ttianal Bureau of Standards E-Ql_i*nin. Bfajg, NBS TECHNICAL NO

442

An Empirical Formula

of the Coherent Scattering

Cross Section of Gamma Rays

z

/

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(A

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U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Ml.iVVS fSlil : .':.'."
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NBS TECHNICAL NOTE 442

ISSUED APRIL 1968

An Empirical Formula

of the Coherent Scattering

Cross Section of Gamma Rays

A. Nath and A. M. Ghose

Bose Institute, Calcutta 9, India

Radiation Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C, 20402 - Price 20 cents

An Empirical Formula for the Coherent Scattering Cross

Section of Gamma Rays

A. Nath and A. M. Ghose

An empirical formula has been developed for estimating

coherent scattering cross sections of y-rays of energies below 1.5 MeV and in the range of - 2.5mc units of momentum

transfer. The formula has been compared with the experimental

data available in the literature in the case of Pb, Sn and

Cu scatterers for different photon energies between 0.279 - 1.33 MeV. Experimental data, in general, have been found to be

in good agreement with the results of the empirical formula,

the deviation being within ±10% in most of the cases. The

empirical formula has also been compared with the form factor

results of Nelms and Oppenheim and with the accurate theoretical data of the Birmingham group. The latter is in excellent

agreement with the empirical formula.

Key Words: coherent scattering, Delbruck, empirical formula,

gamma rays, nuclear Thomson, Rayleigh

1. Introduction

Coherent scattering of gamma rays occurs mainly through the non- resonant processes of

Rayleigh scattering, nuclear Thomson scattering and Delbruck scattering. Nuclear resonance

scattering, though a coherent process, is usually a very rare event and can be observed

only under specially adopted experimental situations. The intensity of the nuclear Thomson

scattering process can be easily calculated by extending Thomson's classical relation for

the electron to the case of the nucleus. Unless q, the momentum transfer involved during

the coherent encounters, is rather large (q = 2mc or more) and the atomic number of the

scatterer is low, nuclear Thomson scattering constitutes a negligible fraction of the total

coherent scattering intensity. The lower limit of the energy of the photon, for which the

Delbruck scattering makes barely perceptible contribution to coherent scattering is believed

to be at least 1 MeV. Therefore, for all practical purposes Rayleigh scattering is the most

important coherent scattering process, especially for y-rays obtainable from normally available radioactive sources. It is therefore generally sufficient to compute the coherent scattering intensity by a consideration of the Rayleigh scattering process alone.

*This work was performed under the sponsorship of the U. S. National Bureau of Standards.

Published with partial support from the Office of Standard Reference Data.

1
Theoretical computation of the Rayleigh scattering intensity is, however, a complex

problem. In spite of considerable theoretical development in the last decade, it is as yet

not possible to estimate Rayleigh scattering intensity for all values of y ray energies, atomic number of scatterer and of the momenta transferred. Although the formalism used by

the Birmingham group [l] 1 yields accurate results, up to now it has been used only for the

K-electrons of Hg for only five specific y-ray energies. Even for these limited cases considered, the scattering cross section can be accurately predicted only when q exceeds the

intrinsic momentum of the K-electrons of the atom (a Zmc) , in which cases K-shell electrons

account for the bulk of the Rayleigh scattering and the contribution of L- shell electrons is

approximately known. Of the several form- factor calculations of Rayleigh scattering, only

those due to Nelms and Oppenheim [2] in the non-relativistic (up to q « 0.5mc) . predict the

cross section fairly well in a few particular cases viz. high Z elements, low energy y~ ra ys

and low momentum transfer (q < 0.2mc). Above q = 0.2mc, measured Rayleigh scattering cross

sections deviate from the values given by Nelms and Oppenheim due to the onset of relativistic effects as well as to the effects of binding in the intermediate state [3]. It is

therefore clear that except for a few isolated cases, the Rayleigh scattering cross section

cannot be estimated accurately from theoretical considerations.

