xrd indexing
TRANSCRIPT
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Michael Carton
11352077
MTE 481: Analytical Methods for Materials
Lab 3: X-Ray Diffraction and Peak Indexing
Date of Experiment: 11/13/2014Submission Date: 12/01/2014
Instructor: Dr. Jinhui Song
TA: Chaolong Tang
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Abstract
The purpose of this experiment was to record the diffraction spectra for various samples
and to index their peaks using mathematical and analytical methods. These methods use a
combination of Braggs law and the plane spacing equations in order to determine the lattice
parameters.1A sample of nickel with a cubic structure and a sample of titanium with a hexagonal
structure were analyzed. The peak values, cubic structures, and lattice parameters determined by
the mathematical and analytical methods agree with each other and with the ICDD values
confirming that the materials indexed were FCC nickel and HCP titanium.
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Experimental Apparatus and Procedure
To begin the experiment a sample of pure nickel powder was acquired and placed in a
sample holder. The Ni powder and holder was loaded into the chamber of the Philips MPD XRD
machine. The doors were closed and secured and chiller checked to ensure that cool water was
running through the system. The operating computer was turned on and the Data Collector
program on the desktop was opened.
Once the program was opened, the software was connected to the hardware by clicking
Instrument then Connect and then OK. The Tension button was clicked and the voltage
and current were changed to 45kV/40mA. Test program 4 was run by selecting File
OpenProgramTest program 4. The start angle was set to 20, the end angle to 100, and the step
size to 0.05/step. The program was initiated by clicking Measure Program and giving the
file a unique name.
Once initiated, the shutter opened and scanned the sample according to the parameters set
in program 4. When the scan was complete, the shutter opened and the sample was removed
from the chamber. The data points from the scan were automatically saved by the computer and
exported to a .XRDML file.
To shut tool down, the voltage and current were reset to 30kV/10mA under the Tension
button on the program interface. The hardware was disconnected by clicking Instrument
Disconnect. The .XRDML file was converted to a .ASC file to be read by Excel and plotted.
The same procedure was repeated for the titanium powder. Standard XRD peak files
(ICDD) for both nickel and titanium samples were pulled for comparison. All of the peaks for
both samples were indexed using mathematical and analytical methods.
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Results and Discussion
Once the data was collected, it was plotted in Excel. The Nickel sample was analyzed
first and its peaks can be seen below in Figure 1.
Figure 1:Experimentally obtained XRD peaks for the Nickel sample
As seen above, there are five distinct peaks between the 2values of 20 and 100.
These values and their relative intensities are recorded in Table Ibelow.
Table I:Peaks from the Nickel sample and their relative intensitiesAngle Intensity Normalized Intensity
44.56 15630 10051.88 4820 30.876.38 1940 12.492.90 1380 8.8398.30 460 2.94
Next, the each of the peaks of the Nickel sample was indexed using both analytical and
mathematical methods. Following the methods learned in class, Tables II IVbelow were
constructed and used to determine the crystal structure and lattice parameter of the Nickel.
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Table II. Mathematical method for determining the lattice parameter and structure of a cubic material
Table III. Analytical method for determining the K value for a cubic material
K = 0.04784
Table IV.Analytical method for determining the lattice parameter and structure of a cubic material
PeakNo.
2 sin() sin()^21*sin()^2/sin(min)^2
2*sin()^2/sin(min)^2
3*sin()^2/sin(min)^2
(h^2)+(k^2)+(l^2) hkl a (angstroms)
1 44.56 0.38 0.14 1.00 2.00 3.00 3.00 111 3.52
2 51.88 0.44 0.19 1.33 2.66 3.99 4.00 200 3.52
3 76.38 0.62 0.38 2.66 5.32 7.98 8.00 220 3.52
4 92.90 0.72 0.53 3.65 7.31 10.96 11.00 311 3.52
5 98.30 0.76 0.57 3.98 7.96 11.94 12.00 222 3.53
Average: 3.52
PeakNo.
2 sin() sin()^2 sin()^2/2 sin()^2/3 sin()^2/4 sin()^2/5 sin()^2/6 sin()^2/7 sin()^2/8
1 44.56 0.37913 0.14374 0.07187 0.04791 0.03594 0.02875 0.02396 0.02053 0.01797
2 51.88 0.43743 0.19134 0.09567 0.06378 0.04784 0.03827 0.03189 0.02733 0.02392
3 76.38 0.61827 0.38226 0.19113 0.12742 0.09556 0.07645 0.06371 0.05461 0.047784 92.90 0.72477 0.52530 0.26265 0.17510 0.13132 0.10506 0.08755 0.07504 0.06566
5 98.30 0.75642 0.57218 0.28609 0.19073 0.14304 0.11444 0.09536 0.08174 0.07152
PeakNo.
