Main On the Scattering of Thermal Neutrons by Bound Protons

On the Scattering of Thermal Neutrons by Bound Protons

In 1945, Niels Arley discovered that this seemingly esoteric publication had been used to dimension the Hanford reactor that produced the plutonium for the Nagasaki bomb, and promised never to touch at nuclear physics any more.
Attached to the booklet is a two-page presentation of the research in Danish, presented at the 19. skandinaviska naturforskarmötet, Helsinki 1936.
Year: 1938
Publisher: Levin & Munksgaard
Language: english
Pages: 29
Series: Det Kgl. Danske Videnskabernes Selskab. Mathematisk-fysiske Meddelelser. XVI, l.
File: PDF, 1.76 MB

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Det Kgl. Danske Videnskabernes Selskab.
Mathematisk-fysiske Meddelelser. X V I , l.

ON THE SCATTERING OF THERMAL
NEUTRONS BY BOUND PROTONS
NIELS ARLEY

K.0BENHAVN
L E V IN

& M UNKSGAARD
HJNAH ML'NKSGAAHf)

1938

IN T R O D U C T IO N
4 fter

F e r m i ’s

discoveryx) o f the possibility o f producing

A jL slow neutrons by surrounding a source o f fast neutrons by
hydrogeneous substances such as paraffin wax, the problem
of the mechanism o f the collision between neutrons and
protons has become important for the study o f the proper­
ties o f slow neutrons. The problem has already been treated
by

F ermi

himself2), who describes the slowing-dpwn pro­

cess in the following way. Neglecting first the fact that
the protons in the paraffin are bound chemically, the fast
neutrons which come from the source will make elastic col­
lisions with the protons giving up on the average half of
their kinetic energy at every collision. In this way they will
soon reach thermal energies, where they will remain for a
relatively long time, because now the chance that a neu­
tron w ill get by a collision with a proton some o f the ther­
mal energy of the latter is about the same as that it w ill
lose energy by the collision. The neutron w ill therefore
diffuse round in the paraffin until it is finally captured by
a proton. So long as the neutron energy is large compared
with the oscillation energy of the proton it is legitimate to
consider the latter as free. As the highest oscillation fre­
quency o f the proton in paraffin is o f the order 3000 cm 1
0 E. F e rm i , and coll., Proc. Roy. Soc. 149, 522 (1935).
Prinled in Denmark.
Hianeo Lunos Hogtrvkkeri A/S.

2) E. F e r m i , Ric. scient. VII. 11. 13 (1936). See also H. A. Re t h e , Hev.
of Mod. Phys., 9, No. 2 1937.
1*

4

On the Scattering of Thermal Neutrons by Round Protons.

Nr. 1. N iel s A k i . k y :

corresponding to an energy of 0.37 v o lt1) it will be correct
to treat the protons as free for neutron energies down to

a

crease11 until the energy is small compared with the energy
of the excited state of the deuleron. In this second stage
the cross-section will he independent of the energy and it is

Classically the total cross-section for the scattering should
he the same above and below one volt, as the cross-section
is classically always the geometrical area of the proton. In
a quantum treatment, however, the binding of the proton
has a large influence, as first pointed out by F erm i 21, who
showed that one may use the Born approximation in cal­
culating cross sections for the slow neutrons. In this appro­
ximation the cross-section is proportional to the square of
the reduced mass1’1, and as this is equal to the neutron

found2) to he about 13 X 10 24 cm2 corresponding to a mean
free path o f 1 cm for neutron energies from about 10 000
volts down to resonance energies of the order of some volts,
in the third stage when the energy gets below one volt the
chemical binding becomes noticeable and the cross-section

X

10 24 cm2 for thermal energies2), so

that the mean free path decreases to about 0.3 cm.
For the two first stages F ermi has obtained the energy
distribution of the neutrons2) which in the second stage,

the proton is free, it is seen that the cross-section in the

where the mean free path is a constant, turns out to be
dE
proportional to
. In the third stage, neutron energies
h
below one volt, the problem of the energy distribution has

first extreme case will he four times as large as in the second

neither as yet been solved theoretically, nor is it known

extreme case. For intermediate cases this chemical factor,

accurately from experiments.4)

mass when the proton is hound strongly compared with the
neutron energy but equal to half the neutron mass when

as it is called, will lie between one and four. F ermi found

For this last problem and for further problems connected

by his model for the binding the value 3.3 in the case of

with the slowing-down process, such as temperature effects,

the C-neutrons.

it is of interest to determine theoretically the effect of the

Because o f this quantum effect we have therefore diffe­

chemical binding on the scattering cross-sections. Recently

rent stages in the slowing-down process. In the first stage,

attempts have been made to connect such calculations with

fast neutrons with energies of the order some million volts,

a still more extended range of problems: it has been pro­

the cross-section is experimentally found to be of the order

posed5) to adopt for the cross-section of free protons — which

1 — 2 x 10 24cm2 4) corresponding to a mean free path in

is of considerable importance for the determination of the

paraffin of about 5 cm. Owing to the collisions the energy
will soon decrease and the cross-section will therefore in1} ^ W>volt
2)

2) M. G oi .dharkr and G. H. B riggs , Proc. Roy. Soe. l(>2, 127 (1937)

he
1.59-10“

4) Cf. e. g. H. A. B kthk and R. F. B a c hk r , Rev. of Mod. Pliys., 8, No. 2
(1936) eq. (62).

12 ^ r a ' 1 =

1.233-10-4 (? )ciu- - i .

Selsk. Skr. Mat.-fys. Med. XV, No. 10 (1938).

Loc. cit.

3) Loc. cit.

2) cf. eq. (1) p. 12.
4) J. C h a d w i c k , Proc. Roy. Soc. 142, 1 (1933) and J.
coll., Phys. Rev. 48, 265 0935).

and O. R. F risch , H. v . H aliian jnn. and J. K o c h , Kgl. Danske Vidensk.

R.

D u n n in g and

4) cf. later p. 9.
•’) B kt h k , loc. cit.

7

Nr. 1. N iels A h l e y :

On the Scattering of Thermal Neutrons by Bound Protons.

neutron- and radiation width o f excited nuclear levels1) as

the data needed are accurately known. Simpler molecules,

well as for the theory o f the deuteron and the discussion

like water for instance, have on the other hand so far only

of the relation between proton-proton and proton-neutron

been used in the liquid state, and in this the interaction

forces2) — instead of the direct experimental value which

between the molecules which is o f considerable importance

is not very accurate, the quotient o f the thermal cross-sec­

for our problem cannot easily be treated quantitatively. W e

tion and a calculated chemical factor. It would, however,

shall therefore in the present paper only discuss a very

be much preferable for the above purposes to have a more

schematic model for the binding.

6

I. Instead of the normal vibrations we assume each pro­

exact experimental determination o f the free proton crosssection as it is only possible to base such calculations on

ton to oscillate independently in a harmonic potential, which

very rough models for the binding o f the protons in pa­

we shall assume to be anisotropic, since it can be deduced

raffin and similar hydrogeneous substances. In spite o f this

from molecular spectra that the protons oscillate with lar­

(act it is, as we have seen, of interest to get some rough

ger frequencies in the direction o f the valency-bond than

ideas about the influence of the binding, and we shall in

in the perpendicular directions. For the frequencies we

this paper treat the problem by help of a model for the

shall take v, = 3000 cm-1 = 0.37 volts, vx = vy = y vz with

binding which we shall discuss in § 1.

y

= 0.4 so that v_ = v„ = 1200 cm-1 = 0.148 volts.
II. As we have already mentioned the binding has no

influence classically on the scattering. This is also true if

§ 1. Discussion of a simplified model for the binding of

we do not consider the motion as a whole but only the

the protons.

separate degrees o f freedom. Now we know that the nuclear

1 he scattering cross-section and the energy loss can be

motions in the molecules have also in addition to the larger

calculated exactly if the proper function for the nuclear mo­

frequencies which we have accounted for by the assump­

tion in the molecules concerned is known. Theoretically it

tion I, a spectrum extending to quite small frequencies.

is possible from an analysis o f the molecular spectra to

These small frequencies we w ill take into consideration by

obtain the frequencies of the vibrations and the normal

assuming that the protons and their potentials can move

coordinates which determine the form o f the different normal

freely like gas molecules with a

vibrations. For the more complicated molecules, however,

tion, so that we^ubstitute for the energy exchange between

such as paraffin which is mostly used for the purpose of

the neutrons and the small frequencies the exchange of

slowing down the neutrons, the resulting expressions would

kinetic energy between the neutrons and these “ molecules” .

indeed be very complicated and unmanageable, quite apart

So long as the neutron energy can be considered large com ­

from the fact that for these complicated molecules not all

pared with the energies corresponding to these frequencies

M axw ell

velocity distribu­

we can namely, as we have just seen, consider these sepa­
H. A. B e t hk and G. P l ac z e k , Phys. Rev. 51, 450 (1937).
2* G. B r e i t and J. R. St e h n , Phys. Rev. 52, 396 (1937).

rate degrees o f freedom as unbound, only the fact that

8

9

Nr. 1. N ihi.s A hi.e y :

On the Scattering of Thermal Neutrons by Bound Brotons.

they are connected with the other degrees of freedom with

elusions regarding the influence o f the binding we can draw

the large frequencies must be accounted for. This we do

from the model, and next we shall use the results to esti­

by ascribing an effective mass to the “ molecules” consisting

mate the effect o f temperature variation on the mean free path.

o f proton and potential, and for this effective mass we choose

In order to obtain definite results regarding the last

the value 14 times the neutron mass, which is the mass of a

problem it is necessary to know the energy ranges of the

C//2 group. This figure is rather arbitrary and corresponds

neutrons with which we are dealing. We shall assume these

to the conception that the energy taken up in the neutron

to be the so-called C-neutrons, that is the neutrons which

collision by a proton is transferred to a single carbon atom

are strongly absorbed in cadmium. The range of strong

in the hydrocarbon chain rather than to several of them.1!

absorption in Cd extends from 0 to about 0.3 volts.11 Further

Our two assumptions are o f course very arbitrary and

we must know the energy distribution of the C-neutrons.

certainly not fulfilled in nature. No account is taken of

This is not exactly known; its theoretical determination is

interference effects, and apart from this it is known, for in­

just one of the aims of the theoretical study of the slowing-

stance, that the frequency of the C-C vibrations in iEthan

down process with which we are dealing in the present paper.

