fundamentals of fourier transformcloud.crm2.univ-lorraine.fr/pdf/antwerp2016/nespolo.pdf ·...

Massimo Nespolo, Université de Lorraine

Fundamentals of Fourier transform

International School on Fundamental Crystallography with applications to Electron Crystallography

June 27 - July 2, 2016 - University of Antwerp

Based on “Transform methods in crystallography”, by Jan C. J. Bart


Periodic functions

A Fourier series decomposes any periodic function into the sum of a (possibly infinite) set of simple oscillating functions, (sines and cosines or complex exponentials).

f(x) is a periodic function of a real variable x with period P if:

f(x+P) = f(x)


Fourier series

Let f(x) be a periodic function integrable on the interval (x0, x

0+P), where

P is the period. We can represent f in that interval by a series of sinusoidal functions:

( ) 0

1

2sin , 1

2

N

N n nn

A nxf x A N

P

p j=

æ ö= + + ³ç ÷è ø

å

fN(x) approximates f(x) [x

0, x

0+P], and the approximation improves as N →

∞. The infinite sum, f(x), is called the Fourier series representation of f(x).


Fourier series


Integral transforms

The integral transform of a function f(x) is defined as:

( ) ( ) ( ),b

a

F k K k x f x dx= òwhere K(k,x) is a function of both k and x and is called the kernel of the transform. k and x represent variables (or vectors of variables) in two dual spaces and the transform realizes the “tunnel” between the two spaces.

Depending of the kernel, we have different types of integral transforms.


Integral transforms

Laplace transform:

( ) ( )0

kxF k e f x dx¥

-= òFourier transform:

( ) ( )ikxF k e f x dx¥

±

-¥

= òHankel transform:

( ) ( ) ( )0

nF k J kx f x dx¥

= òJ

n: Bessel function of order n


Fourier transform

If f(x) is a function of x such that:

( ) ( ) ( )1

2ikxF k f x e dx f x

p

¥-

-¥

= = é ùë ûò T

( ) 2f x dx

¥

-¥ò

is finite, then:

( ) ( ) ( )11

2ikxf x F k e dk F k

p

¥-

-¥

= = é ùë ûò T


Fourier transform

( ) ( ) ( ) ( )1

2ikxF k f x e dx A k k

¥-

-¥

= =òp- iB

( ) ( ) ( )

( ) ( ) ( )

1cos

2

1sin

2

A k f x kx dx

B k f x kx dx

¥

-¥

¥

-¥

=

=

ò

ò

p

p

( ) ( ) ( ) ( )( ) ( ) ( ) ( )f x f x F k A k

f x f x F k iB k

= - Þ = ÎÂ

= - - Þ = - ÎÁ


Step function

A step function of a real variable x is a function which changes its value only on a discrete set of discontinuities. The function values at the discontinuities may or may not be defined.

https://upload.wikimedia.org/wikipedia/commons/a/ad/StepFunctionExample.png


Heaviside step function

The Heaviside step function H(x) (or unit step function) is a discontinuous function whose value is zero for negative argument and one for positive argument.

https://upload.wikimedia.org/wikipedia/commons/d/d9/Dirac_distribution_CDF.svg


Boxcar function

A boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, A.

f(x) = A·[H(x-a) – H(x-b)]

a x

1

b x

1

a xb

1

a xb

1

A


Fourier transform of the boxcar function

a xb

A

x

A

2pl

k

AlF(k) = Al·sin(plk)/plk

The widths of f(x) and F(k) are inversely proportional

Fraunhofer diffraction amplitude of a slit of width 2pl


Fraunhofer diffraction

If both the light source and the viewing plane are at infinity with respect to the diffracting aperture, the incident light is a plane wave, the phase of the light at each point in the aperture is the same.

Diffraction of light occurs when a beam of light is partly blocked by an obstacle and some of the light is scattered around the object; light and dark bands are often seen at the edge of the shadow.

W2 Ll

W: aperture size; L: distance from the aperture; l: wavelength

Concretely, Fraunhofer diffraction occurs for the “far field condition”:


Single-slit diffraction

slit of width equal to wavelength slit of width equal to five

times the wavelength

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity. A slit which is wider than a wavelength produces interference effects in the space downstream of the slit.


Dirac delta function

The Dirac delta function, or δ function, is a distribution on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line and can be seen as as the limit of the sequence of zero-centred normal (Gaussian) distributions.

Gaussian function ( ) ( )2expp

px pxd

p= - Dirac delta

0 0

0

for x

for x

¹ì= í¥ =î


Convolution

A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f. It is defined as the integral of the product of the two functions after one is reversed and shifted.

( ) ( ) ( ) ( ) ( ) ( )¥ ¥

-¥ -¥

= - = -ò òt t t t t t*f t g t f g t d f t g d

● the origin of the function f is set in every position of the function g● the value of f in each position is multiplied by the value of g at that point● the sum is taken over all possible points


Convolutions involving the delta function

The convolution of a function f(x) with a delta function d(x -x0) gives the

function f(x0).

f(x)d(x -x

0) f(x

0)


Fourier transform of convolutions

( ) ( ) ( )1

2ikxF k f x e dx f x

p

¥-

-¥

= = é ùë ûò T

( ) ( ) ( )1

2p

¥-

-¥

= = é ùë ûò TikxG k g x e dx g x

( ) ( ) ( ) ( )*f x g x f x g x= ×é ù é ù é ùë û ë û ë ûT T T


Tunnelling dual spaces

Space 1 (direct space) Space 2 (reciprocal space)

Summation: f(x)+g(x) Summation: T [f(x)]+T [g(x)]

Multiplication: f(x)·g(x)

Convolution: f(x)*g(x)

Convolution: T [f(x)]*T [g(x)]

Multiplication: T [f(x)]·T [g(x)]


Dirac comb

A Dirac comb (impulse train, sampling function) is an infinite series of Dirac delta functions spaced at intervals of T

( ) ( )d¥

=-¥= -åT k

C t t kT


Fourier transform of a Dirac delta

( ) ( )2 1pd d¥

-

-¥

= =é ùë û ò ikxx e x dxT


Fourier representation of electron density (1)

Two different ways of addressing the problem of diffraction by a crystal● the classical theoryclassical theory introduces first the lattice (periodic object) and

later the scattering matter● the Fourier transform theoryFourier transform theory follows the inverse route and first

considers the unit-cell content (non-periodic object) and then examines how the diffraction pattern is modified by forming a regular array of cells.


Fourier representation of electron density (2)

In the Fourier transform theory we consider a non-periodic electron distribution r(r). The discrete vector H, pointing to the reciprocal lattice nodes, is replaced by a continuous vector r* which may end at any point of the reciprocal space. The equations used in the classical theory are modified accordingly:

( ) ( ) ( ) *

*

* exp 2 *r p= - ×ò F i dVr

r

r r r r

( ) ( ) ( )* exp 2 *V

F i dV= ×òr

rr r r rr p

The Fourier expansion of a continuous periodic distribution r(r) (classical theory) is given by the Fourier series:

( ) ( ) ( )1exp 2r p= - ×åF i

V H

r H H r

( ) ( ) ( )1exp 2p

== ×å N

j jjF f iH H H r

H = h,k,l; r = x,y,z


Diffraction by non-periodic objects

Discontinuous medium

N point scatterers (d functions of weight aj) at positions r

j

( ) ( )1* exp 2 *

N

j jjF a ip

== ×år r r

Continuous medium

( ) ( ) ( )* exp 2 *F i dVr p= ×ò rr r r r

The wave scattered by a continuous medium can be represented as the Fourier transform of the scattering density r(r), for all coherent scattering processes (optical, X-rays, neutrons, electrons)

N.B. aj is not a function of r*


Fourier transform of the electron density in an atom

Vibrating atomVibrating atom

( ) ( ) ( )* exp 2 *atom

f i dVr p= ×ò rr r r r

r(r): electron density distribution of the atom at restf(r*): atomic scattering factor