2 (rad) sin()sin()^
2sin()^2/
K(h^2)+(k^2)+(l^2) hkl a (angstroms)
1 44.56 22.28 0.39 0.38 0.14 3.00 3.00 111 3.52
2 51.88 25.94 0.45 0.44 0.19 4.00 4.00 200 3.52
3 76.38 38.19 0.67 0.62 0.38 7.99 8.00 220 3.52
4 92.90 46.45 0.81 0.72 0.53 10.98 11.00 311 3.52
5 98.30 49.15 0.86 0.76 0.57 11.96 12.00 222 3.52
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These methods depend on the plane spacing equations and Braggs law in order to
determine the crystal structures and lattice parameters. Specifically, these equations can be
combined to make the relationship: 2()= (
4)(2+2+2). This equation can then be
solved either mathematically or analytically.2
As seen in Table II, the h2+k2+l2for each peak from the mathematical model are 3, 4, 8,
11, and 12. This indicates that the nickel has a face-centered cubic (FCC) structure.3The lattice
parameter was also calculated and found to be 3.52 .
Analytical methods were also used to confirm the crystal structure of the nickel. As seen
in Table IV, the h2+k2+l2for each peak from the analytical method are 3, 4, 8, 11, and 12. This
agrees with the mathematical model and also indicates that nickel has an FCC structure.3The
lattice parameter was once again calculated and found to be 3.52 which also agrees with the
mathematical model. The value of these lattice parameters are very close to the ICDD value for
nickel which is 3.54 . The ICDD data card is attached at the end of the report.
Overall, the peak angles, the FCC structure, and lattice parameter from both the
mathematical and analytical methods of indexing the peaks confirm that the material is in fact
nickel.
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Next, the experimental spectrum data for the titanium sample was plotted in Excel. This
can be seen below in Figure 2.
Figure 2: Experimentally obtained XRD peaks for the Titanium sample
As seen above, there are eleven clear peaks between the 2values of 20 and 100.
These values and their relative intensities are recorded in Table Vbelow.
Table V:Peaks from the titanium sample and their relative intensitiesAngle Intensity Normalized Intensity
34.96 625 22.538.30 835 30.140.06 2770 10052.86 380 13.762.78 380 13.770.52 390 14.176.08 360 13.077.20 290 10.582.16 90 3.2586.60 80 2.89
92.64 70 2.52
These peaks are very close to the expected ICDD peaks and were indexed using the
mathematical method in order to determine the crystal structure and lattice parameters. For
hexagonal structures, Braggs law can be combined with the plane spacing equations to get the
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following relationship: 2()=
4 [43 (
2++2] + (
). This relationship can then
be used to determine c and a, which are the lattice parameters.
First determine the possible values of 43[2++2]using all of the possibilities for h
and k. These values can be seen below in Table VI.
Table VI: Values for43 [
2++2]for all possible values of h and kk
0 1 2 3
0 0.000 1.333 5.333 12.000
h 1 1.333 4.000 9.333 17.333
2 5.333 9.333 16.000 25.333
3 12.000 17.333 25.333 36.000
Next, the possible values of
()
were determined for all possible values of l and using the
lattice parameter ratio (c/a) of 1.5871. These are below in Table VII.
Table VII: Values for
()
for all values of l and c/a = 1.5871
l l^2 l^2/(c/a)^2
0 0 0.0001 1 0.397
2 4 1.588
3 9 3.573
4 16 6.352
5 25 9.925
6 36 14.292
Then the solutions from Tables VI andVIIwere used to determine the43 [
2++
2] + (
)for every allowed hkl value. These were placed in order from smallest to largest in
Table VIII.
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Table VIII: All allowed hkl values and corresponding43[
2++2] + (
)values.
hkl value
100 1.333
002 1.588
101 1.73102 2.921
110 4.000
103 4.906
200 5.333
112 5.588
201 5.73
004 6.352
202 6.921
104 7.685
203 8.906210 9.333
211 9.73
114 10.352
212 10.921
105 11.258
204 11.685
300 12.000
213 12.906
302 13.588
006 14.292
205 15.258
106 16.625
This order of hkl values was then used to assign indices to the peaks from the diffraction
pattern. The avalues for the hk0type reflections was calculates and the cvalues for the 00ltype
reflections were calculated. These values were averaged in order to determine the a, c and c/a
values. This can all be seen below in Table IX.
