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Self-shielding for thick slabs in a converging neutron beam
Nuclear Instruments and Methods in Physics Research A 432 (1999) 399 } 402
Self-shielding for thick slabs in a converging neutron beam D.F.R. Mildner* National Institute of Standards and Technology, US Department of Commerce, Gaithersburg, MD 20899-8395, USA Received 28 October 1998; received in revised form 5 February 1999
Abstract We have previously given a correction to the neutron self-shielding for a thin slab to account for the increased average path length through the slab when irradiated in a converging neutron beam. This expression overstates the case for the self-shielding for a thick (or highly absorbing) slab. We give a better approximation to the increase in e!ective shielding correction for a slab placed in a converging neutron beam. It is negligible at large absorption mean free paths. 1999 Elsevier Science B.V. All rights reserved. Keywords: Converging neutron beam; Highly absorbing samples; Neutron focusing lens; Neutron self-shielding
The count rate for prompt gamma activation measurements is proportional to the incident neutron current density, the area irradiated by the beam, the macroscopic absorption cross section & and the e!ective thickness t of the sample as seen by the incident beam. (We assume that the scattering cross section is negligible compared to the absorption cross section.) In practice, neutron self-shielding by the sample causes a #ux gradient. Consequently, the count rate is proportional to [1!exp(!& t)], rather than & t, and needs to be corrected by the inverse of [1!exp(!& t)] (& t)\, so that measurements are placed on a sim ilar basis. When a converging neutron beam is used, the average length of the paths traversed by neutrons through the sample is increased, giving rise to a small increase in count rate. This alters the self-shielding correction.
* Tel.: 1-301-975-6366; fax: 1-301-208-9279. E-mail address: [email protected] (D.F.R. Mildner)
We consider a convergent monochromatic beam from a neutron focusing lens [1], and assume that the convergent beam is azimuthally symmetric and uniform up to the maximum convergence angle. We have shown [2] that in the thin sample approximation (& t 1) the total rate of interactions with in the sample from a convergent beam is given by N"J A & t g(h ) $ $ +
(1)
where J is the current density at the focus of area $ A . The factor g(h ) accounts for the convergence $ + of the beam, and is given by g(h )"2(1!cos h ) cosech + + +
(2)
where h is the maximum convergence angle of the + beam. This expression correctly reduces to unity for the normal quasi-parallel beam (h "0), and in+ creases monotonically for h '0. In addition, we + have shown that this factor remains valid when the sample is placed at some angle other than normal to the beam, though the e!ective thickness is increased.
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 2 5 6 - 9
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D.F.R. Mildner / Nuclear Instruments and Methods in Physics Research A 432 (1999) 399} 402
The convergence factor is now given by a power series in & t: g(h )" g (h ) (& t)L\ (5) + L + L where the leading term g (h ) is that for the thin + sample approximation (Eq. (2)). The other terms are given by g (h )"log(cos h ) cosec h , + + + and for n53
Fig. 1. A schematic diagram of a convergent beam focused onto a slab sample placed normal to the beam axis.
We now apply the same analysis to the selfshielding for a thick slab irradiated normally in a converging neutron beam (see Fig. 1). Following Ref. [2], the rate of interactions within a sample of thickness t is given by
N"2J A cosec h $ $ +
F+
[1!exp(!& t sec h)]
;sin h cos h dh.
