The Meaning of the Concept of Reciprocal Space in Crystal Structure Analysis

Ivan Vicković Faculty of Science,University of Zagreb [email protected]

CEEPUS H-76 Summer School, Learning and teaching bioanalysis Prague, May 29-June 3, 2005

CEEPUS Summer school, Prague 2005 1 small molecule protein crystallography

CEEPUS Summer school, Prague 2005 2 Number of solved structures 1912 – 1970 cca 4.000 1970 – 1980 cca 40.000 1980 – 2000 cca 200.000

15 NOBEL LAUREATS

HAVE BEEN RECOGNIZED IN THE FIELD OF

DIFFRACTION STRUCTURE ANALYSIS,

INCLUDING

Roentgen and Laue CEEPUS Summer school, Prague 2005 3 •What is it what makes crystal structure analysis so powerful method in structure determination?

•What is the physical phenomenon that makes us possible to obtain so accurate results?

•What are the data to be observed in order to solve a structure?

•What is the mathematical approach which ensures the reliability of the results?

CEEPUS Summer school, Prague 2005 4 Crystal is regularly repeated arrangement of atoms represented by unit cell

Symmetry is one of the main properties of crystals.

Miller indices are used to describe crystal faces. Angles between the faces are constant.

CEEPUS Summer school, Prague 2005 5 Crystalline and amorphous SiO2

Unit cell is repeated No unit cell can be in 3 dimensions defined good diffraction expected very poor or no diffraction expected CEEPUS Summer school, Prague 2005 6 Diffraction data collection

X-rays diffraction on a crystal diffraction data CEEPUS Summer school, Prague 2005 7 Bragg’s approach to the diffraction condition

A d C d –distance between B the lattice planes in the real space

Path difference: AB+BC=δ AB=δ sin θ=AB/d sin θ=δ/2d Bragg’s law: Reflection on the planes 2dsinθ=nλ http://www.bmsc.washington.edu/people/merritt/bc530CEEPUS Summer school, /bragg/Prague 2005 8 Laue’s approach to the diffraction condition

a sinΦ=nλ Constructive interference on a row of atoms CEEPUS Summer school, Prague 2005 9 Diffraction image on a CCD or imaging plate

Information obtained

1. unit cell dimensions

2. set of reflections hkl and their intensities from systematicaly absent reflections the space group and symmetry can be determined

CEEPUS Summer school, Prague 2005 10 Small molecule in the real space cell

CEEPUS Summer school, Prague 2005 11 Representation of the atoms in a small

molecule of (C5H5)3Sb CEEPUS Summer school, Prague 2005 12 Hexamer of 2-Zn insulin type in the rhombohedral crystal CEEPUS Summer school, Prague 2005 13 To calculate To observe

Diffraction wave Chemical Diffraction Crystal synthesis Intensity Phase Wave function Electron power density map hkl I ϕ ρ C f F a ...... O ... .. S ...... Hg ...... ⇒ ... + .. + + ⇒ ⇔ ...... FOURIER ...... TRANSFORMATION

Atomic Structure fator List of Incomplete picture Impossible scattering atoms in reciprocal space to record factors Complete picture Complete picture in reciprocal in REAL SPACE space

PHASE PROBLEM Patterson function 14 CEEPUS Summer school, Prague 2005 Direct methods HOW TO PRESENT THE PHENOMENON OF DIFFRACTION?

F(S(hkl.)) = f(I,ϕ) I(S(hkl ) = F(S)⋅ F(S) = F 2

ϕ = π ⋅ (S) 2 S(hkl ) r(xyz ) How many data can be observed?

N ( )= ⋅ ( ())⋅ π + + F0 S(hkl ) C I S hkl ∑exp( 2 i(hx j ky j lz j )) j=1 N ( ())= ( ())⋅ ( π ⋅ ⋅ ) F j S hkl ∑ f j S hkl exp 2 i S r j j =1

CEEPUS Summer school, Prague 2005 15 N (())= (())⋅ (π ⋅ ⋅ ) F c S hkl ∑ f j S hkl exp2 i S r j j=1

or F(S)= ∫ρ(r(xyz ))⋅exp(2πi⋅S⋅r)dV V

CEEPUS Summer school, Prague 2005 16 F(S)= ∫ ρ(r(xyz ))⋅exp (2πi ⋅ S ⋅r)dV V

ρ(r)= ∫F(S(hkl ))⋅exp (−2πi⋅S⋅r)dV* V*

A pair of Fourier transforms represents the same phenomenon ρ(r)⇐⇒F(S) in two different spaces (or, strictly mathematicaly, in two different coordinate systems.

CEEPUS Summer school, Prague 2005 17 Fourier transformation

ρ(r)˜F(S)

The maximums in the electron density map are interpreted as coordinates of atoms in the cell. 18 CEEPUS Summer school, Prague 2005 Electron density map in the real space can not be calculated without concept of the reciprocal space CEEPUS Summer school, Prague 2005 19 From direct to reciprocal lattice

In the reciprocal lattice the point hkl is drawn direct reciprocal • at a distance 1/ dhkl from the origin 000

• in direction of the normal to the set of planes (hkl) in the real space.