● The time of a diffraction experiment is much longer than the time of the atomic vibrations

● Approximation: the vibration of an atom is independent from that of its neighbours

p(r1): probability that the centre of an atom be at position r

1

ra(r): electron density of the atom at equilibrium in r

rav(r): electron density of the atom vibrating around the position r



Scattering factor for the vibrating atom

( ) ( ) ( ) ( ) ( )1 1 1p * pva adr r r= - =òr r r r r r r

( ) ( ) ( ) ( ) ( ) ( )1 1* *p * *v va a a af f qr r- -é ù= = é ù = ×ë ûë ûr r r r r rT T

q(r*): atomic displacement factor (Debye-Waller factor)

fa(r): scattering factor of the atom at rest

fav(r): scattering factor of the vibrating atom



Isotropic (spherical) vibration

( ) ( )* exp * *q = -r r Br%

B = BI (I = identity matrix, B = constant)

Harmonic vibration

B: second rank tensor describing an ellipsoid

( ) ( ) ( )2

2

1* exp * * exp * exp

hkl

q B B Bd

æ ö= - = - = -ç ÷

è ør r Ir r%

l = 2dhkl

sinJhkl

( )2

2

sin* exp hklq B

æ öJ= -ç ÷lè ø

r


Effect of translation of Fourier transform

rT(r) = r(r-t) t = translation vector F

T(r*) = T -1[r

T(r)]

( ) ( ) ( )* exp 2 *TF i dVr p= - × =ò rr r t r r

( ) ( ) ( )exp 2 * exp 2 *i i dVp r p= × - × - =ò rr t r t r r t

( ) ( )* exp 2 *F ip= ×r r t

Translation of r(r) by a vector t in direct space is equivalent to modifying the Fourier transform by the phase factor exp(2pir*·t) in reciprocal space, without change in the modulus |F(r*)|, but the real and imaginary part of F(r*) are multiplied by cos(2pr*·t) and sin(2pr*·t) respectively. A description of a transform is thus origin dependent.

The Fourier transform of a centrosymmetric body is real only if the origin is taken at the centre of symmetry (origin choice 2 of the International Tables for Crystallography).


Fourier transform of a group or molecule

rj: vector specifying the position of the j-th atom with respect to an arbitrary

originfj(r*): Fourier transform of the j-th atom referred to its centre as origin

Fourier transform of a set of atoms

FT(r*) = F(r*) exp(2pir*·t)

( ) ( ) ( )1* * exp 2 *

N

j jjF f ip

== ×år r r r

fj is a function of r* (it was not the case for point scatterers)

For spherically symmetrical atoms fj(r*) = f

j(r*) and f

j varies as a

function of r* = |r*| = 2sinJ/l, due to its finite size (not point scatterer).


Fourier transform of a finite lattice

Set of delta functions with weight fj at lattice nodes

( ) ( ) ( ) ( ), ,L j j u v w

j

f f f uvw u v w= - = - + +å år r r r a b cd d

( ) ( )1* exp 2 *

N

j jjF a ip

== ×år r rPoint scatterers

( ) ( ), ,

* exp 2 *L u v wF i u v wp= é × + + ùë ûår r a b c

Fourier transform of a set of delta functions with weight f

j

Lattice (fj = 1)

( )* exp 2 *j jjF f ié ù= ×ë ûår r rp

lattice vectors: r(u,v,w) = ua+vb+wc


Fourier transform of a finite lattice

r*·a = h r*·b = k r*·c = l

( ) ( ) ( ) ( )* exp 2 * exp 2 * exp 2 *Lu v w

F iu iv iw= × × ×å å år r a r b r cp p p

( ) 31 2 sin *sin * sin **

sin * sin * sin *L

NN NF

p ×p × p ×= × ×

p × p × p ×c ra r b r

ra r b r c r

N1, N

2, N

3: number of equally spaced delta functions along a*, b*, c*

The maxima of FL(r*) correspond to the conditions:


Fourier transform of an infinite lattice

( ) ( ), ,

* exp 2 *L u v wF i u v w

¥

=-¥= é × + + ùë ûår r a b cp

r*·a = h r*·b = k

The discrete vector r*hkl

defines the reciprocal lattice.

( ) ( ) ( ) ( )* exp 2 * exp 2 * exp 2 *Lu v w

F iu iv iw¥ ¥ ¥

=-¥ =-¥ =-¥

= × × ×å å år r a r b r cp p p

( )1 2 3

31 2

, ,

sin *sin * sin ** lim

sin * sin * sin *LN N N

NN NF

®¥

p ×p × p ×= × ×

p × p × p ×c ra r b r

ra r b r c r

FL(r*) is non-zero only when:

r*·c = l

( ) ( ), ,

1* * *L hkl

h k l

FV

¥

=-¥

= d -år r r


Fourier transform of the content of the crystal

A crystal (periodic structure) is described as consisting of two functions:1. the real (direct) lattice function f

L(r), which is unity at each lattice node

and zero elsewhere:2. the electron density function r(r), describing the content of a unit cell.

The content of the crystal is the convolution of the periodic function fL(r) and

the finite function r(r)

fC(r) = f

L(r)*r(r)

The Fourier transform of the content of the crystal is the product of the Fourier transforms of the lattice and of the unit cell content

FC(r*) = F

L(r*)·F(r*)

( ) ( ) ( )*

1* exp 2 *

N

hkl j jjF f i

== ×år r r rp( ) ( )

, ,

1* * *L hkl

h k l

FV

¥

=-¥

= d -år r r


Fourier transform of the content of the crystal

FC(r*) = F

L(r*)·F(r*)

( ) ( ) ( )1* * exp 2 *

N

j jjF f i

== ×år r r rp( ) ( )

, ,

1* * *L hkl

h k l

FV

¥

=-¥

= d -år r r

( ) ( ) ( )

( ) ( )*

1

* * * *

* exp 2 *

hkl hkl

N

j hkl hkl jj

F dV F

f i=

d - = =

= p ×

òå

rr r r r

r r r


A note on domain structures

G

H

group of the parent phase

group of the daughter phase

translationengleiche subgroup (same lattice, lower point group)

klassengleiche subgroup (sublattice, same point group)

Twin domains (differing by orientation)

Antiphase domains(differing by position)

Visible in the diffraction pattern Invisible in the diffraction pattern (effect on the phase)

Visible by electron microscopy

( ) ( ) ( )* * exp 2 *TF F i= ×r r r tp

I(hkl) |F(hkl)|2




Fundamentals. Crystal patterns and crystal structures.Lattices, their symmetry and related basic concepts


Ideal crystal:

Perfect periodicity, no static (vacancies, dislocations, chemical heterogeneities, even the surface!) or dynamic (phonons) defects.

Real crystal:

A crystal whose structure differs from that of an ideal crystal for the presence of static or dynamic defects.

Perfect crystal:

A real crystal whose structure contains only equilibrium defects.

Imperfect crystal:A real crystal whose structure contains also non-equilibrium defects (dislocations, chemical heterogeneities...).

What follows describes the structure of an ideal crystals (something that does not exist!), whereas a real crystal is rarely in thermodynamic equilibrium.

Ideal vs. real crystal, perfect vs. imperfect crystal


What do you get from a (conventional) diffraction experiment?

Time and space averaged structure!“Time-averaged” because the time span of a diffraction experiment is much larger that the time of an atomic vibration.

The instantaneous position of an atom is replaced by the envelop (most often an ellipsoid) that describes the volume spanned by the atom during its vibration.

“Space-averaged” because a conventional diffraction experiment gives the average of the atomic position in the whole crystal volume, which corresponds to “the” position of the atom only if perfect periodicity is respected.

Importance of studying the “ideal” crystalNon-conventional experiments (time-resolved crystallography,

nanocrystallography etc.) allow to go beyond the ideal crystal model, but also some information that are often neglected in a conventional experiment (diffuse scattering) can give precious insights on the real

structure of the crystal under investigation.