(C 2Hfi) and other heavy carbon molecules is of the order

T w o methods of investigation have been used to determine

o f 1000 cm-1, which is about five times the energy of

the energy distribution of the C-neutrons experimentally.

thermal neutrons at room temperature21, so that these v i­

First the method of the mechanical velocity selector21. By

brations cannot at all be considered small. The model de­

this method it is found that at room temperature the energy

scribed is on the other hand the next simplest after that

distribution has a maximum for an energy of the order of

chosen by F e r m i 31, the isotropic oscillator with infinite mass,

kT. Second the method of absorption in B o ro n 31. As the

and it is certainly a better approximation than his41. Taking

capture cross-section in Boron is assumed to follow the
1
v
law 41 it is possible hv absorption experiments in this ele­

now our model for granted, we shall first see which con1) It must be emphasized that this model is in no way identical with
a gas of CHo groups. Firstly, in a C H 2 group the positions of the hydro­
gen atoms depend on each other; this gives rise to important interference
effects which we do not consider in our model; secondly, the slowing-

ment to compare the mean value of * for different kinds
v
of neutrons. If for instance the C-neutrons were in thermal
equilibrium with the slowing-down medium this mean value

down process by free CHo groups would — apart from the slowing-down

and hence the Boron absorption should vary with the ab­

by clastic collisions — take place by energy transfer to the three proper

solute temperature of the medium as 7’ ~ 2. W hile between

vibrations of the group and the three rotations of the group as a whole,
while

in our case we have two times three vibrations and no rotation.

2) For T — 290° abs we have k T =

0.025 volts —

203 cm

31 Loc. cit.
b After the conclusion of our calculations a discussion of the effect
of the anharmonic binding on somewhat similar lines has been published
by B k t h e , loc. cit., where, however, the influence of the thermal motions
arc not considered (cf. the §§ 4— 6 of the present paper).

D Cf. e. g. J. G. H of fman and H. A. B e t h e , Phys. Bev. 51, 1021, (1937).

-) J. B. D u nn ing and coll. Phys. Bev. 48, 704 11935). Cf. also B et hi -:,
loc. cit.
3) For a survey of the literature cf. F iu sc h , H ai .han and K och loc. cit.
1) R. F kisch and G. P i.ac zek , Nature 137, 357 (1936). I). F. W eekks ,
M. S. L iv in gs t on and H. A. B k t h e , Phys. Bev. 41), 471 (1936).

10

On the Scattering of Thermal Neutrons by Bound Protons.

Nr. 1. N iels A r l e y :

400° and room temperature no deviation from this T ~ * law

I. The de B roglie wave-length, X,X) for the neutron

has been found the increase of the Boron absorption be­

relative to the proton must be large compared with the

tween room- and liquid air temperature, and still more

range of the neutron-proton force, q :

between liquid air and liquid hydrogen temperature, is much
less than would follow from a

X»

law. This proves that

at least for temperatures of liquid air and downwards the

II. The total cross-section, (), must be small compared
with the square o f the wave-length:

energy distribution of the C-neutrons cannot be represented by
a

M axw ell

Q«

distribution with the temperature o f the slowing-

down medium. The question how far their energy distri­
bution can be represented by a

M axw ell

distribution corre­

sponding to a higher temperature or by a mixture between

q.

X\

III. For I to be satisfied one can deduce2^ that the di­
mension o f the proton wave function, a, must be large
compared with the range of the neutron-proton force:
a»

a maxwellian and a non-maxwellian part shall not be dis­

q.

cussed here. In view of these possibilities, however, it re­

For slow neutrons and protons bound in paraffin all these

mains interesting to investigate the energy dependence of

conditions are certainly satisfied, as for such neutrons X

beam of neutrons.

is of the order of 10~ 9cm or more, Q is of the order of

W e shall therefore for the purpose of the following calcula­

4 8 x l 0 —24cm2 and we further know that q and a are

tions assume the C-neutrons to obey the

respectively o f the order o f 10

the scattering cross-section for a M

ax w ell

M axw ell

law

cm and at least 10

cm.

For the differential cross-sections per unit solid angle

throughout. A consequence o f this assumption together wilh

do, I

(0, (p), where I

(6,ip)dw is defined as the number

however, that we cannot expect a direct comparison o f the

of neutrons which are scattered, after having excited the

results o f our calculations with experiment to give a quanti­

proton from its in th into its n th state, into the solid angle

tative agreement.

dot in the direction

per unit time and per scatterer, if

there in the incident beam is one neutron crossing unit
area per unit time at the place of the scatterer, we have
now in the B orn approximation the well known expressions^

§ 2 . General theoretical remarks.

As first proved by

F e r m i^

it is possible to find a

“ rectangular hole” potential V with radius q' « X

and depth

I)', which substituted for the neutron-proton potential will
give correct cross-sections in the

P

B orn

approximation so

long as the following conditions are satisfied:

—

...
h
This is for non-relativistic energies given bv U ) cm —
1
(2 m N E N) -

2.85 X 10“ 9 EJ t ' - when E s =

m N vlel is measured in volts* vrel

being the velocity of the neutron relative to the proton.
2) For instance by F o u ri e r analyzing the wave function of the pro­
ton in respect to velocity.
3) Cf. e. g. M o t t and M assey , “ Theory of Atomic Collisions”, p. 100,
eq. (21). (The equation is erroneous, the factor

D Loc. cit. Cf. also Re t h e , loc. cit. Part B p. 123.

^

missing). It will be

seen that in this approximation I depends on ft only, not on <p.

12

Nr. 1. N ik ns A ki.k y :

On the Scattering of Thermal Neutrons by Bound Protons.

>...(«) =
2

2M
4

1,1 W u v

( i k l i r ) Y ' ( \ r s - r i>\Wn(r M

k"in n = k()0
k~

Q'n

E„,))>

H

< (k
0,> k fiui)/
v U

where ifjm and tpn are llie wave functions of the proton
before and after the collision, kQ and kmn the initial and
final wave vectors l) of the neutron, and A/v , E0 the reduced
mass and energy of the neutron.
In this expression V' only depends on the distance be­
tween the neutron and the proton, so taking r y

rise to a little confusion we shall briefly give the definitions here,
the transformation formulae being derived in Note 2. In the theory
for two-body collisions three different coordinate systems are used. O
First the system where the one particle is at rest before the col­
lision, which we shall call the rest system and denote by R 2)
(All variables denoted by capital letters). Next the system where

2 MXK0
ti1 ’

k
, 0kl
mn’

2 AL
-2 ( 7V

m( r J

the center o f gravity of the two particles is at rest both before
and after the collision, which we shall call the center of gravity
system and denote by C. (All variables denoted by small letters with
an asterisk). Finally the system which has its origin in the center
of gravity of the one particle both before and after the collision,
which we shall call the relative system and denote by r. (All vari­
ables denoted by small letters). Let the two particles have masses
ni\, m -2 and coordinate vectors/?], /?o, then the center of gravity,
Rr , is defined by

r p as a

new variable in the drN integration we can at once perform

ni\R\ -}- m^R-2 = (uh

7;,,n (0) = <1■ r

HVn

g we get

eXP (i kn,nr p)

nio) Rc

(5)

The coordinates referred to the center of gravity are next defined by

this and using that the exponential is equal to unitv by
this integration due to X »

13

r i* = Ri — Rc,

r.,* = R2— /?,

(6)

Putting ( 6) into (5) we get

,n O'/.) d ' />

( 2)

mi *
r* or
mi

r*

i’i

llH r2*,
mi

= a — 02*, 'fi* = <f-2* + 71 0 )

if we introduce polar coordinates. Finally the relative coordinates

4 ir2,h4

^ V dr

4
9 - R t W e 'r

(3)

are defined by

r, - Ri

Ri = rS — rf,

rx = 0

(8)

Equation (3 ) we can write in the following way using the

the particle with index one being taken as the particle initially
resting in the R system. Using (5) we then have, introducing the

expression for the total cross-section for scattering between

reduced mass

a neutron and a free proton2) which we shall denote by Qfl.cc

■t <l =

m x-

• 0 free.

(4)

We emphasize here that the expression ( 1) or ( 2) is calculated
in coordinates relative to the center o f gravity of the system in
which the proton is hound and as this fact sometimes gives
' ) The wave vector is just the momentum vector divided by h.
-) Cf. Note 1, eq. (N 5).

