Graef M. Introduction to conventional transmission electron microscopy

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68 Basic crystallography

1.9.4 Graphical representation of direct and reciprocal space

There are many different ways to represent crystal structures: amongst the more

commonly used ones are the ball-and-stick representation, the space-ﬁlling repre-

sentation, and polyhedral models to emphasize local coordinations. All of these

representation methods can be used with both orthogonal and perspective projec-

tions. All of these methods consist of two basic steps:

(i) computation of the Cartesian coordinates of each atom in the unit cell(s);

(ii) computation of the projected (or screen) coordinates of each atom.

The latter step will not be discussed in detail in this book. We refer to the extensive

literature on computer graphics libraries (e.g. [NDW93, SS87] and many others).

This computation essentially involves perspective or parallel projections, for a given

location and orientation of the object (the crystal) and the observer. Note that the

4 × 4 matrices used to represent symmetry operations are also used for perspective

projections; it turns out that the elements on the fourth row of a 4 × 4 matrix can

be used to describe various geometric projections in a very compact way.

The ﬁrst step above can easily be completed with the tools derived earlier in this

chapter. After generating the orbit of each atom in the asymmetric unit (

CalcOrbit),

Pseudo Code PC-5 Draw a crystal structure or reciprocal space.

Input: crystal data ﬁle [*.xtal]

Output: PostScript ﬁle(s)

Load crystal data {

PC-2 }

if Real Space Drawing then

compute atom coordinates for each atom in asymmetric unit {

CalcOrbit}

how many unit cells in the drawing?

complete unit cell faces, edges and corners

convert all coordinates to a Cartesian reference frame {

TransSpace}

else

how many reciprocal unit cells in the drawing?

for each reciprocal lattice point do

compute structure factor {

PC-9 }

convert all coordinates to a Cartesian reference frame {

TransSpace}

end for

end if

repeat

construct Cartesian projection matrix for given viewing direction

create PostScript drawing (using atom radii or |structure factor|

)

until no more drawings needed

1.9 Crystallographic calculations on the computer 69

the coordinates are converted from crystal space to the standard Cartesian reference

frame (

TransSpace). The resulting Cartesian coordinates are then entered into the

graphics system pipeline, where they are combined with the projection information

to produce a 2D drawing. The viewing or projection direction can usually be deﬁned

in two ways: either with respect to a Cartesian reference frame, or, more commonly,

by means of the direction indices [uvw]. If the latter mechanism is used, care must

be taken to also transform the direction indices to the standard Cartesian reference

frame, using the direct structure matrix a

(TransSpace). The details of drawing the

atom (radius, color, texture, etc.) are left to the reader; using the program

drawcell.f90

a simple PostScript drawing can be generated for an arbitrary unit cell. The pseudo

code for this program is shown in

PC-5 . An example drawing for the structure of

rutile (TiO

, space group P4

/mnm (#136), a = 0.4594 nm, c = 0.2958 nm, Ti at

2a (0, 0, 0),Oat4f(x, x, 0) with x = 0.3) is shown in Fig. 1.24.

Drawings of the reciprocal space lattice can be made in the same way: ﬁrst we

determine the set of reciprocal lattice points to be included in the drawing, then

the reciprocal structure matrix b

is used to convert the (hkl) triplets to Cartesian

coordinates (

TransSpace

), and ﬁnally these coordinates are passed on to the graphics

Crystal Structure Drawing

Filename: rutile.xtal

a [nm]

b [nm]

c [nm]

alpha [deg]

beta [deg]

gamma [deg]

: 0.4594

: 0.2958

: 90.0000

Viewing Direction [ 0 0 1]

Drawing of direct space

Fig. 1.24. Perspective drawing of the structure of rutile, created with the drawcell.f90

program.

70 Basic crystallography

pipeline for display. The main advantage of using the direct and reciprocal structure

matrices deﬁned in Section 1.8 is that the relative orientation of direct and reciprocal

space is properly represented and, in addition, the algorithm is independent of the

particular choice of Bravais lattice. We will postpone discussion of a program for

drawing electron diffraction patterns until Chapter 4. The program

drawcell.f90 can

also be used to create a perspective or parallel projection drawing of the reciprocal

lattice; the “size” of each reciprocal lattice point is determined by the value of the

corresponding structure factor, as discussed in Chapter 2.

