Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials

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Chapter

Figure

1.38.

Selected crystallographic directions in the orthorhombic lattice.

1.15

Reciprocal lattice

The concept of a reciprocal lattice' was first introduced by Ewald2 and it

quickly became an important tool in the illustrating and understanding of

both the diffraction geometry and relevant mathematical relationships. Let a,

and c be the elementary translations in a three-dimensional lattice (called

here a direct lattice), as shown for example in

Figure

1.4.

A second lattice,

reciprocal to the direct lattice, is defined by three elementary translations a*,

and c*,~ which simultaneously satisfy the following two conditions:

All products in Eqs. 1.12 and 1.13 are scalar (or dot) products. The dot

product of the two vectors,

and

is defined as a scalar quantity, which is

equal to the product of the absolute values of the two vectors and the cosine

of the angle

between them:

For additional information see IUCr teaching pamphlets: A. Authier, The reciprocal

lattice,

http://www.iucr.org/iucr-top/comm~cteach/pamphlets/4/index.html

and the

International Tables for Crystallography, vol. A and vol.

Peter Paul Ewald (1888-1985). German physicist, whose work [P.P. Ewald, Das reziproke

Gitter in der Strukturtheorie,

Kristallogr.

56,

129 (1921)l is considered a landmark in

using reciprocal lattice in x-ray diffraction.

Symbol with an asterisk always refers to a parameter of reciprocal lattice.

Fundamentals of crystalline state

Conversely, the vector (or cross) product of the same two vectors (vlxv2)

is defined as a vector v3 in the direction perpendicular to the plane of vl and

vz, whose magnitude is equal to the product of the absolute values of the two

vectors and the sine of the angle

between them, or

other words, the length of the vector v3 is equal to the area of the

parallelogram formed by the vectors vl and vz (shaded in

Figure

1.39),

and

its direction is perpendicular to the plane of the parallelogram.

Considering Eqs. 1.12 to 1.15, it is possible to show that the elementary

translations in the reciprocal lattice are defined as

bxc cxa axb

a*=__ b*=-,

c*=-

v v

and that the inverse relationships are also true, i.e.

Eqs.

1.16

and

1.17,

the two scalar quantities

and

are the volumes

of the unit cell in the direct and reciprocal lattices, respectively. Hence,

perpendicular to both

and

c; b*

is perpendicular to both

and

perpendicular to both

and

terms of the interplanar distances,

perpendicular to the corresponding crystallographic planes, and its length is

inversely proportional to

i.e.

Figure

1.39.

Vector (cross) product of two vectors. The orientation

is determined using

the right-hand rule: thumb of the right hand is aligned with

vl,

index finger with

vt,

then

aligned with the middle finger. Tails

all vectors face the middle

the palm.

Chapter

An important consequence of Eq. 1.18 is that a set, which consists of an

infinite number of crystallographic planes in the direct lattice, is replaced by

a single vector or by a point at the end of the vector in the reciprocal lattice.'

Furthermore, Eqs. 1.16 and 1.17 can be simplified in the orthogonal crystal

systems to

lla, b*

llb, c*=llc (1.19)

The two-dimensional example illustrating the relationships between the

direct and reciprocal lattices (or spaces), which are used to represent crystal

structures and diffraction patterns, respectively, is shown in

Figure

1.40.

important property of the reciprocal lattice is that its symmetry is the same as

the symmetry of the direct lattice. However, in the direct space atoms can be

located anywhere in the unit cell, whereas diffraction peaks are represented

only by the points of the reciprocal lattice, and the unit cells themselves are

"empty" in the reciprocal space. Furthermore, the contents of every unit cell

in the direct space is the same, but the intensity of diffraction peaks, which

are conveniently represented using points in the reciprocal space, varies.

Reciprocal space and

diffraction pattern

a,,

'30

CP41

'31

@42

*32

0S43

@33

Xrect space and

$12)

\ \ \ \ \

crystal structure

Figure

1.40.

Example of converting crystallographic planes in the direct lattice into points in

the reciprocal lattice. The corresponding Miller indices are shown near the points in the

reciprocal lattice.

Hence, any vector in a three-dimensional reciprocal lattice is determined as

d*hkl

ha*

kb*

lc*.

Also see

Eq.

2.35

and

Figure

2.40.

