Indexing X-Ray Diffraction Patterns - Laboratory Module | MTE 481, Lab Reports of Materials science

Download Indexing X-Ray Diffraction Patterns - Laboratory Module | MTE 481 and more Lab Reports Materials science in PDF only on Docsity! Laboratory Module Indexing X-Ray Diffraction Patterns LEARNING OBJECTIVES Upon completion of this module you will be able to index an X-ray diffraction pattern, identify the Bravais lattice, and calculate the lattice parameters for crystalline materials. BACKGROUND We need to know about crystal structures because structure, to a large extent, determines properties. X-ray diffraction (XRD) is one of a number of experimental tools that are used to identify the structures of crystalline solids. The XRD patterns, the product of an XRD experiment, are somewhat like fingerprints in that they are unique to the material that is being examined. The information in an XRD pattern is a direct result of two things: (1) The size and shape of the unit cells determine the relative positions of the diffraction peaks; (2) Atomic positions within the unit cell determine the relative intensities of the diffraction peaks (remember the structure factor?). Taking these things into account, we can calculate the size and shape of a unit cell from the positions of the XRD peaks and we can determine the positions of the atoms in the unit cell from the intensities of the diffraction peaks. Full identification of crystal structures is a multi-step process that consists of: (1) Calculation of the size and shape of the unit cell from the XRD peak positions; (2) Computation of the number of atoms/unit cell from the size and shape of the cell, chemical composition, and measured density; (3) Determination of atom positions from the relative intensities of the XRD peaks We will only concern ourselves with step (1), calculation of the size and shape of the unit cell from XRD peak positions. We loosely refer to this as “indexing.” The laboratory module is broken down into two sections. The first addresses how to index patterns from cubic materials. The second addresses how to index patterns from non-cubic materials. PART 1 PROCEDURE FOR INDEXING CUBIC XRD PATTERNS When you index a diffraction pattern, you assign the correct Miller indices to each peak (reflection) in the diffraction pattern. An XRD pattern is properly indexed when ALL of the peaks in the diffraction pattern are labeled and no peaks expected for the particular structure are missing. Intensity (%) 2 θ (°) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 0 10 20 30 40 50 60 70 80 90 100 (27.28,100.0) 1,1,1 (45.30,69.2) 2,2,0 (53.68,39.7) 3,1,1 (65.99,9.7) 4,0,0 (72.80,13.7) 3,3,1 (83.66,17.4) 4,2,2 (90.05,9.4) 3,3,3 5,1,1 (100.73,5.5) 4,4,0 (107.30,10.1) 5,3,1 (118.86,9.6) 6,2,0 (2θ, I/Io) h,k,l This is an example of a properly indexed diffraction pattern. All peaks are accounted for. One now needs only to assign the correct Bravais lattice and to calculate lattice parameters. How to we correctly index a pattern? The correct procedures follow. PROCEDURE FOR INDEXING AN XRD PATTERN The procedures are standard. They work for any crystal structure regardless of whether the material is a metal, a ceramic, a semiconductor, a zeolite, etc… There are two methods of analysis. You will do both. One I will refer to as the mathematical method. The