Analytical Methods for Materials

Transcription

1 Analytical Methods for Materials Laboratory Module # Crystal Structure Determination for Non-Cubic Crystals Suggested Reading 1. Y. Waseda, E. Matsubara, and K. Shinoda, X-ray Diffraction Crystallography, (Springer, New York, NY, 011), Ch. 4, pages , Chapter 14 in Pecharsky and Zavalij ( Very useful! ). 3. C. Suryanarayana and M.G. Norton, X-ray Diffraction A Practical Approach, (Plenum Press, New York, 1998), pages B.D. Cullity and S.R. Stock, Elements of X-ray Diffraction, 3 rd edition, (Prentice Hall, Upper Saddle River, NJ, 001), Ch. 10, pages H.P. Klug and L.E. Alexander, X-ray Diffraction Procedures For Polycrystalline and Amorphous Materials, nd Edition, (John Wiley & Sons, 1974) pp

2 Indexing a scan/pattern Methodology All powder patterns can be indexed by comparing observed d-spacing's with those computed using the appropriate formulas. All powder patterns can be indexed by comparing observed values of sin θ and theoretical values of sin θ. Methods are not necessarily successful due to the possibility of peak/line superposition. These methods tend to be more successful with hexagonal, tetragonal crystals rather than more complex orthorhombic, monoclinic, or triclinic crystals. Crystal structure determination for non-cubic crystals 605

3 Methodology - cont d Sometimes XRD patterns do not have all of the peaks that they should. Is specimen textured? What is the particle size? Think! How you process or handle a material can make this happen. This can make it tricky to index patterns correctly. Crystal structure determination for non-cubic crystals 606

4 Recall/Remember d sin 4d sin sin 4d 4 1 d Crystal structure determination for non-cubic crystals 607

5 Interplanar Spacing CUBIC: HEXAGONAL: TETRAGONAL: RHOMBOHEDRAL: ORTHORHOMBIC: MONOCLINIC: TRICLINIC*: 1 d 1 4h hk k l d 3 a c 1 h k l d a c 1 d h k l a h hk k sin hk kl hlcos cos a 1 3cos cos 3 1 h k l d a b c 1 1 h k sin l hlcos d sin a b c ac 1 1 S11h Sk S3l S1hk S3klS13hl d V *See Appendix 1 in the text for a complete listing and definitions of symbols 608

6 For cubic crystals sin 4a ( h k l ) For tetragonal crystals sin h k l 4 a c Crystal structure determination for non-cubic crystals 609

7 Problem with tetragonal symmetry Lower symmetry Results are dependent on c/a ratio (and b/a,,, ). There are no general tables. Nonequivalent indices , etc... Similar problems with hexagonal crystals. Crystal structure determination for non-cubic crystals 610

8 Indexing Hexagonal Structures Plane spacing equation: 1 d 4 3 h hk a k l c Crystal structure determination for non-cubic crystals 611

9 Indexing Hexagonal Structures Combine with Bragg s law ( = dsin): 61 4sin c l a k hk h d Crystal structure determination for non-cubic crystals

10 Indexing Hexagonal Structures Rewrite as: sin c l a k hk h OR / sin a c l k hk h a Crystal structure determination for non-cubic crystals

11 Indexing Hexagonal Patterns We need to know the c/a ratio. We can determine this graphically as was done in the old days. We can do this mathematically using the full power of modern calculators or computers. Skip graphical method Crystal structure determination for non-cubic crystals 614

12 GRAPHICAL METHOD Hexagonal Crystal structure determination for non-cubic crystals 615

13 Graphical Method for Indexing Hexagonal Patterns We need to know the c/a ratio. The Hull-Davey chart will tell us the c/a ratio and provide us with peak identification (i.e., proper Miller indices). Crystal structure determination for non-cubic crystals 616

14 617 Indexing With Hull-Davey Charts Rewrite the following equation: 4sin c l a l k h d In the following form ) / ( 3 4 log log log a c l l k h a d Crystal structure determination for non-cubic crystals

15 Hull-Davey Charts cont d log d 4 l loga log 3 ( c / a) h k l logd loga log s Crystal structure determination for non-cubic crystals 618

16 Hull-Davey Charts cont d Now plot the variation of log [s] with c/a h 3 k 3 l 3 h k l log s h 1 k 1 l c/a ratio Crystal structure determination for non-cubic crystals 619

