Come calcolare legge Bragg diffrazione raggi X cristallografia

Calcola legge Bragg n*lambda = 2*d*sin(theta) diffrazione raggi X XRD cristallografia struttura proteine semiconduttori Si wafer. William e Lawrence Bragg UK 1913 Premio Nobel 1915. Synchrotron Elettra + ESRF + DESY + Politecnico.

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Come calcolare la legge di Bragg (n*lambda = 2*d*sin(theta)) equazione fondamentale diffrazione di raggi X cristallografia che descrive la condizione di interferenza costruttiva dei raggi X riflessi dai piani cristallini paralleli a distanza d: n * lambda = 2 * d * sin(theta). n ordine diffrazione (1 primario; 2 3 secondari armoniche); lambda lunghezza d'onda raggi X nm o Angstrom (Cu Kalfa 0.15418 nm = 1.5418 A; Mo Kalfa 0.07107 nm; Co Kalfa 0.17890 nm; Cr Kalfa 0.22897 nm; synchrotron 0.05-0.3 nm tunable; neutroni termici 0.1-0.3 nm); d spacing piani cristallini nm o A (NaCl 200 0.282; Si 111 0.314; Si 220 0.192; KBr 200 0.329; cubic crystals d_hkl = a/sqrt(h^2+k^2+l^2)); theta angolo diffrazione gradi rispetto al piano cristallino; 2*theta angolo deviazione tipicamente reportato in XRD pattern.

Generalizzazione Laue conditions: k_out - k_in = G_hkl + reciprocal lattice vector + 3D crystal indexing.

Sistemi cristallini Bravais: 14 reticoli + 7 sistemi (cubico + tetragonal + orthorhombic + hexagonal + trigonal + monoclinic + triclinic) + 230 gruppi spaziali; cubic d_hkl = a/sqrt(h^2+k^2+l^2); tetragonal 1/d^2 = (h^2+k^2)/a^2 + l^2/c^2; hexagonal 1/d^2 = 4*(h^2+hk+k^2)/(3*a^2) + l^2/c^2.

Sorgenti raggi X: Cu Kalfa 0.15418 nm tipica laboratorio + tube anode 30-50 kV; Mo Kalfa 0.07107 nm piccoli atomi; synchrotron radiation tunable 0.05-0.3 nm + brilliance 10^10-10^21 photons/s/mm^2/mrad^2 + Elettra Trieste + ESRF Grenoble + DESY PETRA III + APS Argonne + SPring-8 Japan world-leading; X-ray Free Electron Laser XFEL European XFEL Hamburg + LCLS Stanford + femtosecond pulses world-leading; neutron sources ILL Grenoble + ISIS Oxford + SNS Oak Ridge + neutroni termici lambda 0.1-0.3 nm.

Tecniche XRD: powder XRD Debye-Scherrer 1916 + crystalline phase identification + Rietveld refinement 1969; single crystal XRD Laue 1912 + 4-circle diffractometer + Eulerian goniometer + structure determination protein 50000+ structures Protein Data Bank PDB; thin film XRD + Si MOSFET wafer characterization + STMicroelectronics; small-angle X-ray scattering SAXS nanoparticle 1-100 nm; X-ray reflectivity XRR thin film thickness 1-100 nm; grazing incidence GIXRD.

Mass spectroscopy XRF X-ray fluorescence: elemental analysis + non-destructive + complementary XRD + ED-XRF WD-XRF; X-ray photoelectron XPS surface chemistry + 1-10 nm depth.

Numeri correlati: parameter cell a b c; structure factor F_hkl = sum f_j*exp(2*pi*i*(h*x_j+k*y_j+l*z_j)); scattering factor f_j(theta); Debye-Waller B-factor temperature; multiplicity factor m_hkl; Lorentz polarization factor LP.

Software cristallografia: Mercury + VESTA visualization; SHELX 1990 Sheldrick + Coot 2010 protein + Phenix + CCP4 + Olex2; powder Topas Bruker + FullProf + GSAS-II; simulation Materials Studio Accelrys. Storia: William Henry Bragg (Westward UK 1862 - London 1942) Adelaide Univ Australia 1885-1908 + Univ Leeds 1909-1915 + UCL London 1915-1923 + Royal Institution London 1923-1942 director; 1913 paper Proc Roy Soc A 88:428 con figlio William Lawrence Bragg 'The reflection of X-rays by crystals' classico Bragg's law + 1913 paper Proc Roy Soc A 89:248 + 1915 'X-rays and Crystal Structure' Bell London classico textbook + Premio Nobel Fisica 1915 shared with figlio Lawrence + 'for their services in the analysis of crystal structure by means of X-rays' + Member Royal Society + Director Royal Institution London 1923-1942.

William Lawrence Bragg (Adelaide Australia 1890 - Ipswich UK 1971) Cambridge Cavendish 1909-1914 + Univ Manchester 1919-1937 + NPL 1937-1938 + Cambridge Cavendish 1938-1953 director + Royal Institution 1954-1966; 1913 paper Proc Cam Phil Soc 17:43 'The diffraction of short electromagnetic waves by a crystal' classico Bragg's law derivation 25 years old + Premio Nobel Fisica 1915 youngest Nobel laureate ever 25 years 28 days + Member Royal Society + Director Cavendish Cambridge 1938-1953 + Director Royal Institution 1954-1966 + 1962 supervised Watson-Crick DNA double helix Nobel 1962.

Max von Laue (Pfaffendorf Germany 1879 - Berlin 1960) Berlin TH + Univ Frankfurt 1914 + Univ Berlin 1919-1943 + Max-Planck-Institut Berlin 1951-1959 director; 1912 paper Annalen der Physik 41:989 'Eine quantitative Prufung der Theorie fur die Interferenzerscheinungen bei Rontgenstrahlen' classico Laue diffraction discovery + Friedrich-Knipping experiment ZnS sphalerite 1912 + Premio Nobel Fisica 1914 'for his discovery of the diffraction of X-rays by crystals' + Member US National Academy + Anti-Nazi resistance.

Walther Friedrich (Salbke Germany 1883-1968) + Paul Knipping (1883-1935) + Munich; 1912 con Laue Friedrich-Knipping experiment ZnS sphalerite copper Karbsruhe Munich X-ray diffraction first observation.

Hugo Marie Rietveld (Netherlands 1932-2016) Petten Reactor Centrum Nederland; 1969 paper J Appl Crystallogr 2:65 'A profile refinement method for nuclear and magnetic structures' classico Rietveld refinement powder XRD + Rietveld eponym modern crystallography. Eredita: International Tables for Crystallography 9 volumi IUCr 1983-2016 + Cullity-Stock 'Elements of X-ray Diffraction' 3rd 2001 Prentice-Hall 678 pages classico textbook + Pecharsky-Zavalij 'Fundamentals of Powder Diffraction' 2nd 2009 Springer 720 pages Warren 'X-ray Diffraction' 1969 Dover 381 pages classico + Hammond 'The Basics of Crystallography and Diffraction' 4th 2015 OUP 423 pages + Rupp 'Biomolecular Crystallography' 2010 Garland 836 pages protein.

Italia ricerca Bragg + XRD + cristallografia: Elettra Sincrotrone Trieste Basovizza 350 + 3rd generation synchrotron world-leading + 28 beamlines XRD SAXS XPS XAS Italian; FERMI Free Electron Laser Trieste 100 + XFEL world-leading; CNR-IC Istituto Cristallografia Bari + Roma + Firenze + Milano + Trieste 250 + Bragg legacy world-leading Italian crystallography; CNR-IOM Istituto Officina Materiali Trieste 200; CNR-NANO Modena + Pisa + Trento 200 + nanostructures; CNR-ISC Istituto Sistemi Complessi 200; CNR-SCITEC 200; Sissa Trieste 270; ICTP Trieste 200; BIST IIT Genova 1700; FBK Trento 480; S3 Modena 60; INFN-LNF Frascati DAFNE 800 + light source Italian; INSTM 200 + materials.

