Structure Factors and Systematical Absences

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SYSTEMATIC

ABSENCES INCRYSTAL DATA 5

By the end of this chapter you should be able to:

Understand how absences in crystal data relate to lattice types and symmetryelements; Recognize how intensities in crystal data are associated with structure factors; Gain an appreciation for atomic scattering factors and its influence on structure

factors;

Identify the rules governing general and systematic absences.

5.1 INTRODUCTION

When a diraction pattern is collected either on photographic lm or on an X-raydiractometer, each rame o data (see Fig. 5.1) contains reections (spots) o dieringintensities, while at certain points on the rame o data some reections are missing orabsent.

Some o these absent reections have intensities close to zero because only very ewelectrons in the crystal structure are contributing to diraction rom the associated plane,while other reections are precisely zero because o the destructive intererence o theincident X-rays. Tese absences are inuenced by the positions o symmetry-related atomsor molecules within the crystal structure.

In this chapter, we will examine how absences (absent reections) can be used todetermine the space group o a crystal lattice, relating directly to the various lattice typesand the dierent translational symmetry elements that can occur within a crystal lattice.

During the course o a data collection, multipleframes o data are collected, in order toaccumulate suf cient data to cover the reciprocal space o the entire crystal lattice. Withineach rame o data, each reection is related to a specic Miller plane (h, k, l) with a specic

value o observed intensity, Ihklobs

used to dene it. An absence occurs when Ihklobs

0.Once the complete diraction data are collected, the data are processed. Te processing o

the diraction pattern into a useable ormat is also known as data reduction. Te integrationo data occurs as part o the data-reduction process. During integration, the diraction

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5.2 STRUCTURE FACTORS 67

rom each Miller plane within the single crystal lattice is integrated across the multiplerames o diraction data. Tis ensures that each collected diraction spot is linked to aMiller index and has an associated value o intensity. Each integrated value o intensity isproportional to the square o the observedstructure factor (Fhkl

obs) or the

associated Miller

index;

I Fhkl hkl corr obs

( ) .2

Aer integration, scaling, and various corrections or background (Lorentz, polarization,and absorption corrections) are applied, culminating in the output o a text-based computerle o diraction data representative o the single crystal. Tis is known as the .hklle. Te.hklle (see Fig. 5.2) contains Miller indices (hklvalues) in the rst three columns, ollowedby a numerical value o intensity (I) and a value o standard deviation (s).

5.2 STRUCTURE FACTORS

Section learning outcomes

o be able to:

Understand the contributions o atomic scattering actors to structure actors; Calculate structure actors and derive calculated intensities.

From experimental diraction data, we are able to obtain observed and corrected intensitiesIhklcorr , directly related to the square o the observed structure actors(Fhkl

obs )2. Te diraction

The computer file

name extensions

used in this text,

such as .hkl, .res,

and .ins, refer to

those used by the

SHELX suite ofprograms.

AbsencesReflections

FIGURE 5.1 A frame of diffraction data

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5 SYSTEMATIC ABSENCES IN CRYSTAL DATA68

pattern produced is a result o the total scattering rom all unit cells and represents the aver-age content o a single unit cell. Structure actors are, in essence, a mathematical representa-tion o the interaction between molecules in a crystal lattice with X-rays and are inuencedby atomic scattering actors.

FIGURE 5.2 Part of an .hklfile

2 22 124.99 6.26 1

2 12 971.57 38.65 1

2 72 1176.49 100.16 1

2 82 0.59 0.89 1

1 107 443.52 26.96 1

1 117 5.52 4.61 1

1 46 696.96 40.66 1

1 85 5.52 2.91 1

1 111 44.36 3.73 1

1 20 1043.94 69.79 1

1 20 1622.48 66.06 1

1 81 85.56 5.36 1

1 102 1.59 1.29 1

1 13 530.38 21.65 11 143 451.99 35.72 1

1 124 47.33 6.74 1

0 513 113.42 13.21 1

0 712 298.94 23.86 1

0 1010 70.73 9.42 1

0 119 0.10 6.68 1

0 78 312.58 16.27 1

0 37 1.21 1.10 1

0 16 159.26 8.08 1

0 16 138.30 6.35 1

0 35 758.45 38.01 1

0 04 1710.65 82.72 1

0 92 1.80 1.90 1

0 31 649.74 37.22 1

0 40 673.40 36.33 1

0 40 1202.01 52.00 1

0 61 509.40 27.08 1

0 72 900.60 55.22 1

0 04 1443.24 73.70 1

0 35 776.18 37.33 1

1 07 1147.18 47.42 1

hkl values

, standard deviation

I, intensity

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5.2.1 Atomic scattering factors