17 Hull-Davey Plot for HCP hkl=100 hkl=00 hkl=101 hkl=10 hkl=110 hkl=103 hkl=00 hkl=11 hkl=01 hkl=004 hkl=0 hkl=104 hkl=03 hkl=10 To determine the c/a ratio: (i) collect an XRD pattern, (ii) 0.1 identify the XRD peak locations in terms of the Bragg angle, (iii) calculate the d-spacing for each peak, and (iv) construct a single range d-spacing scale (log s d) that is the same size as the logarithmic [s] scale (you can use sin instead if you prefer) See the diagrams on the c/a ratio next couple of pages for guidance. 60

18 h 1 k 1 l h k l h 3 k 3 l 3 log [s] sin -scale d scale h 4 k 4 l c/a ratio log[d] Plot log[d] on the same axis as log[s] with both starting at the same origin (i.e., log [1] = 0 should line up for each) Crystal structure determination for non-cubic crystals 61

19 h 1 k 1 l h k l h 3 k 3 l 3 log [s] sin -scale d scale h 4 k 4 l c/a ratio log[d] Calculate the d-spacing or sin values for the observed peaks and mark them on a strip laid along side the appropriate d-or sin - scale Crystal structure determination for non-cubic crystals 6

20 h 1 k 1 l 1 h k l h 3 k 3 l 3 log [s] d scale h 4 k 4 l c/a ratio This is our c/a ratio for the pattern! 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 the c/a ratio 63

21 1.000 c /a 1.59 EXAMPLE Hull-Davey Plot for Titanium "See it really does work!" log [s] hkl=100 hkl=00 hkl=101 hkl=10 hkl=110 hkl=103 hkl=00 hkl=11 hkl=01 hkl=004 log[d] c/a ratio Crystal structure determination for non-cubic crystals 64

22 Final note on graphical methods The graphical methods work, but are tedious. They aren t really used that much anymore. Modern computers allow one to determine c/a ratios more rapidly via an iterative process. 65

23 Indexing and Lattice Parameter Calculations Mathematical method Limitations if peaks are missing from your XRD pattern. Analytical method Works for any material, no matter how many peaks you have (or don t have). The following will demonstrate the use of each. Go back to graphical method Crystal structure determination for non-cubic crystals 66

24 MATHEMATICAL METHOD Hexagonal Crystal structure determination for non-cubic crystals 67

25 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Recall sin 4a 4 l 3 c / a h hk k a A 4 4 a B The parameters highlighted in red are constant for any given diffraction pattern. We can now index a pattern by considering the parameters [A] and [B] individually. Crystal structure determination for non-cubic crystals 68

26 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. 4 3 and h hk k l c/ a [A] [B] [A] depends only upon h and k. We can calculate allowable values of term [A] by modulating h and k as is illustrated on the following viewgraph. Crystal structure determination for non-cubic crystals 69

27 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. We can calculate the values of term [A] by modulating h and k as is illustrated below.* Due to symmetry relationships, h and k are interchangeable STEP 1: Prepare list for term [A] and sort in increasing order List Sorted List h k 4/3(h +hk+k ) h k 4/3(h +hk+k ) * You could also have used your chart of quadratic forms of the Miller indices to generate the appropriate list. 630

28 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Term [B] can be determined by substituting in the known c/a ratio as is illustrated below for zinc (c/a = 1.856). STEP : assume a c/a ratio and prepare a list for term [B] let c/a = l1.856 l l /(c/a) Crystal structure determination for non-cubic crystals 631

29 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. The next step is to add terms [A] and [B] together, to sort them in increasing order, and to eliminate the combinations of hkl that yield a structure factor of zero Step 3: add terms [A] and [B] together and sort in increasing order. Initial List h k l [A] + [B]

30 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. The next step is to add terms [A] and [B] together, to sort them in increasing order, and to eliminate the combinations of hkl that yield a structure factor of zero Step 3: add terms [A] and [B] together and sort in increasing order. Sorted List h k l [A] + [B]

31 For hexagonal crystals, RECALL!!! F hkl 0 when h k 3 n and l odd fi when h k 3n 1 and l even 3 fi when h k 3n 1 and l odd 4 fi when h k 3 n and l even THUS IF l is odd, see whether or not h + k = 3n. If h + k = 3n and l is odd, then there is no XRD peak! 634

