Reduced masses and frequency ratios

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Factors

Constants

N[I]

N[π]
N[Pi]

N[E]

[c] = m/s

c=299792458.;

pi=3.14159265; (*N[Pi,9]*)
invpi= 0.318309886; (*1/N[Pi,9]*)

Prefix Transformation

centi

base2centi=10.^(2.);
centi2base=10.^(-2.);

milli

base2milli=10.^(3.);
milli2base=10.^(-3.);

micro

base2micro=10.^(6.);
micro2base=10.^(-6.);

nano

base2nano=10.^(9.);
nano2base=10.^(-9.);

pico

base2pico=10.^(12.);
pico2base=10.^(-12.);

femto

base2femto=10.^(15.);
femto2base=10.^(-15.);

Kilo

base2kilo=10.^(-3.);
kilo2base=10.^(3.);

Mega

base2mega=10.^(-6.);
mega2base=10.^(6.);

Giga

base2giga=10.^(-9.);
giga2base=10.^(9.);

Tera

base2tera=10.^(-12.);
tera2base=10.^(12.);

Conversion factors

Angular frequency transformations

Angular frequency to frequency ([ω]= rad/s  to [ν ] = 1/s = Hz)
ω = 2πν
ν = (ω/(2π))

rads2Hz= 1./(2.*pi);

Angular frequency to wavenumbers ([ω]=rad/s to [(OverTilde[ν])]= (cm^(-1)))
ω=2πν= 2π c (OverTilde[ν])
(OverTilde[ν])=(ω/(2π c))

rads2wn= (1./(2.*pi*c))*(1./base2centi);

Frequency transformations

Frequency to wavenumber ([ν]=1/s=Hz  to [(OverTilde[ν])]= (cm^(-1)))
ν=  c (OverTilde[ν])
(OverTilde[ν])=(ν/c)

Hz2wn=(1./c)*(1./base2centi);

Frequency to angular frequency (Hz to rad/s)
ω = 2πν

Hz2rads= 2.pi;

Wavenumber transformations

Wavenumber to angular frequency ((cm^(-1)) to rad/s)

wn2rads=2.*pi*c * (1./centi2base)

Wavenumber to frequency ((cm^(-1)) to 1/s=Hz)

wn2Hz= c* (1./centi2base);

Formulas

Harmonic oscillator

For a harmonic oscillator, the oscillation frequency is given by ω=(Sqrt[(k/m)]), with k the force constant and m the mass. In a two particle system, the mass is replaced by the reduced mass, μ=(((Subscript[m, 1])*(Subscript[m, 2]))/((Subscript[m, 1])+(Subscript[m, 2]))).
To predict a frequency change from a change in the mass of the oscillator, the ratio of the frequencies can be used: ((Subscript[ω, A])/(Subscript[ω, B]))=(Sqrt[((Subscript[k, A])/(Subscript[μ, A]))])*(Sqrt[((Subscript[μ, B])/(Subscript[k, B]))])=(Sqrt[((Subscript[k, A])/(Subscript[k, B]))])*(Sqrt[((Subscript[μ, B])/(Subscript[μ, A]))])
If the change in the force constant k is assumed to be insignificant, and as such (Subscript[k, A]) ≈ (Subscript[k, B]), this can be further simplified:
((Subscript[ω, A])/(Subscript[ω, B]))=(Sqrt[((Subscript[μ, B])/(Subscript[μ, A]))])

redmass[m1RM_,m2RM_]:=(m1RM*m2RM)/(m1RM+m2RM);

freqratio[rm1FR_,rm2FR_]:=Sqrt[rm2FR/rm1FR];

Phonon

Optical at 0 k

Assuming a cubic crystal lattice with two different sites, the equation of motion close to the limit of k=0 is:
(ω^2)≈ 2C ((1/(Subscript[M, 1]))+(1/(Subscript[M, 2])))
The ratio for two different crystals, assuming similar force constants, is then:
((Subscript[ω, A])/(Subscript[ω, B]))≈(Sqrt[(((1/(Subscript[M, 1,A]))+(1/(Subscript[M, 2,A])))/((1/(Subscript[M, 1,B]))+(1/(Subscript[M, 2,B]))))])

freqratioPO0[m1aFRP_,m2aFRP_,m1bFRP_,m2bFRP_]:=Sqrt[((1/m1aFRP)+(1/m2aFRP))/((1/m1bFRP)+(1/m2bFRP))];