Te electrons in atoms are able to interact with and subsequently scatter X-rays. Atomicscattering actors, f

n, also known as atomic orm actors, are the amplitude measure o

X-ray waves scattered rom an atom.As X-rays are scattered by the electrons within an atom, the atomic scattering actor is

dependent on the electron density exhibited by the atom or the atomic number within anatom. able 5.1 lists the atomic scattering actors or some o the elements and how theyare related to the Bragg angle. Reerring to able 5.1, we can see that the atomic scatteringactor o hydrogen (one electron) is signicantly dierent rom that o tin (50 electrons).

From able 5.1, we can also notice that in the heavy elements, the reduction at high Braggangles occurs more slowly. Tis indirectly allows the heavy elements to dominate the struc-ture actors, making it more dif cult to locate the lighter atoms within the diraction map.

Te atomic scattering actor is a unction o the Bragg angle. For example, i we reerto Fig. 5.3, at low Bragg angles (e.g., when sin u/l 0), the atomic scattering actor,fn, is

directly proportional to the atomic number. In contrast, this value tails o at higher angles,as at these high angles not all o the electrons within an atom are scattering in phase.

5.2.2. Calculating structure factors and intensities

Te calculated structure actors Fhklcalc relating to a given Miller index (h, k, l) or a crystal

containing Natoms is given by the ollowing equation:

F f fhx ky lz xh ky lz hkl n n n n

N

n n n n

Ncalc

cos i sin22

11

pp( ) ( ), (5.1)

where xn,yn, and zn denote the ractional coordinates o the atoms in the structure whilef

ndenotes the atomic scattering actor or atom type on.

i is a complex number, 1

[ Learning aid breaking down the equation

In reality what seems to be a complex equation can actually be broken down into several amiliarsections. Within the brackets, h k landx y zare amiliar terms relating to specic Miller indicesand the ractional coordinates o an atom, and fn is the atomic scattering actor.

The atomic scattering

factor is influenced

by both the scatteringfrom valence

electrons, fvalence, and

core electrons, fcore,

given by:

fnfvalence +fcore.

Scattering of the

valence electrons

occurs most efficiently

at low Bragg angles,

while at high Bragg

angles, the scattering

of the core electrons

is more significant. Asthe heavy elements

contain more core

electrons, the atomic

scattering factor tends

to diminish more

slowly at higher

Bragg angles.

TABLE 5.1 Atomic scattering factors for some elements

Prince, E. (ed.) (2004). International Tables for Crystallography Volume C: Mathematical, Physical and Chemical Tables, 3rd edn. Kluwer

Academic.

sin / H C O O Na Al Ca Sn W

0 1 5 8 9 11 13 20 50 74

0.1 0.811 5.13 7.245 7.836 11.0 11.23 17.33 46.36 69.78

0.2 0.481 3.58 5.623 5.623 9.76 9.16 14.32 40.30 62.52

0.5 0.071 1.69 2.338 2.313 4.29 5.69 8.26 29.10 43.70

1.0 0.007 1.11 1.377 1.376 1.78 2.32 5.19 16.38 25.58

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F f hx ky lz i f hx ky lz hkl n n n n

N

n n n n

Ncalc

co ss in22

11

pp( ) ( ),

Te calculation o the structure actor, Fhklcalc can then be divided into two sections: the

right-hand side o the equation splits easily into two halves, the summation with the cosine

unction and the summation with the sine unction, which consider the atomic scatteringactors o the atoms within the unit cell; with the sine unction having an imaginarycomponent i = 1.

