Remarks on the slow neutron scattering by organic molecules

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Remarks on the slow neutron scattering by organic molecules

V. Ardente

To cite this version:

V. Ardente. Remarks on the slow neutron scattering by organic molecules. Journal de Physique, 1964,

25 (5), pp.641-647. �10.1051/jphys:01964002505064101�. �jpa-00205844�

distribution ^{is at first} sight ^{sirnilar to} (A2), ^but

with Ar given by ^{(A8) :}

where

It can be shown that ro j2 ^{for X -~ 0 and}

So the dispersion ^{law can} be measured directly

from (A8) and the magnon life time from (Al0) :

REFERENCES [1] BROCKHOUSE (B. N.) and WATANABE (H.)

^Inelastic

Scattering of Neutron in Solids and Liquids ", II, p. 297, IAEA, Vienna, ^1963.

[2]

^GENNES (P. G.), ^J. Phys. ^Chem. Solids, 1958, 6, ^43.

[3]

^GENNES (P. G.) and VILLAIN (J.), ^J. Phys. ^Chem.

Solids, 1960, 13, ^10.

[4] ^GINZBURG (V. L.) ^{and FAIN} (V. M.), ^Zh. Ekspe

Teor. Fiz., 1960, 39, 1323.

[5] ^IZYUMOV (J. A.), Usp. ^Fiz. Nauk, 1963, 80, ^41.

[6] ^JACROT (B.), KONSTANTINOVIC (J.), ^PARETTE (G.) ^and

CRIBIER (D.),

^Inelastic Scattering of Neutrons in Solids and Liquids ", II, p. 317, IAEA, Vienna,

1963.

[7] ^KEFFER (F.) ^{and LONDON} (R.), ^J. Appl. Physics, 1961, 32, 2S.

[8] ^MORI (H.), Prog. ^Theor. Physics, 1963, 29, 156.

[9] ^MORI (H.) and KAWASAKI (K.), Prog. ^Theor. Physics, 1962, 27, 529.

[10] ^RISTE (T.), ^J. Phys. ^Chem. Solids, 1961, 17, ^308.

[11] ^RISTE (T.), ^J. Phys. Soc., Japan, 1962, 17, SBIII, ^60.

[12] ^RISTE (T.), ^BLINOWSKI (K.) ^{and JANIK} (J.), ^J. Phys.

Chem. Solids, 1959, 9, ^153.

[13] ^RISTE (T.) and WANIC (A.), ^J. Phys. ^Chem. Solids, 1961, 17, 318.

[14] ^{VAN HovE} (L.), Phys. Rev., 1954, 95, 1374.

[15] ^WANIC (A.), ^J. Physique (mémoire précédent, p. 627).

REMARKS ON THE SLOW NEUTRON SCATTERING

BY ORGANIC MOLECULES

By V. ARDENTE,

C. C. R. Euratom, Ispra, Italy.

Résumé. 2014 Dans le cadre des interactions d’un neutron avec un noyau moléculaire et en s’atta- chant plus particulièrement ^{aux molécules} organiques (polyphényls), ^l’auteur fait dans cette note

une analyse préliminaire ^{de la diffusion} par ^des protons ^{liés dans} la molécule de benzène (on _suppose que cette molécule est une unité dynamique fondamentale).

Pour évaluer l’influence des dynamiques moléculaires sur la diffusion des neutrons thermiques, plusieurs hypothèses physiques ^{ont été} essayées.

L’objet principal ^{de ces} suppositions ^{est de nous} permettre ^{en se} donnant de manière explicite

la dynamique ^du proton, d’étudier à la fois l’influence des différents aspects individuels de la

dynamique moléculaire et l’effet des diverses approximations mathématiques ^sur la considération

explicite ^{de tels} aspects dynamiques.

Ceci

été réalisé en se référant à des résultats expérimentaux portant à ^{la fois sur les} quantités intégrales (par exemple ^la section efficace totale de diffusion) ^et différentielles.

Dans le cadre du modèle gazeux, nous avons d’abord étudié l’effet d’un seul paramètre ^de structure, ^{à savoir la masse} effective Mo. Nous

considéré alors un

gaz moléculaire

introduisant deux paramètres (Krieger-Nelkin) ^{l’un du} type

^effective Mo, ^tenant compte

des rotations et translations et l’autre tenant compte des transitions vibrationnelles élastiques ^à partir du niveau fondamental. Nous discutons la possibilité de calculer la meilleure valeur appro- chée du paramètre ^{Mo à} partir ^{des valeurs} expérimentales, la valeur du paramètre vibrationnel n’étant déterminée qu’une ^{fois connue} la distribution des fréquences ^du proton ^{dans la molécule.}

L’aspect particulier ^{de ce} spectre vibrationnel nous amène à discuter un « modèle bi-vibrationnel » analogue ^au modèle de Nelkin pour les molécules de H2O ^dans lequel, pour tenir compte des différents

degrés ^{de liberté,}

remplace ^le spectre vibrationnel complexe par ^un autre isotrope, ^très simplifié.

Pour finir, nous avons traité comme prédominant l’aspect purement vibrationnel (en ^restant

dans le cadre d’une approximation harmonique) ^en considérant explicitement l’anisotropie ^du spectre vibrationnel et

évaluant d’une façon plus correcte le rapport des orientations molé- culaires.

Dans cet esprit, ^nous

^{tout d’abord} négligé (en première approximation), ^les degrés ^de liberté de rotation et de translation ; ceux-ci seront introduits plus ^{loin de} façon approchée ^au

moyen d’une correction du spectre vibrationnel dans la région des basses énergies.

Nous avons fait enfin une comparaison entre cette analyse qui, dans le cadre d’une analogie

avec un ensemble polycristallin ^de microcristaux de graphite, adopte ^le formalisme de Schofield et Hassit et l’ensemble de suppositions plus précises ^et mathématiquement plus ^{correctes contenues}

dans le code

Summit » qui traduit le point ^{de vue de Parks.}

LE JOURNAL DE PHYSIQUE ^TOME 25, ^MAI 1964,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002505064101

Abstract.

In the framework of the interaction of a neutron with molecular nuclei and with

a

special ^{view to} organic ^molecules (polyphenyls),

preliminary analysis ^is made in this paper

the scattering by ^bound protons in the benzene molecule (this ^{molecule is} assumed ^{to be}

fundamental dynamic unit).

In evaluating the effect of the molecular dynamics ^{on the} scattering ^{of thermal} neutrons, ^several physical hypotheses have been tried.

The main purpose ^{of these} assumptions is to enable us, when considering explicitly ^the dynamics

of the proton, ^to investigate ^{both the influence of the} different individual features of the molecular

dynamics and the effect of different mathematical approximations ^{on the} explicit consideration of such dynamical ^features.

This has been performed with reference to experimental results both ^on integral quantities (e. g. the total scattering cross-section) ^{and on} differential ones.

In the framework of the gas model ^we have first investigated the effect of a single " structure "

parameter, namely ^{the effective mass Mo.}

We have then considered a " molecular gas " by introducing ^two parameters (Krieger-Nelkin) :

one, of the effective mass Mo type, accounting ^for rotations and translations and a second one for-

mally taking ^into account the elastic vibrational transitions from the ground ^{state. The} possi- bility ^{is here discussed of} calculating ^{the best fitted value of the} parameter ^{Mo from} experimental data, ^{the value of the} vibrational parameter being determined once the frequency distribution of the proton ^{in the} molecule is known.

The particular ^{structure of this} vibrational spectrum ^{induced us to discuss a}

two-vibrational model " in analogy ^{to the Nelkin} model for the H2O molecule, where, ^{in the} explicit consideration of the various degrees ^of freedom, ^the complicated vibrational spectrum ^{has been} replaced by ^an isotropic, ^very simplified ^one.