second is known as the analytical method. The details are covered below. Mathematical Method Interplanar spacings in cubic crystals can be written in terms of lattice parameters using the plane spacing equation: Index this pattern and determine the lattice parameters. Steps: (1) Identify the peaks. (2) Determine 2sin θ . (3) Calculate the ratio 2sin θ / 2 minsin θ and multiply by the appropriate integers. (4) Select the result from (3) that yields 2 2h k l 2+ + as an integer. (5) Compare results with the sequences of 2 2h k l 2+ + values to identify the Bravais lattice. (6) Calculate lattice parameters. Here we go! (1) Identify the peaks and their proper 2θ values. Eight peaks for this pattern. Note: most patterns will contain α1 and α2 peaks at higher angles. It is common to neglect α2 peaks. Peak No. 2θ sin2θ 2 2 min sin1 sin θ θ × 2 2 min sin2 sin θ θ × 2 2 min sin3 sin θ θ × h2+k2+l2 hkl a (Å) 1 38.43 2 44.67 3 65.02 4 78.13 5 82.33 6 98.93 7 111.83 8 116.36 (2) Determine 2sin θ . Peak No. 2θ sin2θ 2 2 min sin1 sin θ θ × 2 2 min sin2 sin θ θ × 2 2 min sin3 sin θ θ × h2+k2+l2 hkl a (Å) 1 38.43 0.1083 2 44.67 0.1444 3 65.02 0.2888 4 78.13 0.3972 5 82.33 0.4333 6 98.93 0.5776 7 111.83 0.6859 a|_ 116.36 | 0.7220 | (3) Calculate the ratio 2sin θ / 2 minsin θ and multiply by the appropriate integers. Peak No. 2θ sin2θ 2 2 min sin1 sin θ θ × 2 2 min sin2 sin θ θ × 2 2 min sin3 sin θ θ × h2+k2+l2 hkl a (Å) 1 38.43 0.1083 1.000 2.000 3.000 2 44.67 0.1444 1.333 2.667 4.000 3 65.02 0.2888 2.667 5.333 8.000 4 78.13 0.3972 3.667 7.333 11.000 5 82.33 0.4333 4.000 8.000 12.000 6 98.93 0.5776 5.333 10.665 15.998 7 111.83 0.6859 6.333 12.665 18.998 8 116.36 0.7220 6.666 13.331 19.997 (4) Select the result from (3) that most closely yields 2 2h k l 2+ + as a series of integers. Peak No. 2θ sin2θ 2 2 min sin1 sin θ θ × 2 2 min sin2 sin θ θ × 2 2 min sin3 sin θ θ × h2+k2+l2 hkl a (Å) 1 38.43 0.1083 1.000 2.000 3.000 2 44.67 0.1444 1.333 2.667 4.000 3 65.02 0.2888 2.667 5.333 8.000 4 78.13 0.3972 3.667 7.333 11.000 5 82.33 0.4333 4.000 8.000 12.000 6 98.93 0.5776 5.333 10.665 15.998 7 111.83 0.6859 6.333 12.665 18.998 8 116.36 0.7220 6.666 13.331 19.997 (5) Compare results with the sequences of 2 2h k l 2+ + values to identify the miller indices for the appropriate peaks and the Bravais lattice. Peak No. 2θ sin2θ 2 2 min sin1 sin θ θ × 2 2 min sin2 sin θ θ × 2 2 min sin3 sin θ θ × h2+k2+l2 hkl a (Å) 1 38.43 0.1083 1.000 2.000 3.000 3 111 4.0538 2 44.67 0.1444 1.333 2.667 4.000 4 200 4.0539 3 65.02 0.2888 2.667 5.333 8.000 8 220 4.0538 4 78.13 0.3972 3.667 7.333 11.000 11 311 4.0538 5 82.33 0.4333 4.000 8.000 12.000 12 222 4.0538 6 98.93 0.5776 5.333 10.665 15.998 16 400 4.0541 7 111.83 0.6859 6.333 12.665 18.998 19 331 4.0540 8 116.36 0.7220 6.666 13.331 19.997 20 420 4.0541 Bravais lattice is Face-Centered Cubic 2 24 K a λ⎛ = ⎜ ⎝ ⎠ ⎞ ⎟ OR 2 a K λ = If we divide the 2sin θ values for each reflection by K, we get the 2 2h k l 2+ + values. The sequence of values can be used to label each XRD peak and to identify the Bravais lattice. 2 2h k l+ + 2 Let’s do an example for the Aluminum pattern presented above. Steps: (1) Identify the peaks. (2) Determine 2sin θ . (3) Calculate the ratio 2sin θ /(integers) (4) Identify the lowest common quotient from (3) and identify the integers to which it corresponds. Let the lowest common quotient be K. (5) Divide 2sin θ by K for each peak. This will give you a list of integers corresponding to . 