Italia università cristallografia + materiali: Bari DG Geomineralogiche + DCh 1080 + Bragg cristallografia world-leading + Carmelo Giacovazzo IUCr Annarita Giacovazzo + Lia Loconto; Politecnico Milano DCMC + DENG + DEIB 7500 + Carlo Maria Maierna materiali + Roberto Frassine polymers; Politecnico Torino DET + DENERG 5000; Sapienza Roma DCh + DSMC 4000; Federico II Napoli DSF + DICMaPI 5000; Pisa DCCI + DFA 2500 + Beltram nanostructures; Bologna DCh + DIFA 4000; Padova DCh + DEI 3000; Trieste OGS + DMG 800; Catania DFA + DSC 2000; Palermo DIFI + Stebicef 2000; Genova DCCI + IIT 2000; Milano Bicocca DSCO 3500; SISSA 270; ICTP 200.

Italia industria XRD + analisi materiali: Bruker Italy Milano 200 + D2 D8 XRD instruments world-leading; PANalytical Italy 100 + X'Pert Empyrean; Rigaku Italy 100 + ZSX Primus; Thermo Scientific Italy 200 + ARL EQUINOX 100; Anton Paar Italy 50 + SAXSpoint; Malvern Panalytical 100; Saes Getters Lainate 1100 + getter pumps cryogenic XRD samples.

Italia industria semiconduttori XRD + Si wafer: STMicroelectronics Catania + Agrate Brianza + Milano 50000 + 12-inch wafer Si XRD MOCVD CVD characterization world-leading; LFoundry Avezzano AQ 800; InnoLux LCD Italy 200; Vishay Italy 200; NXP Italy Catania 200; SAES Getters 1100.

Italia industria farmaceutica + protein crystallography: Chiesi Farmaceutici Parma 6000 + drug discovery XRD + protein crystallography; Menarini Firenze 17000; Bracco Imaging Milano 3500 + contrast agents Imaging XRD; Recordati 4500; Italfarmaco 4000; Angelini Pharma 5800; Alfasigma 3500; Zambon 2900; Dompe 1900.

Italia industria materiali + cementi + ceramiche XRD: Italcementi Bergamo 4400 + cement clinker XRD characterization world-leading; Buzzi Unicem 9700; Mapei 11000; SACMI Imola 4000 + ceramic tiles XRD; Atlas Concorde 700 + ceramic; Florim 1600; Marazzi 4000; Iris Ceramica 2000; Saint-Gobain Italy 8000 + glass XRD.

Formula legge Bragg

Definizione: n * lambda = 2 * d * sin(theta). n ordine diffrazione (1 2 3...) + lambda lunghezza d'onda + d spacing piani cristallini + theta angolo Bragg + 2*theta angolo deviazione observed.

Sistemi cristallini cubic: d_hkl = a / sqrt(h^2 + k^2 + l^2) + a parametro reticolo + h k l Miller indices.

Tetragonal: 1/d^2 = (h^2 + k^2)/a^2 + l^2/c^2; Hexagonal: 1/d^2 = 4*(h^2+hk+k^2)/(3*a^2) + l^2/c^2.

Sorgenti raggi X: Cu Kalfa 0.15418 nm + Mo Kalfa 0.07107 nm + synchrotron 0.05-0.3 nm tunable + neutroni 0.1-0.3 nm; energia E = h*c/lambda Cu Kalfa = 8.04 keV + Mo Kalfa = 17.5 keV.

Coefficienti tipici: NaCl d_200=0.282 nm + 2theta=44.1° con Cu Kalfa; Si d_111=0.314 nm + 2theta=28.4°; Si d_220=0.192 nm + 2theta=47.3°; protein d=2-10 nm + 2theta basso 1-5°.

Storia e contesto

William Henry Bragg (Westward UK 1862 - London 1942) Univ Adelaide Australia 1885-1908 + Univ Leeds 1909-1915 + UCL London 1915-1923 + Royal Institution London director 1923-1942; 1913 paper Proc Roy Soc A 88:428 con figlio William Lawrence Bragg 'The reflection of X-rays by crystals' classico Bragg's law derivation + 1915 'X-rays and Crystal Structure' Bell London classico textbook + Premio Nobel Fisica 1915 shared with figlio Lawrence + 'for their services in the analysis of crystal structure by means of X-rays' + Royal Society + Director Royal Institution + Copley Medal 1930 + Rumford Medal 1916.

William Lawrence Bragg (Adelaide 1890 - Ipswich 1971) Cambridge Cavendish 1909-1914 + Manchester 1919-1937 + NPL 1937-1938 + Cambridge Cavendish director 1938-1953 + Royal Institution 1954-1966 director; 1913 paper Proc Cam Phil Soc 17:43 'The diffraction of short electromagnetic waves by a crystal' classico Bragg's law derivation 25 anni + Premio Nobel Fisica 1915 youngest Nobel laureate ever 25 anni 28 giorni + Member Royal Society + Director Cavendish 1938-1953 + Director Royal Institution 1954-1966 + 1962 supervised Watson-Crick DNA double helix Nobel 1962 Cavendish + Copley Medal 1966.

Max von Laue (Pfaffendorf Germany 1879 - Berlin 1960) Univ Frankfurt 1914 + Univ Berlin 1919-1943 + Max-Planck-Institut Berlin director 1951-1959; 1912 paper Annalen der Physik 41:989 'Eine quantitative Prufung der Theorie fur die Interferenzerscheinungen bei Rontgenstrahlen' classico Laue diffraction discovery + Friedrich-Knipping experiment ZnS sphalerite 1912 + Premio Nobel Fisica 1914 'for his discovery of the diffraction of X-rays by crystals' + Member US National Academy + Anti-Nazi resistance + Max Planck Medal 1939.

Walther Friedrich (Salbke Germany 1883-1968) + Paul Knipping (Germany 1883-1935) + Munich Univ; 1912 con Laue Friedrich-Knipping experiment ZnS sphalerite copper sulfate Munich + first X-ray diffraction observation.

Hugo Marie Rietveld (Netherlands 1932-2016) Petten Reactor Centrum Nederland 1959-1992; 1969 paper J Appl Crystallogr 2:65 'A profile refinement method for nuclear and magnetic structures' classico Rietveld refinement powder XRD + Rietveld eponym modern crystallography + IUCr Aminoff Prize 2010. Eredita: International Tables for Crystallography 9 volumi IUCr 1983-2016 + Cullity-Stock 'Elements of X-ray Diffraction' 3rd 2001 Prentice-Hall 678 pages classico + Pecharsky-Zavalij 'Fundamentals of Powder Diffraction' 2nd 2009 Springer 720 pages Warren 'X-ray Diffraction' 1969 Dover 381 pages classico + Hammond 'The Basics of Crystallography and Diffraction' 4th 2015 OUP 423 pages + Rupp 'Biomolecular Crystallography' 2010 Garland 836 pages protein + Giacovazzo et al 'Fundamentals of Crystallography' 3rd 2011 OUP 842 pages classico Italian.

Italia ricerca: Elettra Sincrotrone Trieste 350 world-leading + 28 beamlines Italian; FERMI XFEL Trieste 100 world-leading; CNR-IC Istituto Cristallografia 250 world-leading + Bragg legacy + Carmelo Giacovazzo IUCr; CNR-IOM Trieste 200; CNR-NANO 200; CNR-ISC 200; CNR-SCITEC 200; SISSA Trieste 270; ICTP Trieste 200; BIST IIT 1700; FBK 480; S3 Modena 60; INFN-LNF DAFNE 800; INSTM 200.

Universita: Bari 1080 + Carmelo Giacovazzo cristallografia world-leading IUCr; Politecnico Milano 7500; Politecnico Torino 5000; Sapienza Roma 4000; Federico II 5000; Pisa 2500 + Beltram; Bologna 4000; Padova 3000; Trieste 800; Catania 2000; Palermo 2000; Genova 2000; Milano Bicocca 3500; SISSA 270; ICTP 200.

Industria XRD analisi: Bruker Italy 200 world-leading D2 D8; PANalytical 100; Rigaku 100; Thermo Scientific 200; Anton Paar 50; Malvern Panalytical 100; SAES Getters 1100.

Semiconduttori: STMicroelectronics 50000 world-leading + 12-inch wafer Si XRD; LFoundry 800; NXP Italy 200; SAES Getters 1100.

Farma cristallografia: Chiesi 6000; Menarini 17000; Bracco Imaging 3500; Recordati 4500; Italfarmaco 4000; Angelini 5800; Alfasigma 3500.

Materiali cementi ceramiche: Italcementi 4400 world-leading; Buzzi Unicem 9700; Mapei 11000; SACMI 4000; Atlas Concorde 700; Florim 1600; Marazzi 4000; Iris Ceramica 2000; Saint-Gobain Italy 8000.

