Finite-temperature equation of state. T ln 2sinh h" '-

Transcription

1 Finite-temperature equation of state * $ F(V,T) = U 0 + k B T ln 2sinh h" '- #, & )/ + % 2k B T (. 0 # Compute vibrational modes, frequencies Evaluate at a given volume V Compute F at various temperatures T with above Need the Hessian!

2 Start again with fcc lattice Atoms should be relaxed (zero force on atoms) Evaluate at a given volume V Compute F at various temperatures T and volumes V

3 Generate non-primitive basis-- 4 atoms! start with non-primitive fcc basis r1(1)=0.0d0 r2(1)=0.0d0 r3(1)=0.0d0 r1(2)=0.5d0 r2(2)=0.5d0 r3(2)=0.0d0 r1(3)=0.5d0 r2(3)=0.0d0 r3(3)=0.5d0 r1(4)=0.0d0 r2(4)=0.5d0 r3(4)=0.5d0

4 Some parameters for lattice INTEGER, PARAMETER :: nn=4 INTEGER, PARAMETER :: Prec14=SELECTED_REAL_KIND(14) INTEGER, PARAMETER :: natoms=4*nn**3 INTEGER, PARAMETER :: nx=nn,ny=nn,nz=nn REAL(KIND=Prec14), PARAMETER :: sigma=1.0d0,epsilon=1.0d0 4x4x4 lattice of non-primitive units 64x4=256 atoms System has 256x3=768 degrees of freedom! The Hessian matrix needs to be 768x768

5 Generate the full lattice n=0 do ix=1,nx do iy=1,ny do iz=1,nz do i=1,4 n=n+1 rx(n)=(r1(i)+dble(ix-1))*sigma ry(n)=(r2(i)+dble(iy-1))*sigma rz(n)=(r3(i)+dble(iz-1))*sigma enddo enddo enddo enddo! scale lengths by box size rx=rx/nx ry=ry/ny rz=rz/nz

6 We need more than forces this time! Definition of Hessian matrix elements H iµ, j" = # 2 U #r iµ #r j" For N atoms, this means a 3Nx3N matrix Many elements will be zero due to cutoff First derivative is just related to force on j: "U "r j# = % k ( ) "U r $ r j# k# "r kj r kj But need higher order derivatives!

7 " 2 U = $ " 2 U 2 "r iµ "r j# "r ij We need more than forces this time! H iµ, j" = # 2 U #r iµ #r j" For i,j referring to different atoms Also need terms where i=j ( r iµ $ r )( jµ r i# $ r ) j# + "U &( r $ r )( iµ jµ r i# $ r ) j# $ % ) µ,# ( r ij "r ij '( r ij r ij * + " 2 U "r iµ "r i# = - k,i " 2 U "r ik 2 ( ) ( r iµ $ r ) kµ r i# $ r k# $ 2 r ik - k,i "U &( r $ r ) iµ kµ ( r i# $ r k# ) $ % ) µ,# ( + 3 "r ik '( r ik r ik * + Notice that for each pair of components µ,ν, we have # H j iµ, j" = 0 or H iµ,i" = #% H iµ, j" j$i Can be easily tested in code! Also Hessian is symmetric

8 How does this relate to the phonon frequencies? Assume harmonic motion, so that the displacements are u iµ (t) = 1 # a 1/ 2 " $ iµ," e i% " t m i " And the potential energy is quadratic in displacements U = U " 2 U $ u iµ u j# 2 "r iµ r iµ, j# j# 0 And the equation of motion for an atom is, m i d 2 u iµ dt 2 = " % j$ # 2 U #r iµ r j$ 0 u j$

9 Eigenvalue equation Assuming a single normal mode excited, we find from Eq. of motion $ " % = ' 2 % iµ, j# j#,& & iµ,& j# The force-constant matrix is given by " iµ, j# = 2 U 1 $ ( m m ) 1/ 2 $r r i j iµ j# 0 Related to Hessian, just including the masses of the atoms