1.9.5 Stereographic projections on the computer

There are several computer programs available that allow the user to draw stereo-

graphic projections of direct and reciprocal space. While these programs are often

sufﬁcient for regular use, there may be situations when the TEM user will need to

perform a certain operation that is not a feature of any of the existing programs. In

such a situation it may be necessary to write a dedicated program to generate certain

patterns. In this subsection, we will describe a straightforward method to compute

stereographic projections for arbitrary crystals, in both direct and reciprocal space.

We already know from Section 1.8 how to convert crystal or reciprocal coordi-

nates into Cartesian coordinates. Since the SP represents directions, the next step

is to normalize the Cartesian components so that the direction corresponds to a

point on the unit sphere. Let us assign the coordinates (x, y, z) to this point P,as

illustrated in Fig. 1.25. If we connect the point P with the south pole S, then the

Fig. 1.25. Schematic representation of the relation between 3D Cartesian coordinates and

stereographic projection coordinates.

1.9 Crystallographic calculations on the computer 71

triangle PQS is congruent with P



OS, from which we derive:



1 + z

;



1 + z

If the projection circle has radius R, then the above equations are simply multiplied

by R to arrive at the proper 2D coordinates.

A projection of the reciprocal space directions (or plane normals) is easily gener-

ated by ﬁrst transforming the coordinates of each reciprocal lattice point to Cartesian

coordinates, and then proceeding in the same way as described above. We will now

illustrate the use of stereographic projections in three programs: the ﬁrst program

draws the projection of all plane normals or directions (with indices in a given

range) for an arbitrary crystal in an arbitrary orientation. The second program does

the same but only for selected families of planes or directions, and the third pro-

gram allows for superposition of the projections from two crystals, with a given

orientation relation between them.

The pseudo code for the ﬁrst program is shown in

PC-6 . The program can be

found on the

website under the name stereo.f90. Multiple projections can be made

for a given crystal structure, and a [111] projection of a cubic crystal is shown

in Fig. 1.26a. The second program,

family.f90, is described in

PC-7 , and is only

slightly different from

stereo.f90. One member of each family must be speciﬁed

and the point symmetry operators are then used to compute the entire family

Pseudo Code PC-6 Stereographic projection of a range of directions/plane

normals.

Input: PC-2 , PC-3

Output: stereographic projection in PostScript format

repeat

determine projection direction

real space or reciprocal space projection?

open a new PostScript ﬁle and draw projection circle

repeat

which range of directions/plane normals should be drawn?

transform all members to Cartesian space and project

draw (and label) the points

until drawing complete

close PostScript ﬁle

until no further patterns required

72 Basic crystallography

( 1 01)

( 2 11)

( 2 12)

( 1 10)

( 2 21)

( 1 11)

( 1 1 2)

( 1 02)

( 1 10)

( 1 11)

( 1 12)

( 1 21)

( 1 20)

( 1 21)

( 1 22)

( 0 1 1)( 0 1 1)

( 0 1 2)

( 0 0 1 )( 0 0 1 )

( 0 1 1 )

( 010)

( 0 1 1 )

( 012)

( 0 1 1 )

( 0 2 1 )

( 010)

( 021)

( 011)

( 1 2 1)

( 1 2 2)

( 1 1 0)

( 1 1 1)

( 1 1 2)

( 1 0 1 )

( 100)

( 1 0 1 )

( 102)

( 112) ( 1 1 1 ) ( 1 1 0 ) ( 111) ( 112)

( 1 2 2 )

( 1 2 1 )

( 1 2

( 121)

( 1 2 2 )

( 1 1 0)

( 2 2 1)

( 1 1 1)

( 2 1 1 )

( 2 1 0)

( 2 1 1)

( 2 1 2)

( 2 0 1 )

( 100)

( 201)

( 101)

( 2 1 2 )

( 2 1 1 )

( 210)

( 211)

( 2 1 2 )

( 1 1 1 ) ( 221) ( 1 1 0 ) ( 221) ( 111)

[112]

[ 211]

[ 112]

[ 121]

[121]

[112]

[121]

[211]

[121]

2[ 11]

(a)

(b)

Fig. 1.26. (a) Stereographic projection of the plane normals of a cubic crystal viewed along

the [111] direction (created with

stereo.f90); (b) stereographic projection of the 112family

in a cubic crystal (created with

family.f90).

