point kernel calculation of dose fields from line sources using

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Radiat. Phys. Chem. Vol. 51, No. 2, pp. 121–128, 1998 7 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0969-806X(97)00128-X 0969-806X/98 $19.00 + 0.00

POINT KERNEL CALCULATION OF DOSE FIELDS FROM LINE SOURCES USING EXPANDED POLYNOMIAL FORM OF BUILDUP FACTOR DATA: GENERALIZED SECANT INTEGRAL-SERIES REPRESENTATION IVAN MICHIELI Rud-er Bosˇ kovic´ Institute, Department of Technology, Nuclear Energy and Radiation Protection, Bijenicˇka 54, P.O. Box 1016, 10000, Zagreb, Croatia

(Received January 1997; accepted July 1997) Abstract—Recently, point source buildup factor data approximation, in the form of an expanded polynomial set, was successfully introduced. In this paper, it is demonstrated how that function fits into the integrand of the Point Kernel for line source with slab shield, leading to the generalized Secant integrals of the form:

g

c

Ia (c,b) = b a

e−b

sec 8

(sec 8)ad8, for a r 0, b q 0, 0 Q c R p/2.

0

Two rapidly convergent series representations are introduced for such integrals: one for the 0 Q c R p/4, and another for p/4 Q c R p/2. An upper bound of the absolute error assures significant improvement of accuracy for every additional term of the series. 7 1998 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

In a quest for an adequate functional representation of new point isotropic gamma-ray buildup factor data (ANSI/ANS, 1991 6.4.3 standard data) a function based on an expanded polynomial set was recently introduced (Michieli, 1994). The approximation function has the form: 3

B(x,E0) = 1 + exp[bX ]x s Aix , a

i

i=0

where x represents shield thickness (in mfps), a and b are constants for selected shield material and Ai are independent parameters of the function. This function was validated as a possible choice, besides the well known G-P function (Harima et al., 1986) that was adopted by ANSI/ANS—6.4.3, for point kernel calculations. Results of approximation for the exposure buildup factors in water, concrete, iron beryllium and lead, showed that the maximum error was within 4% for all five materials over the standard data domain. Interpolation error for intermediate energies was generally within 10%, even in the vicinity of the K-edge energy in lead. In this paper it is demonstrated how the new function fits into the integrand of the point kernel for line source geometries shielded by infinite slab shields. The results are expressible in terms of

generalized Secant integrals that could be defined as:

g

c

Ia (c,b) = b a

e−b

(sec 8)ad8 ...

sec 8

(1)

0

with a r 0, b q 0, and 0 Q c R p/2. The same definition is given by Hungerford, 1962 for Secant integrals of the nth kind (integer values of parameter a). Besides the numerical integration approach that was applied by Hungerford and APDA’s Computer Group for computing the value of Secant integrals of the first and second kind (a = 1,2), various expansions and rational approximations are in use for computing the value of ‘‘Sievert’’ integral (a = 0) and as a special case of the ‘‘Bickley’’ function (a = 0, c = p/2) depending on the values of integral parameter b and c (Sievert, 1921, 1930; Abramowitz and Stegun, 1972; Wood, 1982). Also, extensive data for Sievert integral values are available in the form of tables and plots (Rockwell, 1956; Shure and Wallace, 1975). Applicability of the above mentioned expansions and approximations, in practice, is limited to different interval values of integral parameters (Wood, 1982). Besides, when Secant integral values are available in tabulated form, interpolation for intermediate values is always difficult (Shure and Wallace, 1975; Chilton et al., 1984).

