R. I. Badran Solid State

The Reciprocal

Two types of lattice are of a great importance:

1. 2. Direct lattice (which is the that determines a given reciprocal lattice).

What is a reciprocal lattice? A reciprocal lattice is regarded as a geometrical abstraction. It is essentially identical to a " vector" k-space.

Definition:  Since we know that R may construct a set of points of a Bravais lattice, thus a reciprocal lattice can be defined as: - The collection of all wave vectors that yield plane with a period of the Bravais lattice.[Note: any vector is a possible period of the Bravais lattice]      - A collection of vectors G satisfying eiGR  1orG  R  2n ,

where n is an integer and is defined as: k1n1  k2n2  k3n3 . Here , is a reciprocal lattice vector which can be defined     as:G  k b  k b  k b , where k , k and k are integers. 1 1 2 2 3 3 1 2  3  [Note: In some text books you may find thatG  K ]. - The reciprocal lattice vector which generates the reciprocal lattice is constructed from the linear combination       a2  a3 of the primitive vectorsb ,b , andb , where b1  2 1 2 3 V  cell and b2 and can be obtained from cyclic permutation of 1 2 3.

Notes:   1. Since , this implies thatbi  a j  2 ij , where

 ij 1if i=j and  ij  0 if ij. 2. The two lattices (reciprocal and direct) are related by the above definitions in 1. 3. Rotating a means rotating both the direct and reciprocal lattices.

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4. The direct crystal lattice has the dimension of [L] while the reciprocal lattice has the dimension of [L-1].

Why do we need a reciprocal lattice? Reciprocal lattice provides a simple geometrical basis for understanding: a) All things of "wave nature" (like behavior of and lattice vibrations in . b) The geometry of x-ray and electron patterns.

Reciprocal lattice to simple cubic (sc) lattice:

The simple cubic primitive lattice, which has the primitive    vectors a1  axˆ , a2  ayˆ and a3  azˆ , has a volume of cell equal 3 toVcell  a . The corresponding primitive vectors in the reciprocal lattice can be obtained as:  a 2 (yˆ  zˆ)  2 b  2  b  ( )xˆ , 1 a3 1 a  a2 (zˆ xˆ)  2 b  2  b  ( )yˆ and 2 a3 2 a  a2 (xˆ  yˆ)  2 b  2  b3  ( )zˆ . 3 a3 a

2 (2 )3 The corresponding volume in reciprocal lattice is ( )3  . a Vcell It must be noted that the reciprocal lattice of a sc is also a sc but 2 with lattice constant of ( ) . a

Reciprocal lattice to bcc lattice:

When a set of primitive vectors for the bcc lattice are given by    a a  axˆ a  ayˆ a  (xˆ  yˆ  zˆ) 1 , 2 and 3 2 , as shown in figure 11, where a is the side of the conventional cell, the primitive lattice vectors of the reciprocal lattice are found as:

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a 2 [ yˆ (xˆ  yˆ  zˆ)]  2  2 b1  2  b  ( )(xˆ  zˆ) , a 3 1 a 2 a 2 [ (xˆ  yˆ  zˆ) xˆ]  2  2 b2  2  b  ( )(yˆ  zˆ) , a3 2 a 2  [a 2 (xˆ  yˆ)]  2 b3  2  b  ( )(2zˆ) . a3 3 a 2 You can easily show that the volume of primitive reciprocal lattice 2 is 2( )3 . This can be compared to the volume of primitive direct a a3 latticeVcell  . 2

Notes: a) The bcc primitive lattice vectors in the reciprocal lattice are just the primitive vectors of an fcc lattice.     b) The general reciprocal lattice vector G  k1b1  k2b2  k3b3 has a special expression for bcc primitive reciprocal lattice  2 as:G  [(k  k )xˆ  (k  k )yˆ  (k  k )zˆ]. a 2 3 1 3 1 2

Reciprocal lattice to fcc lattice:

We know that the primitive vectors of fcc primitive lattice may  a  a  a be defined by: a  (yˆ  zˆ) , a  (xˆ  zˆ) and a  (yˆ  xˆ) , 1 2 2 2 3 2 [see figure 10]. Thus the primitive vectors in the reciprocal lattice are:  2  2  2 b  ( )(xˆ  yˆ  zˆ) ,b  ( )(xˆ  yˆ  zˆ) andb  ( )(xˆ  yˆ  zˆ) . 1 a 2 a 3 a It must be noted that these latter vectors are the primitive lattice vectors of a bcc lattice.

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The volume of the primitive cell of the reciprocal lattice 2  is 4( )3 . [Try to findG for the fcc primitive reciprocal lattice, for a

example, when k1=1, k2= -2 and k3 =3]. Reciprocal lattice to simple hexagonal lattice:  Recalling the primitive vectors of a simple hexagonal a1  axˆ ,  a  a2  (xˆ  3yˆ) a  czˆ 2 and 3 , as shown in figure 15. The corresponding primitive vectors can simply be determined          a2  a3 a3  a1 a1  a2 by using: b1  2 , b2  2 andb3  2 . Vcell Vcell Vcell

3 The volume of primitive cell for the direct lattice isV  a2c . cell 2 a  (xˆ  3yˆ)  czˆ 2  2 1 b1  2 b  ( )(xˆ  yˆ) 3  1 , a2c a 3 2  czˆ  axˆ  2 2 b2  2 b  ( )( yˆ) 3  2 and a2c a 3 2 a  axˆ  (xˆ  3yˆ) 2  2 b3  2 b  ( )zˆ 3  3 . a2c c 2 Using a common origin for both reciprocal and direct lattices, we can build up the primitive reciprocal lattice by determining

the magnitudes of and and

the angle between them. This will give you a clue how the base of the new reciprocal lattice looks like. Thus both vectors

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  4 have the magnitude of b  b  ( ) . Obviously the 1 2 3a  2 magnitude of the third vector is b  ( ) . These can be 3 c  compared to the magnitudes of the primitive vectors, a1  a ,   a2  a and a3  c , respectively, as shown in figures 27 and   28). The angle between the two vectorsb1 and b2 can be   b  b directly obtained as  cos 1 ( 1  2 )  120 . b1 b2

Now what is left is to determine the angle between the vectors  a1 and which is obtained by applying the relation   bi  a j  2 ij and it is (/6). This implies the necessity to

rotate the plane of the two reciprocal vectors and by /6. (See figure 27).

yˆ

  a b2 2   a1 xˆ 30  b1

Figure 27: The primitive vectors of base of simple hexagonal conventional cell in direct space lattice as compared to their corresponding vectors in reciprocal space lattice.

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 2 b3   c a3  c  4  b1  a1  a 3a

60 60  a  a 2  4 b  2 3a

Figure 28: The three primitive vectors of simple hexagonal conventional cell in direct space lattice as compared to their corresponding vectors in reciprocal space lattice.

Conclusion: This shows that the reciprocal lattice to a simple hexagonal lattice with lattice constant a and c is also another simple 4 2 hexagonal lattice, with lattice constants and , but 3a c rotated through /6 about the c-axis with respect to the direct lattice (in a clockwise direction). [See problem 5.2.a in Solid State Physics by N. Ashcroft & N. Mermin].

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