The geometric construction of Ewald sphere and Bragg condition:

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The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied.

1 The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in the diection of x-ay beam. It is chosen to stat fom a cetain oigin which epesents a ecipocal lattice point. π ii) Daw a sphee of adius k = about this chosen oigin. λ iii) A diffacted beam of wave vecto k (with a magnitude π of k = ) will be fomed if the sphee intesects at any λ othe ecipocal lattice point (at the tail of this vecto). iv) A ecipocal lattice vecto G that connects the ecipocal lattice point (oigin) to the ecipocal lattice point (at the tail of diffacted beam vecto) such that G = k k, as shown in figue 4, whee K = G is chosen. Figue 4: The Ewald sphee constuction. 76

2 Expeimental techniques: The wave length, in most expeiments, is contolled by egulating the enegy of the beam paticles. The wave length λ of any beam paticles can be easily obtained fom the π h 1 elation λ = ( ), whee m is the mass of the paticle and me E is the enegy of the beam paticles. Howeve the wave length πhc λ of photons is λ = ( ), whee c is the velocity of light. E Photons useful fo stuctue analysis have enegies on the ode of 10 kev. While electons have enegies on the ode of 100eV and potons have enegies on the ode of 0.1eV. The wave length λ of the continuous potion of x-ay adiation πhc can be obtained at a minimum λ min = ( ), whee V is the ev acceleating potential of the incident electons that appea as photons in the x-ay instument. This wave length has anothe limit λ max. The wave length λ of the othe potion of x-ay adiation (which epesents a seies of naow, intense peaks with cetain wavelengths) depends on the chaacteistic lines of the typical taget employed in the x-ay instument. Fo example, the aveage of the K α lines is at 1.54 o A and that of K β lines is at 1.39 o A fo a coppe taget. 77

3 Thee main expeimental methods: 1) Laue method ) Rotating cystal method. 3) Powde method. In each of these methods, the main pupose is to make sue that a easonable numbe of peaks can be obtained eithe by using a wide spectum of wave lengths o a wide vaiety of cystal oientations. In paticula, an incident wave vecto k will lead to a diffaction peak (o "Bagg eflection") if and only if the head of this wave vecto lies on k-space Bagg plane. Thus, to seach fo Bagg peaks, we must eithe fix the magnitude of k and keep vaying its diection (vaying the oientation of the cystal with espect to the incident beam diection), o changing the magnitude of k (i.e. changing the wave length of the incident beam). 1. The Laue method: In this method the continuous potion of x-ay adiation is used to illuminate the sample unde study. The wave lengths in the ange λ min < λ < λ max with values 0. o A < λ < 3 o A may b used. Now a single cystal of fixed oientation fom a fixed incident beam diection nˆ with the above-mentioned ange of wave lengths can be used to get Bagg peaks. The Ewald sphees fo the Laue method can be constucted when two vectos ae dawn in the same diection with thei heads at the same ecipocal point, as shown in figue 43. The longe vecto k o has a magnitude π and is the adius of the lage sphee. The λ min 78

4 shote one k 1 has a magnitude π and is the adius of the λ max small sphee. Howeve Bagg peaks can be obseved coesponding to any ecipocal lattice vectos lying within the egion contained between the two sphees. The pojection of the ecipocal lattice vectos G along the unit vecto nˆ in the diection of the incident beam may be obtained fom: nˆ G = G sinθ, nˆ G sinθ = G Also G λ sin θ = = G. k 4π Thus the equied wave length can be found as: nˆ G λ = 4π. G Notes: 1) A given peak may be found when the wavelength fo each ecipocal lattice vecto is evaluated within the ange λ min < λ < λ max and if the stuctue facto does exist. ) The scatteing angle fom the above-mentioned elation can be used to find the diection of G and not its magnitude. 3) Since the ange of the wave lengths in the incident beam is nˆ G limited then the elation λ = 4π can be used to place G limits on the magnitude of ecipocal lattice vectos. 79

5 Figue 43: The Ewald sphees constuction fo the Laue method. The use of Laue method: This method is widely used to detemine lattice symmety. In paticula, it is used to detemine the oientation of a single cystal sample whose stuctue is known. If the incident diection lies along a symmety axis of the cystal (i.e. fou fold symmety fo NaCl stuctue) the patten of spots poduced by the Bagg eflected ays will have the same symmety. This is shown in figue

Figue 44: A Laue patten fo NaCl cystal. This patten shows the fou fold symmety.. The otating-cystal method: In this method a monochomatic x-ay is used with vaying angle of incidence.