32 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Several values for the bracketed quantities are listed below. I have subtracted forbidden peaks and eliminated equivalent ones. Step 4: Eliminate equivalent planes/reflections Due to symmetry relationships, h and k are interchangeable because a = b. Sorted List h k l [A] + [B] Now we can assign specific hkl values for each of the peaks in the hexagonal XRD pattern. The sequence of peaks on the pattern will be the same as indicated in the table. 635

33 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Lattice Parameters - 1 We can calculate a by substituting l = 0 into: sin 4 l 4a 3 ( c/ a) h hk k 4 sin 0 4a 3 h hk k a 3sin h hk k This corresponds to peaks with hk0 type indices (e.g., 110, 10, ) 636

34 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Lattice Parameters - We can calculate c by substituting h = k = 0 into: sin 4 l 4a 3 ( c/ a) h hk k 4 l sin 0 4a 3 ( c/ a) c l sin This corresponds to peaks with 00l type indices (e.g., 00, 004, ) 637

35 q sin hkl 4/3(h +hk+k )+l /(c/a) a c average standard deviation c/a ratio = l hk0 00l hk0 This order matches the ICDD card for Zn (# ). Now you can calculate a and c from hk0 and 00l type peaks, respectively. 638

36 NOTE: You don t need to know the c/a ratio in order to index a pattern. You can assume an initial value for c/a, calculate lattice parameters and see if they yield the assumed c/a ratio. If not iterate the assumed c/a ratio until your calculated c/a ratio matches your guess. 639

37 Worked Example # HCP powder pattern STEP 1: Solve (4/3)(h +hk+k ) for allowed reflections I will tell you what this powder is in a few viewgraphs h k STEP : Solve (l /(c/a) ) for allowed reflections ITERATION #1 UNKNOWN (c/a) Guess (c/a) = l l l /(c/a)

38 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. STEP 3: Add results from steps 1 and together. Establishes order of allowed reflections hkl Pt.1+Pt STEP 4: Re-arrange results from step 3 in increasing order. Establishes the order of reflections in the XRD pattern hkl Pt.1+Pt Use this order to index the collected pattern ITERATION #1 UNKNOWN (c/a) Guess (c/a) =

39 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. ITERATION #1 UNKNOWN (c/a) Guess (c/a) = Peak I/Io sin d (nm) hkl a c h +hk+k l STEP 5: Insert hkl according to order established in step 4. AVG c/a = STEP 6: Use hk0 reflections to calculate a. STEP 7: Use 00l reflections to calculate c. The calculated value is larger than guess. NEED TO REVISE GUESS. 64

40 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Worked Example # HCP powder pattern STEP 1: Solve (4/3)(h +hk+k ) for allowed reflections h k STEP : Solve (l /(c/a) ) for allowed reflections ITERATION # UNKNOWN (c/a) Guess (c/a) = l l l /(c/a)

41 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. STEP 3: Add results from steps 1 and together. Establishes order of allowed reflections hkl Pt.1+Pt STEP 4: Re-arrange results from step 3 in increasing order. Establishes the order of reflections in the XRD pattern hkl Pt.1+Pt Use this order to index the collected pattern ITERATION # UNKNOWN (c/a) Guess (c/a) =

42 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. ITERATION # UNKNOWN (c/a) Guess (c/a) = Peak Intensity sin d (nm) hkl a c h +hk+k l STEP 5: Insert hkl according to order established in step 4. AVG c/a = STEP 6: Use hk0 reflections to calculate a. STEP 7: Use 00l reflections to calculate c. The calculated value is still larger than guess. NEED TO REVISE GUESS. 645

43 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Worked Example # HCP powder pattern STEP 1: Solve (4/3)(h +hk+k ) for allowed reflections h k STEP : Solve (l /(c/a) ) for allowed reflections ITERATION #3 UNKNOWN (c/a) Guess (c/a) = l l l /(c/a)

44 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. STEP 3: Add results from steps 1 and together. Establishes order of allowed reflections hkl Pt.1+Pt STEP 4: Re-arrange results from step 3 in increasing order. Establishes the order of reflections in the XRD pattern hkl Pt.1+Pt Use this order to index the collected pattern ITERATION #3 UNKNOWN (c/a) Guess (c/a) =

45 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. ITERATION #3 UNKNOWN (c/a) Guess (c/a) = Peak I/Io sin d (nm) hkl a c h +hk+k l STEP 5: Insert hkl according to order established in step 4. AVG c/a = STEP 6: Use hk0 reflections to calculate a. STEP 7: Use 00l reflections to calculate c. Calculated value is now smaller than guess. REVISE GUESS DOWN. 648