Affinetransformation

Euclidean transformationor isometry

Symmetryoperation

1 2

34

4 1

23

Transformations


●A crystal pattern is an idealized crystal structure which makes abstraction of the defects (including the surface!) and of the atomic nature of the structure.

●A crystal pattern is therefore infinite and perfect.●A crystal pattern is perfectly periodic and symmetric: periodicity affects symmetry and symmetry affects periodicity.

●A crystal pattern has both translational symmetry and point symmetry; these are described by its space group.

●We have to define the concepts of group and its declinations in crystallography.

Symmetry operations of a crystal pattern


Crystal

Crystal pattern(crystal structure)

Molecule orcoordination polyhedron

Unit cell

Lattice nodes

Crystal structure/pattern vs. crystal lattice

Translations


Unit cell

Asymmetric unit*

*In mathematics, it is called “fundamental region”

The minimal unit you need to describe a crystal pattern


The concepts of chirality and handedness(the left-right difference)

Symmetry operations are classified into first kind (keep the handedness) and second kind (change the handedness).If the object on which the operation is applied is non-chiral, the effect of the operation on the handedness is not visible but the nature of the operation (first or second kind) is not affected!

The determinant of the matrix representation of a symmetry operation is +1 (first kind) or -1 (second kind)


A symmetry group (G,○) is a set G with a binary operation ○ having the following features:● the elements of the set are symmetry operations;● the binary operation ○ is a successive application of two symmetry

operations gi and g

j of the set G, which gives as a result a symmetry

operation gk of the same set G (closure property): g

iG ○ g

jG → g

kG

(gig

j = g

k)

● the binary operation is associative: gi(g

jg

k) = (g

ig

j)g

k

● the set G includes the identity (left and right identical): egi = g

ie

● for each element of the set G (each symmetry operation) the inverse element (inverse symmetry operation) is in the set G: g

i-1g

i = g

ig

i-1 = e

The notion of symmetry group

In the following we will normally speak of a group G: it is a shortened expression for group (G,○) where ○ is the successive application. Rigorously speaking, G is not a group but just a set!


The notion of “order”

Order of a group element: the smallest n such that gn = e

Order of a group: the number of elements of the group (finite or infinite)

A special case of Abelian groups: cyclic groups

A cyclic group is a special Abelian group in which all elements of the group are generated from a single element (the generator).

G = {g, g2, g3, …. , gn = e}

If n = the group is infinite

A cyclic group is Abelian

gngm (n = m+p) = gm+pgm = (gmgp)gm = gm(gpgm) = gmgp+m = gmgn


●The elements of the group (G,○) are operations●The operations are performed about geometric elements

●A geometric element together with the set of symmetry operations performed about it is called a symmetry element

A note of nomenclature


One-dimensional lattices

Symmetry operations

Translations

Identity

Reflections

How many 1D lattices are there? Infinite!

How many types of 1D lattice are there? One

etc. etc.


Coordinates of lattice nodes, direction indices [uv]

[10]

00

10

20

30

01 02 03

11 12 13

21 22 23

31 32 33

[01]

[11]

[13]


Symmetry in E2

Operations that leave invariant all the space (2D): the identityOperations that leave invariant one direction of the space (1D): reflectionsOperations that leave invariant one point of the space (0D): rotationsOperations that do not leave invariant any point of the space: translations

Two independent directions in E2 two axes (a, b) and one interaxial angle (g)

The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 2 here)


Types of symmetry elements and operations in E2 (Hermann-Mauguin symbols)

Operations of the first kind(no change of chirality)

Operations of the second kind(change of chirality)

Element OperationRotation point Rotation

1 2p/12 2p/23 2p/34 2p/46 2p/6

Element OperationReflection line(mirror)

m m

Operations obtained as combination with a translation are introduced later

The orientation in space of a reflection line is always expressed with respect to the lattice direction to which it is perpendicular


primitive (p) cells centred (c) cell

Choice of the unit cell


Definition of conventional unit cell

A lattice (infinite) can be represented by many different (finite) unit cells.For each lattice, the conventional cell is the cell obeying the following conditions:

● its basis vectors define a right-handed axial setting;● its edges are along symmetry directions of the lattice;● it is the smallest cell compatible with the above condition.

Crystals having the same type of conventional cell belong to the same crystal family.


mp

b

a

tp

a1

a2

op

b

a

hp

Conventional cell parameters and symmetry directions in the four crystal families of E2

monoclinicNo restriction

on a, b, gNo symmetry direction

orthorhombicNo restriction

on a, b ;g = 90º

[10] et [01]

tetragonala = b ; g = 90º

10 ([10] and [01])11 ([11] and [11])

hexagonala = b ; g = 120º

10 ([10], [01] and [11])11 ([21], [12] and [11])

g g g

g

Crystal families: monoclinic, orthorhombic, tetragonal, hexagonalType of lattice*: primitive, centred

*Lattice whose conventional unit cell is primitive or centred

g bc

ac

bp

ap

oc

a1

a2


[10] [11]

[01]

[21]

[12][11]

Direction indices [uv] in the hexagonal lattice of E2

a1

a2


Why not 5, then?


Symmetry of the lattice : SL = 4mm

Lattice

« Object » (content of the unit cell)

Pattern

Symmetry of the object : So = 4mm

Symmetry of the pattern : Sp = 4mm

Sp = S

L : holohedry

The concept of holohedry


Symmetry of the lattice : SL= 4mm

Symmetry of the object : So = 3m

Symmetry of the pattern : Sp = m = 4mm 3m

Sp < S

L : merohedry

The concept of merohedry


Symmetry of the lattice : SL= 4mm


Symmetry of the pattern : Sp = 1 = 4mm 3m

Sp < S

L : merohedry



Symmetry of the lattice : SL = 4mm


Symmetry of the pattern : Sp = m = 4mm 5m

Sp < S

L : merohedry

The crystallographic restriction (n = 1, 2, 3, 4, 6) applies to the

lattice and to the pattern, but not to the content of the unit cell!



The inversion – an operation that exists only in odd-dimensional spaces

Inversion centre (point)

Why?


1

1

1

...

1

é ùê úê úê úê úê úê úë û

...

x

y

z

w

é ùê úê úê úê úê úê úë û

...

x

y

z

w

-é ùê ú-ê ú

-ê úê úê úê ú-ë û

1 = -In

det(1) = (-1)nodd-dimension spaces: -1

even-dimension spaces: +1

“1”operation

1 2

34

3 4

12

Example in E2

Two-fold rotation!

Matrix representation of the inversion in En


Symmetry in three dimensions (E3 : the three-dimensional Euclidean space)

Operations that leave invariant all the space (3D): the identityOperations that leave invariant a plane (2D): the reflectionsOperations that leave invariant one direction of the space (1D): the rotationsOperations that leave invariant one point of the space (0D): the roto-inversionsOperations that do not leave invariant any point of the space : the translations

Three independent directions in E3 three axes (a, b, c) and three interaxial angles (a, b, g)

The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 3 here)


Types of symmetry elements and operations in E3 (Hermann-Mauguin symbols)

Operations of the first kind(no change of chirality)

Operations of the second kind(change of chirality)

Element OperationDirect axis Rotation

1 2p/12 2p/23 2p/34 2p/46 2p/6

Element OperationInverse axis Rotoinversion 1 (centre) inversion 2 (m) reflexion 3 2p/3 + inversion 4 2p/4 + inversion 6 2p/6 + inversion

Operations resulting from a combination with a translation are introduced later

The orientation in space of a rotation axis or mirror plane is always expressed with respect to the lattice direction to which these are parallel

or perpendicular respectively


c

b

a

ab

g

Labelling of axes and angles in E3


Equivalence of 2 and m

db

q

I

IIII


Combination of 2 and 1 gives m

db

q

I

IIII

p

I

II

Applies to even-fold rotations as well, because they all “contain” a twofold rotation


Types of unit cells in E3

a

b

c

P a

b

c

a

b

c

C B a

b

c

A

S

a

b

c

Ia

b

c

F

The unit cell of type R is defined later


Crystal families and types of lattices in E3: the triclinic (anorthic) family

mp

b

a

g+ a third direction inclined to the plane the rotation points in each 2D lattice do not overlap to form a rotation axis in the 3D lattice.

