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Stato Solido Capitoli:
Appendici |
Note08MONATOMIC CHAIN. (Needs animation for longitudinal monatomic chain at k~0 and k=BZ. A skeleton version is from Wolfram Player A static picture for aliasing and two picture for H-BOND (ICE and DNA) Recap bonds. This lecture strictly one dimensional. [Draw two atoms define x their distance, draw separately V(x) vs x as chemistry describe attractive vs repulsive if atoms too close, very steep Draw quadratic expansion parabola, draw xeq, write V(x) = V0 + 1/2 (x-xeq)^2 + ... Are we throwing the baby with the hot water? what is lost? thermal expansion how does it happen? distance between atoms like particle in bottom of well low T vs higher T, that probes xmin-xeq<xeq-xmin] [hence (xmin+xmax)/2 > xeq potential softer for going out, harder for moving in this gives thermal expansion, Apart from it it is ok to truncate to quadratic simple model of vibration MONATOMIC HARMONIC CHAIN (most important model in the course, come frequently in exercise) A lot of ideas introduced] [dram chain : atoms, identical masses, equal spring constant k, equilibrium distance a [ new slide known as a = LATTICE CONSTANT is DISTANCE BETWEEN IDENTICAL ATOMS draw again on chain x1 x2 x3 ... and under a = write xn = POSITION OF ATOM n x0n = EQUILIBRIUM POSITION OF ATOM n = na (if x0=0)] [ we are interested in deviation fom equilibrium dxn = xn - x0n per cominciare fisica classica e vediamo cosa cambia con MQ dopo Newton eq per ogni massa Fn = m \ddot dxn = kappa (dnx+1 - dxn) + kappa(dxn-1 + dxn) (spring on right plus spring on left, draw to show!) = kappa (dxn+1 + dxn-1 - 2 dxn) again under chain drawing WANT NORMAL MODES] [arrow under NORMAL MODES are ALL ATOMS OSCILATE AT COMMON FREQUENCY this is an eigenvalue problem with a matrix of dimension the number of masses infinity number of masses seems difficult but ther is a trick: guess the answer. under forces WAVE ANSATZ (German for approach, starting point)] [ dxn = A e^{i omega t - i k x0n} where k is WAVEVECTOR, at the end of the day we take the real part because of the Real part we can as well use a minus sign in the exponent Therefore we can assume omega >= 0 as long as k>0 means progressive (right going) and k<0 means regressive (left going), substitute dxn with na in equation substitute in equation -m omega^2 A e^{i omega t - kna} = kappa A e^{i omega t} ( e^{-ik(n+1)a}+e^{-ik(n-a)a -2e^{-ikna})] [simplify -m omega^2 = kappa (e^{-ika} + e^{ika} -2) omega^2 = kappa/m (2-2cos ka) omega^2 = 4 kappa/m sin^2 ka/2 omega = 2 sqrt(kappa/m) |sin ka/2|] [ plot the answer omega vertical vs k horizontal put points +- pi/a draw two domes of |sin ka/2| indicate max at +- pi/a with values 2sqrt(kappa/m) label as DISPERSION CURVE (means omega(k))] we should expect sound waves somewhere in it (vibration of solids poroduce sound waves) wavelengths are cm to several meters whereas 2pi/a is the wavelength of atom spacing, a shade area close to zero in dispersion, small argument expansion of sine [omega = \sqrt(kappa/m) a |k| (remember that k can have both signs) compare with omega = v |k| therefore v = sqrt(kappa/m) a go to new slide From thermodynamics and waves 1st year the so called hydrodynamic limit you should know that (check Solzi) [v = sqrt( B/mu) with B = BULK MODULUS inverse of compressibility, mu MASS DENSITY for chain mu = m/a B = - V dp/V and in 1 dim = - L dF/dL = -a (-kappa) = kappa a] go back to top definition v = ... = sqrt(kappa/(m/a)) = a sqrt(kappa/m) so Mono Chain agrees with hydrodynamic limit (very long wavelength can ignore atoms, but it must agrees quantitaively with atomic chain prediction). go back to dispersion drawing: at long wavelegth small k the dispersion is linear but at short wavelength large k it si not linear any more remember Debye! he assumed that the dispersion stayed linear and than he imposed a cutoff freq: waves cannot exceed a certain value. Well it is not linear but indeed above some freq there is no oscillations All of the normal modes of the system are within a maximum let's look at the highest freq normal mode at k= pi/a it looks dxn = A e^{i omega t - i (pi/a) an} = A e^{i omega t} (- 1)^n every other atom