The acoustic phonon close to k=0 is nearly zero anyway due to a k vector dependence

Optical at π k, zone boundary

If we go to the Brillouin zone boundary for the same lattice, we find the motion to be:
ω≈(Sqrt[((2C)/(Subscript[M, 2]))])
The ratio is then ((Subscript[ω, A])/(Subscript[ω, B]))≈(Sqrt[((Subscript[M, 2,B])/(Subscript[M, 2,A]))])if the force constants are similar. 
(Subscript[M, 1]) seems to be heavier usually, because then the phonon branches don’t cross.
So this would be simply the root of the ratio of the anion mass in our case?

freqratioPOPI[m2aFRPP_,m2bFRPP_]:=Sqrt[m2bFRPP/m2aFRPP];

Acoustic at π k, zone boundary

If we go to the Brillouin zone boundary for the same lattice, we find the motion to be:
ω≈(Sqrt[((2C)/(Subscript[M, 1]))])
The ratio is then ((Subscript[ω, A])/(Subscript[ω, B]))≈(Sqrt[((Subscript[M, 1,B])/(Subscript[M, 1,A]))])if the force constants are similar
We probably wouldn’t see a shift in those then, because the heavier cations are extremely similar in mass.
It would be in line with the fact that our methods should excite only the optical phonons in zerost approximation due to the interaction of the motion with the electrical field. Acoustic modes would not have the charge displacement to do that.
Acoustic phonon in the middle goes to zero because of the k vector dependence.

Masses

Ion masses

mCl=35.453;
mBr=79.904;
mBu4N=242.463;
mBu3NMe=200.387;
mBu3NH=186.357;

mEt4N=130.25;
mEt3NMe=116.227;
mEt3NH=102.197;

Sqrt[mCl/mBr]
Sqrt[mBu3NMe/mBu3NH]
Sqrt[mBu4N/mBu3NMe]
Sqrt[mBu4N/mBu3NH]

Sqrt[mEt3NH/mBu3NH]
Sqrt[mEt3NMe/mBu3NMe]
Sqrt[mEt4N/mBu3NH]

Reduced masses

muBu4NCl=redmass[mCl,mBu4N]
muBu4NBr=redmass[mBr,mBu4N]
muBu3NMeCl=redmass[mCl,mBu3NMe]
muBu3NMeBr=redmass[mBr,mBu3NMe]
muBu3NHCl=redmass[mCl,mBu3NH]
muBu3NHBr=redmass[mBr,mBu3NH]

muEt4NCl=redmass[mCl,mEt4N]
muEt3NMeCl=redmass[mCl,mEt3NMe]
muEt3NHCl=redmass[mCl,mEt3NH]

Frequency Ratios Harmonic oscillator

Anion Change Cl-Br

frBu4N=freqratio[muBu4NCl,muBu4NBr]
frBu3NMe=freqratio[muBu3NMeCl,muBu3NMeBr]
frBu3NH=freqratio[muBu3NHCl,muBu3NHBr]

1/frBu3NMe

1/frBu3NH

1/frBu4N

Cation Change Cl

frClBu4NBu3NMe=freqratio[muBu4NCl,muBu3NMeCl]
frClBu4NBu3NH=freqratio[muBu4NCl,muBu3NHCl]
frClBu3NMeBu3NH=freqratio[muBu3NMeCl,muBu3NHCl]

frClBu4NEt4N=freqratio[muBu4NCl,muEt4NCl]
frClBu3NMeEt3NMe=freqratio[muBu3NMeCl,muEt3NMeCl]
frClBu3NHEt3NH=freqratio[muBu3NHCl,muEt3NHCl]

Actually took the inverses (divided by Bu4N frequency) bc I took the one that gave a ratio above 1, so divided by lower frequency.

frClBu3NHBu4N=1/frClBu4NBu3NH
frClBu3NMeBu4N=1/frClBu4NBu3NMe
frClBu3NHBu3NMe=1/frClBu3NMeBu3NH

frClEt4NBu4N= 1/frClBu4NEt4N
frClEt3NMeBu3NMe=1/frClBu3NMeEt3NMe
frClEt3NHBu3NH=1/frClBu3NHEt3NH

