máster en cristalografía y cristalización sevilla, 12 de...
TRANSCRIPT
Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Diffraction powder pattern indexing &
Intensity extraction
Máster en Cristalografía y cristalizaciónSevilla, 12 de Diciembre de 2012
Vicente Esteve CanoDpto. de Química Inorgánica y OrgánicaUniversitat Jaume I de Castellón
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
1.- Introduction. What is indexing? Basic equations
2.- Figures of Merit. Indexation methods. Accuracy o f data
3.- Consideration about space group determination
4.- Checks on the validity of indexation
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
1.- Introduction. What is indexing? Basic equations
2.- Figures of Merit. Indexation methods. Accuracy o f data
3.- Consideration about space group determination
4.- Checks on the validity of indexation
3
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
powder pattern sample
angular position of diffraction peaks
crystal lattice
intensity of the peaks
unit cell content phase content
peak shape microstructure
Information contained in a powder pattern
λλλλ = 2 dhkl sin θθθθ
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
What is indexing?
Unit cell parameters
( a ,b, c, αααα, ββββ, γγγγ )
Positions 2 θθθθ of diffraction peaks
lattice symmetry
systematic absences
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Difraction powder pattern indexing...
Basic equation
Qhkl =[d*] 2 = h2 A + k2 B + l 2 C + hk D + hl E + kl F(triclinic general case)
A = a*2 D = 2 a* b* cos αααα*
B = b*2 E = 2 a* c* cos ββββ*
C = c*2 F = 2 b* c* cos γγγγ*
- One equation per diffraction line
- h k l are integers (Miller indices)
Optimal number of equations ?
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Difraction powder pattern indexing...
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Qhkl for special lattices
cubic
tetragonal
hexagonal
rhombohedral/triclinic
orthorhombic
monoclinic
Qhkl = (h2+k2+l2)A
Qhkl = (h2+k2)A + l 2C
Qhkl = (h2+k2+hk)A + l 2C
(with -h+k+l=3n+1 absent)
Qhkl = h2A + k 2B + l 2C
Qhkl = h2A + k 2B + l 2C + hlE
Number of observations N: 15 + number of parameters to determine
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Difraction powder pattern indexing...
Cubic system
Qhkl = (h2+k2+l2) A
sin 2θhkl = (λ2/4a2) ( h2 + k2 + l2 )
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Difraction powder pattern indexing...
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
The observational data
λλλλ = 2 dhkl sin θθθθhkl
Qhkl = 4 sin 2θθθθhkl / λλλλ2
Qhkl = 1 / dhkl2(a,b,c, αααα,ββββ,γγγγ)
Accuracy?
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Difraction powder pattern indexing...
Complexity = f (number of reflections N)
N ≈≈≈≈ (4ππππ/3) V m-1 dN-3
V is the volume of the cell
m is the multiplicity factor of Bragg reflections: 2 for triclinic; 4 for monoclinic, 8 for orthorhombic; 16 for tetragonal; 24 for hexagonal; 36 for trigonal; 48 for cubic (de Wolff, 1961)
Actually the number of observed reflections is smaller than N. We lost weak (or not observed) reflections and systematic extinctions
Large N means severe peak overlap
Compare triclinic and cubic situations!
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
How to estimate the unit cell volume of a triclinic unit cell ?
V = 13.39 d320 (Smith, 1977)
N ≈≈≈≈ (4ππππ/3) V m-1 dN-3
because of the overlap N ≈≈≈≈1.4·Nobs
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
1.- Introduction. What is indexing? Basic equations
2.- Figures of Merit. Indexation methods. Accuracy o f data
3.- Consideration about space group determination
4.- Checks on the validity of indexation
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Some historical milestones- Runge (1917) finds Q relations useful for constructing the reciprocal space
- Ito (1949) and de Wolff (1957) report a more general method for the geometrical reconstruction from Q relations
- Visser (1969) Computer program using this algebraic procedure ITO
- Werner (1964) Method based on the permutation of Miller indices TREOR
- Louër & Louër (1972) Method based in successive dichotomies DICVOL
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Difraction powder pattern indexing...