Tabella valori Bragg

  • n*lambda = 2*d*sin(theta): definizione
  • cubic: d=a/sqrt(h^2+k^2+l^2)
  • tetragonal: 1/d^2=(h^2+k^2)/a^2+l^2/c^2
  • hexagonal: 1/d^2=4*(h^2+hk+k^2)/(3*a^2)+l^2/c^2
  • Cu Kalfa: lambda=0.15418 nm
  • Mo Kalfa: lambda=0.07107 nm
  • NaCl 200: d=0.282 nm 2theta=44.1° Cu
  • Si 111: d=0.314 nm 2theta=28.4° Cu
  • Si 220: d=0.192 nm 2theta=47.3° Cu
  • Synchrotron Elettra: tunable 0.05-0.3 nm

5 esempi numerici

  1. NaCl 200 con Cu Kalfa: lambda=0.15418 nm + d=0.282 nm + n=1 + theta=15.86° + 2theta=31.7° reflections.
  2. Si 220 con Cu Kalfa wafer XRD: lambda=0.15418 nm + d=0.192 nm + theta=23.65° + 2theta=47.3° + STMicroelectronics 12-inch wafer characterization.
  3. Protein crystal SAXS synchrotron: lambda=0.1 nm + d=5 nm + n=1 + theta=0.57° + 2theta=1.15° + Elettra Trieste low-angle protein PDB structure determination.
  4. d-spacing inverse from 2theta: lambda=0.15418 nm + theta=22.05° + n=1 + d=0.205 nm calculated + classical XRD pattern indexing.
  5. Neutron NaCl ILL Grenoble: lambda=0.108 nm thermal neutrons + d=0.282 nm + n=1 + theta=11.06° + 2theta=22.1° + classical neutron diffraction.

Limiti e ipotesi

Kinematic approximation: thin crystal scattering single + dynamical theory thick crystal Pendellosung + classical Authier 2001.

Powder vs single crystal: random orientation Debye-Scherrer rings vs single Laue spots + classical Cullity-Stock 2001.

Thermal Debye-Waller: B-factor temperature attenuation peak intensity + classical Warren 1969.

Multiple scattering extinction: large perfect crystal + dynamical correction.

Structure factor cancelation: certain hkl forbidden if F_hkl = 0 + space group symmetry + classical Rupp 2010.

Errori comuni Bragg

  • theta vs 2theta: theta angolo riflessione + 2theta angolo deviazione observed in XRD pattern distinction Cullity-Stock 2001.
  • Order n harmonics: n=1 primario fundamentally + n>1 secondary rispetta n*lambda condition + Bragg's law generalized.
  • Wavelength units: lambda in nm SI vs A Angstrom 1 nm = 10 A distinction.
  • d_hkl formula crystal system: cubic vs tetragonal vs hexagonal vs orthorhombic distinct International Tables.
  • Forbidden reflections: F_hkl = 0 systematic absences + space group symmetry + classical Rupp 2010.

Applicazioni Bragg

  1. Synchrotron Elettra Trieste: 28 beamlines XRD SAXS XPS XAS world-leading Italian; FERMI XFEL ultrafast.
  2. Semiconductor Si wafer: STMicroelectronics + LFoundry + 12-inch wafer Si 111 220 + crystal quality assessment + thin film stress.
  3. Cement industry XRD: Italcementi + Buzzi Unicem + Mapei + clinker phase identification C3S C2S C3A + Rietveld quantitative.
  4. Ceramic tiles: SACMI + Atlas Concorde + Florim + Marazzi + Iris Ceramica + crystalline phase analysis.
  5. Pharmaceutical drug discovery: Chiesi + Menarini + Bracco + protein crystallography PDB + drug-target binding.
  6. Materials characterization XRD: Bruker Italy world-leading + PANalytical + Rigaku + Anton Paar + Malvern Panalytical.
  7. Protein crystallography PDB: 50000+ structures + drug design Rupp 2010.
  8. Glass amorphous SAXS: Saint-Gobain Italy + amorphous halo + radial distribution function.
  9. Geology mineral identification: powder XRD + International Centre for Diffraction Data ICDD 1.5M+ patterns.
  10. Battery cathode characterization: Italvolt + FIAMM + Solvay Specialty + LFP NCM XRD + Rietveld.

