It is instructive however for you to do a simple structure by hand. Rocksalt Structure. Quite common in nature. KCl, NaCl, MgO
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1 Today the structure determinations etc are all computer -assisted It is instructive however for you to do a simple structure by hand Rocksalt Structure Quite common in nature KCl, NaCl, MgO 9-1
2 Typical XRD scan Pea k No 2θ angle Absolute Intensity Relative Intensity d spacing X-ray wavelength used: CoKα = Å counts 5 % 5.07 Å counts 173 counts 7 % 29 % 4.98 Å 4.89 Å λ = 2 d sinθ counts 32 % 3.68 Å counts 100 % 3.16 Å λ/(2 sinθ) = d 9-2
3 A comparison of the X-ray powder diffraction patterns of NaCl (bottom) and KCl (top). Peaks in the KCl diffraction pattern are labeled with Miller indices, h k l, indicating the set of lattice planes responsible for that diffraction peak. The KCl peaks are shifted to lower angles relative to the NaCl pattern due to the larger cubic unit cell of KCl. 9-3
4 Systematic absences Sometimes particular hkl reflections are "missing" this can be useful let's see why and how 9-4
5 Systematic absences in XRD We have seen that in a PXRD experiment a crystalline solid gives peaks at certain angles (2θ) only. These peaks occur when there is constructive interference between the X-ray scattered from the crystal. As given by Bragg s law. However some of the expected peaks could be absent due to additional symmetry in the crystal Eg. For a BCC crystal, X-rays scattered from the atom at the centre of the cube can interfere can interfere with those scattered by the corner atoms. 9-5
6 Systematic absences for BCC & FCC BCC - Reflections absent if h+k+l = ODD (100) (110) (111) (200) (210) (211) (220) FCC - Absent UNLESS h,k,l are all ODD or all EVEN (100) (110) (111) (200) (210) (211) (220) 9-6
7 More: Consider alpha - Iron which is boby centre cubic Reflections from the 100 planes would be expected but the reflection from the body centre atom is 180 out of phase with these, (midway between planes) in a bulk crystal there are equal numbers of corner and body centre atoms and so the beams diffracted by each cancel exactly In contrast the 200 beam is present because there are no atoms between the 200 planes 9-7
8 Lattice Type Rule for reflection to be observed Primitive P None Body Centred, I hkl: h+k+l =2n Face Centred, F hkl: hkl either all odd or all even Side Centred hkl: h + k = 2n Rhomohedral hkl: -h + k + l = 3m or (h-k + l) =3n also see lect 16: Bravais Latices 9-8
9 For a systematic absence two conditions must be simultaneously met waves are exactly 180 degrees out of phase 2. Waves must be exactly the same amplitude 2. is determined by the scattering factor Good example: rocksalt structure "face centre cubic" so using Table above hkl must be all odd or even for reflection to be observed: so : 110 reflection is systematically absent 9-9
10 But look at the 111 planes Some have Na on them some have Cl and they are interleaved : looks OK but the two planes have different scattering powers X rays are scattered by electrons so in the series KCl KBr KF KI the intensity of the 111 for KCl is zero the trend in intensity is: KCl < KF < KBr < KI 9-10
11 Phase purity. In a mixture of compounds each crystalline phase present will contribute to the overall powder X-ray diffraction pattern. In preparative materials chemistry this may be used to identify the level of reaction and purity of the product. The reaction between two solids Al 2 O 3 and MgO to form MgAl 2 O 4 may be monitored by powder X-ray diffraction. 9-11
12 PXRD line-broadening and crystallite size Several effects could change PXRD linewidths: crystallite size, overlap of peaks, microstrain, lattice and stacking faults Particle = one crystallite Particle = five fused crystallites NB : PXRD measures crystallite size!! 9-12
13 Determination of crystallite size In order to observe sharp diffraction maxima in the PXRD pattern, the crystallites need to be of sufficient size to ensure that slightly away from the 2Θ maximum, destructive interference occurs. The effect of crystallite size on peak width in PXRD (a) (b) (c) (d) (e) Instrumental broadening 1μm particles 100 nm 10 nm 5 nm 9-13
14 XRD and the nano-world: see page 159 In order to produce a nice sharp diffraction pattern you need many many planes to Reflect from. The more planes there are, the more rigid is the Bragg condition. Peaks are sharper and Easier to resolve. For small particles the peaks broaden. From the broadening you can get a fix on the Particle size. Let s see how. Scherrer (1918) first observed that small crystallite size could give rise to peak broadening. He derived a well-known equation for relating the crystallite size to the peak width, which is called the Scherrer formula: t = Kλ/(B cosθ) where t is the averaged dimension of crystallites; K is the Scherrer constant 9-14
15 Scherrer Formula t = 0.9λ 2 2 B m B s cosϑ Paul Scherrer T -- crystallite thickness λ -- wavelength θ -- Bragg Angle B m, B s Peak widths of sample and Standard in Radians π ϑ rad = ϑ deg 180 B s - Also compensates for instrumental broadening 9-15
16 example bismuth zinc niobate nanoparticles SEM and TEM 9-16
17 ok enough of X-rays for now. You will learn more in nano2100 Next.particle sizes and surface areas 9-17
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