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In Section 3 of this paper two rapidly convergent series based on different Taylor expansions, in terms of incomplete Gamma functions, are proposed for representing generalized Secant integral [equation (1)]. It is demonstrated in Section 4 (Figs 2 and 3), that in the worst case for such expansions (a = 0, c r p/4), approximately two new members of the series are needed for each new exact decimal digit of the integral value.

it follows RL = R(E0)

3

R(E0,r) =

g6

7

B(r,r',E0)R(E0)e − Smixi/4p=r' − r=2 S(r')dV,

V

where xi are respective shield components thickness (in the line from the differential volume source element dV to the detector), R(E0) is the appropriate flux to detector response conversion factor and S(r') is the source strength. The integrand in brackets is by definition the point kernel function. For an isotropic line source of uniform strength SL [phot cm − 1 s − 1] with a slab shield (Fig. 1.) the differential detector response at point P is given as: dRL =

SLd1 e − mt 4p(h sec 8)2

R(E0)B(mt sec 8),

sec 8

since d1 = h sec28d8 and B(mt sec 8) = 1 + e b(mt

sec 8)

3

s Ai (mt sec 8)a + i,

$g

i=0

c

e − mt

sec 8

d8

0

+ s Ai

g

c

e − mt

%

(mt sec 8)a + id8 .

sec 8(1 − b)

0

In accordance with the definition of generalized Secant Integrals [equation (1)], this result could be presented as:

2. EXTENSION OF POINT KERNEL TECHNIQUES TO INCORPORATE BUILDUP

Within point kernel integration techniques the solution for total detector response at point P(r) of a given monoenergetic source–shield configuration, could be presented by the following equation (Wood, 1982):

SL 4ph

RL = R(E0)

6

SL I (c,mt) 4ph 0 3

+ s Ai (1 − b) − a − iIa + i [c,mt(1 − b)] i=0

7

(2)

The above equation is the sum of two parts i.e. the unscattered and scattered components of the total dose rate. I0(c, mt) is commonly known as the ‘‘Sievert Integral’’. If the detector response of interest is the absorbed energy rate in the material or the absorbed dose rate it is common practice to approximate the first with the kerma rate, and the second with the closely related quantity called ‘‘exposure’’ that is proportional to the kerma rate in air (Chilton et al., 1984). For kerma calculations the appropriate flux to kerma conversion factor is R(E0) = E0 men /r, where men /r is the mass energy absorption coefficient for given material and source energy (ANSI/ANS–6.4.3. ANSI/ANS, 1991; Hubbell and Seltzer, 1995). It depends only on material and source energy and consequently doesn’t influence on the point kernel integration process. For example if one chooses R(E0) = 1, detector response becomes the total flux density FL. Of course, for each particular choice of detector response, the respective buildup factors must be used.

i=0

3. REPRESENTATION OF THE GENERALIZED SECANT INTEGRAL

Using the the substitution sec8 = x, equation (1) can be written as:

g

sec c

Ia (c,b) = b a

e − bx

1

xa dx xzx 2 − 1

(3)

For a = 0, this integral becomes the Sievert Integral I0(c,b) that could be presented in terms of Exponential Integrals (Abramowitz and Stegun, 1972) as: a

I0(c,b) = I0(p/2,b) − s ki (cos c)2i + 1E2i + 2(b sec c) i=0

Fig. 1. Line source with slab shield.

(4)

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where I0(p/2,b) is known as the ‘‘Bickley’’ function (Wood, 1982), and with coefficients of expansion given, in double factorial notation, as: ki = (2i − 1)!!/ (2i)!!. Expression (4) is obtained by expanding (x 2 − 1) − 1/ 2 in powers of (1/x) and knowing that the validity of such expansion is limited to =(1/x)= Q 1 (Gradshteyn and Ryzhik, 1980). The disadvantage of such a representation is the slow convergence of the expansion for small c values and the need for different rational approximations of the Bickley function depending on the value of the parameter b (Wood, 1982). 3.1. General solution To simplify the notation we introduce the auxiliary definition:

Using a Taylor expansion of x a − 1(x + 1)−1/2 in powers of (x − 1) we obtain:

g

Ia (co ,b) = b a

Ia (c,b) = Ia (co ,b) + Ix a (co ,c,b), where c q co q 0. Using the same expansion of (x 2 − 1) − 1/2 in powers of (1/x) we can write:

g

Ix a (co ,c,b) = b a

sec c

a

01

e − bxx a − 1 s ki

sec co

i=0

1 x

valid for; p/2 r c q co q 0, and with coefficients of the expansion given in double factorial notation as: ki = (2i − 1)!!/(2i)!!, (note that ( − 1)!! = 1). After integration and with the definition of the Incomplete Gamma function (Abramowitz and Stegun, 1972): G(m,y) =

g

a

e − tt m − 1dt,(y q 0),

y

equation (5) becomes:

$

Pa,i =

1 z2

i

s ( − 1)j j=0

g

1

e − bx

x a − 1(x + 1)−1/2 zx − 1

a − 1 (2j − 1)!! 1 − j (2j)!!2j

(9)

In general, the following cases are present: 3.1.1. a = 0 (Sievert integral). Since

0 1

−1 = ( − 1)i − j, i−j

then we can write: P0,i =

( − 1)i z2

i

s j=0

(2j − 1)!! (2j)!!2 j

The sum in the above expression consists of terms that are identical to the first i terms of the binomial expansion of (1 − x) − 1/2 and consequently converges to z2. Therefore =p0,i = R 1 and sign p0,i = − sign p0,i + 1 . It follows that =p0,i (x − 1)i= in equation (7) is dominated by a geometric sequence =(x − 1)i= and hence the series in equation (8) alternates and converges for 0 Q co Q p/3. 3.1.2. a = 1.

(6)

0 =0 i−j

except for j = i, we can write: P1,i =

(−1)i (2i − 1)!! z2 (2i)!!2i

From the above expression it is obvious that =p1,i + 1= Q =p1,i = R 2 − i − 1/2, and hence =p1,i (x − 1)i= in equation (7) is dominated by the geometric sequence =[(x − 1)/2]i=. Therefore, for a = 1, the series in equation (8) alternates and converges at least for 0 Q co Q ArcSec[3] 1 70.5°. It is also evident that we can write the general equation for pa,i in the following form: i

dx

0 1

0 1

For an appropriate choice of co , the sum in the above equation presents a rapidly convergent series for Ix a (co ,c,b). It is easily recognized that with a = 0, c = p/2, co = c and using the functional relation En (y) = y n − 1G(1 − n,y), the above sum is the same as the one presented previously for the Sievert integral in equation (4). The final step is to find an adequate representation for Ia (co , b). For that purpose we rewrite equation (3) as follows: sec co

%

where g(m,y) = G(m) − G(m,y) = f0ye − tt m − 1dt, for m q 0 (Abramowitz and Stegun, 1972). Applying Leibniz’s rule for nth derivative of a product (Gradshteyn and Ryzhik, 1980) it can be easily verified that the coefficients of the above expansion are given as:

Since

i=0

Ia (co ,b) = b a

s pa,i (x − 1)idx

a 1 1 Ia (co ,b) = e − b s pa,ib a − i − 2 g i + ,b(sec co − 1) (8) 2 i=0

a

Ix a (co ,c,b) = s kib 2i + 1[G(a − 2i − 1,b sec co ) −G(a − 2i − 1,b sec c)]

a

e − bx

(7) zx − 1 i = 0 generally valid for 0 Q co Q p/3 (brief analysis follows). After integration we get:

2i + 1

dx (5)

sec co

1

Ix a (c1,c2,b) = Ia (c2,b) − Ia (c1,b), c2 q c1 q 0 The Generalized Secant integral [equation (1)] can be presented as:

123

Pa,i = s

0 1

j=0

a−1 P1,i − j j

(10)

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Since the binomial coefficients satisfy the relation

01 0 1 0 1

It is also evident that ra,i satisfies the functional relation:

m m−1 m−1 = + , j j−1 j ra,i =

using equation (10) we can readily establish the general recurrence formula: Pa,i = Pa − 1,i − 1 + Pa − 1,i

(11)

3.1.3. a = 2,3,4,...,n. From the general expression (10) it follows that we can relate pn,i to p1,i by the simple equation:

0 1

n−1

Pn,i = s

u=0

n−1 p1,i − j , j

0 1

a−1 1 1 r + = =. i 2 a,i − 1 z2

Using the above functional relation it can be shown that for a q 1 and for i q a − 1, ra,i presents a decreasing sequence. Since =pa,i = are bounded above by ra,i , the series in equation (8) converges at least for 0 Q co Q p/3. Finally, from the preceding analysis follows that for 0 Q c R p/2, the generalized Secant Integral can be represented as: 3.1.5. Representation for 0 Q c R co Q p/3.

and accordingly; n−1

0 1

1 n−1 3n − 1 =Pn,i = Q i − n + 1 s j = i + 1/2 j 2 2 z2 j = 0 2 1

i.e. the coefficients are bounded above by the geometric sequence similar as in the previous case. So for a = 2,3,..n, the series in equation (8) also converges at least for 0 Q co Q ArcSec[3]. Furthermore, successively applying the recurrence relation equation (11) it can be easily shown that beginning with the i = n − 1, coefficients pn,i start to alternate i.e.: sign[pn,i + 1] = − sign[pn,i ] and also =pn,i + 1= Q =pn,i = for i r n − 1. That feature is important for the error bound analysis that follows (Section 4). 3.1.4. a q 0, (non integer). We distinguish two cases:

a

1

i=0

3.1.6. Representation for co Q c R p/2. a

Ia (c,b) = Ia (co ,b) + s kib 2i + 1[G(a − 2i − 1,b sec co ) i=0

−G(a − 2i − 1,b sec c)]

$

1 ,b(z2 − 1) 2

a−1 Q 1, j

0 1

a−1 = ( − 1)j j

and consequently sign [pa,i + 1] = − sign[pa,i ]. So, for 0 Q a Q 1 the series in equation (8) alternates and converges at least for 0 Q co Q p/3.

a

+ s kib 2i + 1G[ − 2i − 1,bz2]

3.1.4.2. a q 1

i

0 10 1

a−1 1 s= = =pa,i = R ra,i = j 2 j = 0 z2

(14)

The values of the incomplete Gamma functions G(m,y) and g(m,y) can be easily obtained from known rapidly convergent series expansions or a continued fraction representation (Luke, 1969; Press et al., 1988). The advantage of the above representation of the generalized Secant Integral is in rapid convergence (appropriate choice of co ) with expansion coefficients that don’t depend on the value of parameter b. It converges fast enough for all practical applications. Besides, one can use functional relations for calculating successive incomplete Gamma functions (Gradshteyn and Ryzhik, 1980) such as:

This situation is not so simple, but it is evident that 1

%

i=0

then =pa,i = Q =p0,i = R 1. It is also clear that for a Q 1 sign

1

i=0

b0 1b

(13)

It turns out that co = p/4 is a good choice for both series in equation (12) and equation (13) to converge fast enough for all practical cases. As a special case for the Bickley function (a = 0, c = p/2) and since secp/4 = z2 and secp/2 = a, we can write: a

If we compare pa,i to p0,i it becomes apparent that,

%

1 ,b(sec c − 1) (12) 2

I0(p/2,b) = e − b s p0,ib − i − 2 g i +

3.1.4.1. a Q 1

since

$

Ia (c,b) = e − b s pa,ib 1 − i − 2 g i +

i−j

g(m + 1,y) = mg(m,y) − y me − y G(r + 1,y) = rG(r,y) + y re − y

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Fig. 2. Absolute error bounds, for series representation of Ia (p/4,0.01), defined by inequality equations (15) – (17).

From the above relations we can readily obtain:

0

g

1

$0 1

1 1 (2i − 1)!! + i,y = ,y g 2 2 2i

in equation (7) start to alternate as soon as i r n − 1, and accordingly for i r n − 1 we can write: 3

=Erri (In )=c R p/4 Q e − bb n − i − 2 =pn,i + 1=

%

i−1

$

1 2k + 1 y 2 + k and k = 0 (2k + 1)!!