Rayleigh scattering is not only a fundamental mode of interaction of photons with matter demanding a careful investigation in its own right, but the estimate of coherent scattering intensity is also essential under many practical situations, e.g., in the measurement

of nuclear resonance scattering and Delbruck scattering, in shielding calculations, etc. In

the last few years a large number of experimental cross section values of reasonable accuracy have been accumulated (for a complete reference see [3]). These cross sections, as well

as the accurate theoretical values wherever available, might be used as the basis for the

development of an empirical formula for the total coherent scattering cross section. In

this report we have presented an empirical formula, valid up to momentum transfer q = 2.5mc,

which can be used to estimate the Rayleigh scattering cross section in a simple and straightforward manner. We have shown how a suitable form of the empirical formula was obtained

from an analysis of the available data especially by studying the dependence of the cross

sections on the atomic number of absorber, energy of the photon and finally on the momenta

1 Figures in brackets indicate the literature references at the end of this paper.

2
transferred. A procedure for the evaluation of the empirical constants in the formula lias also been indicated. The formula has then been compared with the experimental data and with

the accurate theoretical estimates available, We have limited the range of validity of our

formula from q = to q = 2.5mc, with the energy of the photon remaining below 2 MeV

2. Procedure for the Derivation of the Empirical Formula

Form factor approximations of Rayleigh scattering cross section R (9) can be written in

the form

,n(q)

o- (e) = i(i + cos s e)z"^ ; f( R

q- (i)

where is the angle of scattering, q is the momentum transferred in units of mc, Z is the

atomic number of the scatterer; n(q) and f(q) are functions of q only. If equation (l) is

universally valid then the reduced cross section defined by

d R (q) =

R( e V!U + cos2 G)

(2)

should be a function of q only and its functional dependence on the energy E of the photon

will be implicit through q and not explicit at all. In Fig. 1 we have plotted experimental

values of o'(q) for various values of q and E. It will be seen that the trend predicted by

R the form factor formalism is observable only for low energy y-rays (up to 0.411 MeV as shown

in Fig. l) . For Sn and Pb, data for 0.279 and 0.411 MeV may be represented by a single

smooth curve but in both of these cases higher energy results fall systematically below this

curve, the deviations increasing with E and q monotonical ly . Examination of the data for Sn

and Pb further reveals the fact that the relative deviations behave approximately in the

same manner in both the cases. It therefore follows that the accuracy of (l) can be improved

by the inclusion of an explicitly energy dependent factor <v>(E,q) which becomes important

when E exceeds 0.4 MeV. On the assumption that m(E,q) is independent of Z and by examining

the relative deviation of the cross section data from the mean curve, it was found that if a

factor of the form

cp(E,q) = (13. 5E - 4.4)"°* 35q for E > 0.4 MeV , N (3) =1 for E < 0.4 MeV
+ 0-279 MeV

O 0-411 MeV

X 0-622 MeV

• 1-33 MeV

T RESULTS AFTER BROWN AND MAYERS (1958)

Pb, 0-279, 0-411 MeV

0-5 1-0 1-5 — <l(IN trie UNIT)-

2-0 2-5

Fig. 1. Experimental d

1 (q) data

of Pb and Sn for different y-ray

energies plotted as function of q,

is included on the right hand side of Equation (l) then the values of d' (q) / m(E,q) for

different values of E considered in this report, can be made to lie on a smooth curve. The

quantity

d^(q) = d^(q) / cp(E,q)

U)

= d R

(9) / |(1 + cos2 0) cp(E,q)

might be termed the normalized cross section. Its value is

aj'(q) =z"^f(q) = (Z/Z

o )^^f(q)Z^ )

=Zn(q)f (q) r r

(5)

where Z is the atomic number of a standard absorber,

Z = z/z

r o (6s
and

fr

(q) = f(q)Z

Q

n(q)

(6b)

are respectively the relative Z of the absorber and the reduced value of f(q). The standard

absorber chosen in the present investigation is Pb, which being a high Z element, has relatively high coherent scattering cross sections, which have been measured within small error

limits by several workers. To find the form of f (q) we have first plotted the normalized

cross sections of Pb, obtained both from experimental data as well as from the theoretical

computations of the Birmingham group. Then a smooth curve was drawn through the points taking into account the errors of the experimental points and attaching larger weight factors

in favor of the accurate theoretical values. It was found by trial that for q ^ O.lmc, the

curve can be represented by a simple relation of the type

f r (q) = a

q- m

(7a)

while for < q < O.lmc, a suitable form for f (q) is

f (q) = (b+cq)" P

, < q < O.lmc (7b)

The constants a and m were determined by the method of least squares. The constant b

was found by plotting f (q) against q in a log-log graph and displacing each of the points

by constant amount along q axis until the resultant curve was a straight line. Having obtained b, the constants c and p were found by the method of least squares. The final expression for f (q) was derived as

f

r (q) = (3.15q)" 2,5

°, q > O.lmc (8a)

and

-? 20

f (q) = (O.K + 2.65q) for < q < O.lmc (8b)

The function f (q) is shown in Fig. 2.

