Nitrogen Oxygen
¿J ciC&mha.
1 1 X 1 0 - 2 4 ^ » 5 X 1 0 - 2 4 : cm 2 0
Thus the total effective cross sections for air are
= - ^ yrhere is the no. of nitrogen atoms/cm^
- SKicr^/Ylm^. +11* - S-a. 7xlO'^ /CW-'
The effective where » — *.jx /S *
^ d £ »
£■
..t
( y i [ e ) V- -C l - _ .1 4 / ■TW o>n jtuk.unii £ IX.IOa.
•C/ir
= I.JS^I&^Cl Thus at thermal energy the neutron flu x is 1 / 7 .7 of what it is at 1 0 ° ev.
4 . Mean Distance from a Point Soui-oe vs. Energy
In this section ve shall calculate R ^ ( E ) , the mean square distance to all the neutrons of energy E generated by a mono- energetic point source.
The total number of collisions ^Inl-I* ) by I X .B .
I f the neutrons travelled isotropically the exact distance ^ between co llisio n s, this would be the ordinary random walk T5roblem where the mean square distance travelled is given by
However > is so defined that the probability of having a co llisio n in a distance dr is - ^ r .
Let P (r ) be the probability of not having had a c o llis io n after travelling a distance r.
Then ^
186 Distance from a Ppint Source vs. Energy Ch. I I
- _ _
^ ~ ~ which ia the mean free path by definition.
For isotropic scattering after n collisions,
However, in t h e Hab s y s t e m , t h e f o r w a r d ¿ I r e c t i o n is p r e f e r r e d . This p e r s i s t e n c e of v e l o c i t y m o d i f i e s as f o l l o w s : *
^
Using H . 2 it is easily seen that Ctra 5 (0 is now used as
lab system angle
For A as a slowly varying function of E, rather than 0^ )
* Let Tj be the path after the jth collision. " ■ • I £3
R2 . f I · / . ^ ^ r j · ^ __________
^ = n + J : 2 rj r-i(j cos9,*
Since the average of the product of independent quantities equals the product of the averages,
flj < ^ 6 f4t.
It has been shovm (11.67, page 51) that the average cosine of
the resultant angle ) which is the aim of (k - j) d e fle ctio n s
is ____ ___________
c<«e
Since the distribution of 0 is the same after each c o llisio n ,
the above eouation is seen to be a recursion formula. Since
eqch particular © is independent of the others,
^ 9
it.=
Oh.
11
Spacial Distributions of Neutrons 187I X . 12 is not quite exact. However, for particles of equal
mass (neutrons in hydrogen) there exists an exact but lengthy
expression. It is Riven in section L5 of Ch.TI of ♦^Neutrpn
Physics” (L .A . 2 5 5 ).
We shall nov. discuss experimental methods for determining spatial
distributions of neutrons. A foil
of indium sandwiched between two foils of cadmium makes a sensitive detector for 1 .4 4 ev neutrons. This is easily seen from the absorption cross section
curves given in FIG 4 . A measurement
of the activity of the indium (13s and 54m half lives) gives the value
of the neutron flux . A compilation
of cross section curves of the
elements is given in Goodman and also in the Rev. Mod. Phys., Oct. 1947.
T W
^
f^CJ.T
i.'a
I.fJUrFIG. I X . 4
The experimental distribution of neutrons around a mono- energetic point source is a gaussian-
like distribution as shown in FIG· 5. A description of experimental tech niques is given in a paper by Amaldi and Fermi, Phys. Rev. 5^, 699 (1936)* The theoretical verification of these results v;ill novv be demonstrated.
FIG. I X . 5
C> DIFFUSION THEORY *
1. The Age Equation
The two following assumptions n*ust be made for this theory:
1
. A(E) Is slowly varying.2. A large number of collisions take place. This theory w ill be found more applicable to collisions with heavier nulcei rather than with
protons for tv;o reasons. First, A in hydrogen is much larger
for the first few collisions (see FIG. I X .0 , page 194 ) than for the others. Secondly, the law of Inrge numbers is more applicable in the case of heavier nuclei (
-^100
collisions for carbon), than for hydrogen where 1 Mev neutrons become thermal in about 17 collisions.High Energy Neutron Path in Hydrogen
We shall nov^ derive the neutron diffusion equation, which is also known as the Fermi age equation.
Let n (r ,£ ) be the number of neutrons per unit volume per unit e. , This is the previous n ( & ) , per unit volume.
Let q (r,£ ) be the current density along thee axis. This is the
previous g i g ). per unit volume, it is also called the slowing down aehsTty:
q (r ,c ) = n(r,£)- ^>
_________ ^
^ ________________________________________
♦A review article based on lectures of H.E. Marshalc is in the May - Aug. ±949 issues of Nucleonics,
Consider all neutrons between £ and c + de. I f V n ( r , c ) i s n 't zero, there w ill be a flow from the more concentrateH" regions
to the less concentrated. Kinetic theory tells us that the
number of neutrons in the energy interval d£ flowing thru a
surface element ds per second =- D ^(nd t) . as_ . This relation
defines the d iffusion coefficient, Dj which can be shown to be “ . In our case the persistence of'‘velocity increases the mean
square distance by a factor ·
1
''" w il l soon be shown thatD and are related (see I X . 1 7 ) . ’’ since I X . 12 contains the factor , then D must also contain it when there is persistence of
velocity. .
The mjjnber of neutrons In the energy interval (i£ which accumulate per second in a volume element Al/ is
r A S
•iv
Applying the divergence theorem, one obtains iv
Thus the Increase in neutrons in the interval d£ per unit time
per unit volume by diffusion is D V^ndi . However there is
another mechanism to contribute to this increase. It is the
neutrons from a higher energy region dropping down to the energy
region under consideration. The number of neutrons crossing
the value £ per second is the e current density. Thus the
number acc\:miulatlng per second per unit volume in d£ = - q(c)
since in a steady state there oan be no total accumulation, D F ) + | i - <?
Let
I X . 13
= for constant A
?· Is called the "age" of the neutron. At £ = f. , 0, and
as increases, i" increases. Note that t has the dimensions
Using I X . 13, ic = ^ U = § f a 3 / ^ ^ ]
Thus 9T “
^'^<^Ciy,Tr) s The Age Equation IX. 14 Mathemetlcally this Is the same as the heat equation:
188
Diffusion Equation
Ch. IX
Discussions of its solutions for various in it ia l and boundary conditions are given in books on heat and diffusion.
We shall consider two such solutions here. The first w ill
be the seneral case of an infinite medium given the in it ia l condition that q (r ,0 ) => E ( r ) . The result can immediately be
to check with the experimental results which have been previously discussed.
Consider a particular solution of the age eq.uatlon: Substituting into I X . 14 gives
where
Jb
is the separation constantAny solution can be expressed as a sum of these q.n'3 using Fourier integrals.
( Ct(A)is the FourTer transform of F(r) )
= ( m "
d x 'd y '^ ^ ' I X .1 5 For a mono-energetic point source of strength Q , F (£ ') is the delta function
.*. t) c ^ ^ »
4
. ^ ^ point source at r=0 I X . 16 From this we obtain as the mean square distanceThis agrees with the previous result, I X . 1 2 , i f the factor is Included in D, the diffusion c oefficient.