Looal Bureau ^i' l^tt^aaaro;;; Library, N.R. Bldg

AUG 3 i ljj^

Reference book not to be

^ '61564

taken from the library.

^ecltnicciL ^^^ote 63

SINGLE SCATTERED NEUTRONS

FROM AN ISOTROPIC

POINT SOURCE

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Pubhc Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods for making measurements consistent with these standards; the determination of physical constants and

properties of materials; the development of methods and instruments for testing materials, devices,

and structures; advisory services to government agencies on scientific and technical problems; in- vention and development of devices to serve special needs of the Government; and the development

of standard practices, codes, and specifications. The work includes basic and applied research,

development, engineering, instrumentation, testing, evaluation, calibration services, and various

consultation and information services. Research projects are also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the

Bureau's miique competence is required. The scope of activities is suggested by the listing of

divisions and sections on the inside of the back cover.

Publications

The results of the Bureau's work take the form of either actual equipment and devices or published papers. These papers appear either in the Bureau's own series of pubhcations or in the journals

of professional and scientific societies. The Bureau itself publishes three periodicals available from

the Government Printing Office: The Journal of Research, published in four separate sections,

presents complete scientific and technical papers; the Technical News Bulletin presents summary

and preliminary reports on work in progress; and Basic Radio Propagation Predictions provides

dai|;a for determining the best frequencies to use for radio communications throughout the world.

There are also five series of nonperiodical pubhcations: Monographs, Applied Mathematics Series,

Handbooks, Miscellaneous Pubhcations, and Technical Notes. Information on the Bureau's pubhcations can be found in NBS Circular 460, Pubhcations of the

National Bureau of Standards ($1.25) and its Supplement ($1.50), available from the Superintendent

of Documents, Government Printing Office, Washington 25, D.C.
NATIONAL BUREAU OF STANDARDS

technical ^Tlcte

63

JULY 1960

SINGLE SCATTERED NEUTRONS FROM AN

ISOTROPIC POINT SOURCE

E. R. Mosburg, Jr., and W. M. Murphey

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. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D. C.

Price 50 cents

Single Scattered Neutrons From an Isotropic Point Source

E. R. Mosburg, Jr. , and W. M, Murphey

A calculation was made to determine the number of

particles from an isotropic source which will reach a detector after scattering once in the medium surrounding

the detector and source. The simple result obtained is

applicable when the separation of the detector and the source is much less than the mean free path of the

particle in the medium. The particular case of 1^.1 Mev

neutrons in air was considered. It was found that

kk percent of the single scattered neutrons had scattering

angles of less than 30 degrees.

In a time-of-flight scattering cross section measurement using

an isotropic source, the number of air scattered neutrons reaching

the detector may be comparable to the number of neutrons from the

scattering sample. Since rather thick shields are needed to appreciably

reduce the background flux from the air scattered neutrons, it is

desirable to have information about the angular distribution of the air

scattered neutrons. The general problem of air scattering has been

studied for various situations previously^ ? ^ a ?/ . Here, for cases in

which only the first scattering is important, a particularly simple

form is obtained.

The number of neutrons reaching the detector from a specific

infinitesimal volume element, dV, in space is given by the solid angle,

g, subtended by the detector at dV, times the probability per unit

volume, P, of scattering in the direction of the detector from the

position of the volume element, times, n, the number of neutrons

reaching the volume element from the source, times the size of the

volimie element, dV. This product is then integrated over all space,

to give the total ninnber, N, of neutrons reaching the detector;

N =

J gPn dV. (1)

The probability per unit volume of a neutron scattering in the

direction of the detector from the position of the voltme element is
2.

the cross section -^^^-L^ in millibarns per steradian as a function

of scattering angle, <p , divided by the area element dA in cm subtended

by the volume element as seen from the source, times the number of target

atoms per unit volume, n^.. So,

h ri dA ''t- ^2j

For convenience the geometry as shown in fig. 1 will be used.