2 2h k l+ + 2 2(6) Select the appropriate pattern of 2 2h k l+ + values and identify the Bravais lattice. (7) Calculate lattice parameters. Here we go again! (1) Identify the peaks. Peak No. 2θ 2sin θ 2sin 2 θ 2sin 3 θ 2sin 4 θ 2sin 5 θ 2sin 6 θ 2sin 8 θ 1 38.43 2 44.67 3 65.02 4 78.13 5 82.33 6 98.93 7 111.83 8 116.36 (2) Determine 2sin θ . Peak No. 2θ 2sin θ 2sin 2 θ 2sin 3 θ 2sin 4 θ 2sin 5 θ 2sin 6 θ 2sin 8 θ 1 38.43 0.1083 2 44.67 0.1444 3 65.02 0.2888 4 78.13 0.3972 5 82.33 0.4333 6 98.93 0.5776 7 111.83 0.6859 8 116.36 0.7220 (3) Calculate the ratio 2sin θ /(integers) Peak No. 2θ 2sin θ 2sin 2 θ 2sin 3 θ 2sin 4 θ 2sin 5 θ 2sin 6 θ 2sin 8 θ 1 38.43 0.1083 0.0542 0.0361 0.0271 0.0217 0.0181 0.0135 2 44.67 0.1444 0.0722 0.0481 0.0361 0.0289 0.0241 0.0181 3 65.02 0.2888 0.1444 0.0963 0.0722 0.0578 0.0481 0.0361 4 78.13 0.3972 0.1986 0.1324 0.0993 0.0794 0.0662 0.0496 5 82.33 0.4333 0.2166 0.1444 0.1083 0.0867 0.0722 0.0542 6 98.93 0.5776 0.2888 0.1925 0.1444 0.1155 0.0963 0.0722 7 111.83 0.6859 0.3430 0.2286 0.1715 0.1372 0.1143 0.0857 8 116.36 0.7220 0.3610 0.2407 0.1805 0.1444 0.1203 0.0903 (4) Identify the lowest common quotient from (3) and identify the integers to which it corresponds. Let the lowest common quotient be K. Peak No. 2θ 2sin θ 2sin 2 θ 2sin 3 θ 2sin 4 θ 2sin 5 θ 2sin 6 θ 2sin 8 θ 1 38.43 0.1083 0.0542 0.0361 0.0271 0.0217 0.0181 0.0135 2 44.67 0.1444 0.0722 0.0481 0.0361 0.0289 0.0241 0.0181 3 65.02 0.2888 0.1444 0.0963 0.0722 0.0578 0.0481 0.0361 4 78.13 0.3972 0.1986 0.1324 0.0993 0.0794 0.0662 0.0496 5 82.33 0.4333 0.2166 0.1444 0.1083 0.0867 0.0722 0.0542 6 98.93 0.5776 0.2888 0.1925 0.1444 0.1155 0.0963 0.0722 7 111.83 0.6859 0.3430 0.2286 0.1715 0.1372 0.1143 0.0857 8 116.36 0.7220 0.3610 0.2407 0.1805 0.1444 0.1203 0.0903 K = 0.0361 (5) Divide 2sin θ by K for each peak. This will give you a list of integers corresponding to . 2 2h k l+ + 2 Peak No. 2θ 2sin θ 2sin K θ 2 2 2h k l+ + hkl 1 38.43 0.1083 3.000 2 44.67 0.1444 4.000 3 65.02 0.2888 8.001 4 78.13 0.3972 11.001 5 82.33 0.4333 12.002 6 98.93 0.5776 16.000 7 111.83 0.6859 19.001 8 116.36 0.7220 20.000 (6) Select the appropriate pattern of 2 2h k l 2+ + values and identify the Bravais lattice. Peak No. 2θ 2sin θ 2sin K θ 2 2h k l 2+ + hkl 1 38.43 0.1083 3.000 3 111 2 44.67 0.1444 4.000 4 200 3 65.02 0.2888 8.001 8 220 4 78.13 0.3972 11.001 11 311 5 82.33 0.4333 12.002 12 222 6 98.93 0.5776 16.000 16 400 7 111.83 0.6859 19.001 19 331 8 116.36 0.7220 20.000 20 420 Sequence suggests a Face-Centered Cubic Bravais Lattice (7) Calculate lattice parameters. 1.540562 A 2 2 0.0361 a K λ = = = 4.0541 Å These methods will work for any cubic material. This means metals, ceramics, ionic crystals, minerals, intermetallics, semiconductors, etc… PART 2 PROCEDURE FOR INDEXING NON-CUBIC XRD PATTERNS The procedures are standard and will work for any crystal. The equations will differ slightly from each other due to differences in crystal size and shape (i.e., crystal structure). As was the case for cubic crystals, there are two methods of analysis that involve calculations. You will do both. One I will refer to as the mathematical method. The second I will refer to as the analytical method. Both the mathematical and graphical methods require