Cation Change Br

frBrBu4NBu3NMe=freqratio[muBu4NBr,muBu3NMeBr]
frBrBu4NBu3NH=freqratio[muBu4NBr,muBu3NHBr]
frBrBu3NMeBu3NH=freqratio[muBu3NMeBr,muBu3NHBr]

Actually took the inverses (divided by Bu4N frequency) bc I took the one that gave a ratio above 1, so divided by lower frequency.

frBrBu3NHBu4N=1/frBrBu4NBu3NH
frBrBu3NMeBu4N=1/frBrBu4NBu3NMe
frBrBu3NHBu3NMe=1/frBrBu3NMeBu3NH

Frequency Ratios Phonons Zone boundary

Anion Change Cl-Br

Only one because cation mass change doesn’t influence the ratio

frzb=freqratioPOPI[mCl,mBr]

1/frzb

Cation Change Cl

Shouldn' t see the cation change bc anion, which should be the only determining factor, stays the same

Cation Change Br

Shouldn' t see the cation change bc anion, which should be the only determining factor, stays the same

Frequency Ratios Phonons middle/0k

Anion Change Cl-Br

fr0kBu4N=freqratioPO0[mBu4N,mCl,mBu4N,mBr]
fr0kBu3NMe=freqratioPO0[mBu3NMe,mCl,mBu3NMe,mBr]
fr0kBu3NH=freqratioPO0[mBu3NH,mCl,mBu3NH,mBr]

1/fr0kBu3NH

1/fr0kBu4N

1/fr0kBu3NMe

Cation Change Cl

fr0kBu3NHBu4N=freqratioPO0[mBu3NH,mCl,mBu4N,mCl]
fr0kBu3NMeBu4N=freqratioPO0[mBu3NMe,mCl,mBu4N,mCl]
fr0kBu3NHBu3NMe=freqratioPO0[mBu3NH,mCl,mBu3NMe,mCl]

fr0kEt4NBu4N=freqratioPO0[mEt4N,mCl,mBu4N,mCl]
fr0kEt3NMeBu3NMe=freqratioPO0[mEt3NMe,mCl,mBu3NMe,mCl]
fr0kEt3NHBu3NMH=freqratioPO0[mEt3NH,mCl,mBu3NH,mCl]

1/fr0kBu3NHBu4N
1/fr0kBu3NMeBu4N
1/fr0kBu3NHBu3NMe

1/fr0kEt4NBu4N
1/fr0kEt3NMeBu3NMe
1/fr0kEt3NHBu3NMH

Cation Change Br

fr0kBu3NHBu4N=freqratioPO0[mBu3NH,mBr,mBu4N,mBr]
fr0kBu3NMeBu4N=freqratioPO0[mBu3NMe,mBr,mBu4N,mBr]
fr0kBu3NHBu3NMe=freqratioPO0[mBu3NH,mBr,mBu3NMe,mBr]

Force Constants

The force constant of a motion (if approximated as a harmonic oscillator) can be calculated as k=((mω)^2)

k1OVERk2[m1_,o1_,m2_,o2_]:=(m1* o1^2)/(m2* o2^2);

Compare this to the intensity changes expected from the mass change after ((Subscript[μ, 01])^2)=((planck)/2)(1/(√m*k))(((((δμ)/(δq))))^2)=A which would be ((Subscript[A, Br])/(Subscript[A, Cl]))=√(((Subscript[m, Cl])(Subscript[k, cl]))/((Subscript[m, Br])(Subscript[k, Br])))

Frequencies

fABu3NHBr=38.46;
fCBu3NHBr=121.18;
fABu3NHCl=61.21;
fCBu3NHCl=174.78;

fABu3NMeBr=25.49;
fCBu3NMeBr=77.62;
fABu3NMeCl=45.14;
fCBu3NMeCl=105.78;

fABu4NBr=32.53;
fCBu4NBr=84.47;
fABu4NCl=47.13;
fCBu4NCl=88.99;

Amplitudes

AABu3NHBr=0.97;
ACBu3NHBr=0.56;
AABu3NHCl=0.77;
ACBu3NHCl=0.62;

AABu3NMeBr=0.73;
ACBu3NMeBr=0.33;
AABu3NMeCl=0.66;
ACBu3NMeCl=0.53;

