- Email: [email protected]
Temperature dependence of the effective resonance absorption integral
Letters to the editors
162
The function r stays almost constant if the phase velocity in the wave-guide is increased from 0.5 c to 1 c by increasing the diameter of the aperture in the diaphragm. We take az = @25, so that equation (5) becomes &ljfXt. (6) 1 - $.+q”x~;; “)1 = 1.58 8 &exit J[i 16 14 F 122
lo-
I .
0-
f t: 9
6 42I 0.2
0
I I 0.4 06
I 08
I FO
I I I 1.2 194 1.6
I, a
FIG. 3.-The
dependence
of optimum final electron energy, Uexit opt., on beam current, for various powers, P,(az = 0.25).
We can then easily determine is
the value of UeIit for which U~X,t,,,.=q-$-
&exit
Z,
is a maximum.
UO)
This (7)
&inject. Fig. 1 shows - A~ex,t as a function of Uerit for P, = 2 MW, calculated from (6). The curves &inject. on the beam current for various kal particle energies. of Fig. 2 show the dependence of A~,,it Fig. 3 shows the dependence of the optimum value of the final energy on the current; A&,mt. for which - A~exit has a maximum value at a given current.
this is the energy G. I. ZHILEKO
REFERENCES 1. 2. 3. 4.
Trans. Inst. Radio Engrs. No. 3, 374 (1956). MoTzH., SLATERJ. C., Rev. Mod. Phys. 20, 473 (1948). CHODOROWM. et al., Rev. Sci. Znstrum. 26, 134 (1955). MIRIMANOVR. G. and ZHILEIKOG. I., Radiotekh. Elektro 2, 137 (1957).
Temperature dependence of the effective resonance absorption integral* (Received 18 March 1957) ACCORDINGto GLASST~NEand EDLUND,w the effective resonance absorption integral for a resonance absorber dispersed uniformly throughout a neutron moderating agent can be expressed as m C, dE Jeff =
07 (E)
s0
c, + C(E) . T
* Translated by R. D. LOWDEfrom Atomnaya Energiya 3, No. 9,252 (1957).
(1)
163
Letters to the editors
E is the neutron energy, u,(E) the resonance absorption cross-section, o,,(E) the resonance scattering cross-section and p the number of atoms cm-3 of the absorber; C(E) = pb,(E) + a,,(E)1 is the macroscopic resonance cross-section and C, the macroscopic cross-section for potential scattering of the medium. A knowledge of Jeff makes it possible to calculate the resonance escape probability p = exp (- pJeff/{Ca), 5 here being the mean logarithmic energy loss per collision. Equation (1) is applicable when the appropriate resonance absorption either is slight (C, < C,) or takes place in a narrow energy interval (AE,ff < 5 E); the energy interval is defined by a “danger zone” AE,f within which AE,, C > C,. In this letter we discuss Jeff for the case of an isolated resonance, since the overall resonance integral is merely the sum of the integrals for the resonances separately. The interference between poten‘tial and resonance scattering will be ignored. If rv, I’, and I? = rv + l?, are respectively the radiation, neutron and total widths of the resonance, then or = (l?,/r)u; the energy dependence of the resonance cross-section then follows from the Breit-Wigner formula
E,*
0
a(E) = u,, E
1
-
1 + X2 ’
in which x = (E - E,)/$I’, and uO is the resonance cross-section at the energy E, of the relevant nuclear level. When the nuclei of the scatterer are executing thermal motion the apparent energy dependence of the cross-section is different, due to the Doppler effect. To determine it in this more general case, the nuclear absorption cross-section for any given neutron energy, measured in the laboratory system of co-ordinates, must be replaced by the average over all the possible values resulting from the relative motion of the neutron and nucleus. Assuming that the temperature T”K of the medium is greater than the Debye temperature 0 of the absorbing material, it is foundta) that 0, (x,8 = oo’r (x,0,
s
I
mexp [-(P/4(x
E = r/A,
In (3), k is Boltzmann’s constant.
Y(x,Q
4,
1 +y2
-m
if
- yN
(3)
A = 2
may be regarded as the real part of a complex function = u (x*9*) + iv(x*,y*)
w(x*,y*) as follows : wx,o
= 5$
where x* = E/2x, y* = E/2, and
u(x*,y*h
cc
y*e& dt
U(x*,y*) = .!
rr s --oo(x* -
t)Z + y*z
.