some knowledge of the crystal structure that you are dealing with and the resulting lattice parameter ratios (e.g., c/a, b/a, etc…). This information can be determined graphically using Hull-Davey charts. We will first introduce the concept of Hull-Davey charts prior to showing how to proper index patterns. Hull-Davey Charts Hull-Davey Plot for HCP 0.1 1.0 10.0 0.501.001.502.00 c/a ratio s hkl=100 hkl=002 hkl=101 hkl=102 hkl=110 hkl=103 hkl=200 hkl=112 hkl=201 hkl=004 hkl=202 hkl=104 hkl=203 hkl=210 100 002 101 110103 102 200 201 To determine the c/a ratio, one only needs to collect an XRD pattern, identify the peak locations in terms of the Bragg angle, calculate the d-spacing for each peak and to construct a single range d-spacing scale (2⋅log d) that is the same size as the logarithmic [s] scale (you can use sin2θ instead if you prefer). I know this is confusing so I have schematically illustrated what I mean in the next set of figures. h1k1l1 h2k2l2 h3k3l3 h4k4l4 c/a ratio log [s] + d scale sin 2θ-scale 1.0 1.0 1.0 + Next, you need to calculate the d-spacing or sin2θ values for the observed peaks and mark them on a strip laid along side the appropriate d- or sin2θ - scale. h1k1l1 h2k2l2 h3k3l3 h4k4l4 c/a ratio log [s] + sin 2θ-scale d scale 1.0 1.0 1.0 + The strip should be moved horizontally and vertically across the log [s] – c/a plot until a position is found where each mark on your strip coincides with a line on the chart. This is illustrated schematically on the next figure. Please keep in mind that my illustrations for the Hull-Davey method are SCHEMATIC. This method is very difficult to convey. You should consult the classical references to find out more information about this technique. h1k1l1 h2k2l2 h3k3l3 h4k4l4 c/a ratio log [s] + + 0.10 10.0 10.0 1.0 d scale 1.0 1.0 This is our c/a ratio for the pattern! This method really does work as I showed you in class. Once you know your c/a ratio, you can index the XRD pattern. As we noted above, there are two ways to do this. The first is the mathematical method. Mathematical Method for Non-Cubic Crystals Recall the following equation: ( ) 2 2 2 2 2 2 2 4sin 4 3 ( / ) lh hk k a c λθ ⎛ ⎞ ⎡ = + + +⎜ ⎟ a ⎤ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ Note that the lattice parameter a and the ratio of lattice parameters c/a are constant for a given diffraction pattern. Thus, 2 24a λ⎛ ⎜ ⎝ ⎠ ⎞ ⎟ is constant for any pattern. The pattern can now be indexed in by considering the terms in brackets: ( )2 24 3 h hk k+ + 2 2( / ) l c a Let’s start with term 1. This term only depends on the indices h and k. Thus its value can be calculated for different values of h and k. This is done below for various hk values. 2sin c lλ θ = Worked Example Consider the following XRD pattern for Titanium, which was collected using CuKα radiation. 20 30 40 50 60 70 80 90 100 Titanium Powder (-325 mesh) Two Theta In te ns ity ( co un ts ) CuKα radiation λ = 1.540562 Å Index this pattern and determine the lattice parameters. Steps: (1) Identify the peaks. (2) Determine values of ( 243 h hk k+ + ) 2 for reflections allowed by the structure factor. (3) Determine values of 2 2( / ) l c a for the allowed reflections and the known c/a ratio (4) Add the solutions from parts (2) and (3) together and re-arrange them in increasing order. (5) Use this order to assign indices to the