The value of n(q), the exponent of Z in Equation (5) has been measured by several

workers for various values of q. In Fig. 3, we have shown these results from q near zero to

2.5mc. It will be seen that n(q) gradually increase with q, the rate of increase being

faster near q = but slows down around q = O.lmc. At q = 0, we have assumed that n(0) takes

the value 2, as predicted by the form factor formalism. A close examination of the experi-
mental data on n(q) shows that there exists a wide difference in the trends of the values of

n(q) in the range extending from approximately q = 0,5mc to q = O.Smc. relative to their

values elsewhere. This range constitutes, from the experimental point of view, the most

1000

0-00!

1-0 1-50 y(IN tne UNIT) -

Fig. 2. Variation of the function

f r (q) with q.

2-50

Fig. 3. Fit of the experimental n(q) data with

Equation 10.

4 -

o^ ^

e

1-0 1-5 -q( IN trie UNIT) - 2-5
difficult region for accurate measurement of the coherent scattering cross section. In this

region, the hard noncoherent part of the scattered photons poses a serious problem to the

experimental measurements. The conventional procedure of subtracting the noncoherent fraction from the scattered spectrum by using a low Z scatterer is not valid for correcting the

hard component of the noncoherent radiation. There is therefore a tendency to overestimate

the coherent scattering cross section in this momentum transfer region [3]. The necessary

correction is relatively more important for low Z elements than for high Z elements due to

the fact that the Rayleigh scattering cross section falls off more rapidly with decreasing

Z than the cross section corresponding to the hard noncoherent component. In Fig. 3 this

tendency is reflected in the large dispersion in the measured values of n(q) as well as

their grouping well below the values expected from the trend of values on both sides of this

region. In fitting the Z-dependence curve, therefore, greater weight was attached to the

low q data as well as to data for q > 0.8mc. It was found that the observed data could be

satisfactorily expressed by a simple polynomial in q for values of q > O.lmc

n(q) = 2.90 + 0.08q + 1 .27q2

- 0.36q3 (9)

The coefficients of the above polynomial have been obtained again by the method of least

squares. To account for the sharp drop of n(q) below q = O.lmc, an additional term -0.90

exp(-500q2

) was found necessary. This term has negligible value above q = O.lmc, so that,

the value of n(q) for the entire range of momentum transfer can be written as

n(q) = 2.90 + 0.08q + 1.27q3 - 0.36q3 - 0.90 exp(-500q2

) ( 10)

The final expression for the Rayleigh scattering cross section is, therefore

d R

(e) = |(i + cos^e) cp (E,q)(z/82 ) n(q)

f r

( q ) (11)

where cp(E,q) is given by Equation (3), n(q) by Equation ( 10) and f (q) by Equation (8).

d (9) is expressed in b/sr.

3. Comparison with the Experimental Data

In Figs. 4-9 we have compared the experimental coherent scattering cross section data

for the scatterers Pb, Sn and Cu with the values predicted by the empirical formula given in
Equation ( 1 1) .

In the case of Cu (Fig. 4) coherent scattering cross section is small except at small q

and the experimental data are scarce and we have presented here only the measurements carried

out with 0.662 MeV photons [8], Within the limits of experimental uncertainty the experimental results are in agreement with the empirical formula, as can be seen in Fig. 4«

10

/s

1-0

0-1

Cu, 0-662 MeV

i i i i I i i i i I I' ii I i i i i I 0-05 0-1 0-15

<— ciClNmc UNIT)

0-2

Fig. 4. Comparison of the empirical

formula with the experimental data

in the case of copper and 0.662 MeV

photons

Fig. 5. Comparison of the empirical formula

with the experimental data in the case of

Pb and Sn scatterers for 0.279 and 0.411 MeV

photons. Also shown in the diagram are the

modified form factor results after Brown and

Mayers [ l]

