Crystal dataFormula sum K0.5 N0.5 O1.5

Formula weight 50.55Crystal system orthorhombicSpace group P n m a (no. 62)Unit cell dimensions a = 6.4360

b = 5.4300 c = 9.1920

Cell volume 321.24 Å3

Z

Crystal dataFormula sum Cs3 Cl OFormula weight 450.17Crystal system orthorhombicSpace group P n m a (no. 62)Unit cell dimensions a = 9.4430 b = 4.4520 c = 16.3200

Cell volume 686.10Å3

Z

Atomic coordinates

Atom Wyck. Occ. x y zK1 4c 0.5 0.25510 1/4 0.41640N1 4c 0.5 0.41560 1/4 0.75510O1 4c 0.5 0.40980 1/4 0.89070O2 8d 0.41290 0.45010 0.68640

TITL *Niter-K(NO3)-[Pnma]-Holden J R, Dickinson C W CELL 1.54180 6.436 5.430 9.192 90.0 90.0 90.0SYMM P n m a (62) UNIT 8 60 24 4SFAC K N OK1 1 0.25510 0.25000 0.41640 10.5000 = 0.03110 0.23800 0.02480 0.00000 0.00110 0.00000N1 2 0.41560 0.25000 0.75510 10.5000 = 0.01920 0.02500 0.02970 0.00000 0.00160 0.00000O1 3 0.40980 0.25000 0.89070 10.5000 = 0.05000 0.04070 0.02720 0.00000 -0.0080 0.00000O2 3 0.41290 0.45010 0.68640 11.0000 = 0.04930 0.02680 0.03920 -0.0038 0.00520 0.00640END

p. 58

Cell unchanged but with lower crystal class

Cell changed with a different Bravais Lattice

Cell changed with the same symmetry

 

Space groups (and enantiomorphous pairs) that are uniquely determinable from the symmetry of the diffraction pattern and from systematic absences are shown in bold-type.

Point groups w/o inversion centers or mirror planes are emphasized by boxes.

Space groups (and enantiomorphous pairs) that are uniquely determinable from the symmetry of the diffraction pattern and from systematic absences are shown in bold-type.

Point groups w/o inversion centers or mirror planes are emphasized by boxes.

*

**

* *

=60o*

**

* *

a*a*

A complete data set covers all 8 octants of r.l. points.

Equivalent planes(hk) {hk}[hk] <hk>

p.61

(hk) (hk) (hk) (hk) (hk) (hk) (hk) (hk)

- - - - - -- - -- - - - - -- - -

- - -

Hexagonal Axes vs. Rhombohedral Axes

Two ways to relate rhombohedral indices to hexagonal indices,the obverse and reverse relationship.

Obverse hexagonal axes:

a1 = r2 – r3

a2 = r3 – r1

c = r1 + r2 + r3

Reverse hexagonal axes:

a1 = r3 – r2

a2 = r1 – r3

c = r1 + r2 + r3

The hexagonal cell : a1, a2, c

Rhombohedral cell: r1, r2 , r3

Hexagonal Axes vs. Rhombohedral Axes

Obverse hexagonal axes:

a1 = r2 – r3

a2 = r3 – r1

c = r1 + r2 + r3

h = n – p k = -m + p = m + n + p

-h + k + = 3p

Reverse hexagonal axes:

a1 = r3 – r2

a2 = r1 – r3

c = r1 + r2 + r3

h - k + = 3p

The hexagonal cell : a1, a2, c and indices (h k .)