− e−y s

G(r − i,y) =

×g i +

G(r − i) r − k − 1 y i = 1,2,3,... G(r − k) k=0 i−1

− e−y s

g

=Erri (Ia )=c R p/4 R b a

sec co

1

4. ERROR ESTIMATE

4.1. 0 Q c R co = p/4 For this argument value interval, the generalized Secant Integral Ia (c,b) is represented by the infinite series given in equation (12). As demonstrated in Section 3, we generally distinguish three cases: 4.1.1. 0 R a R 1. In this case the terms pa,i (x − 1)i in equation (7) (x = secc R z2) present an alternating decreasing sequence and according to the Leibniz theorem (Sokolnikoff and Redheffer, 1984) the error in the series expansion in equation (8) or (12) is less than absolute value of the first omitted term, that is: =Erri (Ia )=c R p/4 Q e

3

b

2

$

3 ,b(sec c − 1) 2

%

(16)

a

e − bx

s =pa,k =(x − 1)kdx

zx − 1 k = i + 1

In Section 3 we established the relation =pa,i = R ra,i and since for i q a − 1, ra,i presents a decreasing sequence, the following inequality is valid: a

a

k=i+1

k=i+1

s =pa,k =(x − 1)k Q ra,i + 1 s (x − 1)k, for i q a − 1.

Computing the sum of the remaining geometric series, and knowing that 1 Q x R z2 we can write: a

ra,i + 1 s (x − 1)k k=i+1

= ra,i + 1

=pa,i + 1=

×g i+

%

4.1.3. a q 1 (non integer). If the series in equation (7) is terminated after i terms, the absolute value of the introduced error satisfies following relation:

G(r − i) G(r,y) G(r)

−b a−i−

3 ,b(sec c − 1) 2

(x − 1)i + 1 Q 2ra,i + 1(x − 1)i + 1 2−x

and after integration, for i q a − 1 we can write: (15)

4.1.2. a = n = 2,3,4,..... This is a similar situation to the previous one except that the terms pn,i (x − 1)i

3

=Erri (Ia )=c R p/4 Q 2e − bb a − i − 2 ra,i + 1

$

×g i+

3 ,b(sec c − 1) 2

%

(17)

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In Fig. 2, the absolute error-bound curves, resulting from the above expressions, for different a values (b = 0.01, c = p/4), are plotted in a logarithmic scale up to i = 15. Adding more terms of the series results in an exponential decrease of the error bound. For different a values the slope of the respective curve changes in accordance with the behaviour of the associated =pa,i = (or ra,i ) values. It is worth mentioning that for higher b values and with b q a, the major part of the Ia (c,b) value originates from the interval near c = 0(x = 1) because of the higher exponential attenuation for greater c values. Consequently for high b values we can obtain a desired accuracy (prescribed relative error) with fewer terms in the expansion (12), regardless of the c value. As an example of such behavior, for b q 10 and with a Q 2 we can supply the value of the integral Ia (c r p/4,b) within 1%, using only two terms (i = 0,1) of the expansion (12) (for c q p/4 we put c = p/4). Of course, for smaller c values, convergence of the series (12) becomes more rapid and for c Q 10° (keeping a Q 4), single-term expansion assures approximation with relative error below 1% regardless of b value. It is also important to note that for higher a values (with constant b) a few more terms, proportional to a value, are needed to supply the same percentage of the integral value as in the case of single-term approximation for a = 0. Nevertheless, as soon as more terms are added, the approximation with higher a value becomes superior to that with a = 0. Such behavior is a direct consequence of the expansion coefficient’s features that were analyzed in the previous section. Besides, in the realm of buildup factor polynomialbased approximations, higher a values ( q 5) are hardly expected.

4.2. p/4 Q c R p/2 For this argument value interval, the generalized Secant Integral Ia (c,b) is presented with the sum of the two infinite series defined in equation (13). The error bound for the first series is defined in the previous analysis. The second term in equation (13) is the infinite series representation for Ix a (co ,c,b). The error bound for that part can be quantified in following manner: If the sum in equation (5) is terminated after i terms, the introduced error satisfies the relation:

g

Erri (Ix a )co = p/4 = b a

sec c

a

sec co

j=i+1

e − bxx a − 1 s kj

01 1 x

2j + 1

dx.