(CalcFamily). Example program output is shown in Fig. 1.26b for the {112} family

in a cubic crystal.

The third program,

orient.f90, is described in PC-8 . The program asks for two

crystal structure ﬁles, and a complete orientation relation (two parallel planes and

1.9 Crystallographic calculations on the computer 73

Pseudo Code PC-7 Stereographic projection of a family.

Input: PC-2 , PC-3

Output: stereographic projection in PostScript format

repeat

determine projection direction

real space or reciprocal space projection?

open a new ﬁle and draw projection circle

repeat

which direction/plane normal should be drawn?

compute members of its family {

CalcFamily}

transform all members to Cartesian space and project

draw (and label) the points

until drawing complete

close PostScript ﬁle

until no further patterns required

Pseudo Code PC-8 Stereographic projection of orientation relation for two

crystals.

Input: PC-2 , PC-3 {for two crystals}

Output: stereographic projection in PostScript format

obtain orientation relation

real space or reciprocal space projection?

compute transformation matrix α

{1.67}

determine range of directions/plane normals

transform all quantities to reference frame of crystal A

transform all quantities to the standard Cartesian frame

repeat

determine projection direction and projection matrix

proceed as in

PC-6 for both crystals

until no further patterns required

two parallel directions in those planes), and then draws the two patterns superim-

posed on the same drawing. This program is a bit more complex than the other two

because it combines nearly all aspects of crystallography discussed in this chapter.

The orientation relation (OR) between the two crystals consists of the following

74 Basic crystallography

information:

= (hkl)

 (hkl)

= g

;

= [uvw]

 [uvw]

= t

The direction t must lie in the plane g, and hence the zone equation t · g = 0 must be

satisﬁed for each crystal. All crystallographic information must now be converted

to one single reference frame, say that of crystal A. We must therefore determine

the transformation matrix α

, which expresses the orientation relation above. This

can be done as follows.

(i) Construct a Cartesian reference frame in each crystal; the z-direction of this reference

frame is taken parallel to the direction vector t

A,B

, the x-direction is parallel to the

plane normal g

A,B

, and the y-direction completes the

right-handed orthonormal frame:

A,B

;

A,B

= e

A,B

× e

A,B

;

A,B

(ii) Rewrite the equations above in matrix form:

= E

;

= E



(iii) Equate the Cartesian reference frames of both crystals, so that the proper OR is ob-

tained:

= E



(iv) Solve this

equation for

= E

−1



= α

, (1.67)

and this deﬁnes the transformation matrix α. We can now apply the transformation

relations in Table 1.6 to convert quantities between the properly oriented crystals A

and B.

Now that the transformation matrix for the OR is known, all remaining compu-

tations proceed as before and the stereographic projection of the two crystals can

be produced for an arbitrary viewing direction. The same method can be used to

compute diffraction patterns for two crystals with a given OR. An example of the

output of the

orient.f90 program is shown in Fig. 1.27 for two cubic crystals with a

standard twin orientation relationship (111)

 (111)

and [11

 [

12]

1.9 Crystallographic calculations on the computer 75

A B

(11 1 )

(22 1 )

(11 0)

(22 1)

(11 1)

(21 2 )

(21 1 )

(21 0)

(21 1)

(21 2)

(101)

(201)

(100)

(201)

(101)

(212)

(211)

(210)

(211)

(212)

1 11)

(221)

(110)

(221)

(111)

(12 2 )

(12 1 )

(12 0)

(12 1)

(12 2)

(11 2 )

(11 1 )

(11 0)(11 0)

(11 0)

(11 1)

(11 2)

(102)

(101)

(100)

(101)

(102)

(112)

(111)

(110)

(111)

(112)

(122)

(1 21)

(120)

(121)

(122)

(011 )

(021 )

(010)

(021)

(011)

(012 )

(011 )

(010)

(011)

(012)

(001)

( 001)

(001) ( 001)

(001)

( 001)

(012)

(011)