Rhombohedral cell: r1, r2 , r3 and indices (m n p) Obverse H vs. R:

Parameters in intensity data collectionChoice of wavelengthChoice of wavelength

Resolution: 2/ |hmax| = dhk-1 dhk ½

Resolution: 2/ |hmax| = dhk-1 dhk ½

Data Processing

11. Data ReductionData Reduction Preliminary manipulation of intensities—their conversion to a correct, more usable form

Decay correctionDecay correction

Cause 1: due to unstable crystal – decomposing or slippingCause 2: due to unstable X-ray source – instrument misalignment or instability in tube voltagePoint-detector case: Corrections can be made based on a certain standard reflectionsPoint-detector case: Corrections can be made based on a certain standard reflections

Typical behavior of the relative intensities of threereflections of a crystal monitored with a diffractometer at different times after exposure begins

LpLp correction correction

“L” stands for Lorenz factor:

Cause : the r.l. point have a non-negligible volume so that it will have different angular speed when passing through the Ewald sphere (i.e. a higher intensitywhen in diffracting position for a longer time).

L time = / (|r*|cos)|r*| = 2sin / L = 1/sin2

(the simplest possible form)

angular velocitylinear velociy at

point p

Different forms of the L factor may be given for different experimental arrangements.

“p“ stands for polarization factor

Case 1: no monochromator : p = (1+ cos22θ)/2

Case 2: with monochromator : The beam is further polarized when monochrator(s) is (are) used.

(a) when s0, s1, and s2 are co-planar,

p = (1+ cos22θcos22θM)/ (1+ cos22θM)

(b) when s0, s1, and s2 are not co-planar,

p = (cos22θ + cos22θM)/ (1+ cos22θM)

The diffracted X-ray beam is polarized relative to the incident beam.

Lp factor: (1+ cos22θ)/2sin2

and Irel = Iobs/Lp ; I

rel = I/Lp

Diffraction circle

E+ E//

E+ E//cos2θ

Io E

2+ E// 2

unpolarizedrays

I E2+ (E// cos2θ)

2

p = I/Io

s0

s1

AbsorptionAbsorption correction correction

The absorption effect depends on the crystal’s shape, size, and density. This effect ismuch more severe atlow 2θ angles.

I = Ioe-t

ABSORPTION Correction (i) applied before refinement (in the data reduction stage)(ii) applied during refinement by an input of the precise description of the crystal shape

The path lengths of the beams reflected from the two small elementsof the crystal, A and B, are different for different reflections

(1) Relative transmission factor plotted as a function of angle for a reflection chosen with a value close to 90o. The rotation curve can be used to make an absorption correction.

(2) Empirical absorption correction may also be applied based on the intensity variations in symmetry-equivalent reflections.

Tra

nsm

issi

on f

acto

r

rotation

22. Data AveragingData Averaging

The intensity data are averaged over all symmetry-equivalent reflections.

Friedel’s Law indicating that Ihk Ihk ( Ihk Ihk )

Eleven Laue Symmetry Groups

1 2/m mmm 4/m 4/mmm 3 3m 6/m 6/mmm m3 m3m– – – – –

Iiave = N Ii /N ( i = 1 to n, n = no. symmetry equivalents )

Rint = (Ii - Iiave)/ Ii

HW: List the intensities and their esd’s for all symmetry equivalents of the reflections (3 2 6), (0 2 6), (9 0 6) and (3 3 0) in Xtal01. Calculate their average I and sigma I. What is the Rint just for this group of reflections?

The value of I is used to decide whichdatamis a real signal or just a noise.

Pattern DecompostionPattern DecompostionExtract Bragg-peak intensity from powder pattern

Electron-density function

(x,y,z) {Fhkl}

FT

FT

phase anglehkl = tan-1(B/A)

Complex form Fhkl= Ahkl + iBhkl = |Fhkl|exp(ihkl)

(1)  The phase problemTo solve a crystal structure is to solve the phase problem.

Why? Simply because the “phase” of the diffracted wave is missing in diffraction intensity measurements, i. e., only the amplitudes of the diffracted waves are measured in experiments:

Ihkl FhklF*hkl → Fhkl

(x,y,z) Fhklexp(ihkl)

FT

FT

Chapter 8 Structure Solution

How to Solve the Phase Problem?

