Scattering by a single nucleus

Willis, B. T. M.

doi:10.1107/97809553602060000600

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 593-594

Section 6.1.3.2. Scattering by a single nucleus

B. T. M. Willis^d

6.1.3.2. Scattering by a single nucleus

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The nuclear forces giving rise to the scattering of neutrons have a range of 10⁻¹⁴ to 10⁻¹⁵ m. This is much smaller than the wavelength of thermal neutrons, and so (from elementary diffraction theory) the neutron wave scattered by the nucleus is spherically symmetrical. Unlike magnetic scattering, there is no `form-factor' dependence of nuclear scattering on the scattering angle.

The incident neutron beam can be represented by the plane wave $[\psi _{0}=\exp(i{\bf k}_0\cdot {\bf r}),]$ with $[{\bf k}_0]$ denoting the wavevector of the neutron and r its position relative to the nucleus. Then, for a nucleus of zero spin, the wavefunction of the scattered neutron is $[\psi _{s}=- {b\over r}\exp (ik_0r).]$ b is the bound nuclear scattering length or nuclear scattering amplitude, and the negative sign ensures that b is positive for hard-sphere or potential scattering .

If the nucleus is free to recoil under the impact of the neutron, as in a gas, the scattering must be treated in the centre-of-mass system. The free scattering length is related to the bound scattering length b in condensed matter by $[ b_{\rm free}= {M\over m_n+M}b,]$ where M is the nuclear mass and [m_n] the mass of the neutron. For hydrogen, ¹H, the free scattering length is one half the bound scattering length, but the difference between the two rapidly diminishes for heavier nuclei.

In general, b is a complex quantity: $[b=b_{0}+b^{\prime }+ib^{\prime \prime }. \eqno (6.1.3.1)]$ $[b_{0}]$ is the scattering length associated with potential scattering , i.e. scattering in which the nucleus behaves like an impenetrable sphere. b′ and b′′ are the real and imaginary parts of the resonance scattering that takes place with the formation of a compound nucleus (nucleus plus neutron). Resonance scattering is only significant when the excitation energy of the neutron is close to an energy level of the compound nucleus. This occurs for relatively few nuclei, e.g. ¹¹³Cd, ¹⁴⁹Sm, ¹⁵⁷Gd, ¹⁷⁶Lu, and b then varies rapidly with wavelength (Fig. 6.1.3.1 ). The phenomenon of resonance scattering has been used to phase neutron reflections (Schoenborn, 1975), but one difficulty is the strong absorption arising from the imaginary component b′′. For the majority of nuclei, the compound nucleus is not formed near resonance: the imaginary component is small, and the scattering length is independent of the neutron wavelength.

Figure 6.1.3.1| top | pdf |

Dependence on neutron wavelength of the coherent scattering length of ¹¹³Cd. b₀ is the potential scattering component, and b′ and b′′ the real and imaginary components of the resonance scattering . The resonance wavelength is 0.68 Å.

There is confusion in the literature regarding the appropriate signs for the real and imaginary parts of the scattering amplitude (Ramaseshan, Ramesh & Ranganath, 1975). The scattering-length curves in Fig. 6.1.3.1 have been drawn to be consistent with the structure-factor formulae in Volume A (IT A, 2005).

Consider now the scattering from a nucleus with non-zero spin I. The neutron has spin $[{1\over2}]$ , and the spin of the combined nucleus–neutron system is either $[I+{1\over2}]$ or $[I-{1\over2}]$ . Each spin state has its own scattering length, b₊ or b₋, and the weights of these states (for scattering unpolarized neutrons) are $[w_{+}={I+1\over2I+1} \eqno (6.1.3.2a)]$ and $[w_{-}={I\over{2I+1}}. \eqno (6.1.3.2b)]$ Values of b₊ and b₋ have been determined experimentally for just a few nuclei with non-zero spin: ¹H, ²H, ²³Na, ⁵9Co, ….

References

International Tables for Crystallography (2005). Vol. A, edited by Th. Hahn. Heidelberg: Springer.Google Scholar

Ramaseshan, S., Ramesh, T. G. & Ranganath, G. S. (1975). A unified approach to the theory of anomalous scattering. Some novel applications of the multiple-wavelength method. Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 139–161. Copenhagen: Munksgaard.Google Scholar

Schoenborn, B. P. (1975). Phasing of neutron protein data by anomalous dispersion. Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 407–421. Copenhagen: Munksgaard.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 593-594