Graef M. Introduction to conventional transmission electron microscopy

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58 Basic crystallography

where (see Appendix A1)

F(α, β, γ ) = cos α cos β − cos γ.

The matrix a

is known as the direct structure matrix and it transforms crystal co-

ordinates to Cartesian coordinates. Note that its elements depend both on the direct

and reciprocal metric tensors and, thus, on the lattice parameters {a, b, c,α,β,γ},

as shown by the second equality in (1.64). The inverse transformation is given by

the inverse matrix:

= a

−1

with a

−1







−1

a tan γ

bcF(γ,α,β)

 sin γ

b sin γ

acF(β,γ,α)

 sin γ

ab sin γ









. (1.65)

The direct structure matrix is particularly useful if one wants to create a drawing

of a crystal structure, and an outline of a computer program for crystal structure

drawings is presented in the next section.

Example 1.17 Compute the Cartesian coordinates of the lattice point (2, 3, 1) in

the tetragonal lattice of Example 1.1 on page 6.

Answer: From the lattice parameters a =

and c = 1 we ﬁnd for the direct struc-

ture matrix:





001





Hence the Cartesian components of the vector (2, 3, 1) are (1,

, 1).

One can use the same formalism to determine the Cartesian coordinates of a re-

ciprocal lattice point; such coordinates would be used to draw a diffraction pattern

or any other representation of reciprocal space. To preserve the relative orientation

of crystal and reciprocal space, we look for a second structure matrix b

, which

represents the transformation from the reciprocal reference frame to the same Carte-

sian reference frame. Consider the reciprocal space vector k, with components k

with respect to the reciprocal basis vectors a

∗

and Cartesian components q

k = q

= k

∗

This can be rewritten in terms of the direct basis vectors a

k = q

= k

∗

= r

with r

= k

∗

1.9 Crystallographic calculations on the computer 59

We can now use the direct structure matrix a

to relate q

to r

= a

∗

= a

∗

= b

The reciprocal structure matrix b

is thus deﬁned by

= a

∗

. (1.66)

This matrix converts reciprocal space coordinates into Cartesian coordinates. The

inverse relation is given by

= b

−1

One can use the fact that the length of a vector must be independent of the reference

frame to show that the transpose of the reciprocal structure matrix b is equal to the

inverse of the direct structure matrix a,or

= a

−1

The reciprocal structure matrix is thus given by the transpose of the matrix in

equation (1.65). Note that only the lattice parameters are used to compute the

structure matrices. In addition, the lattice parameters are used to compute the direct

and reciprocal metric tensors. In a computer implementation it is thus convenient

to have a single routine which computes all four matrices.

Example 1.18 For the tetragonal crystal in the previous example, compute the

Cartesian components of the reciprocal lattice point (221).

Answer: The Cartesian components q

require the reciprocal structure matrix b

which is the transpose of the inverse of a





200

020

001





from which the Cartesian components of (221) follow as (4, 4, 1).

1.9 Crystallographic calculations on the computer

1.9.1 Preliminary remarks

In this section we will brieﬂy discuss how one can implement the crystallographic

calculations of the preceding sections in a computer program. The subroutines

discussed below are grouped in a library that can be used for most computations

throughout this textbook, and can be downloaded in ASCII format from the

website.

All routines were written in Fortran-90. Fortran compilers are widely available for

60 Basic crystallography

most computer platforms so that the reader should have no major problems imple-

menting the routines. While there are many other, more advanced programming

languages, Fortran was selected because it is easy to use, and because it is very well

suited for the type of computations presented in this textbook.

