VELOCITY RELATIONS FOR NUCLEAR EVENTS

VELOCITY RELATIONS FOR NUCLEAR EVENTS

T h e laws governing the elastic scattering of neutrons have been used on several occasions in the text. W e wish to derive these laws for reference purposes here. Elastic scattering is distinguished from inelastic scattering in that in the former the target nucleus is left in the same nuclear state as that in which it was found. N o third particle, such as a photon, is emitted in an elastic scattering that carries off energy. T h u s , the elastic scattering event is a two particle one, a fact that makes the laws describing it particularly simple.

T h e laws for elastic scattering are derived from the conservation of linear momentum and energy. W e shall derive these laws first and then go to an application of them to a calculation of transfer probabilities. T h e laws will relate the initial and final speeds of the neutron, the angle through which it is scattered, and the initial and final energies.

B.1 Kinematical Relations

Before proceeding further, it will be necessary to define the so-called center-of-mass system (References 1 and 2 ) . T o this end, let the velocity of this system relative to the laboratory system be denoted by Vc ; let the velocity of the incident neutron be denoted by ν ' ; let the mass of the target nucleus relative to that of the neutron be denoted by M . T h e center-of-mass system is then defined to be that system in which the momentum of the target nucleus equals that of the neutron:

M VC = ν ' - Vc. ( B . l )

T h e velocity ν of the scattered neutron in the laboratory system is the vector sum of the velocity vc of the scattered neutron in the center-of-

301

mass system and the velocity Vc of the center-of-mass system relative to the laboratory system.

v = vc + Vc. (B.2)

L e t the scattered neutron emerge from the scattering at an angle θ0 with respect to the direction of travel of the incident neutron in the laboratory system and at an angle θα in the center-of-mass system. T h e relationships among these various quantities is displayed in Fig. B.l.

Velocity of the neutron Λ Velocity of the neutron in the laboratory system ^ ν in the center-of-mass

system Angle of scattering / Π

. I ll /βλ / "(

in the laboratory system

Velocity of the center-of-mass

Angle of scattering in the center-of-mass system

F I G . B . l . Vector diagram relating the velocity vectors in the laboratory and center-of-mass systems.

T h e initial velocity of the neutron in the center-of-mass system is the difference between the velocity ν ' of the incident neutron and the velocity Vc of the center of mass

by Eq. ( B . l ) .

i = V - Ve , (B.3)

= [ M / ( M + 1 ) ] ν ' (B.4)

B.2 Conservation of Momentum

T h e conservation of linear momentum in the center-of-mass system requires that

vc = M Vn , (B.5)

where Vn is the velocity of the nucleus after scattering in the center-of- mass system. T h e nucleus is assumed stationary with respect to the laboratory system before being hit by the incident neutron.

B.3 Conservation of Energy

Since the target nucleus is assumed to be at rest in the laboratory system, its velocity in the center-of-mass system is equal and opposite to

that of the center of mass itself. W e can now apply the conservation of energy in the center-of-mass system:

1 Μ ι Μ

2 ^νό² + Τ ^{( - V}<^{) 2 =} 2 ^V< + Τ ^{( V}-^{) 2} ' · ^{6 ) ( Β} W e can find a simple relation of use to us later between the speed of

the incident neutron referred to the laboratory system and the speeds of the emergent neutron and of the target nucleus after the collision both referred to the center-of-mass system as follows: Equations ( B . l ) and (B.4) are used to express, respectively, Vc and in the left-hand side of Eq. (B.6) in terms of Μ and ν ' . Equation (B.5) is used to express Vn in terms of Μ and vc in the right-hand side of Eq. ( B . 6 ) . W e are left with a relation between the speed of the incident neutron referred to the laboratory system and that of the emergent neutron referred to the center- of-mass system:

( M v ' )² = [ ( M + 1) vc]² = [M(M + 1) Vn]². (B.7)

B.4 Relation Between the Initial and Final Speeds and the Angle of Scattering

A relation between the speed of the incident neutron and the speed of the emergent neutron and the angle of scattering can be found by squaring Eq. (B.2). T h e result is then expressed in terms of ν and cos θ0 by means of Eqs. ( B . l ) and (B.7) to yield:

v* ⁼ [ μ Τ τ Γ ^{[ M}* + 2 M C O S E* + ^{1 ]} · ( B , 8 )

T h e maximum 2 ?ma and minimum Em\ energies that a neutron can have after an elastic scattering are

ί Μ — 1 \²

£ - = ( μ + τ ) ^£ · ^(Β·⁹⁾

as may be seen from Eq. (B.8).

