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Asymptotic formulae for slow neutron scattering by bound atoms
Journal of Nuclear Energy Parts A/B, 1966, Vol. 20, pp. 313 to 317. Pergamon Press Ltd. Printed in Northern Ireland
ASYMPTOTIC FORMULAE FOR SLOW NEUTRON SCATTERING BY BOUND ATOMS* V. F. TURCHIN and V. A. TARASOV (Received 3 October
1963)
Abstract-Using the Placzek-Wick method, asymptotic formulae have been obtained at high neutron energies for the mean cosine of the scattering angle, energy loss moments and conjugate energy loss moments. The latter differ from the ordinary moments in that integration is conducted not over the final neutron energy but the initial neutron energy. FOR calculating the spectra of epithermal neutrons and neutrons of higher energies, use can be made of methods of the age-approximation type. However, the presence of chemical binding of the atoms substantially changes the differential scattering cross sections of epithermal neutrons compared with scattering on free atoms; thanks to this, such integral characteristics as the mean energy loss, the mean cosine of the scattering angle and so on are also changed. These characteristics must be known in order correctly to allow for the effect of the chemical bond on neutron spectra in the epithermal energy region. PLACZEK’~) has shown that the asymptotic behaviour of the cross sections for neutron scattering on bound atoms with increase in the initial neutron energy can be expressed in terms of the mean values of certain operators characterizing the motion of the scattering atoms, therefore for this case it is not essential to know all the parameters of the dynamics of the atoms. Placzek obtained formulae for the total scattering cross section, the first moment of energy loss by a neutron on scattering and the differential angular scattering cross section do/dQ in the form of an expansion in powers of the inverse initial neutron energy E,,- l. The coefficients in the expansion contain the mean values of the kinetic energy of the scattering atom KAv = (K),’the square of the kinetic energy (GQA, and of two operators:
P2V; ((w2:;
BAV = cA,
=
where M is the mass of the scattering atom, V the potential of the forces acting on it. Placzek’s formulae can be used for the case when the initial neutron energy E,, exceeds the characteristic energy of motion of the atom. For a simple crystal this is reduced to the condition E,, > max (T, O), where T is the temperature, 0 the Debye crystal temperature. In addition to this, the coefficients in the expansion in powers of E,-l in Placzek’s formulae have the form of an expansion in powers of m/M, i.e. of the inverse atomic mass in neutron mass units, for which reason these formulae can only be used when M > m. Subsequently, WICK t2) showed that Placzek’s approximation corresponds physically to the case when the scattering time of a neutron on an atom is small compared with the characteristic time of motion of the atom (approximation of small scattering * Translated by J. J. 5
CORNISH
from Atomnaya Energiya 18,118 (1965). 313
V.
314
F. TURCHIN
and
V. A.
TARASOV
times). It then became clear why Placzek’s formulae are only suitable in the case of M > m. In actual fact, the recoil energy of an atom on scattering is of the order (m/M)_&,. Consequently, the corresponding characteristic time is (/z/E& . (m/M). Allowing for the fact that it must be much greater than the characteristic neutron scattering time, i.e. h/E,, we get the condition (M/m) > 1. Thus, in order to obtain formulae similar to these of Placzek but also applicable to light nuclei, the recoil energy of the atom must be correctly allowed for, which in fact was what WICK didc2). The following formula was obtained by him for the total scattering cross section:
(1)
.
where ,u = m/M; a0 is the cross section for scattering on a free atom. The interference of neutron waves scattered by neighbouring atoms was not taken into account; i.e. equation (1) is derived in an incoherent approximation. Henceforth we shall also limit ourselves to an incoherent approximation, since it is quite sufficient for epithermal neutrons. Having denoted twice the differential scattering cross section in calculation for the differentials of the final energy dE and the solid angle dL? by o(E, + E, I$), where 0 is the scattering angle, we shall define the energy loss moments as (TV(E,) =
CT(E, -+ El f3)(Eo - E)” dE da.
s For the first moment TURCHIN’~)obtained:
VP - 8
KAY
3(p + 1)” * E, +
p+i
4/Q
BAG ---.