6 Figue 44: A Laue patten fo NaCl cystal. This patten shows the fou fold symmety.. The otating-cystal method: In this method a monochomatic x-ay is used with vaying angle of incidence. The diection of the x-ay beam is kept fixed and the oientation of the cystal vaied. As the cystal otates the ecipocal lattice points will otate about the fixed axis. Thus the Ewald sphee (which is detemined by the fixed incident wave vecto k ) is fixed in k-space, while the entie ecipocal lattice otates about the axis of otation of the cystal. Howeve, as ecipocal lattice otates, diffeent ecipocal lattice points coss the suface of the Ewald sphee and when a point is on the suface, the coesponding intensity peak is poduced, as shown in figue

Figue 45: The Ewald constuction fo the otating-cystal method. The use of otating-cystal method: It is used to detemine the shape and size of the unit cell. 3.

7 Figue 45: The Ewald constuction fo the otating-cystal method. The use of otating-cystal method: It is used to detemine the shape and size of the unit cell. 3. The powde method (o Debye-schee method): This method is simila to the pevious method in () (the otating-cystal method), but has in addition the axis of otation is changed ove all possible oientations. In this method a polycystalline sample (o a powde of lage numbe of small andomly oiented cystals) is illuminated by a monochomatic beam. The Bagg peaks will be found by fixing both the incident beam wave vecto k and the Ewald sphee and then allowing the ecipocal lattice to otate though all possible angles about the oigin so that each ecipocal lattice vecto K geneates a sphee of adius k about the oigin. Such a sphee will intesect the Ewald sphee in a cicle povided that K is less than k, as shown in figue 46. The vecto joining any point on such a cicle with the head of k is a wave vecto k, fo which scatteed adiation will be obseved. Each ecipocal lattice vecto of 8

8 length less than 4π/λ and non-vanishing stuctue factos will have a coesponding scatteed peak. φ K = k sin. Figue 46: The Ewald constuction fo the powde method. A cystal is oiented such that an intense scatteed beam occus with an angleθ. If the cystal is otated about the diection of the incident beam, the scatteed beam otates aound the suface of a cone with apex at the cystal and with an angle of apex equal twice the scatteing angle (i.e φ = θ ), as shown in figue

9 k k θ Powde sample φ K Figue 47: The scatteed beam otates aound the suface of a cone with apex at the sample with φ = θ. The indexing of powde patten: The powde patten on a film may appea as a seies of concentic ings, one fo each possible scatteing angle. If we conside a cubic cystal of edge a, the value of a can be detemined expeimentally. Now if the simple cubic lattice is used as a basis fo the indices, the sepaation of the (hkl) a planes is given by: d =. By substituting this h + k + l λ N expession into Bagg law we getsin θ =, 4a whee N = n ( h + k + l ). The scatteing angle θ is measued fo each ing and a value of N is assigned fo each ing. Values of N ae selected so that thei atios ae the same as the atios of the oesponding values of sin θ. If moe than one set of integes has the same atios, the one with the smallest values is usually selected e.g. fo bcc lattice N must be even because h + k + l is even and the squae of even is even. Also the squae of odd is odd. Fo fcc lattice the indices must be all even o all odd. 84

c 2003, Michael Marder

c 2003, Michael Marder Expeimental Detemination of Cystal Stuctues 1 8th Januay 003 c 003, Michael Made Histoy Expeiments and theoy in 191 finally evealed locations of atoms in cystalline solids. Essential ingedients: Theoy