Dose estimation

It is sometimes useful to estimate the radiation dose received by the specimen. This can be done by selecting Dose estimation from the the Post analysis button at the top of stack_analyze. This calculation only becomes active when you have selected both $I$ and $I_{0}$ spectral regions, and you have entered a compound name as described below.

If $n$ photons of energy $E$ are incident upon a sample area $\Delta^{2}$ and the sample has a density of $\rho$ and a linear absorption coefficient of $\mu$, the skin dose ${\cal D}$ (absorbed energy per mass at the outer surface exposed to the beam) is given by

$${\cal D} = \frac{nE\mu}{\rho\Delta^{2}}.$$ (2)

We will write the number of photons $n$ as a flux $\phi$ multiplied by an exposure time $\tau$, and divided by a detector efficiency $\eta$. Using the notation of Henke et al. [4,5], the linear absorption coefficient $\mu$ can be written as

$$\mu = 2 \frac{\rho N_{A}}{A} r_{e} \lambda f_{2}$$ (3)

where $N_{A}$ is Avogadro's number, $A$ is the atomic weight of the substance, $r_{e}$ is the classical radius of the electron, $\lambda$ is the x-ray wavelength, and $f_{2}$ is the imaginary part of the complex, energy-dependent number of oscillators $(f_{1}+if_{2})$ per atom. Using $\lambda=hc/E$, we can rewrite Eq. 2 as

$${\cal D}=2N_{A} r_{e} hc \frac{\phi \tau f_{2}}{\eta \Delta^{2}A}$$ (4)

If we accumulate dose from a series $i=1,\ldots$ images at photon energies with various values of incident flux $\phi_{i}$ in $10^{3}$ photons/second, dwell time $\tau_{i}$ in milliseconds, and pixel size $\Delta$ in nanometers, the total dose ${\cal D}$ over the image sequence can be written as

$${\cal D} (\mbox{Gray}) = 2 N_{A} r_{e} hc \frac{1}{\eta \Delta^{2} A} \sum_{i}\phi_{i} \tau_{i} f_{2,i}$$

$$= \frac{6.74\times 10^{5}} {\eta [\Delta (\mbox{nm})]^{2}\cdot A\... ...i} (10^{3}\mbox{ photons/sec})\cdot \tau_{i} (10^{-3}\mbox{sec})\cdot f_{2,i}$$ (5)

where the numerical factor is arrived at in Eq. 9 of Sec. A. As can be seen from Eq. 5, the information that we need about the sample is contained in the ratio of $f_{2}(E)/A$. We thus wish to obtain an estimate of the energy-dependent oscillator strength $f_{2}(E)$ and the effective atomic weight $A$. Since the optical density $\mbox{OD}=\mu t$ scales like $\mbox{OD}(E) \propto \lambda f_{2}(E) \propto f_{2}/E$, we will write $\mbox{OD}(E)=x f_{2}(E)/E$. We can then determine $x$ at a particular energy $E_{0}$ and use it to write

$$f_{2}(E)= f_{2}(E_{0}) \frac{E}{E_{0}} \frac{\mbox{OD}(... ...} \frac{E_{i}}{E_{j}} \frac{\mbox{OD}_{i}}{\mbox{OD}_{j}}.$$ (6)

where in the second case we use a notation of energy indices $i$ relative to a particular energy indexed by $j$. To obtain an estimate of $f_{2,i}$ and $A$, the user must enter the compound's stoichiometric formula in the form H2O for H$_{2}$O. The determination of $f_{2}(E_{0})=f_{2,j}$ is done at the high energy end of the spectrum by comparison with the tabulation of Henke et al. [5].

The entering of the stoichiometric formula involves some flexibility. In fact, a chemically correct stoichiometric formula is not required; all one needs to do is to indicate the correct ratio of elements. That is, for (CH$_{3}$)$_{2}$S one can enter either (CH3)2S or C2H6S. Decimal fractions are accepted, such as C.5H2 instead of CH4; you will get the same result for each of these example formulas, since changing the number of atoms participating does not change the ratio $f_{2}(E)/A$. Finally, the file compound.dat in the IDL library directory idl_local/henke or in your working directory allows you to make definitions of particular mixtures by name, so that you can simply enter protein or ice, for example.