+ 1) ’ Eo2
P +
8
15 &
t
(K2>A,
+ 1)s * F
(2)
I *
The following can be obtained in the same way for the second moment: ru(ru2- 6~ + 3) (p + 1)” -
2,4P - 1) (p +
BAV
1)s *E,2+
16$,~
-
$1
K2>A,-
(p + 1)4 ’E,2_
KAV
* E,
(3)
In equations (2) and (3) terms proportional to Eo-3 were not computed. The differential angular cross section was obtained by WICK(~)in the follow ing form : do dn~ a(E,-+E,O)dE s
(4
Asymptotic formulae for slow neutron scattering by bound atoms
315
Here,
where
v,=
x-i
X”= An important characteristic the scattering angle
p cos 8‘2
PLl
i
+ sin2 8;
(p2 - sin2 0)li2 Pus-1
(7)
.
of the differential
cross section is the mean cosine of
(8) where
@#,,) =
s
$cosBdfi
=2x
s
l do _IdncosOd(cosO)
is the first angular moment of the differential cross section. The value of @‘l(E,), like all the odd moments, can be computed very simply since, with the exception of ~,,JB), all the functions v,,,(0) en t erin g ex p ression (4) are found to be even with respect to E = cos 8. In order to reveal this, we shall note that V, is a polynomial of the second power with respect to x and [, which possess the following property: the sum of powers x and 5 in each term is an even number. This property is retained when V, is raised to any power m and when multiplied by the function which enters equationt5) 2 x2 + ---xx++ Pu+l
l2 (P + 1)2’
since this function also possesses this property. After dividing by x, the sum of the powers in each term becomes odd. Thus, equation (5) can be written in the form
(10) where F(x) =
5 + a, X
+ a,x + . . . +
UC&&l
X2n” +l
(10
has the property that the coefficients in front of even powers of x are odd with respect to 5, and coefficients in front of odd powers of x are even with respect to 5. From expression (7) it can be seen that the value of variable x substituted in (10) after differentiation is an even function of E; consequently, the parity of the i-th term is entirely determined by the parity .of coefficient a,. Therefore, all terms with odd powers of x in expression (11) will give even functions of E and with integration over da will not contribute to the first moment.
316
V. F. TURCHIN and V. A. TARASOV
We shall examine the even terms. The maximum even power in expression (11) is x2m = (x~)~. Th us, if differentiation is carried out more than m times in equation (10) the corresponding term will vanish. It can be seem from (4) that for all terms except the first n > m. And so the only term which is odd with respect to [, which gives P~.~(@,is 2P (fQ+ 1)2 Es Substituting it in expression (4) and integrating with respect to da we get
Thus, in the small scattering-time approximation the first angular moment of the differential cross section is energy independent and is equal to its maximum value corresponding to scattering on a free stationary nucleus. This circumstance makes it possible to compute the mean cosine of the scattering angle and the transport scattering length & in the epithermal energy region using only the total scattering cross section which, as we know, can easily be measured by the transmission method. From equations (8) and (12) we get 2 00 case=-.-. 3P o@o)
(13)
The transport scattering length is determined from 1
J& = - 1 = fJotr
P~.@oN1
’ p~,@o) - -3P go 1 1
-= - cos e>
2
(14)
C
where p is the number of atoms per unit volume. On the basis of hbLKONL4N’s(4) data on the total cross section for neutron scattering by water we computed the value of 1tr for water at room temperature as a function of neutron energy in the 05-10 eV region (see Fig. 1). The transport cross section for
Eo,
eV
FIG. I.-Transport scattering length At, of water at room temperature as a function of ne’utron energy E. (--- maximum value at E,, - co).
