Extinction¶

Extinction reduces strong single-crystal reflection intensities relative to the kinematic model. It is distinct from sample absorption. The correction below is a factor \(y\) on intensity \(F^2\), not a factor to apply unchanged to an amplitude.

Single-crystal intensity¶

The scalar nuclear calculation compares each measured reflection with

\[F_{c,h}^2=s\,y_h\,|F_{N,h}|^2.\]

Here \(s\) is the linked structure's scale and \(y_h\) the selected extinction factor. A CW experiment supplies one wavelength; a TOF reflection list supplies a wavelength per reflection. This is a reflection model, with no powder-profile convolution or background term. Keep the structure factor, scale and reflection-statistics convention consistent with the observed \(F^2\) data.

Supported types¶

Type	Physical/empirical model	Active inputs
`none`	Identity, \(y=1\)	—
`becker-coppens-type-1`	Mosaic-spread-limited secondary extinction	Crystal radius, mosaic spread, mosaic distribution
`becker-coppens-type-2`	Domain-size-limited secondary extinction with primary extinction	Crystal radius, domain radius
`becker-coppens-mixed`	Primary and secondary extinction with both finite terms	Crystal radius, domain radius, mosaic spread, mosaic distribution
`larson-isotropic`	Empirical isotropic factor	`coefficient`
`larson-anisotropic`	Direction-dependent empirical coefficient	Six `coefficient_ij` terms

Becker–Coppens mosaic distributions are gaussian and lorentzian. These are extinction distributions, not the powder peak-profile selector.

Units and common factor¶

Use \(c=\cos2\theta\), \(q=\sin2\theta\), wavelength \(\lambda\) in Å and cell volume \(V\) in Å³. The extinction formula consumes \(F^2\) in \((10^{-12}\,\mathrm{cm})^2\); a structure-factor value in fm must be converted before inserting it.

domain_radius \(r\) is in µm. crystal_radius \(R_c\) is in mm and sets the spherical path \(\ell=1000R_c\) in µm. It is a descriptor, not a fitted domain radius. mosaic_spread \(g\) is an inverse width in rad⁻¹: larger values describe a narrower mosaic distribution. For the mixed Å/µm convention in the expressions below define \(\tilde g=10^{-4}g\).

Define

\[Y(x;a,b,k)=\left[1+kx+\frac{ax^2}{1+bx}\right]^{-1/2}.\]

Do not insert an arcminute mosaic width or a crystal diameter in place of these quantities.

Primary Becker–Coppens factor¶

\[x_p=\tfrac32F^2(\lambda r/V)^2,\quad a_p=0.20+0.45c,\quad b_p=0.22-0.12(0.5-c)^2,\quad y_p=Y(x_p;a_p,b_p,2).\]

Secondary Becker–Coppens factor¶

The effective domain \(D\) is

\[D_G=\frac{r\tilde g}{\sqrt{\tilde g^2\lambda^2+\tfrac98r^2q^2}},\qquad D_L=\frac{r\tilde g}{\tilde g\lambda+rq}.\]

Type I takes the unlimited-domain limit: \(D_G=\tilde g/(\sqrt{9/8}\,q)\) and \(D_L=\tilde g/q\). Type II takes the unlimited-mosaic-spread limit \(D=r/\lambda\). The mixed model uses both finite inputs. Then

\[x_s=\tfrac32F^2\lambda(\lambda/V)^2\ell D.\]

For a Gaussian distribution,

\[a_s=0.58+0.48c+0.24c^2,\quad b_s=0.02-0.025c,\quad y_s=Y(x_s;a_s,b_s,2.12).\]

For a Lorentzian distribution,

\[a_s=0.025+0.285c,\qquad b_s=\begin{cases}0.15-0.2(0.75-c)^2,&c>0,\\-0.45c,&c\le0,\end{cases}\qquad y_s=Y(x_s;a_s,b_s,2).\]

Type I uses \(y=y_s\). Type II and mixed use \(y=y_p y_s\). The implemented identity/degenerate limits must be retained when evaluating zero radii, zero spread or vanishing angle factors; do not manufacture infinities by cancelling terms outside their valid domain.

Larson factors¶

For the isotropic coefficient \(x\),

\[y=\left[1+0.001\,xF^2\lambda^3/\sin2\theta\right]^{-1/2}.\]

For the anisotropic form,

\[x_{\rm aniso}=x_{11}h^2+x_{22}k^2+x_{33}l^2+x_{12}hk+x_{13}hl+x_{23}kl,\qquad x_{\rm eff}=\frac{x_{\rm aniso}}{4(\sin\theta/\lambda)^2},\]

and substitute \(x_{\rm eff}\) for \(x\) in the same intensity factor. The cross coefficients are entered in the polynomial as written, without an additional factor of two. The implementation returns the identity at vanishing \(\sin2\theta\) or a nonpositive empirical radicand; such a result is not evidence for a physically justified coefficient choice.

Applicability¶

These are spherical/isotropic Becker–Coppens approximations and the stated empirical Larson forms, not a general dynamical-diffraction solution. Parameter values can be correlated with scale and structure; inspect the strong-reflection residuals and the reliability factors.

References¶

Becker, P. J. & Coppens, P. (1974). Acta Crystallographica A, 30, 129–147. DOI. Extinction theory and spherical approximations.
Becker, P. J. & Coppens, P. (1974). Acta Crystallographica A, 30, 148–153. Application of the extinction treatment.
Larson extinction convention, as used in SHELXL and the Mag2Pol manual, §5.1.3. The operative intensity factor and units for this reference are stated above.