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Reciprocal Lattice-1.Pdf R. I. Badran Solid State Physics The Reciprocal Lattice Two types of lattice are of a great importance: 1. Reciprocal lattice 2. Direct lattice (which is the Bravais lattice 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 "wave 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 waves with a period of the Bravais lattice.[Note: any vector is a possible period of the Bravais lattice] - A collection of vectors G satisfying eiGR 1orG R 2n , 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 ij. 2. The two lattices (reciprocal and direct) are related by the above definitions in 1. 3. Rotating a crystal means rotating both the direct and reciprocal lattices. 40 R. I. Badran Solid State Physics 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 electron and lattice vibrations in crystals. b) The geometry of x-ray and electron diffraction 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: 41 R. I. Badran Solid State Physics 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 has a specialG k b expressionk b k b for bcc primitive reciprocal lattice 21 1 2 2 3 3 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. 42 R. I. Badran Solid State Physics The volume of the primitive cell of the reciprocal lattice 2 is 4( )3 . [Try to find 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 , 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 2 The volume of primitive cell for the direct lattice isVcell a c . 2 a G (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 a1 axˆ 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 43 R. I. Badran Solid State Physics 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. 44 R. I. Badran Solid State Physics 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]. 45 .
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