317
Asymptoticformulae for slow neutron scatteringby bound atoms
scattering by oxygen, which makes a small contribution to the total transport cross section otr, was taken as constant and equal to its value for a free atom. In the l-10 eV energy region the total scattering cross section a,(E,J was extrapolated in accordance with equation (1) starting from its value obtained at 1 eV. TURCHIN(~) made use of quantities a,(E) =
u (E, ---fE, O)(E,, - ~5)~ dE,, dQ f
(where n = 0,1,2), known as conjugate energy loss moments, for computing the thermal neutron spectrum. Computing conjugate moments in the small scatteringtime approximation according to WICK(~), which proved to be extremely clumsy, leads to the following results :
4 + 3(p -
KAV 1)2 * E
So(E) = o. +
3p + 1 BA” -+ 12&u + 1) * ly
(p + 1)2 ln p + 1 + L &iv 3,~’ E 2P P--l
WP + 11
E2
= a,
(p +
2p
*E21; (~)a,
1)21np
- + p-1
BAv
3P + 1
.
1
+ 8p (1u2+ 2p + 3) KAY .E 3(p - 1)4
(15)
12/&J + 1) * E2
*E21’ (K2)~v
15/&(p + 1)2 ~22(Jf3
2(4p + 1)
15&J + 1)s
(16) C/J2-
-
4p + l)(lu + 1)2 (/J - 1)4
+
A/J2
B
2(p + l)(p - 1)s. s
&v 1G2b2 + 5) -. + (p - 1)4(p i_ 1)2 ’ E2
(17)
TURCHIN(~)has obtained formulae for ordinary and conjugate energy loss moments in the case of crystals with cubic symmetry. The method used was equivalent as regards its physical assumptions to Placzek’s method, therefore the coefficients in the expansion in front of the inverse energy powers have the form of an expansion in powers of l/p. It is easy to convince oneself that if in equations (l)-(3) and (15)-( 17) the aforesaid coefficients are expanded in powers of l/p the initial terms of the expansion (up to l/p3 inclusive) will coincide with the coefficients obtained by TURCHIN@). REFERENCES 1. PLACZEK G. Phys. Reo. 86,377 (1952). 2. WICK G. Phys. Rev. 94,1228 (1954). 3. T~RCHIN V. F. Slow Neutrons, Moscow, Gosatomizdat (1963). 4. MELKONIAN E. Phys. Rev. 76,175O (1949). 5. Tmcm V. F. CoIZection: Neutron Physics, p. 66. Moscow, Gosatomizdat (1961). 6. TURCHIN V. F. CofIection: Neutron Physics, p. 74, Moscow, Gosatomizdat (1961)
ASYMPTOTIC FORMULAE FOR SLOW NEUTRON SCATTERING BY BOUND ATOMS* V. F. TURCHIN and V. A. TARASOV (Received 3 October
1963)
Abstract-Using the Placzek-Wick method, asymptotic formulae have been obtained at high neutron energies for the mean cosine of the scattering angle, energy loss moments and conjugate energy loss moments. The latter differ from the ordinary moments in that integration is conducted not over the final neutron energy but the initial neutron energy. FOR calculating the spectra of epithermal neutrons and neutrons of higher energies, use can be made of methods of the age-approximation type. However, the presence of chemical binding of the atoms substantially changes the differential scattering cross sections of epithermal neutrons compared with scattering on free atoms; thanks to this, such integral characteristics as the mean energy loss, the mean cosine of the scattering angle and so on are also changed. These characteristics must be known in order correctly to allow for the effect of the chemical bond on neutron spectra in the epithermal energy region. PLACZEK’~) has shown that the asymptotic behaviour of the cross sections for neutron scattering on bound atoms with increase in the initial neutron energy can be expressed in terms of the mean values of certain operators characterizing the motion of the scattering atoms, therefore for this case it is not essential to know all the parameters of the dynamics of the atoms. Placzek obtained formulae for the total scattering cross section, the first moment of energy loss by a neutron on scattering and the differential angular scattering cross section do/dQ in the form of an expansion in powers of the inverse initial neutron energy E,,- l. The coefficients in the expansion contain the mean values of the kinetic energy of the scattering atom KAv = (K),’the square of the kinetic energy (GQA, and of two operators:
P2V; ((w2:;
BAV = cA,
=
where M is the mass of the scattering atom, V the potential of the forces acting on it. Placzek’s formulae can be used for the case when the initial neutron energy E,, exceeds the characteristic energy of motion of the atom. For a simple crystal this is reduced to the condition E,, > max (T, O), where T is the temperature, 0 the Debye crystal temperature. In addition to this, the coefficients in the expansion in powers of E,-l in Placzek’s formulae have the form of an expansion in powers of m/M, i.e. of the inverse atomic mass in neutron mass units, for which reason these formulae can only be used when M > m. Subsequently, WICK t2) showed that Placzek’s approximation corresponds physically to the case when the scattering time of a neutron on an atom is small compared with the characteristic time of motion of the atom (approximation of small scattering * Translated by J. J. 5
CORNISH
from Atomnaya Energiya 18,118 (1965). 313
V.