M =

mi •m2
mx+ m-2
—

M
To
mo '

or

*=

M
------ r<>
mi “

or

i\** =

M r-i,
m2

0O* =

M /■•>, 0]* =
mi

00,

if 2

1-2

0)

U -- 0o,

The following also applies to the case wher
two particles are complex, consisting of more parts. In this case the mass
is the total mass and the coordinatevector is the one of the center of gravity.
2) It should be noted that this system it not always identical with
the coordinatesystem in which we make the observations, cf § 4.

14

Nr. 1. N iels A r l e y :

On the Scattering of Thermal Neutrons by Bound Protons.

We see from (9) the important fact that the angle of the colliding
particle is the same in the C system and in the r system, due to

one for a free proton when the proton is strongly bound,

15

which means that the space in which the proper function
of the proton is different from zero is very small compared
with the wave-length of the neutron. W e can then put the
exponential equal to one, so that we get quite independent
of the form o f the proper function of the proton

L n W

=

< r - jr - dmn =

h0

<?*<*„,„

( 10)

which means that only elastic scattering can occur and
that this is spherical symmetric in the relative system just
as is the case for scattering by a free proton.1* In the rest
system, however, we w ill no longer get the cos 0 law 2* due
to the mass o f the scatterer being now larger than the neu­
tron mass. In

F ig .

1 we have plotted in units o f q the curve

for ( 10) transformed to the rest system3* for the mass of the
scatterer, ms, equal to 14
For the total elastic cross-sections we get from (10)

( „ * ; ) «/ve.

(m

using (4). For the case MN = mN i. e. ms = oc the factor
o f Qfree in (11) reduces to the factor 4 first obtained by
Fm. 1. A ngular distribution o f scattered neutrons in the rest system cor­

responding to isotropic distribution in the center o f gravity system, fo r
m s = 1 4 mN.

which circumstance the formula (1) is often said to be derived
in the C system in spite of the fact that it is really derived in
the /■ system.
From the formulae (2 ) and (4 ) we can at once deduce
that, as was already mentioned in the introduction, the
total cross-section w ill be nearly four times as large as the

F e r m i . 4*

W e have in this work taken ms = 14 mN throughout

so that
(M N\Z
4 — - \mN!

( 14\2
4— ] V15/

4-0.871 = 8.48

which makes a considerable difference.
o Of. Note 1.
2) Of. Note 2 ecj. (N 19).
:5> Of. Note 2 eq. ( N 18).
4> loc. cit.

(12)

16

On the Scattering of Thermal Neutrons by Hound Protons.

Nr. 1. N iels A heey :

§ 3. The anisotropic oscillator.

V,
h «)„

Vo
h (oz

W

We now in ( 2) put the wave functions for our ani­

e_
m
1+

m

F,

m

NVl

17

( 15)

'

sotropic oscillator and as these are products o f three wave
functions for a one-dimensional harmonic oscillator, the

where VN is the velocity of the neutrons in the rest system,

matrix element will be a product of the matrix elements

and then we can write (14) in the following form, due to

o f the type given by eq. (N 24) in Note 3. Using the for­
mulae (N 32) and (N 35) in Note 3 we have at once for
the 0

9

Kmat = 4 w

sin2 2

(by 0 3 ))

0 transition, which is the only one we shall treat

here

exp (( —
ho — (/•exp

a„ —

Kox + ^no(/)
y (/V

(\MJp (»_ !)’

' 9 •)
no-/t- —

9

<1• / .

^nnt j

•
4 \\T
W sirr
t.

*4

Ju

; 1

, ( L— t V /22 h'1-sin 0

/

~~ \ exp ( t2) dt

2
( y < 1). (16)

*«•

(13)
For y = l we get the cross-section for the isotropic oscillator1’

1
' =

i/ ’
iu i>

E*

loo = 7 ' exp I — 4 W sin ^

(17)

Mp being the reduced mass o f the proton, o>„ = 2 tcv,, v„
in the direction o f the

In F in. 2 we have plotted in units o f q the curve (1 6 )2)

--axis and 6 the angle between k0 and k00 i. e. the scat­

transformed to the rest s y s t e m fo r two different values of

the frequency o f the oscillation

IT, IT = 0.0697 (full line) and IT = 0.0156 (dotted line)

tering angle o f the neutron.
Further we must take the mean value of (13) over all

which correspond to /uty = 0.37 volts, y = 0.4, ms = 14 mK

directions o f the oscillator. This we do by taking the axes

and E w equal to the effective energy of neutrons at room

of the oscillator as coordinate system and averaging over all

respectively at liquid air temperature, i. e. 90° abs.4) It is

directions o f k'0'0 in respect to this system, the length o f k'0'0

seen that even at liquid air temperature there is still a

being kept constant. In

considerable deviation from the spherical symmetry which

this way

we get,

denoting the

mean value by /00|osc2’

is always assumed in calculations about the diffusion of
thermal neutrons.5’

/0 0 „ s c = 9 ' e X P

O Cf. F e r m i , loc. cit., and Note 3 eq. (N 34).
2) The function ^ e x p it2) dt is tabulated in Ja h n k e - E mdk “ Tables of

W e introduce as new variable the dimensionless quantity
0 Cf. eq. (N 36).
2) A mean value we shall in this paper always denote by this symbol.

Functions”, p. 106.
•f Cf. Note 2 eq. (N 18)
b Cf. § 6 p. 38.
Cf. F erm i and He t h k , loc. cit.
VidensU.Selsk. Math. fvs. Metkl. XVI.1.

2

18

The curves in F ig . 2 can also be represented by the
function
oo

—

2

r* 4_I ^ 7
r00’
'*00

/

19

On the Scattering of Thermal Neutrons by Bound Protons.

Nr. 1. N iels A l l e y :

00, ‘ w ],

(18)

. o q
4 IF sin2
2
r
1
• 9 8'
4 W sin2
2
y
1

' g-exp
•is’'

/on = q •exp

! + . ' [ ] + 110 n 2+ ^ n ;' + " ( 11>)

i + J n + ; . [ ? + i 18 [ r ,+ - "

[] = ( * — lj-tV V 'sin 2 *
so that the two curves have the same starting point and
starting tangent and the difference comes first in the second
power o f W.
From (14) we can now by integrating over 8 and <y
get the mean value o f the total cross-section. The result is
^00 |ose —

(#11— exp

4 U ’ ;/ i

(i

y )^
(it. ( 20)

sin 8 d8 = n q •
w - Li — ( i — ;0 h
r

« - 'o

For y = 1 we get the well-known formula for the iso­
tropic oscillator ^
is

Q00 = n q ’

1 — exp (— 4 W )
W

( 21 )

In F ig . 3 we have in the full curve plotted (20) in units of
()f

2) for y = 0.4 and ms = 1 4 mN. Also we have in the

same figure in the dotted curve plotted the curve analog­
ous to (18),
F i g . 2. A ngular distribution in the rest system o f neutrons scattered by

anisotropic oscillator. F u ll line corresponds to W =
to W =

0.0697, dotted line

where /qS0 is given by (17) and I q0 stands for the same
function with yoo substituted for w, which we can write
as in (18) with

<>£ = 3 Oor, + fo iT ..

Ooo = Ooo ( ’ w ) ,

( 22)

0.0156, W given by (15).

W given by (15). The reason why the

As we know from (19) the two curves have the same starting
point and starting tangent. This can also be seen by direct
expanding in powers of W

curves (16) and (18) are so like is easily seen analytically

0 Cf. F eiimi , toe. cit. Cf. also Note 4.

by expanding in powers of IF. W e then get

2) Cf. eqs. (4) and (12).

2*

20

Nr. 1. N iels A kley

21

On the Scattering of Thermal Neutrons by Bound Protons.

(h

1 <1

15
4W
,f q • 4 \

Qis" -

1

v 00

15
(2 3 )

1\"

:, c

( n + 1)

i)

n l 1
3 ''

«>/

,r q •4
Since y < 1

(1 — ;0 /2< 1 (due to y — 0.4 < 1) in the inte­

gration range of t in ( 20) we can for large IV neglect the
exponential and we find then after elementary integration
I

Qv

O
0

r » i ) (24)

is"

'■ 00

For y — 0.4 the two coefficients are respectively 0.531 and
0.6. That I q0 and (/0S0 are very nearly equal to /^josc and
QooLsc

a^so physically plausible. l'0s0 and

we can namely

interprete as the average cross-sections for scattering in a
substance consisting to one third of oscillators with energy
hoK and to two thirds o f oscillators with energy yhoo^,
while we by /0()|oscand Q00|os(, are averaging over all directions
of one oscillator with one degree of freedom oscillating with
an energy hw, and two degrees of freedom oscillating with
an energy yhw„, so that one would think that the two kinds
of averaging would give nearly the same result, which is
V This series is, as is easily verified, identical with B f.t h e
Part B eq (4(>3), if we put ;/i s =
by B e t h e .

loc. cit.

oo , as then our ;t </ —r oq, IP > t-\, IT

-

fo

23

Nr. 1. N iei .s A iiley :

On the Scattering of Thermal Neutrons by Bound Protons.

