Modeling the Transmission Data and Determining an X-ray Transmission Function.

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,  

$${{I}\over{I_{o}}} = {\prod{ e^{-\mu_{i}{ \rho_{i} }}}},$$ (40)

where $\mu_{i}$ is the mass absorption coefficients of constituent compound i and $\rho_{i}$ is the mass per unit area of the constituent compound i. $\mu_{i}$ can be expressed as a function of the compound's constituent elements j as,

$${\mu_{i}} = {\sum{{\mu_{j}}w_j}},$$ (41)

where $\mu_{j}$ is the mass absorption coefficients of element j, and w_j is the fraction by weight of element j. Tabulated values for the mass absorption coefficients for elements with atomic weights ranging from Z = 1 to Z = 92 may be found in Henke et al. (1993).

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 Al₂O₃/Al:Si/Polyimide/Al:Si/Al₂O₃. Free parameters for this fit were the mass per unit area of Polyimide (C₂₂H₁₀O₄N₂) and Al. The large values for $\chi^{2}$ 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 $\chi^{2}$ of $\sim$ 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 $2.734 \times 10^{-05}$ gr cm^-2 and Al of $4.071 \times 10^{-05}$ gr cm^-2 and for the Spectroscopy filter Polyimide of $2.632 \times 10^{-05}$ gr cm^-2 and Al of $3.068 \times 10^{-05}$ 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 $\chi(k)$ is defined as the oscillating part of the mass absorption coefficient and is given by,

$${\chi(k)} = {{\mu - {\mu_0}}\over{\Delta\mu_0}}$$ (42)

where $\mu_0$ is the smoothly varying part of the mass absorption coefficient corresponding to an isolated atom, $\Delta\mu_0$ is the change in the mass absorption coefficient over the absorption edge, and k is the wavenumber of the scattered photoelectron given by,

$${k} = {{\sqrt{2m(E-E_{edge})}}\over{\hbar}}$$ (43)

The model used to fit the oscillatory component of the transmission has the form,

The term $a_2(E-E_{j,edge})^{-\frac{1}{2}}$ represents the phase shift of a photoelectron as it traverses the distance 2R, where R is the interatomic separation, a₃ and $\exp{(-a_{1}(E-E_{j,edge}))}$ 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 ${a_{4}}\sin{({a_{5}}E^{\frac{1}{2}} + a_6)}$. 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,

$$step(E_{j,1},E_{j,2}) = 0, \mbox{for $E_{j,1} \gt E$\space or $E_{j,2} < E$}$$ (44)

$$step(E_{j,1},E_{j,2}) = 1, \mbox{for $E_{j,1} < E < E_{j,2}$}$$

In Figure 5.2 and Figure 5.3 we show fits of our simple EXAFS model to the regions above the Al-K and C-K edges. Because of the limitations of the multilayer monochromator the EXAFS above the N-K and O-K edge were not resolved. In Figure 5.2 we also show the relevant energy boundaries used in our model. E_j,edge defines the energy of the absorption K edge, (E_j,2, E_j,4) , (E_j,1,E_j,2), (E_j,3,E_j,4), define the boundaries of the first, second and third term of equation (5.6) respectively.

  

Figure 5.3: Top Panel; C-K EXAFS with model fit. Lower Panel; Difference between data and model  . Figure 5.2: Top Panel; Al-K EXAFS with model fit. Lower Panel; Difference between data and model  .

The total transmission function of the ACIS filters is finally described by the following expression,

 

$$f(E) = \prod{e^{-{\mu_{j}}{\rho_{j}}}} + \sum{\xi_{j}(E)step(E_{j,1},E_{j,4})}$$ (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 f₁ and f₂ 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.

 

$$\xi_{al} \xi_{c}$$ Filter Fitted Parameters
for Smooth Component
gr cm^-2

$$\rho_{Polyimide}=2.73400 \times 10^{-5}$$ Imager, 6765-8 a₀ = 0.00391 a₀ =0.00009895

$$\rho_{al}=4.07111 \times 10^{-5}$$ a₁ = 8.58072 a₁ = -39.49626

$$\rho_{Al_{2}O_{3}}=7.2 \times 10^{-7}$$ a₂ = 71.89980 a₂ = 97.56914

a₃ = 3.48600 a₃ = 3.02784

a₄ = 0.05499 a₄ = 0.05998

a₅ = 400.001 a₅ = 150.00030

a₆ = 0.39964 a₆ = 4.46030

a₇ = 0.04000 a₇ = 0.199918

a₈ = -0.07724 a₈ = -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

$$\rho_{Polyimide}=2.63186 \times 10^{-5}$$ Spectrometer,6732-8 a₀ = 0.00263 a₀ = 0.00013

$$\rho_{al}=3.06826 \times 10^{-5}$$ a₁ = 5.00026 a₁ =-39.99043

$$\rho_{Al_{2}O_{3}}=7.2 \times 10^{-7}$$ a₂ = 71.79978 a₂ = 97.99200

a₃ = 3.49500 a₃ =3.00020

a₄ = 0.04000 a₄ = 0.08000

a₅ = 400.00003 a₅ = 145.00028

a₆ = 0.40503 a₆ = 0.88018

a₇ = 0.03200 a₇ = 0.34000

a₈ = -0.06001 a₈ = -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.

  

Figure 5.5: Top Panel; X-ray transmission data of Spectrometer filter with best fit transmission model. Lower Panel; Percent difference between model fit and data.   Figure 5.4: Top Panel; X-ray transmission data of Imager filter with best fit transmission model. Lower Panel; Percent difference between model fit and data.