Present trends
Increase of the robustness against:
- the presence of small impurities
- zero-point displacement errors
Active inclusion of systematic absences due to space group symmetry
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Difraction powder pattern indexing...
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
The figures of merit
Q20 = Q value of the 20th line. The larger unit cell, the smaller Q 20. The higher symmetry, the larger Q 20
Ncalc = number of calculated reflections with Q < Q 20(systematic absences not taken into account). It is good signal if N calc is only slightly larger than 20
<∆∆∆∆Q>= <|Qobs -Qcalc |> The discrepancy should be as small as possible. Q calc refers to the nearest line
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M20 (de Wolff, 1968)
M20 = Q20 / (2 <∆∆∆∆Q> Ncalc )
For M20 > 10 the indexing is substantially correct
With right measurements in good diffractometersM20 >20 in routine work
Principal problem: Presence of a dominant zone
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Difraction powder pattern indexing...
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
FN (Smith & Snyder, 1979)
FN = (N/Nposs ) / <|2θθθθobs -2θθθθcalc |>
If <∆∆∆∆2θθθθ> and N/ Nposs are identical for a cubic and a triclinic indexing, the cubic indexing is obviously to be preferred (Werner).
In the programs it is given in the format:
FN (<∆∆∆∆2θθθθ>, Nposs )
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Difraction powder pattern indexing...
ITO-method (Visser, 1969)
• A triclinic unit cell is assumed
• The smallest Q is taken as A: A= Q1 (one checks for the presence of h2A= 4A, 9A…
• Next small Q: B= Q2 (check for harmonics)
• Search for hk0 reflections: One looks for that D value explaining the maximum number of Qh±k0= h
2A+k2B±hkD
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Difraction powder pattern indexing...
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Difraction powder pattern indexing...
• Next unexplained Q: C= Q3
• Repeat the process carried out for hk0, to reflexions h0l (E) and to 0kl (F)
• Once A, B, C, D, E, F are known, the remaining Qhkl should be explained in terms of them
• If not, start again the procedure, assuming that Q1=4A (the first order could be absent)
• Useful for symmetries down to monoclinic and unit cells less than 1000A3
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Difraction powder pattern indexing...
Permutation method (TREOR)
a) Set-up of a system of linear equations
Qi = h i2 A + k i
2 B + l i2 C
for the base reflections (with smallest 2 θθθθ values). Different least square trials are performed with permutations of h k l from 0 to 4.
b) Only the best solutions are tested with the Q values of the rest of N lines (used passively).
Werner (1964) Z.Kristallogr. 120, 275
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
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Difraction powder pattern indexing...
TREOR (4)- 84 10 02
s40-va2 7.995000 12.232000 15.670000 15.997000 17.542000 18.492001 20.915001 21.722000 24.254999 24.582001 25.202999 25.565001 26.427999 27.124001 27.267000 27.834999 28.863001 29.186001
FIGURE OF MERIT REQUIRED= 10 MAX NUMBER OF UNINDEXED LINES= 1THE 7 FIRST LINES ADJUSTED BY THEIR HIGHER ORDERSCUBIC,TETRAGONAL,HEXAGONAL AND ORTHORHOMBIC SYMMETR YMAX CELL EDGE= 60.0 MAX CELL VOLUME= 2500.0D1= 0.000200 SSQTL= 0.050000 D2= 0.000400 WAVE= 1.540598NUMBER OF TEST LINES= 18 IQ REQUIRED= 15
Input data (TREOR90):
1) Peak positions 2 θθθθ (or d or Q)
2) Maximum cell volume
3) Maximum cell edge
4) Indicate monoclinic (maximum ββββangle) and triclinic tests
5) Possible errors on observed data
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Difraction powder pattern indexing...
Successive dichotomy method (DICVOL)
Louër & Louër (1972) J.Appl.Cryst. 5, 271
Exhaustive search of solutions in nD parameter space, working from the cubic side of the symmetry sequence: n = 1 (cubic) to 6 (triclinic), according to a dichotomy algorithm
Computer program: DICVOL06 ( J. Appl. Cryst. 37, 724, 2004; Z. Krist. 26, 191, 2007)
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
# Input file for WDICVOL06 (created by WinPLOTR)! DATA FILE: BARBITAL.DAT! INSTRM: 0! WAVE: 1.54060 1.54439 0.5
DICVOL input parameters:20 2 1 1 1 1 0 0 ! N,ITYPE,JC,JT,JH,JO,JM,JTR 25.0 25.0 25.0 0.0 2500.0 90.0 125.0 ! AMAX,BMAX,CMAX,VOLMIN,VOLMAX,BEMIN,BEMAX1.54060 0.000 0.000 0.000 ! WAVE,POIMOL,DENS,DELDEN
0.030 10.000 0 0 1 0 2 ! EPS, FOM, N_IMP, ZERO_search, ZERO_refinem ent, DV06_option, ISUP10.39893 14.61714 14.82710 15.12112 16.53855 16.80712 19.71762 20.86397 21.09645 21.20449 23.49739 24.74766 25.70824 25.97648 26.60000 26.76568 27.53413 28.53317 29.44824 29.80722
...
...
...
Input data (DICVOL06):
1) Peak positions 2 θθθθ (or d or Q)
2) Symmetry system to check
3) Maximum cell edges
4) Maximum ββββ angle if monoclinic test
5) Possible error on observed data
6) Minimum FOM (M 20)
7) Maximum number of impurity lines
8) Zero error and possibility zero error refinement
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
How to use DICVOL?
1. Run DICVOL with the first 20 lines. These lines should be not too weak to avoid the introduction of one peak belonging to an impurity. Select the solution with the best figure of merit M 20
2. Perform DICVOL runs by adding sequentially the weak lines. If the weak lines are indexed, this represents a further confirmation that the solution is correct.
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Hull method (hexagonal)
sin 2θ = A (h2+k2+hk)+ C l 2
log sin 2θhkl = log A + log [h 2+k2+hk + C/A l 2 ]
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
What makes indexing difficult?Large asymmetrical unit = severe peak overlap
Shifted zero point
Lines measured with poor accuracy
Presence of impurities
Pseudosymmetry
Unequal lattice parameters:
1) but equal within experimental error;
2) one much longer than the other two;
3) one much shorter than the other two (dominant zo ne)
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Accuracy of data
The absolute error value for every diffraction line should be <0.03º2 θθθθ (CuK αααα radiation) in order to achieve meaningful FoM’s values.
In practice, <| ∆∆∆∆2θθθθ |> use to be less than 0.01º2 θθθθfor well crystallised materials
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Difraction powder pattern indexing...
D. LouërProceedings of the Int. Conference “Accuracy in powder Diffraction II” held in Gaithersburg (1992)
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
How to improve the accuracy?- adding an internal standard for position
- adjusting the zero-2 θθθθ position (<0.005º2 θθθθ) or using parallel beam geometry
-adjusting the θθθθ/2θθθθ angular range using highly oriented crystallites of a layer-type material, deposited in a thin layer on a silicon-plate to reduce transparency eff ects (e.g. fluorophlogopite mica, 10 harmonics)
- with a correct adjustment no internal standard is required (very smooth front surface of the sample a nd thickness control for materials with only light ato ms; no problem if transmission, flat sample or capillar y geometries)
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
1.- Introduction. What is indexing? Basic equations
2.- Figures of Merit. Indexation methods. Accuracy o f data
3.- Consideration about space group determination
4.- Checks on the validity of indexation
30
Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
PseudosymmetryIt happens when the apparent lattice symmetry is
higher than the true one
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Crystal data: a=6.8576(3) b=6.8831(5) c=19.988(2) Å
Confirmation when space group was found,
P212121 with Z = 4
Tetragonal as lattice pseudosymmetry
True symmetry is orthorhombic
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Difraction powder pattern indexing...
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Considerations about space group determination
Lattice symmetry and 14 Bravais lattices
Triclinic
Monoclinic
Orthorhombic
Trigonal
Hexagonal
Tetragonal
Cubic
P
P, C, ( I )
P, C, (A, B), I, F
R
P
P, I
P, F, I
These absences extend uniformly in the reciprocal space
-1
2/m
mmm
-3m
6/mmm
4/mmm
m-3m
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Reflection conditions for lattice centerings
P: no systematic absences
C: only with h+k = 2n
B: only with h+l = 2n
A: only with k+l = 2n
I: only with h+k+l = 2n
F: only if conditions for A, B and C are simultaneously fulfilled.
