Anomalous dispersion of slow neutrons in crystals

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Anomalous dispersion of slow neutrons in crystals

H.G. Smith, S.W. Peterson

To cite this version:

H.G. Smith, S.W. Peterson. Anomalous dispersion of slow neutrons in crystals. Journal de Physique,

1964, 25 (5), pp.615-617. �10.1051/jphys:01964002505061500�. �jpa-00205840�

615.

ANOMALOUS DISPERSION OF SLOW NEUTRONS IN CRYSTALS

By ^{H. G. SMITH and S. W. PETERSON} (1),

Oak Ridge ^National Laboratory (2), ^Oak Ridge, Tenneessee.

Résumé. 2014 Les études de la diffusion anormale des neutrons par des monocristaux de CdI2

ont été poursuivies ^{dans la} région d’énergie voisine de la raie de résonance 0,178 eV du cadmium.

Toutes les réflexions de la zone (h ⁰ l) ont été mesurées en utilisant des neutrons de longueur

d’onde 1,075 Å afin d’obtenir l’affinement de la structure, ^ce qui ^{donnerait en même} temps ^la

mesure précise ^de l’amplitude complexe ^{de diffusion} pour cette longueur ^d’onde. Certaines ré- flexions ont été mesurées aussi avec d’autres longueurs d’onde. De plus, les intensités de plusieurs

réflexions de Bragg ont été mesurées pour la longueur ^d’onde 1,075 Å ^à partir d’un monocristal de 113CdI2.

L’affinement de la zone (h ⁰ l) par la méthode de moindres carrés a été effectué en tenant

compte ^des paramètres thermiques anisotropes ^{et de} l’amplitude complexe ^de diffusion du

cadmium. ^Cette dernière est en excellent accord avec la valeur calculée à l’aide de la formulation à niveau unique ^de Breit-Wigner ^en utilisant les largeurs déterminées par Brockhouse pour ^le cadmium.

Abstract.

Studies of anomalous scattering ^{of neutrons from} CdI2 single crystals ^{were conti-}

nued over an energy range including ^{the cadmium resonance} peak at 0.178 eV. The complete (h ⁰ l) ^zone of reflections was measured at one wavelength (1.075 A) in order to permit ^{a struc-}

ture refinement which would simultaneously give ^a precise measurement of the complex scattering amplitude ^{of cadmium at this} wavelength. Selected reflections were also measured at several other wavelengths. ^In addition, ^a single crystal ^of 113CdI2 ^was grown, and Bragg intensities were

measured for several reflections at 1.075 A.

A complete least-squares refinement of the (h ⁰ l) ^{zone was} computed ^which included aniso-

tropic ^thermal parameters ^{as well as the} complex scattering amplitude ^{of cadmium. The latter}

is in remarkably good agreement ^{with that} calculated with the aid of the Breit-Wigner single ^level

formulation using the neutron and gamma widths determined for cadmium by Brockhouse.

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

Introduction.

The theory of thermal neutron

scattering by nuclei and crystals ^{has been} fairly

well understood for many years. An incident

plane ^{waive scattered} by ^{a nucleus can} be repre- sented by

where b is the scattering amplitude, ^and depends

on the neutron-nucleus interactions. In general,

the scattering amplitude ^{can be} expressed ⁱⁿ complex form, ^b

^oc + ip, ^{to include the} absorp-

tion process ^as well as the scattering process ;

[3 ^is directly ^{related to the total cross section} by p

Although only ^the magnitudes

and not the phases ^{of the scattered waves are} directly observable, ^the phase change ^{of the scat-}

tered wave can be detected through interference effects in crystals. ^Such experiments ^{have been} reported recently by the authors [1, 2] ^{in the}

studies of anomalous dispersion ^{in oc}

CdS,

and BP. The anomalous scatterers in the above compounds ^{are the} 113Cd, ^{6 Li and}

lOB nuclei. CT for the 6Li and 1°B nuclei varies

as 1 jv in the thermal region, ^{so that the} imaginary

(1) Present address : Chemistry Department, Washington

State University, Pullman, Washington, ^{U. S. A.}

(2) ^Oak Ridge ^National Laboratory ^is operated by

Union Carbide Corporation for the United States Atomic

Energy Commission.

,

component ^{of their} scattering amplitudes ^{is a}

constant in this region ; however, ^{the 113Cd nu-}

cleus exhibits a very large ^{cross section in the}

thermal region ^{which is} strongly ^energy dependent.

