Issuu on Google+

Crystallography - Molecular architecture • Symmetry (molecule, stacking objects) point groups, plane groups, space groups

• Crystal structure determination using X-ray diffraction • Cambridge Structural Database


What is: crystallography – crystal ? • Krústallos (Gr. Frozen water) • solid state of matter • regular external faces geometrical law - Steno (1669) angle between two corresponding faces is constant • regular internal structure Keppler (New year’s day 1611) • Hauy (1784): identical building blocks


What is: crystallography – crystal ? • originaly part of geology - mineralogy • discovery X-rays (Röntgen, 1895, NP 1901) • diffraction experiment crystal (von Laue, 1912, NP 1914) • structure determination simple salts (Bragg, 1913, NP 1915) • since then more and more complex structures

Crystallography evoluates from description of external structure to internal structure!


Relation to other sciences… Structure and properties of… • minerals • metals, alloys • (in)organic compounds • polymers • biomolecules

mineralogy material science chemistry, pharmacy solid-state physics polymer science biochemistry, molecular biology, medecine

With the help of… mathematics informatics


Relation to other sciences… Some examples…

CHEMISTRY

C60, fullerene Proven by X-ray (derivative)


Relation to other sciences… Some examples…

BIOCHEMISTRY

NP Chemistry 2003 – structure determination water- ion channels


Relation to other sciences… Some examples…

PHARMACY


Relation to other sciences… Some examples… POLYMER SCIENCES

Polyethylene

Importance crystallinity


Importance symmetry

External order of a crystal reflects an internal submicroscopic order.

Symmetry is important in almost all disciplines of science, art and society.


Some important realisations crystallography • theory electrostatic bond in inorganic ion structures • proof tetrahedral carbon atom • structure elucidation benzene ring • proof existence H-bond • -helix, -plate in proteins • DNA-model • determination absolute configuration


Crystallisation techniques •

Principle: bring the system slowly toward a state of minimum solubility (supersaturation)

• •

Driving force ? Large number of variables (purity, concentration, pH, temperature, buffer, additives, precipitants…)


•

Some methods -

Cooling

-

Concentration (evaporation volatile solvent, NMR tube)

-

Solvent diffusion solvent

Anti-solvent


-

Vapour diffusion

-

Reactant diffusion (gel) Seed crystals Sublimation


Sparse matrix screen - samples a finite number of variables, e.g. pH and [(NH4)2SO4] - uses very small samples - usually gives preliminary conditions (to be optimised) Grease

Precipitant in reservoir

1-5 L sample 1-5 L precipitant Mix

Invert, seal


Crystallisation robot liquid handling arm with tips (0.25-250 µl)

Mosquito


XtalScreens Database of crystallisation screens http://xray.bmc.uu.se/markh/php/xtalscreens.php


Part 1. Symmetry - Introduction (a) Description of symmetry • symmetry element symmetry operation

rotate 90° • symmetry figure

• equivalent positions


• symmetry matrix (2 x rotation)

• mathematical group – multiplication table (Caley) - order product - Abels E X-1 associative

A = 90 B = 180 C = 270 E = 360/0

• Schoenflies and Hermann-Mauguin notation C4 4 cyclic (A=generating element)


(b) What is a crystal ? Precisely ordered three dimensional array of ions, atoms or molecules

(1)

(2)

(3)

(1) Asymmetric unit (building block) (2) Build unit cell by symmetry operations (inversion, rotation axis, mirror plane) (3) Lattice translation symmetry


Chapter 1. Point symmetry •

Symmetry elements

Inversion centre 1

Ci

Mirror plane m

σ Cn

n-fold rotation axis n Rotation inversion axis -n Rotation reflection axis Sn

(n=2, 3, 4 or 6) Rotation of (360/n)° Proper rotation Point symmetry

Chirality!

Hermann-Mauguin or International Tables Schoenflies


Point groups (32)  set of symmetry operations that leave a point fixed and form a mathematical group  describe the symmetry of an object (e.g. a molecule!)