AABu4NBr=17.52;
ACBu4NBr=0.46;
AABu4NCl=7.26;
ACBu4NCl=2.87;

kCl/kBr - Reduced Mass

Bu3NH

kArmBu3NHCl=k1OVERk2[muBu3NHCl,fABu3NHCl,muBu3NHBr,fABu3NHBr]
kCrmBu3NHCl=k1OVERk2[muBu3NHCl,fCBu3NHCl,muBu3NHBr,fCBu3NHBr]

Bu3NMe

kArmBu3NMeCl=k1OVERk2[muBu3NMeCl,fABu3NMeCl,muBu3NMeBr,fABu3NMeBr]
kCrmBu3NMeCl=k1OVERk2[muBu3NMeCl,fCBu3NMeCl,muBu3NMeBr,fCBu3NMeBr]

Bu4N

kArmBu4NCl=k1OVERk2[muBu4NCl,fABu4NCl,muBu4NBr,fABu4NBr]
kCrmBu4NCl=k1OVERk2[muBu4NCl,fCBu4NCl,muBu4NBr,fCBu4NBr]

kCl/kBr - Anion Mass

Bu3NH

k1OVERk2[mCl,fABu3NHCl,mBr,fABu3NHBr]
k1OVERk2[mCl,fCBu3NHCl,mBr,fCBu3NHBr]

Bu3NMe

k1OVERk2[mCl,fABu3NMeCl,mBr,fABu3NMeBr]
k1OVERk2[mCl,fCBu3NMeCl,mBr,fCBu3NMeBr]

Bu4N

k1OVERk2[mCl,fABu4NCl,mBr,fABu4NBr]
k1OVERk2[mCl,fCBu4NCl,mBr,fCBu4NBr]

ABr/ACl

Bu3NH

measured

AABu3NHBr/AABu3NHCl
ACBu3NHBr/ACBu3NHCl

theoretical - reduced mass

Sqrt[(kArmBu3NHCl*(muBu3NHCl/muBu3NHBr))]
Sqrt[(kCrmBu3NHCl*(muBu3NHCl/muBu3NHBr))]
(kArmBu3NHCl*(muBu3NHCl/muBu3NHBr))
(kCrmBu3NHCl*(muBu3NHCl/muBu3NHBr))

theoretical - anion mass

Sqrt[(kArmBu3NHCl*(mCl/mBr))]
Sqrt[(kCrmBu3NHCl*(mCl/mBr))]
(kArmBu3NHCl*(mCl/mBr))
(kCrmBu3NHCl*(mCl/mBr))

Bu3NMe

measured

AABu3NMeBr/AABu3NMeCl
ACBu3NMeBr/ACBu3NMeCl

theoretical - reduced mass

Sqrt[(kArmBu3NMeCl*(muBu3NMeCl/muBu3NMeBr))]
Sqrt[(kCrmBu3NMeCl*(muBu3NMeCl/muBu3NMeBr))]
(kArmBu3NMeCl*(muBu3NMeCl/muBu3NMeBr))
(kCrmBu3NMeCl*(muBu3NMeCl/muBu3NMeBr))

theoretical - anion mass

Sqrt[(kArmBu3NMeCl*(mCl/mBr))]
Sqrt[(kCrmBu3NMeCl*(mCl/mBr))]
(kArmBu3NMeCl*(mCl/mBr))
(kCrmBu3NMeCl*(mCl/mBr))

Bu4N

measured

AABu4NBr/AABu4NCl
ACBu4NBr/ACBu4NCl

theoretical - reduced mass

Sqrt[(kArmBu4NCl*(muBu4NCl/muBu4NBr))]
Sqrt[(kCrmBu4NCl*(muBu4NCl/muBu4NBr))]
(kArmBu4NCl*(muBu4NCl/muBu4NBr))
(kCrmBu4NCl*(muBu4NCl/muBu4NBr))

theoretical - anion mass

Sqrt[(kArmBu4NCl*(mCl/mBr))]
Sqrt[(kCrmBu4NCl*(mCl/mBr))]
(kArmBu4NCl*(mCl/mBr))
(kCrmBu4NCl*(mCl/mBr))

Comparison with Ludwig Paper

Additional masses

mCH3SO3=94.98;
mCF3SO3=148.95;
mNTf2=279.92;
mC2mim=111.09;

