Influence of the characteristics of ESRF X-Ray beams on diffraction

in perfect crystal V. MocellaO, J .P. GuigayO, y. Epelboin', A FreW1do

°European Synchrot ron Radiat.ion Facility B.P. 220.38043, Grenoble Cedex, Fra nce

.t.Laboratoire de Minéralogie-Cristallographie, URA 009 CNRS,

Universités P.M. Curie et D. Diderot, case 115, 75252, Paris Cedex 05, France

ABSTRACT On a beam hne of a tlurd generation synchrotron facility, such as the ESRF, the smoll Source size of the source and the large distance from the source to the sample are responsible for tbe remarkable coherence properties of the beam at the sam pie position. In Bragg diffraction by a perfect crystal. the incident beam is to be considered as a coherent sphericaJ wave, with a ngular aperture larger t han the angular reAectio n range of the crystal. We show theoreticaUy tbat, in such concUtion, the intensity profile of a Bragg diffraction to pograph, in transmissÎon geometry (Laue case). shouJd reproduce the anguJar Bragg reBection pro6.1e (rocking corve) of tbe crystal. We discuss the conditions to be fulfilled conceming the wavelength dispersion of the incident beam. An experimental verification of the theoreticaJ precliction is presented.

1

Introduction

Third generation synchrotron so urces pro duce X-ray beams of hlgh intensity chn.racterised by a long source­ to-sample distance and by a very smalJ SOurce sÎze. The spatial coberence at the sample position is therefore very ltigh for an X-Ray beam. This is the basis of the recent development of X-Ray phase radiogra phy. using a synchrotron X-Ray beam monochromated by a crystal rnonocmomator and placing the recording film sufficiently far from tb e sample in o rder to t ransform the phase IDoduJation into întensity modlÙation by Fresnel d.i1fTaction. 1 The inBuence of spatial and temporal coherence properties of synchrotron X-Ray beams in Bragg diffraction topography bas been discussed by Carvalho & Epelboin 2 ln the present paper we describe a conceptuaUy simple experirnent which wOlÙd be impossible, Or at least e.xtremely difficult, witb a conventionsl X-Ray generator and with an X-Ray beam Iroro a synchrotron of older generation. We obtain a Bragg diffraction profile (usuaUy named "rocking curve") from a perfect silicon crystal on a diffraction topograph, th8.IÙVOLo (X -1)

,2] T (1)d1) , ,

(5)

-1

where the functîon T is th e relevant propagator. For our prnpose, it is not necessary ta know the analytical expression of the propagator. Il is sufficient ta recaU that it is a function of (x -1]) having nonzero values oruy in the interval lx -1)1 < l, with 1 = t tan BB, t being the crys tal thl ckness. The integral is thus limited to an interval of amplitude 2l along the entrance s urface, which is the Bormann triangle basis and represents the influence domain' of each point P on the exit surface (Figure 1).

3

Locally plane wave regime

Consider nOw th e effective divergence Ct, defi.ned 2 as the incident bearn di vergence secn by the part of entranœ surface which influences the values of tbe diffracted beam in P (Figure 1) :

IcosBB Lo

(6)

Œ= -

OUI

Instead of condition (2) we get, by the same approach, the foUowing condition for a, more closely related to experiment:

(lcosBB)'

no

À

«~ => t « 4

(ÀbJl, ) 1

-'-:-~-

SinBB

(7)

If (7) is fulfiUed, we