FADDEEVA and TERENTEW give a table of u(x*,y*) as a function of x* and y*. As a result of the averaging process the resonance line is observed to be broader than the line for a single nucleus, and to have a lower peak value. The true resonance integral
s co
JR =
%(E) $
0
is unchanged, but Jeff increases due to the widening of the “danger zone” AE,,. Substituting into (1) the expressions obtained above, and ignoring the weak variation of r,(E) and E-l across the interval AE,, one obtains Jeff=_
TV ._x8 24
*
V&Q
P s -,W&
h
+ Wpo)
The integration in (4) may be taken between the limits -cc AE,, x > x,ff = :r
.
(4)
and co without significant error, since for
164
Letters to the editors
the integrand decreases with x sufficiently strongly. Equation (4) shows that the effective resonance integral depends on the temperature of the medium through E(T), and depends also on the potential scattering cross-section IS, = &/p appropriate to an isolated nucleus. If the medium were to be imagined to become indefinitely rarefied, so that p -+ 0, then Jeff+ JR = &T (o,I’,)/E, since m Y(x,E) du = x. s --m In the general case it is convenient to write Jeff as the product
Jeff = JR,f’(E,W). Here l/a = C,/C,, C, = poO. A magnitude f < 1 indicates self-screening of the resonance due to large absorption and scattering cross-sections. At T = 0 (5 = co), there is no Doppler effect; moreover y + l/(1 + x*), and so
f(->:)=fm(i) =d(l:l,a). Putting (5) one defines a function
(6) which contains the temperature T+
dependence of Jeff. In particular,
~0 (A+
~0, l-
a),
~(E,l/a)-t
at T = 0, 7 = 1; and for
~‘(1 + l/a),
whilef+ 1 because at high temperatures the effective resonance cross-section self-screening does not occur. Under the condition xE2 < 1, Y (x,E)
N
q
5 exp
becomes so small that
- $ (
1
and the apparent shape of the resonance is determined by the Doppler effect alone. If E/u < 1 the dominant contribution to the integral in (6) comes from values of x satisfying the condition x5” < 1, so that in this case
The explicit determination of 7 normally calls for numerical integration. For this purpose it is convenient to change to the variable y = x/x,fr, x,rf being given by the condition Y(x,rr, 5) = a. Then
The integral in this expression is approximately unity and depends only weakly on a and E. In the actual computations a further change of variable to z = (1 + y)-I, giving
is of assistance. The integral may alternatively be reduced by proceeding on the following lines. In the expression $I!‘) dx, F is essentially an arbitrary function which falls to zero sufficiently fast as Y + 0 s (x -+ h-z). Changing the variable of integration to r = ‘I’(x,t), Y(O,8 Jeff = s0
Jeff =
Letters to the editors
165
0.5
t=e,t)
FIG. 1.-f@&. .&0*04
3.63*43*2302*82*6-TJ 4 ;
2.42.2FO+81*61*4l-2-
_1.
FIG. 2.-q
withf(t,t)
in which
= - (aY/ax)/Y.
Y’(Y,l)
(&z),
may be represented by the series in which
166
Letters to the editors
ForE-+O,
For t -+ hXl andfort+O f(t,t)
N 2t3/a 1/(1 - t).
Graphs off(Q) and 7 (6,1/a) are shown in Figs. 1 and 2. The former is included for illustrative purposes only, and in practical calculations of the temperature dependence of resonance absorption integrals the curves of ~7should be employed. I. V. GORDEEV v.
v.
ORLOV
T. KH. SEDEL’NIKOV REFERENCES 1. GLASSTONES. and EDLUND M., The Elemenis of Nuclear Reactor Theory. Macmillan, London (1952). 2. AKH~EZERA. and POMERANCHUK I., Topics in Nuclear Theory. Gostekhizdat, Moscow (1950). 3. FADDEEVAV. N. and TERENTEVN. M., Tables of the ProbabilityIntegral withComplex Argument (Edited by Academician V. A. FOCK). Gostekhizdat, Moscow (1954).
