Legacy

The Legacy of P.M. de Wolff

1. A triple jubileum 1964 1974 1984

2. Precursory works L. Weber, F.C. Frank

3. Crystallographic scaling Quasicrystals, Crystals, Molecules

4. Molecular crystallography Bio-macromolecules

5. The legacy An incommensurable passion

Budapest, 29 August 2004, ECM22 A. Janner

. – p.1/25

dewolff

Prof.Dr.Ir. P.M. de Wolff

1911 - 1998

1951 Ph.D.: Guinier-de Wolff camera

1959 Full Professor in Delft

1964 γ Na2Co3: Non-indexable satellites

1972 Kyoto: 4-dim. space group of γ Na2CO3

1974 Acta Cryst. A : World-lines for modulated

atomic positions

—— ————————————————–

1984 Retirement from the University:

"An incommensurable passion"

1992 Int. Tables, Vol C : Superspace groups

1998 The Royal Swedish Academy of Sciences :

Gregori Aminoff Prize

. – p.2/25

anomaly

Non-indexable Satellites

E. Brouns, J.W. Visser and P.M. de Wolff, Acta Cryst. (1964), 17, 614. – p.3/25

anomaly

Non-indexable Satellites

E. Brouns, J.W. Visser and P.M. de Wolff, Acta Cryst. (1964), 17, 614

. – p.3/25

Kyoto

Kyoto : The Four-Dimensional Space Group of γ-Na2CO3

P.M. de Wolff and W. van Aalst, Acta Cryst. (1972), A28, S111

. – p.4/25

Kyoto

The Four-Dimensional Space Group of γ-Na2CO3

Selection rule for the glide reflection

. – p.4/25

Lattice vibrations2

1969 : Symmetry groups of lattice vibrations

. – p.5/25

Multimetrical

1969 : Multimetrical crystallography

. – p.6/25

space-time diffraction. – p.7/25

Pseudo-symmetry. – p.8/25

Pseudo-symmetry2. – p.8/25

weber-frank

Precursive 4D crystallographyL. Weber, Z. Kristallogr. 17 (1922) 200. F.C. Frank, Acta Cryst. 18 (1965) 862

(frank2)

1922: Weber0001

1000

0100

0010

K = (hkl) = (hkil) r = [mnp] = [uvtw]

1965: Frank

[1000]

[0100]

[0010]

- -[2110]/3ϕ

cos ϕ = √(2/3)

Hexagonal four-indicesKr = hm + kn + lp = hu + kv + it + lw

i = - k - l w = p

u = 2m−n

3 v = −m+2n

3 t = −m−n

3

4D cubic

[001] = [0001]

c2 c2

[100] = 1

3[21̄1̄0]

a2 6

9c2

c

a=

√

32

. – p.9/25

Cryst.Scal.

Crystallographic Scaling

Scaling with scaling factor λ

1D (linear) Xλ(x, y, z) = (λx, y, z)

2D (planar) Pλ(x, y, z) = (λx, λy, z)

3D (isotropic) Iλ(x, y, z) = (λx, λy, λz)

Higher dimensional .......

. – p.10/25

Cryst.Scal.

Crystallographic Scaling

Scaling with scaling factor λ

1D (linear) Xλ(x, y, z) = (λx, y, z)

2D (planar) Pλ(x, y, z) = (λx, λy, z)

3D (isotropic) Iλ(x, y, z) = (λx, λy, λz)

Higher dimensional .......

Crystallographic transforming a lattice into a lattice

SλΛ = Λ Sλ integral invertible

in general: SλΛ = Σ Σ ⊆ Λ or Λ ⊆ Σ

Sλ rational invertible. – p.10/25

Star Pentagon

Pentagonal Case

(sgk1b)

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

-1 -1 -1 -1

-2 0 -1 -1

1 -1 1 0

0 1 -1 1

-1 -1 0 -2

2 1 1 2

Polygrammal Scaling

Star Pentagon:Schäfli Symbol {5/2}

Scaling matrix: (planar scaling)

2̄ 1 0 1̄

0 1̄ 1 1̄

1̄ 1 1̄ 0

1̄ 0 1 2̄

Scaling factor:-1/τ2 = −0.3820...

(τ = 1+√

5

2= 1.618...)

. – p.11/25

AlMn Fourier-Scaling

Al78Mn22: Fourier Map of a Decagonal Plane

W. Steurer, J.Phys: Condens. Matter 3 (1991) 3397-3410

. – p.12/25

AlMn Fourier-Scaling

Al78Mn22: Pentagonal Lattice Points

Comparison of the Fourier map with model positions (Low Indices only)

(deca01a,Indexed Fourier points AlMn,U1=60,N0=70)

-1-1-1-1 0-2-2 0 -3-4-4-3-2-2-4-1

0-2-1-1 1-3-2 0 -2-5-4-3 1 0-1 1 -1-3-4-1

1 0 0 0

-2-2-3-2

-1-3-3-2-1 0-2-1 0-1-3 0

0-4-3-2 0-1-2-1 1-2-3 0 -2-4-5-3

0-1-1-2 1-2-2-1 1 1-1 0 -1-2-4-2

2 0-1 0 -1-2-3-3 0-3-4-2 0 0-3-1 -2-3-6-3

2 0 0-1 0-3-3-3 0 0-2-2 1-1-3-1 -2-3-5-4

1-1-2-2 2-2-3-1 -1-4-5-4 2 1-2 0 -1-1-4-3 0-2-5-2

1-1-1-3 2 1-1-1 -1-1-3-4 0-2-4-3

0 1-3-2 3 0-1-1 0-2-3-4 1-3-4-3

1 0-3-2 2-1-4-1 -1-3-6-4

1 0-2-3 2-1-3-2 -1-3-5-5 2 2-2-1 0-1-5-3

2-1-2-3 0-4-5-5 3 1-2-1 0-1-4-4 1-2-5-3

. – p.12/25

AlMn Fourier-Scaling

Al78Mn22: Planar Scaling (Star Pentagon)

