neutronpy.crystal.Material.calc_optimal_thickness

Material.calc_optimal_thickness(energy=25.3, transmission=0.36787944117144233)

Calculates the optimal sample thickess to avoid problems with extinction, multiple coherent scattering and absorption.

Parameters:

energy : float, optional

The energy of the incident beam in meV. Default: 25.3 meV.

transmission: float, optional

The transmission through the material in decimal percentage, \(0 < T < 1.0\). Default: \(1/e\).

Returns:

thickness : float

Returns the optimal thickness of the sample in cm

Notes

The total transmission of neutrons through a material is defined as

\[T = \frac{I}{I_0} = e^{-\Sigma_T d},\]

where \(\Sigma_T\) is the total scattering cross-section, i.e.

\[\Sigma_T = \Sigma_{coh} + \Sigma_{inc} + \Sigma_{abs},\]

and \(d\) is the thickness in cm.

Scattered intensity is thus defined to be

\[I_s \propto dT\left(\frac{d\Sigma_{coh}}{d\Omega}\right) \propto d e^{-\Sigma_T d}.\]

\(I_s\) is therefore a maximum when \(d=1/\Sigma_T\), resulting a transmission of approximately \(T=37\%\). This is valid when the coherent cross-section is much less than the total cross-section, i.e.

\[\Sigma_{coh} \ll \Sigma_T \approx \Sigma_{inc} + \Sigma{abs}.\]

However, if the coherent cross-section of the material is the dominant part of the total scattering cross-section, i.e. \(\Sigma_T \approx \Sigma_{coh}\), then \(d = 1/\Sigma_T\) is too large and there will be a problem with multiple scattering. Therefore a transmission of \(T\geq90\%\) is desirable.