Finally ^we have treated as predominant ^the purely vibrational features (still ^{in the} frame of the harmonic approximation), considering explicitly ^the anisotropy ^{of the} vibrational spectrum ^and evaluating ^{in a more correct} way the relation of the molecular orientations. In view of this, ^we started by neglecting (as ^{a first} approximation) ^the rotational and translational degrees ^{of freedom;}

these will be appropriately introduced later by ^{means of a} correction in the vibrational spectrum

in the low energy region.

We also made ^a comparison between this analysis, which2014in the frame work of an analogy ^{with a} polycrystalline assembly ^of graphite microcrystals

adopts the formalism of Schofield and Hassitt and the set of more refined and mathematically ^{more correct} assumptions ^contained in the Summit code which translates the Parks’ viewpoint.

1. Introduction.

In the framework of the interaction of a neutron with molecular nuclei and with a special ^{view to} organic ^molecules (poly- phenyls), ^a preliminary analysis ^{is made on the} scattering by ^the proton bounded in the benzene molecule.

We confine here our attention to the total scat-

tering ^{cross section so as to arrive at some basis}

for determining ^the validity of different models.

In evaluating the effect of the molecular dyna-

mics on the scattering ^{of slow} neutrons, ^{a series of} fundamental assumptions ^and physical hypothesis

have been tried. In this way informations ^can be collected about the limits of sensitivity ^{of the slow}

neutron scattering ^{to the details of the} dynamics

and to the structural features of the benzene molecule. A model which would be able to predict successfully ^the experimental ^{data without the}

need of a very detailed ^account of molecular dyna- mics, ^{should show} really ^{a weak} sensitivity ^{to some}

characteristic benzene features, and could be extended to other polyphenyls ^{with reasonable}

confidence.

The fundamental assumptions ^{in the} present description ^{are that the} single C 6H 6 molecule is the basic dynamical ^unit (without any explicit ^inclu-

sion of intermolecular eff ects and those specific ^to

the liquid ^{state) ; and} the fact that we neglect ^the

interference scattering ^{and the nuclear} spin ^corre-

lations.

The analysis ^of simplified molecular models

(the Krieger-Nelkin ^{one and a here} proposed " ^two

vibrational oscillators " model) ^{and of a "} poly-

cristalline " one should be useful for two reasons :

to show the real possibility ^of using ^{some rather} simple ^{model to} calculate the scattering ^behavior

and to define a starting point ^{for a more detailed} study ^{of the} proton ^motions.

2. Scattering ^models.

^We begin ^{with an} expression for the differential cross section,

6 (Eo 2013~.E, 6), ^for scattering ^{of neutrons of inci-}

dent _energy Eo ^{in the} energy interval dE at E and through ^an angle ^{0 into the solid} angle ^dQ.

The operator formalism introduced by ^Wick [1]

and Zemach and Glauber [2] gives for the direct

energy-transfert differential cross section the follo-

wing general analytic expression (1) :

where sb is the bound atom n - p ^cross section

(6b

^{/M+1B 2 af M bein} g the roton mass

and M at the free asymptotic ^{value of the total} scattering ^cross section), ^{is the}

(1) ^{We use a} system ^{of units in which A}

1, ^{and the} neutron mass is 1. Temperatures ^{are measured in units}

of energy.

energy transfer, ^x

^{K -} Ko ^is the momentum

transfer, ^{T is the} absolute temperature ^{and the} function x, which contains all the dependence ^of

the scattering ^{on the} dynamics ^{of the nuclei in the} molecule, ^is given by

i T) = eiHt eix,rv e-iHt e-ix.

> _T,n

The symbol > _TO means that an average is to be taken ^over the equilibrium distribution ^of the initial states of the molecule at the tempe-

rature ~’ and over the molecular orientations Q.

H is the molecular Hamiltonian, ^{rv is the} position-

vector operator ^{of the vth} proton.