314
F. TURCHIN
and
V. A.
TARASOV
times). It then became clear why Placzek’s formulae are only suitable in the case of M > m. In actual fact, the recoil energy of an atom on scattering is of the order (m/M)_&,. Consequently, the corresponding characteristic time is (/z/E& . (m/M). Allowing for the fact that it must be much greater than the characteristic neutron scattering time, i.e. h/E,, we get the condition (M/m) > 1. Thus, in order to obtain formulae similar to these of Placzek but also applicable to light nuclei, the recoil energy of the atom must be correctly allowed for, which in fact was what WICK didc2). The following formula was obtained by him for the total scattering cross section:
(1)
.
where ,u = m/M; a0 is the cross section for scattering on a free atom. The interference of neutron waves scattered by neighbouring atoms was not taken into account; i.e. equation (1) is derived in an incoherent approximation. Henceforth we shall also limit ourselves to an incoherent approximation, since it is quite sufficient for epithermal neutrons. Having denoted twice the differential scattering cross section in calculation for the differentials of the final energy dE and the solid angle dL? by o(E, + E, I$), where 0 is the scattering angle, we shall define the energy loss moments as (TV(E,) =
CT(E, -+ El f3)(Eo - E)” dE da.
s For the first moment TURCHIN’~)obtained:
VP - 8
KAY
3(p + 1)” * E, +
p+i
4/Q
BAG ---.
+ 1) ’ Eo2
P +
8
15 &
t
(K2>A,
+ 1)s * F
(2)
I *
The following can be obtained in the same way for the second moment: ru(ru2- 6~ + 3) (p + 1)” -
2,4P - 1) (p +
BAV
1)s *E,2+
16$,~
-
$1
K2>A,-
(p + 1)4 ’E,2_
KAV
* E,
(3)
In equations (2) and (3) terms proportional to Eo-3 were not computed. The differential angular cross section was obtained by WICK(~)in the follow ing form : do dn~ a(E,-+E,O)dE s
(4
Asymptotic formulae for slow neutron scattering by bound atoms
315
Here,
where
v,=
x-i
X”= An important characteristic the scattering angle
p cos 8‘2
PLl
i
+ sin2 8;
(p2 - sin2 0)li2 Pus-1
(7)
.
of the differential
cross section is the mean cosine of
(8) where
@#,,) =
s
$cosBdfi
=2x
s
l do _IdncosOd(cosO)
is the first angular moment of the differential cross section. The value of @‘l(E,), like all the odd moments, can be computed very simply since, with the exception of ~,,JB), all the functions v,,,(0) en t erin g ex p ression (4) are found to be even with respect to E = cos 8. In order to reveal this, we shall note that V, is a polynomial of the second power with respect to x and [, which possess the following property: the sum of powers x and 5 in each term is an even number. This property is retained when V, is raised to any power m and when multiplied by the function which enters equationt5) 2 x2 + ---xx++ Pu+l
l2 (P + 1)2’
since this function also possesses this property. After dividing by x, the sum of the powers in each term becomes odd. Thus, equation (5) can be written in the form
(10) where F(x) =
5 + a, X
+ a,x + . . . +
UC&&l
X2n” +l
(10
has the property that the coefficients in front of even powers of x are odd with respect to 5, and coefficients in front of odd powers of x are even with respect to 5. From expression (7) it can be seen that the value of variable x substituted in (10) after differentiation is an even function of E; consequently, the parity of the i-th term is entirely determined by the parity .of coefficient a,. Therefore, all terms with odd powers of x in expression (11) will give even functions of E and with integration over da will not contribute to the first moment.
