Working through the equations

I am not entirely sure where this all came from but seems like it may be helpful.

$\rho z=$mass thickness $mg/cm^2$

$\phi (\rho z)=$the depth distribution of electron excited x-rays

$\delta^{f1} =$surface film with u elements thickness

$\delta^{f2} =$subsurface film(buried layer) with v elements thickness

$R=K(E_0^n-E_c^n)/\rho$ – The X-ray range according to Anderson & Hassler

$I_{generated}=\rho(\Delta \rho z)\int_0^\infty \phi(\rho z)d(\rho z)$ – dz in thickness

approximated to(gaussian function)

params derived from random walk

$\exp(-\chi\rho z)=$Xray absorption factor

Experimental k-ratio - $k_{i,exp}=\frac{I_{i,obs}(sp)}{I_{i,obs}(st)}$ sp-sample, st-standard, obs-observed treating elements individually and ignoring absorption effects, $\rho z$ thickness can be estimated:

but $\phi(\rho z)$ can only be arrived from the main iteration so estimate with: triangular aprox fig 8.

$q+\frac{1}{2}\left[q-\left(q \frac{\rho z_i^{f1}}{\rho z_{r,i}}\right)\right]=\frac{1}{2}\left(k_{i,exp}\cdot q\cdot\rho z_{r,i}\right)$ $\rightarrow \rho z_i^{f1}=\rho z_{r,i}\left(1-\sqrt{1-k_{i,exp}}\right)$

$\rho z_{ri}=$x-ray generation range for element i, = $\rho z_{r}=7.0\left(E_0^{1.65}-E_C^{1.65}\right)\left[\frac{\mu g}{cm^2}\right]$

$\delta_0^ {f1}=\sum_1\rho z_i^{f1}$ for i from 1 to u elements

Need to work through the derivation for a buried layer…

Compositions - $C_{i,0} = \frac{k_{i,exp}}{\sum k_{i,exp}}$

thickness estimates - $\delta_{m+1}^{f1}=\delta_{m}^{f1}\frac{\sum k_{i,exp}}{\sum k_{i,m}}$m is iteration

depth-distribution function, φ(ρz), which is a histogram showing the number of X-rays generated in layers of the specimen, each with a thickness of dz, relative to the number of X-rays the beam would produce in a freestanding layer of the same material of thickness dz.

where, φ(Δρz) is the intensity generated in a freestanding layer of thickness Δρz. The

Value of φ(ρz) at the surface, φ0; the maximum value of φ(ρz), φm; the depth at φm, Rm; and the maximum depth where φ(ρz) is zero, Rx are the parameters used to describe a φ(ρz) distribution. Both Rm and Rx decrease with atomic number (Z) and increase with beam energy (E0). φ0 increases with E0.

GMRfilm Procedure:

1. Setup
2. Calculate Initial Thicknesses
   1. Surface Film $\delta_{0}^{f1}=\sum_{i}R_i\left(1-\sqrt{1-k_{i,exp}}\right)$
   2. Subsurface Film (Buried Layer)
      • From the MAS 88 Paper $\delta_{0}^{fn}=\sum_iR_i\sqrt{1-k_{i,exp}-\frac{2\Delta_n}{R}+\frac{\left(\Delta_n\right)^2}{R^2} }^{1/2}$

      • From GMRfilm $\delta_{0}^{fn}=\sum_iR_i \left(1.0-\sqrt{1-k_{i,exp}-\frac{2\Delta_n}{R}+\frac{\left(\Delta_n\right)^2}{R^2} }\right)-\Delta_n$

        Where:

        i is the current element in the layer

        $$R_i=7.0(E_{i,0})^{1.65}$$

        $Delta_n=/sum_{m=1}^{n-1}\delta_0^{fm}$ is the depth of layer n

3. Next?

Spatial Distribution

depth

lateral spread

spatial resolution

determined by:

1. Diameter and intensity of impinging beam
2. Size, shape, and density of the specimin the penetration, deceleration and scatering of e- in the specimin (electron diffusion)***
3. Absorption of X-rays in the specimin
4. effect of secondary radiation by continuos or characteristic X-rays

Diffusion of electrons

See Schumacher [2.2]

Everhart and Hoff [13.1] - total path length cna be represented by (derived from Bethe stopping power [Eq 9.2.2]): - eq 13.1.1 - for Al-Si for 5-25KeV: k=4e-6, n=1.75

Cosslett and Tomas [13.2]

• depth can not be derived directly from Bethe(loss along trajectory) becasue of scattering (not straight path)

Depth range of X-ray Generation:

need to add critical excitation to the eq 13.1.1

E-E eq

Castaing [13.3} - big old eq - similar eq by [12.23] look closer are derivation from the book on page 419 - this is the One used in GMRfilm

find refss 13, 14, 15