On the mass tensor approximation of slow neutron scattering

Physica 32 16-26

Kosaly, G. Solt, G. 1966

ON THE MASS TENSOR APPROXIMATION NEUTRON SCATTERING by G. KOSALY Central

Research

Institute

OF SLOW

and G. SOLT

for Physics,

Budapest,

Hungary

Synopsis A systematic analytical and numerical investigation of the different approximation methods of slow neutron scattering is given. The mass-tensor results are compared with the exact theory. The large recoil character of the mass-tensor approximation is shown. It is found that the Krieger-Nelkin procedure does not reproduce the masstensor results even if the momentum transfer goes to zero. A modification of the Krieger-Nelkin procedure is suggested. The results are sufficient to account for some recent numerical and experimental results.

1. luttroductiort. The treatment of slow neutron interactions with rotating molecules is an important chapter in slow neutron scattering theory. The first paper on the problem was that of Sachs and Tellerr) in which they formulated the mass-tensor (M.T.) concept for the evaluation of the cross section. Krieger and Nelkins) rederived the M.T. treatment, by use of the exact formalism of Zemach and Glaubers) and concluded the M.T. approximation to be valid for large incoming neutron energies (Ec) and high target temperatures (T). These authors developed, in addition, a simplified version of the M.T. theory, the so-called Krieger-Nelkin approximation (K.N.). The papers referred to above deal with the theory of the integral and differential cross section. The exact formalism of the double differential cross section was given by Rahmand). Applying the results of the latter to gaseous methane, Griffings) found the applicability of the M.T. approach depending on the scattering angle as well. A numerical investigation of the K.N. approximation in the case of methane was carried out by McMurry e.a. 6). Griffing’s work, together with some experimental results, indicates that the validity criteria given originally by Krieger and Nelkin are not sufficient to explain all the features of the M.T. and K.N. approximations. As a matter of fact, this is to be expected since the double differential cross section involves more quantities than Ee and T. In what follows a systematic study of the validity conditions of the M.T. and K.N. approximations is given 7). -

16 -

MASS TENSOR

APPROXIMATION

OF SLOW NEUTRON

SCATTERING

17

In the case of incoherent scattering the double diffrential cross section is given, except for some trivial factors, by the Fourier transform of a correlation function 3) *) 2. Basic formulas.

dso

2

-&

dlR==

S(K,

4

+-

S(K, E) =

-&

s --m

emiet F(K,

t)

E=E-E.

K=k--0;

S(K, E) goes under the name of “scattering law”, while mediate correlation function” F(K,

(2.1)

dt

t)


=

F(K,

t) is the “inter-

e-iKr(0)>T

(2.2)

r(t) is the coordinate-operator of the scattering atom. For simplicity, we consider the case of the model molecule having the b

0

m

M

Fig. 1.

form and assume the scattering to occur on an end point atom. The molecule is taken to be rigid. Generalization of the treatment to the case of vibrating molecules is straightforward. Coupling between translation and rotation is neglected. Under these assumptions the correlation function readss) F(K,

t) =

FTR

* FROT

FTR(K,

t) =

eX$--r(it

+ Tt2)}, Y =

FROT(X, t) =
v=

&

K[b

+

2m)

V + W)})T

El

+ ib

2(M =

(2.3) X L]

*) Throughout the paper a system of units is employed in which both the Planck’s and the Boltzmann’s constant are to be unity.

18

G. KOSiiLY AND

G. SOLT

Here T is the temperature in energy units, HRor is the rotational Hamiltonian, El = BZ(Z + 1) is the I-th energy eigenvalue, L is the angular momentum operator and b is the position vector operator of the nucleus with respect to the molecular centre of mass. The bracket stands for the thermodynamical averaging scattering

over the rotational quantum law is also known4) as

E) =

&, +=!~Jo + 1)exp r

S(K,

states.

I-

The exact

for the

l)-J2 *

(2.4)

1) *

ZO(~O +

+-

expression

0,

1 *

1/4nrT l&l

*

c

n=

-&[a+r+W+

exp (2% +

1)

j;(Kb)

l)--Jo(lo+

C(&d;

oo)2

IL-11

2~0~

~o(2Zo + 1)exp

=1 0

1G

-b(zO + l)

-

j,&) is the Iz-th spherical Bessel function Gordan coefficientss). It is easy to write down the same quantities

1

and the C’s are the

Clebsch-

in the M.T. approximations)

as

FMT = FTR *FE&! +1

FMT ROT = 4

s

exp I1 -

1 (it + Tt2)j dx,

-._ 2;;X)

-1

x=-

(2.5)

/c-b K.b

El(x)

=

&

(1 -

X2)

+1

- -!f-

2M(N

SMT(&&)

The integral

=

4

M(x)

___

%cK~T

means averaging

I

exp \-

(it+

M(x) 2K2T

~

over molecular

TP), 1 dx,

E+ &

orientations.