R: only with -h+k+l =3n (hexagonal axes and obversedsetting)
Considerations about space group determination
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Present in all the crystalline systems but triclinic.
In triclinic crystal possible space groups symmetry, P1 and P-1, belong to the same diffraction class P1
Considerations about space group determination
Reflection conditions due to the presence of translational symmetry: Bravais centered lattices; glide planes and helicoidal axes
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Reflection conditions. Monoclinic system
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Reflection conditions. Orthorhombic system
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Reflection conditions. Trigonal system
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Reflection conditions. Hexagonal system
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Reflection conditions. Tetragonal system
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Reflection conditions. Cubic system
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
1.- Introduction. What is indexing? Basic equations
2.- Figures of Merit. Indexation methods. Accuracy o f data
3.- Consideration about space group determination
4.- Checks on the validity of indexation
42
Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Checks on the validity of the indexation1) Reviewing all the discrete data
DICVOL run
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Checks on the validity of the indexation1) Reviewing all the discrete data
SEARCH OF ORTHORHOMBIC SOLUTION(S)**********************************
VOLUME DOMAIN BEING SCANNED :===========================
LOWER BOUND = 0.00 A**3 HIGHER BOUND = 400.00 A**3
O R T H O R H O M B I C S Y S T E M DIRECT PARAMETERS : A= 7.97263 B= 5.74605 C= 4.96433 VOLUME=
227.42STANDARD DEVIATIONS : .00039 .00038 .00037REFINED ZERO-POINT SHIFT : 0.0163 deg. 2-theta
.
.
.
* NUMBER OF LINES.- LINES INPUT = 20.- LINES INDEXED = 20.- LINES CALCULATED = 31
* MEAN ABSOLUTE DISCREPANCIES<Q> =0.2501E-04
<DELTA(2-THETA)> =0.2990E-02MAX. ERROR ACCEPTED (DEG. 2-THETA) =0.4500E-01
* FIGURES OF MERIT1.- M( 20) = 181.82.- F( 20) = 215.8(0.0030, 31)
THE SOLUTION IS NOW USED TO TRY TO INDEX ALL INPUT 111 LINES :==============================================================DIRECT PARAMETERS : A= 7.97179 B= 5.74459 C= 4.96251 VOLUME=
227.26STANDARD DEVIATIONS : .00023 .00012 .00016REFINED ZERO-POINT SHIFT : 0.0283 deg. 2-theta
.
.
.
* NUMBER OF LINES.- LINES INPUT = 111.- LINES INDEXED = 111.- LINES CALCULATED = 229
* AVERAGE 2-Theta DIFFERENCE FOR INDEXED LINES =-0.0001* MEAN ABSOLUTE DISCREPANCIES
<Q> =0.1005E-03<DELTA(2-THETA)> =0.7640E-02
MAX. ERROR ACCEPTED (DEG. 2-THETA) =0.4500E-01* FIGURES OF MERIT
1.- M( 20) = 105.22.- F( 20) = 122.8(0.0053, 31)3.- F(111) = 63.4(0.0076, 229)
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Checks on the validity of the indexation2) Use of observed (or known) density to check
whether a solution corresponds to an integer number of chemical formula in the cell (Z)
cAVN
MZ=ρ
3) Diffraction symbol (space group) determination
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Difraction powder pattern indexing...
Checks on the validity of the indexation4) Pattern matching (e.g. Le Bail method). Visual
inspection of the quality of the fit.
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Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Checks on the validity of the indexation5) Structure determination
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Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
This presentation is entirely based on the presentation prepared
by
Aknowlegments and References
Jordi Rius PalleiroInstitut de Ciència de Materials de Barcelona (CSIC)&Xavier AlcobéCentres Cientítics i Tecnològics, Universitat de Barc elona
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And the paper entitled “Automatic Indexing: Procedures and Applications”, by Daniel Louër, 1992.
Difraction powder pattern indexing...
Máster en Cristalografía y cristalización
Máster en Cristalografía y cristalización Sevilla, 2012
Difraction powder pattern indexing...
Thanks so much for yourattention!
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