B.5 Relation between the Scattering Angles in the Laboratory and the Center-of-Mass System

T h e relation between the scattering angle θ0 in the laboratory frame and that of 6C in the center-of-mass frame can be found by observing from the definition of a scalar product that

cos 0O = v ' · vjv'v . (B.10)

Equation (Β.8) is used to express the scattered velocity of the neutron in the laboratory system in terms of ν', Μ, and θ0 in the denominator of Eq. (B.10). T h e scattered velocity ν of the neutron in the laboratory is replaced by the velocity vc in the center-of-mass system by means of Eq. (B.2). T h e resulting expression is then further reduced by application of Eqs. ( B . l ) and (B.7).

1 + Μ cos Sc

Vl +M² + 2M cos 0C '

cos flp = , ^c ( B . l l )

V /•§ H /TO Λ 11 /Τ Λ * '

B.6 Relations among the Scattering Angles and the Initial and Final Energies

T h e Eq. (B.8) may be solved for cos 0C in terms of the kinetic energy, E'y of the incident neutron and the kinetic energy, E> of the scattered neutron

COSθ

°

m

I f this result is substituted into Eq. ( B . l l ) , one finds that

m , A 1 - M + ( 1 + M ) (Ε/Ε')

C O S ( >« ⁼

WJEVU

T h e lethargy, u, is defined by

« = l n ( £r/ E ) , (B.14)

where Er is some reference energy. T h e lethargy may be positive or negative, in contrast to the energy. T h e cosine of the scattering angle may be expressed in terms of lethargy by means of this definition.

a ( 1 - M ) /u'-u\ (l+M) iv! -u\

cos θ0 = ±——-L exp - y j + Λ —— J - exp ^ j . (B.l5)

B.7 Relations between the Direction Cosines of the Velocity of a Scattered Neutron in the Laboratory System and in a Center-of-Mass System

T h e relations between the direction cosines of the velocity vector of a scattered neutron in the laboratory system and in the center-of-mass system whose axes are parallel to corresponding axes of the laboratory

system are to be worked out presently. Because the corresponding axes of the two coordinate systems are parallel,

i = ic,

j= j c , (B.16)

k = kc.

T h e velocity vector of the incident neutron may be expressed in terms of its components in the laboratory frame

v' = v'[*'di + β% + Y'dk] 9 (B.17) and the velocity vector vc of the scattered neutron in the center-of-mass

system may be expressed in terms of its components Μ

V c = Ί Τ + Τ * '^{( a c ic + Pc +c i n) K ) ( B , 1 8} by use of Eq. (B.7).

F r o m Eqs. ( B . l ) , (B.2), (B.8), (B.17) and (B.18), it is found that

ν = + Μ * , ) i + ( f t + Μβ0) j + (/d + M yc) k ]

χ [1 + M² + 2MM + β,β'α + ycyd)Y^ . (B.19) T h e velocity ν of the scattered neutron may be expressed in terms of its components:

ν = v[oLdi + + ydk ] . (B.20)

F r o m the last two relations the direction cosines are seen to be

ccd = Κ + M « J [ 1 + M² + 2M(occoc'd + ββ'Λ + ycy i ) ] "^{l /2} , βα = [β'α + Μβ,][1 + Μ² + 2M(occ«'d + β£'Α + ycy ^ ) ] -^{1 /2} ,

Ύα = [γ'α + Μγ,][1 + Μ² + 2Μ(<ν*£ + βΰβ'α + ycy i ] "^{1 /2} . (Β.21) I n words, the direction cosines of the scattered neutron with respect to the laboratory system can be found from the known direction cosines of the incident neutron referred to the laboratory system and the direction cosines of the scattered neutron with respect to the center-of-mass system whose axes are parallel to the corresponding axes of the laboratory system.