0-5 1-0

* qCINiiic UNIT)-
10 -

1-0

\? -1 10

10 -

10

3

-\ • STANDING AND

—\ JOVANOVICH, 1962

:

I

x NATH AND GHOSE , 1964 EXTRAPOLATED RESULTS

[1]

— \

'-

\ N yPb , 1-33 MeV

V

\

\

1 1i I i i i i I i i i i I i i i i T 0-50 1-0 1-5 q in mc unit —

2-0 2-5 0-5 1-0 1-5 2-0

- 1(IN irtc UNIT) "

Fig. 6. Comparison of the empirical

formula with the experimental data

for Pb and 0.662 MeV photons. Also shown in the figure are the ac- curate theoretical results calculated

after Brown and Mayers [l].

Fig. 7. Comparison of the empirical

formula with the experimental data

for Pb and 1.33 MeV photons. Also

shown in the figure are the accurate

theoretical results calculated after

Brown and Mayers [l].

In the case of Pb we have considered the cross sections for 0.279, 0.411 (Fig.5-Curve i)

0.662 (Fig. 5), and 1.33 MeV (Fig. 7) photons. The experimental data agree quite well with

the values predicted by Equation (ll) throughout the range of q considered here. As q tends

to zero, the formula (ll) gives results which are practically identical to those given by

form factor formalisms.

Experimental data considered in the case of low momentum transfer (q up to about 0.5mc)

are mainly due to [3]. In the earlier measurements ([ll], [ 12] , [ 13] , [ 10] ) , the effect of

the energy degeneracy of the Compton scattered photons on the observed cross sections [ 14]

was not taken into account and hence, accuracy of some of these data is questionable. In

the case of 1.33 MeV photons, a disquieting fact reveals itself, namely, the large number of
experimental results reported in the literature differ from each other by factors which are

several times their error limits [3J. Of all the measurements in this field, we consider

those by Standing and Jovanovich [7] to be the most accurate set, since in this investigation

the hard noncoherent component has been properly taken into account by adopting an ingenious

technique. For large values of q, therefore, we have considered only the data given by these

authors. These experimental results generally agree with the Equation (ll) within ± 10$.

For 0.411 and 0.662 MeV photons we have included the data given by Mann [4-J 5 Bernstein and

Mann [5], and by Anand and Sood [9]. For several values of q, the set of values given by the

first two measurements differ from those given by the third measurements, by several times

the quoted error limits despite the fact that all the three measurements were carried out by

the same technique. In these cases the empirical formula gives values which are generally

intermediate between the two sets of measurements. We may conclude, therefore, that the agreement of the experimental data for Pb in the energy range of 0.279 to 1.33 MeV with the

empirical formula is entirely satisfactory and within the limits of experimental uncertainty.

To test the accuracy of the formula for intermediate Z elements, we have presented in

Figs. 5 II, 8, and 9, the predicted and measured values of 3

R (q) for Sn, which were corrected

2x10

10 \\ •

X

Sn, 0-662 MeV

MANN , 1956 ANAND et at , 1965

NATH AMP GHOSE , 1964

\j

T m.f.f. RESULTS , [11 1-0 r

-1 10

-2 10

16

3 - » \^x

r<si

I*4 1

1 1 , 1 , i i i i i i i i 1 1 , 0-5 1-0 1-5

* £ UN mc UNIT)

2-0

Fig. 8. Comparison of the empirical formula with the experimental data for Sn

and 0.662 MeV photons. Also shown in the

diagram are the modified form factor re- sults after Brown and Mayers [l].

10
Fig. 9. Comparison of the empirical formula

with the experimental data for Sn

and 1.33 MeV photons.

0-5 1-0 1-5 2-0 1CIN mc UNIT)

for nuclear Thomson scattering where necessary. Here it is to be noted that although there

is general agreement of the experimental data with Equation (ll) within the range of experimental uncertainty, there is a systematic trend of the cross section data in the range q =

0.5mc to about q = lmc to be higher than the values given by the formula. If we bear in mind

the experimental difficulty of measuring the coherent scattering cross sections in this range

of q, due to the presence of relatively large amounts of harder noncoherent radiation in the

beam scattered by elements with not too high Z as described in the previous section, such a

trend is to be expected. Elsewhere in the entire range of momentum transfer, the agreement

between the predicted and the measured cross sections is quite satisfactory. It is possible

to restore the agreement also in the intermediate region by a suitable choice of n(q) . but

such values will be rather unrealistic. It is expected that more careful experiments will

remove this anomaly in the future.

