handout5 - Handout 5 The Reciprocal Lattice In this lecture...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 5 The Reciprocal Lattice In this lecture you will learn: • Fourier transforms of lattices • The reciprocal lattice • Brillouin Zones • X-ray diffraction • Fourier transforms of lattice periodic functions ECE 407 – Spring 2009 – Farhan Rana – Cornell University Fourier Transform (FT) of a 1D Lattice Consider a 1D Bravais lattice: x a a ˆ 1 = r Now consider a function consisting of a “lattice” of delta functions – in which a delta function is placed at each lattice point: x a a ˆ 1 = r x () ( ) = −∞ = n a n x x f δ = = = −∞ = −∞ = + −∞ = m x n a n k i x k i n x a m k a e e a n x dx k f x x π 2 2 The FT of this function is (as you found in your homework): ( ) x f The FT of a train of delta functions is also a train of delta functions in k-space
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice as FT of a 1D Lattice x a a ˆ 1 = r x ( ) x f FT is: x a b ˆ 2 1 π = r x k ( ) x k f a 2 1 The reciprocal lattice is defined by the position of the delta-functions in the FT of the actual lattice (also called the direct lattice) x a a ˆ 1 = r x a b ˆ 2 1 = r Direct lattice (or the actual lattice): Reciprocal lattice: x x k ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice of a 1D Lattice For the 1D Bravais lattice, x a a ˆ 1 = r The position vector of any lattice point is given by: 1 a n R r v = R v The reciprocal lattice in k-space is defined by the set of all points for which the k- vector satisfies, 1 . = R k i e r r for all of the direct lattice R v For to satisfy it must be that for all : , where p is any integer 1 . = R k i e r r k r p R k 2 . = r r a m k p a n k p R k x x 2 2 2 . = = = r r R v where m is any integer Therefore the reciprocal lattice is: x k x a b ˆ 2 1 = r
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Reciprocal Lattice of a 2D Lattice Consider the 2D rectangular Bravais lattice: x a a ˆ 1 = r y c a ˆ 2 = r If we place a 2D delta function at each lattice point we get the function: () ( ) ( ) = −∞ = −∞ = nm c m y a n x y x f δ , The above notation is too cumbersome, so we write it in a simpler way as: = j j R r r f r r r 2 The summation over “ j ” is over all the lattice points A 2D delta function has the property: ( ) ( ) ( ) o o r g r g r r r d r r r r r = 2 2 and it is just a product of two 1D delta functions corresponding to the x and y components of the vectors in its arguments Now we Fourier transform the function : ( ) r f r () () ( ) ∑∑ = = = = −∞ = −∞ = c m k a n k ac e e R r r d e r f r d k f y x j R k i r k i j j r k i j π 2 2 2 2 .
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).

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handout5 - Handout 5 The Reciprocal Lattice In this lecture...

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