Point group 4/mmm (porphyrazin)


 Multiplication table for NH3


 Overview 32 point groups


 Some examples

http://www.molwave.com/software/3dmolsym/3dmolsym.htm


Chapter 2. Translation symmetry •

One-dimensional space groups (7)


•

Two-dimensional space groups or plane groups

Limited possibilities of order of a rotation axis!


Possible two-dimensional lattices (5) Oblique p Rectangular p


Rectangular c

Square p

Hexagonal p


Plane groups (17) Combination of point symmetry elements and two-dimensional lattices: • only n or m orthogonal to plane • 2 in plane = m • rotation inversion axis not possible •i=2 Oblique


Rectangular


Square


Hexagonal


Plane group pg

pg


•

Crystal systems (7)

triclinic

monoclinic (2 or m)

hexagonal (6)

orthorhombic (three 2 and/or m)

rhombohedral (3)

tetragonal (4)

cubic (four 3 and three 2)


•

Bravais lattices (14)

Bravais, 1850

RHOMBOHEDRAL R

Point group 4mm


•

Relation with simple crystal structures of metals

a-Po

W

CN 6 or 8 PC 0.524 or 0.680


Mg

CN 12 PC 0.74

ABABAB…

HCP


Pt

CCP

ABCABC…

O T

CN 12 PC 0.74


Why cubic?

Cubic F lattice


Chapter 3. Space groups • a b c n

Glide mirror plane translation component

IN

OUT

½a ½b ½c half diagonal n


Screw axis nb: rotation 360°/n + translation b/n Name

order

translation

symbol


Space groups (230, listed in International Tables vol. A)  set of symmetry operations that describe the symmetry of a crystal and form a mathematical group  combination of 32 point groups and 14 Bravais lattices  space group symbol:

Lijk

Some examples…

L ijk

= Bravais lattice (P, C, I, F) = minimal symmetry elements for different directions


Note the "halfway marks" on each unit cell as visual guide!

a

P1 No.1 Asymmetric unit

c

( x, y , z )

+

Number of equivalent positions in unit cell or multiplicity Z

+

+

Gridline for a/2

Z=1;

(1  x, y, z ) Gridline for c/2

Height

Equivalent position (fractional coordinates)

( x, y , 1  z )

+

( x, y , z )

Non-centrosymmetric

(1  x, y, 1  z ) General position


a -

c

,

_

-

P1 No.2

,

+

+

Symbol: 1 = 1-bar = inversion center = origin! -

,

-

,

+

Z=2;

+

( x, y , z )

( x, y , z )

Centrosymmetric


Perspective view of a unit cell in P1bar with all centers of symmetry shown. The eight independent centers of symmetry (i. e., those NOT related by any symmetry operation) are shown in blue.


Monoclinic, combine P with point group 2

a c

P2 No.3 +

+

+

+

+

+

+

Z=2;

+

( x, y , z )

( x, y , z )

Non-centrosymmetric Special positions on 2-fold axes: (0, y, 0)

(0, y, ½)

(½, y, 0)

(½, y, ½)


Change 2-fold axis into 2-fold screw axis

a

P21 No.4 ½+

½+

c

+

½+

+

½+

+

Z=2;

( x, y , z )

+

( x, y  1 2 , z )

Non-centrosymmetric

Special positions on 2-fold screw axes?


Change P lattice into C lattice

a c

+

C 2 No.5 ½+

+

+

+

½+

+

Z=4;

½+

( x, y , z )

(1 2  x,1 2  y, z )

+

+

½+

+

( x, y , z )

(0, 0, 0) 

(1 2  x,1 2  y, z )

(1 2 ,1 2 , 0) 

Non-centrosymmetric

Only special positions on 2-fold axes


Next combinations for monoclinic system? P

C

2

P2, P21, C2

m

Pm, Pc, Cm, Cc

2/m

P2/m, P21/m, P2/c, P21/c, C2/m, C2/c

Crystal System

Bravais Lattices

Point Groups

Triclinic

P

1, -1

Monoclinic

P, C

2, m, 2/m

Orthorhombic

P, C, F, I

222, mm2, mmm

Trigonal

P, R

3, -3, 32, 3m, -3m

Hexagonal

P

6, -6, 6/m, 622, 6mm, -62m, 6/mmm

Tetragonal

P, I

4, -4, 4/m, 422, 4mm, -42m, 4/mmm

Cubic

P, F, I

23, m-3, 432, -43m, m-3m


Do you know what this group is called?