Diff erent expressions ^and approximations ^to

evaluate the function (concerning ^{both the H}

function and the T, Q averages) ^will correspond ^to

the diff erent models examined. We shall assume

,

that a classification of the atomic motions into

vibrations, rotations and translations of the mole- cule gives ^an adequate description ^{of the proton}

motions in the scattering system. We shall take

as our starting point ^the approximation ^{that the}

various degrees of freedom carry ^out simple ^har-

monic oscillations.

" MOLECULAR" MODELS.

We begin ^with

a simple translational model : the " free gas model "

for free protons ^{at the scatterer} physical tempe-

rature. Figure ^{2 shows the} strong discrepancy

between the experimental ^{data and the} theoretical

curve calculated on this extreme basis.

The same figure ^{2 allows to evaluate} the effect of

introducing ^a single " structure " parameter, ^a

kind of " effective mass " introduced in the

mass tensor approximation ^of Sachs and Teller [3]

or in the quantum Krieger and Nelkin reformu- lation [4, 5]. ^{The basis of} these methods is the

concept ^{of a nuclear mass tensor whose} properties

contain the effects of chemical binding : namely

the effects on slow-neutron scattering ^{of the} dyna-

mics of the rotational and translational motions.

Without a real reference to this translational and rotational concept (2) ^{we would} just point ^out

the possibility ^of fitting ^the experimental ^values

of the total cross section with a fictitious mass

adjusted ^{in some limited} energy interval, ^disre- garding ^the discrepancies outside this interval.

The 5.5 effective mass

for instance

could represent ^{a kind} of upper limit likely ^{to describe} fairly ^well the low thermal region, though ^the agreement ^is bad in the higher ^energy region. ^In principle, by using ^the gas model ith the bound

(2) ^{See the note in the} following page.

FIG. 1.

Shows the fundamental vibration frequencies ^for benzene,. ^The frequencies associated with the planar

and perpendicular ^{modes are} separately indicated. The height ^{of a line indicates} the average of the squared

amplitude ^{of the} protons ^{in each} normal mode.

cross section ab = ~ + 1 ^{2 cf taken as constant}

instead of _6free, an energy-dependent para- meter should be used ( ).

The failure of the simple freely rotating rigid-

molecule point of view is caused by neglecting ^the

internal vibrational features, ^{which are} very effec- tive on thermal neutron scattering by ^molecules

as complex ^{as the} polyphenyls. Figure ^{1 shows}

the normal vibrations of benzene, ^the simplest ^of

the polyphenyls, ^{based on} the extensive studies available [6, 8] ^{on this} planar ^{molecule. There are}

3N-6

30 normal modes of vibration. Of these 2N-3

21 are of the parallel type (vibrations ^for

which the amplitude ^{vectors are} parallel ^{to the} plane ^{of the} molecule). ^{In the} remaining ⁹ modes,

the amplitude ^{vectors are} perpendicular ^{to the} plane ^{of the molecule.} Because of the high degree

of symmetry of the benzene molecule (point group

D6h) ^{7 of the} parallel ^{and 3 of the} perpendicular

vibration frequencies ^are doubly degenerate.

To see how strongly ^{the total} scattering ^cross

section depends ^{on the} vibrations, ^{we now turn}

our attention to a more detailed theory : ^{for the}

benzene molecule we consider the " f ree ^{molec ular}

gas

"

point ^{of view of} Krieger and Nelkin [4].

This " K. N. " model, ^{which allows} taking ^into

account the zero-point vibrational motion, ^reduces

the description ^{of the direct} scattering ^{to the con-}

sideration of two molecular parameters : Mo, ^an

effective proton ^mass for translation and rotation,

and a vibrational constant equal ^{to the mean}

square zero-point vibrational displacement ^{of the} proton. Using the Zemach and Glauber formalism,

the following approximations ^are made, ^{at the}

expense ^{of a} restriction in the energy range ^of

validity :

i) ^molecular rotational and vibrational eff ects

are explicitly separated ;

ii) ^the vibrational factor XIrH > ~ ^{is the one} evaluated by ^Zemach and Glauber. If attention is restricted to neutron energies ^{below the vibra-}

tional threshold, and all molecules are assumed to be initially ^{in their} ground vibrational states (i. e.,

T

0 for vibrations), XHH > ~ ^{reduces to} the time

independent ^form

where is the amplitude ^vector corresponding

to the average proton ^{and to the} vibrational

mode, ^{and c~~ is the} angular frequency ^{of the Xth}

mode. For each value of x, ^the amplitude ^vectors

are normalized by ^{the condition}

(3) In the benzene molecule the effective

Mo ^intro-

duced by Krieger and Nelkin turns out to be 21.3 neutron

masses.