46 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. Worked Example # HCP powder pattern STEP 1: Solve (4/3)(h +hk+k ) for allowed reflections h k STEP : Solve (l /(c/a) ) for allowed reflections ITERATION #4 UNKNOWN (c/a) Guess between 1.5 and 1.6 My guess (c/a) = This powder is Ti of unknown purity. I ve intentionally selected the ICDD c/a values for Ti. l l l /(c/a)

47 MATHEMATICAL METHOD FOR NON-CUBIC MATLS. STEP 3: Add results from steps 1 and together. Establishes order of allowed reflections hkl Pt.1+Pt STEP 4: Re-arrange results from step 3 in increasing order. Establishes the order of reflections in the XRD pattern hkl Pt.1+Pt Use this order to index the collected pattern ITERATION #4 UNKNOWN (c/a) Guess (c/a) = This is the c/a ratio from the ICDD card

48 MATHEMATICAL METHOD FOR NON-CUBIC MATLS Peak I/Io sin d (nm) hkl a c h +hk+k l ITERATION #4 UNKNOWN (c/a) Guess (c/a) = STEP 5: Insert hkl according to order established in step c/a = We could refine this further by calculating θ values for each reflection using our guessed c/a ratios and comparing them to the observed ones. This means assuming values for c and a. If the difference is a small fraction of the FWHM for the observed reflections then we can consider our result to be accurate. STEP 6: Use hk0 reflections to calculate a. STEP 7: Use 00l reflections to calculate c. PRETTY GOOD! The ICDD value is nm 651

49 What if material is missing xrd peaks? Makes misinterpretation of peaks possible * Titianium Ti Powder Scan Step Scan 0.1 o /min 1 second Scan * * * Titianium Rolled sheet Step Scan 0.1 o /min 1 second CW Ti Sheet Intensity (counts) * * * * * Two Theta (degrees) * * * * * * Intensity (counts) * * * * 100 * * * Two Theta (degrees) 1 XRD peaks 10 XRD peaks Same material so why are peaks be missing and why are peak intensities different? 65

50 ANALYTICAL METHOD FOR NON-CUBIC MATLS. ANALYTICAL METHOD Hexagonal Works for all patterns whether or not peaks are missing. Crystal structure determination for non-cubic crystals 653

51 ANALYTICAL METHOD FOR NON-CUBIC MATLS. Recall sin 4 l 4a 3 c/ a h hk k a and c/a are constant for any pattern. Thus, we can write sin A h hk k Cl A 3a C 4c 654

52 ANALYTICAL METHOD FOR NON-CUBIC MATLS. Since h k and l are always integers: h hk k 0, 1, 3, 4, 7, 9, 1,... l 0, 1, 4, 9, Calculate sin θ for each peak. Divide each sin θ value by integers 3, 4, 7, 9, 1, (from h +hk+k allowed by the structure factor) 3. Look for lowest common quotient*. The lowest common quotient is the value of sin θ/n that equals one of the observed sin θ values. Crystal structure determination for non-cubic crystals 655

53 ANALYTICAL METHOD FOR NON-CUBIC MATLS. Since h k and l are always integers: h hk k 0, 1, 3, 4, 7, 9, 1,... l 0, 1, 4, 9, Let lowest common quotient = A. 5. Peaks with the lowest common quotient are hk0 type peaks. Assign allowed hk0 indices to those peaks. Crystal structure determination for non-cubic crystals 656

54 Steps to success: 1. Calculate sinθ for each peak. Divide each sinθ value by integers 3, 4, 7 (from h +hk+k allowed by the structure factor) 3. Look for lowest common quotient. 4. Let lowest common quotient = A. 5. Peaks with lowest common quotient are hk0 type peaks. Assign allowed hk0 indices to peaks. For this part of the problem sin sin n h hk k Quadratic form of Miller indices Peak I/Io sin (sin ) (sin ) (sin ) (sin ) (sin ) hkl (sin )A A = Crystal structure determination for non-cubic crystals Indices correspond to: h +hk+k = 1, 3, 4, 7 or hk = 10, 11, 0, 1 657

55 ANALYTICAL METHOD FOR NON-CUBIC MATLS. Rearrange equation to find C sin A h hk k Cl Cl sin A h hk k 6. Subtract (h + k + l ) A from sin θ for each peak (i.e., A, 3A, 4A, 7A, ) 7. Look for the lowest common quotient from Step Identify values of sin θ that increase by factors of l (i.e., l = 1, 4, 9, ). These correspond to 00l peaks. Crystal structure determination for non-cubic crystals 658

