

_ 1 S~ P ~ ( E o ) (A1) M z 7 M; '

the consistency between the first and second moments determine an effective temperature

kar kBT [1 -- p - Mmol + ~z 2MaPZ(E°)] (2nz + l)htoz (A2)


and an effective vibrational factor

P;.(Eo) (2n;~ + 1) (A3) F = ~ M~hco~ " 2

The switching functions Px(Eo) vary from one to zero as the value of Eo goes over each hco~, corresponding to the situations when the neutron cannot excite the 2-oscillator energy levels or when that mode becomes fully excited by the collision process, respectively. At intermediate neutron energies, the variation of Pa depends on the shape of that part of the frequency spectrum associated to the 2-mode, characterized by the value hcoa and a quantity aa, representative of the width of that spectrum portion.

From the above expressions, we see that the effective mass /2 varies from the molecular mass Mmo~ to the atomic mass M, and the effective temperature r tends to T, the system temperature, or T, the free-atom value, according to whether all Pz are one or zero, respectively. Also, the vibrational factor F changes from its maximum value when the neutron is slow enough to see the atoms bound to a quasi-rigid molecule, to zero for epithermal neutrons when the effect of molecular excitations is already contained in the temperature iP of the apparent atomic gas.

With those definitions, the synthetic scattering function which describes a neutron scattering process involving an energy and momentum exchange hco and hQ, respectively, is written as

T"(Q, co, Eo) = S~,,~(Q, co)e -r~z02/2 + C.,~(Q, co).

(A4)

Here, the principal term, S~,dQ, co), denotes the scattering function of an ideal gas of particles of mass /2 at temperature ~, whereas C.,dQ, co) is a correction term given by

c.,dQ, co)

) n : . ~ Su, dQ +, to +) e- r~2el /2

h2Q2 } + (1 + n~) 2 ~ _ Su,,(Q_, ~o_) e -rh2o~-/2 (A5)

where Q± is the modulus of the scattering vector corresponding to an energy exchange hco± = h(co + co~).

The neutron-energy dependent quantities /2, and F were introduced as a simple way to describe the variation of the scatterer's properties, as the different internal degrees of freedom of the molecule become fully excited in the collision process. Admittedly, in our argument we have associated large energy transfers with high neutron energies, and although this is a necessary condition, the contribution to the cross section due to processes involving small energy transfers must also be accounted for. Those are considered - at least par- tially - by the correction term representing one- phonon contributions which may be important in those cases of thermal- or collision-induced excitations, but with a neutron energy not high enough to allow a quasiclassical treatment of the corresponding mode.

Introducing the notation

h2Q , KN (A6) S.,dQ, co)=Su, dQ, co) exp - F 2 J '

Eq. (A4) can be written as

T(Q, co; Eo)

KN )~ ( ~ K N = Su,~ (O, co) - P;~ n:. S.,¢ (O +, co +)

Su, dQ_, co_) (A7)

where pa = P~(Eo)/(Mahcox). Eqs. (A1) through (A7) represent the mathemat-

ical expression of the Synthetic Model. As it was emphasized above, this model was devised to describe a real scattering law in an approximated way, bearing in mind that the full dynamics of the atomic motion is not accounted for in a detailed manner, nor the interference contributions to the scattering process are considered at all. However, the formal simplicity borne into its formulation has been exploited to produce analytical expressions for several magnitudes of interest in neutron thermalization problems, especially for total cross sections and energy-transfer kernels of any order in the Legendre expansion of the double-differential

128 J. Dawidowski et al. / Physica B 203 (1994) 116-128

cross section. In particular, the latter is written in terms of our scattering function T(Q, o~; E0), and for an unpolarized beam of neutrons, as

~20" k 1 - ko ~ av N~ Tv(Q, co; Eo) (AS) ~f2 ~E

where ko and k denote the (modulus of) incident and scattered neutron wave vectors, respectively, v runs over the species of nuclides at equivalent molecular positions, each with a number N~ of them and with a bound scattering cross section a~. Furthermore, the isotropic scattering kernel defined by

f~ ~2° ao(Eo -~ E) = 2n cos(0) ~ sin(0) dO (A9)

is the magnitude which controls the energy distribution of scattered neutrons, and it is given by a rather compact, analytical expression when the Synthetic Model is used [12].

References

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Documents

Multiple scattering and inelasticity corrections in thermal neutron scattering experiments on molecular systems