MOMENTS METHOD FOR NEUTRONS

MOMENTS METHOD FOR NEUTRONS

T h e moments method has been extensively applied to the calculation of neutron densities in shields, and w e shall treat this application here.

T h i s application has the same advantages and disadvantages as in problems involving photons. O f the various terms of the Boltzmann equation, only the scattering integral for neutrons differs from that for photons. T h e scattering integrals differ for several reasons: First, neutron scattering is largely isotropic in the center-of-mass system, whereas photon scattering is rather anisotropic. Second, reasonably simple analytical expressions exist for the energy dependence of gamma ray scattering, whereas only very complicated graphical, empirical data exist describing the energy dependence of neutron scattering. T h i r d , the dependence of gamma ray scattering upon atomic properties is provided by relatively simple analytic expressions, whereas each nuclide must be considered separately in determining neutron scattering.

T h e moments method for neutron attenuation proceeds exactly like that for gamma rays. Since neutrons of different speeds are now being taken into account, the flux depends upon the speed of the neutrons, and the scattering integral includes an integral over all possible speeds of the incident neutrons. I n place of Eq. (5.1.1), we then have

Ω - V<f>(r, ν, Ω ) + at( r , ν) φ(ν, ν, Ω ) = T ( r , ν, Ω ) + S(r, ν, Ω ), (C.1) where the scattering integral ^ r , ν, Ω ) is given by

1 ( r , ν, Ω ) = j dv'dSl'a8(r, Ω ' - > Ω , ν' ν) φ(τ, ν', Ω ' ) ; (C.2)

ν and ν' are the speeds of the scattered and incident neutrons, and the other notation is as before. T h e quantity as( r , Ω ' — • Ω , ν' -> ν) is the 315

probability that a neutron of speed ν' going a unit distance in the direction Ω ' is scattered to produce a neutron of speed ν per unit speed going in the direction Ω per unit solid angle. T h e probability as( r , Ω ' —• Ω , ν' —• ν) may be expressed as the product of the prob­

ability as( r , ν') that a neutron of speed v' in going a unit distance is scattered and the probability F(v\ ν, 0O) that the scattered neutron has a speed ν per unit speed and is scattered through an angle 0O in the laboratory system per unit solid angle, the scattering being induced by a neutron of speed ν'. T h i s latter probability F(v', vy 0O) is in turn the product of the probability/(z/, 0O) that the neutron is scattered through the angle 0O into a unit solid angle and the probability

dE dv

that the neutron scattered into a unit speed has a speed v. By Eq. (B.40) . , . 1 - Μ ν' , 1 + Μ ν . _ „,

£ ( * » = _ _ _ _ + _ ΐ _ (C.3)

T h e Dirac delta function expresses the deterministic constraint between v, v\ and 0O imposed by the laws of energy and momentum conservation, the quantities v, v\ and 0O being regarded as independent above. T h e product [//(cos eo)ldE][dE/dv] of derivatives merely provides the transformation needed between a unit increment in g and a unit speed. Finally, for reasons that appear below, it is convenient to use the angle 0C of scattering in the center-of-mass system, instead of that 0O

in the laboratory system. Accordingly,

i/(cos 0O)

By combining all these independent probabilities together, we have as( r , Ω ' —> Ω , ν' —> ν) =

so that

asl(r, ν', ν) = a8( r , v')f[v\ ec{g)] Pt(g) ^{( M}2

^

7

upon using Eqs. (5.1.5) and (B.12). W e may now expand the flux in Legendre polynomials as in Eq. (5.1.5). Proceeding as from Eq. (5.1.1) to (5.1.11), we find for one dimensional geometry that

(2Γ+Ί) dz— + \-2f+T) dz— + ^at(*> ^V) +**> ^V)

= j dv' asl(z, ν, ν') φχ{χ, ν') + ν), (C.5) where the notation is as in Chap. 5.

In our subsequent work, the limits of the scattering integral frequently come into consideration. It simplifies these limits if the cosine of the angle 0C through which the neutron is scattered in the center-of-mass system is used as the independent variable, instead of the speed v' of the incident neutron in the laboratory system. Equation (B.12) provides the necessary transformation.