1. Patterson manipulation methods

Ihkl → P(u,v,w) → (xH,yH,zH) → {hkl}FT

Bragg intensity Patterson Function some located atoms

{Fhkl} calcal

initially derived phases

Heavy-Atom methods; Superposition methods

Heavy-Atom methods;

To find the position of a heavy atom, one must utilize the “Harker vectors”, which correspond to vectors formed between t symmetry-related atoms. For example, in the space group P21/c, there are three kinds of Harker vectors, namely, (u,v,w), (u,½,w), and (0,v, ½).

The two chlorine atoms are at (0.113, 0.912, 0.080) and (0.295, 0.731, 0.383). The first 23 strongest Patterson peaks are shown to the right: Harker lines of (0,v, ½) type are: peaks #2, #8, #16 Harker planes of (u,½,w) type are: #3, #10, #11, #15It is clear to see that from peaks #2 and #3, the atomic coordinate of the first chlorine atom, Cl1, could be derived; and from peaks #8 and #10, the coordinates of the second chlorine atom , Cl2, could be obtained.

2. Direct methods

(x,y,z) Fhklexp(ihkl)

FT

FT

obs cal

Crystal Structure Determination and Refinement Using the

Bruker AXS SMART APEX System

Flowchart for Method

Interpret the results

Complete and refine the structure

Solve the structure

Data reduction

Measure intensity data

Evaluate crystal quality; obtain unit cell geometryand preliminary symmetry information

Select, mount, and optically align a suitable crystal

Adapted from William Clegg

“Crystal Structure Determination”

Oxford 1998.

Select and Mount the Crystal

• Use microscope

• Size: ~0.4 (±0.2) mm

• Transparent, faces, looks single

• Epoxy, caulk, oil, grease to affix

• Glass fiber, nylon loop, capillary

Goniometer Head

Goniometer

Goniometer Assembly

project database

default settings

detector calibration

SMART

setup

sample screening

data collection

ASTRO

data collection strategy

SAINTPLUS

new project

change parameters

SAINT: integrate

SADABS: scale & empirical absorption correction

SHELXTL

new project

XPREP: space group determination

XS: structure solution

XL: least squares refinement

XCIF: tables, reports

                       

                            

George M. Sheldrick Professor, Director of Institute and part-time programming technician

1960-1966: student at Jesus College and Cambridge University, PhD (1966)    with Prof. E.A.V. Ebsworth entitled "NMR Studies of Inorganic Hydrides"1966-1978: University Demonstrator and then Lecturer at Cambridge University; Fellow of Jesus College, CambridgeMeldola Medal (1970),  Corday-Morgan Medal (1978)1978-now: Professor of Structural Chemistry at the University of GoettingenRoyal Society of Chemistry Award for Structural Chemistry (1981)Leibniz Prize of the Deutsche Forschungsgemeinschaft  (1989)Member of the Akademie der Wissenschaften zu Goettingen (1989)Patterson Prize of the American Crystallographic Association (1993) Author of more than 700 scientific papers and of a program called SHELX Interested in methods of solving and refining crystal structures (both small molecules and proteins) and in structural chemistry

email:  gsheldr@shelx.uni-ac.gwdg.defax:  +49-551-392582

(1) Concept of the least-squares refinements

Mathematical basis of Least Squares methodMathematical basis of Least Squares method

• A series of unknowns: X1, X2, …., Xm

• A series of observations: f1, f2, …., fn

a11X1 + a12X2 + …+ a1m Xm = f1

• the coefficients a’s are knowna’s are known and (i) more equations than unknowns, i. e. n > m, (ii) the observations are not perfect (iii) these n equations are not fully consistent Need “Least squares” method !