b a

Z=4;

+

-

+

-

-

+

-

+

+

-

+

-

-

+

-

+

( x, y, z ) ( x, y, z ) ( x, y ,z ) ( x, y ,z )


Bovine Pancreatic Trypsin Inhibitor (BPTI)

Four molecules in unit cell related by 2-fold screw axes Space group P212121


P212121 No.19

b a

½-

14

14 +

+

½+

½+ 14

14

½-

14

14 +

Z=4;

( x, y, z ) ( x , 1 2  y ,1 2  z )

+

( 1 2  x , y ,1 2  z ) ( 1 2  x ,1 2  y , z )


All possible combinations of symmetry operations result in 230 space groups describing the symmetry of a crystal. Some conventions for nomenclature: • Priority for parallel symmetry elements: m > a > b > c > n > rotation axis > screw axis • Origin on inversion centre (centrosymmetric) or point with highest symmetry • Nomenclature of centering and glide planes depends on choice axes!

C

A

a

c


For biomolecules only space groups with only rotation axes or screw axes are possible! As a consequence only 65 space groups are possible (known as chiral or Sohncke space groups). Space groups statistics for:

biomolecules P212121 P21 C2

organic molecules 24% 13% 9%


Part 2. Diffraction

Chapter 4. X-ray sources •

X-ray tube

0.1 Å <  < 100 Å


Continuous radiation V.e = h min (Å) = 12400/V (V)

Characteristic radiation (Sieghbahn notation) Intensity ratios K  K  K      

M

L K

K

K

K

K


â&#x20AC;˘

Rotating anode


â&#x20AC;˘

Microfocus source solid anode

air cooled ! high intensity (4.5 x rot. anode)

microsource 50kV, 0.6mA, 30W rot. anode 50kV, 80mA, 4kW sealed tube 50kV, 40mA, 2kW


• Metal jet anode (Ga) K = 1.34 Å high intensity (140 x rot. anode) Metaljet

70kV, max 4.3mA, 200W


Synchrotron radiation (SR)

electrons emitted by an electron gun accelerated in a linear accelerator (linac) - 200 MeV transferred into a circular accelerator (booster) - 6 GeV injected into storage ring (844m circumference) where they circulate at constant energy for many hours at 350.000 revolutions per second  the life time depends on quality of vacuum (between 10-9 and 10-12 mbar)  the storage ring includes both straight and curved sections    


 Bending magnets: electrons are deflected from their straight path by several degrees. This change in direction causes them to emit SR

 Undulator:

• • • •

complex array of small magnets wavy trajectory beams of radiation overlap and interfere with each other much more intense beam!


Advantagesâ&#x20AC;Ś High brightness (extremely intense, highly collimated) Wide energy spectrum (choose ď Ź with monochromator) Highly polarised Very short pulses (less than a nanosecond) Brillance = Number of photons per seconde per mm2 (source area) per mrad2 (opening angle)


Absorption Absorption by an element – absorption edge

M

L K

K

K

K

K


Kβ-filter

Reduction K:  Reduction K: 

Monochromator is used nowadays!


Absorption coefficient ()

I0

I

x How to calculate ? Use mass absorption coefficient m (see Internat. Tables) ( is density)

(gi is mass fraction) Example Calculate how much CuK radiation is absorbed by a thin plate of 0.1 mm NaCl. ( is 2.2 g/cm-3 for NaCl, m is 30.1 cm2/g for Na and 106 cm2/g for Cl)


Chapter 5. X-ray diffraction •

Scattering by electrons (Rayleigh scattering)

Waves as vectors A cos[2π(νt-x/λ)]

Summation of two waves by vector addition


Scattering by a row of atoms

When in phase? x = y or ď ąin = ď ąout

Scattering by an array of atoms When in phase? x = y Bragg planes as mirrors! Use Miller indices


â&#x20AC;˘

Miller planes

Families of parallel planes with specific orientations and periodic spacings can be drawn through a crystal. Miller index = 1/intercept

(110)

(011)

(120)

d-value

dhkl ď Ą-Fe2O3


OP = dhkl First representative of (hkl) intersects: X- axis at a/h Y-axis at b/k Z-axis at c/l For orthogonal systems:

1 = _ h2 + __ 2 dhkl a2

k_2 + l_2 b2 c2

Note (hkl) versus [hkl] !