Next, we introduce the flux per unit energy to replace the flux per unit speed. W e then find

/ Βφ^ζ,Ε) / / + ! \ 0* »+i ( * . E)

( 2 7+ 1 ) —d z — + —dz~ + V

= Sl(z, E) + j d(cos ee) Pfe) ^f[E\ 6c(g)] σ8(ζ, Ε') φ fa Ε'). (C.6) Finally, we multiply both sides of this equation with z^j and integrate from — o o to + o o . W i t h the definition (5.7.5) of the moments and the condition (5.7.13), we find that

+ SU(E) + j d(cos e9) Pfc) ^f[E\ 6c(g)] σ8(Ε') φ,^Ε') = Ο , (C.7)

where σ and σ8 are independent of position. I n this expression the scattering integral is the chief source of difficulty.

T h r e e approximations have been used to treat the scattering integral.

I n the first, the energy loss of the neutrons at collisions is neglected, in which case the scattering integral reduces to σ8(Ε)/ι(Ε)φ^ι(Ε). T h i s approximation is good for the small contributions to moderation due to oxygen when hydrogen is present; when hydrogen is absent, then the approximation is valid for nuclides heavier than iron.

In the second approximation, the scattering integral is expanded in powers of M^{- 1}. W e leave the details of the development as an exercise for the reader. T h e result is

IW-ZpUl*. Μ)] Ε^(ΕΊΦί,ι{ΕΊ][Ρι{μ) - J * = i M +

(C.8) Certaine has developed a more accurate method of dealing with the scattering integral (References 1-3). T h e method consists of approx­

imating the energy dependent part of the integrand by a sequeiite of continuous straight line segments. T h e integral over each such segment can be computed analytically; to this end, recurrence relations for, and starting values of, two different quantities must be developed. T h e complete integral is then the sum of the subintegrals whose integrands have been approximated by'straight lines and whose values have been computed analytically. In other words, the method is very closely akin to Simpson's approximation.

W e shall describe only one complete variation of Certaine's method to a degree sufficient to enable the reader to use and understand it. Other variations may be found in References 1-3. Although the algebra is quite tedious, the method is really quite straightforward. T h i s method is applicable to even the lightest nuclides, whereas the other two described above are not. One of the difficulties is that for large /, P^E) oscillates rapidly.

Certaine's method starts by expanding the collision transfer prob­

ability in Legendre polynomials:

/[£'. β&)] = % QL+A g {(Ε') ρ,'οο, ( c . 9 )

where μ = cos θ'0 and &\>{E') are the expansion coefficients. T h e moment of the scattering integral Ί is then given by

Ί

'.' = %

--τ-- ί,

*w>

of energy E. Because the average lethargy gain of a neutron upon scat­

tering is independent of the lethargy of the incident neutron, we choose subintervals so that they are of equal length Δ in lethargy. I f the lethargy of the scattered neutron is u, then the lethargy of the incident neutron may be as high as u or as low as u — 2 l n ( M + l)/(^f — 1)· L e t us next define the Kth subinterval by the relation

uK+i<u - 2 1 n ( M + \)I(M - \)<uK. ( C . l l ) T h e cosine of the scattering angle in the center-of-mass system then

satisfies the relation

μ0 = 1 .

Further, the lethargy uk of the kth subinterval is given by

uk = u -kA. (C.12)

T h e part of the integrand of the scattering integral "7 dependent upon the energy of the incident neutron is then approximated by a series of continuous straight line segments:

6r(t ) os(t )φ]Λ(Ε ) ^ 2 '

where

Uk = u' -uk. (C.13)

If this approximation is substituted into the scattering integral, we find that

co K+l

= ^ ^ &Ι',^ΖΛΦΙ,Ι',^i.i'.k > (C.14)

where

- ^{21'2a ^{0 ekA} |^{[ 1} J7

+ [1 - Kol j"" *μ υ} :_λ( μ ) Ρι{μ) Pfe)

j .

T h e quantity Uk may be regarded as a function of μ by ( B . l 2 ) , Μ² + 2Μμ + 1 "

υ,(μ) = In

Μ² + 2Μμΐί + 1 (C.16)

If w e knew the \t l\k , then the moments of the scattering integral could be found and the scattering integral itself computed. F r o m this, we can find the moments of the neutron flux from the relation

7~T\Ti-l,l-l,i I /Λ/ ι ι\ ^σθ9ί-1.ί+1.< ~ " ^σΐΨΐΛΛ < ^3,l,i ( 2 / + Ι ) ™ -¹·¹"¹· · ¹ ( 2 / + 1)

oo # + 1

+ 2/ 5/^1'.Λ+»^σ8.Λ:+ίl.l'.fc+i^i.l'.fc+t — 0 (C.17)

by numerical analysis. One starts from the lethargy of the source, which is the lowest lethargy and works upward.

