X-ray Dif f ract ion • Basic aspects of x- ray crystallography and powder dif f raction • Dif f raction f rom nanocrystalline materials Paolo. Scardi@unitn. it Special thanks to: Luca Gelisio, Alberto Leonardi, Luca Rebuf f i, Cristy L. Azanza Ricardo, Mirco D’I ncau, Andrea Troian, Emmanuel Garnier, Mahmoud Abdellatief
FROM SI NGLE CRYSTAL TO POWDER DI FFRACTI ON 1. Tradit ional reciprocal space approach : sum & average D perf ect (inf init e) cryst al ( ) ∝ ∑∑ ( ) π ⋅ 2 i S r * I s f f e mn sc m n m n ( ) ∝ ∫ Ω I s d { } ( ) ( ) ( ) ( ) ( ) ( ) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ sc = 2 IP S D F APB C GRS ( ) ( ) ( ) ... I s F I s I s I s I s I s I s I s I s π PD 2 4 s 62 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DI FFRACTI ON FROM NANOCRYSTALLI NE MATERI ALS L real nanocryst als are complex obj ect s non-cryst allographic (e.g. mult iply t winned) nanopart icles, 2D and highly disordered layer syst ems: Debye formula � t ranslat ional symmet ry: not verif ied (Direct Space) � large st rain / misf it – complex local at omic arrangement CdS-CdSe OCTAPODS I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DI FFRACTI ON FROM NANOCRYSTALLI NE MATERI ALS 2. Direct (real) space approach : average & sum ∑∑ ( ) ∫ π ⋅ Ω 2 2 i s r f e d mn ( ) = m n I s r mn r mn π PD 2 4 s I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DI FFRACTI ON FROM NANOCRYSTALLI NE MATERI ALS 2. Direct (real) space approach : average & sum ∑∑∫ ( ) π ⋅ Ω 2 2 i s r f e d mn ( ) = m n I s r mn r mn π PD 2 4 s ( ) π π sin 2 sr 1 ( ) ∫ π ⋅ = π φ π φ φ = 2 i s r 2 isr cos 2 mn e e 2 r sin d mn mn π π mn 2 4 2 r sr mn mn 0 r mn ( ) π sin 2 sr ∑∑ ( ) = 2 mn I s f π PD 2 sr m n mn Debye Scat t ering Equat ion (DSE) I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DSE APPLI CATI ON TO NON-CRYSTALLOGRAPHI C NPs Debye Scat t ering Equat ion (DSE) ( ) π sin 2 sr ∑∑ = 2 mn I ( ) s f π PD 2 sr m n mn 66 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DSE APPLI CATI ON TO GRAPHENE AND RELATED MATERI ALS Debye Scat t ering Equat ion (DSE) ( ) π sin 2 sr ∑∑ = 2 mn I ( ) s f π PD 2 sr m n mn L. Gelisio et al., J . Appl. Cryst . 43 (2014) 647 67 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DSE APPLI CATI ON TO GRAPHENE AND RELATED MATERI ALS Debye Scat t ering Equat ion (DSE) ( ) π sin 2 sr ∑∑ = 2 mn I ( ) s f π PD 2 sr m n mn L. Gelisio, PhD Thesis, Univ. of Trent o, 2014 Carbon nanot ubes 68 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DSE CALCULATI ON BY ATOMI C DI STANCE HI STOGRAM Debye Scat t ering Equat ion (DSE) ( ) ( ) π π sin 2 sr sin 2 sr ∑∑ ∑ = ≡ 2 2 mn mn I ( ) s f f B π π PD mn 2 sr 2 sr m n mn mn mn 8 1000 800 (a) 700 16000 (b) 100 600 7 300 Intensity 14000 10 6 200 B n 12000 1 5 Intensity 100 0.1 10000 100 110 4 0 B mn 0.01 210 8000 0.00.1 4.4 4.6 4.8 5.0 5.2 0 2 4 6 8 10 12 14 16 18 20 2 θ (degrees) r n (nm) 211 221 3 410 6000 300/ 321 330/ 411 310 111 222 322 421 2 311 320 4000 200 220 420 331 400 1 2000 0 0 0 1 2 3 4 5 0 20 40 60 80 100 120 140 160 2 θ (degrees) r mn (nm) Atomic distance histogram (B mn ) f or a cubic cr yst al wit h 8x8x8 sc unit cells (a) and cor r esponding powder pat t er n accor ding t o I PD (s), wit h f =1, unit cell par amet er , a 0 =0.361 nm (b). P. Scar di & L. Gelisio, “Dif f r act ion f r om nanocr yst alline mat er ials”, Chapt er XVI I I in Synchr ot ron Radiat ion, ed. S. Mobilio et al. Spr inger 2015. I n the coming months, look f or a special issue of Acta Crystallographyca A, edited by Billinge, Cervellino, Neder & Scardi Total Scattering methods – the 100 Years of the Debye Scattering Equation (DSE2015 conf erence, Cavalese (I ) June 2015) 69 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PAI R DI STRI BUTI ON FUNCTI ON (PDF) Zernike & Prins (1927): f or amorphous specimens, volume V , N at oms, t he radial dist ribut ion f unct ion (RDF) is: ∞ ( ) I s ( ) ( ) ( ) ∫ RDF r = π ρ ≅ π ρ + π − π 2 2 4 r r 4 r 8 r s 1 Sin 2 sr ds 0 2 Nf 0 int ensit y in absolut e unit s: ( ) ( ) − − − 2 I s N f a d c → = 2 f c 2 f ( ) ( ) i M Compton 70 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PDF AND SYNCHROTRON RADI ATI ON SR is mandat ory t o improve resolut ion! 1950 1999 � S. J . L. Billinge, Z. Krist allogr. 219 (2004) 117 71 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PAI R DI STRI BUTI ON FUNCTI ON (PDF) ( ) ( ) = π ρ 2 RDF r 4 r r radial dist ribut ion f unct ion ( ) ( ) = π ρ − ρ G r 4 r r reduced radial dist ribut ion f unct ion 0 ( ) ( ) = ρ ρ g r r pair dist ribut ion f unct ion - PDF 0 ( ) ∞ I s 2 ( ) ( ) ( ) ∫ = + − π = 1 1 2 s S s Sin sr ds S s ρ 2 r Nf 0 0 � S. J . L. Billinge, Z. Krist allogr. 219 (2004) 117 72 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PDF AND SYNCHROTRON RADI ATI ON SR is mandat ory t o improve resolut ion! � Court esy of R. Neder 73 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PDF OF NANOPARTI CLE SYSTEMS Ef f ect of f init e size and shape of t he nanopart icle � Court esy of R. Neder 74 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PDF OF NANOPARTI CLE SYSTEMS I ndicat ion of st acking f ault s � Court esy of R. Neder 75 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PDF ANALYSI S OF NANOPARTI CLE SYSTEMS Au nanopart icle + ligand � Court esy of R. Neder 76 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
PDF ANALYSI S OF NANOPARTI CLE SYSTEMS Au nanopart icle + ligand Bottom-up modelling DISCUS DIFFEV � Court esy of R. Neder 77 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
TOTAL SCATTERI NG TECHNI QUES PDF approach Debye Scat t ering Equat ion P. Scar di et al., Phys. Rev. B91 (2015) 155414 K. Page et al., J .Appl.Cr yst . 44 (2011) 327 78 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
TOTAL SCATTERI NG TECHNI QUES PDF approach Debye Scat t ering Equat ion K. Page et al., J .Appl.Cr yst . 44 (2011) 327 P. Scar di & L. Gelisio, Nat . Sci. Repor t s 6, 22221 (2016) 79 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DI FFRACTI ON FROM NANOCRYSTALLI NE MATERI ALS 1. Tradit ional reciprocal space approach : sum & average ( ) ∝ ∫ Ω I s d { } ( ) ( ) ( ) ( ) ( ) sc = 2 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ IP S D F APB C GRS I s F I s I ( ) s I ( ) s I ( ) s I s I s I s ... π PD 2 4 s 2. Tot al Scat t ering met hods Direct (real) space approach: average & sum Debye Scat t ering Equat ion (DSE) ( ) π sin 2 sr ( ) ∑∑ = 2 mn I s f π PD 2 sr m n mn Pair Dist ribut ion Funct ion (PDF) ( ) ρ ∞ r 1 ( ) ( ) ( ) ∫ = = + − g r 1 Q S Q 1 Sin Qr dQ ρ π ρ 2 2 r 0 0 0 1 ( ) ( ) ( ) ∫ = 2 + π ρ − ρ π I s N f 1 4 r r Sin 2 sr dr π 0 2 s V 80 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
DI FFRACTI ON FROM NANOCRYSTALLI NE MATERI ALS Current research / f ut ure t rends � t oward an int egrat ion bet ween at omist ic modelling and dif f ract ion analysis: real st ruct ure of nanopart icle syst ems Debye Scat t ering Equat ion ( ) π sin 2 sr ∑∑ = 2 mn I ( ) s f π PD 2 sr m n mn geometrical relaxed (energy minimization) L. Gelisio, K.R. Beyerlein & P. Scardi, Thin Solid Films (2012). In press. 81 I CTP School - Trieste, 04. 04. 2016 P. Scardi – Dif f raction f rom nanocrystalline materials
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