Based on scattering within a unit cell:

F f e f ehkl ni

n

i hx ky lz NN

n n n ncalc f p2

11

( )

e cos sin ,i x x x i

so, substituting cos x i sin x or eix,

F f fh hx ky lz x ky lz hkl nN

n n n n

N

n n n

calc

11

22 cos ( ) sin ( ).pp i

In a diraction pattern, an absence occurs when the intensity o a reection rom aspecic set o Miller planes is equal to zero, I 0. As the intensity, I, is equal to the squareo the structure actor, F, then by denition, in an absence, Fcalc,is also equal to zero.

0

0 0.2 0.4 0.6

sin/

Plot of sin/ vs.fo

fo

0.8 1 1.2

1

2

3

4

5

6

7

8

9

FIGURE 5.3 The atomic scattering factor of oxygen as a function of the Bragg angle

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Considering a centrosymmetric unit cell (a unit cell that possesses a centre osymmetry), the sine part o the equation disappears. So, Fhkl

calc

or a centrosymmetric unit

cell becomes just

F f hx ky lz hkl n n n n

Ncalc

cos21

p ( )+ + (5.2)

Tis is because, in a centrosymmetric unit cell, atoms will be located at x, y, zand atx, y, z. Substituting the values into Eqn 5.1, we nd that sin(x)sin(x). We alsond that cos2po an odd number is always equal to1 while cos2p o an even number isalways equal to +1.

Consider the ollowing example o how structure actors are calculated in a simplecentrosymmetric unit cell.

WORKED EXAMPLE: CALCULATING STRUCTURE FACTORS

Scenario

A unit cell contains two atoms o carbonwith ractional coordinates (0.10, 0.10, 0.10), (0.90,0.90, 0.90) and two atoms o oxygen with coordinates (0.20, 0.85, 0.30), (0.80, 0.15, 0.70).

Te parameters o the unit cell are a b 3, c 5 , andabg90.Te scat-tering actors or the two types o atom are given below as unctions o(sin u)/l.

(sin )/ fC fO

0 6 8

0.1 5.13 7.25

0.2 3.58 5.63

0.3 2.5 4.09

0.4 1.95 3.01

0.5 1.69 2.34

0.6 1.54 1.94

With the inormation provided calculate the structure actor, F112, and subsequentlydetermine the equivalent intensity value, I.

Strategy for solution

With these data, we can calculate the Fhkl values or all reections, such thath2k2l2 < 8 and all Miller indices are positive. As the structure is centrosymmetric

cos2p(odd

number) 1

cos2p(even

number) +1

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(each atom pair is related by an inversion centre at (, , )) all Fare real (i.e., thesine unction o the equation disappears) leaving:

F f hx ky lz hkl n n n n

Ncalc

cos ( ).21

1. First, we need to determine each atomic scattering actor (fx) at the relevant (sin)/or the given Miller index, in this case (1, 1, 2).

2. Next, determine cos ( )2p hx ky lz n n n or each atom pair.

3. Finally, calculate F112by substituting the values into the equation.

Calculate (sin)/for Miller index (1, 1, 2) and subsequently determine the

atomic scattering factors

From Braggs law;

22d n

n

d

sinsin

.

=

=

Hence,4 2

2

2

2

2

2

2

2

sin.

h

a

k

b

l

c

For F112, 4 13

1

3

1

25

0 3822

2

2

sin

. .

Tereore,(sinu)/l = (0.3822)/2 = 0.6182/2 = 0.31.

From graphs ofC andfO versus (sinu)/l,

we nd that when (sinu)/l 0.31,fC2.5 andfO3.97.

00 0.2 0.4 0.6

sin/

Plot of sin/ vs.fC

0.8 1 1.2

1

2

3fC

4

5

6

7

Plot of sin/ vs.fO

0

0 0.2 0.4 0.6

sin/

fo

0.8 1 1.2

1

2

3

4

5

6

7

8

9

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Determine cos2(hx + ky+ lz) for each atom pair

For the rst atom pair:hxC + kyC + lzC0.40; hxO + kyO + lzO= 1.65;

cos2p(hxC + kyC + lzC) 0.81; cos2p(hxO + kyO + lzO) 0.59.While or the second atom pair:

hxC + kyC + lzC3.6; hxO + kyO + lzO2.35;

cos2p(hxC + kyC + lzC) 0.81; cos 2p(hx0 + ky0 + lz0) 0.59.

