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A method commonly used to model the X-ray transmission of filters assumes that the absorption through a multilayer filter with constituent compounds i is described by the equation,
(40) |
(41) |
We initially performed a least squares fit to the transmission data of the Imager and Spectroscopy filters using the function given in equation 5.2 while considering a multilayer layer filter structure of the form Al_{2}O_{3}/Al:Si/Polyimide/Al:Si/Al_{2}O_{3}. Free parameters for this fit were the mass per unit area of Polyimide (C_{22}H_{10}O_{4}N_{2}) and Al. The large values for obtained in these fits make them formally unacceptable. The residuals to these fits indicate that most of the discrepancy between the model and data occurs above the absorption edges of C, N, O, and Al. The transmission data clearly show oscillations above the absorption edges that extend up to several hundred eV. Such structures are commonly known as extended X-ray absorption fine structure and occur only when atoms are in condensed matter. The oscillations arise from interference of the scattered electron wavefunction outgoing from a central atom, i, with the backscattered electron wavefunctions from nearby atoms, j. A detailed review of EXAFS theory and applications may be found in Stern and Heald (1983).
In our next attempt to fit the transmission data we excluded energies corresponding to EXAFS and used the same function as in the previous model. The values obtained for of 0.2 (for 442 degrees of freedom) indicate a significant improvement in the fit. We obtain values for the mass per unit area for the Imager filter, for Polyimide of gr cm^{-2} and Al of gr cm^{-2} and for the Spectroscopy filter Polyimide of gr cm^{-2} and Al of gr cm^{-2}.
For the purpose of determining a function that fits the transmission data well, we used a very simplistic EXAFS model that incorporates features from the independent particle model developed by Stern 1978, Lee and Pendry 1975, Stern et al. 1975. Our model considers only interference effects from the nearest atomic shell.
The EXAFS component is defined as the oscillating part of the mass absorption coefficient and is given by,
(42) |
(43) |
The model used to fit the oscillatory component of the transmission has the form,
The term represents the phase shift of a photoelectron as it traverses the distance 2R, where R is the interatomic separation, a_{3} and account for phase shifts in the presence of potentials, disorders and thermal vibrations of atoms about their average distance R from the central atom. Near edge structure in our transmission data is modeled with the term . Equation 5.6 does not take into account the nonlinear dependence of the phase shift and the dependence of the backscattering amplitude on k. A more physical model will be presented in a future publication.
For our modeling purposes j takes the values j=0 for the Al-K edge and j=1 for the C-K edge. We define the function step(E_{j,1},E_{j,2}) as follows,
(44) |
The total transmission function of the ACIS filters is finally described by the following expression,
(45) |
For energies excluding the EXAFS regions the transmission function f(E) is given by the fit of our model given by equation (2) that incorporates the atomic scattering factors f_{1} and f_{2} as tabulated by Henke et al., (1993). For energies within the EXAFS regions our model function f(E) includes in addition the best fit model to the EXAFS regions. The values for all the relevant parameters that enter equation (5.8) are listed in Table 5.2.
Filter | Fitted Parameters for Smooth Component gr cm^{-2} |
Fitted Parameters for(E) Component |
Fitted Parameters for (E) Component |
Imager, 6765-8 | a_{0} = 0.00391 | a_{0} =0.00009895 | |
a_{1} = 8.58072 | a_{1} = -39.49626 | ||
a_{2} = 71.89980 | a_{2} = 97.56914 | ||
a_{3} = 3.48600 | a_{3} = 3.02784 | ||
a_{4} = 0.05499 | a_{4} = 0.05998 | ||
a_{5} = 400.001 | a_{5} = 150.00030 | ||
a_{6} = 0.39964 | a_{6} = 4.46030 | ||
a_{7} = 0.04000 | a_{7} = 0.199918 | ||
a_{8} = -0.07724 | a_{8} = -0.06410 | ||
E_{0,1} = 1.556 | E_{1,1} = 0.29 | ||
E_{0,2} = 1.5775 | E_{1,2} = 0.3095 | ||
E_{0,3} = 1.558 | E_{1,3} = 0.3095 | ||
E_{0,4} = 1.89 | E_{1,4} = 0.3600 | ||
E_{0,edge} = 1.56 | E_{1,edge} = 0.2842 | ||
Spectrometer,6732-8 | a_{0} = 0.00263 | a_{0} = 0.00013 | |
a_{1} = 5.00026 | a_{1} =-39.99043 | ||
a_{2} = 71.79978 | a_{2} = 97.99200 | ||
a_{3} = 3.49500 | a_{3} =3.00020 | ||
a_{4} = 0.04000 | a_{4} = 0.08000 | ||
a_{5} = 400.00003 | a_{5} = 145.00028 | ||
a_{6} = 0.40503 | a_{6} = 0.88018 | ||
a_{7} = 0.03200 | a_{7} = 0.34000 | ||
a_{8} = -0.06001 | a_{8} = -0.11108 | ||
E_{0,1} = 1.556 | E_{1,1} = 0.29 | ||
E_{0,2} = 1.5775 | E_{1,2} = 0.3098 | ||
E_{0,3} = 1.5580 | E_{1,3} = 0.3098 | ||
E_{0,4} = 1.89 | E_{1,4} = 0.36 | ||
E_{0,edge} = 1.56 | E_{1,edge} = 0.2842 |
The transmission data for the Imager and Spectroscopy filters together with the model transmission function and the percent difference between the model and data are presented in Figure 5.4 and Figure 5.5.
Mark Bautz