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One & Two-Dimensional Inverse Problem

Let us imagine a situation: someone, Mr A, comes to unknown person, Mr B, to take measurements of Mr B's body for a new sweater knitting which has been ordered by Mr B's friend. Mr B has never been seen by Mr A, but Mr A relies on an obvious (how dangerous it can be!) guess that Mr B is a human. Mr A enters the Mr B's office and approaches Mr B's room door. There is a sign on the door: "Mr B". Mr A opens the door and comes inside. We can safely (relatively only!) suppose that in ninety nine cases of one hundred Mr A will see a man, Mr B, in the room (unless Mr B is a hard smoker or Mr A has come after hours). Then Mr A can take measurements quite precisely and knit a nice well-fitted sweater for Mr B.

However, there is a chance that Mr A will see a Dragon in the Mr B's room which just has eaten Mr B, or it is turned out that Mr B's is not a human at all! If this event happens, the life and career of Mr A are in a very serious danger. He is not able to take measurements any more, simply because Dragon does not want to be measured or it is too big and ugly to be touched.

What do Mr A have to do to avoid such a situation? We would suggest Mr A to knock the door first or, even better, to look into the room through a window from outside. We would want to encourage him do not rely only on the door sign, but to try to get information about the room content without a priori knowledge - door sign.

Such a tale could seem a bit speculative, however, it explicitly reflects a correspondence between two major approaches to the inverse problem solution: least square fitting (LSF) of calculated function to the experimentally observed data on one side, and a model-independent reconstruction (MIR) of inner crystal structure on the other.

The page presents well-developed and experimentally proven approaches to the model- independent reconstruction of crystalline material internal structure using high-energy radiation. The method allows one to characterise one (1D) and two-dimensional (2D) crystal- lattice strain profiles in single crystals. The 2D case for the moment is presented only for the case of deformations are periodic in one direction. This is very common situation of semiconductor superlattices used in modern microelectronics technology.

In details the theory is presented in the following publications

  1. T. E. Goureev, A. Y. Nikulin and P. V. Petrashen'. Phys.Stat.Sol.(a), 130, 263 (1992).
  2. A. Y. Nikulin, O. Sakata, H. Hashizume and P. V. Petrashen, J. Appl. Cryst., 27, 338 (1994).
  3. A. Y. Nikulin, T. E. Gureyev, A. W. Stevenson et. al., J. Appl. Cryst., 28, 803 (1995).
  4. A. Y. Nikulin, A. W. Stevenson and H. Hashizume, Phys. Rev. B, 53, 8277 (1996).

Phase retrieval formalism of an X-ray diffracted wave for one-dimensional (1D) case

The possibility of the retrieval of phase of the electromagnetic wave scattered by an object in the case of one-dimensional modulation distribution relies on the assumption that the wave-amplitude function R(q) and its logarithm ln{R(q)} are analytical. It means that the Cauchy-Riemann relations must be valid: where R(q)=u(qr,qi) (1) and qr,q2 are the real and imaginary parts of the reciprocal coordinate. In the case R(q) has no zeros in the Upper Complex Half-Plane (UCHP) it is possible to reconstruct the phase of the scattered wave from the measured intensity I(q)=|R(q)|^2 via the logarithmic-dispersive relation (LDR): (2), where P is the Cauchy principal value. If R(q) has zeros am (m=1, 2, ..., M) (where M is the total number of points in depth on the imaged strains) in UCHP, then the term 2Σmarg(qi - αm) should be added to the minimal phase solution (2).

In the kinematical approximation, for the case of 1D lattice distortions, the amplitude reflection coefficient of diffracted x-rays by single-crystal can be written in the form (3), where z is the depth below the crystal surface, is proportional to the modulus of the crystal susceptibility such that arg= 2pih u(z), h is the reciprocal vector, u(z) is the displacement field and mu is the linear absorption coefficient. We will consider the normalised amplitude ~R(q): (4). The Fourier transform of ~R(q) is non-zero from the surface to z=T and zero for z>T. As a result ~R(q) can be analytically continued into the whole complex plane. This fact allows us to obtain the phase of ~R(q) via a logarithmic-dispersion relation (2) which has the following form in this case: .

For more information and example figures see also Software

Phase retrieval formalism of an X-ray diffracted wave for two-dimensional (2D) case Actually, two-dimensional phase reconstruction for the case of the crystal structure modulation is periodic in one direction deals with the individual harmonics of the crystal susceptibility expended into a Fourier series. Then, the one-dimensional phase-retrieval formalism can be applied to each Fourier harmonic of two-dimensional intensity distribution in the vicinity of the Bragg point.
Gray-scale representation, on logarythmic scale, of intensity distributions in reciprocal space for the Si(111) sample without periodic SiO2 surface layer. The total range covered in both qx and qz directions was 4.7x10^-3 nm^-1 and 6.2x10^-2 nm^-1, respectively.
Crystal-truncation rod (CTR) distributions (extracted from 2D intensity profile) associated with the fundamental (n=0) and satellite (n=1 to 11) reflections, observed from the ion-implanted Si sample with periodic SiO2 surface layer. The satellite curves are vertically shifted to avoid overlaps.
Crystal-truncation rod (CTR) distributions (extracted from 2D intensity profile) associated with the fundamental (n=0) and satellite (n=1 to 11) reflections, observed from the ion-implanted Si sample without periodic SiO2 surface layer. The satellite curves are vertically shifted to avoid overlaps.
For more information and results see also Two-Dimensional Crystal-Lattice Strain Reconstruction with Nanometre Spatial Resolution and High-Resolution X-ray Diffraction Experiments Using Synchrotron Radiation