Aggiornato a luglio 2026

Domande frequenti

Cosa misura la legge di Bragg?

Legge di Bragg n*lambda = 2*d*sin(theta) descrive la condizione di interferenza costruttiva dei raggi X riflessi dai piani cristallini paralleli a distanza d. La differenza di percorso tra raggi adiacenti deve essere multiplo intero della lunghezza d'onda. Cu Kalfa lambda=0.15418 nm + Si 111 d=0.314 nm + 2theta=28.4° tipica. William Henry Bragg + figlio William Lawrence Bragg UK 1913 Proc Roy Soc A 88:428 classico + Premio Nobel 1915 youngest 25.

Perché' Lawrence Bragg ha vinto il Nobel a 25 anni?

William Lawrence Bragg vinse Premio Nobel Fisica 1915 a 25 anni 28 giorni con suo padre William Henry Bragg per 'la analisi della struttura cristallina mediante raggi X'. Lawrence derivò la legge di Bragg n*lambda=2*d*sin(theta) nel 1913 a Cambridge Cavendish a 23 anni mentre era ancora studente. Resta youngest Nobel laureate ever in any category. Successivamente Director Cavendish 1938-1953 + supervised Watson-Crick DNA double helix Nobel 1962.

Differenza tra theta e 2theta in XRD?

theta angolo di Bragg tra raggio incidente e piano cristallino + half-angle; 2theta angolo deviazione tra raggio incidente e raggio diffratto + full-angle observed in diffractometer. XRD pattern tipicamente plotta intensità vs 2theta in gradi. Esempio: NaCl 200 con Cu Kalfa theta=15.86° + 2theta=31.7° observed scan range. Confondere theta e 2theta errore comune principianti Cullity-Stock 2001 'Elements of X-ray Diffraction' Prentice-Hall 678 pages.

Cosa sono Cu Kalfa Mo Kalfa Co Kalfa?

Linee caratteristiche raggi X emesse da anodo metallico bombardato con elettroni 30-50 kV in tube X-ray laboratorio. Cu Kalfa lambda=0.15418 nm = 1.5418 A energia 8.04 keV più comune; Mo Kalfa lambda=0.07107 nm energia 17.5 keV piccoli atomi penetranti; Co Kalfa lambda=0.17890 nm + Cr Kalfa 0.22897 nm. Classical reference. Synchrotron tunable 0.05-0.3 nm + APS Argonne + ESRF + Elettra Trieste world-leading + brilliance 10^21.

Cos'e' Rietveld refinement powder XRD 1969?

Rietveld refinement Hugo Marie Rietveld Netherlands 1932-2016 paper J Appl Crystallogr 2:65 1969: tecnica per analizzare quantitativamente XRD pattern di campioni policristallini fittando il pattern observed con quello calculated dalla struttura cristallina. Permette quantitative phase analysis + structure refinement + precision lattice parameters + IUCr Aminoff Prize 2010 + software Topas Bruker + FullProf + GSAS-II modern crystallography.

Cos'e' la Protein Data Bank PDB?

Protein Data Bank PDB founded 1971 Brookhaven National Laboratory + Rutgers + RCSB-PDB world-leading: archivio mondiale di strutture 3D di macromolecole biologiche (proteine RNA DNA) determinate principalmente tramite X-ray crystallography Bragg + cryo-EM + NMR. > 200000 strutture deposited 2024 + open access + drug design Rupp 'Biomolecular Crystallography' 2010 Garland 836 pages + 50000+ structures via Bragg's law XRD.

Applicazioni industriali Bragg's law XRD?

Applicazioni: Synchrotron Elettra Trieste + FERMI XFEL world-leading 28 beamlines XRD SAXS XAS Italian; semiconductor Si wafer STMicroelectronics 12-inch wafer characterization world-leading; cement industry Italcementi + Buzzi + Mapei clinker phase identification Rietveld; ceramic tiles SACMI + Atlas Concorde + Florim; pharmaceutical Chiesi + Bracco + Menarini protein crystallography PDB; materials Bruker world-leading + PANalytical + Rigaku XRD instruments.

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