in fact found to be the case as we have just seen. Due to

only measure the velocity o f the neutrons relative to our

the expression ( 20) being far more complicated than the

observing system, VN, and not the one relative to the scatterer,

expression ( 22), we shall in the following use Q'0S0 instead

©rel, and so we must define an experimental cross-section

of QooLe’ tbe error being negligible especially as we shall

Qexp by the equation

22

only be interested in that part of ( 20) which belongs to

p

=

(25)

qi>n Qexp

small values o f W.
so that the experimental cross-section is given in terms of
§ 4. Influence o f the temperature motion of the

the usual one bv
(26)

scattering centers.

iY

W e must now take the second feature o f our binding
model into consideration. At the same time we shall define

Now we can take our second assumption about the

a new scattering cross-section which can be directly measured.

binding model into consideration, the velocity Vs o f the

The cross-section is as a rule determined experimentally

scatterer

by measuring the absorption in varying thicknesses of

to some probability law, F ( v s) , the probability for finding

paraffin. 0 If now the scatterer does not rest but moves

the scatterer with a velocity between Vs and Vs -{-dV being

not

being constant,

but distributed

according

with a velocity Vs relative to the coordinate system in which

just equal to T (®s) dVs. So on the average we shall find

we are measuring, it is clear that another number per unit

the scattering probability, which we shall denote by P|s,

time o f neutrons w ill be turned out o f the beam and so

equal to

we shall find another absorption coefficient. This number

= j! P E (o s) rf»„.

of neutrons expelled from the beam we can easily get by

( F ( v s) ( l v s

using the fact that the total cross-section is the same in
all G a l i l e i

systems2^ and so the total number scattered per

unit time and per scatterer or the probability for a scatter­

and so the average experimental cross-section, ()

|s, will

be given by

ing process is just
\ ° re] Q F ( v s) d V ,
Q cxpl.S

where q is the density of the neutron beam, i. e. number

(27)
\ F (V .)d V s

per unit volume, ©rel the velocity of the neutrons relative
to the scatterer and Q the total cross-section calculated in
the relative system.

M axw ell

distribution

In an experiment, however, we can

Cf. e. g. E. A ma ldi and E. F e r m i , Phys. Rev. 50, 899 (1936).
2) Cf. Note 2.

For F ( v s) dVs we have assumed the

F ( V S) dVs = (?^) exp (— /*!>“,) dvs,

(t =

(*28)

24

On the Scattering of Thermal Neutrons by Bound Protons.

Nr. 1. N ikls A r l e y :

25

where ms is the mass of the scatterer (which we in this
paper have chosen to be equal to 14 times the neutron mass),
k is the B oltzm ann constant1), r s is the absolute temper­

ature, and the constant is chosen so that \ F ( v s)d V s = 1.
For Q we ought to take the expression (20), but as we
are not interested in temperatures much higher than room
temperature, the main part of the integral in (27) will
come from that part of Q which belongs to values o f the
energy not much higher than -^kT which means that our
variable W

will be o f the order 0.1 due to the value of h<»_

having been chosen equal to 0.37 volts. For small values
of W, however, we have seen that (20) can be approxim­
ated by ()!0S0 defined in ( 22), so that we can safely put
Q1
q0 instead o f the Q from (20) into (27). W e have there­

j

fore first to put (21) into (27) and we get then using (15)

ijj!|j|jlra

and (28)
rl

is

Qexpls

e * P ( - V | ® s-® v| 2)

X

®.vp
, _

X exp (— ii up dV s ,

““

i

-r .

(29)

'V

2 ho)

Taking Vs — VN = V as new variable and choosing a polar
coordinate system with VN as polar axis the integration
can be worked out and we get3)
US
<✓4c,J,=
-T

X O iji

uv) — exp

4 it

r '4.

ji

'

lliiiiliiiiji

l/\.2 X
A.8
p

4 //+ ii

H v,

(30)
(4 fi/ “T /*) /.

where f/>(.r) is the Gauss error function defined in eq. (N 49).
O k =

1.571 X 10 16 erg grad 1 — 8.623 X 10 r’ volts grad F

room temperature, T — 290° abs, we have k T =
2> Cf. eq. (15).
2) Cf. Note 5.

0.0250 volts.

For

.02

W
F ig . 4.

. 0*

.06

-OS

.12 .It- .16 .18

.22

.2*

.26

.18

Energy dependence o f scattering cross-section fin units o f 0 fr(,(J

fo r scattering centers at room temperature, i. e. 290° abs ( fu ll line). D ol­
led line corresponds to resting scattering centers.

26

Nr. 1. N iki .s A rl k y :

On the Scattering of Thermal Neutrons by Bound Protons.

W e introduce now the new dimensionless variables W ])

(32). For x greater than 2.5 <D(x) = 1 and so we get, due

and s defined by

to s «
1
2
2 m N VN

f

W - /#■' v%
1+
/
/1

k Ts

t<

h (o
m
' 1+
mv

£
€

1

is

Qexp S

m Kr
m.

1(31)

27

7r q - W

H 1 - e x p ( — 4 W )) = ()'\

> 2.5

s

and so under the same condition

Mn
(cf. eq. (N 36);
Mp

Qis’>

Qls
^ exp

This result we also get if we take the temperature of
Putting (31) into (30) we tinally find, due to

^W

the scatterer Ts = 0° which means that the scatterer is
resting, and we should therefore as cross-section find just
ioft

v ex

7Tq • W

Q'

X

1

For neutrons of room temperature F kin = k T — 0.025

w

(32)

volts we have W = 0.063 and we see from the curve that
is'f

®ll>)

X

which is in fact the case.

the corresponding Q
£ kto =

For ( ) ‘s i we find the same formula only with
j

r

J

exPls

y

«, is equal to 2.76 •P Jree. If we take

we get W = 0.095 and

= 2.46 •0 fr(.e.<>

W and

s substituted for W and s. In F ig . 4 we have in the full
§ 5. M axwell distribution of the incident neutrons.

curve plotted
is"

Qexp S

-

xp S

(33)

J-J v exp

in units of ()free2) os a function of W for ms = 14 mN ,
hwz = 0.37 volts, y = 0.4 and Y’s = 290° abs which makes
s — 222’

From the formulae (32) and (33) we can already draw
conclusions about the temperature effects. In order to be
able, however, to compare the results with experiments,
we must take into consideration that the beams of thermal
neutrons which can be produced in praxis, e. g. by slowing

we ^aye ploded the curve for Qls 3) and it

down fast neutrons in paraffin, are never homogeneous

is seen that for W > 0.1 the two curves are identical. The

but have some energy distribution. As discussed in § 1

reason for this can easily be seen analytically from eq.

this is not known quite exactly, but we shall here ap­

The \V here is formally equal to W in (15) only the E y there is
now the kinetic energy in the observation system
system as in (15). Only for T s =
two systems are identical.
2) Cf. eqs. (4) and (12).
3> Cf. eq. (22).

and not in the

rest

0°, i. e. resting scattering centers, these

proximate it by the
M axw ell

P As will
temperature,

M axw ell

distribution. If e ( E ) is the

distribution for the current, that means that the
be seen later, the
cf.

§ 6 especially

2.69•Qfree (cf. also T a b u -:

effective energy

p. 38.

1, Ts =

The correct

7\. = 290°).

is 1.103 k T at room
value

is

therefore

On the Scattering of Thermal Neutrons by Bound Protons.

29

Nr. 1. N iels A h l k y :

28

probability for the neutron which hits the scatterer having
an energy between E and E + dE is e ( E ) dE, then the

where a is some constant characteristic for the detector
used. So we get that

cross-section which would be measured should just be the

e ( E ) 1( E ) dE -

« F (E ) dE,

j[ F ( E ) dE = 1

average value of ()exp|s
and as the factor a drops out in (34), what we have to
calculate is in fact only the mean value in regard to the
M axw ell

e (E ) dE

distribution for the density

In praxis, however, this is not the value measured due to

Q exp.s x

+

.....

the fact that the Boron detector which is mostly used to
measure the intensity o f the neutron beam is not equally

For ()exp|s we have now to put Qe L given by (32)

sensible for all neutron energies, but absorbs according to

and (33), and we must therefore first calculate Q^x])|s . If

the ^ law. If we then by 1 (E ) denote the sensibility of the

we define a new dimensionless quantity, n, by

v

detector, that means the fraction of the neutrons hitting

*Tk

the detector which it records, then what is really measured

1) a)

Mr

mv
:

is obviously the following average value of the cross-section
Oexjs which we shall denote by ()exp|Jv

f

(cf. eq. (N 36))

(36)

1+

we can write
F ( E ) d E = G ( W ) d\V = 2 n v*/i

M T’ e x p j -

W ) dW

jjO a ^ c ( E ) I ( E ) d E
0'•exp

(34)

s

and putting this and (32) into (3o), we get

\ e (E )I(E )< iE

The

M axw ell

= /r q •2 n ~ ' 2

0 exp

and this we shall now calculate.
distribution for the current, e (E ), is pro­

portional to vx F ( v N) d V s , F ( v n) dVN being given in (28)

vi

—i

vv;

(1 + 4 s)

X

4W
exp( - i + 4 + ©

if we substitute N for S, or transformed from velocity to
X

energy, proportional to
E v ' 2 it

Due to the

n

"(A-7V)'

E^exp

l r J dE>

law we have further that

i

( e ) = uf:

W

exp ( —

W )d W

Both integrals are here of the same type
QC

^ VVr~' exp ( — re2 W ) © (fi w ' ") dW = 2 tT ' :s+ ' Arctg
which formula is proved in Note 6.

x

(

30

Nr. 1.