The

website provides the source code for all Fortran routines used in this book. In

addition to individual programs, the

website also provides a number of libraries con-

taining routines that are frequently used. The following library ﬁles are available:

local.f90 This module deﬁnes parameters that are speciﬁc to the location where

the programs are executed; the parameters in this ﬁle may be customized

by the user.

io.f90 This

module contains most of the

input and output routines

used by

other routines. All user interactions occur via routines deﬁned in this

module, so that ideally this is the only module that should have to be

modiﬁed if

the programs are implemented on a different platform.

error.f90 This module contains some simple error reporting routines.

constants

.f90

This module contains the deﬁnitions of various physical constants and

parameters used by many different programs.

math.f90 This module groups a number of basic mathematical routines, such as

matrix multiplication and inversion.

crystalvars.f90 This module deﬁnes the variable types for lattice parameter informa-

tion, direct and reciprocal metric tensors and structure matrices.

symmetryvars.f90 This module deﬁnes all variables needed to use the 3D space groups

and related group theoretical objects.

crystal.f90 This module contains all routines that are related to geometrical crystal-

lographic computations, such as dot and cross products and coordinate

transformations, and also basic routines for creating crystal data ﬁles.

symmetry.f90 This module implements the 3D space groups and related group theo-

retical objects.

ﬁles.f90 This module contains all ﬁle operations, such as opening and closing

of ﬁles, including crystal data ﬁles.

postscript.f90 Crystallographic computations generally produce graphical output,

and, unfortunately, there is no standard graphics language or library

that will work on every computer platform. For this reason graphics

output from the programs in this book is provided in PostScript format,

so that it can be printed on paper or displayed on the screen using a

PostScript viewer. PostScript is platform independent, so it should work

on UNIX, Windows, and Macintosh platforms. The ﬁle

postscript.f90

groups all routines that produce PostScript output. Additional routines

can easily be created as needed, based on the templates in this ﬁle.

tiff.f90 This module provides a simple routine to create a TIFF image ﬁle

containing a grayscale image.

1.9 Crystallographic calculations on the computer 61

diffr

action.f90

All routines related to elastic scattering are grouped in this library. This

includes computation of the relativistic electron wavelength, Fourier

coefﬁcients of the electrostatic lattice potential, extinction distances,

excitation errors, and so on. In addition this module provides many of

the support routines for multi-beam and systematic row computations.

others.f90 In this module all routines provided by other researchers are grouped

together.

graphics.f90 This module contains a number of routines for the creation of graphics

output in the form of 2D plots, shaded surfaces and contour plots. The

routines make extensive use of the

postscript.f90 routines.

All library ﬁles are combined into a single object code library during the compila-

tion process. This object library

libtem.a is then linked with the individual program

as needed. Detailed instructions for compilation of all routines are available from

the

website.

In the interest of brevity we will not reproduce any Fortran source code in this

book. Instead, we will use a simple form of pseudo code to describe the outline of

the most important computations. All pseudo code will be labeled by symbols of

the type

PC-0 . Pseudo code is a textual description of the logic of a program, and it

should be rather straightforward to follow the various steps in the actual code. In the

example

PC-0 a simple pseudo code section is shown. The top line, beginning with

Input:, states which other sections of code must be completed before executing

the current section. The next line brieﬂy describes the type of output produced

by the code, if any. The remaining lines are the pseudo code statements (on the

left) with the corresponding subroutines or functions indicated on the right. When

appropriate the pseudo code will also state which equation from the text is used

in the Fortran code. The program ﬂow statements (if, repeat until, while, ...) are

printed in bold-face characters.

Pseudo Code PC-0 Example of pseudo code.

Input: –

Output: no output

compute something

if this something is positive then

use equation 1-15 {

subroutine CalcMatrices}

else

use equation 2-11

end if

display result on screen

62 Basic crystallography

1.9.2 Implementing the metric tensor formalism

Nearly all crystallographic computations require that the user enter the lattice pa-

rameters, the space group, and the atom positions for the atoms in the asymmetric

unit. From the lattice parameters we can compute the direct and reciprocal met-

ric tensors and structure matrices, which contain all geometrical information about

crystal space and reciprocal space in a form useful for numerical computations. The

following routines are provided for user input of the crystallographic information

(their use is described in pseudo codes

PC-1 and PC-2 and the source code can

be found in the

crystal.f90 and symmetry.f90 modules).

