CALCULATIONS OF DOSE ENHANCE~NT. IN DEVICE STRUCTURES. J.C. GARTH. Using a semi-empirical electron transport model and the solution of the ...
Nuclear Instruments and Methods in Physics Research B40/41 (1989) 1266-1270 North-Holland, Amsterdam
CALCULATIONS
OF DOSE ENHANCE~NT
IN DEVICE STRUCTURES
J.C. GARTH
Using a semi-empirical electron transport model and the solution of the onegroup S, transport equation for a multilayered system, we have developed a fast algorithm for calculating X-ray dose enhancement in device structures. The program MULTILAYER, written in BASIC for an IBM-PC compatible microcomputer, calculates (I) the dose profile across a multilayered structure including electron transport, (2) the absorbed dose neglecting electron transport and (3) the ratio of these two quantities: the “dose enhancement” profile. The dose as a function of position can be found for a structure with l-10 layers. MULTILAYER uses monoenergetic photons and is valid over the 100-1250 keV photon energy range. The output compares well with much longer electron Monte Carlo and discrete-ordinates transport calculations.
1. htroduetion In this article we describe a simple method for calculating the dose distribution in plane geometry in materials irradiated by gamma rays and X-rays. In particular, using a microcomputer, we can find the dose across a multilayer structure, such as a complex electronic device, so that the high-energy electron transport between any number of layers can be readily handled without lengthy Monte Carlo calculations, This method has been incorporated into a BASIC program known as MULTILAYER, written for an IBM-PC compatible ~cr~omputer [I]. We want to calculate the “dose enhancement” profile, the position-dependent ratio of the dose including electron transport effects to the dose due to X-ray absorption alone (the “equilibrium dose”). For an interface between two thick material layers of high and low atomic number, the dose in the lower atomic number (Z) material at X-ray energies below 300 keV is greater than its equilibrium value due to X-ray-produced electrons emitted from the higher-z material [2]. For example, for silicon next to gold, the dose enhancement at the interface can be as much as a factor of 30 at 100 keV 121. At 500 keV and above the dose enhancement ratio depends on the X-ray photon direction and can be either greater or less than 1 [2,3]. In any case, the dose enhancement profile is a rapidly varying function of the distance from the interface. Althougb our numerical technique was developed for electronics applications, the method is also valid for dose calculations for medical irradiations, food processing, dosimeters and other radiation applications. In general, the greater the difference in Z between adjacent materials, the greater will be the dose enhancement effect.
2. Mathematical
model
A systematic study of dose profiles as a function of photon energy at two-medium interfaces, using the Monte Carlo method, was performed by Garth et al. [2]. Based on curve fits to these profiles, a semi-empirical model for the dose in the low-Z material next to a high-Z material was developed by Burke and Garth [4]_ The formula they obtained for the dose profile in medium 2 due to a uniform isotropic source of X-ray-produced electrons in medium 1 is Q,(X)
= CA,
exp( -BiX),
(1)
0168-583X/89/$03.50 0 Elsevier Science Publishers B.V. (North-Homed
Physics Publishing
Division)
J.C. Garth / Calculations of dose enhancement in device structures
1267
where
and
2O+P*) O+n,)
B,=
’
(1
-P2)
1
n2
R,(4)
’
where x is the distance in medium 2 from the interface, Z,, Z, the atomic numbers and &, fi2 the electron backscatter coefficients for medium 1 and 2, respectively (where the empirical formula of Darlington [5], j3 = 0.1865 Z”.3185 - 0.25,
(4)
