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Posts Plotly: 1D Diatomic Chain Dispersion
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Plotly: 1D Diatomic Chain Dispersion

1D Diatomic Chain Dispersion

Consider a 1D chain with two atoms in the unit cell

1D diatomic chain. two sublattices. Generated by TikZ/LaTeX.

The coordinates of each atom in the cell

Rjs(t)=xj+ds+ujs(t);s=1,2

where xj is the vector of the j-th cell, ds is the relative vector of the s-th atom in the cell, ujs(t) is the displacement of the s-th atom in the j-th cell from its equlibrium position.

For the atoms in the j-th cell, Newton’s law yields these euqations of motion

md2uj1dt2=K(uj2+uj122uj1)Md2uj2dt2=K(uj1+uj+112uj2)

where m and M are the masses of atom 1 and 2, respectively.

To find the solutions to the equations, we assume that all of the atoms move with the same frequency,

uj1(t)=Aqmei(qxjωt)uj2(t)=BqMei(qxjωt)

Substitute this into the equations of motion, we have

(2Kmω2KmM(1+eiqa).KmM(1+eiqa)2KMω2)(AqaBq)=0

The equations will have a solution when the determinant of the matrix equals zero, i.e.

(2Kmω2)(2KMω2)2K2mM(1+cos(qa))=0

This equation is quadratic and the solution can be easily found out

ω2±=KMm[(m+M)±m2+M2+2mMcos(qa)]

where the subscript +/ denote the optical/acoustic mode, respectively.

The maximum frequency is found ath q=0 of the optical mode

ωmax=2K(m+M)mM
ω+m/M = 0.500.000.100.200.300.400.500.600.700.800.901.00

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