● One symmetry element: the inversion centre ● No symmetry direction● The conventional unit cell is not defined –

no reason to choose à priori a centred cell● Symmetry of the lattice: 1● No restriction on a, b, c, a, b, g

triclinic crystal family (anorthic)

aP

2D 3D

2 1 1 = 1


monoclinic crystal family

● One symmetry direction (usually taken as the b axis).● The conventional unit cell has two right angles (a and g)● Symmetry of the lattice: 2/m● Two independent types of unit cell respect these conditions

mP mS

mp

b

a

g

Crystal families and types of lattices in E3: the monoclinic family

+ a third direction perpendicular to the plane the rotation points in each 2D lattice do overlap to form a rotation axis in the 3D lattice.

2D 3D

2 2 1 = 2/m


A lattice of type mP is equivalent to mB but not to mC

cB

bBa

B

aB = (a

P-c

P)/2 ; c

B = (a

P+c

P)/2

mB mP

bP b

B

cP

aP

cC

bCa

C

bC

aC

Show that unit cells of type mC and mP do not represent the same type of lattice.

bP

aP


aA = -c

C

cA = a

C

mA mC

Lattices of type mC, mA, mI and mF are all equivalent (mS)

mC mF

cF = -a

C+2c

CmC mIa

I = a

C-c

Cc

I

aI

bI b

A

cF

bF b

C

aF a

C

cC

bCa

C

aA

cA b

A b

C


Three monoclinic settings

b-unique c-unique a-unique

b unrestricted by symmetry

g unrestricted by symmetry

a unrestricted by symmetry

mB = mPmA = mC = mI = mF

mC = mPmA = mB = mI = mF

mA = mPmB = mC = mI = mF

The symbol of a monoclinic space group will change depending on which setting and what type of unit cell you choose, leading to up to 21 possible symbols for a monoclinic space group.


op

ba

b

a

oc

g g

oP oI oS oF

orthorhombic crystal family

● Three symmetry directions (axes a, b, c)● The conventional unit cell has three right angles(a, b, g)● Symmetry of the lattice: 2/m 2/m 2/m● Four types of cell respect these conditions● The unit cell with one pair of faces centred can be equivalently described as A, B or C by a

permutation of the (collective symbol : S)

Crystal families and types of lattices in E3: the orthorhombic family

+ a third direction perpendicular to the plane 2D 3D

2mm 2mm 1 = 2/m2/m2/m


ab

c

a

b

c

a

bc

oC oB oA

oS

Three possible setting for the oS type of lattice in E3


tp

a

bg

tP tI

tetragonal crystal family● Five symmetry directions (c, a&b, the two diagonals in the a-b plane)● Symmetry of the lattice: 4/m 2/m 2/m● The conventional cell has three right angles and two identical edges● Two independent types of unit cell respect these conditions: tP (equivalent to tC) and tI

(equivalent to tF)

tP

tC

½

tF

½

½

+ a third direction perpendicular to the plane

½

tI

½

½

½

Crystal families and types of lattices in E3: the tetragonal family

2D 3D

4mm 4mm 1

= 4/m2/m2/m


Why unit cells of type tA et tB cannot exist?

0 0

0 0

0

0

0 0 0

00

1/2

1/2

1/2

1/2

1/2 1/2

1/2 1/20

4-fold axis!

a

b



0 0

0 0

0

0

0 0 0

00

1/2

1/2

1/2

1/2

1/2 1/2

1/2 1/20

1/21/2 1/2

1/21/2

1/21/21/2

1/2

a

b


00

00

00


0 0

0 0

0

0

0 0 0

00

1/2

1/2

1/2

1/2

1/2 1/2

1/2 1/20

1/21/2 1/2

1/21/2

1/21/21/2

1/2

a

b

tF!


00

00

00


0 0

0 0

0

0

0 0 0

00

1/2

1/2

1/2

1/2

1/2 1/2

1/2 1/20

1/21/2 1/2

1/21/2

1/21/21/2

1/2

a

b

tF! tI!


tp

a

bg

cP cI cF

cubic crystal family

● Thirteen symmetry directions (the 3 axes; the 4 body diagonals ; the six face diagonals)● Symmetry of the lattice: 4/m 3 2/m● The conventional unit cell has three right angles and three identical edges● Three types of unit cell respect these conditions : cP, cI et cF

+ a third direction perpendicular to the plane AND c = a = b

Crystal families and types of lattices in E3: the cubic family

2D 3D

4mm 4mm 3

= 4/m32/m


hp

a

bg

hexagonal crystal family

A1

A2

C

+ a new type of unit cell!

Crystal families and types of lattices in E3: the hexagonal family

+ a third direction perpendicular to the plane

2D 3D

6mm 6mm 1

= 6/m2/m2/m


Uniqueness of the hexagonal crystal family in E3

A1 A

2

C

a1

a2

a3

A1

A2

a1

a2

a3

Symmetry directions

A1, A

2, C hexagonal axes parallel to the symmetry directions

a1, a

2, a

3 rhombohedral axes NOT parallel to the symmetry directions

Two types of lattice with different symmetry in the hexagonal crystal family


[110]R

[101]R

[011]R

[101]R

[011]R

[110]R

a1

a2a

3

A3

A1

A2

[100]H

[010]H

[110]H

[110]H

[010]H

[100]H

[210]H

[120]H

[110]H

[110]H

[210]H

[120]H

z = 0

z = 1/3

z = 2/3

Symmetry difference between hP and hR types of lattice

6/m2/m2/m

32/m


Lattice systems: classification based on the symmetry of the lattices

aPmP mS tP tI

oP oI oS oF

hP hRcP cI cF

1 triclinic 2/m monoclinic 4/m 2/m 2/m(4/mmm)

tetragonal

2/m 2/m 2/m (mmm) orthorhombic

6/m 2/m 2/m(6/mmm) hexagonal rhombohedral

cubic3 2/m(3m)

4/m 3 2/m(m3m)


Crystal systems: morphological (macroscopic) and physical symmetry

A1

1 or 1

A2

2 or m or 2/m3´A

2

Three twofold elements

A4

A3

A6

4´A3

triclinic monoclinic

tetragonal

orthorhombic

hexagonaltrigonal

cubic


hexagonal

Crystal families, crystal systems, lattice systems and types of Bravais lattices in E3

6 crystal families

aP

mP ( mB )

mS ( mC, mA, mI, mF )

oP

oS ( oC, oA, oB )

oI

oF

tP ( tC )

tI ( tF )

hR

hP

cP

cI

cF

no restriction on a ; b ; c, a , b , g

no restriction on a ; b ; c ; b.a = g = 90º

no restriction on a ; b ; c.a = b = g = 90º

a = b ; a = b = g = 90ºaucune restriction sur c

a = b = ca = b = g = 90º

a = b ; a = b = 90º, g = 120º no restriction on c

conventional unit cell

a = anortic*(triclinic, asynmetric, tetartoprismac...)

m = monoclinic(clinorhombic, monosymmetric, binary,hemiprismatic, monoclinohedral, ...)

o = orthorhombic(rhombic, trimetric, terbinary, prismatic, anisometric,...)

t = tetragonal(quadratic, dimetric, monodimetric, quaternary...)

h = hexagonal(senaiey, monotrimetric...)

c = cubic(isometric, monometric, triquaternary, regular, tesseral, tessural...)