Nuclear energy levels of l@Tu* (Received 15 April 1957) YTTERBIUMof natural isotopic composition was irradiated with slow neutrons from the RPT pile for a month.“’ After thirty days the ytterbium was separated chromatographically from other rare earth elements and the *sOYbradiation (half life 30.6 day@) was examined, using a double focusing /3-spectrometer,‘a1 a scintillation counter and a proportional counter filled with heavy gases (Ar, Kr). We paid special attention to the soft part of the /3-spectrum and to the soft y-rays from laeTu. Using the &Xctrometer we first determined the following conversion lines: K - 63.13 keV; MI -N - 8.42 keV; L,M - 20.74 keV. With the proportional counter we observed lines corresponding to the following atomic transitions, Lal; Lpl; K,,,,; Kbs 1 (transition energies 7.18; 8.10; 504 and 57.5 keV respectively) and also the y-transitions of lasTu w&h energies 21.74 and 63.13 keV. Altogether 13 y-transitions were detected by the three methods; the multipolarities are shown in brackets.7 8.42(Ml + E2); 20.74(Ml); 63.13@1): 93.62(0.9 Ml + 0.1 E2); 109.67(Ml); 118.20@?2); 130.48@2); 156?; 177.21(0.75 Ml f 0.25 E2); 197.97(Ml); 240.6@1?); 307.7@2) keV. Examination of the y-rays enabled the following nuclear levels of lssTu to be established: 8.42; 118.20; 13890; 316.06; 379.19 and 472.8 keV. The ground state of lasTu has spin I, = +:I61 With the multipolarities shown for the y-transitions, the first three levels can be attributed to rotational levels with spins 3/2+, 5/2& and 7/2&. The last three levels have spins 7/2 + ,7/2F and 9/2F . The elucidation of the parity of these levels requires separate consideration. The first is evidently a one particle level. We can write the Bohr-Mottelson formula for the energy levels of a nucleus with ZO= + in the form: El = E0 + u t2”[Z(Z+ 1) + u(-~)I+~‘~(Z + 1))l + AMI + 8 + d-l)I+l’*
(Z + &)I2
* Translated by R. BERMAN from Atomnaya Energiya 3, No. 9, 256 (1957). t The multiplicity is determined by comparing the experimental and theoretical conversion coefficients. The theoretical values were obtained by interpolation and extrapolation of the data tabulated by BAND(~) and by ROSE.(~)
S~rv
and
162
The function r stays almost constant if the phase velocity in the wave-guide is increased from 0.5 c to 1 c by increasing the diameter of the aperture in the diaphragm. We take az = @25, so that equation (5) becomes &ljfXt. (6) 1 - $.+q”x~;; “)1 = 1.58 8 &exit J[i 16 14 F 122
lo-
I .
0-
f t: 9
6 42I 0.2
0
I I 0.4 06
I 08
I FO
I I I 1.2 194 1.6
I, a
FIG. 3.-The
dependence
of optimum final electron energy, Uexit opt., on beam current, for various powers, P,(az = 0.25).
We can then easily determine is
the value of UeIit for which U~X,t,,,.=q-$-
&exit
Z,
is a maximum.
UO)
This (7)
&inject. Fig. 1 shows - A~ex,t as a function of Uerit for P, = 2 MW, calculated from (6). The curves &inject. on the beam current for various kal particle energies. of Fig. 2 show the dependence of A~,,it Fig. 3 shows the dependence of the optimum value of the final energy on the current; A&,mt. for which - A~exit has a maximum value at a given current.
this is the energy G. I. ZHILEKO
REFERENCES 1. 2. 3. 4.
Trans. Inst. Radio Engrs. No. 3, 374 (1956). MoTzH., SLATERJ. C., Rev. Mod. Phys. 20, 473 (1948). CHODOROWM. et al., Rev. Sci. Znstrum. 26, 134 (1955). MIRIMANOVR. G. and ZHILEIKOG. I., Radiotekh. Elektro 2, 137 (1957).
Temperature dependence of the effective resonance absorption integral* (Received 18 March 1957) ACCORDINGto GLASST~NEand EDLUND,w the effective resonance absorption integral for a resonance absorber dispersed uniformly throughout a neutron moderating agent can be expressed as m C, dE Jeff =
07 (E)
s0
c, + C(E) . T
* Translated by R. D. LOWDEfrom Atomnaya Energiya 3, No. 9,252 (1957).
