Temperature Dependence of Filter Properties

An issue that may be relevant to the accurate determination of the filter transmission on orbit is the temperature dependence of the amplitude of EXAFS. According to EXAFS theory the amplitude of the EXAFS oscillations are a function of temperature. This dependence arises from the fact that thermal vibrations of the atoms in a solid produce a phase mismatch of the backscattered electron wave function. The transmission properties of the ACIS filters were measured at room temperatures $\sim$ 20 C while the on orbit filter temperature is expected to be about -60 C. The temperature dependence of the EXAFS component is incorporated in a Debye-Waller type term Q(k,T) (Stern et al., 1975). For thermally induced disorders of atoms and for deviations about the average shell distance of R_j which follow a Gaussian distribution, Q(k,T), is given by,

$$Q(k,T) = e^{-2k^{2}{\sigma}^{2}}$$ (49)

where $\sigma$ is the mean square deviation about the average value R_j. For the Einstein model of lattice vibrations, where motions between adjacent atoms are uncorrelated $\sigma^{2}$ has the form,

$${\sigma^2} = {{\hbar}\over{M_{r}{\omega}}}{1\over{e^{ { {\hbar}{\omega} } \over {kT} }-1}}$$ (50)

where M_r is the reduced mass and $\omega$ is the frequency of vibration of the atoms and is related to the Einstein temperature of the solid through the expression,

$$\omega = {{T_{ein}k}\over{\hbar}}$$ (51)

To derive the temperature dependence of the filter transmission we assume that $\chi(k) = C(k)Q(k,T)$. If we define f_sm as the smooth component of the transmission function, (as defined in section 5), and f_tot as the total transmission function then using the above assumption and equation (4) we obtain,

$${{f_{tot}(T)}\over{f_{sm}(T)}}\:=\:{{e^{-{\rho}{(\mu}-{\mu_{... ...{\rho}{\mu_{0}}{\chi(k)}}}}\:=\: {e^{-{\rho}{\mu_0}C(k)Q(k,T)}}$$ (52)

and solving for C(k) we have,

$${C(k)} = -{{1}\over{{\rho}{\mu_0}}}{{\ln{({{f_{tot}(T_{gr})}\over{f_{sm}}})}}\over{Q(k,T_{gr})}}$$ (53)

The on orbit transmission at temperature T_orb is expressed as a function of the transmission as measured on ground at temperature T_gr by the expression,

$${{f_{tot}(T_{orb})}\over{f_{tot}(T_{gr})}} = e^{-C(k){\rho}{\mu_{0}}[Q(k,T_{orb})-Q(k,T_{gr})]}$$ (54)

In Figure 5.12 we show the percent change in filter transmission above the Al-K absorption edge for an expected on orbit filter temperature of -60 C.

 

Figure 5.12: Top Panel; Filter transmission of Spectrometer filter above the Al-K absorption edge at 20 and -60 C. Lower Panel; Percent change in filter transmission between 20 and -60 C