A neutron may pass by a nucleus with a probability determined by the nuclear interaction distance, or be absorbed, or undergo scattering that may be either coherent or incoherent.[1] The interference effects in coherent scattering can be computed via the coherent scattering length of neutrons, being proportional to the amplitude of the spherical scattered waves according to Huygens–Fresnel theory. This scattering length varies by isotope (and by element as the weighted arithmetic mean over the constituent isotopes) in a way that appears random, whereas the X-ray scattering length is just the product of atomic number and Thomson scattering length, thus monotonically increasing with atomic number.[1][2]

The scattering length may be either positive or negative. The scattering cross-section is equal to the square of the scattering length multiplied by 4π,[3] i.e. the area of a circle with radius twice the scattering length. In some cases, as with titanium and nickel, it is possible to mix isotopes of an element whose lengths are of opposite signs to give a net scattering length of zero, in which case coherent scattering will not occur at all, while for vanadium already the opposite signs of the only naturally occurring isotope's two spin configurations give a near cancellation. However, neutrons will still undergo strong incoherent scattering in these materials.[1]

There is a large difference in scattering length between protium (-0.374) and deuterium (0.667). By using heavy water as solvent and/or selective deuteration of the probed molecule (exchanging the naturally occurring protium by deuterium) this difference can be leveraged in order to image the hydrogen configuration in organic matter, which is nearly impossible with X-rays due to their small sensitivity to hydrogen's single electron.[4] On the other hand, neutron scattering studies of hydrogen-containing samples often suffer from the strong incoherent scattering of natural hydrogen.

element protons isotope neutron scattering length
bcoh (fm)		 coherent
cross-section
σcoh (barn)		 incoherent
cross-section
σinc (barn)		 absorption
cross-section
σa (barn)
Hydrogen 1 1[2][5] -3.74[1][2][5][6] 1.758[1] 79.7,[6] 80.27[1] 0.33,[6] 0.383[1]
Hydrogen 1 2 6.67[1][2][5][6] 5.592[1] 2.0,[6] 2.05[1] 0.0005[1][6]
Boron 5 natural 5.30[1] 3.54[1] 1.70[1] 767.0[1]
Carbon 6 12 6.65[1][2][5][6] 5.550[1] 0.0,[6] 0.001[1] 0.0035,[6] 0.004[1]
Nitrogen 7 14 9.36,[1] 9.40,[2] 9.4[5][6] 11.01[1] 0.3,[6] 0.5[1] 1.9[1][6]
Oxygen 8 16 5.80,[2] 5.8[1][5][6] 4.232[1] 0.0,[6] 0.000[1] 0.00019,[6] 0.0002[1]
Aluminum 13 natural 3.45,[1] 3.5[6] 1.495[1] 0.0,[6] 0.008[1] 0.23,[6] 0.231[1]
Silicon 14 natural 4.2[6][7] 0.0[6] 0.17[6]
Phosphorus 15 30 5.10[2]
Sulfur 16 32 2.80,[2] 2.8[5]
Titanium 22 natural -3.44,[1] -3.4[6][7] 1.485[1] 2.87,[1] 3.0[6] 6.09,[1] 6.1[6]
Vanadium 23 natural -0.38[1] 0.018[1] 5.07[1] 5.08[1]
Chromium 24 natural 3.64[1] 1.66[1] 1.83[1] 3.05[1]
Manganese 25 55 (natural) -3.73[1] 1.75[1] 0.4[1] 13.3[1]
Iron 26 natural 9.45,[1] 9.5[6] 11.22[1] 0.4[1][6] 2.56,[1] 2.6[6]
Nickel 28 natural 10.3[1] 13.3[1] 5.2[1] 4.49[1]
Copper 29 natural 7.72[1] 7.485[1] 0.55[1] 3.78[1]
Zirconium 40 natural 7.16,[1] 0.72[6] 6.44[1] 0.02,[1] 0.3[6] 0.18,[6] 0.185[1]
Niobium 41 93 (natural) 7.054[1] 6.253[1] 0.0024[1] 1.15[1]
Molybdenum 42 natural 6.72[1] 5.67[1] 0.04[1] 2.48[1]
Cadmium 48 natural 4.87[1] 3.04[1] 3.46[1] 2520[1]
Tin 50 natural 6.23[1] 4.87[1] 0.022[1] 0.626[1]
Cerium 58 natural 4.8[6] 0.0[6] 0.63[6]
Gadolinium 64 natural 6.5[1] 29.3[1] 151[1] 49700[1]
Tantalum 73 natural 6.91[1] 6.00[1] 0.01[1] 20.6[1]
Tungsten 74 natural 4.86[1] 2.97[1] 1.63[1] 18.3[1]
Gold 79 197 7.60[2]
Lead 82 natural 9.41[1] 11.115[1] 0.003[1] 0.171[1]
Thorium 90 232 (natural) 9.8[6] 0.00[6] 7.4[6]
Uranium 92 natural 8.42[1][6] 8.903[1] 0.00,[6] 0.005[1] 7.5,[6] 7.57[1]

More comprehensive data is available from NIST[8] and Atominstitut of Vienna.[9]

References

  1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 M.T. Hutchings; P.J. Withers; T.M. Holden; Torben Lorentzen (Feb 28, 2005). Introduction to the Characterization of Residual Stress by Neutron Diffraction. CRC Press. ISBN 9780203402818.
  2. 1 2 3 4 5 6 7 8 9 10 Dmitri I. Svergun; Michel H. J. Koch; Peter A. Timmins; Roland P. May (Aug 8, 2013). Small Angle X-Ray and Neutron Scattering from Solutions of Biological Macromolecules. OUP Oxford. ISBN 9780199639533.
  3. ↑ Amparo Lopez-Rubio & Elliot Paul Gilbert (2009). "Neutron scattering: a natural tool for food science and technology research" (PDF). Trends in Food Science & Technology: 1–11.
  4. ↑ Fong Shu; Venki Ramakrishnan & Benno P. Schoenborn (2000). "Enhanced visibility of hydrogen atoms by neutron crystallography on fully deuterated myoglobin". PNAS. 97 (8): 3872–3877. Bibcode:2000PNAS...97.3872S. doi:10.1073/pnas.060024697. PMC 18109. PMID 10725379.
  5. 1 2 3 4 5 6 7 Oliver C. Mullins; Eric Y. Sheu, eds. (Nov 11, 2013). Structures and Dynamics of Asphaltenes. Springer Science & Business Media. p. 161. ISBN 9781489916150.
  6. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 N.K. Kanellopoulos, ed. (Sep 26, 2000). Recent Advances in Gas Separation by Microporous Ceramic Membranes. ISBN 9780080540320.
  7. 1 2 F. Rodríguez-Reinoso; Jean Rouquerol; KK Unger; Kenneth S.W. Sing, eds. (Aug 26, 1994). Characterization of Porous Solids III. Elsevier. ISBN 9780080887371.
  8. ↑ "Index of /resources/n-lengths/elements".
  9. ↑ "Neutron Scattering Lengths".
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