11
4. Comparison of the Empirical Formula with the Form Factor

Results of Nelms and Oppenheim

We have compared the form factor results of Nelms and Oppenheim in Figs. 10 and 11 in

the case of mercury and arsenic with our formula. In the case of Hg, the two results agree

well up to q = 0.2mc, although values given by Equation (ll) are systematically lower than

the Nelms-Oppenheim values in the region of very low q. This difference is, however, well

within 10%. Above 0.2mc, the difference of the two results increases systematically. This

is expected because we have placed greater emphasis on the experimental data, which, for this

case, are known to give this type of deviation.

-Empirical formula Nelms and Oppenheim [2]

0-5 0-1 0-01

« %ONme UNIT)

100

10 £

- 1

0-001

- Empirical formula

Nelms and Op/benheim [2]

10

1-0 -a

0-1

0-01 0-5 0-1 0-01

* qCINMc UNIT)

0-001

Fig„ 10. Comparison of the empirical formula with the form factor results of

Nelms and Oppenheim in the case of Hg.

Fig. 11. Comparison of the empirical formula with the form factor results of

Nelms and Oppenheim in the case of As.

In the case of As (Fig. ll), also, the agreement is quite good up to q = O.lmc, the maximum deviation being less than 8%. After this, Nelms and Oppenheim values fall systematically

above the values given by our formula. This trend also confirms the experimental trend of

the low Z data.

12
5. Comparison with the Results of Refined Formalism

A comparison of the two results has been presented in Figs. 6 and 7 in the case of Pb

for 0.662 and 1.33 MeV y-rays. The results of the refined formalism presented here have been

obtained from the results of Brown and Mayers [l] for the K-electrons of Hg at 1.28 and

2.56mc2

. L-shell contribution to these results has been estimated after Bernstein and Mann

[5]. Minor correction has also been included to account for the difference of the atomic

number of Pb and Hg and for the small difference in the energies of 1.28mc2 and 0.662 MeV

photons and of 2.56mc2 and 1.33 MeV photons. All these corrections should not, however, introduce an uncertainty of more than 3%. We can see from Figs. 6 and 7 that in both the

cases, the agreement between the two results is quite satisfactory. In the case of 1.33 MeV,

the agreement up to q = l.Smc is excellent; above q = l.Smc, calculated values of Equation

(ll) tend to be systematically higher than the accurate theoretical data. In the case of

0.662 MeV, however, the two results are quite close to each other right up to q = 2.5mc.

The good agreement between the two results is particularly satisfying, because the refined

formalism is expected to give Rayleigh scattering cross section quite accurately.

In a number of cases (e.g. Figs. 5 II and 8) we have also presented the theoretical results calculated on the basis of the modified form factor as suggested by Brown and Mayers

[l]. Although the accuracy of the modified form factor formalism is not yet clear, the data

presented here show that the modified form factor results are generally not appreciably different from the results given in Equation (ll) although the agreement is certainly poorer

than in the case of the refined formalism.

6. Discussion

We thus find that except in the case of Sn for a small region of q near 0.5mc (where

the experimental data themselves are considerably uncertain) there is, in general, good agreement of the formula developed here both with the experimental data up to 1.33 MeV in the

range of momentum transfer considered here and with the available accurate theoretical data.

A few experimental data above 1.33 MeV, e.g., 2.62 and 2.76 MeV, have been reported in the

literature. We have not considered these data here for two reasons. Firstly, in most of

these measurements, the hard noncoherent component (which is relatively more important for

13
harder photons) has not been adequately corrected for and hence these data cannot be relied

upon [15]. Secondly, the Delbruck scattering contribution is not known, but in all probability it cannot be neglected at the higher energies. Also, as can be seen from Equation (l),

we have retained the form factor expression for the polarization factor, i.e. (l + cos2 G) .