Niels

A isley

On the Scattering of Thermal Neutrons by Bound Protons.

Putting in the correct values for

mental value for the free proton cross-section measured

an elementary calculation

with resonance neutrons is, however, very inaccurate, as

o
v eisxp|!.s N = ^ q *4 >t

, .i
Arctg

(1 — 4 (n + s))

ms

n
s

*

fc —

Ts ms

n

1
' Arctg ^
l + 4(n + s)

kTs

£
Ti m
m '
‘ 1F— ■
ms

31

n
Sj

kTx
U

Ti m

(38)

ms

*

ms

ml
Ms
2 (cf. eq. (N 36) (
Mp ~ ms m%

For Ts = 0° we have foundl) that (?eXp|<j = ()'s so that
we can obtain () is|a: by putting s — 0 in (38)
lira

Qexp

rrq-2n

H i — (1 + 4 ri)

k)

due to Arctg oc = ^ .
For Qexp|s^we 8et

sanie f ° rmula with

y
substituted for s and n respectively, and so finally
q ;exp

In

F ig . 5

V-

^ Qis
''■exp v + o Q.ex p;

A

and ^ n
y

(39)

JN2 0' e x p ;

we have plotted the curve (39) in units of

( ) tree2^ as a function o f T n for various values o f 7’s with
nis — 14 mN, Ti m _ = 0.37 volts, and y — 0.4. The values
are also given in

T able

1.

W e see that for Ts

=

TN

=

290°

the cross-section is 2.7 times larger than the free crosssection.

A m aldi

and

F e r m i :{)

find experimentally for the

ratio of the two cross-sections the value 3.7. The experi1) Cf. p. 27.
2) Cf. eqs. (4) and (12).
•b loc. cit.

W e see further that for liquid air temperature the crosssection is 34 °/« higher than for room temperature, the scat-

32

33

Nr. 1. Nines A ri .f.y :

On the Scattering of Thermal Neutrons by Bound Protons.

T able 1. The total elastic scattering cross-section in units

calculation o f ()^.JS, the only new formula needed being

of Qfrvr as a function o f Ts and 7’v given by eq. (,W) with

given in Note 7,

ms = U ms i. e. iq = 0.871-Qfn,e (cf. p. lb).
20°

Ts

90°

1

0 ° .......
2 0 °.......
90°
29 0 °.......

290°

s

^ cons

1
Tx

v‘

7c7’ volts

s " V ■2 W

rU

J\ /

K|+

2 W\ , 2 /wx'11
* ) + , " ( . )

{

\v\]
(40 )
s )\
exp(

1

3.39 ; 3.11 1 2.58
3.59 s 3.17 i 2.58
4.26 3.30 2.61
5.49 3.61 2.69

0
0.00172
0.00776
0.0250

liquid hydrogen temperature
liquid air
—
room
—

terer being kept at room temperature. The agreement with
the experimental value of 26 % found by

F i n k 1*

where W and s are given by eqs. (31). It is seen that
W
(40) is only a function o f
as it must be, since fiw. does
not enter into the problem considered here. Also it is seen
/w \ !that, due to <P(oc) = 1, we get for s = 0 or lor (
) »
1
that, as is physically obvious,

is even better

than can he expected in view o f the rough assumptions of

o.

Qexp

our model2*. The values for liquid hydrogen temperature
( 20° abs) are only given for the sake of illustration, as for

Putting further (40) into (35) we get, after elementary

temperatures as low as these our model loses every justi­

calculations, the only new integral needed being given in

fication. In this case, infinite effective mass would be the

Note 8,

more appropriate approximation.
In order to see how much of the variation in our cur­

^const
Wexp S

0-2 / V

(ns) " + (/i + s) Arctg

(41)

ves comes from the special form of the cross-section of
the anisotropic oscillator and how much from the motion

n being given by (36). Also here Hm , drops out, as it must,

of the scattering centers |the factor —-1 in (26)j we have

since (41) only depends on

Further we get for 7’s — 0°

to compare the curves with the curve for Ts = 0, as the
lim

latter contains only the first influence. W e see that the

const

Q exp SK

Q

difference is negligible for room temperature but gets im ­
portant for liquid air temperature. Another way of studying
the influence of the motion of the scattering centers con­
sists in calculating ()exp|<Jv for Q equal to a constant. Put­
ting this into (27) we find, proceeding exactly as in the

const, and const
as we must get, since lim ()c“
s-> 0 exp |S
const. In F ig . 6 we have plotted the curves (41) as func­
tion of the two temperatures for ms = 14/nv and Q being
taken equal to

1) G. A. F i n k , Pins. Kev. 50. 738 (1936).
by F risch , H alban and K o c h , loc. cit.

3) Cf. § 1.

A similar value was

found

f ° r Ts =

anc*

= 290°, that is

Q = 2.58-Ofrcj,1^ so that we can directly compare these
1) Cf. T abi.k 1.
Vidensk. Selsk. Math.-IYs. M edil. XVI. 1.

3

34

N r. 1. N 1k i.s A k lky :

On the Scattering of Thermal Neutrons by Bound Protons.

curves with the curves in Fid. 5. It is seen that the general

35

Ibis we mean that energy, E, which a homogeneous beam

character of the curves is the same, coming from the com­

of neutrons must have in order that the scattering cross-

mon integrations, but that the curves in F ig . 5 have another

section shall be equal to that of a

M axw ell

beam of tem­

perature Ts . For E we therefore have the equation
(42)

OeJs ( E ) = Qexp S

Now our expressions for the ()’s are not given directly as
functions of the energies but o f the variables W, n and s.
W e have, however,1)
IF = a E ,

a

1
li <o_

ms

= 2.52 volts

n = u E v,

x = ft E s

---- — 0.180 volts
n (o
m
' 1+ ~ s
ms

l , ft

ms
for
hw_ = 0.37 volts

and

ms --= 14/ny

where E is the energy o f the homogeneous neutron beam
in the coordinate system of the observer and E s , E s are
equal to kTN and kTs respectively. So we can solve the
equation (42) in terms of W, n and s. This can, however,
only he done analytically in a few special cases.
Fit., (i. Same as F i g .

fo r a scattering cross-section which does not depend

on neutron energy in the center o f gravity system.

asymptote coming from the special function chosen for Q
in (27).

I. case

n
s

1.

As the highest temperature we are interested in is less
than say 1000° = 0.0862 volts, we see that both n and s

§ 6. Effective neutron energy.
W ith the help o f the curves in F ig . 4 and F ig . a we can
now treat the problem of the effective neutron energy. By

are small compared to unity and so we can in this case
expand everything in (38) with the result that
B Cf. eqs. Oil) and (3(i).

:s*

(43)

On the Scattering ol Thermal Neutrons hv Bound Protons.

Nr. 1. N iei .s A l l e y :

36

16

n ,s"
W‘\p s

(44)

7T

37

x being a numerical constant depending on 7\., so that
the curve is seen to decrease at the beginning when k'T-^
is increased.

From this formula we see that Q is large for

<< 1
II. case:

and from the curve for (/tfxp|s we can conclude that the

In

1.

corresponding value of ( --j is also small. By expanding

This we cannot fulfil for all values of ,v if we still want

in eq. (32) we then lind

both ,s and n to be small compared to unity. If, however,
all three conditions are fulfilled, we can put Q and Arctg

is'r

(45)

Q exp S

equal to 1 and ^ respectively and we get then from (32)
and (38)

Putting (44) and (45) into (42) we readily find that the

C „.s ”

- T < / - 4 (i- 2 ir ),

.Tty-4(1 — 3 /i). (48)

q exp

effective energy is given by

^ 01 Oexp we Ket the same expressions with ^ W and ^ n
E = ^ kTs

(for Q

«

1, /1 «

1, s «

1j

(46)

substituted for W and n respectively. From (42) we then
easily obtain

independent of the temperature of the scattering centers.
This value is also the effective energy of a

M axw ell

beam

E =

9 kTS

for

>:

1, n «

1, .s < < 1

(49)

in regard to absorption in Boron ^ (while we define the
effective energy in regard to scattering) because the crosssections in both cases vary as

v

This is the classical relation that the mean energy of a
M axw ell

(cf. eq. (4 5 )).

If we take also higher powers in the expansions into

beam is equal to

also expect to turn out under the conditions stated above.

consideration, we are able to get information about the
starting tangent of the curve

~E
k1v

= f ( k T s) . The calcula-

tions are however lengthy and we shall therefore only

k rI\ which result we would

n

III. case

- > 1, n

1.

In this extreme case we would find independently of s

give the result found, namely that for small values of E
rr— t

('•re \,p LS, = 7T(j‘
W
i

(47)
1) Cf. e. g. H. H. G o l d s m it h and F. H a s k t i , Phys. Hev. 50, 328 (1936).
Cf. also B e t h e , loc. cit. Part B p. 136.

\’

f .is

q
'- e x p

.r q •l n

(50)

and so from (42)
E = I kTs

(for ( ” ) »

1, n »

1

(51)

On the Scattering of Thermal Neutrons by Hound Protons.

Nr. 1. N iels A u l e y :

38

This case is, however, not of much physical importance,
as we cannot neglect the inelastic scattering lor energies
which make n »

1.

39

Note 1.
For our potential V ' we have assumed 7. > > « ' O, so we know
that all the phases will be negligible except the first one, this
being given b y - ’
arctg ( ^ tg k ' o ' )

( Nl )

kn o .