The neutron diffraction studies [1] ^{on a}

^{CdS as}

a function of energy showed conclusively, ^{as did}

earlier total scattering ^{cross section} measurements

by Brockhouse [3J, ^{that the} Breit-Wigner single-

level formulation was an adequate description ^of

the 0.178 eV resonance in 1,3Cd.

It was desirable to have a more detailed compa- rison of the experimental data with the Breit-

Wigner expression ^{than the CdS data} allowed,

because the absorption corrections required ^were large ^{and the} crystals ^{were of somewhat} irregular shape. ^{It was also} of interest to demonstrate that the structure parameters ^of crystals containing highly absorbing ^{nuclei could be} accurately ^deter-

mined by ^neutron diffraction.

For these purposes crystals ^of Cdl, ^were grown from aqueous solution by ^slow evaporation ^{at room} temperature. ^The crystals grown by ^{this method} crystallize ⁱⁿ well-developed hexagonal platelets

and belong ^{to the} noncentrosymmetric space group C6mc (not ^the centrosymmetric space group D3d typical of normal Cd’2)-

Experimental.

The size and shape ^{of the}

selected crystal ^were accurately ^{measured to}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002505061500

616

insure reliable absorption corrections and also to obtain an absolute scale factor by comparison

with a standard NaCl crystal. ^It weighed 23 . 5 mg and was 0.52 mm thick.

The crystal ^{was oriented on the neutron} spectro-

meter to record the (h ⁰ 1) reflections, since all the structure parameters ^of CdI2, including ^individual anisotropic temperature factors, ^{and the} complex scattering amplitude ^{of Cd are} determinable from this zone. The wave length ^{of the neutron beam}

was 1.075 A, although selected reflections were

also measured at other wave lengths, including

that corresponding ^{to the cadmium resonance} peak

at O. 178 eV. In addition, ^a single crystal ^contai- ning ^87.3 % ^{113Cd was} grown ^and Bragg ^inten-

sities were measured at 1.075 A for several reflections.

The data were corrected for absorption ^{with the}

aid of the Busing ^and Levy [4] program ^on the

computer ^{ORACLE with the} following absorption

coefficients for the normal Cdl, (12.26 % 1,3Cd) crystal : 03BC

^24.9 em-1, ^58.3 cm-1, ^and

73.3 cm-1 at the wave lengths ^1.075 A, ^0.77 A,

and 0.68 A, respectively. ^The absorption ^coeffi-

cient for the enriched crystal is 177 cm-1 at 1.075 A.

Analysis of the data.

The crystal ^structure

of this form of Cdl, (type C27) had been deter-*

mined by ^Hassel [5] ⁱⁿ 1933 with the following

results :

- - - Q . - - - ,

A weighted, anisotropic, least squares refinement of the (h ⁰ 1) ^{zone data was} calculated with the

Busing ^and Levy full-matrix program modified to include the imaginary part ^{of the Cd} scattering amplitude ^{as a variable} parameter. ^{The re-}

sults are listed in Table I. (The ^coherent scattering amplitude of iodine was assumed to be 0.52 X 10-12 cm and was not varied in the least squares refinement.)

TABLE I

LEAST-SQUARES ^{ANALYSIS OF} Cd’2 (12.26 % 113Cd) fcd

^{0.38 ±} 0.01 ± 0.120i ± O.OOi

f j = ^0.52 (fixed)

Analysis ^{of the measured} reflections at the seve-

ral wave lengths resulted in the imaginary compo- nents of the scattering amplitude for cadmium which are shovTn in figure 1, given for the 113Cd

FIG. 1.

Energy dependence ^of imaginary component

of the amplitude of neutron scattering ^{from 113Cd.}

isotope ^{as a function of neutron} energy. The

scattering amplitude ^is expressed ^{in the form} (3)

fCd1l3

fo + + l~f ~~

where f o, ^the nonresonant scattering amplitude,

is given approximately by ^{the nuclear} radius ;

the real resonant increment, Af’, ^{and the} imagi-

nary ^or 900 phase ^{shifted resonant} increment, A/",

are

Here _g is the spin weighting factor, w ^{the iso-} topic ^abundance factor, £o ^{the wave} length ^at

resonance divided by 2n, ^{r n the neutron width at}

resonance, r the total width, .~ the energy of measurement, ^and Eo ^{the resonance} energy.

