MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.62 Spring 2008 Lecture #22 Page 1 Einstein and Debye Solids Reading: Hill, pp. 98-105, 490-495 The Einstein (quantum) model (all vibrational modes have the same frequency) gave much better agreement with experiment than the Dulong-Petit (classical) model (equipartition). But as T decreases, the Einstein model CV decreases too fast relative to the experimentally observed (approximately T+3 dependence) behavior of CV. Perhaps it would be more realistic to allow the vibrational frequencies to follow a plausible, computationally convenient, but non-constant probability distribution, ρ(ν) Debye Treatment Debye derived an improved model for the thermodynamic properties of solids by assuming that the distribution of normal mode frequencies is equivalent to that for sound waves. CV= ki=13N !6"h#ikT$%&'()2e!h#ikT1! e!h#ikT( )2CVDebye= k0#max*d#+( #)h#kT$%&'()2e!h# kT1! e!h# kT( )2density of vibrational frequencies For sound waves traveling in a three-dimensional solid ! "( )# "2[see Nonlecture derivation, below]. One way of seeing this is that ! =c2"k~=c2"kx2+ ky2+ kz2( )1/ 2, where k~is the wave vector such that the wave is described by ei k~• r~and ! =2"|!k |. The density of states with a given ν is the number of ways of choosing k~with the corresponding magnitude, ⎟k~⎟. Just as the degeneracy for a given speed state is proportional to c2 in the kinetic k~theory of gases (as you'll see later in 5.62), the number of ways of picking with magnitude k = | k~| is proportional to k2. revised 3/21/08 9:01 AM5.62 Spring 2008 Lecture #22 Page 2 So ρ(ν) ∝ k2 ∝ ν2 This figure is supposed to show a spherical shell of radius !k|k| and thickness dk. The number of states with |k| between |k| and |k| + dk is proportional to the volume of this shell, 4π|k|2. Problem: What is the distribution of acoustic frequencies in an elastic solid? We are interested in the 3N lowest frequencies. Solution: Find the harmonic frequencies which satisfy the boundary condition that the displacements are zero at the surface of a crystal of volume V. The wave equation for this problem is very similar to the Schrödinger Equation for a particle in a 3D infinite cubical well. Consider the initial wave at t = 0, with displacements as function of position, x: Φ0(x) = Φ(x,t = 0). At t ≠ 0, the initial wave has moved in the +x direction by vst, where vs is the speed of sound in this medium Φ(x,t) = Φ0(x - vst). Making the harmonic approximation: !0(x) = A cos2"x#$%&'()!(x,t) = A cos2"(x * vst )#+,-./0 where vs#= 1= A cos2"1xvs* 2"1t+,-./0Find the values of ν for which Φ(x,t) = 0, where x is at the surface of the crystal of volume, V. revised 3/21/08 9:01 AM5.62 Spring 2008 Lecture #22 Page 3 Schrödinger Equation: !2" x, y, z( )=#2m$!2" x, y, z( ).For 3-D infinite cubical box of length a on each edge 2m!nxnynz!2="2a2nx2+ ny2+ nz2( )where satisfy the boundary condition that ψ(x,y,z) = 0 on all six surfaces of the cube. Now use the required values of {ε} to solve for allowed ν's: !nxnynz{ }d2!(x,t)dx2=d2dx2A cos2"#xvs$ 2"#t%&'()*+,-./0= $2"#vs%&'()*2!(x,t).Generalize to 3 dimensions !2"(x, y, z ,t) = #2$%vs&'()*+2"(x, y, z ,t).Comparing prefactors for Schrödinger equation and wave equation 2!"vs#$%&'(2=!2a2nx2+ ny2+ nz2( ).So now we know how ν depends on the number of standing waves in each of the three crystal directions. We want to know the density of vibrational modes as a function of frequency, ρ(ν), but it is easier to derive the density of modes as a function of n, ρ(n), where n2! nx2+ ny2+ nz2( ).This is the equation for a sphere. So the number of modes between n and n + dn is given by the volume of one octant (nx, ny, nz, and n are all positive) of a spherical shell of radius n and thickness dn !(n)dn =184"n2dnrevised 3/21/08 9:01 AM5.62 Spring 2008 Lecture #22 Page 4 We want ρ(ν) !(") = !(n)dnd".To find the value of the Jacobian, dnd!, for the transformation between n and ν as the independent variable, n = nx2+ ny2+ nz2( )1 / 22!"vs=!an =!anx2+ ny2+ nz2( )1/ 2n =2a"vsdnd"=2avs#(") = #(n)dnd"=184!n2( )2avs=4!a3"2vs3a3= V.There are 3 polarizations (x, y, or z) for each lattice mode, thus !(") = 34#Vvs3"2This is the frequency distribution function that goes into the Debye model. So we have a physically reasonable model for the density of vibrational states as a function of frequency, ρ(ν). But Debye had one more trick up his sleeve before inputting ρ(ν) to a statistical mechanical calculation of macroscopic thermodynamic properties. There cannot be an infinite number of modes; only 3N–6 ≈ 3N. So Debye cut off the mode distribution arbitrarily at νmax to give the correct number of modes. ρ(ν) = Aν2 where A is determined by Debye's cutoff at νmax 3N =0!max"#(!)d! = A0!max"!2d! =A!max33$ A =9N!max3$ # !( )=9N!max3!2NOTE: We still don't know what νmax is, only that the mode distribution is normalized to this parameter. revised 3/21/08 9:01 AM5.62 Spring 2008 Lecture #22 Page 5 Now calculate some bulk properties: Cv= k0!"h# / kT( )2e$h#/ kT1$ e$h#/ kT( )2%(#)d#CVDebye= k0#max"9N#max3#2h#kT&'()*+2e$h#/ kT1$ e$h#/ kT( )2d#Debye Temp. , -D,h#maxk x =h#kTd# =kThdxChange of variable from ν to x CVDebye= k0!DT"9N#max3x2kTh$%&'()2x2e*x1 * e*x( )2kTh$%&'()dx= 9NkkTh#max$%&'()30!DT"x4e*x1 * e*x( )2dx= 9NkT!D$%&'()30!DT"x4exex*1( )2dxy=!DT+ ,++9Nky30y"x4exex*1( )2dxnot a generallytabulated function Integrate by parts u = x4v =!1ex! 1( )du = 4 x3dx dv =exex! 1( )2dxCVdebye=9Nky3!x4ex!10y+4 x3dxex!10y"#$%%&'((= 3Nk •3y340y"x3dxex!1!y4ey!1#$%&'(= 3Nk • 4 •3y30y"x3ex!1dx !3yey!1#$%&'(CVDebye= 3Nk • 4D(y) !3yey!1"#$%&'Debye Einstein Function Function revised 3/21/08 9:01 AM5.62 Spring 2008 Lecture #22 Page 6 Check high and low temperature limits of C VDebye: high T limit CvDebye3Nk=12y30y!x3ex"1dx "3yey"1x =h #kTy =$DT=h #maxkT%12y30y!x31+ x +…"1( )dx "3y1+ y + … "1( )T ! $D& x, y ' 0=12y30y!x2dx " 3 =12y3•y33" 3 = 4 " 3 = 1 ! CVDebye" #" 3Nk for T ! $D(agrees with classical and Einstein treatments) low T limit CvDebye3Nk=12y30y!x3ex"1dx "3yey"1T # 0 $ x, y # %&12y30%!x3ex"1dx "