7 crystal systems(morphological symmetry)

trigonal (ternary...) (***)

triclinic

monoclinic

orthorhombic

tetragonal

hexagonal

cubic

7 lattice systems(lattice symmetry)

triclinic

monoclinic

orthorhombic

rhombohedral

cubic

tetragonal

14 types of Bravais lattices(**)

(*) Synonyms within parentheses. (**) S = one pair of faces centred. Within parentheses the types of lattices that are equivalent (axial setting change – see the monoclinic example). (***) Crystals of the trigonal crystal system may have a rhombohedral or hexagonal lattice


[110] [011] [101] 110

[100] [010] [110] 100

Symmetry directions of the lattices in the three-dimensional space(directions in the same box are equivalents by symmetry )

Lattice system First symmetry direction

triclinic

monoclinic

orthorhombic

tetragonal

rhombohedral

hexagonal

cubic

[010]

[100] [010] [001]

[100] [010] 100[001]

[111]

[001]

[001] [100] [010] 001

[110] [110] 110

rhombohedral axes

hexagonal axes [001]

[100] [010] [110] 100

[111] [111] [111] [111] 111

[110] [110] [011][011] [101] [101]

110

[110] [120] [210] 110

a ; b ; ca , b , g any

a ; b ; ca = g = 90º, b any

a ; b ; ca = b = g = 90º

a = b ; ca = b = g = 90º

a = b = ca = b = g = 90º

a = b ; ca = b = 90º, g = 120º

a = b ; ca = b = 90º, g = 120º

a = b = ca = b = g

Parameters Second symmetry direction

Third symmetry direction


Lattice system lattice plane lattice direction

triclinic --- ---

monoclinic (b-unique) (010) [010]

orthorhombic (100)(010)(001)

[100][010][001]

tetragonal (001)(hk0)

[001][hk0]

rhombohedral and hexagonal (hexagonal axes)

(0001)(hki0)

[001][2h+k,h+2k,0]

Cubic (hkl) [hkl]

Condition of lattice plane/direction mutual perpendicularity in the seven lattice systems of the three-dimensional space (without metric specialisation)


Point groups and morphological symmetry. Introduction to the stereographic projection




A brief analysis of the symmetry of the square in E2


Find the symmetry elements, the symmetry operations and the symmetry group of a square

m[10]

m[11]

m[01]

m[11]

a1 (a)

a2 (b)

Symmetry elements:

Symmetry operations:

4, m[10]

, m[01]

, m[11]

, m[11]

41, 42=2, 43=4-1, 44=1, m[10]

, m[01]

, m[11]

, m[11]

Check that the above symmetry operations do form a group

● The identity is included (and obtained as 44 or m2)● The inverse of each operation is included (1-1=1; m-1=m;

2-1=2, 4-1=43)● For the binary operation and the associativity, build the

multiplication table

1

2

4

3


Multiplication table of the symmetry operations of a square

m[10]

m[11]

m[01]

m[11]

1

2

4

3

1 2 41 43 m[10]

m[11]

m[01]

m[11]

11

241

1

431

m[10]

m[11]

m[01]

m[11]

11

241

1

431

m[10]

m[11]

m[01]

m[11]

2 41 43 m[10]

m[11]

m[01]

m[11]

1 41 41 m[01]

m[11]

m[10]

m[11]

43 2 1 m[11]

m[01]

m[11]

m[10]

41 1 2 m

[11] m

[10] m

[11] m

[01]

m

[01] m

[11] m

[11] 1 43 2 41

m

[11] m

[10] m

[01] 41 1 43 2

m

[10] m

[11] m

[11] 2 41 1 43

m

[11] m

[01] m

[10] 43 2 41 1

a1 (a)

a2 (b)


H G

Subgroups and cosets

A subgroup (H,○) of a group (G,○) is a subset of the elements of the set G that under the same binary operation ○ still forms a group (i.e. it is associative and contains the identity and the inverse of each element).

m[10]

m[11]

m[01]

m[11]

1

2

4

3

Group G Subgroup H

1 2 41 43 m[10]

m[11]

m[01]

m[11]

11

241

1

431

m[10]

m[11]

m[01]

m[11]

11

241

1

431

m[10]

m[11]

m[01]

m[11]

2 41 43 m[10]

m[11]

m[01]

m[11]

1 43 41 m[01]

m[11]

m[10]

m[11]

43 2 1 m[11]

m[01]

m[11]

m[10]

41 1 2 m

[11] m

[10] m

[11] m

[01]

m

[01] m

[11] m

[11] 1 43 2 41

m

[11] m

[10] m

[01] 41 1 43 2

m

[10] m

[11] m

[11] 2 41 1 43

m

[11] m

[01] m

[10] 43 2 41 1


H GC3H

C2

C1

C4

Subgroups and cosets

m[10]

m[11]

m[01]

m[11]

1

2

4

3

Group G Subgroup H Coset C

1 2 41 43 m[10]

m[11]

m[01]

m[11]

11

241

1

431

m[10]

m[11]

m[01]

m[11]

11

241

1

431

m[10]

m[11]

m[01]

m[11]

2 41 43 m[10]

m[11]

m[01]

m[11]

1 43 41 m[01]

m[11]

m[10]

m[11]

43 2 1 m[11]

m[01]

m[11]

m[10]

41 1 2 m

[11] m

[10] m

[11] m

[01]

m

[01] m

[11] m

[11] 1 43 2 41

m

[11] m

[10] m

[01] 41 1 43 2

m

[10] m

[11] m

[11] 2 41 1 43

m

[11] m

[01] m

[10] 43 2 41 1

A subgroup (H,○) of a group (G,○) is a subset of the elements of the set G that under the same binary operation ○ still forms a group (i.e. it is associative and contains the identity and the inverse of each element).


Lattice planes and Miller indices


Planes passing through lattice nodes are called “rational planes”

a

c

b

p-th nodeq-th node

r-th node

Parametric equation of the plane:x/p + y/q + z/r = 1

(qr)x + (pr)y + (pq)z = pqrhx + ky + lz = m

First plane of the family (hkl) for m = 1hx + ky + lz = 1

Intercepts of the first plane of the family (hkl) on the axesp = pqr/qr = m/qr = 1/hq = pqr/pr = m/pr = 1/kr = pqr/pq = m/pq = 1/l

Making m variable, we obtain a family of lattice planes, (hkl), where h, k and l are called the Miller indices.

The values h, k and l are called the Miller indices of the lattice plane and give its orientation.All lattice planes in the same family have the same orientation → (hkl) represents the whole family of lattice planes.


Why the reciprocal of the intersection (1/p) rather than the intersection (p) itself?

a

c

b

Consider a plane parallel to an axis – for example c

What is the intersection of this plane with the axis c?

What is the l Miller intersection of this plane? 1/ = 0


Example: family (112) in a primitive lattice

a

c

b

1/11/1

Intercepts of the first plane of the family:on a: 1/1on b: 1/1on c: 1/2

1/11/2

Intercepts of the second plane of the family:on a: 2/1on b: 2/1on c: 2/2


Example: family (326) in a primitive lattice

a

c

b1/3 1/2

Intercepts of the first plane of the family:on a: 1/3on b: 1/2on c: 1/6

1/6

Intercepts of the sixth plane of the family:on a: 6/3on b: 6/2on c: 6/6


Miller indices for a primitive lattice are coprime integers

a

c

b1/2 1/2

Intercepts of the first plane of a hypothetic family (222):on a: 1/2on b: 1/2on c: 1/2 1/2

The first rational plane of this family has intercepts:on a: 1/1on b: 1/1on c: 1/1

This plane does not pass through any lattice node – it is an irrational plane

In a primitive lattice, the Miller indices of a family of lattice planes are coprime integers: (111)


Miller indices for different types of lattice : (h00) in oP and oC (projection on ab)

oP

First plane of the familyIntersection a/1

h = 1family (100)

First plane of the familyIntersection a/2h = 2

family (200)

oC

In morphology, we do not see the lattice and thus the Miller indices of a faceface are usually coprime integers


The stereographic projection: how to get rid of accidental morphological

features of a crystal


Spherical projection and spherical poles


Building the stereographic projection: from the spherical poles (P) to the stereographic poles (p, p')

N

O

P

p

S

N

O

P

S

p

N

O

P

S

p'p


S

R

N

OQ

r

r

r/2r

pr tan r/2

r

r/2

p'

r cotan r/2

P'

p/2-r/2

r/2P'

p'

pp'

Q RpN

P

S

R

N

OQ

r

S

R

N

OQ

r

P

r

r/2

Building the stereographic projection: from the spherical poles (P) to the stereographic poles (p, p')


p'Q R

pN

O

N

S

r

r/2

S

Q R

P

N

O

p

P'

p'

r

r/2

N

O

S

Stereographic projection: poles and symmetry planes


{001} {111} {110}

OctahedronCube Dodecahedron

a

b

c

a b

c

a

b

a

b

a

b

a

b

a b

c

Example of decomposition of the morphology of a crystal


N

O

P

p

S

N

O

S

pP

Stereographic projection Gnomonic projection

Be careful - some textbooks exchange the two terms!