A: Linear case:A: Linear case:

aa1111XX11 + a + a1122XX22 + …+ a + …+ a11mm X Xmm = f = f11

aa2211XX11 + a + a2222XX22 + …+ a + …+ a22mm X Xmm = f = f22

… …

aann11XX11 + a + ann22XX22 + …+ a + …+ annmm X Xmm = f = fnn

{a{aijij} are known} are known and n > mand n > m

The error:The error:ee11 = = aa1111XX11 + a + a1122XX22 + …+ a + …+ a11mm X Xmm– f– f11

ee22 = = a a2211XX11 + a + a2222XX22 + …+ a + …+ a22mm X Xm m – f– f22

… …

eenn = = a ann11XX11 + a + ann22XX22 + …+ a + …+ annmm X Xm m – f– fnn

A A XX = = FF

n x mn x m m x1m x1

n x1n x1

n equationsn equations for n for nobservations and toobservations and tosolve solve m unknownsm unknowns

We want to get XWe want to get Xii’s when ’s when SS = e = e1122 + e + e22

22 + … + e + … + enn22 is minimum is minimum

S is the sum of squares of what you calculate minus what you observedS is the sum of squares of what you calculate minus what you observed

i. e. i. e. min (S) = min (min (S) = min (i i wwi i eeii2 2 ))weighting factorweighting factor

Minimizing “S”

(a) substitute equations for S

(b) find the minimum

(c) then we obtain the normal equation

the normal equation

(d) Solve the m simultaneous eqn for x, i.e. the estimate of

n

i

n

i

m

jjjijiii fxawewS

1 1 1

22 ][

0

jX

SFor all j = 1 , 2 , 3 , · · · · · · , m

n

kkkkij

n

kkkjki

m

j

wfaxwaa11

m

jijij cxb

1i = 1 , 2 , · · · · · · · , m

jx

* must calculate matrix B

B-1 x̂mm

or

WFAWAAX

WFAXWAA

WFXWA

TT

TT

1ˆ

ˆ

ˆ

B Normal eqns

the soln: WFABX T1ˆ

Question: How good are X’s?

*must estimate the “precision of the derived unknowns ( parameters )” Define : variance - covariance matrix

221

22221

11221

...........

............

............

nnn

n

n

V

(ij= ji)

Correlation coefficient:

)1(

21

22 ii

ji

ijij

common variance 1 AAV T 2

Common variance i

ii mnew 22

mn

ewBxor iiij

j 21

2

mnB

ewxx

mnB

ewx

ik

iiki

ij

iij

1,

1

2

22

Example: x1 = 2 x2 = 4 m = 3 x3 = 6 n = 5 2x1+3x2+x3 = 21 x1+2x2+x3 = 17

33.6

00.4

67.1

44

101

61

3,5,3

5,14,0

3,2,6

121

231

001

010

100

00111

01032

10021

17

21

6

4

2

;;

121

231

001

010

100

3

2

1

3

2

1

X

X

X

XFA

AA

F

x

x

x

XA

T

T

If accept x1,x2,x3 = 2 ,4 ,6 at first 22ie

The L.S yields 67.1167.033.0033.0 2222 ie

60.0,31.0,07.0

31.0,31.0,31.0

07.0,31.0,38.0

83.035

67.12

V

And the correlation function is

00.1,45.0,11.0

45.0,00.1,72.0

11.0,72.0,00.1

B: Non-linear case

let niXXXff mii ,1,,, 21

01011

1

1223

22

21

0 ,,, mmj

imi XX

X

fXX

X

fXXXXf

ji

j

iij

j

m

j j

ii

jj

m

j j

iii

XXfF

X

faAAXF

XX

ff

XXX

fff

;

;

1

0

1

0

if =

Single-crystal case: the Structure Factor

2sin2

1

0 Bjlzkyhxi

n

jj eefF jjj

Unknowns : (xj,yj,zj), Bj,·······etc

p. 115

The function to be minimized is

m

j j

cco

h

co

h

P

FFF

FFwR

1

2

p.117

The normal equation

co

h

c

hk

jj

h

c

j

c

hhjk

FFP

FwC

PX

P

F

P

FwB

L. S. Procedure for non-linear case :(i) guess Xj

o

(ii) form F: fi = fi - fio (fi

obs - fical)

