Standing waves

Authier, A.

doi:10.1107/97809553602060000569

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 5.1, p. 541 | 1 | 2 |

Section 5.1.4. Standing waves

A. Authier^a ^*

^a Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.4. Standing waves

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The various waves in a wavefield are coherent and interfere. In the two-beam case, the intensity of the wavefield, using (5.1.2.14) and (5.1.2.24), is $[\eqalignno{|D|^{2} &= |D_{o}|^{2} \exp (4\pi {\bf K}_{{\bf o}i} \cdot {\bf r})\cr &\quad \times [1 + |\xi|^{2} + 2C|\xi | \cos 2\pi ({\bf h} \cdot {\bf r} + \Psi)], &(5.1.4.1)}]$ where Ψ is the phase of ξ, $[\xi = |\xi | \exp (i\Psi). \eqno(5.1.4.2)]$

Equation (5.1.4.1) shows that the interference between the two waves is the origin of standing waves. The corresponding nodes lie on planes such that $[{\bf h} \cdot {\bf r}]$ is a constant. These planes are therefore parallel to the diffraction planes and their periodicity is equal to $[d_{hkl}]$ (defined in the caption for Fig. 5.1.2.1a). Their position within the unit cell is given by the value of the phase Ψ.

In the Laue case, Ψ is equal to $[\pi + \varphi_{h}]$ for branch 1 and to $[\varphi_{h}]$ for branch 2, where $[\varphi_{h}]$ is the phase of the structure factor, (5.1.2.6). This means that the nodes of standing waves lie on the maxima of the hkl Fourier component of the electron density for branch 1 while the anti-nodes lie on the maxima for branch 2.

In the Bragg case, Ψ varies continuously from $[\pi + \varphi_{h}]$ to $[\varphi_{h}]$ as the angle of incidence is varied from the low-angle side to the high-angle side of the reflection domain by rocking the crystal. The nodes lie on the maxima of the hkl Fourier components of the electron density on the low-angle side of the rocking curve. As the crystal is rocked, they are progressively shifted by half a lattice spacing until the anti-nodes lie on the maxima of the electron density on the high-angle side of the rocking curve.

Standing waves are the origin of the phenomenon of anomalous absorption, which is one of the specific properties of wavefields (Section 5.1.5). Anomalous scattering is also used for the location of atoms in the unit cell at the vicinity of the crystal surface: when X-rays are absorbed, fluorescent radiation and photoelectrons are emitted. Detection of this emission for a known angular position of the crystal with respect to the rocking curve and therefore for a known value of the phase Ψ enables the emitting atom within the unit cell to be located. The principle of this method is due to Batterman (1964, 1969). For reviews, see Golovchenko et al. (1982), Materlik & Zegenhagen (1984), Kovalchuk & Kohn (1986), Bedzyk (1988), Authier (1989), and Zegenhagen (1993).

References

Authier, A. (1989). X-ray standing waves. J. Phys. (Paris), 50, C7–215, C7–224.Google Scholar

Batterman, B. W. (1964). Effect of dynamical diffraction in X-ray fluorescence scattering. Phys. Rev. A, 133, 759–764.Google Scholar

Batterman, B. W. (1969). Detection of foreign atom sites by their X-ray fluorescence scattering. Phys. Rev. Lett. 22, 703–705.Google Scholar

Bedzyk, M. J. (1988). New trends in X-ray standing waves. Nucl. Instrum. Methods A, 266, 679–683.Google Scholar

Golovchenko, J. A., Patel, J. R., Kaplan, D. R., Cowan, P. L. & Bedzyk, M. J. (1982). Solution to the surface registration problem using X-ray standing waves. Phys. Rev. Lett. 49, 560–563.Google Scholar

Kovalchuk, M. V. & Kohn, V. G. (1986). X-ray standing waves – a new method of studying the structure of crystals. Sov. Phys. Usp. 29, 426–446.Google Scholar

Materlik, G. & Zegenhagen, J. (1984). X-ray standing wave analysis with synchrotron radiation applied for surface and bulk systems. Phys. Lett. A, 104, 47–50.Google Scholar

Zegenhagen, J. (1993). Surface structure determination with X-ray standing waves. Surf. Sci. Rep. 18, 199–271.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 5.1, p. 541