Absorption¶

Absorption reduces the intensity leaving the sample. The cylinder models below apply a multiplicative transmission \(A\) to each reflection's intensity; they do not move the reflection or define a single-crystal extinction correction.

Types and active fields¶

Beam mode	Saved type	Active field	Meaning
CW	`none`	—	\(A=1\)
CW	`cylinder-hewat`	`mu_r`	Hewat cylinder expansion
CW	`cylinder-lobanov`	`mu_r`	Two-branch Lobanov cylinder fit
TOF	`none`	—	\(A=1\)
TOF	`cylinder`	`abscor1`	Wavelength-dependent Hewat cylinder; canonical model token `cylinder-hewat`

For a cylinder, \(\mu R\) uses the radius, not the diameter. It is dimensionless when \(\mu\) and \(R\) use reciprocal length units consistently.

Hewat cylinder¶

Let \(u=\mu R\) and \(s=\sin\theta\):

\[A(u,\theta)=\exp[-(1.7133-0.0368s^2)u+(0.0927+0.375s^2)u^2].\]

In CW the angle is each reflection's Bragg angle and \(u\) is the mu_r parameter. The polynomial is a small-to-moderate-absorption approximation. At large \(u\) its quadratic exponent is not a general physical transmission law; use an appropriate model and inspect the resulting transmission rather than treating extrapolation as validated physics.

For TOF at a fixed bank angle,

\[\lambda_h=2d_h\sin\theta_{\rm bank},\qquad u_h=\text{ABSCOR1}\,\lambda_h.\]

The same Hewat expression uses that bank angle and reflection-specific wavelength. ABSCOR1 therefore carries the inverse-wavelength scale needed to produce \(u_h\).

ABSCOR2 is read and saved, but not yet used in the calculation (planned).

Lobanov cylinder¶

For CW cylinder-lobanov, put \(s=\sin\theta\) and \(u=\mu R\). For \(u\le3\):

\[A=\exp[-t_0u-t_1u^2-t_2u^3-t_3u^4],\qquad t_0=16/(3\pi),\]

\[\begin{aligned} t_1={}&(25.99978-0.01911\sqrt{s})e^{-0.024551s^2}+0.109561s-26.04556,\\ t_2={}&-0.02489-0.39499s^2+1.219077s^3-1.31268s^4+0.871081s^5-0.2327s^6,\\ t_3={}&0.003045+0.018167s^2-0.03305s^4. \end{aligned}\]

For \(u>3\):

\[A=0.01\left[\frac{t_1-t_4}{(1+t_2(u-3))^{t_3}}+t_4\right],\]

\[\begin{aligned} t_1={}&1.433902+11.07504s^2-8.77629s^4+10.02088s^6-3.36778s^8,\\ t_2={}&(0.013869-0.01249s^2)e^{3.27094s^2}+\frac{0.337894+13.77317s^2}{(1+11.53544s^2)^{1.555039}},\\ t_3={}&\frac{1.933433}{(1+23.12967s^2)^{1.686715}}-0.13576s+1.163198,\\ t_4={}&0.044365-\frac{0.04259}{(1+0.41051s^2)^{148.4202}}. \end{aligned}\]

The high-\(u\) expression is tabulated as a percentage before the \(0.01\) conversion to a transmission fraction. The two empirical branches have a small seam mismatch; do not impose an invented continuity correction when reproducing this model.

References¶

Hewat, A. W. (1979). Acta Crystallographica A, 35, 248. Cylinder absorption approximation.
Lobanov, N. N. & Alte da Veiga, L. (1998). Cylinder-transmission parameterisation, as used by the GSAS-II absorption implementation. The coefficients and branch conventions reproduced above identify the implemented approximation.