(3)
We expand the exponential in a power series to give N"!2J A cosec h $ $ +
F+ [(!& t sec h)L/n!] L
;sin h cos h dh "2J A cosec h [(!& t)L/n!] $ $ + L F+ ; secL\ h d(cos h)
(4)
(6)
g (h )"(!1)L2/[n!(n!2)] L + ;(1!secL\ h ) cosec h . (7) + + Note that using L'Ho( pital's rule, these expressions reduce at h "0 such that + N"J A [(1!exp(!& t)], (8) $ $ which is the usual expression for the interaction rate for a thick sample in a quasi-parallel beam. Table 1 shows values of g (h ) for n45 evaluL + ated for convergence angles h "0.53, 103, 153 and + 203. We observe that the ratios g (h )/g (0) given in L + L Table 2, where g (0)"(!1)L\/n! appear to L increase systematically with n. That is, if we write g (h )"1# f (h ), we may also write + + g (h )/g (0)"1#n f (h ) to a good approximation. L + L + This approximation becomes progressively worse at large convergence angles and at large n. However at h "203, the value of g (h ) for n"8 di!ers by + L + only 0.5% from this formula, and at h "403, the + value of g (h ) for n"4 di!ers only by 3.8%. L + The approximation allows us to express Eq. (5) as g(h )" g (0)[1#n f (h )](& t)L\ + L + L "[1!exp(!& t)](& t)\ # f (h ) exp(!& t). (9) + Hence, the neutron shielding correction factor for a thick slab in a converging beam is now given by Eq. (9), where the values of f (h ) are given in + Table 3. Note that for the quasi-parallel beam (h "0), f (h )"0 and the correction is as before. + + For a thin sample, Eq. (9) reduces to g(h )"1# f (h ), as determined earlier [2]. + +
D.F.R. Mildner / Nuclear Instruments and Methods in Physics Research A 432 (1999) 399} 402
401
Table 1 Values of the coe$cients g (h ) for n45 L + h (deg) +
g (h ) +
g (h ) +
g (h ) +
g (h ) +
g (h ) +
0 5 10 15 20
1.0 1.00191 1.00765 1.01733 1.03109
!0.5 !0.50191 !0.50769 !0.51753 !0.53175
0.16667 0.16762 0.17053 0.17554 0.18288
!0.041667 !0.041986 !0.042962 !0.044658 !0.047186
0.0083333 0.0084132 0.0086589 0.0090901 0.0097433
Table 2 Normalized values of the coe$cients g (h ) for n45 L + h 3 +
g (h )/g (0) +
g (h )/g (0) +
g (h )/g (0) +
g (h )/g (0) +
g (h )/g (0) +
0 5 10 15 20
1.0 1.00 1.00 1.01 1.03
1.0 1.00 1.01 1.03 1.06
1.0 1.00 1.02 1.05 1.09
1.0 1.00 1.03 1.07 1.13
1.0 1.00958 1.03906 1.09081 1.16919
191 765 733 109
382 539 507 349
573 320 328 726
765 109 180 247
Table 3 Values of the coe$cients f (h ) + h (deg) +
f (h ) +
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 0.00008 0.00031 0.00069 0.00122 0.00191 0.00275 0.00374 0.00489 0.00619 0.00765 0.00927 0.01105 0.01298 0.01508 0.01733 0.01975 0.02234 0.02509 0.02800 0.03109
For small values of the convergence angle the correction to the self shielding is small and decreases with increasing & t. Fig. 2 shows the correc tion to the self shielding formula for slabs caused by
Fig. 2. The correction to the self-shielding for a slab in a quasiparallel beam to account for a convergent beam with h "133 + as a function of & t. The thin slab approximation 1# f (h ) is + independent of & t, whereas the small convergence approxima tion goes from 1# f (h ) at & t"0 to 1 for large & t. +
a convergent beam with h "133, using both the + thin slab approximation and the better approximation given by Eq. (9). For in"nitesimally thin samples these two expressions are equivalent, given by 1# f (h ), but for & t'0 they diverge consider+ ably. We observe that the correction for small convergence angles is negligible for large & t.
402
D.F.R. Mildner / Nuclear Instruments and Methods in Physics Research A 432 (1999) 399} 402
This result can be understood by noting that the thin slab approximation considers only the leading term in the expansion of exp(!& t sec h)
1!& t sec h around & t"0. Instead, a better ap proximation considers the leading term of the expansion around h"0, the trajectory polar angle of the beam axis, or sec h"1, that is, 1!exp(!& tsec h) 1!exp(!& t) #& t exp(!& t)(sec h!1). (10) Substituting this expression into the integral of Eq. (3) results in N"J A [+1!exp(!& t),#& t exp(!& t) $ $ ;+2(1!cos h ) cosec h !1,], (11) + +
from which Eq. (9) follows. Clearly the small convergence approximation becomes increasingly worse for larger convergence angles. When the thick slab is placed such that the normal is at an angle to the beam axis, both Eqs. (9) and (11) and Fig. 2 hold, except that & t is now replaced by & t sec . References [1] Q.F. Xiao, H. Chen, V.A. Sharov, D.F.R. Mildner, R.G. Downing, N. Gao, D.M. Gibson. Rev. Sci. Instr. 65 (1994) 3999. [2] D.F.R. Mildner, H.H. Chen-Mayer, Nucl. Instr. and Meth. A 422 (1999) 21.