Determining the structure factor (F112)associated with Miller index (1, 1, 2)

F f hx ky lz n n n n

N

112

1

2

2 5 0 81 2 5 0 81

cos ( )

[( . ( . )) ( . ( . ))]

[( . ( . )) ( . ( . ))].3 97 0 59 3 97 0 59

8.70 (correct to two deecimal places)

Te (almost) completed table ollows (to two decimal places):

SELF-TEST QUESTIONS

1. Describe the term absence, in terms o intensity, I, and the diraction map.

2. Briey describe the process that leads to the production o an .hklle.

3. Explain how the atomic scattering actors are inuenced by Braggs angle.

4. Why is it sometimes dif cult to locate the light atoms in an electron-density mapwhen heavy elements are present in the crystal structure?

5. Calculate the remaining our Fvalues, and subsequently work out the calculatedintensities (prior to applied corrections) or each set o Miller indices. (ip: yourcalculator has to be in the radian mode.)

h k l F I = F2

1 0 0 10.36 107.32

0 0 1 3.90 15.17

1 1 0 11.46 131.22

0 1 1 9.10 82.87

1 0 1 9.21 84.85

1 1 1 7.21 51.96

2 0 0 4.63 21.450 2 0 0.85 0.72

0 0 2 7.01 49.08

2 1 0 1.31 1.70

h k l F I = F2

1 0 2 1.09 1.19

1 2 0 4.14 17.17

0 1 2 10.56 111.51

0 2 1 5.59 31.25

2 1 1 9.19 84.51

1 1 2 8.70 75.77

1 2 12 2 0

2 0 2

0 2 2

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5.3 SPACE GROUPS AND ABSENCES

Section learning outcomes

o be able to: Recognize and identiy general absences and systematic absences; Understand and explain how general absences are derived; Determine space groups rom general and systematic absences; Identiy absences in a dataset.

An essential part to solving a crystal structure relies on determining the lattice typeand symmetry elements (or space groups) within the crystal lattice. Tis inormation iscontained within the diraction pattern o each crystal lattice, where both reections andabsences contribute. As we saw rom the earlier sections, the structure actors contain

inormation relating to the Miller planes and the location (ractional coordinates) o eachatom type within the unit cell. By examining the diraction patterns reections and theabsences within the unit cell, one is able to determine the location o all the atoms andmolecules within it.

Absences in diraction patterns (when I 0), can be divided into two categories; general absences, pertaining to lattice types, and systematic absences, relating to transla-tional symmetry elements.

5.3.1 General absences

On a diraction pattern, destructive intererence o X-rays can occur, resulting in absencesor non-primitive lattice types (or example, I, F, or C). Te rules dening these absencesaect all the reections in the diraction pattern (or all Miller index values ohkl). Teseabsences are known as general absences.

Apart rom the primitive lattice type (P) all other lattice types can be determined by generalabsences that occur within the diraction pattern. Tese absences are dened by rules relat-ing to certain sets o Miller indices within the diraction data, as given in able 5.2.

For the body-centred (I) lattice type, absences occur when the sum o the Miller indices(h + k + l) is an odd value, while or the ace-centred (F) lattice types, absences occur whenall indices (h, k, and lvalues) are either all even or all odd.

For general absences relating to the specic ace-centred lattices, absences occur whenthe sum o the two non-corresponding Miller indices are an odd value. For example, or

I F C

FIGURE 5.4 Some non-primitive lattice types

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the A-ace centred lattice, absences occur when the sum ok and lis an odd value. For the

B-ace centred lattice, the absences occur when the sum oh and lis an odd value, whileor the C-ace centred lattice, absences occur when h + k is an odd value.

By calculating the structure actors and considering the coordinates o the unit celllattice points, we are able to show how the rules or general absences are derived.

GENERAL ABSENCE SELECTION RULE FOR AN A-TYPE BRAVAIS LATTICE AN EXAMPLE

In the example o an A-type Bravais lattice, assuming that each lattice point is equiva-lent to a single atom o typeJ, the coordinates o atoms at the eight corners o the unitcell are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (0, 1, 1), (1, 0, 1), and (1, 1, 1). Eachatom makes a 1/8 contribution to the unit cell.