( 010)

( 011)

( 012)

(011)

(021)

( 010)

( 021)

( 011)

(122 )

(121 )

(1 2 0)

(121)

(122)

(112 ) (111) (110) (111) (112)

(102)

(101)

( 100)

( 101)

( 102)

(112)

(111)

( 111)

( 110)

(110)

( 111)

(111)

(112)

( 122)

(121)

( 120)

( 121)

( 122)

(111 ) (221) (110) (221) (111)

(212 )

(211 )

(210)

(211)

(212)

(101)

( 201)

( 100)

( 201)

( 101)

(212)

(211)

( 210)

( 211)

( 212)

(111)

( 111)

(221)

( 221)

( 110)

(110)

( 221)

(111)

Fig. 1.27. [1

10] stereographic projection of two cubic crystals in twin orientation (cre-

ated using

orient.f90); the size of the projection circle was reduced to ﬁt on the page, and

consequently some of the indices overlap.

Example 1.19 Consider the orientation relation (111)

 (111)

and [11



[

12]

in a cubic crystal. Compute the transformation matrix α

Answer: For the ﬁrst basis vector e

we have

g = g

∗

= g

∗

which after normalization becomes



76 Basic crystallography

This relation is valid for each crystal A and B. For g = (111) we ﬁnd

A,B

√

)

A,B

+ a

A,B

+ a

A,B

The third basis vector e

is given by



and therefore

√

)

A,B

+ a

A,B

− 2a

A,B

; e

=−e

The vector e

is then determined from the cross product:



ijk



from which we ﬁnd

√



− a



; e

=−e

The matrices E and E



are given by

√





√

3 −

√

11−2





; E



√





√

−

√

−1 −12





The inverse of E is readily shown to be equal to

−1

= a

√







√

−1

√

−2







The matrix α = E

−1



is then given by





−12 2

2 −12

22−1





Note that this is a symmetric unitary matrix, and therefore it is equal to its own

inverse and to its transpose. The same matrix can therefore be used for all the

transformations listed in Table 1.6.

Exercises 77

1.10 Recommended additional reading

The reader may ﬁnd useful additional information on crystallography and symmetry

in the following books:

Crystal Structures: A Working Approach, H.D. Megaw, W.B. Saunders Company

(Philadelphia, 1973).

Essentials of Crystallography, D. McKie and C. McKie, Blackwell Scientiﬁc Publications

(Oxford, 1986).

Manual of Mineralogy, C. Klein and C.S. Hurlbut, Jr., 20th edition, John Wiley & Sons

(New York, 1985).

Space Groups for Solid State Scientists, G. Burns and A.M. Glazer, 2nd edition, Academic

Press (Boston, 1990).

Crystallography, W. Borchardt-Ott, 2nd edition, Springer (Berlin, 1995).

Point Groups, Space Groups, Crystals, Molecules, R. Mirman, World Scientiﬁc (Singa-

pore, 1999).

A good introduction to vector calculus in non-Cartesian coordinate frames can be

found in:

Vector and Tensor Analysis with Applications, A.I. Borisenko and I.E. Tarapov, Dover

Publications (New York, 1979).

Introduction to Vector

and Tensor Analysis

, R.C. Wrede, Dover Publications

(New York,

1972).

A basic introduction to programming in Fortran-77 and Fortran-90 can be

found in:

Numerical Recipes in FORTRAN: the Art of Scientiﬁc Computing, W.H. Press,

B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 2nd edition, Cambridge University

Press (New York, 1992).

FORTRAN 77 for Engineers and Scientists: with an Introduction to FORTRAN 90,L.

Nyhoff and S. Leestma, 4th edition, Prentice Hall (New Jersey, 1996).

Fundamentals of Engineering Programming with C and Fortran, H.R. Myler, Cambridge

University Press (New York, 1998).

Fortran 90/95 for Scientists and Engineers, S.J. Chapman, McGraw-Hill (Boston, 1998).

Exercises

1.1 A monoclinic crystal has lattice parameters {1,

, 2, 90, 60, 90} (distances in nm).

(a) Write down the explicit expression

for its metric tensor.

(b) Compute the distance between the origin and the point





(d) Compute the reciprocal metric tensor.