Self-shielding for thick slabs in a converging neutron beam D.F.R. Mildner* National Institute of Standards and Technology, US Department of Commerce, Gaithersburg, MD 20899-8395, USA Received 28 October 1998; received in revised form 5 February 1999
Abstract We have previously given a correction to the neutron self-shielding for a thin slab to account for the increased average path length through the slab when irradiated in a converging neutron beam. This expression overstates the case for the self-shielding for a thick (or highly absorbing) slab. We give a better approximation to the increase in e!ective shielding correction for a slab placed in a converging neutron beam. It is negligible at large absorption mean free paths. 1999 Elsevier Science B.V. All rights reserved. Keywords: Converging neutron beam; Highly absorbing samples; Neutron focusing lens; Neutron self-shielding
The count rate for prompt gamma activation measurements is proportional to the incident neutron current density, the area irradiated by the beam, the macroscopic absorption cross section & and the e!ective thickness t of the sample as seen by the incident beam. (We assume that the scattering cross section is negligible compared to the absorption cross section.) In practice, neutron self-shielding by the sample causes a #ux gradient. Consequently, the count rate is proportional to [1!exp(!& t)], rather than & t, and needs to be corrected by the inverse of [1!exp(!& t)] (& t)\, so that measurements are placed on a sim ilar basis. When a converging neutron beam is used, the average length of the paths traversed by neutrons through the sample is increased, giving rise to a small increase in count rate. This alters the self-shielding correction.
* Tel.: 1-301-975-6366; fax: 1-301-208-9279. E-mail address: [email protected] (D.F.R. Mildner)
We consider a convergent monochromatic beam from a neutron focusing lens [1], and assume that the convergent beam is azimuthally symmetric and uniform up to the maximum convergence angle. We have shown [2] that in the thin sample approximation (& t 1) the total rate of interactions with in the sample from a convergent beam is given by N"J A & t g(h ) $ $ +
(1)
where J is the current density at the focus of area $ A . The factor g(h ) accounts for the convergence $ + of the beam, and is given by g(h )"2(1!cos h ) cosech + + +
(2)
where h is the maximum convergence angle of the + beam. This expression correctly reduces to unity for the normal quasi-parallel beam (h "0), and in+ creases monotonically for h '0. In addition, we + have shown that this factor remains valid when the sample is placed at some angle other than normal to the beam, though the e!ective thickness is increased.
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 2 5 6 - 9
400
D.F.R. Mildner / Nuclear Instruments and Methods in Physics Research A 432 (1999) 399} 402
The convergence factor is now given by a power series in & t: g(h )" g (h ) (& t)L\ (5) + L + L where the leading term g (h ) is that for the thin + sample approximation (Eq. (2)). The other terms are given by g (h )"log(cos h ) cosec h , + + + and for n53
Fig. 1. A schematic diagram of a convergent beam focused onto a slab sample placed normal to the beam axis.
We now apply the same analysis to the selfshielding for a thick slab irradiated normally in a converging neutron beam (see Fig. 1). Following Ref. [2], the rate of interactions within a sample of thickness t is given by
N"2J A cosec h $ $ +
F+
[1!exp(!& t sec h)]
;sin h cos h dh.