(iii) calculate or approximate aij, i.e. fi/ Xi

(iv) set up normal equations and solve for Xj

(v) Xj‘ = Xjo + Xj

(vi) go back to (ii) unless Xj << (Xj), i. e.

convergence obtained when { Xj / (Xj)} << 0.05

Powder indexing: having series of powder lines knowning their Bragg angles at which the lines occur

hlbhkaclbkahdhkl

2*cos**2***(1sin2 222222

21

*)cos**2*cos** cklbca 2

1q When a=b , = = = 90°

22222 ** clakhq

Find a* c* for series of hkl powder lines at position q

We need 222 *2*

);(*2*

lcc

qkha

a

q

Powder case

Normal equation: calabs

iii qqwq

aqq

waa

qw

*

0

**0

*0

2

*0

,*

calabsiii qqw

c

qc

c

qwa

c

q

a

qw

*

0

**0

**0

*0

,

where 2*0

22*0

22

lckhaqcal

deficiences of the model, cal

deficiences in obs: experimental inaccuries (in |Fobs| ) errors in phase angles (in cal ; true ? ) and “termination- of -series” error (no. observed reflections)

mis-placed atoms, missing atoms, superfluous atoms, errors in atomic scattering model and temp. factors

rhFFv

eFFv

h

calobscalobs

h

rhical

h

obs

h

calobscalh

2cos2

1 2

“Fobs and Fcal should be on the the same scale”

*noncentrosymm case:

Fcal

Fobs F

c

truecalobs 2

1

*centrosymm case: signs are either right or wrong,

for good model obs ~ true

FourierFourier Synthesis --searching for atoms which are missing or mis-interpriated in the structure model

structure model

diffraciton data

{|Fcal|} {|Fobs|}

(x,y,z)

F.T.

all atoms found and refined

L.S. refinements

+

{cal}+

find missingatoms

detailed structure

. Structure refinement based on F--single-crystal case Refinement is a method of adjusting the parameters that define the propsoed (model) structure to obtain optimal agreement between the calculated data and observed data.

The agreement factor: R = ||Fobs|-|Fcal|| / |Fobs|

Is R everything? A lower R value may not indicate an acceptable or correct structure

• Using Least-squares methods to minimize the quantity ||Fobs|-|Fcal|| or w||Iobs|-|Ical||

• Structure parameters includes (excuting in sequence) (1) atom type (fj) (2) atomic coordinates (xj,yj,zj) (3) thermal parameters(Uij) -- from iso- to aniso-tropic (4) site-occupancy factor (5) secondary extinction, weighting, and others.

A random structure noncentrosymmetric R ~ 0.59centrosymmetric R~ 0.83

Fouriersynthesis

Structure Refinement —from model to detailed structure

structure model

(initial structure parameters: f, atomic coordinates, and fixed temperature factors)

refined {(xj,yj,zj)}

L. S. methods

find missing atoms

F.T. of {|Fobs|, cal}

complete structure

R1 = ||Fobs|-|Fcal|| / |Fobs|R2 = w||Iobs|-|Ical|| / |Iobs|

• Indexing the powder pattern to find if it belongs to a known-structure type• For a totally unknown structure type, generally a two-stage method is applied.

Stage 1. Non-structural profile-fitting method

step-scan data

Bragg reflections

Patten decompositon(without reference tostructure model)

How to obtain a structure model from powder data for Rietveld refinement?

step-scan data

detailed structure

Rietveld refinement(with reference tostructure model)

Stage 2. Structural solution from Bragg reflections

X-ray diffraction profiles are more complicated

Name Function

1. Gaussian

2. Lorentzian

3. Mod. 1 Lorentzian

4. Mod. 2 Lorentzian

5. Pseudo-voigt

6. Pearson VII

2BxeAH

2

1

'1'

xBHA

22

1

"1"

xBHA

5.12

1

'"1'"

xBHA

212

11

'1' BxeAHr

xBHAr

mDx211

H : full width at half maximum(function of tan)

Hx B 22

Definitions of R’s used in Rietveld Analysis

I

(i) Collection of a highly resolved powder pattern.

