J. Appl. Cryst. (2003). 36, 129±136 V. Mocella et al.  Takagi±Taupin equations 129 research papers Journal of Applied Crystallography ISSN 0021-8898 Received 12 September 2002 Accepted 11 November 2002 # 2003 International Union of Crystallography Printed in Great Britain ± all rights reserved A new approach to the solution of the Takagi± Taupin equations for X-ray optics: application to a thermally deformed crystal monochromator V. Mocella, a * W.-K. Lee, a G. Tajiri, a D. Mills, a C. Ferrero b and Y. Epelboin c a Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA, b ESRF, BP 220 F-38043 Grenoble Cedex 09, France, and c Laboratoire de Mine Âralogie- Cristallographie, 4, place Jussieu Case 115, F-75252 Paris Cedex 05, France. Correspondence e- mail: mocella.v@irea.cnr.it A new computational approach to the Takagi±Taupin equations for the analysis of X-ray optics is presented and applied to heat-loaded synchrotron monochromators. The new numerical approach is based on the assumption that the incoming wave is completely incoherent (incoherent point source). The simulation avoids any kind of simpli®cation: it takes into account the thermal deformation of the crystals calculated by the ®nite-element method, the transverse beam direction and the beam spatial distribution. As an application example, the theoretical rocking curves for low-energy low-order and high- energy high-order re¯ections are compared with results of measurements carried out at the Advanced Photon Source (APS) on a cryogenically cooled double-crystal monochromator. The good agreement shows that the adopted approach succeeds in simulating very closely the actual experimental setup while exhibiting satisfactorily short execution times. 1. Introduction Diffraction by crystals forms the basis of many X-ray optical elements. The ability to predict the performance of these elements is an important issue in many experiments. This is especially true at synchrotron facilities where the success of a beamline may hinge on how well the overall optics of the beamline perform. Although the diffraction properties of perfect crystals are well understood, unfortunately in practice the environment under which they are employed often introduces undesired distortions to the crystal element. The most common of these are mounting-induced and X-ray beam thermal-load-induced strains. Attempts to simulate the performance of an optical element under stress usually begin with ®nite element analysis (FEA). This provides information on the structural deformations of the crystal due to the thermal load or mounting mechanism. However, in the past, the diffraction analysis has often been limited to a geometrical approach which assumes that the crystal behaves as a mirror and deviations from the perfect crystal are taken into account simply by looking at the slope error along the crystal surface (Chrzas et al. , 1990; Zhang, 1993; Freund et al., 1997). This simple approach can give useful information, but is quantitatively in agreement with experi- ments only for small deformations. When strong deformations are involved a more accurate approach is necessary (Mocella, Ferrero et al. , 2001; Freund et al., 2000). Diffraction by a perfect crystal can be described using dynamical diffraction theory (see Authier, 2001). This theory has been generalized for deformed crystals in the two-beam case (only one diffracted beam in addition to the forward diffracted beam) via the Takagi±Taupin equations (TTE) (Takagi, 1962, 1969; Taupin, 1964): @D 0 @s 0  ik h D h s 0 ; s h  ; @D h @s h  ik h D 0 s 0 ; s h   2i   sin 2 B    1 k @ h  u   @s h  D h s 0 ; s h  ; 1 where k = 1/,  is the wavelength of the incident beam, h is the reciprocal-space vector of the considered re¯ection,  h ,  h are Fourier components of the electric susceptibility,  B is the kinematic Bragg angle and  is the angular deviation from the Bragg condition. The crystal deformation is represented by u, which describes the displacement vector of the lattice positions with respect to the undeformed crystal lattice. This couple of hyperbolic partial differential equations has analy- tical solutions only for special cases of simple deformations (Litzman & Jana  cek, 1974; Katagawa & Kato, 1974; Chukhovskii et al., 1978). In the general case, a numerical solution is necessary. Numerical solutions have been used extensively for the interpretation of the contrast in X-ray topography images (Epelboin, 1985, 1977). Up to now, only limited applications of the numerical solutions of the TTE have been reported for X-ray optics (Gronkowski & Malgrange, 1984a; Uschmann et al., 1993). Recently, a numerical approach to the TTE, coupled with