peaks in your diffraction pattern. (6) Look for hk0 type reflections and calculate a for these reflections. (7) Look for 00l type reflections. Calculate c for these reflections. Here we go! (1) Identify the peaks. Peak I/Io sin2θ d 35.275 21 0.0918 2.542 38.545 18 0.1089 2.334 40.320 100 0.1188 2.235 53.115 16 0.1999 1.723 63.095 11 0.2737 1.472 70.765 9 0.3353 1.330 74.250 10 0.3643 1.276 76.365 8 0.3821 1.246 77.500 14 0.3918 1.231 82.360 2 0.4335 1.170 86.940 2 0.4733 1.120 92.900 10 0.5253 1.063 (2) Determine values of ( 243 h hk k+ + ) 2 for reflections allowed by the structure factor. k 0 1 2 3 0 0.000 1.333 5.333 12.000 1 1.333 4.000 9.333 17.333 2 5.333 9.333 16.000 25.333 h 3 12.000 17.333 25.333 36.000 (3) Determine values of 2 2( / ) l c a for the allowed reflections and the known c/a ratio. Titanium: c/a = 1.587 l l2 l2/(c/a)2 0 0 0.000 1 1 0.397 2 4 1.588 3 9 3.573 4 16 6.352 5 25 9.925 6 36 14.292 (4) Add the solutions from parts (2) and (3) together and re-arrange them in increasing order. hkl Pt.1+Pt.2 hkl Pt.1+Pt.2 002 1.588 100 1.333 100 1.333 002 1.588 101 1.730 101 1.730 102 2.921 102 2.921 103 4.906 110 4.000 110 4.000 103 4.906 004 6.352 200 5.333 112 5.588 112 5.588 200 5.333 201 5.730 201 5.730 004 6.352 104 7.685 202 6.921 202 6.921 104 7.685 203 8.906 203 8.906 105 11.258 210 9.333 114 10.352 211 9.730 210 9.333 114 10.352 211 9.730 212 10.921 204 11.685 105 11.258 006 14.292 204 11.685 212 10.921 300 12.000 106 15.625 213 12.906 213 12.906 302 13.588 300 12.000 006 14.292 205 15.258 205 15.258 302 13.588 106 15.625 (5) Use this order to assign indices to the peaks in your diffraction pattern. Peak I/Io sin2θ d (nm) hkl a c h2+hk+k2 l2 35.275 21 0.091805 2.5423 100 38.545 18 0.108941 2.3338 002 40.320 100 0.118779 2.2351 101 53.115 16 0.199895 1.7229 102 63.095 11 0.273744 1.4723 110 70.765 9 0.335278 1.3303 103 74.250 10 0.36428 1.2763 200 76.365 8 0.382132 1.2461 112 77.500 14 0.39178 1.2307 201 82.360 2 0.433526 1.1699 004 86.940 2 0.473309 1.1197 202 92.900 10 0.525296 1.0628 104 AVG (7) Look for the lowest common quotient. From this we can identify 00l type peaks. Recall, that 001 is not allowed for hexagonal systems. The first 00l type peak will be 002. We can calculate C from: 2 2 2 2sin ( )C l A h hk kθ⋅ = − ⋅ + + 2in (8) Look for values of s θ that increase by factors of 4, 9… (this is because l = 1, 2, 3… and l2 = 1, 4, 9…). Peaks exhibiting these characteristics are 00l type peaks, which can be assigned the indices 004, 009, etc…). Also note that the values of 2sin θ will be some integral number times the value observed in (7) which indicates the indices of the peak (9) Peaks that are neither hk0 nor 00l can be identified using combinations of our calculated A and C values. (10) Calculate the lattice parameters from the values of A and C. Confused yet? You could be. I was the first time I learned these things. Let me show you an example that should make all things clear. Here we go! Consider the diffraction pattern for Titanium as shown below. This one is a little different than the specimen that we analyzed above. Intensity (%) 2 θ (°) 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 10 20 30 40 50 60 70 80 90 100 105 110 115 120 (35.10,25.5) (38.44,25 (40. (53.02,12.8) (62.96,13.4) (70.70,13.0) (74.17,1.8) (76.24,13.1) (77.37,9.3) (82.35,1.7) (86.79,2.1) (92.79,1.8) (102.41,4.4) (105.82,1.4) (109.08,8.3) (114.35,5.4) (119.30,2.7) .6) 