Fundamentals

crystalline state

1.16

Crystallographic space groups

Similar to finite symmetry elements, which can be combined in point

groups (see section 1.10), various combinations of the same symmetry

elements plus allowed translations, while obeying the rules described in

section 1.7, result in the so-called crystallographic space groups. As follows

from the differences in their names, symmetry elements in a space group are

spread over the space of an infinite (continuous) object, contrary to point

groups, where all symmetry elements have at least one common point.

Therefore, each point of a continuous object can be moved in a periodic

fashion through space by the action of symmetry elements that

form a space

group symmetry, whereas at least one point of the object remains unmoved

by the action of symmetry elements that belong to a point group symmetry.

Given the limitations on the allowed rotations and translations (see

sections 1.4 and 1.13), there is a total of 230 three-dimensional

crystallographic space groups,' which were derived and systematized

independently by Fedorov2 and S~honflies.~ A complete list of space groups

is found in

Table

1.1

where they are arranged according to seven crystal

systems (see section 1.8) and

crystallographic point groups (see section

1.10).

1.16.1

Relationships between space and point groups

The symbols of 230 crystallographic space groups, which are found in

Table

1.1

are known as short international or short Hermann-Mauguin

symbols. They are based on the symbols of the corresponding point groups.

The orientation of symmetry elements with respect to the three major

crystallographic axes in a space group is the same as in the parent point

group (see

Table

1.8) and it depends on the position of the element in the

symbol. The rules that govern space group symbolic are quite simple:

When vector directions (e.g. magnetic moments) are included into consideration, the

number of space groups increases dramatically. Thus, a total of 1651 dichromatic (or

Shubnikov) symmetry groups are used to treat symmetry of magnetically ordered

structures. See A.V. Shubnikov and V.A. Koptsik, Symmetry in science and art, Plenum

Press (1974) for a brief description of color symmetry groups. A complete treatment of

dichromatic groups is found in

V.A.

Koptsik, Shubnikov groups, Moscow University

Press (1966).

Evgraf Stepanovich Fedorov (1 853-1919). Russian mineralogist and crystallographer who

by applying the theory of finite groups to crystallography derived 230 space groups in

1891. Fedorov's last name is spelled as Fyodorov or

Fedoroff in some references.

Arthur Moritz Schonflies (1853-1928). German mathematician who derived

230

space

groups independently of E.S. Fedorov in 1891. See

http:Nwww-gap.dcs.st-and.ac.uk/

-history/Mathematicians/Schonflies.html

for a brief biography.

Chapter

Table

1.17.

The

230

crystallographic space groups arranged according to seven crystal

systems and

crystallographic point groups as they are listed in the International Tables for

Crystallography, vol. A. The centrosyrnmetric groups are in

bold,

while the non-

centrosymmetric groups that do not invert an object are in

italic.

The remaining are non-

centrosymmetric groups that invert an object (contain inversion axis or mirror plane).

Crystal system

(total number of Point

Space groups based on a given point group symmetry (the

superscript indicates the space group number as adopted in the

space groups

symmetry)

International Tables for Crystallography, vol. A)

Triclinic

(2)

Monoclinic

(13)

Orthorhombic

(59)

Tetragonal

(68)

Trigonal

(25)

2/m

222

mrn2

mmm

4/m

422

4mm

4m2

Wmmm

Fundamentals

crystalline state

Crystal system

Space groups based on a given point group symmetry (the

(total number of Point

superscript indicates the space group number as adopted in the

space groups

symmetry)

International Tables for Crystallography,

vol.

P3m1 '56, P3 ~m'~~, P3c1

15',

~31c'~~, R3mI6O, R3cI6'

~31m'~', P~ICI~~, ~?ml'~~, ~?cl'~~, ~3m~~~,

~5'~~

Cubic

(36)

The international crystallographic space group symbols begin with a

capital letter designating Bravais lattice, i.e.

(see

Table

1.13

and

Table

1.14).

The point group symbol, in which rotation axes and mirror planes can be

substituted with allowed screw axes or glide planes, respectively, is

added as the second part of the international space group symbol.

some cases, the second and the third symmetry elements of the point

group can be switched, e.g. point group symmetry am2 produces two

different space groups without introducing glide planes

andlor screw axes

in the space group symbol, i.e. ~42m and ~-4rn2.

These rules not only reflect how the symbol of the space group symmetry

is constructed, but they also establish one of the possible ways to derive all

230 crystallographic space groups:

First, consider the Bravais lattices (see

Table

1.14)

and point groups (see

Table

1.8)

which are allowed in a given crystal system.