GetLatParm This routine asks the user to enter the crystal system, followed by the

independent lattice parameters in nanometers. If the crystal system

is trigonal, the user is asked to specify hexagonal or rhombohedral

lattice parameters.

GetLatParm then calls CalcMatrices.

CalcMatrices This routine computes the direct and reciprocal metric tensors, and the

direct and reciprocal structure matrices. Direct and reciprocal lattice

parameters and all matrices are stored in a user-deﬁned variable type

(the equivalent of the Fortran-77 “common block”) for use by other

routines.

GetSpaceGroup This routine lists the space groups for the selected crystal system,

and asks the user to select by number. For space groups with two

origin settings, the user is also

asked which setting should be used.

GetSpaceGroup then calls GenerateSymmetry.

GenerateSymmetry From the space group number and setting a recursive algorithm cre-

ates all 4 × 4 symmetry matrices, as described in Section 1.9.3. The

matrices are stored in a special variable type for use by other routines.

GetAsymPos The user is prompted for the coordinates of the atoms in the asym-

metric unit, as well as the site occupation parameters and the Debye–

Waller factors.

The vector dot and cross products are then readily implemented using the met-

ric tensor formalism. The following general purpose routines are available in the

Pseudo Code PC-1 Deﬁne a new crystal structure.

Input: –

Output: crystal data ﬁle [*.xtal]

prompt for crystal system and lattice parameters {

GetLatParm}

prompt for space group number and setting {

GetSpaceGroup}

prompt for contents of asymmetric unit {

GetAsymPos}

save data in ﬁle {

SaveData}

1.9 Crystallographic calculations on the computer 63

Pseudo Code PC-2 Load crystal structure data in memory.

Input: –

Output: crystal data stored in common blocks

if this is a new crystal then

PC-1

else

read data from ﬁle [*.xtal] {

LoadData}

compute metric tensors and structure matrices {

CalcMatrices}

generate all symmetry matrices {

GenerateSymmetry, PC-3 }

end if

crystal.f90 module (the variables between square brackets are the arguments of the

function or subroutine calls).

CalcDot [p, q, s

]

Computes the dot product between two vectors. The last argument

of the function call, s

, is a switch; when the switch is “d”, the dot product is

computed using the direct metric tensor, “r” uses the reciprocal metric tensor,

and “c” is a dot product between a direct space vector and a reciprocal space

vector, which implies using the Kronecker delta as a metric tensor.

CalcCross [p, q, r, s

, s

, v] Computes the cross product

between two vectors. There are

two switches, one to indicate the space in which the input vectors are deﬁned

and the other to select the space in which the output vector will be expressed.

The last parameter indicates whether or not the cross product should be scaled

by the unit cell volume.

CalcLength [p, s

] Computes the length of a vector. This routine calls CalcDot and uses

the same switch values.

CalcAngle [p, q, s

] Computes the angle between two vectors. This routine calls CalcDot

three times, using the same value for the switch.

TransSpace [p, q, s

, s

] Transforms the components of a vector p from space s

to space

, and returns the result in the variable q.

TransCoor [p, q,α,s

, d] Transforms a vector in space s

from the old to the new refer-

ence frame (d = “on”) or from new to old (d = “no”), using the transforma-

tion matrix α. This implements all vector coordinate transformations listed

in Table 1.6.

MilBrav [p, q, d] Transforms the components of a direction vector in a hexagonal

system from the 3-index (p) to the 4-index (q) version when the switch

d = 34, and the reverse when d = 43.

