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A parametrization of the mass attenuation coefficients for elements with Z=1 to Z=92 in the photon energy range from approximately 1 to 150 keV
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1991 Phys. Med. Biol. 36 987 (http://iopscience.iop.org/0031-9155/36/7/007) View the table of contents for this issue, or go to the journal homepage for more
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Phys. Med. Bioi., 1991, Vol. 36, No 7, 987-999. Printed in the U K
A parametrization of the mass attenuation coefficients for elements with 2 = 1 to Z =92 in the photon energy range from -1 to 150 keV Robert G Ouellett and L John Schreinerf
t Departemem de Physique BiomCdicale, HBpital Notre-Dame, 1560 est rue Sherbrooke Montreal, QuCbec, HZL4M1, Canada f Department of Oncolagy (Radiation Oncology), McCill University, 1650 avenue der Cedrer Montreal, Quebec, H3G lA4, Canada Received 31 May 1990. i n final form 12 March 1991 Abstract. Parametric equations which require only 148 constants to generate mass attenuation coefficients are presented. They are valid for elements ( Z from I to 92) in the diagnostic radiology photon energy range (from 1 keV, or the L, absorption edge if Z 2 11, to I50 keV). The deviation from standard tabulated data is in general 40 keV for aluminium) and the mass attenuation coefficients are better fit by the sum of two-power laws, p / p ( Z , h v ) = A ( Z ) ~ V " ~C(Z)huD'"'. '+ The added term is required to best parametrize the p / p for energies greater than the K edge energy for Z between 32 and 54, inclusive (region V). The fitting routines also give improved results of the p / p parametrization in the photon energy range between the L, and K edges for Z =21 to Z = 37 (region 11) if the sum of two-power laws is used to generate the fitted attenuation coefficients. For the lighter elements ( Z < 30) at low energies ( h v < 15 keV) the dependence of the photoelectric cross sections on the photon energy differs considerably from the power law, particularly at energies in the immediate vicinity of the K edges. This deviation is believed to be caused by shifts in the electronic wave functions which result from secondary ionization induced by Auger electrons (Loi er al 1977). To account forthe shift from a power law dependence, an exponential term, F ( Z ) eh"'G'z), is introduced into the formulation. The parameters F ( Z ) and G ( Z ) are negative so that the term diminishes the total p / p near the K edge energy but has little effecton the calculations at higher photon energies. Therefore, the parameterization for region IV is fixed throughout the energy range up to 150 keV. It should be stressed that the discussion above is not meant to indicate that the different terms in equation ( 1 ) can be identified directly with particular photon interactions with the elements. Indeed this is not the case. While a single-power law term expresses the tabulated data well in the regions in which the photoelectric effect dominates the interactions, the addition of other terms in the other photon energyatomic number regions is not based a priori o n the added contribution from Compton or coherent scattering to the total p / p values. They are required to improve thegoodness of the fit of the model to the tabulated total mass attenuation data. Models which are based on the physical interactions are known to be considerably more complex
R G Ouellet and L J Schreiner
Parametrization of mass attenuation coefficienf
993
R G Oueller and L J Schreiner
994
Table 8. The polynomial parameters required for the evaluation of the mass attenuation coefficients in region VI: p / o [ Z , h u ) = A ( Z ) h v s ' Z ' . Deviation of calculated from tabulated attenuation coefficients in region VI:
mean =0.8%, maximum=3.0%.
K S hv s 150 keV
Region VI A[Z)=
1
o,Z'
55s z s 9 2
B(Z)=
i=o
1
biz'
i-0
-1.664E+ 06
bo
-2.423
b,
-9.872E-03
(I,
5.977E+04 -4.395E+02 0.6004
ba b,
1.195E - 04 -1.01SE-07
ad
-2.124E-03
a. a, o2
(McCullough 1975, White 1977, Hawkes and Jackson 1980, Jackson and Hawkes 1981) than the parametrization presented here. The ability of the equations given in tables 1-8 to accurately generate tabulated mass attenuation coefficients is illustrated in figures 2 and 3. Figure 2 shows the percentage deviation of the calculated p i p for aluminium for photon energies from the K absorption edge to 150 keV. The deviations of the attenuation coefficients calculated using a number of other models are included for comparison. The parametrization proposed compares quite favourably with the results of the other models. The percentage deviation from the tabulated data of the attenuation coefficients calculated by our model are reviewed for all elements in figure 3. The tabulated data are reproduced with an overall mean error of 1.4% and a maximum deviation of 6.9%. There are eight elements (Na, AI, Si, S, Ar, Mn, Ni, Te) for which the mean error is greater than 3% (but less than 4%); however, the large deviations are restricted in general (except for Ar) to photon energies below the K absorption edge of the elements. For photon energies between the K edge and 150 keV the calculated pip values have mean deviations of