10 General procedure 1. Choose a volume V 2. Compute Hessian --> force constant matrix 3. Diagonalize real-symmetric matrix 4. Compute free energy F(V,T) for various temperatures 5. Return to 1. This should allow us to generate F(V,T) curves Plot F(V,T) vs. volume V for various temperatures Find minimum by fitting to Eq. of state Thermal expansion. Anharmonicity through dependence of modes on V Usually called quasi-harmonic approximation

11 Declarations, Equation of state code! Program to compute the free energy for a lennard jones solid IMPLICIT NONE INTEGER, PARAMETER :: nn=4 INTEGER, PARAMETER :: Prec14=SELECTED_REAL_KIND(14) REAL(KIND=Prec14), PARAMETER :: L0=2.0d0**(2.0d0/3.0d0)*nn,temp=100.0d0 INTEGER, PARAMETER :: nvol=25 INTEGER, PARAMETER :: natoms=4*nn**3,ndim=3*natoms INTEGER, PARAMETER :: nx=nn,ny=nn,nz=nn INTEGER :: i,j,ix,iy,iz,mu,nu,m,n,matz,m1,n1,ierr REAL(KIND=Prec14), PARAMETER :: sigma=1.0d0,epsilon=1.0d0 REAL(KIND=Prec14), PARAMETER :: amass=1.0d0 REAL(KIND=Prec14), PARAMETER :: rcut=3.0d0*sigma REAL(KIND=Prec14) :: epot,dvdr,dvdr2,pi,fac,free,dvol,vmin,vmax,l REAL(KIND=Prec14) :: sxij,syij,szij,rij,sovr,volume,nv REAL(KIND=Prec14), DIMENSION(4) :: r1,r2,r3 REAL(KIND=Prec14), DIMENSION(3) :: s REAL(KIND=Prec14), DIMENSION(natoms) :: rx,ry,rz REAL(KIND=Prec14), DIMENSION(ndim,ndim) :: hess,z REAL(KIND=Prec14), DIMENSION(ndim) :: w,fv1,fv2 REAL(KIND=Prec14), PARAMETER :: mass=39.948d0,eps=120.0d0,sar=3.4d0

12 Some preliminaries Need to compute at several volumes, one temperature Also need to zero Hessian for calculations pi=4.0d0*datan(1.0d0)! Compute volume volume=l0**3 vmin=volume*0.85d0 vmax=volume*1.05d0 dvol=(vmax-vmin)/(nvol-1) Then put the atoms on the fcc lattice as in MD code (see several slides previous)

13 Do loop on the volumes, several volumes do nv=1,nvol! Clear the hessian out hess=0.0d0 epot=0.0d0! set potential energy to zero volume=vmin+(nv-1)*dvol L=volume**(1.0d0/3.0d0)

14 Loop on all atom pairs, compute for rij<rcut do i=1,natoms-1 do j=i+1,natoms sxij=rx(i)-rx(j) syij=ry(i)-ry(j) szij=rz(i)-rz(j) sxij=sxij-dble(anint(sxij)) syij=syij-dble(anint(syij)) Added a 1/rij here compared to forces szij=szij-dble(anint(szij)) sxij=sxij*l; syij=syij*l; szij=szij*l rij=dsqrt(sxij**2+syij**2+szij**2) s(1)=sxij/rij; s(2)=syij/rij; s(3)=szij/rij if(rij.lt.rcut) then sovr=sigma/rij epot=epot+4.0d0*epsilon*(sovr**12-sovr**6) dvdr=-(24.0d0*epsilon/rij**2)*(2.0d0*sovr**12-sovr**6) dvdr2=(24.0d0*epsilon/rij**2)*(26.0d0*sovr**12-7.0d0*sovr**6) New lines, first and second derivative of V with respect to r

15 Identifying atom numbers with matrix elements 3N degrees of freedom (x,y,z, corresponding to µ=1,2,3) Atom #1 1,2,3 Atom #2 4,5,6 Etc. Then for atom number i, component mu: m=3*(i-1)+mu For atom j, component nu: n=3*(j-1)+nu Then we can fill matrix element hess(i,j)