As the lattice points (in this case atoms o typeJ) at the corners o the unit cell are all

equivalent, any one o these can be selected to count as the total contribution rom allthe corners o the unit cell.

Let us select the atom at (0, 0, 0).

Te atoms at the centres o the two A aces o the unit cell have ractional coordinates(0, , ) and (1, , ). Each o these atoms makes a contribution o to the unit cell.As these atoms are also equivalent, either o these can be selected as the total contribu-tion rom the corners o the unit cell. Let us select the atom at (0, , ).

Substituting into Eqn 5.2, we get an expression or the general structure actor or thisBravais lattice.

F f f l khkl J J cos ( ) cos .2 0 21

2

1

2

( )Simpliying the equation:

F f f l khkl J J cos ( ).p

Keeping in mind that when an absence occurs, the intensity, I, is equal to zero (I 0).(Remember also that IF2)

TABLE 5.2 Rules governing general absences

Lattice type Conditions for general absences

P NoneA k+ l 2n + 1 (i.e., the sum of kand lis odd)

B h + l 2n + 1 (i.e., the sum of h and lis odd)

C h + k 2n + 1 (i.e., the sum of h and kis odd)

F Reflections must have either all even or all odd indices

to be observed

Mixed odd and even indices are not allowed

I h + k+ l 2n + 1 (i.e., the sum of the indices is odd)

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From the equation, it becomes clear that the structure actors will have a value o zerowhen cosp(k + l) 1. Tis will only occur when k + lis an odd number. Tus, an A-typeBravais lattice will be identied when all reections in a dataset or which k + lis an odd

number are absent, i.e., have no intensity. Tis is oen written as k + l

2n + 1.A similar method can be used or proving other lattice types. It can also be extendedto more complex cases in which the lattice point environment is a molecule or a groupo molecules.

Examples o absences corresponding to A-type lattices are or hklvalues o: (1, 0, 1),(1, 2, 1), (2, 3, 2), and (3, 2, 5).

GENERAL ABSENCE SELECTION RULE FOR AN I-TYPE BRAVAIS LATTICE AN EXAMPLE

In the example o an I-type (body-centred) Bravais lattice, assuming that each latticepoint is equivalent to a single atom o type J, the coordinates o atoms at the eight

corners o the unit cell are (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (0, 1, 1), (1, 0,1), and (1, 1, 1). Each atom makes a 1/8 contribution to the unit cell.

As the lattice points (in this case atoms o type J) at the corners o the unit cell are allequivalent, any one o these can be selected to count as the total contribution rom allthe corners o the unit cell.

Let us select the atom at (0, 0, 0).

Te atom in the centre o the unit cell has a ractional coordinate o (, , ). Tisatom makes a contribution o 1 to the unit cell.

Substituting into Eqn 5.2, we get an expression or the general structure actor or thisBravais lattice.

F f hx ky lz

F f f

hkl n n n n

N

hkl J J

calc

cos

cos ( ) cos

2

2 0 21

2

1

p

p

( )

hh k l 1

2

1

2( ).

Simpliying the equation:

F f f kh lhkl J J cos ( ).

Similarly, the intensity, I, is equal to zero (I 0) when an absence occurs. (Rememberalso that IF2.)

From the equation, it becomes clear that the structure actors will have a value o zero

when cos(h + k + l) 1. Tis will only occur when h + k + lis an odd number.Tereore, an I-type Bravais lattice will be identied when all reections in a datasetor which h + k + lis an odd number are absent, i.e., have no intensity. Similarly, thisis oen written as h + k + l 2n + 1.

Examples o absences corresponding to I-type lattices are or hklvalues o: (1, 1, 1),(1, 3, 1), (2, 3, 2), and (3, 2, 6).

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5.3.2 Systematic absences

Like general absences, systematic absences in diraction data arise when destructiveintererence o X-rays occurs in relation to specic translational symmetry elements (screwaxes and glide planes). Te absences also take place according to rules relating to specicsets o Miller indices within the diraction data.