Nlv being ^{the mass of the vth nucleus} of the mole- cule. The dependence ^{of on the mole-}

cular orientation is contained in the quanti-

ties

iii) the averages ^over molecular orientations are

approximated by inserting average values of func- tions of the Eulerian angles ^wherever they appear.

In this way ^an expression ^{in closed form for the}

total cross section for direct scattering ^{is obtained :}

where

m _{VR 0} ,- o .z ^{e-t’ dt : p and C are functions}

of MO (the " rotational " mass which contains the combined effects of rotation and translation), ^of

the temperature ^{of the} sample ^{T and of} Eo, ^the

neutron initial energy. Figure ^{3 shows the com-} parison ^with experimental data : the general ^out-

line is notably ^better than in the gas model. The K. N. extension of the mass-tensor approximation,

to include to a certain extent vibrational effects,

appears ^to allow

in organic ^molecules

^{a not}

too bad description ^for energies lower than 0 .1 eV.

On the other hand the fact that the agreement

with experiment ^is not yet ^so good ^{means that the} Krieger-Nelkin theory ^{is not realistic} enough : ^the discrepancies ^are reasonably attributed

beside the mathematical simplifications

^{to the use of}

the mass-tensor concept ^{and to the} neglect ^of

inelastic transitions. The effect of putting Me equal ^{1, as shown in} figure 3, gives ^{an idea of the}

" weight " ^{of this} rotational parameter. ^A possi- bility ^exists in the frame of this model of fitting

the rotational mass Mo on experimental data, independently of his tensorial origin. ^{This should}

be a quite realistic and useful simple two-parameter

model (~).

Proceeding ^to analyse ^{more and more elaborate}

models we now discu,ss ^{a two} regions, ^two-vibra-

tional model, suggested by ^the particular ^structure

of the vibrational spectrum ^of the average proton

in the benzene molecule. The idea is to replace

this very complicated ^and anisotropic vibrational spectrum by ^a very simplified isotropic ^one.

As it appears in figure 1, ^a general feature of the benzene vibration spectrum ^{is the} separation ^{of the}

fundamental frequencies ^{in two} parts : ^a relatively

small number of modes, all very closely spaced

(4) ^As

matter of fact there exists also a possibility ^of having ^{some direct} experimental information about the _YH$

parameter.

645 around 0.38 eV and other more numerous and

more spread ^out low energy modes, about from 0 . 05 to 0 .2 eV. We simply ^{assume two ;16; a-}

tional oscillators at 0.12 eV and 0 . 38 eV.

With this very simplified spectrum

^{which we} expect ^{to cover the} general ^{features of all the} polyphenyls

^{we consider} the benzene molecule from the ^same point ^{of view as the} light ^water

Nelkin one [9], for which numerical codes are

available (5). ^{That means that the} following assumptions ^{are made :}

i) ^The differential energy

transfer cross sec-

tion is calculated with the intermediate Z ^function approximated by

X ~ Xt Xr Xv

with _xt for free molecular translations (the ^hin-

drance of the translations is neglected), xr for hindered rotations of the molecule in the presence

of its neighbours and ^for the interactions with the two harmonic vibrational oscillators inside the molecule. _{xt, Z,} and Z, ^are separately averaged

over the initial states of the system ; ^{the hindered} rotations are formaly replaced by ^{a torsional iso-} tropic oscillation with a single frequency ^cor put tentatively equal ^{to 0.02 eV.}

ii) To evaluate a(Eo -~ ~’, 0) the further appro- ximation is made of averaging ^{over molecular}

orientations in the same manner already ^used by Krieger and Nelkin to calculate the scattering by

free polyatomic ^molecules.

iii) For convenience in numerical calculations the excitation of only ^{one mode} oscillation is treated rigorously ^{for a} given incident energy range.

