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lecture7 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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2.57 Fall 2004 – Lecture 7 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 7 In the last lecture, we have talked about atoms in a one-dimensional chain. We find the solution as u j = A exp[ i ( ω t kja )], where the frequency is = 2 K m sin ka 2 . Note: When k approaches zero for large wavelength, the frequency becomes a linear function of the wavevector, i.e. 2 2 Kka K ka mm ≈= . We can calculate the sound velocity by sound d v dk = . In last two lectures, we have derived the allowed k values as 2 k n Na π = (n=0, ± 1, ± 2,. ..). In the above figure, k= π /a corresponds to λ =2a. However, k> π /a is meaningless in physics because there is no atoms vibrating between one period. Thus the allowable Na ja (j-1)a (j+1)a π /a ω k ATOM

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2.57 Fall 2004 – Lecture 7 2 wavevector for a lattice vibration is naturally confined to the first Brillouin zone ( |k|< /a π ). Therefore, we totally have N allowable wavevectors (also wavelength) in between – π /a<k< π /a. Each of these wavevectors corresponds to one mode of the vibration of the lattice. This mode is called a normal mode . For a harmonic oscillator, quantum mechanics gives 1 (1 / 2 ) ; 2 n K Eh n m νν =+ = , in which the frequency is consistent with classical mechanics. Similarly, quantum mechanics gives the energy levels for a chain with N atoms as E n = h ν ( n + 1 2 ) or 1 () 2 n En ω = (n=0,1,2,3,…), in which = 2 K m sin ka 2 . The basic vibrational energy quanta, h ν , is called a phonon. Comparison between electrons, phonons, and photons: (1) Electrons obey the Pauli exclusion principle, which says that each quantum state can only have at most one electron. Photons and phonons are not limited by the Pauli exclusion principle. Each quantum state, which corresponds to one set of wavevectors, can have many phonons and photons. (2) Unlike electrons, phonons and photons at rest do not have mass though they have momentum and energy. They are also called fictitious particles since they are the quantization of the normal mode of a field. 3.3.3 Polyatomic lattice chain a a 1 , k 1 a 2 , k 2 m M
2.57 Fall 2004 – Lecture 7 3 Now let us consider a chain with two atoms per period, in which two types of motions exist. In the first case, the adjacent two atoms vibrate in phase, while in the second figure the two atoms are moving out of phase. Clearly, the out-of-phase modes require more energy. Lower frequency (energy) branch is called the acoustic branch and the higher frequency one is called the optical branch, because the high frequency phonons in the optical branch can interact with electromagnetic waves more easily. In general, if there are m atoms in a basis and N lattice points in the chain, there are one acoustic branch with N acoustic modes, and (m-1) optical branches with (m-1) N optic modes.

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lecture7 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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