International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.3, pp. 67-68

## Section 2.3.2.2. Neutron scattering lengths

C. J. Howarda* and E. H. Kisia

aSchool of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
Correspondence e-mail:  chris.howard@newcastle.edu.au

#### 2.3.2.2. Neutron scattering lengths

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The scattering lengths of most interest in neutron powder diffraction are those for coherent elastic scattering, bcoh, often abbreviated to b. As already mentioned, there is no angle (Q) dependence, since the scattering from the nucleus is isotropic. A selection of scattering lengths for different isotopes and different elements is given in Table 2.3.2.

 Table 2.3.2| top | pdf | Coherent scattering lengths and absorption cross sections (for 25 meV neutrons) for selected isotopes
 Data are taken from Section 4.4.4 of Volume C (Sears, 2006). Where not stated, the values are for the natural isotopic mix. The X-ray atomic form factors, f, evaluated at Q = 1.2π Å−1, are included for comparison.
ElementIsotopebcoh (fm)σs(tot) (10−24cm2)σa (10−24cm2)fIsotopic abundance (%)
H   −3.7390 (11) 82.02 (6) 0.3326 (7) 0.25
1 −3.7406 (11) 82.03 (6) 0.3326 (7)   99.985
2 6.671 (4) 7.64 (3) 0.000519 (7)   0.015
3 4.792 (27) 3.03 (5) 0

B   5.30 (4) − 0.213 (2)i 5.24 (11) 767 (8) 1.99
10 −0.1 (3) − 1.066 (3)i 3.1 (4) 3835 (9)   20.0
11 6.65 (4) 5.78 (9) 0.0055 (33)   80.0

C   6.6460 (12) 5.551 (3) 0.00350 (7) 2.50
12 6.6511 (16) 5.559 (3) 0.00353 (7)   98.90
13 6.19 (9) 4.84 (14) 0.00137 (4)   1.10

O   5.803 (4) 4.232 (6) 0.00019 (2) 4.09

Ti   −3.370 (13) 4.06 (3) 6.43 (6) 13.2
46 4.725 (5) 2.80 (6) 0.59 (18)   8.2
47 3.53 (7) 3.1 (2) 1.7 (2)   7.4
48 −5.86 (2) 4.32 (3) 8.30 (9)   73.8
49 0.98 (5) 3.4 (3) 2.2 (3)   5.4
50 5.88 (10) 4.34 (15) 0.179 (3)   5.2

V   −0.3824 (12) 5.10 (6) 5.08 (2) 14.0

Ni   10.3 (1) 18.5 (3) 4.49 (16) 18.7
58 14.4 (1) 26.1 (4) 4.6 (3)   68.27
60 2.8 (1) 0.99 (7) 2.9 (2)   26.10
61 7.60 (6) 9.2 (3) 2.5 (8)   1.13
62 −8.7 (2) 9.5 (4) 14.5 (3)   3.59
64 −0.37 (7) 0.017 (7) 1.52 (3)   0.91

Cu   7.718 (4) 8.03 (3) 3.78 (2) 19.9
63 6.43 (15) 5.2 (2) 4.50 (2)   69.17
65 10.61 (19) 14.5 (5) 2.17 (3)   30.83

Zn   5.680 (5) 4.131 (10) 1.11 (2) 20.8

Zr   7.16 (3) 6.46 (14) 0.185 (3) 27.0

Gd   6.5 (5) 180 (2) 49700 (125) 45.9
155 6.0 (1) − 17.0 (1)i 66 (6) 61100 (400)   14.8
157 −1.14 (2) − 71.9 (2)i 1044 (8) 259000 (700)   15.7

Pb   9.405 (3) 11.118 (7) 0.171 (2) 60.9

The first thing to note is the variation in scattering length from element to element and indeed from isotope to isotope. The scattering lengths are in most cases positive real numbers, in which case there is a phase reversal of the neutron on scattering, but for some isotopes the scattering lengths are negative, so there is no change in phase on scattering. The scattering lengths are determined by the details of the neutron–nucleus interaction (Squires, 1978).5 In the event that the neutron–nucleus system is close to a resonance, such as it is for 10B, 155Gd and 157Gd, scattering lengths will be complex quantities and the scattered neutron will have some different phase relationship with the incident one. The imaginary components imply absorption, which is reflected in the very high absorption cross sections, σa, for these isotopes.

The total scattering cross section, σs, is given by when only coherent scattering from a single isotope is involved, which is very nearly the case for oxygen since 99.76% of naturally occurring oxygen is zero-spin 16O. In most cases there is a more substantial contribution from incoherent scattering, which may be either spin or isotope incoherent scattering. Spin incoherent scattering arises because the scattering length depends on the relative orientation of the neutron and nuclear spins, parallel and antiparallel arrangements giving rise to scattering lengths and , respectively. Isotope incoherent scattering arises because of the different scattering of neutrons from different isotopes of the same element. In almost all circumstances (except, for example, at the extraordinarily low temperatures mentioned in Section 2.3.1) the distributions of spins and isotopes are truly random, which means that there is no angle dependence in this scattering: this is sometimes described as Laue monotonic scattering.

When b varies from nucleus to nucleus (even considering just a single element), the coherent scattering is determined by the average value of b, that is , , and the average incoherent cross section is given by . The total scattering cross section σs is the sum of the two cross sections (Squires, 1978; see also Section 2.3.2 in Kisi & Howard, 2008). For the particular case of a nucleus with spin I, the states I + 1/2 and I − 1/2 give scattering determined by and , respectively, and have multiplicities 2I + 2 and 2I, respectively, from which it follows that

More information, including a comprehensive listing of scattering lengths, can be found in Section 4.4.4 of International Tables for Crystallography Volume C (Sears, 2006). This listing presents the spin-dependent scattering lengths via bcoh and binc as just defined. Other compilations can be found in the Neutron Data Booklet (Rauch & Waschkowski, 2003), and online through the Atominstitut der Österreichischen Universitäten, Vienna, at http://www.ati.ac.at/~neutropt/scattering/table.html . In addition, the majority of computer programs used for the analysis of data from neutron diffraction incorporate, for convenience, a list of bcoh values for the elements.

### References

Kisi, E. H. & Howard, C. J. (2008). Applications of Neutron Powder Diffraction. Oxford University Press.Google Scholar
Rauch, H. & Waschkowski, W. (2003). Neutron scattering lengths. Neutron Data Booklet, 2nd ed., edited by A. J. Dianoux & G. Lander, §1.1. Grenoble: Institut Laue–Langevin.Google Scholar
Sears, V. F. (2006). Scattering lengths for neutrons. International Tables for Crystallography, Volume C, Mathematical, Physical and Chemical Tables, 1st online ed., edited by E. Prince, pp. 444–454. Chester: International Union of Crystallography.Google Scholar
Squires, G. L. (1978). Introduction to the Theory of Thermal Neutron Scattering. Cambridge University Press.Google Scholar