Further

1
(exp (2 /i,0) — l )
2 i A’o
since

(N 2)

< < 1. As in our ease k ' g' and A*0 g' are both smalt, we

can expand the tg and arctg in (N 1) with the result

v> -

1
3 kng'Hk'Z

-A*n2) -

1
3 A',,

m x D'

.

(N 3)

Putting (X 3) into (X 2) we find

/=

/i4

ur

(X 4)

which shows that I is independent of both the angle and the
velocity of the neutrons so that the scattering is spherically
symmetric. 0
Since Q = (/dm we finally get for the total cross-section for
scattering of slow neutrons by free protons, that this is a con­
stant given by

4a
9

( X 5)

( D' g' zj -

Note 2.
F ig . 7. Effective neutron energy as function o f neutron temperature.

As the transformation formulae between different coordinate
systems are often used but seldom given in full, we shall here

In Fin. 7 we give the curve for

(

k 1x

as a function of

k'l\- for V' — 290° found numerically from the curves in

compile them for reference. Firstly let us consider two coordinate
systems K and K* so that K* has axes parallel to the axes o f K and
further moves along the positive .r-axis of K with constant velo-

F igs . 4 and 5. This “ pipe” like curve we have already used

>) Cf. ]). 10.

in § 2 !) to obtain that

2) Cf. e. g. M o t t and M a ss e y , “ Theory of atomic collisions” eq. (30/,

the effective

neutron energy

at

room and at liquid air temperature is equal to 1.103 kTs
and 0.795 kTx respectively.
D Cf. p. 17.

/

p. 21). (T he mass there is equal to the reduced mass,

mv\
' j.

•!> M ot t and M asse y , loe. cit. eq. (17), p. 24.
b It should he remarked that this means that in the rest system the
differential cross-section is proportional to cos <■) cf. Note 2, eq. (N 19).

Nr. 1. Xiia.s A iu.k y :

On the Scattering of Thermal Neutrons by Hound Protons.

city //. A particle is moving with velocity v* in the system K*
forming an angle d* with the ,r*-axis. In the non-relativistic ease
which we are here considering, u < < e, we have then that in the
K system the particle moves with velocity v — u-+-v*, the angle
d between v and the .r-axis being determined by

during the collision. This energj' can be positive or negative; for
m — 0 it is positive for all //. Using (7) and v * = — u, v2* = V)— u,
we get from (X 8) due to (N 7)

40

u
cos d*-ja
V*
=
, COS 0 =
u V
/. , n'1 , n //
1-f-j- 2 cos d
cos d*-jV
\
V*p*
sin d*

ty d

sin d =

b
V

(NO)

t)ni i ( I'-n— Rm )

mi
/?ii -f- /n-2

17,2

41

(X 9)

- nu ( n\\ + nu)

(N 6) is now' fully determined by (N 7) and (N 9), but only in the
case of elastic scattering, E n— Em — 0, we get simple analytic ex­
pressions. In this case wTe get, using (7)

sin d*
. \17
//- , .. n
= ( ‘” + v*.,- + 2 p * cos d* !j

• To,
mi + m2 '

/V* = 77,2n./* = //
mi "

(X 10)
v

so that ( X (1) becomes, independently of Vo, using (9)
which formulae are at once deduced from fid . 8.

, mo

tg <-)■>=

----------,
, m -2
cos dH----

cos

cos do -4- ----- ,, ,
I . , ms2 , nu
Vr14---- , + 2
“ cos do

=

/?ii

\

1+

01 =
For the K- and //‘ -system we now take the R and C systems')
and shall obtain the transformation formulae (N 6) for this case,
when the particle with mass nu moves with constant velocity Vo
along the .r-axis before the collision. From (5) we lind, due to

nii

(N il)

Sill do

sin <9-2 =

\$*■ X , X ’

nil-

(n — do),

—2 ; - f-

//?iJ

2

2 cos do

nii

"1

C
Pi = c/2-f- a,

so 0 < ®o < n when nii 4= /?/o, but 0 < 0.2 <

7T

wdien nii =

nu be­

cause wTe then simply get

= 0
u =

V-

(X7)

Wx

and as we assume that no outer forces are acting, this velocity
is the same before and after the collision. To obtain V\'* and v-<'*
(the dashes referring to the state after the collision) and so the
transformation formulae for the two scattering angles, dj* and d2*,

1

HI1/’!*-

1
9 mo m*

( R r

EJ

1
9

nnvi'*:

1
9

//Jojo'*

Solving (X 11) for do we find
sin (do — @o) =

we only need to use the conservation law for the energy

9

which, combined with (X 11), gives the well-known relation

( X 8)

m~ sin 0o

nii

(X 12)

which for nii > > ni2 reduces to

.

i mo .
mi

„

tfa — 0 o“p — sin ©o.
where K n— Em is the excitation energy given up by the particle
2 in order to excite the particle 1 from its in'th to its n’th state
i) cf. p. i:v

-

Further we can, using the conservation laws, deduce the formulae
for the energies before and after the collision in the rest system:

42

Nr. 1. N iels A h l h y :

Ho — ^ in.2 1+ ,

4 nii nii
H>,
}

Hi

cos2 &i

,

(nii + m2)2

,,

(nii + nn)2

43

The function g ( o ) we can transform to ft) by (N 11) and we
lind then

= 0
4 nit ni*

E, (1

Ef =

On the Scattering of Thermal Neutrons by Bound Protons.

( m2 cos ft)

(N 13)

1—

\nii

//(»)

1—

cos2 (+

/nr

/)+

sin2 ft) '

i

(N 20)

sin2 ft)’

which for up > > /m just reduces to 1 + 2 /?*2 cos ft).

‘ ‘

By dciinilion we have for the cross-sections that

I ,nn(®> * ) dP. = I mn (d*, y *) </W* =

(d, y) rfr„

(N 14)

(X 15)

readily seen from the definition. Galling the probability for the
scattering process P, that is the number per unit time and per
scatterer scattered out of the beam, the cross-section Q is defined
as the ratio between P and the number in the incident beam

n, we get

crossing unit area per unit time at the place o f the scatterer, so
that we have

(dropping the index 2) which we can write, due to <P = y* = y
r

,7 ,

,

.

. sin d do

‘„ „ A » . * ) = i ...» ' w ) sin (,) T & For the special case of elastic scattering, m =
from (N 11)
sin d
. . ni‘2~ , , , m o
i
1+
„+ 2
cos o
sin ft)
mi2
nii
.<
1 i i}l2

I d ------ COS

nii

dQ

COS

(N 17)

mx

Putting (N 16) and (N 17) into (N 15) we linally get using (N 11)

,
( 1 +

'
For mi > >

' COS

mi)

'

=

tive to the scatterer. Since P, o and urel are the same in all Ga ­
l i l e i systems, Q is at once seen to be invariant.-)
From (N 14) we can also deduce the transformation formulae
for the differential cross-sections from one Ga l il e i system K to

I„„ (»,./ )■ »(»)

,

(X 18)

0

\

(N 21)

another K*. The formulae for the angles are, when u is the velo­
city of the system K * , measured in K

1 + m \ + 2 m- cos o'
>nn <®> • " ) = / „ „ ( » , . / ) '

P
ft + e l

where « is the density of the beam, that is the number of par­
ticles per unit volume, and nr(ll is the velocity of the beam rela­
do.

H+

Q=
(N 16)

0

c o s 2 ft)

/«i

From (N 14) we can at once deduce that the iolal cross-sec­
tions in the two coordinate systems are identical. This is, how­
ever, only a special case of the more general theorem that the
total cross-section is the same in all Ga i .ilki systems.1) This is

COS f t ) * +

cos ft)

II-

2

Vi*-

nii
sin ■b

we get

u
* COS
Vo*

+

u-

-- * COS do

Vi

.cos hi’

* .,+

Vi*

v*1

2

V* V-2
ll
* cos do
Vo

•(X 22)

sin (y,i*— y.2* ) sin do
(I - cos'- <->)' ■■(1 +

in°
„ i ,, iii-i
q (o ) = 1 + 2 — cos o = 1 + 2 — cos ft).
/»i
n?i

* | //

0*

+ 2

cos »,<

/\

') By a G a l i l e i

In the special case /?p = /n2 we find the well-known formula

system is understood a coordinate system which

moves with constant velocity along a straight line.
-> W e have only considered the non-relativistic case. In tire relativ­

*7+)

cos

o

4 cos ft).

(N 19)

istic case 0 is also invariant so long as the coordinate systems move in
the direction of the current.

44

Nr. 1. Nines A r l k y :

(with the expression lor eos 0 inserted from the above formula)
« 1. 2* =

<

(® i, 2* , « ) .

0* =

<

(z»i*, V * ),

<
&* =

<

4.)

On the Scattering of Thermal Neutrons by Hound Protons.

([c,*X B ],

Bv definition we have1)

[v-* X a ] )

H jy ) =

(— 1)"' exp (if-) (

) exp (— if2).

(X 26)

([^ X C o*], [ v f x u )

(and analogous formulae for the quantities without stars) which
formulae are obtained from the general formulae of spherical
trigonometry, using (N 0), the direction u being taken as the polar
axis in a polar coordinate system. Theoretically we can from
(N 14) and (N 22) obtain the transformation formulae for the
differential cross-sections, but in praxis the resulting expressions
are so complicated that they are quite unmanageable, with the
exception of the special case where u has the same direction as
one of the Z)*’s, in which case (N 22) reduces to (X 6). Further,
if ti lies in the plane of v * and v * we get <f> = </>* = I), the for­
mula for © being the same.

