Process with digital signal processing dsp where a is

This preview shows 4 out of 6 pages.

process with digital signal processing (DSP), where a is in analog to the sampling period, k corresponds to the detected frequency. The highest measurable frequency is inversely proportional to a . Now we will extend our discussion to three-dimensional cases. With a translation lattice vector 1 1 2 2 3 3 T n a n a n a = + + G G G G , we can reach any crystal point from the origin. For a function u( r )= u( r+T ), we will first give the following answer and then show that the given Fourier expansion indeed satisfies the required periodicity, ( ) i u u e = r G G G r where G and the inverse transformation are given by G =m 1 b 1 +m 2 b 2 +m 3 b 3 and ( b 1 , b 2 , b 3 ) are conjugated to the primitive lattice vectors ( a 1 , a 2 , a 3 ) through V / ) ( 2 3 2 1 a a b × = π , V / ) ( 2 1 3 2 a a b × = π , V / ) ( 2 2 1 3 a a b × = π where ( ) 3 2 1 a a a × = V is the volume of the primitive unit cell in real space. For the one-dimensional case, we have G =2 π n/a, r =x. With the above definitions, we can show that u( r ) is indeed invariant with any translational lattice vector in the real space, T (=n 1 a 1 +n 2 a 2 +n 3 a 3 ), where n 1, n 2 n 3 integers, ( ) 1 1 2 2 3 3 2 ( ) ( ) ( ) r T G r G T G G G G G r G r G G G G G r T r i i i i i n m n m n m i u u e u e u e u e u π + + + + + + = = = = = Thus, we see that the new set of vectors introduced, ( b 1 , b 2 , b 3 ), which has a unit of m -1 , are the corresponding Fourier conjugate to the real space lattice vector ( a 1 , a 2 , a 3 ). We can use ( b 1 , b 2 , b 3 ) to construct a new lattice called the reciprocal lattice . Previous definitions on real space lattices, such as unit cells and the Wigner-Seitz primitive unit cell, are equally applicable to such a reciprocal lattice. This reciprocal space is the Fourier conjugate of the real space. Although a very abstract concept, the reciprocal lattice can actually be easily mapped out with diffraction experiments. When electron waves or X-rays (electromagnetic waves) with proper energy are shone onto a crystal, the reflection or transmission occurs only along specific directions, as shown in the following figure.
Image of page 4

Subscribe to view the full document.

2.57 Fall 2004 – Lecture 8 5 Consider an incident wave from the source along direction k . The incident wave is proportional to ( ) i s i e k r r , where r is any point in the sample and r s is the location of the source relative to the origin of coordinates. The wave scattered into the detector is then proportional to ( ) ( ) f d i n e k r r r , where k f is the propagation direction of the scatter a wave, n( r ) is the nuclei density, and r d is the position of the detector. Because each of the atoms
Image of page 5
Image of page 6
You've reached the end of this preview.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern