Seminar chm804

SUBMITTED BY: Aysha fatimaM.Sc

TOPIC:SYSTEMATICALLY

ABSCENT REFLECTIONS

SYSTEMATICALLY ABSCENT REFLECTIONS

• Each set of plane in a crystal should diffract X rays but in many cases interference of waves causes the resultant intensity to be zero.

• These abscent reflections may be divided into two groups:

a) those that are abscent due to some quirk in the structure

b)And those that are abscent due to symmetry or type of lattice possessed by structure.

The latter are known as systematic abscences.

• The presence of translational symmetry elements and centering in the real lattice causes some series of reflections to be absent

• Translational symmetry of an object means that a particular translation does not change the object

• Translation repeats objects by movement along a line at specific distances and angles

Translational symmetry elements include glide lines and planes, and screw axes. 

• Glide Planes: a combination of mirror operations and translation.

• Screw Axes: a combination of rotation axes and translation.

A screw axis is a translational symmetry element and consists of a rotation followed by a translation.

Screw axes exist for each rotation axis. The presence of systematic

absences can be understood in a simple way from Bragg’s Law

• If a set of lattice planes occupy a position such that they reflect X- rays completely out-of-phase with another set of lattice planes, then no reflection will be observed. i.e. although the Bragg condition is satisfied for the sets of planes ,

• the destructive interference “extinguishes” the reflection.

• This situation only arises if there are translational symmetry elements or centering in the crystal lattice

e.g. the (001) reflection in a cubic lattice (BCC) is absent.Consider the additional path lengths vs. beam “1”:For “2” it is 2d sin(q); for “3” it is 2(d/2) sin(q), thus the rays from “3” will be exactly out-of-phase with those of “2” and no reflection will be observed.

• Systematic abscences arise if either the lattice type is non primitive (BCC ,FCC)or if elements of space symmetry (screw axis,glide planes) are present

• These systematic absences (or “systematic extinctions”) thus indicate the presence of centering and/or specific symmetry elements in the lattice and provide us with information about the space group of the crystal.

• the conditions for reflections or absences are reported as simple equations in which “n” indicates any integer.

• E.g. if for reflections of the type (h00), h = 2n + 1 are absent (this means that if h is odd, then the reflection will not be observed)

• Conversely, this means that the limiting condition for such reflections to be observed is: for (h00), h = 2n (i.e. reflections are only observed when h is even)

E.g. for C centered cells, such as the one pictured above, (hkl) reflections are systematically absent when: h + k = 2n + 1 (if the sum of h and k is odd)

Symmetry Element reflection absence conditions

• A centered Lattice (A) hkl k+l = 2n+1• B centered Lattice (B) h+l = 2n+1• C centered Lattice (C) h+k = 2n+1

• face-centered Lattice (F) hkl h+k = 2n+1 h+l = 2n+1 k+l = 2n+1

• Body centered Lattice (I) hkl h+k+l = 2n+1

EXAMPLE(absence due to lattice type) Iron (-Fe)• Iron is BCC.

• Reflections from (100)has zero intensity & is systematically abscent.

• Strong 200 reflections is observed because all atoms lie on (200) planes & there are no atoms lying between (200) planes to cause destructive interference.

• 110 reflection is observed whereas 111 is systematically abscent in -Fe.

• For each non primitive lattice type there is a simple characteristic formula for systemic abscences .

• For BCC reflections for which (h+k+l) is odd are abscent such as 100,111,320 etc are systematically abscent

Iron (bcc)

0 1

1

0 0

2 1 1

2

0 2

2

0 1

3

2 2

2

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

alpha-Iron (Hull, A.W. (1917) Phys Rev 10, 661)Lambda: 1.54180 Magnif: 1.0 FWHM: 0.300Space grp: I m -3 m Direct cell: 2.8660 2.8660 2.8660 90.00 90.00 90.00

Example: NaCl

• NaCl is face centred cubic.

• Only those reflection may observed for which hkl are either all odd or all even .

• 110 is systemetically abscent but 111 may be observed.

:

Comparison: NaCl vs KCl

Fhkl = 4fNa + 4fCl if h,k,l all even

Fhkl = 4fNa - 4fCl if h,k,l all odd

1 1

1

0 0

2

0 2

2

1 1

3

2 2

2

0 0

4

1 3

3

0 2

4

2 2

4

1 1

53

3 3

20 30 40 50 60 70 80 90 100

NaCl (Hull, A.W. 1919)Lambda: 1.54178 Magnif: 1.0 FWHM: 0.200Space grp: F m -3 m Direct cell: 5.6400 5.6400 5.6400 90.00 90.00 90.00

NaCl

1 1

1

0 0

2

0 2

2

1 1

3

2 2

2

0 0

4

1 3

3

0 2

4

2 2

4

1 1

53

3 3

0 4

4

1 3

5

0 0

62 4

4

20 30 40 50 60 70 80 90 100

KCl (Hull, A.W. 1919)Lambda: 1.54178 Magnif: 1.0 FWHM: 0.200Space grp: F m -3 m Direct cell: 6.2800 6.2800 6.2800 90.00 90.00 90.00

KCl , K+ and Cl- are isoelectronic

KCl

• 111 intensity is zero since K+ & Cl- are isoelectronic

• Intensity should decrease in the order KCl <KF < KBr< KI

Example: CsCl• CsCl is primitive cubic.• if difference between Cs and Cl is

ignored The atomic positions are same as in BCC

α Fe • 100 is abscent in α Fe but is an

observed reflection with CsCl because scattering powers of Cs and Cl are different

So weak/strong reflections

Example: Copper

• Copper is face centred cubic. • Atoms at (0,0,0), (½,½,0), (½,0,½),

(0,½,½)Three cases to consider

h,k,l all odd

h,k,l all even

h,k,l mixed (2 odd, 1 even or 2 even, 1 odd)Thus, reflections present when …

Generally true for all face centred structures

THANK YOU