(1)
163
Letters to the editors
E is the neutron energy, u,(E) the resonance absorption cross-section, o,,(E) the resonance scattering cross-section and p the number of atoms cm-3 of the absorber; C(E) = pb,(E) + a,,(E)1 is the macroscopic resonance cross-section and C, the macroscopic cross-section for potential scattering of the medium. A knowledge of Jeff makes it possible to calculate the resonance escape probability p = exp (- pJeff/{Ca), 5 here being the mean logarithmic energy loss per collision. Equation (1) is applicable when the appropriate resonance absorption either is slight (C, < C,) or takes place in a narrow energy interval (AE,ff < 5 E); the energy interval is defined by a “danger zone” AE,f within which AE,, C > C,. In this letter we discuss Jeff for the case of an isolated resonance, since the overall resonance integral is merely the sum of the integrals for the resonances separately. The interference between poten‘tial and resonance scattering will be ignored. If rv, I’, and I? = rv + l?, are respectively the radiation, neutron and total widths of the resonance, then or = (l?,/r)u; the energy dependence of the resonance cross-section then follows from the Breit-Wigner formula
E,*
0
a(E) = u,, E
1
-
1 + X2 ’
in which x = (E - E,)/$I’, and uO is the resonance cross-section at the energy E, of the relevant nuclear level. When the nuclei of the scatterer are executing thermal motion the apparent energy dependence of the cross-section is different, due to the Doppler effect. To determine it in this more general case, the nuclear absorption cross-section for any given neutron energy, measured in the laboratory system of co-ordinates, must be replaced by the average over all the possible values resulting from the relative motion of the neutron and nucleus. Assuming that the temperature T”K of the medium is greater than the Debye temperature 0 of the absorbing material, it is foundta) that 0, (x,8 = oo’r (x,0,
s
I
mexp [-(P/4(x
E = r/A,
In (3), k is Boltzmann’s constant.
Y(x,Q
4,
1 +y2
-m
if
- yN
(3)
A = 2
may be regarded as the real part of a complex function = u (x*9*) + iv(x*,y*)
w(x*,y*) as follows : wx,o
= 5$
where x* = E/2x, y* = E/2, and
u(x*,y*h
cc
y*e& dt
U(x*,y*) = .!
rr s --oo(x* -
t)Z + y*z
.
FADDEEVA and TERENTEW give a table of u(x*,y*) as a function of x* and y*. As a result of the averaging process the resonance line is observed to be broader than the line for a single nucleus, and to have a lower peak value. The true resonance integral
s co
JR =
%(E) $
0
is unchanged, but Jeff increases due to the widening of the “danger zone” AE,,. Substituting into (1) the expressions obtained above, and ignoring the weak variation of r,(E) and E-l across the interval AE,, one obtains Jeff=_
TV ._x8 24
*
V&Q
P s -,W&
h
+ Wpo)
The integration in (4) may be taken between the limits -cc AE,, x > x,ff = :r
.
(4)
and co without significant error, since for
164
Letters to the editors
the integrand decreases with x sufficiently strongly. Equation (4) shows that the effective resonance integral depends on the temperature of the medium through E(T), and depends also on the potential scattering cross-section IS, = &/p appropriate to an isolated nucleus. If the medium were to be imagined to become indefinitely rarefied, so that p -+ 0, then Jeff+ JR = &T (o,I’,)/E, since m Y(x,E) du = x. s --m In the general case it is convenient to write Jeff as the product
Jeff = JR,f’(E,W). Here l/a = C,/C,, C, = poO. A magnitude f < 1 indicates self-screening of the resonance due to large absorption and scattering cross-sections. At T = 0 (5 = co), there is no Doppler effect; moreover y + l/(1 + x*), and so
f(->:)=fm(i) =d(l:l,a). Putting (5) one defines a function
(6) which contains the temperature T+
dependence of Jeff. In particular,
~0 (A+
~0, l-
a),
~(E,l/a)-t
at T = 0, 7 = 1; and for
~‘(1 + l/a),
whilef+ 1 because at high temperatures the effective resonance cross-section self-screening does not occur. Under the condition xE2 < 1, Y (x,E)
N
q
5 exp
becomes so small that
- $ (
1
and the apparent shape of the resonance is determined by the Doppler effect alone. If E/u < 1 the dominant contribution to the integral in (6) comes from values of x satisfying the condition x5” < 1, so that in this case
The explicit determination of 7 normally calls for numerical integration. For this purpose it is convenient to change to the variable y = x/x,fr, x,rf being given by the condition Y(x,rr, 5) = a. Then
The integral in this expression is approximately unity and depends only weakly on a and E. In the actual computations a further change of variable to z = (1 + y)-I, giving
is of assistance. The integral may alternatively be reduced by proceeding on the following lines. In the expression $I!‘) dx, F is essentially an arbitrary function which falls to zero sufficiently fast as Y + 0 s (x -+ h-z). Changing the variable of integration to r = ‘I’(x,t), Y(O,8 Jeff = s0
Jeff =
Letters to the editors
165
0.5
t=e,t)
FIG. 1.-f@&. .&0*04
3.63*43*2302*82*6-TJ 4 ;
2.42.2FO+81*61*4l-2-
_1.
