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

International Tables for Crystallography (2010). Vol. B, ch. 5.1, p. 633   | 1 | 2 |

Section 5.1.4. Standing waves

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: AAuthier@wanadoo.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)[link] and (5.1.2.24)[link], is[\eqalignno{|D|^{2} &= |D_{o}|^{2} \exp (4\pi {\bf K}_{{\bf o}{\rm 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)[link] 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[link]). 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)[link]. 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[link]). 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[link], 1969)[link]. For reviews, see Golovchenko et al. (1982)[link], Materlik & Zegenhagen (1984)[link], Kovalchuk & Kohn (1986)[link], Bedzyk (1988)[link], Authier (1989)[link], Zegenhagen (1993)[link], Patel (1996)[link], Patel & Fontes (1996)[link], Lagomarsino (1996)[link], Zegenhagen et al. (1997)[link], Vartanyants & Zegenhagen (1999)[link] and Vartanyants & Koval'chuk (2001)[link].

References

Authier, A. (1989). X-ray standing waves. J. Phys. (Paris), 50, C7–215, C7–224.
Batterman, B. W. (1964). Effect of dynamical diffraction in X-ray fluorescence scattering. Phys. Rev. A, 133, 759–764.
Batterman, B. W. (1969). Detection of foreign atom sites by their X-ray fluorescence scattering. Phys. Rev. Lett. 22, 703–705.
Bedzyk, M. J. (1988). New trends in X-ray standing waves. Nucl. Instrum. Methods A, 266, 679–683.
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.
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.
Lagomarsino, S. (1996). X-ray standing wave studies of bulk crystals, thin films and interfaces. In X-ray and Neutron Dynamical Diffraction: Theory and Applications. NATO ASI Series, Series B: Physics, Vol. 357, edited by A. Authier, S. Lagomarsino & B. K. Tanner, pp. 225–234. New York, London: Plenum Press.
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.
Patel, J. R. (1996). X-ray standing waves. In X-ray and Neutron Dynamical Diffraction: Theory and Applications. NATO ASI Series, Series B: Physics, Vol. 357, edited by A. Authier, S. Lagomarsino & B. K. Tanner, pp. 211–224. New York, London: Plenum Press.
Patel, J. R. & Fontes, E. (1996). X-ray standing waves: thermal vibration amplitudes at surfaces. In X-ray and Neutron Dynamical Diffraction: Theory and Applications. NATO ASI Series, Series B: Physics, Vol. 357, edited by A. Authier, S. Lagomarsino & B. K. Tanner, pp. 235–248. New York, London: Plenum Press.
Vartanyants, I. A. & Koval'chuk, M. V. (2001). Theory and applications of X-ray standing waves in real crystals. Rep. Prog. Phys. 64, 1009–1084.
Vartanyants, I. A. & Zegenhagen, J. (1999). Photoelectric scattering from an X-ray interference field. Solid State Commun. 113, 299–320.
Zegenhagen, J. (1993). Surface structure determination with X-ray standing waves. Surf. Sci. Rep. 18, 199–271.
Zegenhagen, J., Lyman, P. F., Böhringer, M. & Bedzyk, M. J. (1997). Discommensurate reconstructions of (111)Si and Ge induced by surface alloying with Cu, Ga and In. Phys. Status Solidi B, 204, 587–616.








































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