16 Implement. do mu=1,3 do nu=1,3 m=(i-1)*3+mu n=(j-1)*3+nu if(mu.eq.nu) hess(m,n)=-dvdr2*s(mu)*s(nu)+dvdr*(s(mu)*s(nu)-1.0d0) if(mu.ne.nu) hess(m,n)=-dvdr2*s(mu)*s(nu)+dvdr*s(mu)*s(nu) hess(m,n)=hess(m,n)/amass hess(n,m)=hess(m,n)! Matrix needs to be symmetric, and we are only using eac. Need to add then terms for H iµ,jν m1=(i-1)*3+mu n1=(i-1)*3+nu hess(m1,n1)=hess(m1,n1)-hess(m,n) m1=(j-1)*3+mu n1=(j-1)*3+nu hess(m1,n1)=hess(m1,n1)-hess(m,n) Diagonalize using the subroutine rs, once hess is filled

17 What do you get? call rs(ndim,ndim,hess,w,matz,z,fv1,fv2,ierr) Eigenvalues are the ω 2 for each of the 3N modes First three should be nearly zero might be slightly negative Next ones should all be greater than zero (in general) Units can be found in THz (for example) w(n)=dsqrt(w(n))! Find frequencies from eigenvalues! Convert frequency to a unit in THz fac=dsqrt((1.381d0*6.022d0*eps)/(10.0d0*mass*sar**2)) w(n)=w(n)*fac/(2.0d0*pi)

18 What do you get? Eigenvalues are the ω 2 for each of the 3N modes First three should be nearly zero might be slightly negative Next ones should all be greater than zero (in general) Units can be found in THz (for example) w(n)=dsqrt(w(n))! Find frequencies from eigenvalues! Convert frequency to a unit in THz fac=dsqrt((1.381d0*6.022d0*eps)/(10.0d0*mass*sar**2)) w(n)=w(n)*fac/(2.0d0*pi) fac=(1.381d-4/1.602d0)! kb in ev/k epot=epot*eps*fac free=epot do n=4,ndim! sum on eigenvalues free=free+fac*temp*dlog(2.0d0*dsinh(w(n)/(2.0d0*fac*temp))) enddo free=free/natoms write(6,100) volume*sar**3/natoms,free

19 Birch-Murnaghan equation of state Determine equilibrium volume V 0, Bulk modulus Murnaghan: $ p = "& #F %#V ' ) ( T $ B = "V & #p %#V ' ) ( T Experimentally, it is found that B varies with pressure B = B 0 + " B p Combining these and integrating, we get p(v ) = B 0 " B *#,% + $ V 0 V B & 0 " - ( )1/ '.

20 Birch-Murnaghan equation of state A slightly more sophisticated expression is: Birch-Murnaghan: p = 3B 0 2 )" + $ *# V 0 V % ' & 7 / 3 " ( V % 0 $ ' # V & 5 / 3, ( B 0 / ( 4) )" + $ *# V 0 V % ' & 2 / 3, 42 ( After integrating we get F = F 0 + 9V 0 B )" 1 + $ *# 32 V 0 V % ' & 2 / 3, ( )" B 0 / + V % 0 + $ ' *# V & 2 / 3, ( ) " 6 ( 4 V % 0 + $ ' * # V & 2 / 3, Equilibrium (p=0) volume is V 0 B 0 is bulk modulus at p=0 F 0, V 0, B 0 are fitting parameters Monte Carlo code to do fitting exists

21 Free energy vs. volume T=5K T=10K T=50K T=100K Thermal expansion, V 0 increases with temperature Curvature related to bulk modulus B Vibrational energy/entropy

22 ! get initial direction h h=g! output current potential energy After Hessian and intial direction g found write(6,*) 'Hessian computed, potential energy=',epot! compute lambda lambda=0.0d0 do n=1,ndim lambda=lambda+g(n)*g(n) enddo vec=0.0d0 do m=1,ndim do n=1,ndim vec(m)=vec(m)+hess(m,n)*h(n) enddo enddo denom=0.0d0 do m=1,ndim denom=denom+h(m)*vec(m) enddo