20 CEEPUS Summer school, Prague 2005 Direct and reciprocal cell in the monoclinic crystal system 21 CEEPUS Summer school, Prague 2005 correspond to nodes sets of lattice planes

For every real unit cell exists a reciprocal unit cell CEEPUS Summer school, Prague 2005 22 Geometric construction of the reciprocal lattice

Unit cell of the reciprocal space

(101) (001) 102 002 d* 102 d* (100) 002 c c (002)

(102) 101 001 β c* d* d* 101 001 a

β* 000 a 100 d* 100 a*

planes (hkl) in the Vectors in the reciprocal space are related ac plane of the real to the planes in the real space: space d*(hkl)= h a*+k b*+l c*= S(hkl) 23 CEEPUS Summer school, Prague 2005 Laue’s approach to the diffraction condition

a sinΦ=nλ Constructive interference on a row of atoms 24 CEEPUS Summer school, Prague 2005 Bragg’s approach to the diffraction condition

A d C d –distance between B the lattice planes in the real space

Path difference: AB+BC=δ AB=δ sin θ=AB/d sin θ=δ/2d Reflection on the planes Bragg’s law: 2dsinθ=nλ http://www.bmsc.washington.edu/people/merritt/bc530CEEPUS Summer school,/bragg/ Prague 25 2005 Ewald’s approach to the diffraction condition

crystal s S(hkl )=s 0-s

s0

2r=2/ λ •The diffraction will occure at a node of the reciprocal lattice when the Ewald sphere touches that node.

•Vectors in the reciprocal space are related to the incident and diffracted beams: S( hkl)=s 0-s= d*( hkl) 26 CEEPUS Summer school, Prague 2005 The application of the concept of reciprocal space 1. Indexing of diffraction maxima (in the early days)

Bernall chart for determination of Weissenberg chart for determination Miller indices on a of Miller indices on a photodiffractogram photodiffractogram 27 CEEPUS Summer school, Prague 2005 2. Data collection in reciprocal space visualised by Ewald sphere

Ewald sphere, r=1/ λ -a detectorX-ray

X-ray crystal source

Rotating reciprocal cell

CEEPUS Summer school, Prague 2005 28 3. Data collection and structure solving

• To determine the unit cell in the real space and to precalculate all possible directions in which the diffraction beams can be expected a, b, c,α , β , γ , set of hkl

• To calculate electron density map as Fourier transform of structure factors, and data processing

ρ(r) F(S) CEEPUS Summer school, Prague 2005 29 Sommerfeld’s Institute for theoretical , München 1912

M. von Laue

A. Sommerfeld P.P.Ewald CEEPUS Summer school, Prague 2005 30 W.Friedrich Roentgen’s and Sommerfeld’s group in Cafe Lutz

31 CEEPUS Summer school, Prague 2005 Active social life in the team

P.P. Ewald

32 CEEPUS Summer school, Prague 2005 EWALD’S DISERTATION 1910 - Sommerfeld: “...to find out the optical characteristics of anizotropic distribution of izotropic rezonators distributed statistically”

Ewald’s assumption:atoms in crystals are regularly distributed rezonators

1911 Christmas: all calculations done

1912 January:dinner with M.von Laue, discussion between colegues:

• von Laue hesitates both the assumption and the results in the case of crystals

• Ewald ensures the generality of the results

PAUL PETER EWALD: Disertation submitted: February 16. 1912., Thesis defended: March 5, 1912 33 CEEPUS Summer school, Prague 2005 Easter holidays: von Laue, Friedrich and Knipping are ready to prove that crystals can reflect but CAN NOT DIFFRACT X-RAYS since the ATOMS ARE DISTRIBUTED STATISTICALY. It has to be proven that Ewald’s calculation in disertation are WRONG, with no Ewald’s participation M. von Laue

But, later on, for decades, all theoretical and experimental development of crystal and molecular structure analysis has been based on Ewald’s idea of periodically distributed atoms in crystals, and idea of reciprocal space !

34 CEEPUS Summer school, Prague 2005 : June 8, 1912 : • Friedrich, Knipping & Laue: “Interferenz-erschinungen bei Röntgenstrahlen”, Sitzungsberichte, pp. 303-322 • M. Laue: “Eine quantitive Prüfung der Theorie für Interferenzerschinungen bei Röntgenstrahlen”, Sitzungsberichte, pp. 363- 373.

PAUL PETER EWALD : • 1913: “Das reziproke Gitter”, Physikalishe Zeitschrift , vol. 14 , pg. 465-472 • 1917: “On the Foundations of Crystal Optics” • 1921 : Das “reziproke Gitter” in der Strukturtheorie, Zeit. f. Krist ., 56 129.

CEEPUS Summer school, Prague 2005 35 Fast development 1914/1915

W. L. Bragg «The structure of some crystals as indicated by their diffraction of X-rays»,

W. L. Bragg & W. H. Bragg «The structure of diamond»

W. L. Bragg «The analysis of crystal by X-ray spectometer».

First solveed structures : KCl, KBr, KI,

ZnS, NaCl, CaF 2, CaCO 3 i FeS.

36 CEEPUS Summer school, Prague 2005 Conclusion: P.P. Ewald (1888-1985) explained the first dinamic theory of diffraction on crystals in the Thesis 1912-befor diffraction device had been made and regular arrangement of atoms in crystals had been recognized. Question: Where is the Ewald’s place?

Wilhem Conrad Röntgen William H. Bragg, Paul PeterEwald Max von Laue, William L. Bragg (or Nikola Tesla before destruction of the laboratory by fire ?) 1895 Diffraction 1921 First structures solved Reciprocal latice X-Rays experiment 1914 1912 1913

37 CEEPUS Summer school, Prague 2005 Thank you and have a safe trip back home

38 CEEPUS Summer school, Prague 2005