Systematic absences can be ound in diraction data only or translational symmetryelements, screw axes, and glide planes. Systematic absences do not occur or non-translationalsymmetry elements.

Rules or systematic absences can be derived and proved using a similar method to theproo or general absences; however, this is beyond the scope o this book.

Let us now consider systematic absences that are characteristic o screw axes and glideplanes.

Screw axes

Te rules determining systematic absences associated with screw axes (see able 5.3) arebased on the axis to which the screw is parallel. For example, or screw axes along a, thecondition or the systematic absence is always a derivative o the (h, 0, 0) Miller index,where an h value is odd and absent.

A 21 screw axis along a would have multiples o (2n + 1) values oh absent (see able 5.4).Tis similarly occurs or the 41 axis. However, reections corresponding to (2n + 1) will alsobe absent as this is a subset o (4n + 1). Tis similarly occurs or the 3n and 6n screw axes.

Similarly, screw axes along b are derivatives o (0, k, 0); with odd values ok being absent,and screw axes along c are derivatives o (0, 0, l) where values olthat are odd are absent.

Glide planes

In determining the systematic absences rules or glide planes, we consider the ollowing:

Te equivalent Miller index o the axes to which the glide is perpendicular is always 0. Itis the axis along which the glide plane is parallel that denes the conditions or the absence.Te rules are given in able 5.5.

For example, a glide plane perpendicular to the a-axis is given by (0, k, l); i the glideplane is along the b-axis, then the values or the absent reections are dened by the k

TABLE 5.3 Systematic absences for screw axes

Symmetry element Conditions for systematic absence

Screw axis along a 21, 4241, 43

(h, 0, 0) h 2n + 1

(h, 0, 0) h 4n + 1

Screw axis along b 21, 4241, 43

(0, k, 0) k 2n + 1

(0, k, 0) k 4n + 1

Screw axis along c 21, 42, 6331, 32, 62, 64

41, 4361, 65

(0, 0, l) l 2n + 1

(0, 0, l) l 3n + 1

(0, 0, l) l 4n + 1

(0, 0, l) l 6n + 1

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TABLE 5.5 Systematic absences for glide planes

Symmetry element Conditions for systematic

absence

Glide plane || (1, 0, 0) bglide perpendicular a

cglide perpendicular a(0, k, l) k2n + 1

(0, k, l) l2n + 1

Glide plane || (0, 1, 0) aglide perpendicular b

cglide perpendicular b(h, 0, l) h2n + 1

(h, 0, l) l2n + 1

Glide plane || (0, 0, 1) aglide perpendicular c

bglide perpendicular c(h, k, 0) h2n + 1

(h, k, 0) k2n + 1

h k l Ihkl hkl

0 0 12 82.5 0.8

0 0 13 0.4 0.2

0 0 14 98.1 1.3

0 0 15 0.2 0.1

0 0 16 101.1 1.2

0 0 17 0.6 0.7

0 0 18 0.9 1.6

0 0 19 0.5 0.2

0 0 20 54.2 0.7

0 0 21 0.7 0.6

0 0 22 98.8 1.2

h k l Ihkl hkl

0 0 1 6.4 0.2

0 0 2 135.0 0.5

0 0 3 0.6 0.1

0 0 4 254.8 1.2

0 0 5 0.1 0.6

0 0 6 194.2 1.4

0 0 7 0.7 0.5

0 0 8 376.1 2.5

0 0 9 0.3 0.7

0 0 10 253.4 1.2

0 0 11 0.6 0.3

TABLE 5.4 A sample dataset systematic absences caused by a 21 screw axis along c

h k l I hkl hkl

6 0 1 253.6 0.1

5 0 1 0.1 0.6

4 0 1 132.8 0.4

3 0 1 0.7 0.3

2 0 1 312.2 1.2

1 0 1 0.2 0.7

0 0 1 156.1 0.9

1 0 1 0.1 0.2

2 0 1 65.1 2.1

3 0 1 1.6 0.5

h k l I hkl hkl

4 0 1 324.7 1.1

5 0 1 2.5 0.5

6 0 2 523.8 0.2

7 0 2 0.8 0.1

6 0 2 34.2 0.7

5 0 2 0.4 0.2

4 0 2 257.2 1.5

3 0 2 1.2 0.1

2 0 2 136.8 1.5

1 0 2 0.3 0.6

0 0 2 112.8 2.1

1 0 2 0.1 0.5

2 0 2 86.1 1.1

TABLE 5.6 A sample dataset systematic absences caused by an aglide perpendicular to b

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values being odd and absent. For a b glide perpendicular to a, systematic absences aredened or (0, k, l), with odd values ok being absent (k 2n + 1).