All points with the same scattering angle lie on a circle drawn

through the source, detector, and one of the scattering points with

that angle (see note 1). The sum over all space will be carried out

by using the cylindrical symmetry about the axis connecting the source and the detector and by integrating over all points with a common

scattering angle and then integrating over scattering angle. Since

there is cylindrical symmetry about the axis connecting the source and

the detector the volume element, dV, can be defined as a ring of volume

dV = (2^rj^sin0^)rj^dO^dr^.

The solid angle, g, for a detector which is a sphere and has a cross sectional area of 1 cm2 is g = ]^ steradians.

SdA

The number of neutrons reaching the volume element is n = 2

where S is the total emission of the isotropic source in number per

second. Since the total cross sections at l4.1 Mev for nitrogen and

oxygen are about I.5 barns (l barn = 10"^^ cm^) , and since there are about 5«3 X lO-'-9 atoms per cm5 in air, the mean free path of neutrons

in air is about 86 meters. The exponential attenuation of the

unscattered flux may be neglected for distances under about 10 meters.

— U. Fano, L. V. Spencer, and M. J. Berger, Penetration and Diffusion

of X-rays, Encyclopedia of Physics Vol. 58/2 p.66l.

2/— A. Langsdorf , Neutron Collimation and Shielding, Chapter IVE of Fast

Neutron Physics, Part I by J. B. Marion and J. L. Fowler, Interscience

Publishers (I960)

^ R. E. Beissner and D. R. Smith, "Studies in Shielding", Convair Report

FZK-9-117, 1958.
5.

Making the above substitutions we have:

N =

1

^.7 S ,. 1 n dV n- dA —< dA "t

7>^f)

\S2«r^sin0dr^d9^

Sn

\^l

^jtr^ r^

N

n.S sinO-dO dr

^i«i

2 r

From fig. 1: a = 180 - 2 (0^ + 90° - (p ) sop=2Cp-'a = 2<p -l8O+2©^+l8O°-2Cp=20^

so R sin 0^ = —r- 2 ,„2 . 2-

r = 4R sin 0. .

N = iN / X n S d0 dr,

^ ^-^ 8 R^ sin0^

Now we transform from integrating over 0^ and r to integrating over

9^ and (n .

3r,

where

®1^1

<^0 ^Tj^

T0^ TeT ^0,

<^r,

T

(3)

(^)

(5)

(6)

(7)

(8)
since and <p are independent variables. Again from fig. 1 we have

(9) r. = - ° . sin (0, - CD ) , 1 sm qp

^1 Y

^ *

where D is the distance from the source to the detector,

d r sin 9

= ^7T::zi> (10)

() (D sin^(7)

n S

'-\[f^t? TJ^-'-rSince R = D

2 sin<p '

1 n s r

e, =<p

? «l = ° T^ ^^i-^^ (1^)

Integrating over we have the simple result.

n S

^<r(f) ^-f-. \f-j^ <>?• (^5)

This integral may be used for any single scattering \rfiere exponential

attenuation of the number of particles may be neglected. Usually

^(T- (Cp)

N ^' is not known as an analytic function and one must integrate

numerically. Equation (I3) actually gives the number of neutrons for

36" (cp) one energy. If —-^ XJ varies greatly over the energies of neutrons

being considered then an integration of (I5) must be made for each

energy.
5.

In our case N was determined for a l^l-.L Mev neutron soiree.

Since the total cross sections of nitrogen and oxygen are about

eqiial at Ik Mev and exact results were not re«iuired it was assumed,

for the purpose of CP (cp) , that the air was all nitrogen. D was

100 cm and the angular distribution (fig. 2) was that given in

BNL-i400-U The results of the integration of (15) are plotted versus scattering angle, CD , in fig. 3. Fig. k shows the fraction of single

scattering angle less than a given angle. Reading from the graph

of fig. k one notices that kk percent of the single scattered neutrons

have scattering angles of less than 50 degrees. It was concluded,

therefore, that the detector shielding for scattering angles less

than 50° should be designed to give about three times as much

attenuation as the shielding for other angles.