Second, consider all permissible substitutions of mirror planes and/or

rotation axes with glide planes and screw axes, respectively.

this step,

it is essential to examine whether or not the substitution results in a new

group or the resulting group can be reduced to the existing one by

permutation of the unit cell edges.

Chapter

For example, think about the monoclinic point group m in the standard

setting, where m is perpendicular to

(Table

1.8). According to

Table

1.14,

the following Bravais lattices are allowed in the monoclinic crystal system: P

and C. There is only one finite symmetry element (mirror plane m) to be

considered for replacement with glide planes (a, b, c, n and d):

The first and obvious choice of the crystallographic space group is Pm.

By replacing m with a, we obtain new space group, Pa.

Replacing m with b is prohibited since the plane is perpendicular to b.

By replacing m with c, we obtain space group PC. This space group

symmetry is identical to Pa, which is achieved by switching

and c

because glide plane, a, translates a point by 112 of full translation along a,

and glide plane, c, translates a point by

112 of full translation along c. PC

is a standard choice (see

Table

1.17).

By replacing m with n, we obtain space group Pn. This space group can

be converted into PC when the following transformation is applied to the

unit cell vectors: anew

-aold, bnew

bold, cnew

aold

cold.

Glide plane, d, is incompatible with the primitive Bravais lattice.

By repeating the same process in combination with the base-centered

lattice, C, two new space groups symmetry, Cm and Cc can be obtained.

Therefore, the following four monoclinic crystallographic space groups

(Pm,

PC, Cm and Cc) result from a single monoclinic point group (m) after

considering all possible translations in three dimensions.

1.16.2

Full international symbols of crystallographic space groups

The 230 crystallographic groups listed in

Table

1.1

are given in the so-

called standard orientation (or setting), which includes proper selection of

both the coordinate system and origin. However, there exist a number of

publications in the scientific literature, where space group symbols are

different from those provided in

Table

1.17. Despite being different, these

symbols refer to one of the same 230 crystallographic space groups but using

a non-standard setting or even using a non-standard choice of the coordinate

system. These ambiguities primarily occur because of the following:

1. The crystal structure was solved using a non-standard setting, since most

of the modern crystallographic software enables minor deviations from

the standard, and the results were published as they were obtained,

without converting to a conventional orientation. It is worth noting that

many, but not all technical journals allow certain deviations from

crystallographic standards.

The crystal structure contains some specific molecules, blocks, layers, or

chains of atoms or molecules, which may be easily visualized or

represented using space group symmetry in a non-standard setting.

Fundamentals

crystalline state

Deviations from the standard are most often observed in the monoclinic

crystal system, because there are many different ways that result in a non-

standard setting in this crystal system. This uncertainty is even reflected in

the International Tables for Crystallography, where there are two different

settings in the monoclinic crystal system. When the unique two-fold axis is

parallel to

(i.e. to Y-axis), this setting is considered a standard choice, but

when it is parallel to

(or to Z-axis), this is an allowed alternative setting. In

addition, the unique two-fold axis can be chosen to be parallel to

(i.e. to

X-

axis), which is considered a non-standard setting.

To reflect or to emphasize a non-standard choice of space group

symmetry, the so-called full international or full Herrnann-Mauguin symbols

can be used. In the full symbol, both the rotation axes parallel to the specific

direction (see

Table

1.8)

and planes perpendicular to them (if any) are

specified for each of the three positions. If the space group has no symmetry

element parallel or perpendicular to a given direction, the symbol of the one-

fold rotation axis is placed in the corresponding position. For example,

P121/al and P1121/a, both refer to the same space group symmetry where the

orientation of the crystallographic axes,

and

is switched. Full

international symbols may also be used to designate space group symmetry

in a standard setting, such as when the short notation Pmmm is replaced by

the full Hermann-Mauguin notation P2/rn2/m2/m. The full symbol in this

case emphasizes the presence of the three mutually perpendicular two-fold

axes and mirror planes that are parallel and perpendicular, respectively, to

the three major crystallographic axes.

Thus, one of the most commonly observed in both natural and synthetic

crystals, space group symmetry

P2Jc (standard notation with the unique

two-fold screw axis parallel to Y, and

90" and

90") can be listed

as follows:

P21hll and P21/cll, when the unique two-fold screw axis is chosen to

be parallel to

Both symbols represent non-standard settings of this

group, in which

90".

