Abstract
This chapter develops elastic scattering from wave concepts. It begins with a review of the elementary mathematics of waves in one dimension, and then reviews plane waves, spherical waves, and phase factors. Coherent and incoherent scattering is explained with some detail. Scattering amplitudes and cross sections are defined, and x-ray scattering from a semiclassical atom is used to explain anomalous scattering and its trends across the periodic table of elements. The scattering of the electron wavefunction is obtained with an integral form of the Schrödinger equation, and the Born approximation is developed. The atomic form factor is defined, and used with some physical interpretations and approximations for electron scattering by atoms.
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Notes
- 1.
We say ψ(kx−ωt) travels to the right with a “phase velocity” of ω/k. The wave ψ(kx+ωt) travels to the left.
- 2.
This is often useful because real scatterers typically emit spherical waves, but Fourier transforms require plane waves.
- 3.
This is why the sky is blue. Visible light is of low energy compared to excitations of electrons in the molecules of the atmosphere.
- 4.
With k≡2π/λ, Δk=(4πsinθ)/λ, and s=sinθ/λ.
- 5.
Compton scattering is incoherent and inelastic.
- 6.
Notice the dog in the lower right, which Gainsborough evidently decided was inappropriate for the portrait. The top of the x-ray image also shows the collar of another person, indicating the canvas itself was used for a previous portrait.
- 7.
An intuitive shortcut from (4.64) to (4.71) is to regard \((\nabla^{2}+\boldsymbol{k}_{0}^{2})\) as a scattering operator that generates a scattered wavelet proportional to U(r′)Ψ(r′). The scattered wavelet must also have the properties of (4.55) for its amplitude and phase versus distance. The scattered wavelet amplitude from a small volume, d3 r′, about r′ is:
$$ \mathrm{d}\varPsi _{\mathrm{scatt}}\bigl(\boldsymbol{r},\boldsymbol{r}^{\prime}\bigr)=U\bigl( \boldsymbol{r}^{\prime} \bigr)\varPsi \bigl(\boldsymbol{r}^{\prime}\bigr) \frac{\mathrm{e}^{\mathrm{i}k\vert \boldsymbol{r}-\boldsymbol{r}^{\prime }\vert }}{\vert \boldsymbol{r}-\boldsymbol{r}^{\prime} \vert } \,\mathrm{d}^{3} \boldsymbol{r}^{\prime}, $$(4.63)which is a spherical wave at r originating at r′. This approach is even more intuitive for x-ray scattering, which is proportional to the number of electrons about the atom. For x-rays, U(r′) becomes ρ(r′), the electron density. The result is the same as (4.83) below, but with a different prefactor and ρ(r′) instead of V(r′).
- 8.
Extending the Born approximation to higher orders is not difficult in principle. Instead of using an undiminished plane wave for Ψ(r′), we could use a Ψ(r′) that has been scattered once already. Equation (4.73) gives the second Born approximation if we use do not use the plane wave of (4.74) for Ψ(r′), but rather:
$$ \varPsi \bigl(\boldsymbol{r}^{\prime}\bigr)=\mathrm{e}^{\mathrm{i}\boldsymbol{k}_{0}\cdot \boldsymbol{r}^{\prime} }+\frac{2m}{\hbar^{2}}\int V \bigl(\boldsymbol{r}^{\prime\prime}\bigr) \varPsi \bigl(\boldsymbol{r}^{\prime\prime}\bigr) G\bigl(\boldsymbol{r}^{\prime},\boldsymbol{r}^{\prime\prime}\bigr) \, \mathrm{d}^{3}\boldsymbol{r}^{\prime\prime}, $$(4.75)where we now use a plane wave for Ψ(r′′):
$$ \varPsi \bigl(\boldsymbol{r}^{\prime\prime}\bigr)\simeq \mathrm{e}^{\mathrm{i}\boldsymbol{k}_{0}\cdot \boldsymbol{r} ^{\prime\prime}}. $$(4.76)The second Born approximation involves two centers of scattering. The first is at r′′ and the second is at r′. The second Born approximation is sometimes used when calculating the scattering of electrons with energies below 30 keV from heavier atoms such as Xe. For solids, however, the second and higher Born approximations are not used very frequently. If the scatterer is strong enough to violate the condition of weak scattering used in the first Born approximation, the scattering may also violate the assumptions of the second Born approximation.
- 9.
If we neglect a constant prefactor, this assumption of |r−r′|=|r| is equivalent to assuming that the scatterer is small compared to the distance to the detector.
- 10.
For neutron scattering, however, the scattering potential originates with the tiny volume of the nucleus (see Sect. 3.9). Nuclear form factors have no dependence on Bragg angle in the energy ranges of materials science.
- 11.
Defining \(U \equiv \mathrm{e}^{-r/r_{0}}\) and dV≡sin(Δkr) dr, we integrate by parts: ∫U dV=UV−∫V dU. The integral on the right hand side is evaluated as: \((\varDelta kr_{0} )^{-1}\int_{r=0}^{\infty}\cos (\varDelta kr ) \mathrm{e}^{-r/r_{0}} \,\mathrm{d} r\), which we integrate by parts again to obtain: \(- (\varDelta kr_{0} )^{-2}\int_{r=0}^{\infty}\sin (\varDelta kr ) \mathrm{e}^{-r/r_{0}}\,\mathrm{d}r\). This result can be added to the ∫U dV on the left hand side to obtain (4.100).
- 12.
The cross-section from all contributions |f el(Δk)|2 therefore decreases somewhat slower than Z 2.
- 13.
In the present usage the angle θ is defined as half the total angle of scattering, consistent with our definition of the Bragg angle.
- 14.
This neglects the effects of anomalous x-ray scattering attributed to atomic resonances (Sect. 4.2.1).
- 15.
This is a consequence of the long-range character of the Coulomb interaction around a non-neutral atom. A real crystal is electrically neutral, however. For a pair of ions, one positive and one negative, it is straightforward to show that the prefactor in (4.118) changes from Δk −2 to Δk −1. Furthermore, the Δk −1 divergence of f el is suppressed if there are alternating chains of +−+− and −+−+, or until Δk is so small that ΔkL≃2π, where L is the size of the crystal along \(\widehat{\boldsymbol{\varDelta k}}\).
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Fultz, B., Howe, J. (2013). Scattering. In: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29761-8_4
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DOI: https://doi.org/10.1007/978-3-642-29761-8_4
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