56 6. Subtract from each sinθ value 3A, 4A, 7A (from h +hk+k allowed by the structure factor); 7. Look for lowest common quotient (LCQ). From this you can identify 00ltype peaks. The first allowed peak for hexagonal systems is 00. Determine C from the equation: C l = sin θ-(h +hk+k )A Since h=0 and k=0, then: C=LCQ/l = sin θ/l ; 8. Look for values of remainders that increase by factors of 1, 4, 9, (because l = 1,,3,4..., l =1,4,9,16...). The peaks exhibiting these characteristics are allowed 00l-type peaks (e.g., 00, ). We identify the 4th peak as 10 because we observe the LCQ for sin θ-1a. Recall that the 1 comes from the quadratic form of Miller indices (i.e., h +hk+k =1). We identify the 8th peak as 11 because we observe the LCQ for sin θ-3a. Recall that the 1 comes from the quadratic form of Miller indices (i.e., h +hk+k =3). We identify the 11th peak as... etc... A = h +hk +k Peak I/Io sin sin -A sin -3A sin -4A sin -7A sin /LCQ h k l C=LCQ/l l =LCQ/C LCQ =

57 ANALYTICAL METHOD FOR NON-CUBIC MATLS. 9. Peaks that are not hk0 or 00l can be identified using combinations of A and C values. This is accomplished by considering: sin Cl A h hk k Cycle through allowed values for h k and l, and compare sin θ value to the labeled peaks. They are the hkl peaks! Crystal structure determination for non-cubic crystals 660

58 A C Peak I/Io sin h k l sin = C l + A (h +hk+k ) Calculated Now that we know A and C, we can calculate lattice parameters. a 3A c 4C a c c /a

59 Before we finish, let s consider the application of the analytical method to both of these experimental XRD patterns * Titianium Ti Powder Scan Step Scan 0.1 o /min 1 second Scan * * * Titianium Rolled sheet Step Scan 0.1 o /min 1 second CW Ti Sheet Intensity (counts) * * * * * Two Theta (degrees) * * * * * * Intensity (counts) * * * * 100 * * * Two Theta (degrees) 1 XRD peaks 10 XRD peaks Will it work? 66

60 Ti Powder Pattern sin sin /3 sin /4 sin /7 sin /9 sin /1 sin /A A h k l A= Sin θ/a = h +hk+k Must match pattern h +hk+k = 1, 3, 4, 7... for hk = 10, 11, 0, 1... sin A sin 3A sin 4A sin 7A sin h k l sin /LCQ C=sin /l LCQ This suggests (001), but 001 is not allowed in HCP. (00) is first. Divide sin by LCQ and look for a pattern of integers that increases 1,4,9,16... (00), (004), (006), (008)

61 Ti Powder Pattern A C calculated Observed sin Difference sin (h +hk+k ) (l ) X+Y h k l obs calc a = c = c/a= Values for hk come from the list of allowed indices (determined from structure factor calculation). Look for combinations of hkl where observed sin values nearly equal calculated values. This tells you the indices for each peak. 664

62 Ti Foil Pattern sin sin /3 sin /4 sin /7 sin /9 sin /1 sin /A A h k l A= Sin θ/a = h +hk+k Must match pattern h +hk+k = 1, 3, 4, 7... for hk = 10, 11, 0, 1... sin A sin 3A sin 4A sin 7A sin h k l sin /LCQ LCQ This suggests (001), but 001 is not allowed in HCP. (00) is first. Divide sin by LCQ and look for a pattern of integers that increases 1,4,9,16... (00), (004), (006), (008)

63 Ti Foil Pattern A C calculated Observed sin Difference sin (h +hk+k ) (l ) X+Y h k l obs calc a = c = c/a= Values for hk come from the list of allowed indices (determined from structure factor calculation). Look for combinations of hkl where observed sin values nearly equal calculated values. This tells you the indices for each peak. LOOKS LIKE THE METHOD STILL WORKS! 666

64 Practice Exercises 1. The XRD data listed to the right was collected for a hexagonal crystal. CuK α radiation was used. 1. Determine the c/a ratio.. Index the diffraction data/pattern. Peak # θ

65 Practice Exercises 1. The XRD data listed to the right was collected for a tetragonal crystal. CuK α radiation was used. 1. Determine the c/a ratio.. Index the diffraction data/pattern. Peak # θ