The Fortran program latgeom.f90 illustrates how these routines can be used to

implement basic lattice geometry computations. We leave it as an exercise for

64 Basic crystallography

the reader to add an option to convert from Miller to Miller–Bravais indices in a

hexagonal system, using the

MilBra

subroutine. The ION-routine latgeom.pro on

the

website implements some of the crystallographic computations described in this

chapter in an interactive module. The user deﬁnes the lattice parameters and selects

two vectors. The routine returns the metric tensors, structure matrices, and the

lengths of and angles between the two vectors in both direct and reciprocal space.

1.9.3 Using space groups on the computer

The section describing the 230 space groups in International Tables for Crystallog-

raphy, Volume A [Hah96], is 606 pages long and contains a wealth of information.

To implement the space groups into a computer program appears to be a formidable

task, considering the amount of information available. It is therefore quite surpris-

ing that it is possible to compile the relevant information on all 230 space groups

into one single ASCII ﬁle which is only 4104 bytes long! Here again, we ﬁnd an

indication of the powerful tools group theory provides.

In this section we describe a simple space group encoding scheme.

†

From the

deﬁnition of a group on page 42 we know that every combination of two symmetry

operations must again be a symmetry operation. A group can then be characterized

by a limited number of operators, known as the generators, from which all other

operations can be derived by repeated multiplication. The Bravais lattice translation

vectors are obviously generators, otherwise one could never generate the inﬁnite

lattice. For the highest symmetry space group (Fm

3m), there are only seven gen-

erators from which all 192 symmetry elements in the fundamental unit cell can be

derived.

Since each generator can be freely chosen, as long as the set of generators gener-

ates the entire group, a judicious choice of the generators for the 230 space groups

results in a list of only 14 different point symmetry operators. In other words, if

we label and store the 3 × 3 matrices of those 14 operations, then for each group

we only need to list the labels of the matrices to be used as generators and the

components of the translation vectors associated with the generators, and this re-

sults in only one line of data per space group. It is then up to the algorithm to take

the generators and compute all possible products between them until no more new

operators are found.

The algorithm creates all the 4 × 4 matrices for the operations inside the funda-

mental unit cell (the cell enclosed by a

). Adding a translation vector t = t

to the

positions generated by the matrices then translates the position to a neighboring

†

The author would like to acknowledge G. Ceder (MIT) for permission to make his source code available.

1.9 Crystallographic calculations on the computer 65

unit cell. For the purposes of this book, it is sufﬁcient to only use the operators in

the fundamental unit cell.

To generate all atom positions inside a unit cell when the positions in the asym-

metric unit are given, we compute all the symmetry matrices ﬁrst, and then apply

all matrices to the coordinates of each input atom. Then we remove identical po-

sitions (which can occur if the input coordinates coincide with one of the special

positions in the space group) to end up with the orbit of the input atom. The coor-

dinates thus generated are fractional coordinates with respect to the unit cell basis

vectors. They can be used in subsequent computations of, for example, the structure

factor or Fourier coefﬁcients of the electrostatic lattice potential (see later chapters).

In Section 1.8 we described a general method to convert the fractional coordinates

to Cartesian coordinates, so that crystal structure drawings or diffraction patterns

can be generated.

The following lines list the ASCII codes for a few selected space groups, together

with the space group symbol and number:

P1 1 01aOOO0

C222 21 03aDDObOOOcOOO0

Cmcm 63 13aDDObOODcOOD0

P4/nbm 125 04bOOOgOOOcOOOhDDO1YYO

3m 227 07aODDaDODbODDcDDOdOOOeFBFhBBB1ZZZ.