B.8 Relations between the Direction Cosines of a Scattered Neutron in Two Center-of-Mass Systems

Calculations involving anisotropic scattering often require the use of two center-of-mass systems: one having its ζ axis along the velocity vector ν ' of the incident neutron and the other with its axes parallel to the corresponding axes of the laboratory frame, as in the previous section.

T h e relation of the two coordinate systems is displayed in Fig. B.2.

Quantities referred to this rotated center-of-mass frame (r-frame) with its ζ axis along the direction of the incident neutron will be distinguished by a subscript r. Quantities referred to the other center-of-mass frame

F I G . B.2. Relation of the two center-of-mass frames and the laboratory frame.

(c-frame) with axes parallel to corresponding ones of the laboratory system will be denoted by a subscript c. T h e r-frame can be generated from the c-frame by two rotations:

1. Rotate the c-frame about its zc axis until its xc axis lies in the plane of ν ' and zc. T h e new y c axis will then lie along the yr axis.

2. Rotate the new frame just found about the yr axis, i.e., the new yc axis, until the new zc axis lies along ν ' , i.e., the zr axis.

It is desired to find the direction cosines of the direction of the scattered neutron with respect to the t-frame in terms of those of the scattered neutron referred to the r-frame and the direction cosines of the incident neutron referred to the laboratory frame.

Point at which scattering occurs

r

(B.22)

(B.23) Our goal in relating quantities referred to two coordinate systems rotated with respect to each other is facilitated by a vector method. T h e key idea of this method is that a vector lying along the axis of rotation is invariant during the rotation of one coordinate system into the other.

T h e velocity vector of the incident neutron can be expressed in terms of its components referred to the c-frame:

< =

< K i

. = v' x k

h ν'

Vl -

and by Eq. (B.22), we learn that

. = — ft* + <*ci

F r o m the definition of the r axes and by Eq. (B.22)

kr = v ; / < = o # + ftj+y;k. (B.24) T h e remaining unit vector follows from the orthogonality of j r and kr

which is a consequence of Eqs. (B.23) and (B.24), and the definition of a vector product:

i

K i

VT^yf

upon squaring Eq. (B.24) and using the fact that kr is a unit vector.

T h e direction cosines of the scattered neutron referred to the c-frame can be related to those of the scattered neutron in the r-frame and to those of the incident neutron in the laboratory frame as follows:

1. By inserting Eqs. (B.23), (B.24) and (B.25) into

vr = vr[ocrir + 0rjr + yrkr] , (B.26) it is found that

' c

+ [ f t ^ L t f f l +

p

] j +

VT=Yf*

2. Since the t w o frames move with the same velocity vr = vc

vr = vc.

By this result and by Eqs. (B.16), w e find that

vr = vr[oici + ftj + yck ] . (B.28)

3. Since corresponding axes of the c and laboratory frames are parallel, and since the c-frame translates in the direction in which the incident neutron moves

Slc = Ω ' or

ac = ^ad y

ft = 01, (B.29)

, ύΌ

=

4 . F r o m Eqs. (B.27), (B.28) and (B.29), w e find the result desired:

+ ^ad 7 r ι

He = /Λ ,9 "Τ ΡαΎτ ι

VI - yd2

yc = -V\ -y'?«r + y'dyr- (B.30) In words, the direction cosines of the scattered neutrons referred to the

ofrarne can be found from those of the scattered neutron referred to the r-frame and those of the incident particle referred to the laboratory frame.