I we examine the sample dataset in able 5.6, we nd that the dataset represents values

o (h, 0, l), which suggests a glide plane perpendicular to the b-axis. I we examine the tableor absences, we nd that absences occur or all values oh that are odd, which suggests ana glide. We can then conclude that the table represents absences, indicating that within thisparticular crystal structure there is an a glide that lies perpendicular to the b-axis.

SELF-TEST QUESTIONS

1. Based on the example or the A-type Bravais lattice in Section 5.3.1, prove thatthe general absences rule or the C-type Bravais lattice is (h + k) absent or all odd

values.

2. Briey list the general absences you would expect to nd i the crystal lattice was a:

(a) B-type Bravais lattice;

(b) F-type Bravais lattice;

(c) I-type Bravais lattice.

3. Determine the space group or a crystal lattice that exhibits the ollowingabsences:

(a) No general absences;

(b) Systematic absences corresponding to:

(0, k, l), where values o k odd are absent;

(h, 0, l), where values o l odd are absent;

(h, k, 0), where values o h odd are absent.4. What absences would you expect to nd or a crystal structure with the

orthorhombic space group P 21 21 21?

CHAPTER SUMMARY

1. An absence occurs when difraction spots or reections are missing rom difraction

data.

2. Absences provide inormation about Bravais lattice types (general absences) and non-

translational symmentry screw axes and glide planes (systematic absences)

3. Te intensity, I, is a unction o the square o the structure actor.

4. Te structure actor is derived rom atomic scattering actors.

5. Atomic scattering actors are dependent on the electron density o the atom and are

inuenced by the Bragg angle o the incident X-ray.

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6 Te rules governing general absences are:

Lattice type Conditions for general absences

P None

A k+ l 2n + 1 (i.e., sum of k+ lodd)

B h + l 2n + 1 (i.e., sum of h + lodd)

C h + k 2n + 1 (i.e., sum of h + kodd)

F Reflections must have either all even or all odd indices to be observed.

Mixed odd and even indices are not allowed.

I h + k+ l 2n + 1 (i.e., sum of indices odd)

7. Te rules governing systematic absences are:

Symmetry element Conditions for systematic absenceScrew axis along a 21, 42

41, 43

(h, 0, 0) h 2n + 1

(h, 0, 0) h 4n + 1

Screw axis along b 21, 4241, 43

(0, k, 0) k 2n + 1

(0, k, 0) k 4n + 1

Screw axis along c 21, 42, 6331, 32, 62, 6441, 4361, 65

(0, 0, l) l 2n + 1

(0, 0, l) l 3n + 1

(0, 0, l) l 4n + 1

(0, 0, l) l 6n + 1

Symmetry element Conditions for systematic absence

Glide plane || (1, 0, 0) bglide perpendicular acglide perpendicular a

(0, k, l) k2n + 1(0, k, l) l2n + 1

Glide plane || (0, 1, 0) aglide perpendicular b

cglide perpendicular b(h, 0, l) h2n + 1

(h, 0, l) l2n + 1

Glide plane || (0, 0, 1) aglide perpendicular c

bglide perpendicular c(h, k, 0) h2n + 1

(h, k, 0) k2n + 1

FURTHER READING

Giacovazzo, C., Monaco, H. L., Artioli, G., et al. (eds.) (2002). Fundamentals of Crystallography,

2nd edn. IUCr exts on Crystallography. Oxord University Press, New York.Glusker, J. P. and rueblood, K. N. (1985). Crystal Structure Analysis A Primer. Oxord

University Press, New York.

Stout, G. H. and Jensen, L. H. (1989).X-Ray Structure Determination: A Practical Guide.John Wiley & Sons, Ltd, New York.

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Structure Factors and Systematical Absences