Scaling factor: λ = −1/τ2

(deca02a,Pentragamma AlMn,U1=60,N0=70)

. – p.12/25

Latt.-Sublatt. sqrt(3/2)

Frank’s ’Cubic’ Hexagonal Lattice: c

a=

√

3

2

3D Hexagonal lattice projection of a 4D Cubic latticeHexagonal basis: a1, a2, a3

Sublattice basis: b1 = [0 3 0], b2 = [2̄ 1̄ 2], b3 = [2 1 1]

Metric tensors:

g(a) =

1 1̄

20

1̄

21 0

0 0 3

2

g(b) =

1 0 0

0 1 0

0 0 1

2

Transformation matrix: Sba =

0 2̄ 2

3 1̄ 1

0 2 1

Lattice-Sublattice transformation: S̃ba g(a) Sba = 9 g(b)

√

3

2-hexagonal

Sba−→ 1√

2-tetragonal (Scaling factor 3)

F.C. Frank, Acta Cryst. 18 (1965) 862-866. – p.13/25

Zn-Mg-Sn sqrt(8/3)

Z-Phase Zn6Mg3Sm:√

3

8-Hexagonal (P63/mmc)

Zone F: [211] ∼ [101̄1] Zone G: [212] ∼ [101̄2]

Singh, Abe and Tsai, Phil.Mag. Lett. 77 (1998) 95-103

Ranganathan, Singh and Tsai, Phil. Mag. Lett. 82 (2002) 13-19

. – p.14/25

R-phycoerythrin 1-hex.

Hexameric R-phycoerythrin (Trigonal 32)

Isometric hexagonal form lattice

a

b

x

y

a = 4r°r°

x

z

4r°

Chang et al., J.Mol.Biol 262 (1996) 721-731 (PDB 1lia) . – p.15/25

Bacteriorhodopsin sqrt(3)-hex

Bacteriorhodopsin trimer√

3-hexagonal form lattice Cubic host lattice (Lipid)

a

re

x

y

a√3r0

z

x

c = a√3

Edman et al., Nature 401 (1999) 822-826 (PDB 1qko)

a = re = 3 r0 = 1√

3c One-parameter form lattice

. – p.16/25

Octagr. Pf1

Inovirus Pf1: Octagrammal scalingD.J. Liu and L.A. Day, Science 265 (1994) 671-674

Asp4

Ser41

Code-Dependent RegionHole Region

Backbone Region

Major Coat Protein

(Bases) (Sugar-phosphates)

Ser41

PP

. – p.17/25

PA-Sm free [bound]

Pyrococcus Abyssi Sm Core (PA-Sm)Heptagonal isometric lattice

H=16u d=2u

h=7u

h=7u

9u

uu

uu

Asp65

Asp44

Gly58

Asp14

r = 8uer°

A3Asp14Gly58Asp44 Asp65

S. Thore et al. J.Biol. Chem. 278 (2003) 1239-1247. – p.18/25

rna-sm complex

PA Sm complex with RNA

0

1

2

3

4

5

6

S. Thore et al. J.Biol. Chem. 278 (2003) 1239-1247. – p.19/25

PA-Sm1,b [free]

PA Sm-RNA complex: PA Sm sub-system

H=18u d=4u

h=7u

h=7u

11u

uu

uu

Asp144

Asp165

Lys115

Asp114

Ca

A3

r = 8uer°

Lys115Asp114Asp144 Asp165

. – p.20/25

PA-Sm1,RNA

PA Sm-RNA complex: RNA Sub-system

H=21u’h’=3u’

8u’

8u’

H’=5u’

h"=2u’

h"=2u’

H"=9u’

r"e

r’e

r’ = 8u’e

A3

. – p.21/25

Int.hex.lattices

ca-Distribution of Hexagonal Crystals

Crystal Data Determinative TablesVol.2, Inorganic compounds, Donnay & Ondik, 1973

(24.000 entries)

B. Constant and P.J. Shlichta, Acta Cryst. A59 (2003) 281-282. – p.22/25

nt.tetr.lattices

ca-Distribution of Hexagonal Crystals

√

32

1√

3√

2

√

2

√

8√

3

√

6

√

15√

22

√

8√

33

√

8√

3

. – p.22/25

Inorganic .hex.lattices

Distribution of Inorganic Hexagonal Crystals

(ICSD) Inorganic Crystal Structures Database (12000 hex. entries)

c/a 0 1 2 3 4 5 0

200

400

600

800

1000

1200

1/2 1/√2 1 √2 √(8/3) 3/√2 √6 √(15/2) 2√(8/3) 2√3 √15 2√6

(in collaboration with R. de Gelder) . – p.23/25

Passion

An incommensurable passion

"How was such a discovery possible ?"

"We were able to index each normal powder diffraction"

"After five years work we were able to publish a first paper"

"... accepted only reluctantly"

"How was it possible for our small group to work without

international competition from 1964 to 1977 ?"

"crystallographers and physicists speak another

language"

P.M. de Wolff, (1984), TH Delft, retirement lecture. – p.24/25

Aminoff. – p.25/25