B.9 Transfer Probabilities for Elastic, Isotropic Scattering and Fission

T h e transfer probabilities for elastically scattered neutrons can be readily calculated from the assumption or approximation that neutrons are isotropically scattered in the center-of-mass system. T h e probability

^ ( 0C) per unit angle of scattering in the range d6c centered at 0C is

&(θ,)άθ, = ^{2 π ! ά}Ι ° °^Μ° . (B.31)

T h e probability ^(E' —• E) of a neutron of energy E' giving rise to a neutron of energy in the range dE centered at Ε is related to the probabil­

ity 0*(6C) by the principle that neutrons are conserved.

0>(E'^E)dE = &{B^c) ; ( A - ) dE, (B.32)

where J(ejE) is the Jacobian (Reference 3, 4, or 5) of θ0 with respect to E. In the present instance

1>(ττ)|-

from the definition of a Jacobian and by

c os e

' = m [

ΐ

<0« {Μ + l )²

dE 2 M £ ' sin ee (B.33)

(B.34)

a relation deduced earlier in Section B. 6. By Eq. (B.31) and (B.32) (Μ + l )²

4ME' (B.35)

In words, the probability of elastically scattering a neutron to an energy between Ε and Ε -f- dE is independent of the energy Ε and dependent only on the mass ratio Μ and the incident energy E' of the neutron.

T h e probability 0*(u' —• w) of scattering from lethargy u' into a lethargy interval ώ centered at u is

^ ( Μ ' - > u) du = ^(£' ->E) J (-^-) (B.36)

F r o m the definition of a Jacobian and of lethargy, one learns that

>(-f )l-

T h e probability ^(u' - > w) is then

[0 , i f w - w' < 0

— II) = ( ^{( M}4] ^ e ^xP ~ (" ~ «0 . i f- l n r > M - w ' ^ 0 (B.38)

0 ^if u — u' > — In Y

where Y is defined to be

Finally the probability 2P(u' -> u, 0O) per unit solid angle of scattering from a lethargy u' to a lethargy between u and u + du at an angle 0O

in the laboratory system is by Eq. ( B . l 5 ) found to be

χ δ jcos 0O - [^{( 1} + ^M\w-um + (1 2 ^{M )}g -^{( M /}-^{M ) / 2}] j - (B.40)

T h e following integral is often useful:

J Λ ^ ( Μ ' -> ii, 0O) = 0>(w' - > w ) . ( B . 4 1 ) T h e transfer probability for fission is easily derived from the pro­

bability &ι(Ε' - > E) that a neutron created in fission by a neutron of energy E' have an energy in the interval dE centered at Ε (see Reference 7 or 2)

0>t(E' -+E)dE = e~^E sinh V2E dE, (B.42)

where for the validity of this semi-empirical formula the units of energy must be M e v . T h i s spectral distribution and the number of neutrons produced per fission are relatively independent of the energy E' of the incident neutron below 1 M e v , since the energy of the incoming neutron is negligible compared with its binding energy. A t energies above 1 M e v , one might expect ν and/or έΡϊ(Ε' —> Ε) to depend on the energy of the incident neutron. Because of the relatively low energy of the neutron emitted as compared with its binding energy, it might be expected that the quantum mechanical probability amplitude would be predominately iS-wave, i.e., that the neutrons are emitted isotropically in the center-of- mass system. Experiment verifies this deduction. T h i s quantum mechan­

ical result can be seen classically by noting that the neutrons emitted have such low speed that their classically computed angular momentum is less than h. Because the fissionable elements are so very heavy as com­

pared with a neutron, the difference between the center-of-mass system and the laboratory system will be neglected. T h e probability that a fission neutron be headed in the directional element d& centered at Ω is

<*Ω/4ΤΓ .

T h e hypothesis that there i s¹ no angular correlation between the direction of emission of the emitted neutron and its energy enables one to calculate the probability that a fission neutron have a speed in the range dv centered at ν going in the element dSl of solid angle centered at Ω by

simply multiplying the t w o independent probabilities together.

&4y* - > ν, Ω ) dv dSl = ^/-^ e~^E sinh VlE ~ , (B.43)

( B - 4 4 ) the last relation being a definition of f(E).