(3)
We expand the exponential in a power series to give N"!2J A cosec h $ $ +
F+ [(!& t sec h)L/n!] L
;sin h cos h dh "2J A cosec h [(!& t)L/n!] $ $ + L F+ ; secL\ h d(cos h)
(4)
(6)
g (h )"(!1)L2/[n!(n!2)] L + ;(1!secL\ h ) cosec h . (7) + + Note that using L'Ho( pital's rule, these expressions reduce at h "0 such that + N"J A [(1!exp(!& t)], (8) $ $ which is the usual expression for the interaction rate for a thick sample in a quasi-parallel beam. Table 1 shows values of g (h ) for n45 evaluL + ated for convergence angles h "0.53, 103, 153 and + 203. We observe that the ratios g (h )/g (0) given in L + L Table 2, where g (0)"(!1)L\/n! appear to L increase systematically with n. That is, if we write g (h )"1# f (h ), we may also write + + g (h )/g (0)"1#n f (h ) to a good approximation. L + L + This approximation becomes progressively worse at large convergence angles and at large n. However at h "203, the value of g (h ) for n"8 di!ers by + L + only 0.5% from this formula, and at h "403, the + value of g (h ) for n"4 di!ers only by 3.8%. L + The approximation allows us to express Eq. (5) as g(h )" g (0)[1#n f (h )](& t)L\ + L + L "[1!exp(!& t)](& t)\ # f (h ) exp(!& t). (9) + Hence, the neutron shielding correction factor for a thick slab in a converging beam is now given by Eq. (9), where the values of f (h ) are given in + Table 3. Note that for the quasi-parallel beam (h "0), f (h )"0 and the correction is as before. + + For a thin sample, Eq. (9) reduces to g(h )"1# f (h ), as determined earlier [2]. + +
D.F.R. Mildner / Nuclear Instruments and Methods in Physics Research A 432 (1999) 399} 402
401
Table 1 Values of the coe$cients g (h ) for n45 L + h (deg) +
g (h ) +
g (h ) +
g (h ) +
g (h ) +
g (h ) +
0 5 10 15 20
1.0 1.00191 1.00765 1.01733 1.03109
!0.5 !0.50191 !0.50769 !0.51753 !0.53175
0.16667 0.16762 0.17053 0.17554 0.18288
!0.041667 !0.041986 !0.042962 !0.044658 !0.047186
0.0083333 0.0084132 0.0086589 0.0090901 0.0097433
Table 2 Normalized values of the coe$cients g (h ) for n45 L + h 3 +
g (h )/g (0) +
g (h )/g (0) +
g (h )/g (0) +
g (h )/g (0) +
g (h )/g (0) +
0 5 10 15 20
1.0 1.00 1.00 1.01 1.03
1.0 1.00 1.01 1.03 1.06
1.0 1.00 1.02 1.05 1.09
1.0 1.00 1.03 1.07 1.13
1.0 1.00958 1.03906 1.09081 1.16919
191 765 733 109
382 539 507 349
573 320 328 726
765 109 180 247
Table 3 Values of the coe$cients f (h ) + h (deg) +
f (h ) +
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 0.00008 0.00031 0.00069 0.00122 0.00191 0.00275 0.00374 0.00489 0.00619 0.00765 0.00927 0.01105 0.01298 0.01508 0.01733 0.01975 0.02234 0.02509 0.02800 0.03109
For small values of the convergence angle the correction to the self shielding is small and decreases with increasing & t. Fig. 2 shows the correc tion to the self shielding formula for slabs caused by
Fig. 2. The correction to the self-shielding for a slab in a quasiparallel beam to account for a convergent beam with h "133 + as a function of & t. The thin slab approximation 1# f (h ) is + independent of & t, whereas the small convergence approxima tion goes from 1# f (h ) at & t"0 to 1 for large & t. +
a convergent beam with h "133, using both the + thin slab approximation and the better approximation given by Eq. (9). For in"nitesimally thin samples these two expressions are equivalent, given by 1# f (h ), but for & t'0 they diverge consider+ ably. We observe that the correction for small convergence angles is negligible for large & t.
402
D.F.R. Mildner / Nuclear Instruments and Methods in Physics Research A 432 (1999) 399} 402
This result can be understood by noting that the thin slab approximation considers only the leading term in the expansion of exp(!& t sec h)
1!& t sec h around & t"0. Instead, a better ap proximation considers the leading term of the expansion around h"0, the trajectory polar angle of the beam axis, or sec h"1, that is, 1!exp(!& tsec h) 1!exp(!& t) #& t exp(!& t)(sec h!1). (10) Substituting this expression into the integral of Eq. (3) results in N"J A [+1!exp(!& t),#& t exp(!& t) $ $ ;+2(1!cos h ) cosec h !1,], (11) + +
from which Eq. (9) follows. Clearly the small convergence approximation becomes increasingly worse for larger convergence angles. When the thick slab is placed such that the normal is at an angle to the beam axis, both Eqs. (9) and (11) and Fig. 2 hold, except that & t is now replaced by & t sec . References [1] Q.F. Xiao, H. Chen, V.A. Sharov, D.F.R. Mildner, R.G. Downing, N. Gao, D.M. Gibson. Rev. Sci. Instr. 65 (1994) 3999. [2] D.F.R. Mildner, H.H. Chen-Mayer, Nucl. Instr. and Meth. A 422 (1999) 21.