We have thus 100% of polarization at G = 90 . Measurements on the polarization of Rayleigh

scattering at 90 as well as the theoretical data on the refined formalism show that for

0.662 MeV, polarization is indeed close to 100%. Also, at 1.25 MeV, deviation of the polarization observed at 90 from 100% is not very large although certainly finite. Theoretical

results at 5.12mc'=; , however, predict a polarization which is considerably different from the

result expected on the basis of the present formula. In the case of high energy Y" ray s

(E » mc2

)

, the present formula can therefore be used only at smaller angles. Up to 1.33

MeV the present formula has been adequately tested and can therefore be safely used for q

up to 2.5mc to estimate the coherent scattering cross section within ± 10%.

Acknowledgment

The authors wish to thank Dr. D. M. Bose, Director of Bose Institute and Dr. S. R.

Khastgir, Head, Department of Physics, for their interest in this work.

U
References

[l] G. E. Brown, R. E. Peierls and J. B. Woodward, Proc. Roy. Soc. A227,

51 ( L954-) ; G. E. Brown and D. F. Mayers, Proc. Roy. Soc. A234 , 387

(1955); G. E. Brown and D. F. Mayers, Proc. Roy. Soc. _A2_4_2, 89 (1957);

and H. Cornille and M. Chapdelaine, Nuovo Cimento _14., 1386 (1959).

[2] A. T. Nelms and I. Oppenheim, J. Res. NBS _5_5, 53 (1955).

[3] A. Nath, Trans. Bose Res. Inst., 29, (4.) 1 ( 1966)

[4] A. K. Mann, Phys. Rev. _101, 4 ( 1956) .

[5] A. M. Bernstein and A. K. Mann, Phys. Rev. _U0, 805 (1958).

[6] E. Hara, J. Benaigs and J. Mey, Compt . Rend. 24-4- , 2155 (1957).

[7] K. G. Standing and J. V. Jovanavich, Can. J. Phys. _40, 622 (1962).

[8] A. Nath and A. M. Ghose, Nucl. Phys. _5_7, 547 ( 1964.) .

[9] A. Anand and B. S. Sood, Nucl. Phys. _73, 368 (1965).

[10] P. P. Kane and G. M. Holzwarth, Phys. Rev. _122, 1579 (l96l).

[ll] A. Storruste, Proc. Phys. Soc. A63, 1197 (1950).

[12] H. Schopper, Z. Phys. ,14-7 , 253 (1957).

[13] A. Nath, Trans. Bose Res. Inst. 27, 33 ( 1964-) .

[14] A. Storruste and P. 0. Tjom, Nucl. Phys. 6, 151 (1958).

[15] A. Nath, Science and Culture 30, 502 (1964); and A. Nath, Science and

Culture 32, 308 ( 1966)

15

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31.50; foreign, 32.25*.

'Difference in price is due to extra cost of foreign mailing.

NBS TECHNICAL PUBLICATIONS

N0NPERI0DICALS

Applied Mathematics Series.

tables, manuals, and studies. Mathematical

Building Science Series. Research results, test methods, and performance criteria of building materials, components, systems, and structures. Handbooks. Recommended codes of engineering

and industrial practice (including safety codes) developed in cooperation with interested industries,

professional organizations, and regulatory bodies.

Special Publications. Proceedings of NBS conferences, bibliographies, annual reports, wall charts,

pamphlets, etc. Monographs. Major contributions to the technical literature on various subjects related to the Bureau's scientific and technical activities.

National Standard Reference Data Series.

NSRDS provides quantitative data on the physical

and chemical properties of materials, compiled from

the world's literature and critically evaluated.

Product Standards. Provide requirements for

sizes, types, quality and methods for testing various

industrial products. These standards are developed

cooperatively with interested Government and in- dustry groups and provide the basis for common

understanding of product characteristics for both

buyers and sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other agency

and NBS-sponsored work) of limited or transitory

interest. CLEARINGHOUSE

The Clearinghouse for Federal Scientific and

Technical Information, operated by NBS, supplies

unclassified information related to Governmentgenerated science and technology in defense, space, atomic energy, and other national programs. For

further information on Clearinghouse services, write:

Clearinghouse

U.S. Department of Commerce

Springfield, Virginia 22151

Order NBS publications from:

Superintendent of Documents

Government Printing Office Washington, D.C. 20402
U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

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