23 Reciprocal Space

8/13/2019 23 Reciprocal Space

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Reciprocal Space

Interplanar distances and angles


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The interplanar distance d hkl is defined to be the distance fromthe origin of the unit cell to the (hkl) plane nearest the origin alongthe normal to the plane, i.e. the perpendicular distance from theorigin to the plane.

The angle between two sets of lattice planes is

defined to be the angle between the normals to the two planes.


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Consider the three vectors, p 1, p2, and p 3

acp

cbp

bap

3

2

1

hl

l k

k h

11

11

11

Since they are parallel to the plane, thecross product of any two is normal(perpendicular) to the plane.

accbba

ppn 21

lhk l hk 111


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If we find the unit normal,n/|n| , and take its dot product with anyvector, t, that terminates on the plane, we get the interplanardistance, d hkl.

We can also take the dot products of a pair of these normals todetermine the angle between the normals.

Computationally this is not that easy,unless the crystal axes are allorthogonal, so we take a differentapproach.


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Orthorhombic System All angles are 90, so the cross products are easily calculated.

jacicbk ba acbcab

and

222

hk

ab

hl

ac

kl

bc

hk

ab

hl

ac

kl

bc

n

k jin


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Orthorhombic System

Now take the simplest vector that terminates on the plane:

itha

222

222

00

hk

ab

hl

ac

kl

bc

hk l

abc

hk

ab

hl

ac

kl

bc

h

a

nt hk

ab

hl ac

kl bc


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Orthorhombic System

Simplifying and squaring

2222 1

cl

bk

ah

d hkl


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accbban lhkl hk

111

Lets look back at our normal to the plane equation

Multiply by hkl

baaccb

accbbanl k h

k hl hk l

Divide by the volume

V l

V k

V h

V

hkl baaccbn


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Consider just the last term of this equation with l = 1.

V ba

a xb is a vector perpendicular to the a-b face with length equal tothe surface area contained in the a- b face. So

001

1

d V

ba


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This is a useful result and from it two quantities can be identified,a reciprocal cell spacing, c*, and a reciprocal lattice spacing,d001*. These two quantities have units of reciprocal ngstroms.

bacba

001

*001

* 1d

d c

And we can the define

cbacb

100

*100

* 1d

d a

acbac

010

*010

* 1d

d b


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Useful Relationships

cos** cccc

By definition

001

* 1d c

By geometry

cos001 cd

So 1* cc

And by analogy

1* aa 1* bb


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Additional Useful Relationships

Because by definition a reciprocal spaceaxis is perpendicular to two of the otherreal space axes, the following are true.

0bc0ac

0cb

0ab0ca

0ba

*

*

*

*

*

*


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These new reciprocal lattice axes can be used to definereciprocal lattice vectors, hkl, just as our regular basis set canbe used to define translation vectors, UVW.

****

cbad l k hhkl

In fact the similarities extend to our calculations ofdistances and angles:

||||cos

1

*

*

2***

21

21

hkl

*

hkl

hkl

*

hkl

hkl

hkl hkl hkl

*

hkl d

dddd

dddd


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Construction of Reciprocal Lattice

1. Identify the basic planes in the direct space lattice, i.e.(001), (010), and (001).

2. Draw normals to these planes from the origin.

3. Mark distances from the origin along these normalsproportional to the inverse of the distance from theorigin to the direct space planes


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Above a monoclinic direct space lattice is transformed (the b-axis is perpendicular to the page). Note that the reciprocallattice in the last panel is also monoclinic with * equal to

180 .The symmetry system of the reciprocallattice is the same as the direct lattice.


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Reciprocal Metric Tensor

The reciprocal metric tensor exists and is defined in an analogous

manner to the direct space metric tensor.

2*******

***2****

******2*

*

coscoscoscoscoscos

ccbca

cbbba

cabaa

G

The good news is that G* can be recovered from G:

1* GG


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Using the reciprocal metric tensor makes calculation of d-spaces remarkably easy.

l

k

h

Gl k hd

hkl

*21

2

2

2

11121

W

V U

GW V U t t

Compare to


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And also interplanar angles:

**2

*1

21

coshkl hkl

d d

l

k

h

Gl k h


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Calcite Example

Trigonala = 4.990, c = 17.061 058613.01

23140.0cos1

*

**

cc

aba

034354.000

0053546.0026773.00026773.0053546.0

*G


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What is the Angle Between (1014) and (1104)

2-*2* 410 108514.0401

401 Gd

*411

-1*410 32941.0 d d

These two spacings are the same because they arerelated by three-fold symmetry.

Calcite Example(continued)


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Calcite Example(continued)

2-** 411* 410 028195.041

1

401 Gd d

94.74

32941.032941.0

028195.0cos

cos

1

*

411

*

410

*

141

*

4101

d d

d d

So


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The following show common direct space reciprocal spacerelationships and equations to calculate d-spaces. These arenormally found in symmetry/crystallography texts and arelisted by crystal system.

However, remembering a couple of simple facts about themetric tensor and its inverse leads to the same results.


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Direct Space to Reciprocal Space


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Reciprocal Space to Direct Space


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Dspace Equations

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23 Reciprocal Space