do

(2.6)

(2.5) and (2.6)

MASS

TENSOR

are approximatedin

APPROXIMATION

OF SLOW

NEUTRON

SCATTERING

19

the K.N. methods) FKN

=

FTR

*

FE&

FzzT = exp

-

-&

(it + Tts)

fl j&l=&

1

z=s

M(x)

(2.7)

3m

-_I

FKN = exp I \

M;l SKN(K,

E) =

= (M + 2m)-l +

B;’

~eXp{-~[E+&]2/

(2.8)

Comparison of (2.7) and (2.5) shows that in the K.N. treatment one averages over molecular orientations in the exponent of the correlation function. and Nelkins) rederived 3. Validity of the mass tensor theory. Krieger the M.T. approach using the exact formalism of Zemach and Glaubers). They found its validity to be restricted to large incoming neutron energies Ee and high target temperatures T. In fact, these criteria might be applied in the theory of the integral cross section only. In the theory of the double differential cross section, however, one deals with more parametersthus the applicability of the M.T. theory has to be reinvestigated. Moreover, on the basis of the original validity conditions, it is difficult to understand the fact that the M.T. treatment gives often excellent results for the integral cross section at extremely low energies too a) lo). For the study of the M.T. treatment we may compare the exact and approximate versions of the first few Placzek moments. With the definition &(K)

=

j

E%(K,

E) d&

(3.1)

-ce

one finds that these moments are essentially the coefficients of tn in the Taylor expansion of F(K, t) that is they can be calculated as thermal averages of the form <(U + V + W)~>T. Using now (2.3), (2.5) and (2.7) f or calculating the exact and approximate moments we get SO(K) = --Sl(K)

=

SfT(K)

-syT(K)

=

(3.2a)

1

= P’ +

R

(3.2b)

20

G. KOSALY

.53(K) SfT(K)

43(K)

$R3

=

+R[4B2 -syT(K)

=

AND

G. SOLT

=

;R2 + 2R[
=

zR2 + 2R[T + r] + ~2 + 2rT

+ R2 [y

+ B + I] + ~2 + 2rT

+ FB

+ +]

+

+ 8B + 6r + 6rB + 3~2 + 6rT] + $$R3 + Rz[yT

+ R[ 12rT + 3rs] +

+ +]

13 +

(3.2~)

~3 +

6r2T

+

(3.2d)

6r2T

Here K2 y

=

REX2

2(M + 2m) ’

(3.3)

2Mo

= B~

(3.4)

r is the recoil energy of the molecule, %!e is the rotational contribution to the Krieger-Nelkin effective mass, R is the average energy change in a scattering process on a fixed (infinite central mass) molecule at zero temperature, for given value of the momentum transfer. It seems reasonable to call R the “rotational recoil”. At this point definition

it seems useful to introduce

A(T)

= +

[Ti-

the function

A(T)

]

by the

(3.5)

Simple calculation shows that A(T) varies between 0 and 0.33*) Let us now investigate the ratios of the exact to the approximate Placzek moments. One finds that 1-

S2

--__1=

A(T)

(3.6)

SF’

It can be readily seen that the left hand side of (3.6) becomes much smaller than unity when R > B or T > B. On the other hand, considering the ratio of the third moments, it can be shown that Ss/SyT - 1 decreases, but does not go to zero with increasing T/B. Conversely it goes to zero with increasing RIB. It is apparent from the above that the M.T. theory holds for large recoil values, that is, at a givenvalue of K the M.T. theory gives the better results for the& dependence of the scattering law, the larger the recoil R. As to the influ*) Using (3.5) in (3.2c), one finds that that the use of the effective temperature even smaller them.

M.T. second moment.

Ss > SZ MT. At the same time it follows from A/T) > 0 T’ =
Thus this procedure

may impair

the results instead

of improving

MASS TENSOR APPROXIMATION

a

OF SLOW NEUTRON SCATTERING

21

b

Fig. 2a, b, c. Scattering law us. energy gain. D equals the standard deviation of the scattering law in the M.T. approximation. Exact curves (dash-cross line) are compared with M.T. curves (solid line).

ence of the temperature, it is seen that the approximation improves with increasing T/B,but does not become exact. Since the above conclusions were drawn from comparison of the first few Placzek moments only, it seems of interest to perform a numerical investigation. Fig. 2 shows a comparison between the exact and the M.T. curves plotted for various values of momentum transfer. The curves are seen to bein excellent agreement for high momentum transfer.

22

G. KOSiiLY

In the foregoing

AND

we have investigated

G. SOLT

the scattering

law for fixed

K

as a

function of E. Now the question arises, how to apply the above statements to practical cases when, as usual, Eo and 19 are fixed, while the final neutron energy varies.

Then one cannot

assign a definite

since the value of the momentum Considering that R

_

m;;*

transfer

recoil value to each curve

changes with E.