The smooth curve in the figure represents ^the Breit-Wigner theory prediction, making ^{use of the}

neutron and _gamma widths determined for cad- mium by Brockhouse [3]. ^The accuracy ^of each of the plotted points, which include data from both CdI ^{and CdS} crystals, is indicated by ^the height ^{of the vertical line.}

In figure 2 the real components ^are plotted ^as

(3) The very small ^term arising from the motion of the neutron magnetic moment in the nuclear Coulomb field, and reported recently by ^Shull [6], ^{has been} neglected ^here

since the effects are expected ^{to be} negligible ^{for an} unpola-

rized beam of neutrons.

,

617

a function of energy. The ^{smooth curve is} again

the theoretical curve based ^on application ^of

the Breit-Wigner formalism. Since the real com-

ponents ^are determined with less certainty ^than

the imaginary ones, only values obtained as a

result of least-squares analyses ^{are used.}

FI G. 2. - Energy dependence ^{of real resonance} component ^{of the} amplitude of neutron scattering ^from

Measurements made on Cdl2 crystals ^enriched

in 113Cd, although ^{difficult to make} because of

extremely high absorption, ^were analyzed prima- rily ^to give ^the scattering amplitude. ^{The resul-} ting ^{value was} f

^0.80 + ^{0 . 75i. This value} is interesting ⁱⁿ that the resonance contribution to the real component ^is large enough ^to outweigh

the positive potential scattering ^{term and} give

a negative ^{real term} contrary ^{to the case with} ordinary ^cadmium.

Anomalous scattering ^{can be a} very ^useful tool in crystal ^structure analysis and, indeed, ^{is neces-}

sary ⁱⁿ the determination of absolute configu-

ration. One can partly ^determine phases ^{of centro-} symmetric structures by ^{the wave} length depen-

dence of b or by isomorphous substitution, ^{but it}

is possible ^{to determine the absolute structure of}

an acentric crystal ^with the sin Patterson func- tion [7].

The normal Patterson function gives ^maxima

that represent the inter-atomic vectors between all the N atoms in the unit cell. There are

l1~(N -1 ) ^{of these} peaks ^{and it is not} always possible ^to deconvolute the Patterson function.

However, ^{the Pa function}

gives peaks only ^{between the} anomalous and non-

anomalous scatterers and should, ⁱⁿ general, ^be simpler ^to deconvolute than the ordinary ^Patterson

function. If there is only ^{one anomalous scatterer}

in the acentric unit cell it gives ^the crystal ^structure directly.

The crystal ^{structure of} Cd’2 ^is relatively simple ^{and could} be solved by ^{most indirect} methods ; nevertheless, ^a s’n Patterson was calcu- lated based on the differences in the observed inten- sities of the (h ⁰ 1) ^{and ( h 0} 1) reflections, ^{and the}

results were readily interpreted ^{in terms of the}

vectors between the anomalous and non-anoma-

lous scatterers.

Conclusion.

The present results thus strongly

confirm the existence of anomalous coherent neu-

tron scattering effects, give substantial support ^to

the Breit-Wigner single ^level formulations, ^and

indicate that the anomalous scattering ^terms may be calculated with considerable confidence from certain parameters. ^In addition, highly ^absorbing

nuclei in crystals ^can be very useful in solving ^the crystal ^{or molecular structure.}

Acknowledgements.

The authors wish to ack-

nowledge helpful discussions with G. T. Trammel and H. A. Levy. They ^{also want to thank}

W. R. Busing ^for modifying ^his least squares program ^to include complex scattering amplitudes.

REFERENCES

[1] ^PETERSON (S. W.) and SMITH (H. G.), Phys. ^Rev.

Letters, 1961, 6, ^7.

[2] ^PETERSON (S. W.) ^{and SMITH} (H. G.), ^J. Phys. Soc., Japan, 1962, 17, Suppl. B-II, ^335.

[3] BROCKHOUSE (B. N.), ^{Can. J.} Physics, 1953, 31, 432.

[4] ^BUSING (W. R.) ^{and LEVY} (H. A.), ^Acta Cryst., 1957, 10, 180.

[5] ^HASSEL (O.), ^Z. Physik. Chem., 1933, ^B 22, 333.

[6] ^SHULL (C. G.), Phys. ^Rev. Letters, 1963, 10, 297.

[7] ^OKAYA (Y.), ^SAITO (Y.) and PEPINSKY (R.), Phys. Rev.,

1955, 98, ^1857.