Bragg’s law Laue – Bragg – 1912 Constructive interference when 2dsin=n Consider only first order diffraction (n=1).  

Intensity Ihkl depends on electron density (xyz) in corresponding Miller plane hkl


d-value ~ ď ą-1

Magenta atom scatters out of phase for black plane


Reciprocal space (*)

Peter Ewald (1888-1985)

Direct space – family of planes  Reciprocal space – one point


Mathematical definition:

Triclinic system


Ewald construction

1 3

= visualisation of Bragg’s law in reciprocal space sin = OP/OB = (1/d)/(2/)

4 2


Rotate crystal to obtain as much reflections as possible…

Direction given by Bragg’s law! Intensity?


Chapter 6. Intensity of diffracted beams â&#x20AC;˘

Scattering by an electron Thomson

â&#x20AC;˘

Scattering by an atom atomic form factor or scattering factor fj (scattering amplitude of an atom expressed in terms of free electrons)

International Tables Crystallography, volume C


Scattering by a unit cell electron density in unit cell is  sum of spherical atoms structure factor F is scattering amplitude of the unit cell expressed in terms of scattering factors fj of free atoms at fractional positions xj, yj, zj) in two dimensions…

OC = x.a OA = a/h

OD = y.b OB = b/k

difference in path length between O and C?

difference in path length between O and P? difference in phase between O and P?


in three dimensionsâ&#x20AC;Ś

Structure factor expressed as sum of individual waves:


2

Intensity of diffracted beam Ihkl = Fhkl

Centrosymmetric case: +Fhkl or –Fhkl For each atom at position xj,yj,zj, also an atom at –xj,-yj,-zj

B = 0, structure factor is real number, phase is 0 or 180° (or sign +1 or -1)


Scattering by a crystal for M unit cells in crystal the total intensity becomes:

K is scale factor A is absorption correction Lp is Lorentz-polarisation factor also a correction for the temperature is necessary!

T = exp – (B sin2 / 2) B is temperature factor (Å-2) isotropic or anisotropic


Data collection

  1Å

Relation between position and intensity of reflections and the electron density in the unit cell Data collection = measure intensities of all possible reflections


Crystal mounted in capillary or loop on goniometer


Crystal Mounting Typically a single crystal, once chosen, will be mounted on a glass fiber, a nylon loop or a MiTeGen mount. Usually the crystal will be attached to the mount with grease, super glue or 5-minute epoxy. We normally cool the crystal to a VERY COLD temperature to immobilize the crystal (100 K).

Glass fiber + epoxy

Nylon loop

MiTeGen mounts


Crystal in drop with a loop mount

Mounted Crystal

Captured crystal in drop with a loop mount


Intensity Ihkl depends on electron density ď ˛(xyz) in corresponding Miller plane hkl


Detectors

•CCD detector

•Imaging plate

BaFBr:Eu2+ (150m) in organic binder Eu2+

X-ray

Eu3+ + e-

e- trapped in Br- vacancies Laser beam scanning: luminiscence Erasing by flash light


Cryocooling

N2 100K

â&#x20AC;˘Procedure

Nylon loop cryoprotectant

â&#x20AC;˘Advantages - reduces radiation damage - improves sometimes resolution - reduces atomic motion

dry air


Friedel’s law -- I(hkl) = I(hkl)

diffraction pattern always centrosymmetric symmetry diffraction pattern given by Laue group (= centrosymmetric point groups)


Chapter 7. Fourier synthesis General expression of structure factor:

Fourier representation of electron density:

Summation is 0, except for h’=-h, k’=-k, l’=-l


Fourier representation of electron density:

Compare with structure factorâ&#x20AC;Ś

Relation? Fourier transform of each other! Analogy with microscope


fj

Key formulas Ihkl ~ F2hkl

sin/

Fhkl = j fj exp 2i(hxj+kyj+lzj) (sum over atoms in unit cell, fj scattering factor)

Fhkl = |Fhkl| exp ihkl (structure factor is complex number)

(xyz) = 1/V    Fhkl exp -2i(hx+ky+lz) h

k

l

(sum over all reflections, Fourier synthesis)

Phase problem: calculation of electron density needs both amplitude and phase of the structure factor To know the answer, you need the answer !

FT


Fourier transform lattice

motif

crystal


Phase problem can not be neglectedâ&#x20AC;Ś

amplitude duck + phase cat

colour = phase

amplitude cat + phase duck


Fourier synthesis

phase is position of wave


One-dimensional example: line Fourier of PbI2 Hexagonal, centrosymmetric structure c = 6.977 Ă&#x2026;


Systematic absences: Consequence of translation symmetry!


Chapter 8. Solving the phase problem (a) Patterson function

(1934)

amplitude is square of structure factor, no phases needed peaks at locations corresponding to vectors between atoms intensity ~ product of number of electrons of both atoms Two-dimensional example:

centrosymmetric N atoms

(N2-N)/2 intramolecular peaks


Patterson map contains image of original molecule!

Check which pair of peaks, along with an atom at the origin, would reproduce the Patterson map? Too complex for large number of atoms!

Use unit cell symmetry! Suppose 21 axis ď źď ź c. For each atom at (x,y,z) also one at (-x,-y,z+1/2). Difference vector (u,v,w) becomes: (2x,2y,1/2) Gives coordinates of heavy atom!

Harker plane


Symmetry of the Patterson function

• centrosymmetric, translation symmetry lost • 24 Patterson groups


Equivalent positions of P21/c from International Tables


Vectors between equivalent positions also the peak positions in Patterson Maps

x y z

-x –y –z

-x ½+y ½-z

x ½-y ½+z

x y z

0 0 0

-2x -2y -2z

-2x ½ ½ -2z

0 ½-2y ½

-x -y -z

2x 2y 2z

0 0 0

0 ½+2y ½

2x ½ ½+2z

-x ½+y ½-z

2x ½ -½+2z

0 -½-2y ½

0 0 0

2x -2y 2z

x ½-y ½+z

0 -½+2y ½

-2x -½ -½-2z

-2x 2y -2z

0 0 0

Same colors indicate overlap positions, but four black coordinates are just four different positions


(b) The heavy atom method (isomorphous replacement) •

add a strong diffractor (so-called heavy atom) to specific sites in unit cell causes slight perturbations in diffraction pattern

Native

Heavy-atom derivative

Underlined pairs show reversed relative intensities!


• prepare after the normal data collection one or more heavy-atom derivatives with e.g. AgNO3, HgCl2, PtCl42-, AuCl42-

• protein crystals are soaked in solutions of heavy ions • collect diffraction patterns • measurable changes in at least a modest number of intensities • derivative crystal must be isomorphous and diffract to a reasonably high resolution • locate heavy atom (x,y,z)H by Patterson methods and calculate FH

FPH = FH + FP Known: |FPH|, |FP|, FH


FPH = FH + FP or FP = FPH - FH • Draw vector -FH • Draw circle with radius |FPH| centered on head vector -FH • FP lies somewhere on this circle

• Add circle with radius |FP| • Two points of intersection or two possible phases


â&#x20AC;˘ Repeat this for second (or moreâ&#x20AC;Ś ) heavy-atom derivative (should bind at a different site, otherwhise same phase information, as the phase depends only on the atom location and not its identity) â&#x20AC;˘ One should agree better with one of the two solutions from the first derivative


â&#x20AC;˘ In general three or more heavy-atom derivatives are necessary to produce enough phase estimates (MIR, multiple isomorphous replacement) â&#x20AC;˘ Sometimes large uncertainty in phase! Use phase probabilities as measure of uncertainty of an individual phase.