Stereographic vs. gnomonic projection


Bravais-Miller indices

In case of hexagonal or rhombohedral lattices, it is useful to use a reference built on four axes, three in the plane normal to the unique axis (A

1, A

2, A

3) and one (c) for the unique axis. Consequently, four indices of lattice planes,

(hkil), are used, called the Bravais-Miller indices. Here h, k, i, l are integers inversely proportional to the intercepts of a plane of the family with the four axes. Only two of the three axes in a plane are linearly independent: A

3 is expressed as A

3 = -A

1-A

2. Analogously, the indices

h, k, i are cyclically permutable and related by h + k + i = 0.

A1

A2

A3

(1012)

(1102)(0112)

(0112)

(1012)

(1102)

(1121)

(1211)

(2111)

(1121)

(1211)

(2111)

A1

A2

A3

(102)

(112)(012)

(012)

(102)

(112)

(111)

(121)

(211)

(111)

(121)

(211)

Bravais-Miller indices Miller indices


Weber indices

For trigonal and hexagonal crystal, an extension to a four-axes axial setting exists also for lattice directions, known as the Weber indices. The Weber indices of the direction perpendicular to a lattice plane are the same as the Bravais-Miller indices of that plane.

Let A1,A

2,A

3,C be the four hexagonal axes, and let be uvw and UVTW the indices of a direction with

respect to A1,A

2,C or A

1,A

2,A

3,C respectively. The relations betwee uvw and UVTW are:

u = 2U+V; v = U+2V; w = W

U = (2u-v)/3; V = (2v-u)/3; T = -(u+v)/3.

The relation T = -U-V holds for U and V but notnot for u and v, whereas for the Bravais-Miller indices the addition of the third axis does not modify h and k so that the relation i = -h-k is applied directlydirectly. For this reason, the Bravais-Miller indices are widely used in crystallography, whereas the Weber indices are more used in fields like electron microscopy and metallurgy but seldom in crystallography.

Miller indices Bravais-Miller indices Perpendicular direction Perpendicular direction (Weber indices)(hkl0) (hki0) [2h+k,h+2k,0] [hki0]Ex. (100) (1010) [210] [1010]Ex. (210) (2110) [100] [2110]


Zones and zone axis




Form {100}: the cube

Form {111}: the octahedron

Multiplicity 6

Multiplicity 8

c

ab

The concept of form: set of symmetry-related faces

c

a b

Example in the cubic crystal system


zone [001]

Zone: set of faces whose intersection is parallel to a same direction, called the zone axis

c

ab

zone [010] zone [100]

Example in the cubic crystal system


Formal definition

A zone axis is a lattice row parallel to the intersection of two (or more) families of lattices planes. It is denoted by [u v w]. A zone axis [u v w] is parallel to a family of lattice planes of Miller indices (hkl) if (Weiss law):

uh + vk + wl = 0

The indices of the zone axis defined by two lattice planes (h1,k

1,l

1), (h

2,k

2,l

2) are given by:

1 1 1 1 1 1

2 2 2 2 2 2

u v wk l l h h k

k l l h h k

= =

Conversely, any crystal face can be determined if one knows two zone axes parallel to it (zone law)

Three lattice planes have a common zone axis (are in zone) if their Miller indices (h1,k

1,l

1), (h

2,k

2,l

2), (h

3,k

3,l

3)

satisfy the relation:

1 1 1

2 2 2

3 3 3

0

h k l

h k l

h k l

=


0 0 1 0 0 1

1 0 1 1 0 1

0 1 0

Compute the zone axis

Let the Miller indices of two lattice planes be (h1,k

1,l

1), (h

2,k

2,l

2).

1 1 1 1 1 1

2 2 2 2 2 2

h k l h k l

h k l h k l

u v w

+ -Remove common factor, if any

Exercise: zone axis for faces (001) and (101) Exercise: zone axis for faces (231) and (362)

2 3 1 2 3 1

3 6 2 3 6 2

0 1 3

Low-index zone axes correspond to lattice rows parallel to several lattice planes

1 1 1 1

2 2 2 2

0 0

0 0

0 0

h l h l

h l h l

n n = l1h

2-l

2h

1


Screw axes and glide planes




● geometric element ： the point, line or plane left invariant by the symmetry operation.

● symmetry element: the geometric element defined above together with the set of operations (called element set) that leave in invariant.

● symmetry operation ： an isometry that leave invariant the object to which it is applied.

Elements and operations

geometric element

element set

symmetry element

symmetry operation

The operations that share a given geometric element differ by a lattice vector. The one characterized by the shortest vector is called defining operationdefining operation.

symmetry operation

symmetry operation


Finding the geometric element● Remove the intrinsic translation part (screw or glide component).● Impose that the coordinates of the geometric element are fixed by the

operation.

1 0 0 0

0 1 0 1 2

0 0 1 1 2

æ öç ÷ç ÷ç ÷è ø

Linear part: twofold rotation about [010]

Intrinsic part: screw component

Location part: the element does not pass through the origin

1 0 0 0

0 1 0 0

0 0 1 1 2

u u

v v

w w

æ öæ ö æ öç ÷ç ÷ ç ÷=ç ÷ç ÷ ç ÷

ç ÷ ç ÷ç ÷è ø è øè ø1 2

u u

v v

w w

- =ìï =íï- + =î

0

1 4

u

v

w

=ìï "íï =î

0,y,¼

≠

Screw axes np (screw component: p/n)

2

period

half periodperiod

Defining operation ： twofold rotationSymmetry element ： rotation axis

Defining operation ： twofold screw rotationSymmetry element ： screw axis

t(001)

t(002)

22

24

Geometric element ： line

t(001)

t(003) 26

21

23

≠ fflffl


Glide planes g

g (½ ½,0) x,x,z

Geometric element ： plane

period

[001]

[110]

g (0,0,0): mg (½,0,0): a

g (0,½,0): b

g (0,0,½): c

g (half-diagonal): n

g (quarter-diagonal): d

t(1,1,0)

g (5/2,5/2,0) x,x,z

g (3/2,3/2,0) x,x,z t(2,2,0)

t(3,3,0)

g (7/2,7/2,0) x,x,z t(4,4,0)

Special cases


Screw axes

2

00

21

01/2

00

0

01/3

2/3

31

02/3

1/3

32

3


0

0

0

0

4

0

1/4

1/2

3/4

41

0

3/4

1/2

1/4

43

0

1/2

0

1/2

42 2

Screw axes


61

2/3

5/6

1/3

1/2

0

1/6

1/3

2/3

2/3

0

0

1/3

0

0

0

0

0

0

6 62 2

00

0

63 3 6

5 64 2

1/3

1/6

2/3

1/2

0

5/6

2/3

1/3

1/3

0

0

2/31/2

1/2

1/2

Screw axes


np

nn-p

Screw axes


m a

a

a/2

Glide planes


c

Glide planes

b

b/2

b

c/2

c


d

Glide planes

diagonal

Diamond structure glide plane 2D glide line; 3D glide plane with “unusual' glide

n

d/2d/4


g

abc

Glide planes

e

d/2 d/2d/2


How can we have a glide of ¼ if the reflection is an operation of order 2 ?