18,100.0) λ = 1.540562 Å Titanium c/a = 1.1.587 Peak I/Io sin2θ (sin2θ)/3 (sin2θ)/4 (sin2θ)/7 (sin2θ)/9 (sin2θ)/12 hkl (sin2θ)/LCQ 35.100 25.5 0.0909 0.0303 0.0227 0.0130 0.0101 0.0076 100 1.0 38.390 25.6 0.1081 0.0360 0.0270 0.0154 0.0120 0.0090 1.2 40.170 100 0.1179 0.0393 0.0295 0.0168 0.0131 0.0098 1.3 53.000 12.8 0.1991 0.0664 0.0498 0.0284 0.0221 0.0166 2.2 62.940 13.4 0.2725 0.0908 0.0681 0.0389 0.0303 0.0227 110 3.0 70.650 13 0.3343 0.1114 0.0836 0.0478 0.0371 0.0279 3.7 74.170 1.8 0.3636 0.1212 0.0909 0.0519 0.0404 0.0303 200 4.0 76.210 13.1 0.3808 0.1269 0.0952 0.0544 0.0423 0.0317 4.2 77.350 9.3 0.3905 0.1302 0.0976 0.0558 0.0434 0.0325 4.3 82.200 1.7 0.4321 0.1440 0.1080 0.0617 0.0480 0.0360 4.8 86.740 2.1 0.4716 0.1572 0.1179 0.0674 0.0524 0.0393 5.2 92.680 1.8 0.5234 0.1745 0.1308 0.0748 0.0582 0.0436 5.8 102.350 4.4 0.6069 0.2023 0.1517 0.0867 0.0674 0.0506 6.7 105.600 1.4 0.6345 0.2115 0.1586 0.0906 0.0705 0.0529 210 7.0 109.050 8.3 0.6632 0.2211 0.1658 0.0947 0.0737 0.0553 7.3 114.220 5.4 0.7051 0.2350 0.1763 0.1007 0.0783 0.0588 7.8 119.280 2.7 0.7445 0.2482 0.1861 0.1064 0.0827 0.0620 8.2 A = 0.0908 = 3 x A = 4 x A = 7 x A Indices correspond to: h2+hk+k2 = 1, 3, 4, 7… or hk = 10, 11, 20, 21 Steps to success: 1. Calculate sin2θ for each peak 2. Divide each sin2θ value by integers 3, 4, 7… (from h2+hk+k2 allowed by the structure factor) 3. Look for lowest common quotient. 4. Let lowest common quotient = A . 5. Peaks with lowest common quotient are hk 0 type peaks. Assign allowed hk 0 indices to peaks. λ 1.54062 Peak I/Io sin2θ sin2θ-A sin2θ-3A sin2θ-4A sin2θ-4A h k l C= LCQ/l 2 l 2 =LCQ/C 35.100 25.5 0.0909 1 0 0 38.390 25.6 0.1081 0.0173 0 0 2 0.0270 4.0 40.170 100 0.1179 0.0271 53.000 12.8 0.1991 0.1083 1 0 2 0.0271 62.940 13.4 0.2725 0.1817 0.0001 1 1 0 70.650 13 0.3343 0.2435 0.0618 74.170 1.8 0.3636 0.2728 0.0911 0.0003 2 0 0 76.210 13.1 0.3808 0.2900 0.1083 0.0175 1 1 2 0.0271 77.350 9.3 0.3905 0.2997 0.1180 0.0272 82.200 1.7 0.4321 0.3413 0.1597 0.0688 0 0 4 0.0270 16 86.740 2.1 0.4716 0.3807 0.1991 0.1083 2 0 2 0.0271 92.680 1.8 0.5234 0.4326 0.2509 0.1601 102.350 4.4 0.6069 0.5161 0.3345 0.2436 105.600 1.4 0.6345 0.5436 0.3620 0.2711 2 1 0 109.050 8.3 0.6632 0.5724 0.3907 0.2999 0.0274 114.220 5.4 0.7051 0.6143 0.4326 0.3418 0.0693 119.280 2.7 0.7445 0.6537 0.4721 0.3812 0.1087 LCQ = 0.1083 6. Subtract from each sin2θ value 3A , 4A , 7A … (from h 2+hk +k 2 allowed by the structure factor) 7. Look for lowest common quotient (LCQ). From this you can identify 00l -type peaks. The first allowed peak for hexagonal systems is 002. Determine C from the equation: C ⋅ l 2 = sin2θ-A (h 2+hk +k 2) since h =0 and k =0, then: C=LCQ/l 2 = sin2θ/l 2 8. Look for values of sin2θ that increase by factors of 4, 9, 16... (because l = 1,2,3,4..., l 2=1,4,9,16...) The peaks exhibiting these characteristics are 00l -type peaks (002...). We identify the 4th peak as 102 because we observe the LCQ for sin2θ-1A. Recall that the 1 comes from the quadratic form of Miller indices (i.e., h 2+hk +k 2=1). We identify the 8th peak as 112 because we observe the LCQ for sin2θ-3A. Recall that the 1 comes from the quadratic form of Miller indices (i.e., h 2+hk +k 2=3). We identify the 11th peak as ... etc... This peak is 004 because sqrt(LCQ/C)=4