Chapter 3 X-ray diffraction • Bragg’s law • Laue’s ...

1 1 Chapter 3 X-ray diffraction Bragg s law Laue s condition Equivalence of Bragg s law and Laue s condition Ewald construction geometrical structure factor2 Bragg s lawConsider a crystal as made out of parallel planes of ions, spaced a distance d apart. The conditions for a sharp peak in the intensity of the scattered radiation are the x-rays should be specularly reflected by the ions in any one plane the reflected rays from successive planes should interfere constructivelyPath difference between two rays reflected from adjoining planes.

2 Sin2dFor the rays to interfere constructively, this path difference must be an integral number of wavelength sin2dn=Bragg s angle is just the half of the total angle by which the incident beam is deflected. 2 There are different ways of sectioning the crystal into planes, each of which will it self produce further same portion of Bravais lattice shown in the previous page, with a different way of sectioning the crystal planes. The incident ray is the same. But both the direction and wavelength (determined by Bragg condition with d replaced by d ) of the reflected ray are different from the previous Laue formulation of X-ray diffraction by a crystal No particular sectioning of crystal planes Regard the crystal as composed of identical microscopic objects placed at Bravais lattice site Each of the object at lattice site reradiate the incident radiation in all directions.

3 diffraction peaks will be observed in directions that the rays scattered from all lattice points interfere constructively RrIncident X-ray :along directionn wavelength wave vectornk 2 =rA scattered wave:directionn wavelength wave vectornk = 2 r) (coscosnnddd = +r Condition for constructive interference mnnd= ) (rfor integer mMultiply 2mkkd 2)(= rrr5mkkR 2)(= rrrThe condition holds for all possiblemkkd 2)(= rrrdrsoOr equivalently1)(= Rkkiervrthereforekkrr is a reciprocal lattice vectororkkGrrr =Laue condition.

4 Constructive interference will occur provided that the change in wave vector is a vector of reciprocal lattice kkGrrr =Alternative formulation of Laue conditionThis formulation involves only incident wave vector , and does not involve krSince is a reciprocal lattice vector, so iskkrr kk rrGkkrrr= Gkkrrr = k rGkkkrrrr = =GkGkkrr +=2222 Squaring both sidesGkGrr =22orGGkGrr =26 The component of the incident wave vector along the reciprocal lattice vector must be half of the length ofkrGrGris interpreted as.

5 GrGr21Gr21An incident wave vector will satisfy the Laue condition if and only if the tip of the vector lies in a plane that is the perpendicular bisector of a line joining the origin of the reciprocal space to a reciprocal lattice k-space planes are called Bragg =27 Equivalence of the Bragg and von Laue formulationsSuppose the incident and scattered wave vectors and , satisfy the Laue condition that be a reciprocal lattice vectorkrk rkkGrrr =kkGrrr =Elastic scattering:kk =rrIt follows that and make the same angle with the plane perpendicular to.

6 Therefore the scattering can beviewed as a Bragg reflection with Bragg angle , from the family of direct lattice planes perpendicular to the reciprocal lattice vector . krk r Gr GrThe distance between successive planes in this family must satisfy:dG 20=rwhere is the shortest wave vector parallel to Gr0 Grmust be an integral multiple of , since reciprocal lattice is a Bravais latticeGr0Gr0 GnGrr=dnGnG 20==rrFrom the figure, sin2kG=r8dnk = sinNote that 2=k sin2dn=Bragg conditionA Laue diffraction peak corresponding to a change in the wave vector given by the reciprocal lattice vector corresponds to a Bragg reflection from the family of direct lattice planes perpendicular to.

7 The order, n, of the Bragg reflection is just the length of divided by the length of the shortest reciprocal lattice vector parallel to GrGrGris the measurement direction. Only structural information along the set of planes perpendicular to , is being Bragg condition: sin2nd=Define dspacing:022),,(GnGndlkhdrr === sin),,(2lkhd=Soh, k, l are the coordinates of the reciprocal lattice vector associated with the diffraction321blbkbhGrrrr++=0 GnGrr=So h, k, l have a common factor n.

8 For example (222) direction peak is actually the the 2ndorder (n=2) diffraction peak of the (111) plane. 9 Experimental geometries suggested by the Laue conditionGeneral observationsAn incident wave vector will lead to a diffraction peak (or Bragg reflection ) if and only if the tip of the wave vector lies on a reciprocal space Bragg plane. Since Bragg planes are a discrete family of planes, a fixed incident wave vector , for a fixed X-ray wavelength and fixed incident direction relative to the crystal axes there will be in general no diffraction peaks at allIf one wishes to search experimentally for Bragg peaks, one must therefore relax the constraint for fixed , either varying the magnitude of ( varying wavelength)

9 Or varying its direction (in practice, varying the orientation of the crystal with respect to incident direction).krkrkrFor cubic system:) (2zlykxhaG++= r2222lkhaG++= r2222),,(lkhaGlkhd++==r Tetragonal system:2222221clakhd++=Hexagonal:2222223 41clakhkhd+ ++=Orthorhombic:22222221clbkahd++=10 The Ewald constructionDraw a sphere in reciprocal space centered on the tip of the incident wave vector of radius (so that it passed through the origin) krkThere will be some wave vector satisfying the Laue condition if and only if some reciprocal lattice point (in addition to the origin lies on the surface of the sphere)

10 In general, a sphere in reciprocal space with the origin on its surface will have no other reciprocal lattice points on its surfaceSo in general, there are no diffraction following methods are used to relax the constrains in order to achieve diffraction peaks1. Laue methodFix the orientation of the single crystal. Search for Bragg peaks by using not a monochromatic X-ray beam, but one containing wavelength for up to .1 0 11 The Ewald sphere will expand into the region, contained between the two spheres determined by and , and Bragg peaks will be observed corresponding to any reciprocal lattice vectors laying within the 200 =rnk 211 =rLaue method is best suited for determining the orientation of a single crystal specimen whose stuctureis The rotating crystal methodFix the wavelength, allow the angle of incidence to vary in practice.