If the unit cell is centred !

Vector centring the unit cell, with a norm d/2

+

,+ +


An historical introduction to the reciprocal lattice




Bravais' polar lattice (1848)A dual lattice of the direct lattice based on face normals

Auguste Bravais (1811-1863)


b

a

c

a´b = S(001)

(001)

d(001)

=c·e(001)

e(001)

=

a´b / |a´b|

V = c·a´b = S(001)d(001)

= S(hkl)d(hkl)



(h1 k

1 l1 )

(h2 k

2 l2 )

(h3 k

3 l3 )

S(h1k

1l1)/V1/3 (Å)

S(h2k

2l2)/V1/3 (Å)

S(h3k

3l3)/V1/3 (Å)

Polar lattice(in vector space)

|rphkl = S(hkl)/V1/3 = [V/d

(hkl)]/V1/3 = V2/3/d

(hkl) (Å)

The vector space Vn is realized with a metric in Å obeying the above conditions



b

a

c

bp (010)

ap (100)

cp (001)

p p p1 3 1 3 1 3 1 3 1 3 1 3

100 010 001; ;

V V V V V V

S S S

b× c c × a a × ba b a

vi·vjp = mij

2 31 3 1 3

VV

V Vj kp

i i im

v v

v v = v

By construction, the vector [hkl]p is perpendicular to the face (hkl)


Journal of Applied Crystallography 43 (2010) 1144-1149http://dx.doi.org/10.1107/S0021889810023915

http://dx.doi.org/10.1107/S0021889810023915


Angles between vectors of the polar lattice are precisely the angles between face normals

(h1k

1l1) (h

2k

2l2)

[h1k

1l1]p

[h2k

2l2]p

j

d = p-j

cosd = r1

p·r2

p /||r1

p||·||r2

p||

p

p p

2

1 1 1 2

2

1 2

1 1 1 1 2 2 2 2

1 2

h

h k l k

l

h h

h k l k h k l k

l l

æ öç ÷ç ÷ç ÷æ ö ç ÷ç ÷ ç ÷è ø ç ÷ç ÷ç ÷è ø

æ ö æ öç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷æ ö æ öç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷è ø è øç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷è ø è ø

=

G

G G

p p p p p p

p p p p p pp

p p p p p p

æ öç ÷ç ÷=ç ÷ç ÷ç ÷è ø

× × ×× × ×× × ×

G

a a a b a c

b a b b b c

c a c b c c


Use of the polar lattice makes indexing of stereographic poles straightforward – monoclinic example

c

a

(001)

(100)

Parallel to a and b axes

cp

ap

On the cp axis

Parallel to c and b axes

On the ap axis


Use of the polar lattice makes indexing of stereographic poles straightforward – hexagonal example

A1

A2

A3

(1010)(1120)

Parallel to A2 and C axes

Same intersection on A1 and A

2 (positive)

Half intersection on A3 (negative)

apbp

On the ap axis

On the bisector of ap and bp


A1

A2

A3

(1010)(1120)

(0110)

(1100)

(1100)

(0110)

(1010)

(2110)

(1120)(2110)

(1210)(1210)

apbp

Use of the polar lattice makes indexing of stereographic poles straightforward – hexagonal example


Wilhelm Conrad Röntgen (1845-1923)

Then, something happened....


1895 : discovery of X-rays


Max von Laue (1879-1960)

1912: crystals diffract X-rays


X-ray diffraction pattern of ZnS by Friedrich, Knipping and Laue

1912: crystals diffract X-rays


Diffractions make a lattice that can be

indexed on the basis of suitably chosen

axes

A diffraction pattern can be indexed with a lattice whose metric is inverse (reciprocal) of the Bravais lattices


Paul Peter Ewald (1888-1985)

1913 : the reciprocal lattice


b

a

c

b* (010)

a* (100)

c* (001)

a* = (bc)/V, b* = (ca)/V, c* = (ab)/V Å-1 !The reciprocal lattice

The reciprocal space is a vector space Vn realized with a metric in Å-1

1913 : the reciprocal lattice


Linear parameters of the reciprocal lattice

a* = (bc)/V, b* = (ca)/V, c* = (ab)/V

a·a*= a·(bc)/V = 1 ; b·b*= b·(ca)/V = 1 ; c·c*= c·(ab)/V = 1

vi·vj* = ij

a* = bcsin/V, b* = casin/V, c* = absin/V

V* = a*·b*c* = (bc)·(ca)(ab)/V3 = (bc)·[(c·ab)a-(c·aa)b]/V3 =

(bc)·[Va-0b]/V3 = V2/V3 = 1/V

a = (b*c*)/V* ; b = (c*a*)/V* ; c = (a*b*)/V*


Angular parameters of the reciprocal lattice

a* = bcsin/V, b* = casin/V, c* = absin/V

a = b*c*sin/V* ; b* = a*c*sin/V* ; c* = a*b*sin/V* (V* = 1/V)

V* VVsin *sin sin* * sin sinV V

aa

ac abb c abc

V* VVsin *

sin sin* * sin sinV V

bb

bc aba c abc

V* VVsin *

sin sin* * sin sinV V

cc

bc aca b abc


Metric tensor of the reciprocal lattice

* * * * * *

* * * * * *

* * * * * *

G*

a a a b a c

b a b b b c

c a c b c c

G* = G-1 ; det(G*) = 1/det(G) V* = 1/V

Angles between the (h1k

1l1) and (h

2k

2l2) faces computed as

angle between r*(h1k

1l1) and r*(h

2k

2l2) vectors

*1 1 1 2 2 2

1 1 1 2 2 2 * *1 1 1 1 1 1 2 2 2 2 2 2

cos ^h k l h k l

h k l h k lh k l h k l h k l h k l

G

G G


a

b

c

a/h

b/k

c/l

d(hkl)e

hkl*

ehkl

* = rhkl

* / |rhkl

*|

d(hkl)

= ehkl

* · a/h

d(hkl)

= ehkl

* · b/k

d(hkl)

= ehkl

* · c/l

* * *

1 0 0 1

hkl hkl hkl

h k l h k ld hkl

h h

r r r

a a* + b* + c* + * +

* * *

0 1 0 1

hkl hkl hkl

h k l h k ld hkl

k k

r r r

b a* + b* + c* + * +

* * *

0 0 1 1

hkl hkl hkl

h k l h k ld hkl

l l

r r r

c a* + b* + c* + * +

(hkl)

Equidistance of lattice planes


Josiah Willard Gibbs(1839-1903)

Reciprocal system of vectors (1881)

Not really Ewald's discovery....


http://www.archive.org/details/117714283


Geometric interpretation of reflection conditions




Classification of reflection conditions

● General: apply to all Wyckoff positions.● Special:： apply to special Wyckoff positions. In

particular, in presence of non-characteristic and extraordinary orbits one gets additional special reflection conditions.

● Integral: appear when a non-primitive unit cell is selected.

● Zonal: appear in presence of glide planes.● Serial: appear in presence of screw axes.

Warning!

Systematic extinctions Systematic absences （ actually, presences! ）

Reflection conditions


(200)

110

010 010000

½½0½½0

100

(110)

C-centred unit cellhkl:h+k=2n

Primiitve unit cellNo conditions

210 210

110

3/2½03/2½0200

½½0 ½½0

100 010

110

120

130

200

210

220310

110100

000

100010

020

(130)(120)

(110)

(100)

When you choose a primitive unit cell you do not When you choose a primitive unit cell you do not see integral reflection conditionssee integral reflection conditions

Integral reflection condition: depend on the choice of the unit cell, not on the structure


t/2 t/2

Zonal reflection conditions: witness of glide planes


+ + +

- -

+ + +

+ + +

- -

- -

Direct spaceIn projection along the a axis the period along b seems halved

b

c

Reciprocal spaceOn the (0kl)*plane the period along the b* axis appears doubled.