â&#x20AC;˘ Correct solution gives interpretable electron density map!


(c) Direct methods •

Nobel Prize for Chemistry 1985: J. Karle, H. Hauptman

Direct = derive the phases of Fhkl by mathematical means using

Basic assumptions:

only intensity information

 (xyz)  0  atomicity  random distribution of equal atoms


• F(hk) = 2  fj cos 2 (hxj + kyj) j

(01)

(summation over N/2 atoms)

centro, 2D Centric case s(H) x s(K) x s(H+K) = +1 and s(H+K) = s(-H-K)

(20)

s(H) x s(K) x s(-H-K) = +1

Sayre, Cochran, Zachariasen, 1952

Acentric case (H) + (K) + (-H-K) = 0 (21)

2-relation or triplet relation


9. Interpretation of electron density density maps Chapter 8. Interpretation of electron maps Calculate electron density map (= Fourier synthesis) â&#x20AC;&#x201C; contour â&#x20AC;&#x201C; grid


Tracing of CA backbone + positioning of side chains


Tracing of CA backbone + positioning of side chains


For tryptophan side chain…

Resolution (Å) of a crystal structure determination (minimum d-value corresponding to maximum 2-value) 1Å

Atoms

Residues, side chains

Sec. structure, molecules

From 0.5 to 6.0 Å resolution…


Chapter 10. Refinement techniques • iterative process of improving agreement between Fo and Fc

 =  whkl (|Fo|-|Fc|)2hkl • possible parameters for least-squares adjustment: - atomic positions (3) - occupancies (1) - temperature factors (1 or 6) • follow the refinement by monitoring the R-value of residual value:


• Locate water molecules, ions … from difference Fourier synthesis

(xyz) = 1/V h k l (Fo-Fc)hkl exp -2i(hx+ky+lz)

Criteria: peak height, distances, H-bonds, co-ordination, …


Fo-Fc map • is difference map

Δ(xyz) = 1/V    (Fo-Fc)hkl exp -2i(hx+ky+lz) h

k

l


â&#x20AC;˘ judging the model using structural parameters - R-value (check also resolution and sigma cut-off) - rms deviations for bond lengths (< 0.02Ă&#x2026;) and angles (< 4°) from ideal values - standard deviations ? - difference Fourier - resolution - thermal parameters


• deposition coordinates and structure factors (PDB, NDB, CSD, …) • Crystallographic Information File (CIF and mmCIF) loop_ _atom_site.group_PDB _atom_site.type_symbol _atom_site.label_atom_id _atom_site.label_comp_id _atom_site.label_asym_id _atom_site.label_seq_id _atom_site.label_alt_id _atom_site.Cartn_x _atom_site.Cartn_y _atom_site.Cartn_z _atom_site.occupancy _atom_site.B_iso_or_equiv _atom_site.footnote_id _atom_site.auth_seq_id _atom_site.id ATOM N N VAL A 11 . 25.369 30.691 11.795 1.00 17.93 . 11 1 ATOM C CA VAL A 11 . 25.970 31.965 12.332 1.00 17.75 . 11 2 ATOM C C VAL A 11 . 25.569 32.010 13.808 1.00 17.83 . 11 3 ATOM O O VAL A 11 . 24.735 31.190 14.167 1.00 17.53 . 11 4 ATOM C CB VAL A 11 . 25.379 33.146 11.540 1.00 17.66 . 11 5 ATOM C CG1 VAL A 11 . 25.584 33.034 10.030 1.00 18.86 . 11 6 ATOM C CG2 VAL A 11 . 23.933 33.309 11.872 1.00 17.12 . 11 7 ATOM N N THR A 12 . 26.095 32.930 14.590 1.00 18.97 4 12 8 ATOM C CA THR A 12 . 25.734 32.995 16.032 1.00 19.80 4 12 9 ATOM C C THR A 12 . 24.695 34.106 16.113 1.00 20.92 4 12 10 ATOM O O THR A 12 . 24.869 35.118 15.421 1.00 21.84 4 12 11 …


Cambridge Crystallographic Data Centre CCDC • established in 1965 • first numerical scientific database • now non-profit organisation (1989) • distributes CSD System to more than 850 universities and 120 companies • related software and research  new building since 1992

The Cambridge Structural Database (CSD) • carbon-containing molecules : organics, organometallics, metal complexes • peptides up to 24 residues (rest in PDB) • mono-, di- and trinucleotides (rest in NDB) • x-ray and neutron diffraction Also : ICSD and CRYSTMET !