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Table IX:Calculations of the c and a lattice parameters for the experimental spectrum
PeakNo.
2RelativeIntensity
sin()^2d
(nm)hkl a() c () (h^2)+hk+(k^2) l^2
1 34.96 22.5 0.0902 2.563 100 2.960 1
2 38.3 30.1 0.1076 2.347 002 4.695 43 40.06 100 0.1173 2.248 101
4 52.86 13.7 0.1981 1.730 102
5 62.78 13.7 0.2713 1.478 110 2.957 3
6 70.52 14.1 0.3333 1.334 103
7 76.08 13 0.3797 1.250 200 2.886 4
8 77.2 10.5 0.3892 1.234 112
9 82.16 3.25 0.4318 1.172 201
10 86.6 2.89 0.4703 1.123 004 4.491 16
11 92.64 2.52 0.5230 1.065 202
As seen in the table above the average values for a is 2.934 , the average value for c is
4.593 , and the average value for c/a is 1.565. These are close to the ICDD values for titanium
which are a = 2.951 , c = 4.670 , c/a = 1.583.
Summary and Conclusions
In conclusion, the experimental spectra of the nickel and titanium match the expected
spectra according to the ICDD data. The mathematical and analytical methods of indexing the
spectra agree with each other and result in crystal structures and lattice parameters that also agree
with the expected results. The experimental results indicate the nickel has an FCC structure and
the titanium has an HCP structure. The experimental value for the lattice parameter of nickel is
3.52 which is very close to the ICDD value of 3.54 . The experimental values for titaniums
lattice parameters are: a = 2.934 , c = 4.593 , c/a = 1.565 which is very close to the ICDD
values of: a = 2.951 , c = 4.670 , c/a = 1.583. The small variations can be accounted for by
the different systems used to generate the spectra and slight rounding errors.
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References
1. Callister, William D., Rethwisch, David G, Fundamentals of Materials Science andEngineering, 3rdEd., John Wiley and Sons, Inc., Hoboken, NJ, 2008
2. Suryanarayana, C.,Experimental Techniques in Materials and Mechanics, Taylor & FrancisGroup, Boca Raton, FL, 2011
3. Leng, Yang,Materials Characterization: Introduction to Microscopic and SpectroscopicMethods, John Wiley & Sons, Ltd., Hoboken NJ, 2008
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00-001-1258
Nov 13, 2014 5:56 PM (CAF User)
Status
Deleted QM:
Blank Pressure/Temperature:
Ambient ChemicalFormula:
Ni EmpiricalFormula:
Ni
Weight :
Ni100.00 Atomic :
Ni100.00 CompoundName:
Nickel
Radiation:
MoK : 0.7093
SYS: Cubic SPGR: Fm-3m (225)Author's Cell [ AuthCella: 3.54 AuthCell Vol: 44.36 AuthCellZ: 2.00 AuthCellMolVol: 22.18]
Density [Dcalc:
4.394g/cm Dmeas:
8.72g/cm] SS/FOM:
F(12) =7.9(0.117, 13)
Temp:
298.000K (Ambienttemperature assigned by ICDD editor) MeltingPoint:
1728 K
SpaceGroup:
Fm-3m (225) MolecularWeight:
58.70
CrystalData[XtlCella: 3.540 XtlCellb: 3.540 XtlCellc: 3.540 XtlCell : 90.00 XtlCell : 90.00XtlCell : 90.00 XtlCellVol: 44.36 XtlCellZ: 2.00] Crystal DataAxial Ratio [ a/b: 1.000 c/b: 1.000]ReducedCell [ RedCell a: 2.503 RedCellb: 2.503 RedCell c: 2.503 RedCell : 60.00
RedCell :
60.00 RedCell :
60.00 RedCellVol: 11.09]
Crystal (SymmetryAllowed):
Centrosymmetric
Pearson: cF2.00 Subfile(s): Common Phase, Forensic, Inorganic, Deleted Pattern, Metals & AlloysLastModificationDate: 01/11/2013 Cross-RefPDF s: 04-001-0091 (Alternate)
References:
Type
PrimaryReferenceOptical DataUnit Cell
DOI Reference
Hull. Phys. Rev. 17, 571 (1921).Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107.The Structure of Crystals, 1st Ed.
