Hi Debasish,

Neutron scattering cross-sections are proportional to the neutron scattering lengths (b). These scattering lengths can be positive or negative depending on whether the interaction between the scattering nucleus and the neutron is attractive or repulsive. Further, neutron scattering from several nuclei can be coherent or incoherent depending on whether the scattered waves from different scattering centers interfere or not.

In the former case of coherent scattering, the scattering length (bcoh) is the average scattering length (average over different nuclei) i. e. bcoh= whereas in the case of incoherent scattering the scattering length (binc) is the root mean squared deviation of scattering lengths, i.e. binc= ( - 2)1/2.

In case of scattering from a single element, the difference in the scattering lengths arises from the different total spin states of the nucleus-neutron system. For example, for hydrogen with a nuclear spin of 1/2, the nucleus-neutron system can either be in triplet (S+=1) or a singlet (S-=0) state with the corresponding multiplicities of 3 and 1. In case of hydrogen the scattering lengths in these two spin states (let's say b+ and b-) have opposite signs (positive for triplet (b+), negative for singlet (b-)). Because of this, the average scattering length (averaged over the two states with weightage according to the multiplicities) (which is same as coherent scattering length bcoh) is negative and small. This is because the average of numbers with opposite signs is smaller than the average of those numbers when they have the same signs. Thus bcoh= for hydrogen is small and negative.

The incoherent scattering length is however the rms deviation, i.e. binc= ( - 2)1/2. Squared quantity always being positive means that is large and so subtracting a small quantity 2 (square of a small quantity () is an even smaller quantity) from that results in a large incoherent length. Therefore the incoherent neutron scattering cross-section of hydrogen is very large.

For deuterium (with nuclear spin of 1, S+=3/2, S-=1/2) on the other hand, as both b+ and b- are positive, their average is relatively large. So the difference between and 2 is not very large. Consequently the incoherent neutron scattering length and hence the incoherent cross-section of deuterium is not very large.

Hope you find this useful.

With best regards,

Siddharth

@Siddharth, thank you very much for answering my question.

I was wondering why the of hydrogen is negative, considering the Singlet state has a smaller probability than Triplet state in this neutron-proton coupled system.

Hi Yicheng,

of hydrogen is negative because b- of the singlet state is negative and more than three times larger in magnitude than b+ of the triplet state. In particular, b- = -47.52 fm, where as b+ = 10.85 fm. So,

=(3*b++1*b-)/4

=(32.55-47.52)/4

= -3.74 fm

Thus bcoh for hydrogen is small and negative.

Using the above values, one can further calculate as

=(3*(b+)2+1*(b-)2)/4

=(3*(10.85)2+(-47.52)2)/4

=652.592 fm2

The incoherent scattering length can then be obtained as

binc = ( - 2)1/2

=25.27 fm