Frequencies

fEt3NHCH3SO3=149.4;
fEt3NHCF3SO3=128.9;
fEt3NHNTf2=105.2;
fC2mimNTf2=83.2;

fBu3NHCl=174.78;
fBu3NHCBr=121.18;

fBu4NCl=88.99;
fBu4NBr=84.47;

Shifts zone center

Assuming a cubic crystal lattice with two different sites, the equation of motion close to the limit of k=0 is:
(ω^2)≈ 2C ((1/(Subscript[M, 1]))+(1/(Subscript[M, 2])))
The ratio for two different crystals, assuming similar force constants, is then:
((Subscript[ω, A])/(Subscript[ω, B]))≈(Sqrt[(((1/(Subscript[M, 1,A]))+(1/(Subscript[M, 2,A])))/((1/(Subscript[M, 1,B]))+(1/(Subscript[M, 2,B]))))])

freqratioPO0[m1aFRP_, m2aFRP_, m1bFRP_, m2bFRP_] := Sqrt[((1/m1aFRP) + (1/m2aFRP))/((1/m1bFRP) + (1/m2bFRP))];

Calculated

freqratioPO0[mCH3SO3, mEt3NH,mCl, mBu3NH]
freqratioPO0[mCF3SO3, mEt3NH,mCl, mBu3NH]
freqratioPO0[mNTf2, mEt3NH,mCl, mBu3NH]

freqratioPO0[mCH3SO3, mEt3NH,mBr, mBu3NH]
freqratioPO0[mCF3SO3, mEt3NH,mBr, mBu3NH]
freqratioPO0[mNTf2, mEt3NH,mBr, mBu3NH]

freqratioPO0[mNTf2, mC2mim,mCl, mBu4N]
freqratioPO0[mNTf2, mC2mim,mBr, mBu4N]

freqratioPO0[mNTf2, mEt3NH,mCH3SO3, mEt3NH]
freqratioPO0[mNTf2, mC2mim,mCH3SO3, mEt3NH]
freqratioPO0[mNTf2, mEt3NH,mNTf2, mC2mim]

Experimental

Divide by lighter anion frequency

fEt3NHCH3SO3/fBu3NHCl
fEt3NHCF3SO3/fBu3NHCl
fEt3NHNTf2/fBu3NHCl

fEt3NHCH3SO3/fBu3NHCBr
fEt3NHCF3SO3/fBu3NHCBr
fEt3NHNTf2/fBu3NHCBr

fC2mimNTf2/fBu4NCl
fC2mimNTf2/fBu4NBr

fEt3NHNTf2/fEt3NHCH3SO3
fC2mimNTf2/fEt3NHCH3SO3
fEt3NHNTf2/fC2mimNTf2

Shifts zone edge

If we go to the brillouin zone boundary for the same lattice, we find the motion to be:
ω≈(Sqrt[((2C)/(Subscript[M, 2]))])
The ratio is then ((Subscript[ω, A])/(Subscript[ω, B]))≈(Sqrt[((Subscript[M, 2,B])/(Subscript[M, 2,A]))])if the force constants are similar. 
Not quite sure how to assign (Subscript[M, 1]) and (Subscript[M, 2,]) though (Subscript[M, 1]) seems to be heavier usually, because then the phonon branches don’t cross.
So this would be simply the root of the ratio of the anion mass in our case?

freqratioPOPI[m2aFRPP_, m2bFRPP_] := Sqrt[m2bFRPP/m2aFRPP];

Calculated

freqratioPOPI[mCH3SO3, mCl]
freqratioPOPI[mCF3SO3, mCl]
freqratioPOPI[mNTf2, mCl]

freqratioPOPI[mCH3SO3, mBr]
freqratioPOPI[mCF3SO3, mBr]
freqratioPOPI[mNTf2, mBr]

freqratioPOPI[mNTf2, mCH3SO3]

Experimental

Divide by lighter anion frequency

fEt3NHCH3SO3/fBu3NHCl
fEt3NHCF3SO3/fBu3NHCl
fEt3NHNTf2/fBu3NHCl

fEt3NHCH3SO3/fBu3NHCBr
fEt3NHCF3SO3/fBu3NHCBr
fEt3NHNTf2/fBu3NHCBr

fC2mimNTf2/fBu4NCl
fC2mimNTf2/fBu4NBr

fEt3NHNTf2/fEt3NHCH3SO3
fC2mimNTf2/fEt3NHCH3SO3
fEt3NHNTf2/fC2mimNTf2