[E. + E -

2dEEo

cos 61

it is obvious that in an energy gain measurement the recoil is relatively small in the vicinity of the elastic peak and relatively large if E > Eo. Thus for a given Eo and 8 the M.T. treatment improves with increasing energy gain. This statement is in complete agreement with the results obtained on water vapour by Glaser e.a.11). (3.7) shows that R increases with increasing initial energy and increasing scattering angle. Thus, when comparing scattering patterns with different initial neutron energies and different scattering angles the M.T. approximation should improve with the increase of the above parameters (See fig. 3 and also the numerical calculations of Grif f i ng 5) for gaseous methane and of Czerlunczakiewicz e.a.12) for gaseous ammonia). As mentioned above there are cases where the M.T. theory gives excellent

Fig. 3. Scattering

law for water from

vapour

the M.T.

as given

approximation

by Glgser

e.a. 11). The prediction

is also shown.

MASS

TENSOR

APPROXIMATION

OF SLOW

NEUTRON

SCATTERING

23

results for the integral cross section at very low energies. It is well known that in the low energy range the integral cross section becomes linear in wavelength. Here the quantity of interest is the slope (da/dl),,=,. For methane at room temperarure e.g. the experiment of Taylor e.u.13) gives the slope as 15.4 b/A in complete agreement with the M.T. theory. It is difficult to understand this fact, considering the original validity conditions. On the other hand at lower temperatures there is no agreement between the M.T. theoretical and experimental slope values. To consider this problem we write the integral cross section in the form us

= 2nJ+&JmdE@o, -1

E, 1~)

(3.8)

0

It is apparent that the M.T. theory gives the better results for am the larger the contributions from intervals connected with large values of R. This statement, together with (3.7) justifies the original validity conditions. At the same time one sees from (3.7) that R becomes large for large energy gains at small values of Eo too. Thus, if the temperature is high enough. leading to large energy gains, the M.T. theory may give good results even for Eo = 0. Consequently it is the large recoil character of the M.T. theory which explains its applicability at low energies. 4. The Krieger-Nelkin firocedure. For most of the molecules even the M.T. theory is too cumbersome for practical purposes. That is why its approximate version, the K.N. procedure is so widely used. For comparison of the M.T. and K.N. methods again the Placzek moments are investigated. From (2.5) and (2.7) one has

_s”N

-syT

=

1

= -.

K2

2Mo

First, we put M = 0. Then sFT(K)

sFN(K)

=

=

$

+ _!f_ 180ma

s

+

1

4;2

(4.3a)

0

(4.3b)

&

0

Putting as usual K2

(44

a=2mT

we get for the second central moment defined as p2

=

s2 -

ST

G. KOSALY AND G. SOLT

24

IS

1-

!

a-o.1

lo-

t

f

-1

! 45-

“\

3

c3

:

-2

e4o]

d

Fig. 4a, b, c, d. Scattering law vs. energy gain in the case of zero central mass. The curves are M.T. (solid line), K.N. (dashed-line) and M.K.N. (dash-dotted line). 2 D =

Z/p$fT.

MASS

TENSOR

APPROXIMATION

OF SLOW

NEUTRON

2.5

SCATTERING

that KN

I% -= ST

1

(4.5)

1 + a/75

Consequently the K.N. approximation gets increasingly worse as the value of tc increases. This effect is clearly apparent from fig. 4. The increasing disagreement between the M.T. and the K.N. curves as the momentum transfer increases, is a simple consequence of the averaging method employed in the K.N. approximation. By this argument one is tempted to think that for a + 0 the K.N. approximation reproduces the M.T. theory. Inspection of the Placzek moments, however, shows that this is not the case. The ratio of the fourth moments does not go to unity even for a -+ 0. This effect becomes more marked if there is a third central atom in the molecule, that is M # 0. This is due to the increased anisotropy of the mass tensor in the presence of a central mass. The effect is clearly apparent from comparison of figs 4 and 5.

\ i

\ &L\

\ \

\\,

‘\ \

‘\ \

‘\

\

J

\ E-S, a

Fig. 5a, b. Scattering law vs. energy gain in the case of nonzero central mass. The curves are M.T. (solid line), coinciding K.N. and M.K.N. (dash-dotted line). D =

1/,ufT.

We have seen that the 0-th and I-st order moments of the K.N. and M.T. methods are identical. This suggests a modification of the K.N. treatment by keeping the simple Gaussian form but with second moment (4.3~~) instead of of (4.3b). Figs 4a, b, c show that there exists a range where the modified K.N. (M.K.N.) curves give a better fit to the M.T. approach than the K.N. approximation.

26

MASS

TENSOR

APPROXIMATION

OF SLOW

NEUTRON

SCATTERING

Acknowledgements. Authors would like to thank Mrs. Elisabeth Nagy for her able help in computer programming. One of the authors (G. K.) is indebted to Dr. J. J. Rush for a valuable discussion. Received

2-4 65

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Report Scattering

No. of