Equations

The model TraTT solves, written out from the solver (src/tratt/core.py, src/tratt/drive.py). The assumptions behind it and the analytic limits the tests assert are in DESIGN.md; this page is the same model in one place, with the equations typeset.

Variables

Everything is dimensionless.

symbol	meaning	in the code
$\zeta =z/L\in [0,1]$	depth in units of the crystal thickness $L$	zeta
$\tilde{t}=t/t_{\mathrm{tr}}$, $t_{\mathrm{tr}}=L/v_s$	time in units of the acoustic transit time	t
$\alpha =L/{\mathrm{\Lambda }}_P$	thickness in Pendellösung lengths	alpha
$\eta$	deviation parameter (wavevector mismatch from the Bragg condition) multiplied by $L$	eta
$S=h\cdot u_0$	acoustic modulation strength	Drive.S
${\alpha }_{\mathrm{ac}}=L/{\mathrm{\Lambda }}_{\mathrm{ac}}$	thickness in acoustic wavelengths	Drive.alpha_ac
$q=w=2\pi {\alpha }_{\mathrm{ac}}$	acoustic phase across the slab, and per transit	Drive.q, Drive.w

The README's $R={\mathrm{\Lambda }}_{\mathrm{ac}}/{\mathrm{\Lambda }}_P$ is $\alpha /{\alpha }_{\mathrm{ac}}$.

The forward susceptibility ${\chi }_0$ does not appear in the equations below. In Laue geometry it enters both beams with the same coefficient and is removed by a common phase substitution. In Bragg geometry it enters the two beams with opposite signs, so no common phase removes it: its real part is absorbed into $\eta$ and its imaginary part is the absorption $\kappa$ (below). In both, $\eta =0$ is the refraction-corrected Bragg angle.

Laue geometry

Both beams travel into the crystal. With $D_0(0)=1$ and $D_h(0)=0$ at the entrance face, the two-beam Takagi–Taupin equations are integrated to the exit face by fixed-step RK4:

$$\frac{d{D}_0}{d\zeta }=i\pi \alpha e^{+i\phi (\zeta ;\tilde{t})}{D}_h,\frac{d{D}_h}{d\zeta }=i\pi \alpha e^{-i\phi (\zeta ;\tilde{t})}{D}_0+i\eta D_h.$$

The diffracted intensity is $I=|D_h(1){|}^2$ (rocking(..., geometry="laue")). A complex $\alpha$ adds the anomalous (Borrmann) part of absorption.

Bragg geometry

The diffracted beam travels back to the entrance face, so the boundary conditions sit on opposite faces ($D_0(0)=1$, $D_h(1)=0$). The solver integrates the ratio $R=D_h/D_0$ backward from $R(1)=0$, which turns the two-point problem into an initial-value problem:

$$\frac{dR}{d\zeta }=-i\pi \alpha e^{-i\phi }-i\eta R-i\pi \alpha e^{+i\phi }{R}^2.$$

The reflectivity is $I=|R(0){|}^2$ (rocking(..., geometry="bragg")). Normal absorption, which does not gauge out in this geometry, enters as $\eta \to \eta +i\kappa$ (solve_bragg(..., kappa=)).

The acoustic drive

The sound wave enters at the surface at $\tilde{t}=0$. The lattice is frozen at each time (the X-rays cross the crystal much faster than one acoustic period), and the phase it imposes is

$$\phi (\zeta ;\tilde{t})=SA(\tilde{t}-\zeta )\sin (q\zeta -w\tilde{t}).$$

$A$ is a causal envelope of the retarded time $s=\tilde{t}-\zeta$: a step ($A=0$ for $s<0$, $1$ after) or, with Drive(rise=r), the smoothstep $3x^2-2x^3$ with $x=min(max(s/r,0),1)$. $A(s<0)=0$ is the wavefront: at time $\tilde{t}$ only the depth $\zeta <\tilde{t}$ is vibrating. For a non-dispersive wave $\mathrm{\Omega }=K{v}_s$, so $w=q$, and the phase is a function of the retarded time alone, $\phi =-SA(s)\sin (qs)$.

Satellite channels

The channel positions and weights come from the kinematic limit (weak diffraction, $\alpha \ll 1$), where the Laue amplitude is a single integral over depth,

$$D_h(1)=i\pi \alpha {\int }_0^1{e}^{-i\phi (\zeta )}{e}^{i\eta (1-\zeta )}d\zeta .$$

Where the crystal is filled ($A=1$; with the step envelope, everywhere once $\tilde{t}\ge 1$), the phase factor expands in Bessel functions,

$$e^{-i\phi }=\sum _n(-1{)}^n{J}_n(S)e^{in(q\zeta -q\tilde{t})},$$

and term $n$ of the integral peaks at ${\eta }_n=nq$ (Drive.eta_channel(n)): satellite $n$ has amplitude $(-1{)}^n{J}_n(S)$ and intensity proportional to $J_n(S{)}^2$. The positions are analytic; nothing is fitted. For integer ${\alpha }_{\mathrm{ac}}$ the response of one channel vanishes at the centres of the others. Outside the kinematic limit the intensity ratios no longer follow $J_n(S{)}^2$; the full equations above give them.

Limits the tests assert

• Laue, real $\alpha$, no sound, $\eta =0$: $|D_h(1){|}^2={\sin }^2(\pi \alpha )$.
• Laue, real $\alpha$ and $\eta$, with or without sound: $|D_0{|}^2+|D_h{|}^2=1$ at the exit face.
• Bragg, real $\alpha$, with sound: $|R{|}^2\le 1$.
• Bragg, no sound, $\alpha =4$: $|R{|}^2>0.99$ for $|\eta |\le 0.6\cdot 2\pi \alpha$ and $|R{|}^2<0.5$ for $|\eta |\ge 1.6\cdot 2\pi \alpha$, the total-reflection plateau of a thick crystal, whose edge at $|\eta |=2\pi \alpha$ is soft at this thickness.
• Step envelope, $\tilde{t}\ge 1$: $\phi$ has its final form, so $I(\eta ,\tilde{t})$ is periodic in $\tilde{t}$ with period $2\pi /w$.

What the model leaves out

Two beams only, symmetric geometry, a plane monochromatic X-ray wave, a fixed polarization factor, and a sound wave that acts only as a displacement of the lattice (no change of ${\chi }_h$, no vibrational Debye–Waller factor). Absorption is off unless $\alpha$ is made complex (Laue) or $\kappa >0$ (Bragg). The full list, with what each assumption discards, is in DESIGN.md.