MultiScat

Formulation

Primary scattering from direct beam

Scattering from direct beam at pixel $p$ from atom $j$ : $$\begin{equation} F_j(\bb u_p) = f_j(\theta_{p})e^{2\pi jk_0\big(z_j+R_{jp}\big)} \end{equation}$$

The intensity at pixel $l$ is then : $$\begin{equation} I(\bb u_p) = \Bigg|\sum_{j=1}^{N} F_j(\bb u_p) \Bigg|^2 \end{equation}$$

The amount of power scattered by atom $j$ is $\sigma_jI_0=f_j^2I_0$ which has to be taken off the power of the incident beam $I_0$.

Secondary scattering

Scattering from atom $i$ due to scattering by atom $j$ : $$\begin{equation} F_{ji}(\bb u_l) = f_j(\theta_{ij})e^{2j\pi k_0\big(z_j+r_{ij}\big)} f_i(\theta_{p}-\theta_{ij})e^{2\pi jk_0R_{jp}} \end{equation}$$

$$\begin{eqnarray} I(\bb u_p) &=&\Bigg|\sum_{j=1}^{N}\Bigg(\tilde F_j(\bb u_p) + \sum_{i\neq j}^{N} F_{ij}(\bb u_p) \Bigg)\Bigg|^2 \\ &=&\Bigg|\sum_{j=1}^{N}e^{2\pi jk_0z_j}\Bigg( f_j(\theta_{p})e^{2\pi jk_0R_{jp}} + \sum_{i\neq j}^{N} f_j(\theta_{ij}) f_i(\theta_{p}-\theta_{ij})e^{2j\pi k_0R_{ip}}\Bigg)\Bigg|^2 \\ \end{eqnarray}$$

Similarly to scattering from direct beam, secondary scattering should reduce the amplitude of primary scattering by the amount scattered b

Path length estimate

The path length may be computed with the following increasing level of approximations: $$\begin{eqnarray} R_{ij} &\underset{Greens}{=}& \sqrt{\left(x-x_0\right)^2+\left(z-z_0\right)^2} \\ &\underset{Fresnel}{\approx}& \left(z_0-z\right) + \frac{\left(x-x_0\right)^2}{2\left(z_0-z\right)} \\ &\underset{Fraunhofer}{\approx}& \left(z_0-z\right) + \frac{x_0^2}{2\left(z_0-z\right)} - \frac{xx_0}{\left(z_0-z\right)} \\ \end{eqnarray}$$

Note that the path length from the source the to atom is $z$ for planar illumination.

$$\begin{eqnarray} R_{il} &\underset{Greens}{=}& \sqrt{\left(x_i-x_l\right)^2+\left(z_i-z_l\right)^2} \\ &\underset{Fraunhofer}{\approx}& \left(z_l-z_i\right) + \frac{x_l^2}{2\left(z_l-z_i\right)} - \frac{x_ix_l}{\left(z_l-z_i\right)} \\ \end{eqnarray}$$

NearBragg James Holton

An algorithm computing the interference patterns of an assembly of scatterers based on their path difference to a detector.

Source file format : x,y,z, O,B,P

Parameters

Type	parameters	default (m)
detector	distance	0.1
	detsize,detpixels,pixel	0.1, 0.1e-3
beam spectral	lambda,dispersion,dispteps	1e-10, 0,-1
beam spatial	divergence,divsteps,hdivrange,vdivrange,hdivsteps,vdivsteps	-1
	source_distance,source_depth,depthstep	10,0,-1
misc	oversample,curvedet,point_pixel, Xbeam,Ybeam
features	coherent, nopolar + occ, bfactor, phase-shift through file parsing
io	file,cosfile,sinfile,floatfile,intifle
	scale, roi
print	noprogress, printout, accumulate

Code structure

double fluence = 1.25932015286227e+29;//incident flux in particles/m^2 */
double r_e_sqr = 7.94079248018965e-30;//classical radius of the electron
double steps = divsteps*dispsteps*depthsteps*oversample*oversample;//total number of sub-steps to normalize over
double airpath = sqrt(pixel_X*pixel_X+pixel_Y*pixel_Y+pixel_Z*pixel_Z);//solid angle subtended by a pixel from the origin: (pix/airpath)^2*cos(2theta)
double omega_Rsqr_pixel = pixel*pixel*distance/airpath;//m^2
    DWF = occ[i]*exp(-Bfac[i]*stol**2)
    Fa+=DWF*cos(phase)/source_to_atom_path/atom_to_pixel_path
    I += Fa**2 + Fb**2;// normalized intensity
floatimage[j] += I/steps*omega_Rsqr_pixel*fluence*r_e_sqr;

Monte-Carlo style

Mutliple and inelastic scattering :

• Stochasticly using the theoretical scattering rate.
• The differential scattering cross section can be used for more accurately account for the scattering factor of individual atoms.
• Inelastic scattering loss of coherency and spectral broadening can be represented through rigorous bookeeping.

mosaicity

[nanobragg]https://bl831.als.lbl.gov/~jamesh/nanoBragg/)

According to wiki, dynamical diffraction is often negligible when high level of mosaicity is involved.

• mosaic_spread=argv[-mosaic]/RadToDeg : isotropic mosaic spread in degrees (use 90 for powder)
• mosaic_domains=argv[-mosaic_dom] : number of randomly-oriented mosaic domains to render
• mosaic_seed=argv[mosaic_seed] : random-number seed for mosaic domain generation (default: 1234567)
• mosaic_umats : rotation matrix
• mosaic_missets and missets_seeds

pyCode

Comparison with Holton

• solid_angle detector
• pixel offset
• X/Z reverse
• no scattering form factor

pattern	Intensity

comparison	error