6.1

\(\renewcommand\AA{\unicode{x212B}}\)

Materials

Neutron scattering lengths and cross sections of the elements and their isotopes have been taken from NIST.

Chemical Composition with Examples

  • H2 O - Isotopically averaged Hydrogen
  • (H2)2 O - Heavy water
  • D2 O - Another way to specify heavy water

Enter a composition as a molecular formula of elements or isotopes. For example, basic elements might be H, Fe or Si, etc. A molecular formula of elements might be H4-N2-C3, which corresponds to a molecule with 4 Hydrogen atoms, 2 Nitrogen atoms and 3 Carbon atoms. Each element in a molecular formula is followed by the number of the atoms for that element, specified without a hyphen, because each element is separated from other elements using a hyphen.

The number of atoms can be integer or float, but must start with a digit, e.g. 0.6 is fine but .6 is not. This can be used to set elemental ratios within a chemical composition. For example 95.1% Vanadium 4.9% Niobium can be expressed as V0.951 Nb0.049. Warning: Using this representation will calculate all properties except for SampleNumberDensity which must be set manually if required

Isotopes may also be included in a material composition, and can be specified alone (as in (Li7)), or in a molecular formula (as in (Li7)2-C-H4-N-Cl6). Note, however, that No Spaces or Hyphens are allowed in an isotope symbol specification. Also Note that for isotopes specified in a molecular expression, the isotope must be enclosed by parenthesis, except for two special cases, D and T, which stand for H2 and H3, respectively.

Cross Section Calculations

Each of the cross sections (\(\sigma\)) are calculated according to

\[\sigma = \frac{1}{N_{atoms}}\sum_{i}\sigma_{i}n_{i}\]

where \(N_{atoms} = \sum_{i}n_{i}\). A concrete example for the total cross section of D2 O

\[\sigma = \frac{1}{2+1}\left( 7.64*2 + 4.232*1\right) = 6.504\ barns\]

Number Density

The number density is defined as

\[\rho_n = \frac{N_{atoms}ZParameter}{UnitCellVolume}\]

It can can be generated in one of several ways:

  1. Specifying it directly with SampleNumberDensity.
  2. Specifying the ZParameter and the UnitCellVolume (or letting the algorithm calculate it from the OrientedLattice on the InputWorkspace).
  3. Specifying the mass density. In this case the number density is calculated as
\[\rho_{n,eff} = f \rho_n = \frac{N_{atoms} \rho_m N_A}{\sum_{i}n_{i}M_i}\]

where \(f\) is the packing fraction, \(\rho_m\) is the mass density, \(N_A\) is the Avogadro constant, and \(M_i\) is the relative molecular mass of the ith atom.

  1. Specifying SampleEffectiveNumberDensity and SamplePackingFraction to calculate it from the equation above.
  2. Specifying only SampleEffectiveNumberDensity (in this case, the packing fraction is assumed to be 1).
  3. A combination of SampleEffectiveNumberDensity or SamplePackingFraction with the parameters from method 2 or 3.

The effective number density, \(\rho_{n,eff}\), should be used for absorption calculations. However, the number density, \(\rho_n\) should be used for refinements and converting between real space representations.

When the number density is provided or generated from one of the above methods, both the packing fraction \(f\) and effective number density \(\rho_{n,eff}\) are computed from the above equation if they were not specified.

Attenuation Coefficients

The attenuation effect is calculated according to the following formula:

\[\exp(-\rho_n(\mu_s+\mu_a)t)\]

where \(\rho_n\) is in unit of \(\AA^{-3}\), \(t\) is the material thickness in cm, the two attenuation coefficients representing scattering and absorption (\(\mu_s\) and \(\mu_a\) respectively) are calculated as follows:

\[\mu_s = \rho_n \frac{1}{N_{atoms}}\sum_{i}s_{i}n_{i} \text{ units of 1/cm}\]
\[s = \sigma_{total scattering}\]
\[\mu_a = \rho_n \frac{1}{N_{atoms}}\sum_{i}a_{i}n_{i} \text{ units of 1/cm}\]
\[a = \frac{\lambda}{\lambda_0} \sigma_{absorption} (\lambda_0) \text{ where } \lambda_0=1.8\AA\]

A detailed version of this is found in [2].

The sum of the two attenuation coefficients can be replaced by an externally measured profile of attenuation versus wavelength if the scattering effect is wavelength dependent eg if a material is crystalline and shows some Bragg edges in its attenuation profile. Mantid supports a space delimited text file format for the externally measured profile containing the following columns:

  • wavelength (in \(\AA\))
  • attenuation factor (in \(mm^{-1}\))
  • error (currently ignored)

The Xray Attenuation coefficients can also be provided by text file with the following columns containing:

  • energy (in \(KeV\))
  • attenuation factor (in \(mm^{-1}\))
  • error (currently ignored)

Any lines not following this format (eg header rows) are ignored. The file must have a .DAT file extension.

Normalized Laue

The low-\(Q\) limit of \(S(Q)\) is \(-L\) where \(L\) is called the normalized Laue term

\[bAverage = <b_{coh}> = \frac{1}{N_{atoms}}\sum_{i}b_{coh,i}\]
\[bSquaredAverage = <b_{tot}^2> = \frac{1}{N_{atoms}}\sum_{i}b_{tot,i}^2\]
\[L = \frac{<b_{tot}^2>-<b_{coh}>^2}{<b_{coh}>^2}\]

References

The data used in this algorithm comes from the following paper.

  1. Varley F. Sears, Neutron scattering lengths and cross sections, Neutron News 3:3 (1992) 26 doi: 10.1080/10448639208218770
  2. J. A. K. Howard, O. Johnson, A. J. Schultz and A. M. Stringer, Determination of the neutron absorption cross section for hydrogen as a function of wavelength with a pulsed neutron source, J. Appl. Cryst. (1987). 20, 120-122 doi: 10.1107/S0021889887087028

Category: Concepts