In our case there ^are two regions ^defined by ^a boundary ^{at 0.32 eV.}

iv) For the rotational mass Mar of the isotropic

harmonic oscillator, ^which approximately replaces

FIG. 2, 3, 4, ^5.

^{Show the} average total scattering ^cross section for

proton ^of CsHs

function of incident neutron energy, ^as calculated by different theoretical models, ^{at room} temperature (T = 0.0258eV).

6free

20.36 barns is the free-proton ^{cross sections.}

FIG. 2.

Refers to the gas model. The full line to

real free gas model (nuclear ^mass equal ^to 1), ^{the dashed}

one to

effective mass " Mo equal ^to 5.5 neutron masses.

FIG. 3.

Refers to the Krieger elkin ^{model. The}

dashed line represents the effect of putting the effective

mass equal ^{to the} physical

(5) ^{In our} calculations ^we have used the Honeck’ Gaker Fortran code (see ^THERNIOS report, BNL-5826).

646

FIG. 4.

Refers to a

two vibrational oscillators "

model, proposed ^to include, ⁱⁿ analogy ^{with the Nelkin} point ^{of view for} water, vibrations, hindered rotations and free translations of the molecule.

the hindered rotations, ^the Krieger-Nelkin ^value

for a freely rotating ^{molecule is taken.}

Therefore the physical ^{model is that of a trans-}

lator of mass Mt

78, ^a hindered rotational oscil- lator of mass Mr == 21.3 and energy

o . 02 eV and two vibrational oscillators with energies

=

0 .12 eV,

0 . 38 eV and ^an eff ective

mass determined from the condition that the

scattering approaches the free-atom scattering ^at high energies :

In the region of incident energies Eo ^{less than}

0.32 eV, ^the 0.38 eV oscillator is treated in the elastic limit, ^a phonon expansion ^{is used for the}

first 0.12 eV oscillator and all transitions are

allowed for the rotational and translational modes.

For incident energies ^{above 0.32} eV, ^{the rotations}

are treated as free (high-energy limit), ^a phonon expansion is used for the 0.38 eV oscillator and all transitions are allowed for the 0.12 eV oscil- lator. Taking ^the time transform of the inter-

mediate scattering function, ^a single expression ^can

be written for all incident energies :

where is the usual Bessel function of ima-

ginary argument ^{and the} occuring parameters ^are given by ^the following ^{two sets :}

Region ^I

Region ^II

The total cross section as a function of the inci- dent energy is computed by determining ^nume- rically ^the differential a(Eo ^- .~, 0). ^{This cross}

section is then integrated numerically ^over

the final energies ^and scattering angles. ^The

results are compared ^{with the} experiments ⁱⁿ figure ^4.

2.2. " POLYCRYSTALLINE "MODEL. -we finally

refer to the extreme vibrational point ^of view, ⁱⁿ

contrast with the translational one. We consider the heavy organic system ^{as a} system ^of nuclei, vibrating ^about equilibrium positions, ^{which are}

fixed in a laboratory system of coordinates. In the frame of this " heavy molecule " approximation

the vibrational features alone are considered (still

in the harmonic approximation) ^and the contri- butions of translation and rotation are neglected.

The advantage ^{of this} approach is the formal equi-

valence with a crystalline assembly, ^{for which}

647

extensively developed mathematical methods are

available.

If one refers to benzene, ^the analogy between the structure of its planar molecule and graphite ^is

very apparent. Taking ^{into account that the}

molecules are oriented at random, ^{the correct} equivalence ^is between benzene molecules and a

polycrystalline assembly ^of graphite microcrystals.

FIG. 5.