DatabaseComments:
Color: White. Deleted Or Rejected By: Deleted by NBS card. Melting Point: 1728 K.
d-Spacings 8 - 00-001-1258 Fixed Slit Intensity - Cu K Avg 1.54184
2
44.407651.640376.1563
d
2.040000
1.770000
1.250000
I
1005040
h
122
k
102
l
100
* 2
92.189298.1913122.3379
d
1.070000
1.0200000.880000
I
60102
h
324
k
120
l
120
* 2
144.2592154.7640
d
0.8100000.790000
I
2016
h
34
k
32
l
10
*
Page 1 / 12014International CentreforDiffractionData.All rights reserved.
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00-001-1197
Nov 13, 2014 3:29 PM (CAF User)
Status
Deleted QM:
Blank Pressure/Temperature:
Ambient ChemicalFormula:
Ti EmpiricalFormula:
Ti
Weight :
Ti100.00 Atomic :
Ti100.00 CompoundName:
Titanium
Radiation:
MoK : 0.7093
SYS: Hexagonal SPGR: P63/mmc (194)Author's Cell [ AuthCella: 2.951 AuthCell c: 4.67 AuthCell Vol: 35.22 AuthCellZ: 2.00AuthCellMolVol: 17.61] Author's Cell Axial Ratio [ c/a: 1.583]Density [Dcalc:
4.517g/cm Dmeas:
4.49g/cm] SS/FOM:
F(15) =5.1(0.162, 18)Temp: 298.000K (Ambienttemperature assigned by ICDD editor) MeltingPoint: 2093 K
SpaceGroup:
P63/mmc (194) MolecularWeight:
47.90CrystalData[XtlCella: 2.951 XtlCellb: 2.951 XtlCellc: 4.670 XtlCell : 90.00 XtlCell : 90.00XtlCell :
120.00 XtlCellVol: 35.22 XtlCellZ: 2.00]
CrystalDataAxial Ratio [ c/a:
1.583 a/b:
1.000 c/b:
1.583]
ReducedCell [ RedCell a: 2.951 RedCellb: 2.951 RedCell c: 4.670 RedCell : 90.00RedCell :
90.00 RedCell :
120.00 RedCellVol: 35.22]
Atomic parameters arecross-referencedfromPDFentry04-001-8963
Crystal (SymmetryAllowed): Centrosymmetric
SG SymmetryOperators:
Seq
1234
Operator
x,y,z-x,-y,-z-y,x-y,zy,-x+y,-z
Seq
5678
Operator
-x+y,-x,zx-y,x,-z-y,-x,zy,x,-z
Seq
9101112
Operator
x,x-y,z-x,-x+y,-z-x+y,y,zx-y,-y,-z
Seq
13141516
Operator
-x,-y,z+1/2x,y,-z+1/2y,-x+y,z+1/2-y,x-y,-z+1/2
Seq
17181920
Operator
x-y,x,z+1/2-x+y,-x,-z+1/2y,x,z+1/2-y,-x,-z+1/2
Seq
21222324
Operator
-x,-x+y,z+1/2x,x-y,-z+1/2x-y,-y,z+1/2-x+y,y,-z+1/2
Atomic Coordinates:
Atom
TiNum
1Wyckoff
2cSymmetry
-6m2x
0.33333y
0.66666z
0.25SOF
1.0IDP AET
12-d
Pearson: hP2.00 Subfile(s): Metals & Alloys, Inorganic, Deleted Pattern, Common Phase, Forensic, ExplosiveLastModificationDate: 01/11/2013 Cross-RefPDF s: 04-001-8963 (Primary)
References:
Type
PrimaryReferenceCrystal StructureOptical Data
Unit Cell
DOI Reference
Phys. Rev. 26, 56 (1925).Crystal Structure Source: LPF .Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107.
The Structure of Crystals, 1st Ed.
DatabaseComments: Color: White. Deleted Or Rejected By: Deleted by NBS. Melting Point: 2093 K.
d-Spacings 15 - 00-001-1197 Fixed Slit Intensity - Cu K Avg 1.54184
2
35.052238.471240.261152.925862.7842
d
2.560000
2.3400002.240000
1.7300001.480000
I
40401004040
h
10111
k
00001
l
02120
* 2
70.243276.156377.623681.585186.0363
d
1.340000
1.2500001.2300001.1800001.130000
I
5040301010
h
11202
k
01000
l
32142
* 2
92.1892102.2847110.1952113.8499122.3379
d
1.0700000.9900000.9400000.9200000.880000
I
2030303010
h
12211
k
00110
l
43145
*
Page 1 / 12014International CentreforDiffractionData.All rights reserved.