is used to find the backscatter coefficient of each medium from the atomic numbers Z, and Z,), pi is the photon interaction coefficient for the ith type of interaction (Compton, photoelectric, Auger) in medium 1, pi the probability that an electron (rather a fluorescent X-ray) is generated in the ith interaction, E, the energy of the photo-, Compton or Auger electron in the ith interaction, R,(E), R,(E) the continuousslowing-down approximation (CSDA) electron range in medium 1 and 2, respectively, for electron energy E [keV], and ni, n2 the exponents for medium 1 and 2 in the range formula Rj(E) = KjE”~ [4]. Here the dose D2i(x) is expressed in keV cm2/g. Adding D2i(x) to a similar dose contribution Dz2(x) in medium 2 from electrons generated in medium 2 (see ref. [6]), the dose enhancement as a function of x in medium 2 is given by b,,(x)
+
42b)1/4~~h
(5)
where hv is the X-ray photon energy and p,,(hv) is the mass-energy absorption coefficient in medium 2. We remark that the dependence of Ai and Bj on the backscatter coefficients & and p2 was obtained in ref. [4] by treating transport as a series of electron reflections back and forth between medium 1 and 2. The S, transport equation that we employ in the next section leads to formulas for the dose with the same functional form. 2.2. One-group S, transport equation To obtain a solution for the multimedium electron transport problem, we employ the mathematical apparatus developed for the neutron transport theory [6,7]. The one-speed (one-group) S2 transport equation for the angular flux qj,,,( x) in the j th medium is given by
qrnz +F
=&(
J
‘kil +
ej2) + qJx),
J
where m = 1,2, the discrete ordinate direction angles are given by Q = -n2 = l/G, Xj and cj are total mean free path and single-scattering albedo, respectively, for the jth medium, and Sj,(x) is source function for particles generated in the m th direction in the j th medium. In the first term on right side of eq. (6) isotropic scattering has been assumed. In ref. [6] we showed that a nearly complete correspondence between the semi-empirical model and S, transport equation could be obtained if we related the cj and the Xj to the aj and R/(E) by equations c.
J
=
1
_
(l- 4j2 C1+
Bj)'
the the the the the
(7)
’
and ‘j=(l-cj)Rj(E).
(8) XI. DETECTORS/CALIBRATION
J.C. Garth / Calculations of dose enhancement in device structures
1268
Also in ref. [6] the solution to the two-medium case, where both regions 1 and 2 are semi-infinite, was obtained. The dose expression corresponding to eq. (1) (neglecting the effects of the directionality of the source electrons, also treated in ref. [6]) is 41(x)
=
PiPiEi
2
R1CE;I.I (1-PI)(~ ’ &) R,(E;,) 0 - PlP2)
ev[-B2(&bl
T
where
6
o+a2>
B2(Ei) =(1 -
&)
R,(E,)
00)
*
Comparing eqs. (9) and (10) with eqs. (l), (2) and (3), we see that the coefficient A, is the same in the expressions for &r(X), but the B,(E,) differ by the ratio 2(1+ n,)/(fin,) (this ratio equals 1.83 for n2 = 1.7 for Si [4])_ To make the MULTILAYER algorithm agree with the semi-empirical model, we have used eq. (3) rather than eq. (10) to determine the B’(E,) coefficients in eq. (11) below. This is justified because the slope of the dose profile near an interface is always steeper than the slope of the angular or scalar flux.
We now write down the form of the solution of eq. (6) for N layers. Consider the layered structure shown in fig. 1. We will assume solutions in the ith medium of the form q,(x)=5
exp[-B,(x-x,_,)]
+/3&
exp[-B,(x-xi_,)]
!Q(X)=fljq
+L”j
exp[-Bj(xi-x)]
+cpil(x),
(IIa)
eXP[-Bj(Xj-X)]
++jz(X),
(lib)
where $,(x) is an inhomogenous term which is nonzero only in the region where the ith electron source term for a given photon interaction in the k th medium is nonzero. (We also take photon attenuation into account in the source term, so that the source is nonu~fo~.) We will assume infinite medium boundary conditions on layers 1 and N by setting C, and I&, = 0, but we assume the X-ray intensity = 1 at x = x0. If the source function for electrons arising from the i th interaction in the k th medium is given by k-l sjmw
=
f$k -(I
+
tisdki)
ev[
-cLk(x
-
xk-,>I
II
ed-MG),
(12)
I=0
where pk is the X-ray attenuation
*..
2
1
coefficient
of the k th layer and
fkiis a factor between - 1 and 1
N
N-l
hv 3