Refers to the vibrational

polycrystalline

model (« phonon " expansion). ^{The full} line shows the effect of a Debye low energy tail (to 0,02 eV) ^added

to the vibrational spectrum.

The experimental ^{values are :}

for benzene at 300 °K by ^{Pauli and} Antonini »see [7]).

for diphyl (73.5 % C12H,ol ^{26.5 %} C12HLOO) ^at

293 OK by ^{the Ris6} experimental group (private ^commu- nication).

On this line, scattering calculations have already

been done by ^{Boffi et al.} [6, 7]. They ^{refer to}

Parks’ [11] very detailed studies ^on the neutron-

graphite interaction, using ^{for the} numerical calcu- lations the SUMMIT code [12], ^{which in} part

translates the Parks’ mathematically very elabo- rated model.

We followed the Schofield-Hassitt [13] ^formalism

for an anisotropic lattice such as graphite. ^This approach also contains the possibility ^{of consi-} dering explicitly ^the anisotropy ^{of the} vibrational

spectrum ^{and of} evaluating ^{in a correct} way ^the average ^over the molecular orientations. The incoherent cross section is given by ^{a "} phonon expansion ", ^{where the} multiple integrals occuring

are approximated using the central limit theorem of statistics. The integral dependence ^{on the}

vibrational spectra ^{enables to introduce the mole-}

cular discrete delta-lines, ^without any artificial

broadening.

The results in the frame of this polycrystalline

model are shown in figure ^{5 :} the full line is obtained by taking ^{into account other than the} simple vibrational effects, ^a Debye ^{tail in the low}

energy region, with an " eff ective " mass of 21.3.

A short collision time calculation should allow to

join smoothly ^{the "} phonon " ^{curve with the} high

energy limit.

3. Final remarks.

On this line we intend now

to investigate ^the scattering models influence both

on other integral quantities ^{of interest in thermal}

neutron problems ^{and on} infinite medium thermal neutron spectra. Nevertheless our analysis ^based

on such an integral quantity ^{as the total cross}

section leads already ^{to some} interesting ^conclu-

sion.

The dependence ^{on the} vibrational anisotropy

and on a correct orientation _average does not

seem too great. ^This gives confidence in the extension of the benzene results to other poly- phenyls. ^{There exists the} necessity ^of including

other than simple vibrational effects. The details of the vibrational spectrum ^{do not} appear of deci- sive importance.

From the experimental point ^{of view it would be}

very ^{useful to} dispose ^{of some} information related to the low energy region (rotational levels).

’

REFERENCES [1] ^WICK (G. C.), Phys. Rev., 1954, 94, 1228.

[2] ^ZEMACH (A. C.) and GLAUBER (R. J.), Phys. Rev., 1956, 101, ^118.

[3] ^SACHS (R. G.) and TELLER (E.), Phys. Rev., 1941, 60,

18.

[4] ^KRIEGER (T. J.) and NELKIN (M. S.), Phys. ^Rev., 1957, 106, ^290.

[5] ^KRIEGER (T. J.) and NELKIN (M. S.), ^KAPL 1597,

1956.

[6] ^BOFFI (V. C.), ^MOLINARI (V. G.) ^{and PARKS} (D. E.),

General Atomic Report ^GA-3024.

[7] ^BOFFI (V. C.), ^MOLINARI (V. G.) ^{and PARKS} (D. E.),

Proc. Chalk River Inelastic Scattering Symposium, September 1963, p. 285.

[8] ^HERZBERG (G.), ^Molecular Spectra ^{and Molecular} Structure, ^2nd ed., ^{vol. II.}

[9] ^NELKIN (M.), Phys. Rev., 1960, 119, ^741.

[10] ^GOLDNIAN (D. T.) and FEDERIGHI (F. D.), ^{KAPL 2225.}

[11] ^PARKS (D. E.), General Atomic Report ^GA-2438.

[12] ^BELL (J.), General Atomic Report ^GA-2492.

[13] ^SCHOFIELD (P.) ^{and HASSIT} (A.), ^{Geneva Conference}

Paper, 1958, P/18.