The following inequalities are also valid: a

s kj j=i+1

01 01 01 1 x

2j + 1

1 x

a

Q ki + 1 s

01

j=i+1

2j + 1

1 x

2i + 3

= ki + 1

1 1− x

2

R 2ki + 1

01 1 x

2i + 3

for x r z2.

Finally, after integration we get: Erri (Ix a )co = p/4 Q 2ki + 1b 2i + 3[G(a − 2i − 3,bz2) −G(a − 2i − 3,b sec c)]

(18)

In Fig. 3, the error-bound curves, resulting from the above expression, for different a values (b = 0.01, co = p/4, c = p/2), are plotted in a logarithmic scale, up to i = 15. An exponential decrease of error bounds is obvious. The final slopes of the curves are the same because the coefficients of expansion ki do not depend on a value.

Fig. 3. Absolute error bounds, for series representation of Ix a (p/4,p/2,0.01), defined by inequality equation (18).

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Table 1. Three-term approximation of Sievert Integral I0(c,b) c [°] 1° 5° 10° 20° 30° 45° 60° 80° 90°

b

0.01

A B' C A B' C A B' C A B' C A B' C A B' C A B0 C A B0 C A B0 C

0.01728 0.01728 0.000 0.08640 0.08640 0.000 0.1728 0.1728 0.000 0.3455 0.3455 0.005 0.5181 0.5185 0.069 0.7766 0.7862 1.228 1.0341 1.0381 0.381 1.3722 1.3757 0.257 1.5136 1.5171 0.233

0.1 0.01579 0.01579 0.000 0.07895 0.07895 0.000 0.1578 0.1578 0.000 0.3152 0.3152 0.005 0.4715 0.4718 0.069 0.7021 0.7105 1.206 0.9238 0.9274 0.391 1.1783 1.1816 0.278 1.2286 1.2319 0.267

1 0.006420 0.006420 0.000 0.03206 0.03206 0.000 0.06388 0.06388 0.000 0.1258 0.1258 0.005 0.1836 0.1837 0.065 0.2574 0.2600 1.008 0.3077 0.3091 0.456 0.3282 0.3296 0.414 0.3283 0.3296 0.414

5

10

20

35

1.176–04* 1.176–04 0.000 5.842–04 5.842–04 0.000 1.147–03 1.147–03 0.000 2.127–03 2.127–03 0.004 2.828–03 2.829–03 0.048 3.320–03 3.335–03 0.434 3.407–03 3.418–03 0.345 3.409–03 3.421–03 0.344 3.409–03 3.421–03 0.344

7.920–07 7.920–07 0.000 3.912–06 3.912–06 0.000 7.536–06 7.536–06 0.000 1.308–05 1.308–05 0.004 1.593–05 1.594–05 0.033 1.697–05 1.700–05 0.153 1.702–05 1.704–05 0.143 1.702–05 1.704–05 0.143 1.702–05 1.704–05 0.143

3.594–11 3.594–11 0.000 1.754–11 1.754–11 0.000 3.261–10 3.261–10 0.000 5.053–10 5.053–10 0.003 5.550–10 5.550–10 0.015 5.609–10 5.610–10 0.028 5.609–10 5.610–10 0.028 5.609–10 5.610–10 0.028 5.609–10 5.610–10 0.028

1.0985–17 1.0985–17 0.000 5.267–17 5.267–17 0.000 9.316–17 9.316–17 0.000 1.272–16 1.272–16 0.002 1.312–16 1.312–16 0.005 1.313–16 1.313–16 0.006 1.313–16 1.313–16 0.006 1.313–16 1.313–16 0.006 1.313–16 1.313–16 0.006

* Read as 1.176 × 10 − 4 A = ‘‘exact’’ values of the Sievert integral I0(c,b) from Shure and Wallace, 1975. B' = three-term approximation by equation (12) of this paper. B0 = three-term approximation by equation (13) of this paper. C = percentage relative error =%=.