is moving in opposite directions, that's how fast you can possibly get- show animation shows transverse and longitudinal waves, shows moving k start from small value (increase velocity to see the motion) the atoms move together backward an dforward increase k and go to maximum pi/a (slow down motion) the atoms are moving in opposite direction (they caannot do better that this, think about that) [ DISPERSION IS PERIODIC IN k it keeps going being prop to |sin| periodicity 2pi/a general law PERIODIC IN REAL SPACE (DIRECT SPACE) delta x = a <=> PERIODIC IN k-SPACE (RECIPROCAL) Delta k = 2pi/a PERIODIC UNIT in k space = BRILLOUIN ZONE Luis de Brilluoin why? dxn = A e^{i omega t -ikna} move it k-> k+2pi/a] [ = A e^{i omega t -i(k+2pi/a)na} = A e^{i omega t -ikna} e^{ -2pi n} = 1] so what is the wavelength? lambda = 2pi/k picture of oscillation of masses with aliasing showing two waves, k and k+2pi/a predicting same displacements on masses. No nmeaning in between masses, (so lambda is the smallest, by convention)] [ REAL SPACE LATTICE is POINTS EQUIV TO x=0, i.e. xn = na RECIPROCAL SPACE LATTICE POINTS IN k-space EQUIVALENT to k=0, let's call them Gm = 2 pi m/a therefore e^ģ Gm xn = 1 for all points in direct and reciprocal space lattice. A similar expression defines the reciprocal lattice in higher dimensions] [HOW MANY NORMAL MODES? only inside the BZ USE PERIODIC BOUNDARIES L = Na in a circle k = 2pi/L p (p Z)] [ NUMBER OF MODES range of k / spacing between modes 2pi/a /2 pi/L = L=a = N there are N normal modes because there are N masses to begin with, same as for I physics few masses] [ classical so far, move to QUANTUM general rule NORMAL MODE OF FREQ w => EIGENSTATE WITH ENERGY En = hbar w (n+1/2) related to amplitude of normal mode: osciellates a lot larger energy, osicllate less smaller energy, key point is that energy is quantized in units of hbar w A QUANTUM OF VIBRATION IS A PHONON (analogous to a quantum of light being a photon) ITS ENERGY is hbar omegak. Bad to specify that a photon is a quantum of light ENERGY) (also of MOMENTUM etc.) No energy in NORMAL MODE -> zero phonons; the next step is 1 phonon, 2 phonons PHONONS are BOSONS (because you can put as many as you want in the same k mode <En> = hbar wk (nB(beta hbar wk) + 1/") energy of the phononS in a particular k mode and nB is Bose factor, expected number of phonons in k mode at a given T] CHAIN is a simplified version of 1 d matter, number of simplifications: only nearest neighbor springs, and perfectly harmonic springs, but apart from these we can calculate exact heat capacity [UTOT = totale energy in chain = Sum_k=-pi/a^pi/a in setps of 2pi/L hbar wk (nB(beta hbar wk) +1/") only inside BZ) Sum -> L int dk/2pi_-pi/a^pi/a so if we want to compute number of modes L/2pi int_-pi/a^pi/a dk 1 = L/a = N] (exactly what Debye did) [WE USE wk = 2 sqrt(kappa/m) |sin ka/2| sum and then differenciate o get the heat capacity DEBYE USED wk=v|k| agrees al small k but not at high kł EINSTEIN USED wk = wE] [ONE MORE Concept k <-> k + Gm (2pim/a) phonons wave-particle carry momentum e.g. phonons can scatter (into other phonons, electrons, light etc.) How can they conserve momentum if momentum is not well defined DEFINE CRYSTAL MOMENTUM hbar [ k modulo 2pi/a] up to additive terms of, eg [12 mod 5] = [7 mod 5] = [2 mod 5] momentum conservation derives from translational symmetry of space (Noether theorem Emmy Noether) not true in lattices,symmetry is only for finite periods, somewhat less than full TS, hence momentum conservation is only up to hbar Gm N theroem applies to continuous symmetries, this is discrete symmetry (add where does the momentum go) Adds HYDROGEN BONDS Draw a big atom F bonds to H. By argument for covalent bonds F sucks most of the H eletcron and leaves a bare p. Not screened by electron around it. If one has another F near. p+ makes very strong polarization of n F and attracts it. This is H-BOND. SHOW classic picture ICE. Much weaker than covalent or ionic but strong enough to make ice solidify at 0 C = 273 K. Anoteyhr classic case in biology is DNA is made of a double strand and in the middle, where they zip together H bonds to O or similar N to H holds together the DNA and the RNA |