1

Characterization: Classification: Long range ordering periodicity unit cell

Symmetry 7 crystal systems 230 space groups

Structural information: Unit cell Miller indices (h, k, l) d spacing

Relative intensity atomic positions

2

X-ray CrystallographyIntroductionCrystal Diffraction

Diffraction Structure

Structure Amplitudes, Fhkl

Atomic scattering factors

Fourier transfer Fhkl (x, y, z)

Least squares refinement

Structure properties --- Distance / angles ; packing etc.

Structure data base

3

Data measurement: h, k, l, dhkl , nhkl , Ihkl ( F2

hkl )

Phase determination heavy atom method; direct method; multiple scattering

Fourier transformation: F2 (r); reciprocal real space

Least squares refinement: (r) xi, yi, zi, ui

Structural model: bond distances; bond angles; atomic thermal vibration etc.

Structural Analysis:

4

d

l

Bragg Diffraction

5

Structure Factor Calculation

90

180

270

360

450540

90

180

270

450540

90

180

270

360

450540

0 0 0

X1 f1cos X2 f2cosXtotal (f1f2)cos

X1 X2

Xtotal

f1 f2

P1 P2f1 f2 Resultant

90

180

450

90

180

270

450

90

180

360

450

0 0 0 X1 X2

Xtotal

540 540 540

360

X1 f1cos(1) X2 f2cos(2)

f1

f2

F

f1

P1

f2

P2

1

2

Resultant

Combination of Wave

Direction (h, k, l)

Amplitude Fhkl

Phase

Xtotal f1cos(1) f2cos(

2) cos (f1cos1 f2cos2)

sin (f1sin1 f2sin2) LetXtotal F cos( )

F cos cos F sin sin

f2sin2

f1sin1

A’f2sin2

f1sin1

Ff2

f1

1

2

B’

F A’ iB’ F ei

1 1 2 2

n

i ii=1

A'=f cos +f cos

= F cosα= f cosDirection (nhkl)

Amplitude fi

Phase i

F sinαB'= =tanα

A' F cosα

-1 B'α=tan

A'

1 1 2 2

n

i ii=1

B'=f sin +f sin

= F sinα= f sin

(A)2 (B’)2 F2cos2 F2sin2 F2

6

Structure Amplitude (Factor)

DetectorR

X1

rnS0

S

X2

1 n 0

2 n

1 2 n 0

X = r S

X = R - r S

X + X = R - r S - S

7

Pathlength difference bet. atom n at rn and the origin at 0

2

02

1exp 2

h nP n

e Ef i t R r

mc R

11

2

0 0

i vt x

E e

h

n n n n

ha kb lc

r x a y b z c

Electric field at rn :

Electric field at P (defection) ：

E0 : electric field amplitude of the incident beam at rn

c

nnnnh lzkyhxr

hnn rSSrRXX )( 021

8

2θ

0a

lcSS

kbSS

lkhhaSS

SSLet hkl

)(

)(

,,)(

//

0

0

0

0

integer

0

~S

S~

aS ~~0

aS ~~

9

D

F

E’

G

E

’

’

a

mth order

mth order

Zeroth order

Incident beam

10

Fh

44

if e

33

if e 22

if e

11

if e

If centrosymmetric and no anomalous scatter: Bh=0; α=0 orπ

Ah

Bh

hh

i

j

rij

calh

jhj

iBA

FiF

eF

ehfF

r

jh

sincos

)(

)(2

2

11

2 2

1

cos 2

sin 2

tan

h

h

ih h

h j j j jj

h j j j jj

hh

hh

h

F F e

A f hx ky lz

B f hx ky lz

F A B

B

A

h

12

Centric case with non-anomalous scatterers

hkl hk l

hkl hk l

I I

F F

0

0

hkl hk l

hkl hk l

hkl

A A

B B

B

or

Zpt. charge

atomic sphere (fixed atom at ri) T=0K

with thermal vibration T as function of sin( )

h h

h h

A A

B B

2sin( )

B : Thermal parameter

B = 0

B > 0

"'sin0 fifff ii

13

Centric case with anomalous scatterers

NA : No. of atom types in an asymmetric unit

NE : No. of symm. elements of the space group

Ri : sym. operatorh : (h, k, l)xj : (xj, yj, zj)

2/

2exp2

)2exp(

NE

iji

NA

ji

NE

iji

NA

jjh

xRhif

xRhifF

i jijjj

i jij jh

i jij j

i jijjjh

jjjj

xRhff

xRhfB

xRhf

xRhffA

fifff

2

2

2

2

0

0

0

sin

cos

sin

cos

'

"

"

'

"'if

then

when at (000)1

14

Considering nuclear thermal vibration

12

2

TTBiso

)( sin

)(exp jjjh rhifF 2thermal vibration electron density smearing

if isotropic spherically symmetric

As a point scatterer

Å2

factoretemperaturToffunctionaisT

offunctionaisfj

:sin

sin

2

2

invibrationofamplitudesquaremeanu

uBiso

2

228

15

Tanisotropic thermal vibration as an ellipsoid

3×3 matrix 6 elements uij

11 12 13

21 22 23

31 32 33

u u u

u u u

u u u

symmetric: u12= u21; u23 = u32; u13 = u31

based on a, b, c,-axis

16

1

2

3

0 00 00 0

u

uu

i ju diagonize

eigen value eigen function

u3

u1

u2

(three principle axes of the ellipsoid)

