Uncertainty on the cross-section

The collimation defects and the wavelength dispersion are such that the measured cross-section must be considered only as a relative quantity. 

Let us calculate the number J(fc0, 0) of particular impinging during a time interval t on the detector whose surface of area Od s at a distance r from the sample, in a direction defined by an angle 0. The incident flux distribution is Jo(ic) [see (6.1.19)] and the mean wave vector is fc0. 

The probability that an incident particle with wave number k will be scattered by the sample with a deviation angle 0 is 

The collimation is characterized by the function (fc, 0, 0 ) which is the density probability that an incident particle with wave number k, deviated in the sample by an angle 0, encounters the detector whose direction is given by 0. In such conditions, the number of scattered particles that encounter the sample in a time interval t is 

When the particles are neutrons, the widths of the distribution J0 (fc) and of the density J(fc,0, 0) are rather large. In this case, to obtain E (fc0, 0) by measuring / (fc0, 0), it is not absolutely necessary to know J0(fc) and J(fc, 0, 0). In principle, one could measure the effects of these widths directly. For this, it would be sufficient to scatter the beam by a perfect monocrystal with known properties. The cross-section of this crystal is different from zero, only if 

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