T h e quantities ^<SCLl'tk must be evaluated. T h i s can be done by means of the recurrence relation

~?l+l.l'.fe — ( j _|_ j )

2)

*v.

=

²

f

where

and by means of a relation to be given later for ^o.r./c ^{t n at} gives its value explicitly. T h e relation (C.18) follows directly from the definitions (C.15) and (C.19). I t is first noted from Eq. (C.19) that

fitful)-Sffi+ilur..^).

T h i s result may be used to reduce integrals of the form found in E q . (C.15)

( 2 /

',

Γ

PvM Pn<g)

= ( t t t ^-) § · * · ' · ' ( ^ ί γ ¹) ^ ^{M PP l ) { g}

- (TFT ) (^Ψ~) C

Application of the present result to Eq. (C.15) proves the recurrence (C.18).

T h e must be evaluated. Various relations exist for the com­

putation of these quantities. A recurrence relation

- ( γ + τ ) ( ^c - ² ° ) facilitates the calculation of the . T h i s result is quickly shown by applying the recurrence relation ( D . 8 ) for m = 0 twice to the definition (C.19). In addition, we need to know that

* νΛ = - - § ± ^ M - ] (C.21)

to calculate « ^ Y, j · T h i s result follows immediately from the generating function ( D . 2 ) for Legendre polynomials, the orthogonality relation ( D . 7 ) for m = 0 for these polynomials, the recurrence relation ( D . 8 ) for them, and, of course, the definition (C.19).

W e return to the evaluation of the J S f}h l- tk . So far we have found a recurrence relation (C.18) to evaluate these quantities, and we have evaluated the that occur in this recurrence relation by means of Eqs. (C.20) and (C.21). W e need the further relations

*o.i:* =

^

'

+

)

Χ*"'

• % m=w I

! ?

'

\ ' " M +

'

<

'

-

and

= 2 ~^{A e} 2M 4^{δ ί}·^!' &/!(/ - ; ) ! I 2 M J ^{g k [ AU} + ^{1 ) ]>}

if A = Κ or X + 1 , (C.23) where

g0(x) = (er> - 1 + x)lx*, gJx) = (cosh x)lx² ,

S K ' ^y ' (C.24)

gK(x) = [1 - 2e«^x + e<«+^1)x + (? - 1) * ] / »² ,

b = — ( M² + 1)/2M 2 , / M + 1

= 0, if/' - / < 0 o r o d d

(-yi'-u/ψ + /)!

(C.25) (C.26) (C.27) (C.28)

T h e proof of these relations is straightforward and tedious, and follows from Eq. (C.15). T h e expression

I )² where

= (1 + 2M/x + M²)/2M ,

, (C.29)

(C.30) may be deduced by integrating the left-hand side once by parts and by substituting the expansion

into the result. T h e observations

UK+l{-\)=A(\ -q),

(C.31)

(1 + Mf

2M ^g-kAV+l)

(1 ± Mf 2M

*^{+ 1}G*»)

rfⁱ⁺¹(± i )

follow from Eqs. (C.12), (C.16), (C.26), and (C.30).

y

= o,

(C.32)

follows from the definition ( D . l ) of Legendre polynomials.

( ± 1 )^{? +1} -b^l+^l

1+ 1 X 2M

J

j=0

may be used to write Eq. (C.29) more compactly for the case ak = μχ, bk = 1. T h e conclusions (C.22) and (C.23) are the consequence of introducing the definitions (C.24) into the result (C.29) for various pairs (ak , i ^ ) ,¹ of multiplication of the results by ΒΧΛ>, of summing over /, and by use of the observations (C.31) through (C.33).

W e have now completed our work. T h e moments may be found from Eq. (C.17), the scattering integrals may be found from Eq. (C.14), and quantities ^l t l't k , Jt'x>%i , J(x> , 0 and JS^0.r.fc from Eqs. (C.15), (C.20), (C.21), (C.22), and (C.23). T h e concepts are simple; the execution tedious.

1. Certaine, J., A solution of the neutron transport equation. Introduction and Part I. NYO-3081 (1954).

2. Certaine, J., A solution of the neutron transport equation. Part II: N D A Univac moment calculations. NYO-6268 (1955).

3. Certaine, J., A solution of the neutron transport equation. Part III: Recon­

struction of a function from its moments. N Y O - 6 2 7 0 (1956).

1 T h e pairs used are (μί, 1), (μ*+1 , ^fc), (μ!ί-1 , μΑ), (μκ-ι, μκ), ( — 1 , μκ) and

References

ί ) .