(ii) Indexing of the powder pattern and determine the space group of the unit cell.

(iii) Integration of reflections to make a list of Bragg reflections, i.e. {hkl , Ihkl}.

(iv) Structure solution using Patterson methods or Direct methods.

(v) Structure refinement. For diffraction patterns with a great number of overlapping reflections, the profile refinement (i. e. Rietveld method) is adequate. For high- resolution diffraction patterns, a realistic approach is to perform structure refinement based on structure factors determined from the resolved diffraction intensities.

Crystal structure analysis from powder data

a general procedure

thermal disorder

(a) (b)

An example of disordered structure static disorder

VIII_6d

VIII_6e

Corrections to be applied during refinement -- when lacking good agreement between |Fcal| and |Fobsl| in the final stages of refinement, it is necessary to inspect the sources of error in the measured intensity

PRIMARY EXTINCTION

ABSORPTION (i) applied before refinement (in the data reduction stage)(ii) applied during refinement by an input of the precise description of the crystal shape

Primary extinction is a weakening of intensity cuased by multiple reflection process as shown in the right. The doubly-reflected ray has a phase difference of relative to the primary beam, not only contributing to the reflected beam, but also causing a decrease in the intensity of the incident beam.

primary beam

reflected beam

doubly-reflected ray

latti

ce p

lane

s

--- an attenuation of both incident and reflected beams

Primary extinction will cause I ~ |F|n with n < 2Ideally perfect crystal: I~ |F| Ideally imperfect crystal: I~ |F|2

In a mosaic crystal, multiple reflections are less probable than in an ideally perfect crystal. Primary extinction is generally not considered.

SECONDARY EXTINCTION

-- the effect of shielding the inner lattice planes by reflection of a fraction of the primary intensity by the outer planes

Observed mainly in high intensity reflections of low sin/value, and increases with the size and perfectness of a crystal, i.e., |Fobs| << |Fcal| for reflections of low indices and high intensity.

ANOMALOUS SCATTERING

When the wavelength of the incident beam is close to the

wavelength k of the K-absorption edge for an atom of the

scattering material, i.e., k , the scattering process will

show an unusual behaviour caused by an anomalous phase-

shift of the scattered wave (anomalous dispersion). Under

this condition, the atomic scattering factor, f, is not a real

number but a complex quantity fA:

fA= f + f’ + if’‘

*The effect of anomalous dispersion increases with . The breakdown of Friedel’s law

--when anomalous scattering occurs

Fhkl = (f + f’ + if’‘)exp[2i(hx + ky + lz)]

Let A’ = G (f + f’) + A and B’ = H (f + f’) + B

Fhkl = (A’ - H f’‘) + i (B’ + G f’‘)

|Fhkl2| = (A’ - H f’‘) + i (B’ + G f’‘)

|F-h - k - l2| = (A’ + H f’‘) + i (B’ - G f’‘)

F

Ihkl I-h-k-l

Determination of the absolute configuration

Using the effect of anomalous dispersion

Assume the atom position vectors of

the left-hand (L) structure: rj(L)

the right-hand (R) structure: rj(R) (j = 1, …., N)

Fhkl (R) = F-h-k-l (L)

Since Friedel’s law does not hold, we get

Fhkl (R) Fhkl* (R) = F-h-k-l (L) F-h-k-l

* (L)

Fhkl (L) Fhkl* (L)

Even the relatively small dispersion effect of oxygen with CuK radiation may be sufficient to determine the absolute configuration