An interesting behavior occurs for small b and larger a values. In this case the major part of the Ia (p/2,b) value, originates from the interval in the vicinity of c = p/2 (x q q 1), because of the strong peak of the integrand value in that region. As a result, for b Q 0.1 and with a q 3 we can supply a value of Ia (p/2b) within 1% with a single term (i = 0) of the expansion in equation (6) for Ix a (p/4,p/2,b).

5. NUMERICAL EXAMPLES

To illustrate the efficiency of the introduced series representation of Generalized Secant Integral, we calculate the values of the Sievert Integral (a = 0) for a wide range of b and c values using equation (12) and equation (13) with only three terms (i = 0, 1, 2) of the series. The results together with the exact values and percentage relative error are given in Table 1.

6. CONCLUSION

Line sources with slab shields are frequently encountered in gamma-ray attenuation problems as an approximation to more complex source–shield geometries. Using buildup data in the form of an expanded polynomial set, point kernel calculation of dose fields from such geometries ends up with the solution that is expressible in terms of generalized Secant integrals.

The introduced series representations in terms of incomplete Gamma functions [equation (12) and equation (13)] turn out to be adequate (beside numerical integration) for computing generalized Secant integral values to a desired accuracy. The main features of the proposed representations are listed below: 1. Expansion coefficients of both series are not dependent on the value of parameter b. 2. Choosing co = p/4 assures convergence that is fast enough for both series, regardless of a and c values. In the worst case (a = 0, c r p/4) approximately two new terms of the series are needed for each new exact decimal digit of the integral value. 3. For all practical applications (a R 5) three-term approximation provides a value of the integral within 1% or better. 4. For smaller c values (c R 20°) a single or two-term approximation is usually sufficient for practical application. 5. Successive values of incomplete Gamma functions can be easily computed using known functional (recurrence) relations. REFERENCES

Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Dover, New York. ANSI/ANS (1991) ANSI/ANS—6.4.3—1991, Gamma-Ray Attenuation Coefficients and Buildup Factors for Engineering Materials. American Nuclear Society, La Grange Park, Illinois.

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Chilton, A. B., Shultis, J. K. and Faw, R. E. (1984) Principles of Radiation Shielding. Prentice-Hall, Inc., New Jersey. Gradshteyn, I. S. and Ryzhik, I. M. (1980) Table of Integrals, Series and Products. Academic Press, New York. Harima, Y., Sakamoto, Y., Tanaka, S. and Kawai, M. (1986) Validity of the geometric-progression formula in approximating gamma-ray buildup factors. Nuclear Science Engineering 94, 24. Hubbell, J. H. and Seltzer, S. M. (1995) Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients 1 keV to 20 MeV for elements Z = 1 to 92 and 48 additional substances of dosimetric interest. In NISTIR, Internal report No. 5632. Hungerford, H. E. (1962) Tables of secant integrals of the first and second kinds. Nuclear Science Engineering 14, 312. Luke, L. Y. (1969) The Special Functions and their Approximations, Vol. I and II. Academic Press, New York. Michieli, I. (1994) The use of an expanded polynomial

orthogonal set in approximations to gamma-ray buildup factor data. Nuclear Science Engineering 117, 110. Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1988) Numerical Recipes. Cambridge University Press, Cambridge. Rockwell, T. III (1956) Reactor Shielding Design Manual. McGraw-Hill Book Co. Inc., New York. Shure, K. and Wallace, O. J. (1975) Compact tables of functions for use in shielding calculations. Nuclear Science Engineering 56, 84. Sievert, R. M. (1921) Die Intensita¨tsverteilung der prima¨ren g-Strahlung in der Na¨he medizinischer Radiumpra¨parate. Acta Radiologica 1, 89. Sievert, R. M. (1930) Die g-strahlungsintensita¨t an der Oberfla¨che und in der na¨chsten Umgebung von Radiumnadeln. Acta Radiologica 11, 239. Sokolnikoff, I. S. and Redheffer, R. M. (1984) Mathematics of Physics and Modern Engineering. McGraw-Hill Book Co. Inc., New York. Wood, J. (1982) Computational Methods in Reactor Shielding. Pergamon Press, New York.