Calculation of x-ray wave fields in perfect crystals by applying Riemann’s method to solve the Takagi-Taupin equations

S.

Takagi

, “,” , – ().

D.

Taupin

, “,” , – ().

U.

Bonse

and

M.

Hart

, “,” , – ().

R.

Deslattes

and

A.

Henins

, “,” , – ().

P.

Becker

,

K.

Dorenwendt

,

G.

Ebeling

,

R.

Lauer

,

W.

Lucas

,

R.

Probst

,

H.-J.

Rademacher

,

P.

Seyfried

, and

H.

Siegert

, “,” , – ().

G.

Basile

,

A.

Bergamin

,

G.

Cavagnero

,

G.

Mana

,

E.

Vittone

, and

G.

Zosi

, “,” , – ().

The quantity $a0$ appears in the relation $NA=8M/(ρa03),$ where ρ and M are the density and the molar mass, respectively, of a single crystal, for example, of silicon. If each quantity could be determined with a relative uncertainty of a few parts per $10−9,$ then the relative combined standard uncertainty $ur$ of $NA$ could be reduced to $1×10−8.$ The effort to achieve this goal is known as the Avogadro project, started in 1991 by the Bureau International des Poids et Mesures. The goal of this project is to relate the international definition of the kilogram, at present a Pt-Ir artifact, to an invariant quantity of nature such as, for instance, the Avogadro constant. In this way we could count the number of atoms in a mole of Si, for instance, and therefore a new definition of the unit of mass would be traced back to an atomic mass unit.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1964), p. 52.

W. Zachariasen, Theory of X-ray Diffraction in Crystals (Dover, New York, 1967).

A. Authier, Dynamical Theory of X-ray Diffraction, 2nd ed. (IUCR/Oxford U.P., Oxford, 2001).

The conditions are sometimes called initial because in the T-T equations the variable z could take the role of time.

P.

Becker

, “,” , – ().

G.

Mana

and

G.

Zosi

, “,” , – ().

The change of variables has been carried out using Mathematica, 〈http://www.wolfram.com)〉.

See EPAPS Document No. E-AJPIAS-73-012505 for our Mathematica notebook.

S. Takagi, “Wave optics—A dynamical theory of diffraction for distorted crystals,” International Summer School on X-ray Dynamical Theory and Topography, 18–26 August 1975, Limoges, France, unpublished notes.

G.

Thorkildsen

, “,” , – ().

G.

Thorkildsen

,

H.

Larsen

, and

E.

Weckert

, “,” , – ().

C. Hammond, The Basics of Crystallography and Diffraction (IUCTC/Oxford U.P., Oxford, 2004).

M. von Laue, Röntgenstrahleninterferenzen (Akademische Verlagsgesellschaft, Frankfurt, a.M., 1960).

S.

Takagi

, “,” , – ().

See Eq. (I.6.4-5) in Ref. 2.

This quantity can depend on the polarization state of the incident wave.

See Eq. (40) in Ref. 21.

T.

Katagawa

and

N.

Kato

, “,” , – ().

F.

Chukowski

,

K.

Gabrielyan

, and

P.

Petrashen

, “,” , – ().

A.

Authier

,

C.

Malgrange

, and

M.

Tournaire

, “,” , – ().

C.

Carvalho

and

Y.

Epelboin

, “,” , – ().

G.

Mana

,

C.

Palmisano

, and

G.

Zosi

, “,” , – ().

G.

Mana

,

C.

Palmisano

, and

G.

Zosi

, “,” , – ().

J. Kevorkian, Partial Differential Equations (Wadsworth & Brooks/Cole, Pacific Grove, 1990), Secs. 4.4.3 and 1.2.1.

A.

Authier

and

D.

Simon

, “,” , – ().

To see in greater detail how the energy propagates into the crystal for different thickness values, see Ref. 15 for four additional figures.

B.

Batterman

and

H.

Cole

, “,” , – ().

The roots of the Bessel functions $J0$ and $J1$ are not generally periodic except asymptotically for large z.

E. G.

Kessler

,

A.

Henins

,

R. D.

Deslattes

,

L.

Nielsen

, and

M.

Arif

, “,” , – ().