3 SVD analysis for the thick-target model

We consider now the case of the thick-target model. Within this framework the determination of the injection rate F₀(E₀) from the photon spectrum can be accomplished in two ways. A first possibility is represented by explicitly solving the inner integral at the right hand side of Eq. (5). If $\tilde{Q}(\epsilon,E_0)$ is the result of this integration, F₀ can be determined by solving the linear inverse problem with discrete data  

$${\vec{g}}=A_0 F_0$$ (46)

with A₀ defined by  

$$(A_0 F_0)(\epsilon_n) =\int_{\epsilon_n}^{\infty} \tilde{Q}(\epsilon_n,E_0) F_0(E_0) {\rm d}E_0$$ (47)

with $n=1,\ldots,N$.Otherwise F₀ can be obtained by consecutively solving the two linear ill-posed inverse problems represented by the discrete version of Eq. (6) and by Eq. (7).

We note that for both these approaches it is necessary to choose an explicit form for the average energy loss rate R(E). In the case of a collisional cold but ionized thick target characterized only by Coulomb losses we have  

$$R(E)=\frac{K}{E}$$ (48)

with $K=2 \pi {\rm e}^4 \Lambda$; e is the electron charge and $\Lambda$ is a parameter which depends on the Debye length and which can be considered constant in the energy range typical of this application.

If the determination of F₀ is addressed by solving problem (5) the singular value decomposition can be performed in the case of the Kramers and the Bethe-Heitler approximations by computing the Gram matrix. In the case of the Kramers cross-section we find  

$$\tilde{Q}(\epsilon,E_0)=\frac{Q_0}{K} \frac{E_0 - \epsilon}{\epsilon}$$ (49)

and in order to deal with square integrable integral kernels we consider the linear inverse problem with discrete data  

$$g(\epsilon_n)=\frac{1}{\epsilon_n} \int_{\epsilon_n}^{\infty... ...lpha}(E_0) \frac{E_0 - \epsilon_n} {E_{0}^{\alpha}} {\rm d}E_0$$ (50)

with $G_{\alpha}(E_0)=F_0(E_0) E_{0}^{\alpha}$ (all the multiplicative constants have been put equal to one). If $\alpha \gt 3/2$ the functions  

$$\phi_n(E_0)= \left\{ \begin{array} {lr} 0 & E \leq \epsilon_... ...silon_n}{E_{0}^{\alpha}} & E \gt \epsilon_n \end{array} \right.$$ (51)

are in $L^2(0,\infty)$ for all $n=1,\ldots,N$. For $n \geq m$ the nm entry of the Gram matrix is given by

$$\begin{eqnarray} G_{nm} & = & \frac{1}{\epsilon_n \epsilon_m} \left[ \frac{1}{2\... ..._m \frac{1}{2\alpha-1} \frac{1}{\epsilon_{n}^{2\alpha-1}} \right] \end{eqnarray}$$ (52)

and for n<m

 

G_nm=G_mn .

If the Bethe-Heitler cross-section is assumed, we have

$$\begin{eqnarray} g(\epsilon_n) &=& \frac{1}{\epsilon_n} \int_{\epsilon_n}^{\inft... ..._n}{E_0}}}{1- \sqrt{1-\frac{\epsilon_n}{E_0}}}\right] {\rm d}E_0 .\end{eqnarray}$$ (54)

We assume again $\alpha \gt 3/2$ so that the functions  

$$\phi_n(E_0) = \left\{ \begin{array} {lr} 0 & E_0 \leq \epsil... ...{E_0}}}\right] \end{array} & E\gt\epsilon_n \end{array} \right.$$ (55)

are square integrable; in this case the entries of the Gram matrix are computed by using numerical integration.

 

Table 2: Condition numbers for the thick-target model in the case of the Kramers (K) and the Bethe-Heitler $\rm (B-H)$ approximations for different numbers N of sampling points. The photon energy range is $\epsilon_1=10$ keV, $\epsilon_N=100$ keV and the sampling is geometric

The condition numbers for the problems (50) and (54) are presented in Table 2 and in Fig. 2 we plot the first four singular functions in the case of N=25 geometrically sampled points in the energy range $\epsilon_1=10$ keV, $\epsilon_N=100$ keV. As in the case of the thin-target model, if the Bethe-Heitler cross-section is adopted the instability of the problem is greater and the resolution limit achievable is poorer for prescribed solution accuracy.

  

Figure 2: Singular functions for the thick-target model in the case of the Kramers (solid) and the Bethe-Heitler (dashes) cross-section: a) first singular function; b) second singular function; c) third singular function; (4) fourth singular function