If we put (X 26) into (X 25) and integrate by parts, we get due to
the fact that exp (— z/2) and all its derivatives vanish in + co faster
than every power o f y
(X27 )
using ll'n = 2«//JI_|, n > 1.-) Using this recurrence formula /
times on itself, we can show' by induction that the result is

11

X T '

S
/-j_.s)! ^J ^ t>n—l s , w—/ ( X 28)

('i

Here l is restricted by the condition that n — l and m - - l must
both be positive or zero, that is l < min (/??, n). Assuming in < n

Note 3. D
For the eigenfunction of the one-dimensional oscillator we
have
n
!,
d~ ■
2 2(; j!)“
'I'n (*’)
, = ex" ( - 2 \$ " •
(X23)
_ / , lg
_ (
( H T 9 ) r<(0
- 1MP („) ’

we can therefore put l -= m. By reducing en_ ni.... s () in the same
way, wre get

M). ( - '|2) ( N 2!>)
due to
el0 — ^ exp ( i b y ) exp ( — if2) dy =
• —

reduced mass of the proton, m the
quency of the proton and H n the n’th H ermite polynomial. With
these wave functions we shall now calculate the matrix element

CO

= exp ( — 4 ) \ exI> ( — (11— Y ) ) dU = CXP ( “
Putting now (X20) into (X 28) with l = m, we get

exP ( ' C ; 1') I m
(X24)
l''n* ' exp •\!>m dx = ;i

-2

4 ) ;f' "•

2 ( nl in l)

V

III

b-(

.7 - 2'" ( i b ) " ~ m nl ml exp

jj //n(y) exp ( i b y ) II„,(y) exp ( ~ y 2)d y

\y

( i b ) ls

4 1——

2s si (in — ,s) ! (n

m-rs)l

(m < n).
(X 25)

b =

uk,„„ ,

n — 1.

For n < in we get the same formula with n and m interchanged.
(X 30) into (X 24) finally gives us

11 '1 he matrix elements given in this note have been given previously
by the author, cf. Nordiska Naturforskarmotet

i Helsingfors

1936. The

reports, p. 248.
-b (,f. e. g. H cark and F i i k y : “Atoms, Molecules and Quanta”, p. 533.

1» Cf. Coi

k a n t - H i i .is k u t :

“ Methoden

der

p. 78.

-> Cf. C o c h a n t - H i m i e r t , loc. cit. p. 78.

mathematischen

Physik”,

( X 30)

46

Nr. 1. N jkls A r i .k y :

On the Scattering of Thermal Neutrons by Bound Protons.

Uy
Mp

f71|exI) ( ' C ; r) |m
n hill
2

2

( nl i

( i />)' n~ m

n

( - 1Y If

\

' exp

s- 0
/=

min (n, m),

b =

(X 31)

2s si (l — s)l ( \n — m |+ s ) !

ak'ninx.■

For the one state being just the groundstate we get
n |exp ( i k'(m x ) |0 ) = ( n !) '-2

H i b f exp

(N 32)

products oi three o f the type (X 23). Due to the states being de­
generate with the multiplicity gn ^ (/l ”

also be obtained by writing the wave functions in polar coordi­
nates. We only give the formulae for reference1)

Xa

(21+ 1) (/ — |m |)!
4a

(l + |m |)!

- / n + (+ l\ - ^
exp
2
j'
n — Z\.

V

\

9n

\

|

exp (inup) P\

2a2/. //
I\
i i\

2

’

(cos o) X

/+ 1 + 1 .
~ 2 ' ’ a2>

(XT37)

') I'

we must form

-l < m < 4- /, /--- n, n —2, n— 1,
1

1 (X30)

where mB is the mass of the binding center m B = nis — nip,
nip yo mN . For our value of m s = 14 /nv , t has the value 1.0051,
which can be safely replaced by unity, so that we apart from
the important factor in q are left with F ermi ' s formula.
It might be of interest to note that the formula (XT 34) can

<P„hlAr,o,t/)
For the 3-dimensional isotropic oscillator we can at once get
the cross-sections from (X 32) since the eigenfunctions are only the

( m \ m s \ /nij> + m
\ms M m s) \ m v ,n IS

47

|l" :l' ( ’K m O |

n iijiiK )

j- (XT33)

o. K.

P"' is the ordinary associated L egendre polynomial2) and ^

the

confluent hypergeometric function.2) We find
[n lm |exp ( ik"mr ) |000)

This is very complicated unless m — 0 in which case we can at
once perform the summations if we only choose the (arbitrary)
coordinate system so that k'l'un is along one of the axes

h>n ~ '/ - V

/X
•)

‘> kona'

(2/4-1)

4V 2

= 11

=

k on 1
i
7 ko ill (( b-\n
2 ) CX|)

M s F, 9 __
Mp h to

t

n— /

|(nJcxi) (iA*;;fl.r)|0) ( « {/|l|0)(/i.|l|0)|3
r

n + l\

n4-/4-l

(/ i-fitl)-

exP _

ak.
( X 34)

Eo
F,

Putting (X 38) into the formula analogous to (N 33) we just lind
the expression in (X 34) for I (m due to the formula proved by the
/

author1)

Fn— FoV

“

F (l

COS

t) ( XT33)

( t ' i ! i
(n -f- /4" 1)!

by (1). This is the formula found by F e r m i 1) apart from the factor
[n

\

in

q

^2j (X 38)

‘

(cf. eq. (4)) and trom the factor

My

in eq. ( X 35) which

by F ermi (and by He t h e ) are both put equal to unity:

i = n. n—2 ••• >U

r

1
2n nl

(X 39)

( 't ' ) !

1> The author, loc. cit.
2> C o u r a n t - H i l b k r t , loc. cit., p. 282.
•!> Cf. e. g. M o t t and M assey , loc. cit., p. 38.

lots cit. Cf. also B e t h k , loc. cit. Part B eq. (45a.). It should be noted
that by the authors quoted m s it put equal to infinity throughout.

4) See Matematisk Tidsskrift, Copenhagen 1937.
on his 50th bi rthda y”, p. 42.

“ To Prof. H. B ohr

48

Nr. ]. N ikls A hi .k v :

In the calculation leading to formula (N 38) we also get the matrix
elements lor the fixed rotator with two degrees of freedom whose

N

l ) l n\' , (a-) = — i

eigenfunctions are just the first part of (N 37) multiplied by
d ( r — ro), r0 being the dimension of the rotator. We find
U m |exp ( / * " r ) 100) -

49

On the Scattering of Thermal Neutrons by Bound Protons.

<l0lll ( 2 ) " ( 2 y + 1) ' 2ij (k ”j r 0 )“ ' V . ; , 2(A'"-r0)

^

( n + l ) (— 1)" i f, : ', <4.v) =
(X 47)

sin 2 i x _
2n

(N 40)

c~x— c 2x
4 7i

X)

1

Putting (N 42)— ( N 47) into ( N 41) we get, using ,} ko2a2 = W 21
•]j j_i „ being the Bessel function of order 7 -f- } .

I^

n\

2 I t m ~x

m NJ

1— e x p ( — 4 W)

.

9

W

y

Note 4.
It may he interesting to note that the formula for the total
elastic cross-section for the isotropic oscillator can also be dedu­
ced by direct calculation o f the B orn phases and their summa­
tion which is indeed very seldom possible. W e have, since the
phases are all small,1)

^0 “

(N 41)

+ 1) L’n
n

=

which is just eq. (21) remembering eq. (4).

Note o.
We first prove the formula

\ exp ( — « 2(.r — ,t)2) — exp (— « 2(.r-j-7) 2)j dx — .1 ' ~a 1'f>( « 7)
•0
where cP ( x ) is the Gauss error function given by

0

<l>( x ) =
CO

‘2 M N k0 f*
t*

*2 " J,

due to o’ «

a.

n<> Ja*,,,-) W

X H

M r

(N 42)

h r)rd r.

Taking in the first part y — x — /?, in the second y — .r -f- 7 as
new variable we get

(N 43)

If we put (N 43) into (N 42) using (N 37), we get

Ms D 4 1/„ pis (

(— «2 y2) dij

so (X 48) follows at once. We can now work out the integral in
(29). The two angle integrations being performed, we are left with

(N 41)

1

^00

! % ( ‘U) exp (— y u2v) i [exp (— a v-) — exp ( — ( u + 4 u’) v’2) j X

Now
00

\ exp (-

(N 49)

i o

7’ dt = 1d'o(rY) \-(— I ) ’) 4 /' o'3

VL

(N 48)

1

P- P) Jr ( « 0 Jr (bt) t dt = 2ff_ exp [ —

a2-f- b- '
ab \ 2)
4
' \2p2

“

*

‘T

*’o
X exp (2 (u vN v) — exp (—-2 a u v v)! dv.

(N 45)

The two integrals here are just of the type (X 48). In the first we
I r (x) =

exp ( — t' ) ij'j ■Jl,( i x )

(N 46)

have a- — a,
1

1

= Rat, in the second « 2 = ( « 4- 4 u ), i =
iv

1

‘

a Us

. ,•

a -j- 4 u

Putting in these values in (X 48) we easily find (30).
P Cf. M o t t and M assey , loc. cit. eq. (a), p. 138 and eq. (12), p. 90.
W a t s o n : “ Bessel functions”, eq. (1). p. 395.
3) W a t so n , loc. cit., eq. (2). p. 77.