P121/cl (or sh0rtP2~/c, whichis standard) andP121/al (or sh0rtP2~/a,

which is also standard except for the choice of the glide plane), when the

unique two-fold screw axis is chosen to be parallel to Y.

P 1 12Ja and P 1 12,h, when the unique two-fold screw axis is chosen to

be parallel to Z (both represent alternative setting, where the unique two-

fold screw axis is parallel to Y, and therefore,

90").

P2,/nll, P121/nl (or short P2Jn) or P1 121/n, when the unique two-fold

screw axis is chosen to be parallel to

or Z, respectively, but the

selection of the diagonal glide plane, n, represents a deviation from the

standard.

Chapter

Since the selection of the crystallographic coordinate system is not

unique, conventionally, a right-handed set' of basis vectors a,

is chosen

in compliance with Table

1.12

and in a way that the combination of

symmetry elements in the space group is best visualized. The selection of the

conventional origin is usually more complicated, but in general the origin of

the coordinate system is selected at the center of inversion, if it is present, or

at the point with the highest site symmetry, if there is no center of inversion

in the group.

Additional information about each of the 230 three-dimensional

crystallographic space groups can be found in the International Tables for

Crystallography, vol. A. It includes their symbols, diagrams of all symmetry

elements present in the group together with their orientation with respect to

crystallographic axes, the origin of the coordinate system and more, for both

the conventional and alternative (if any) settings. The format of the

International Tables for Crystallography and some relevant issues are briefly

discussed in the following section 1.17.

1.16.3 Visualization

space group symmetry in three dimensions

excellent way to achieve a better understanding of both the infinite

symmetry elements and how they are combined in crystallographic space

groups is to use three-dimensional illustrations coupled with computer

animation capabilities. For example, the crystal structure of the vanadium

oxide layer in

tmaVsOzo (tma is short for tetramethylammonium,

[N(CH&]+)~

is shown as a still image in Figure

1.41,

and it is also available

as an electronic figure on the CD, which accompanies this book. The crystal

structure is shown with the unit cell and its symmetry elements in the VRML

(Virtual Reality Modeling Language) file format (Figure 1.41.wrl). The file

may be displayed using a Web browser, which includes a VRML viewer as a

plug-in. Standalone VRML viewers can be downloaded from the Web,

e.g.

from the Web3D repository

(http://www.web3d.org/vrml/browpi.htm).

In addition to visualizing the crystal structure in three dimensions, using

VRML enables one to move and rotate it. It is also possible to virtually "step

inside" the lattice or the structure and to examine them from there. The

pseudo three-dimensional drawing of the trnaVs020 crystal structure on the

was created using the General Structure Analysis System (GSAS),3 and

In a right-handed set the positive directions of basis vectors

are chosen from the

middle of the palm of the right hand towards the ends of thumb, index and middle fingers,

respectively.

Chirayil,

P.Y.

Zavalij, M.S. Whittingham, Synthesis and characterization of a new

vanadium oxide,

tmaVsOzo,

Mater. Chem.

2193 (1997).

A.C. Larson and R.B. Von Dreele, General Structure Analysis System (GSAS), Los

Alamos National Laboratory Report, LAUR 86-748 (2000).

Fundamentals

crystalline state

Figure

1.41.

The three-dimensional view of the crystal structure of

tmaVsOzo

(vanadium

oxide layers and nitrogen atoms from

tma

molecules) with added symmetry elements. The

space group symmetry of the material is C2lm.

the symmetry elements were added later by manually editing the VRML file

to visualize them.

The three-dimensional model of the trnaVs02,, crystal structure located on

the CD (file Figure 1.41.wrl) uses the following color scheme:

Two-fold rotation axes and mirror planes are shown in blue.

Glide planes, a, and 2, screw axes are shown in yellow.

Inversion centers are represented as small red spheres.

The inversion center in the origin of coordinates (0, 0, 0), which is

occupied by nitrogen atom is shown as a blue sphere.

Vanadium and oxygen atoms are shown as pink and green spheres,

respectively.

1.16.4 Space groups in nature

conclusion of this section it may be interesting to note that not all of

the 230 crystallographic space groups have been found in real crystalline

materials. For example, space groups P4222 and P42cm are not found in the

Inorganic Crystal Structure Database,' which at the time of writing this book

The Inorganic Crystal Structure Database.

Karlsruhe, Germany (2001).