T h e quantity

^ «^{( M}' - * ^{U) = ( M}8^ M^{1 )2 eU}'~^U η ( ™ Φ (^{Ρ Β}·^{4 5}) follows immediately from Eq. ( B . l 5 ) and the expansion

&{u' - u, θ0) = £ ^ t_ L ^ ^ u) pn ( c os θ0). (B.46)

w=0

T h e value of cos θ0 that makes the argument of the Dirac delta function zero is denoted by cos θ'0.

T h e integrals

Γ du' = -\nYy (B.47)

J u+lnY

Γ du\u' -u) = -(l/2)(ln Yf, (B.48)

J u+lnY

—Γ du' e^u'~^u = 1 , (B.49)

— Γ A*' — « ) = — ί , (B.50)

1 This "hypothesis" is really a consequence of the more fundamental one that a compound nucleus is formed before fission occurs. Such a hypothesis would be of very questionable validity were the energy of the incident neutron to be greatly larger than the binding energy of a neutron.

where

* = ¹ + "pry > (B.52)

may be readily evaluated. F r o m Eq. (B.50) it follows directly that ξ is the average change in lethargy in a collision. F r o m E q . (B.51) it follows that </x0> is the average cosine of the angle of scattering. Several numerical values for later use are shown in T a b l e B . l .

T A B L E B.l

V A L U E S OF C E R T A I N S L O W I N G D O W N PARAMETERS AS A F U N C T I O N OF A T O M I C W E I G H T

Element Y i r ( i n r )²

2f(l - Y)

A

H¹ 0.000 1.000 1.000

H² 0.111 0.725 0.584

H e⁴ 0.360 0.425 0.309

L i⁷ 0.444 0.268 0.205

Be⁹ 0.640 0.209 0.153

C¹² 0.716 0.158 0.110

O¹⁶ 0.779 0.120 0.084

P b^{2 08} 0.981 0.00958 0.000

B i^{2 09} 0.981 0.00953 0.000

JJ238 0.984 0.00838 0.000

References

1. Glasstone, S., and Edlund, M . C , "Nuclear Reactor Theory." Van Nostrand, Princeton, N e w Jersey, 1952, Chapters 4, 6.

2. Meghreblian, R. V., and Holmes, D . K., "Reactor Analysis." McGraw-Hill, N e w York, 1960, Chapter 4.

3. Franklin, P., " A Treatise on Advanced Calculus." Wiley, N e w York, 1940, Chapters 10, 11.

4. Kaplan, W . , "Advanced Calculus." Addison-Wesley, Reading, Massachusetts, 1953, Chapters 2, 4.

5. Wilson, Ε. B., "Advanced Calculus." Ginn, Boston, 1912, Chapters 5, 9.

Problems

1. (a) Calculate a power series in M^{- 1} for ξ accurate to M ~⁷.

(b) Calculate a power series in M^{- 1} for —In Y accurate to M~⁹.

(c) Calculate a power series in M~^x for 1 — y(ln Υ)²/2ξ(\ — Y) accurate to M ~⁴.

2. In transforming between the c- and r-coordinate frames, both center-of-mass systems, we could generate the c-frame from the r-frame by rotating the r-frame about its yr axis until the zr axis is parallel to the zc axis and then by rotating the new frame about the zc axis until its χ axis is parallel to xc . W e derived the direction cosines of the scattered particle referred to the r-frame at Eqs. (B.30).

Let us try another possible frame; let us merely rotate the r-frame about an axis perpendicular to the plane containing the zr axis and the zc axis until the latter is rotated into the former.

(a) Write an expression for a unit vector pointing along the axis of rotation.

( b ) Find an expression for a unit vector pointing along the intersection of a plane containing the zc and zr axes and a plane containing the xr and yr axes.

(c) Derive expressions for the direction cosines ac, β0 , and yc of the scattered particle in terms of the direction cosines, <χα , β'α , and yd of the incident particle and the direction cosines oc'r, β 'τ, and yr of the scattered particle in the r-coor- dinate frame. The power of the vector method becomes very apparent here.