Reflection conditions ： 0kl : k = 2n

Zonal reflection conditions: witness of glide planes


t

t/4

Direct spaceIn the projection on the c axis the period appear reduced to ¼.

Reciprocal spaceIn the [001]* direction the period appears multiplied by four.

Reflection conditions ：00l : l = 4n

Serial reflection conditions: witness of screw axes


From reflections conditions to space groups


hkl:k+l = 2n0kl:k+l = 2nh0l:l = 2nhk0:h = 2nhk0:k = 2nh00:h = 2n0k0:k = 2n00l:l = 2n

Integral reflection conditions: centred cell

oA

Zonal reflection conditions: glide plane

hk0: perpendicular to [001]

h even ： a glidea

[001]

Extension symbol ： A--a

Possible space-group types ： Am2a, A21ma, Amma

Example 1: orthorhombic


hk0:h+k = 2n0k0:k = 2nhhl:l = 2n00l:l = 2n


hk0: perpendicular to [001]

h+k even ： n gliden

[001]

Extinction symbol ： Pn-c

Possible space-group type: P42/nmc

No integral reflection condition tP


hhl: perpendicular to [110]

l even ： c evenc110

Non-independent

Exemple 2: tetragonal


hkl:h+k+l = 2n0kl:k = 2n0kl:l = 2nh0l:h = 2nh0l:l = 2nhk0:h = 2nhk0:k = 2nhhl:2h+l = 4nhhl:l = 2n00l:l = 4n

Integral reflection condition cI

Zonal reflection condition

hk0: perpendicular to [001],h even a[001]

hk0: perpendicular to [001],k even

h0l: perpendicular to [010], h even

h0l: perpendicular to [010], l even

0kl: perpendicular to [100], k even

0kl: perpendicular to [100], l even

b[001]

c[010]

a[010]

c[100]

b[100]

Zonal reflection condition hhl: perpendicular to [110], 2h+l = 4n ??

Example 3: cubic


hhl:2h+l = 4n

[110]

c

(a+b)/2

c/2

-,

+

+

(a+b+c)/4

Direct lattice: (a+b+c)/4 Reciprocal lattice: h+k+l = 4n(110) h = k

h+h+l = 4n

dd[1[1110]0]


Extinction symbol: Ia-d

Possible space-group type: Ia3d

d110

hkl:h+k+l = 2n0kl:k = 2n0kl:l = 2nh0l:h = 2nh0l:l = 2nhk0:h = 2nhk0:k = 2nhhl:2h+l = 4nhhl:l = 2n00l:l = 4n

Integral reflection condition cI

Zonal reflection condition

hk0: perpendicular to [001],h even a[001]

hk0: perpendicular to [001],k even

h0l: perpendicular to [010], h even

h0l: perpendicular to [010], l even

0kl: perpendicular to [100], k even

0kl: perpendicular to [100], l even

b[001]

c[010]

a[010]

c[100]

b[100]

Zonal reflection condition hhl: perpendicular to [110], 2h+l = 4n

Example 3: cubic


a[001]

b[001]

c [010

] a [010

]

c[100]

b[100]

d110




Hierarchy of classification

Bravais classes (14)Geometric crystal

classes (32)Laue classes (11)

Lattice systems (7) Crystal systems (7)

Space group types with 2nd kind operations (165)

Pyroelectric (46)Bieberbach (4)

Space group types without 2nd kind operations(43)

Pyrolectric (14)Bieberbach (3)

Sohncke space group types(65)


Space groups (infinite)

Symmorphic space group types (73)

Pyroelectric (21)

Point group types (32)

Point groups (136)Harmonic crystal classes (66)

Arithmetic crystal classes (73)

Crystallographic space group types (230)

Proyelectric (68)Bieberbach (13)

Affine space group types (219)

Pyroelectric (64)Bieberbach(10)

Crystal families (6)

Chiral space group types(22)


Achiral space group types (208)


Asymmorphic space group types (103)


Hemisymmorphic space group types(54)

Pyroelectric (22)

mmm Pmmm

Pnnn Pmma Pcca

Point groupSymmorphic space group

Hemisymmorphic space group Asymmorphic space group Asymmorphic space group

site-symmetry group at the origin: mmm

s.-s. group at the origin: 222 s.-s. group at the origin: .2/m. s.-s. group at the origin: 1


P42m P42c P421m P42

1c P4m2 P4c2 P4b2 P4n2 I4m2 I4c2 I42m I42d

Arithmetic

Harmonic

Geometric

42mP 4m2P 4m2I 42mI

42mP 42mI

42m

Example of crystal classes hierarchy

A geometric crystal class which represents a full symmetry of a lattice is called a holohedryholohedry. Otherwise, a merohedrymerohedry.


Crystal systems


H, G: point groups H G

If G acts on lattice, then H to acts on the same lattice

If H acts on lattice, G does not necessarilynot necessarily act on the same lattice

Space groups and crystal structures whose point group act of the same type of Bravais lattice belong to the same crystal system.

Ex. 1: m3m (G) acts on cP, cI, cF. 23, m3, 432, 43m (H) act on the same lattices

23 (H) acts on cP, cI, cF. m3m (G) act on the same lattices. Etc.

Crystals with point groups m3m, m3, 432, 43m, 23 all belong to the cubic crystal system.

Ex. 2: 6/mmm (G) acts on hP. 6, 6, 6/m, 622, 6mm, 62m, 3, 3, 32, 3m, 3m (H) act on hP too.

3m (H) acts on hP and hR but 6/mmm (G) does not act on hR.

Crystals with point groups 6/mmm, 6, 6, 6/m, 622, 6mm, 62m all belong to the hexagonal crystal system. Those with point groups 3, 3, 32, 3m, 3m belong to

the trigonal crystal system.

6 (H) acts on hP. 6/mmm (G) acts on hP too.


Lattice systems


Space groups and crystal structures which correspond to the same holohedry belong to the same lattice system.

Ex. 1: 23P and m3P correspond to the m3m holohedry and belong to the cubic lattice system.

Ex. 3: 32R and 622P correspond to the different holohedries (3m and 6/mmm). 32R belong to the rhombohedral lattice system, whereas 622 belongs to the hexagonal lattice system.

Ex. 2: 32P and 622P correspond to the 6/mmm holohedry and belong to the hexagonal lattice system.

Hexagonal

TrigonalRhombohedral


Crystal families


Space groups and crystal structures whose lattices have the same number of free parameters belong to the same crystal family if the corresponding point groups are in group-subgroup relation.

Ex. 1: Two crystals with holohedries 4/mmm and 6/mmm have lattices with two free parameters (a and c). However, 4/mmm and 6/mmm are not in group-subgroup relation and thus the two crystals belong to different crystal families (tetragonal and hexagonal).

Ex. 2: Two crystals with holohedries 3m and 6/mmm have lattices with two free parameters (a and c). Furthermore, 3m is a subgroup of 6/mmm and thus the two crystals belong to the same crystal family (hexagonal).

Hexagonal

TrigonalRhombohedral


Bravais flocks

1P

1P 2P 2CmP mC

2/mP 2/mC 222P 222C 222I 222Fmm2P mm2C mm2I mm2F

mmmP mmmC mmmI mmmF

4P 4P 4I

422P 422I4mmP 4mmI42mP 42mI4/mP 4/mI

4/mmmP 4/mmmI

m3P m3I m3F

23P 23I 23F

432P 432I 432F43mP 43mI 43mF

m3mP m3mI m3mF

mm2A

4m2P 4m2I

4I

1

2

4

8

16

12

24

48

6

3

Point group order

3P3R

3P3R 321P32R 3m1P3mR 6P 6P

3m1P3mR 622P 62mP6mmP 6/mP

6/mmmP

6m2P

312P 31mP

31mP

1

2

4

8

6

3

16

12

24

Point group order

fundamentals of fourier transformcloud.crm2.univ-lorraine.fr/pdf/antwerp2016/nespolo.pdf ·...

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