316
V. F. TURCHIN and V. A. TARASOV
We shall examine the even terms. The maximum even power in expression (11) is x2m = (x~)~. Th us, if differentiation is carried out more than m times in equation (10) the corresponding term will vanish. It can be seem from (4) that for all terms except the first n > m. And so the only term which is odd with respect to [, which gives P~.~(@,is 2P (fQ+ 1)2 Es Substituting it in expression (4) and integrating with respect to da we get
Thus, in the small scattering-time approximation the first angular moment of the differential cross section is energy independent and is equal to its maximum value corresponding to scattering on a free stationary nucleus. This circumstance makes it possible to compute the mean cosine of the scattering angle and the transport scattering length & in the epithermal energy region using only the total scattering cross section which, as we know, can easily be measured by the transmission method. From equations (8) and (12) we get 2 00 case=-.-. 3P o@o)
(13)
The transport scattering length is determined from 1
J& = - 1 = fJotr
P~.@oN1
’ p~,@o) - -3P go 1 1
-= - cos e>
2
(14)
C
where p is the number of atoms per unit volume. On the basis of hbLKONL4N’s(4) data on the total cross section for neutron scattering by water we computed the value of 1tr for water at room temperature as a function of neutron energy in the 05-10 eV region (see Fig. 1). The transport cross section for
Eo,
eV
FIG. I.-Transport scattering length At, of water at room temperature as a function of ne’utron energy E. (--- maximum value at E,, - co).
317
Asymptoticformulae for slow neutron scatteringby bound atoms
scattering by oxygen, which makes a small contribution to the total transport cross section otr, was taken as constant and equal to its value for a free atom. In the l-10 eV energy region the total scattering cross section a,(E,J was extrapolated in accordance with equation (1) starting from its value obtained at 1 eV. TURCHIN(~) made use of quantities a,(E) =
u (E, ---fE, O)(E,, - ~5)~ dE,, dQ f
(where n = 0,1,2), known as conjugate energy loss moments, for computing the thermal neutron spectrum. Computing conjugate moments in the small scatteringtime approximation according to WICK(~), which proved to be extremely clumsy, leads to the following results :
4 + 3(p -
KAV 1)2 * E
So(E) = o. +
3p + 1 BA” -+ 12&u + 1) * ly
(p + 1)2 ln p + 1 + L &iv 3,~’ E 2P P--l
WP + 11
E2
= a,
(p +
2p
*E21; (~)a,
1)21np
- + p-1
BAv
3P + 1
.
1
+ 8p (1u2+ 2p + 3) KAY .E 3(p - 1)4
(15)
12/&J + 1) * E2
*E21’ (K2)~v
15/&(p + 1)2 ~22(Jf3
2(4p + 1)
15&J + 1)s
(16) C/J2-
-
4p + l)(lu + 1)2 (/J - 1)4
+
A/J2
B
2(p + l)(p - 1)s. s
&v 1G2b2 + 5) -. + (p - 1)4(p i_ 1)2 ’ E2
(17)
TURCHIN(~)has obtained formulae for ordinary and conjugate energy loss moments in the case of crystals with cubic symmetry. The method used was equivalent as regards its physical assumptions to Placzek’s method, therefore the coefficients in the expansion in front of the inverse energy powers have the form of an expansion in powers of l/p. It is easy to convince oneself that if in equations (l)-(3) and (15)-( 17) the aforesaid coefficients are expanded in powers of l/p the initial terms of the expansion (up to l/p3 inclusive) will coincide with the coefficients obtained by TURCHIN@). REFERENCES 1. PLACZEK G. Phys. Reo. 86,377 (1952). 2. WICK G. Phys. Rev. 94,1228 (1954). 3. T~RCHIN V. F. Slow Neutrons, Moscow, Gosatomizdat (1963). 4. MELKONIAN E. Phys. Rev. 76,175O (1949). 5. Tmcm V. F. CoIZection: Neutron Physics, p. 66. Moscow, Gosatomizdat (1961). 6. TURCHIN V. F. CofIection: Neutron Physics, p. 74, Moscow, Gosatomizdat (1961)