**12

2*211

122

11

**12

2*211

24

1exp

2exp

22exp

exp

bhkaBahB

hkh

bhkauahu

uT kijh

ii

aCi

~1i

i

aCi

~2

ii

aCi

~3

Thermal ellipsoid

17

In case of ab-plane mirror plane sym:

U11 , U22 , U33 ， U12 U13=U23=0

x → x

y → y

z → -z

U11 U22 U33 U12 U13 U23

U12=U12

U13 →-U13 ； U23 →-U23 U∴ 13=U23=0

1 0 0

0 1 0

0 0 1

Constraint in Thermal vibration

Cm ~

18

Systematic Absences

ijh xihfF 2expP21 for any atom i at x y z R1 x -x y+1/2 -z R2 x

cos 2 sin 2

cos 2 1 2 sin 2 1 2

h iF f hx ky lz i hx ky lz

hx k y lz i hx k y lz

02

10

100

010

001

0

0

0

100

010

001

21 ;; RR

19

G0k0 cos2ky cos2[ky (k/2)]

i {sin2ky sin2[ky (k/2)]}

cos2ky cos2kycosk sin2kysink

i sin2ky i sin2kycosk i cos2kysink

2(cos2ky i sin2ky)

0

when k 2n

when k 2n1

i.e. for 0k0 reflections

222

kkykyfG ih coscos

222

kkykyi sinsin

When 0 i.e. h 0 and I 0

Let hx + lz

20

Systematic Absences(space group extinction from translational sym. elements)

abscentn

hhR

presentnhhRIf

hRxiT

hxxh

xRih

xRxxihT

j

j

j

jhkl

j

j

jj

NE

jjhkl

2

12

2

2

2

2

1

1

1

1

)(

)(exp

exp

;exp

scalarsince

21

nlkkld

nll

nlhlhn

nllhc

nkk

absent

40

41

41

0

100

010

001

400

41

0

0

100

001

010

4

120

21

0

21

100

010

001

120

21

0

0

100

010

001

1200

0

21

0

100

010

001

2

1

1

/

/;

;

;

;

;

22

12

21

0

21

12

12

12

21

21

021

0

210

21

21

100

010

001

12

21

21

21

100

010

001

nlhhklB

nlk

nkh

nlh

hkl

F

nlkhhklI

CenteringLattice

;

;

23

nll

nkk

nhh

nknhhka

nhnllhc

nknlklc

nkkhklcenterC

conditionsccaCEx

200

200

200

220100

220010

220001

2

;

;

24

Difference in phase

2

X-rayBeam Atom

Atomic Scattering

25

Atomic Scattering

If the atom is a point charge (compared w.r.t the wavelength), it scatters as Z (atomic number)

Scattering amplitude amplitude scattered by atom Eatom

amplitude scattered by free e- Ee(factor)

ρ(r) ： electron density around nucleusdq = ρ(r) dV q: charge V : volume

e

atom

E

Ef

d

r

26

calculatedbecanfchosenisnfwaveonce

er

drkr

krrr

e

kwhere

drdrikrre

f

zfSSeiSSwhen

r

drdvrSSrSS

dvrSS

ie

rdf

e

dvf

r

""'""

)(

sin)(

sin

sincosexp)(

//..

cossin

sincos)(

exp)(

;

2

0

2

2

00

200

0

4

4

221

0

2

2

2

0 0

27

Int’l Table of X-ray CrystallographyVol CΨ mostly from HF p.477

sinθ/λ

fZ

2

2

0

3

21

1

10

0

sin

:

af

ea

Hfor

H

a

r

electroninnerofejectionf

forcebindingelectronfieldmagneticexternalwhenf

fifff

DispersionAnomalous

cibaparameters

ceaf

ii

bi

i

":

:

"'

,,

sinsin

0

0

24

12

0

49

For example

Analytical form : Vol C.p500.

28

f0 for atoms from Z = 1 to 90

29

coefficients of analytical form in atomic scattering factors

30

31

Gd

f”

f’

1705 1710 1715

30

20

10

0

10

20

30

(Å)

30

20

10

0

10

20

30

184 185

(Å)

Sm

f”

f’

Anomalous Scattering

30 20 10 00

10

20

30

Gd f”

f’ 30 20 10

10

20

Sm

f’

30

f”

00

(a) (b)

(c) (d)

Fig. :Anomalous scattering terms f’ and f” for ：(a) gadolinium near the L3 edge ； (b) samarium near the L3 edge

Fig. :Plot in the complex plane of f’ i f” ：(c) gadolinium near the L3 edge ； (d) samarium near the L3 edge

32

Atomic f’ and f”

33

Relationship between Fhkl & ρuvw

dvere

fdvere

df rii

ri

hh 22 11

Crystal scattering general form with continuous ρ(u,v,w)

Corresponds to the intensity of h,k,l refln.

reciprocal space

corresponds to the electron density at u,v,w position

direction space

dvlwkvhuiuvwF

dveuvwF

v

hkl

v

ui

hh

2exp

~~2~

uvwF FT

h~

lwkvhuilkhlkh eF

Vuvw 2

,,,,

1

Atomic scattering