The encoding scheme works as follows: consider a crystal structure described

by the orthorhombic space group Cmcm (space group # 63). This space group

is completely encoded by the character string 13aDDObOODcOOD0. This string

should be read as follows: the ﬁrst digit takes on the value 1 or 0, to indicate whether

or not the space group contains the inversion operator as a symmetry element. In the

case of Cmcm the inversion operator is indeed one of the generators. The next digit

gives the number of additional generator elements, in this case 3. Each generator

is then described by four characters, and there are three such quadruplets in the

string: aDDO, bOOD, and cOOD. Their meaning will be discussed in the next

paragraph. The next character in the string (0) is only different from zero if the

space group has more than one origin setting in the International Tables. For the

space group Cmcm this character is equal to zero, indicating that there is only one

origin setting. For the space group #125 on the other hand, the character following

the four generator quadruplets is equal to one, indicating that a second origin setting

is available. Since a change in origin implies a translation, there are three additional

uppercase characters in the string, which represent the components of the translation

vector that transforms coordinates in the ﬁrst setting to coordinates in the second

setting (using the conventions of the International Tables). For space group #125

the translation vector for the second setting is represented by the character triplet

66 Basic crystallography

YYO which, according to the conversion Table A3.2 in Appendix A3, corresponds

to the vector with components [−

, −

, 0].

Each generator matrix is a 4 × 4 matrix; the 3 × 3 submatrix corresponding to the

point symmetry operation is labeled by one of the 14 lowercase letters a through n.

The actual matrix elements are listed in Table A3.1 in Appendix A3. Since a space

group operator may also contain a translational part, the four-character string aDDO

contains three additional uppercase characters, representing the three components

of the translational part of the symmetry operator. From the table in Appendix A3

one can see that the character D corresponds to the value

, and the character O to

zero. The string aDDO thus represents the 4 × 4 matrix

aDDO ≡







100

010

0010

0001







and the string bOOD represents the matrix

bOOD ≡







−1000

0 −100

001

0001







and so on. The remaining elements of the space group Cmcm are then generated

from the three matrices aDDO, bOOD, and cOOD, and the inversion operator by

ﬁrst computing all the powers of the generators, and then all products between

generators until no further unique symmetry elements can be found (see

PC-3 ).

This computation is performed by the subroutine

GenerateSymmetry, which then

stores all symmetry operator matrices in a special user-deﬁned variable type for

use by other routines. The Fortran program

listSG.f90 can be used to list all the

equivalent point positions for a general point (x, y, z) for any given space group.

Once all symmetry matrices are known it is straightforward to compute the

members of families, orbits, and stars. Pseudo code

PC-4 shows how one can

compute the equivalent points for an atom in the asymmetric unit. The routines

CalcOrbit and CalcStar implement the pseudo code. Note that these routines can also

be used to generate equivalent point positions for the point groups. Both routines

have a switch to turn on or off the reduction of the coordinates to the fundamental

unit cell. If reduction is turned off, then the coordinates of the equivalent points

will represent the point group symmetry with respect to the origin of the unit cell.

The Fortran program

orbit.f90 implements the CalcOrbit routine to create all the

equivalent points for an arbitrary space group; the program prompts the user for the

1.9 Crystallographic calculations on the computer 67

Pseudo Code PC-3 Generate space group symmetry matrices.

Input: space group number {GetSpaceGroup}

Output: array of all non-equivalent symmetry matrices

load and decode the space group code string

determine the generator matrices

repeat

multiply generators amongst themselves and resulting matrices

until no more new matrices found

reduce all translation components to the fundamental unit cell

identify the point operations D

(k)

and store their 3 × 3 matrices

compute the reciprocal point symmetry matrices D

∗(k)

{eqn. 1.42}

store all independent symmetry matrices

Pseudo Code PC-4 Compute the orbit of a given point.

Input: crystal data ﬁle [*.xtal]

Output: equivalent atom positions

load crystal data {

PC-2 }

for each symmetry matrix do

multiply matrix with atom coordinates

if this is a new position then

keep position and increment counter

if reduction required then

reduce position to fundamental unit cell

end if

else

discard position

end if

end for

return list and number of equivalent positions

space group number and an atom position, and reduces all generated coordinates

to the basic unit cell. Members of families of directions or planes can be computed

in the same way, simply by passing the indices of a general direction or plane to

the

CalcFamily routine. The CalcFamily routine is particularly useful to generate

stereographic projections, as described in Section 1.9.5.