Cambridge Structural Database Worldwide repository of validated small-molecule crystal structures Lamotrigine Acta Cryst., Sect.C:Cryst Struct. Commun. (2009), 65, o460 Refcode: EFEMUX01

CSD Growth 1970-2010

Dec 09 – 500,000th structure milestone reached Release 2013 – 631,626 structures


Database plus Access Software Acta Cryst., B58, 380-388 & 389-397, 2002

*free downloads http://www.ccdc.cam.ac.uk/


Organisation of the CSD


Searching the CSD - ConQuest

CSD-5


O

Demonstration Olex2 1688 reflections measured Space group P212121 Start with S position

CH3 S CH3 O


M.C. Escher (1898-1972)


 

 

p1


p2


pg


pmm


pgg


p4g


p3


p31m 1 ď &#x17E; korte diagonaal m ď &#x17E; lange diagonaal


p3m1 m ď &#x17E; korte diagonaal 1 ď &#x17E; lange diagonaal


p4 geen 3-tallige pseudocel


pg


b

a


b

a


b

a


b

a


b

a


b

a


Omringing Pd: verwrongen vierkant

Omringing S: verwrongen tetraheder

Findit


(*) (*)

(*)


Homework 1

Illustrate the difference between the symmetry elements 61 and by completing their symmetry figure below.

Hints: - remember to rotate always counterclockwise - don't forget to add a comma for inversion


Homework 2

Determine the point group (Schoenfliess nomenclature) of the following molecules.

1. CH4, CH3Cl, CH2Cl2, CHCl3, CCl4 2. dichlorobenzene (three isomers) 3. C2H2 and C2HF 4. ethane (eclipsed)

5. ethane (staggered)

6. CH2=C=CH2 (allene)

Hints: - use flow chart from lecture 2


Homework 3

Draw the symmetry elements for the drawings given. Add the unit cell, determine the plane group and draw the asymmetric unit.


Homework 4

1. Complete the symmetry figure by adding the equivalent positions.


2. Complete the symmetry figure by adding the equivalent positions. Give the name of this space group, the coordinates of the equivalent positions, the multiplicity and the asymmetric unit. (a)

(b)


(c)


Homework 5

The Green Fluorescent Protein crystallises in the space group P41212 with unit cell a = b = 89.23, c = 119.77 Å, α = β =  = 90°. Calculate the diffraction angle 2 for reflection (10,0,0) when Cu Kα radiation ( = 1.54 Å) is used for the data collection.


Homework 6 (a) Sodium chloride NaCl crystallises in a cubic unit cell with a = 5.63 Ă&#x2026; and atoms at the following positions:

Calculate the structure factor for the reflections (111) and (210) using the formulas:

Use the symbols fNa and fCl for the scattering factors of Na and Cl.


(b) Cesium iodide CsI crystallises in a cubic unit cell with atoms at the following positions: Cs

0, 0, 0

I

1/2, 1/2, 1/2

Predict the intensities of the reflections 0,0,1 and 0,0,2 as a function of the scattering factors of both atoms.


Homework 7 Crystal data of euphenyl iodoacetate: C32H53IO2, space group P21, Z=2 a = 7;26

b = 11.55

c = 19.22 Å

α = 90.0

β = 94.07

 = 90. 0°

Explain the Harker section P(u, 1/2, w).


Determine x and z of the iodine atoms in the unit cell.

Determine the sign of the following reflections:

The scattering factors are as follows:

Which phases are correct?


Tinh thể học (Crystallography) - QNU, MChem program