FIG. 2.-q
withf(t,t)
in which
= - (aY/ax)/Y.
Y’(Y,l)
(&z),
may be represented by the series in which
166
Letters to the editors
ForE-+O,
For t -+ hXl andfort+O f(t,t)
N 2t3/a 1/(1 - t).
Graphs off(Q) and 7 (6,1/a) are shown in Figs. 1 and 2. The former is included for illustrative purposes only, and in practical calculations of the temperature dependence of resonance absorption integrals the curves of ~7should be employed. I. V. GORDEEV v.
v.
ORLOV
T. KH. SEDEL’NIKOV REFERENCES 1. GLASSTONES. and EDLUND M., The Elemenis of Nuclear Reactor Theory. Macmillan, London (1952). 2. AKH~EZERA. and POMERANCHUK I., Topics in Nuclear Theory. Gostekhizdat, Moscow (1950). 3. FADDEEVAV. N. and TERENTEVN. M., Tables of the ProbabilityIntegral withComplex Argument (Edited by Academician V. A. FOCK). Gostekhizdat, Moscow (1954).
Nuclear energy levels of l@Tu* (Received 15 April 1957) YTTERBIUMof natural isotopic composition was irradiated with slow neutrons from the RPT pile for a month.“’ After thirty days the ytterbium was separated chromatographically from other rare earth elements and the *sOYbradiation (half life 30.6 day@) was examined, using a double focusing /3-spectrometer,‘a1 a scintillation counter and a proportional counter filled with heavy gases (Ar, Kr). We paid special attention to the soft part of the /3-spectrum and to the soft y-rays from laeTu. Using the &Xctrometer we first determined the following conversion lines: K - 63.13 keV; MI -N - 8.42 keV; L,M - 20.74 keV. With the proportional counter we observed lines corresponding to the following atomic transitions, Lal; Lpl; K,,,,; Kbs 1 (transition energies 7.18; 8.10; 504 and 57.5 keV respectively) and also the y-transitions of lasTu w&h energies 21.74 and 63.13 keV. Altogether 13 y-transitions were detected by the three methods; the multipolarities are shown in brackets.7 8.42(Ml + E2); 20.74(Ml); 63.13@1): 93.62(0.9 Ml + 0.1 E2); 109.67(Ml); 118.20@?2); 130.48@2); 156?; 177.21(0.75 Ml f 0.25 E2); 197.97(Ml); 240.6@1?); 307.7@2) keV. Examination of the y-rays enabled the following nuclear levels of lssTu to be established: 8.42; 118.20; 13890; 316.06; 379.19 and 472.8 keV. The ground state of lasTu has spin I, = +:I61 With the multipolarities shown for the y-transitions, the first three levels can be attributed to rotational levels with spins 3/2+, 5/2& and 7/2&. The last three levels have spins 7/2 + ,7/2F and 9/2F . The elucidation of the parity of these levels requires separate consideration. The first is evidently a one particle level. We can write the Bohr-Mottelson formula for the energy levels of a nucleus with ZO= + in the form: El = E0 + u t2”[Z(Z+ 1) + u(-~)I+~‘~(Z + 1))l + AMI + 8 + d-l)I+l’*
(Z + &)I2
* Translated by R. BERMAN from Atomnaya Energiya 3, No. 9, 256 (1957). t The multiplicity is determined by comparing the experimental and theoretical conversion coefficients. The theoretical values were obtained by interpolation and extrapolation of the data tabulated by BAND(~) and by ROSE.(~)
S~rv
and