NOTE:

f

= "i-^a

a Is a constant

©^ = (90 - X) . (90 . 1

)

0. = . Jl + 01

90 - (| - f) 90 - (|)

da a. S S

2 2^22 2

Therefore ®i + ^o =

rc2

= + S a 22 •= a constant,

Therefore CD is a constant independent of 0^^ and 0,

-^ D. J. Hughes and R, S. Carter, Neutron Cross Sections, Angular

Distributions. BNL 400, June 1956.

USCOMM-NBS-DC-24,025
SOURCE

SCATTERING

- POINT-^

DETECTOR

OF Icm^

CROSS SECTION

Figure 1. Geometry used in the calculation

2

r _ 2r '2 2 R^ = R^ + r^ - 2.x^ R cos (9^ + 90 - ©g)

= 0^ . 0^

r = 2R sin

R =

2 sin ?

r^ = distance from source to scattering point

r = distance from scattering point to detector

Q = scattering angle

R = radius of a circle through source, scattering point and

detector
c

b

ow

to

J3E

.4 .1 :2

COS <^, CENTfR OF MASS

1.0

Figure 2. Elastic scattering cross section of 14. 1 Mev neutrons on

nitroge^i (BNL-400).
SlcvJ
•HM

(0

oM

o

Q)

+->

u

0) M

-? -^ P CO

.^«

I—I M

•H

3 = *

*" 081

= <^
0) •Mo

a

u

a

o

0) O

•H
U.S. DKPARTMKINT OK COMMKRCK

Frederick H. Mueller, Secretary NATIONAL BURKAU OF STANDARDS

A. V. Asiin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical workin general, each section carries out specialized research, development, and engineering in the field indi('ated by

its title. A brief description of the activities, and of ihe resultant publications, appears on the inside of the front

cover. WASHINGTON, D.C.

Electricity and Electronics. Resistance and Reactance. Electron Devices. Electrical Instruments. Magnetic Measurements. Dielectrics. Engineering Electronics. Electronic Instrumentation. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Photographic Technology. Length. Engineering

Metrology

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Radicals Research. Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Neutron Physics. Radiation Theory. Radioactivity. X-rays. High

Energy Radiation. Nucleonic Instrumentation. Radiological Equipment.

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Chemistry. Electrodeposition. Molecular Structure and Properties of Gases. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

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Organic and Fibrous Materials. Rubber. Textiles. Paj^er. Leather. Testing and Specifications. Poly- mer Structure. Plastics. Dental Research.

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Physics.

Mineral Products. Engineering Ceramics. Glass. Refractories. Enameled Metals. Constitution and Microstructure.

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Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics.

Data Processing Systems. SEAC Engineering Group. Components and Techniques. Digital Circuitry.

Digital Systems. Analog Systems. Application Engineering.

• Office of Basic Instrumentation. • Office of W eights and Measures. BOULDER, COLORADO

Cryogenic Engineering. Cryogenic I'.quipment. Cryogenic Processes. Properties of Materials. Gas Liquefaction. Radio Propagation Physics. Upper Atmosphere Research. Ionospheric Research. Regular Propagation

Services. Sun-Eartli Relationships. VHl' Research. Radio Warning Services. Airglow and Aurora. Radio

Astronomy and Arctic Propagation.

Radio Propagation Engineering. Data Reduction Instrumentation. Modulation Research. Radio Noise. Tropospheric Measurements. Troposphcric Analy.sis. Propagation Obstacles Engineering. Radio-Meteorology. Lower Atmosphere Physics.

Radio Standards. High Frequency Electrical Standards. Radio Broadcast Service. High Frequency Impedance Standards. I'jiectronic Calibration Center. Microwave Physics. Microwave Circuit Standards.

Radio Communication and Systems. Low Frequency and Very I^)w Frequency Research. High Fre- quency and Very High Frequency Research. Ultra High Frequency and Super High Frequency Research. Modulation Research. Antenna Research. Navigation Systems. Systems Analysis. Fiehl Oj)erations.






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