O W a t s o n , loc. cit. eq. (3). p. 152.
2) Cf. eqs. (1) and (15).
Vidensk. Selsk. M:itli.-fys. Medd. XVI, 1.

4

50

On the Scattering of Thermal Neutrons by Bound Protons.

Nr. 1. Nn:r.s A h f e y :

.)!

Note <>.
( — i* y2 exp (— « 2y2) d y + 2p \ y exp ( - « 2y2)di/ +
We prove here the formula

•’o

•'.

X

i00

\ w

' exp (--,d \ Y ) d‘ ( i W ' ■') (l\v =- 2 , r ' V

A ret”

•o

(X 50)

•'>?

u

where P ( x ) is defined in eq. (X 40).
We put first t =

• (I

,r =

'

.

(X 51)

/* exp (

u

x ~~1

[exp ( —

« V 2) +

a2y2)dy.

2r
i
+ ~i e x p (— « 2;t2)- f '

—

l rr*

•o

If we now differentiate the function f ( x ) we gel

f'(x ) =

«' ■
)

fhe second integral is zero, the two last ones can be performed
at once and the first one by integration by parts. The result is

I F " as new variai)le and get

\ = 2 r l [ exp ( - ^ ) <!>(!) dt = 2 4 1/Or),

'o

,+/*>

+ 4/S V y exp ( — « 2y2) dy + ,t2 1 exp (

'“ y'

P ( ,r i)

which immediately proves (N 52).

“j j [ t P ( t ) } dt.

Note S.
We prove the formula

Integrating by parts we can get the inhomogeneous differential
equation for f ( x )
f (•'*') = X ~ l f ( x ) + ;,~J 1 1

^

\ v ,;s

exp ( ~a*W) <!>{? W 'h d W

+ A rctg'S |

7 i

( X 55)

^
where <P(x) is defined in eq. (N 49).
We take as new variable / = /? W 12 arid get using (X 49).

which hy the ordinary methods can be solved to
f ( x ) - - 7T ' "-.r-Arctg .r-J-(constant-.r).

( — 2 , W i 2 ?f" ' ! ( d/(j d u /2 exp ( — “ J ) exp ( - u2).

Tlie constant can be determined to be equal to zero by expanding
fi’ (0 and integrating term by term. For .r2 < 1 the resultant series
is convergent to just n ~ ' - - x -Arctg x. This in (X 51) then proves
(X 50).

•o

l o *°

X ow

■x

1

I

X

dt ( du -

*0

X

( dll ( dt

*0

* (i

#ol

ao

and so we get, performing the V dt by integrating by parts

Note 7.

•’ll

We prove the formula
X

\ = 24
\ .r2 1exp ( —

<y 2

(.r — ,4)'2) — exp ( —

u -

(a- - f ^)'2)

*o

•’n

<lx
(N 52)

= ,f>

P a + |«»] + J 6XP

where <T>(x) is defined in eq. (N 49).
Putting y — x + ,-f we get

X

2 a ''" i du exp (— u- ) X

vn

j (43 1 j"« n
"21,2
X r « « 2 I X ° X,) i - - 7 - 1 + V ( ' - * (

aU

a

t

Here all integrations can be performed, using ecp (N 50). The re­
sult is
4

52

Nr. 1. N ihi .s A ri .k y :

s

« jS

<<2+ ft2

On the Scattering of Thermal Neutrons by Bound Protons.

cc

53

ture. In the mathematical notes we have further compiled

2 ~ Arctg y.

various formulae for transformation of coordinate systems,

which proves (N 53) because we have the elementary identity

matrix elements and integrals used in the text.
In conclusion I wish to thank Prof.

,y — Arctg | = Arctg x.

N ie ls

B ohr

for

his kind interest in this work and to express my appre­
ciation to Prof. G.

P laczek

for suggesting the problem to

me and for many valuable and helpful discussions in the

Summary.

course o f the calculations. Further I wish to thank Dr. F.
Kalckar,

In the present paper we discuss the scattering o f ther­
mal neutrons in hydrogeneous substances. In § 1 we dis­
cuss the binding model for the protons. W e assume the
protons to be bound independently in an anisotropic os­
cillator taking the largest oscillation energy equal to 0.37
volts, and the others equal to 0.4 times that. Further we
take the lower frequencies into consideration by ascribing
an effective mass, which we have chosen equal to four­
teen times the neutron mass, to the system consisting o f
proton plus potential and assuming these “ molecules” to
move freely like gas molecules with a

M axw ell

distribution.

In §§ 2 and 3 the cross-sections are calculated. In §§ 4
and 5 we discuss the temperature effects. Firstly it is found
that when both the neutrons and the scattering substance
have room

temperatures, the cross-section is 2.7 times

larger than the free cross-section. Secondly it is found that
the cross-section for neutrons at liquid air temperature i. e.
90° abs is 34 °/o higher than at room temperature. These
figures are compared with the experiments. Finally we in
£ 6 discuss which effective energy must be attributed to a
beam o f

M axw ell

neutrons in regard to the scattering cross-

section. It is found that for our model this effective energy
lies between 0.7 k T and 1.1 k T depending on the tempera-

Dr. C.

M oller

stimulating discussions.

and Dr. V.

W e is s k o p f

for many

Pohjoismainen ( 19. skandinaavinen) luonnontutkijain kokous Helsingissa 1936.
Nordiska ( 19. skandinaviska) naturforskarmotet i Helsingfors 1936.
Eripainos. — Sdrtryck.

T A B L E OF C O N T E N T S
Page
Introduction

3

§ 1. Discussion of a simplified model for the binding of the protons
§ 2. General theoretical remarks

6

§ 3. The anisotropic o s c il la t o r ................................................................

10

§ 4. Influence of the temperature motion of the scattering centers

22

§5. M a x w

27

ell

distribution of the incident neutrons

§ 6. Effective neutron e n e r g y ..................................................................

Mag. scient.

N iees A r e e y ,

Kobenhavn:

10

Om spredningen a f neutroner med termiske hastigheder
ved bundne protoner.

34

Note 1...........................................................................................................

39

Note 2...........................................................................................................

39

Note 3 . .........................................................................................................

44

Note 4...........................................................................................................

48

Note 5...........................................................................................................

49

Note 6...........................................................................................................

50

Note 7...........................................................................................................

50

Note 8...........................................................................................................

51

S u m m a r y ......................................................................................................

52

Som det vil blive vist, har Borns approximation 2 gyldighedsomraader:
et for hurtige partikler og et for meget langsomme. Man kan derfor i vort
tilfaelde benytte denne approximation, og vi finder for det differentielle virkningstvcsrsnit (i relative koordinaterdl,) udtrykket:
Tm n W , < P ) =

V*n ( r ) e x P ( i k m n r ) f m ( r ) d r

%

%

exP ( i k m nr

2 n %2 ^

dr

k0 : neutronernes indfaldsimpuls (i /i-enheder)
kmn»
impuls efter
anregung
kmn'
»
impulstab ved stodet (vektorielt)
Wn- protonens n’te egenfunktion. V (r): proton-neutron vexelvirkningsenergien. M: neutronens reducerede masse.
q0 er en konstant, der er lig ~n x det totale virkningstvaersnit for neutron

Afleveret til Trykkeriet den 28. December 1987.
Faerdig fra Trykkeriet den 4. Maj 1988.

-fri proton (hvilket i relative koordinater betyder fast proton.) Den ubekendte
funktion V (r) faar man saaledes ind i en konstant, der kan faas fra experiment,
og regningerne reduceres til beregning af matrixelementerne for exp {ik"mn r ) ,
Disse regninger er gennemfort for tilfaeldene: 1) linear- 2) rumlig oscillator og
Idet vi nojes med 0—x-n overgange, forer regningerne til folgende formler:
1) Dineaer oscillator.
"%

a 1 k,on

^on
d on

kn

%

n!

n

exp

a k

(her kan k"on kun antage 2 vserdier svarende til reflexion og gennemgang, idet
problemet er 1-dimensionalt.)
2) 3-dimensional oscillator.
j

1 on

_

17

' 7 1 \/Tl

? r 1\

^ + x) ( — 2— )
fn —l
r ')

>2l

' •'till

%

1!
(n + l + 1)!

X

7
k"2
i2 non
x 0 2K i f exP ( —
(2 betyder summation over alle de til energikvantet n horende tilstande med
1

impulsmomentkvantet l, hvor l kan antage vserdierne n, n- 2 , n-4, ......... > 0.)
Bortset fra de forskellige faktorer er altsaa vinkel- og energiafhsengigheden
den samme for lineser og for rumlig oscillator. ( a betyder begge steder
oscillatorkonstanten a = ( ^ )-' 11: protonens masse, on oscillatorens frek\/t CO/ ‘
yens x 2 jt.)
?on

ft

I)
(K n r)

/n + i ( C r)

J n -r i-’
2 den n ~r ;;~ ’te Besselfunktion. r: rotatorens arm, der faas fra experi-

ment, idet inertimomentet er u r 2.

Helsingfors 19"56. Finska Litteratursiillskapets Tryckeri Al>.