Investigating Vibrations in NaI Lattice for Potential ILM Formation

Benjamin Agyare, Stockton University & Peter Riseborough (PhD), Temple University

ILMs are persistent vibrations of atoms in a homogeneous lattice which extend only over a finite region of space. It s been reported that these vibrations can be found in NaI, but only exist for wave- vectors at the corner of the 3-D Brillouin Zone. It has been suggested that ILMs occur whenever the van-Hove singularities of the two phonon density of states coalesce at the upper-edge of the continuum.

Objectives: 1. To investigate the one-phonon & two- phonon density of states of NaI and examine its van-Hove singularities in order to identify at which q vectors they may coalesce. 2. To introduce anharmonic interactions and determine if they are enhanced by the van-Hove leading to the formation of ILMS.

Modeled by coupling the atoms located at the lattice sites equivalent to (0,0,0) for I atoms (mass MA = 127) to the Na atoms (of mass MB =23) located at the nearest-neighbor sites ? isotropic harmonic forces. Lattice parameter a a is equal to 6.47 Angstroms. I 21,1,1 by Na + + = =

The primitive lattice vectors are given by ?1=? The volume of the primitive unit cell is = ?1 ?2 ?3 = ?3 The reciprocal lattice vectors are ?1= ? ?2= ? 2? ? Volume of the Brillouin zone of primitive cell is 4 ? 20,1,1 , ?2=? 21,0,1 , ?3=? 21,1,0 4 z 2? 1,1,1 , 2? 1, 1,1 , and ?? ?3= 1,1, 1 ?? y a3 3 3 2? x

Equation of motion, for ? = ????: ? ? Assume ?? ? in equation of motion, we have ? ? ?,? ? ?, ? and substitute = ? ,? ???,???? ?? ?? ?? ???? ? ???+?? ?? ? 1 = ? ??? and ?2??? = ???? The secular determinant ?,? ? ?2?????,? ?,?? ?? ?? ? ?,?2= ??? ?,? ? ? 1 ? ???? where ??? = ? ?,?

? ?,?2 is equal to ?? ?? ?? 0 ?? ???2 ?? ?2,? ?2,? ?1,? ?2,? ?2,? ?1,? 0 0 ?2,? 0 0 ?? ?1,? 0 ?2,? ?? ?? ?? ???2 ?? ?? ?1,? ?2,? ?? ?? ?? ???2 ?? ?? ?2,? 0 ?1,? 0 ?? ???2 ?? ?? ?? 0 0 ?? ?? ?? ???2 ?? ?2,? ?2,? ?2,? ?2,? ?? 0 0 ?? ?? ?? ?? ???2 ?? ?1,? The force constant which couple the lattice displacement of atoms with the same sub lattice indices can be expressed as

??? 2+ cos??? cos??? ??= 2?1+ 2?22 cos??? ?? cos 2 2 ??? 2 2+ cos??? cos??? 2+ cos ??= 2?1+ 2?22 cos ?? 2 ??? 2 ??= 2?1+ 2?22 cos??? ?? 2 Then matrix elements involving the nearest neighbor force constants ?1 which couple A and B displacement are given by ??= 2?1cos??? Then matrix elements involving the next-nearest neighbor force constants ?2 which couple the different components of displacement of atoms on the same sub-lattice are given by ??= 2?1sin??? ?2,? 2 ??? 2, ?1,? ??= 2?1cos??? 2, ?1,? ??= 2?1cos ?1,? 2 ??? 2, ??? 2sin??? ??= 2?1sin??? 2sin??? 2, ?2,? 2sin ?2,? ??= 2?1sin

The one phonon density of state is ?=?3 4 2+? The two phonon density of state is written as ?3? 2?3? ? ?? ? ?? ?3? 2?3? ? ?? ?,?=?3 ? ? ?? ?? 4 2+? 2 ? The integration over delta function is turned into a surface of constant energy ?=?3 4 2?3 ???? 2+? ? ?,?=?3 4 2?3 ???? 2+? 2 ? ? ?2?? 1 ? ?? ?2?? 1 ?? ? ? + ?? Van-Hove singularity occurs ?? ?? =0 or ?? ?? =0 ? ? ? + ?? 2+? 2+? 2 ?

One Phonon DOS. Optic mode ratio K2/K1 = 0.598, -> 0.003. Acoustic One Phonon DOS mode ratio K2/K1 = 0.202, -> 0.001. Approx. area under each curve = 6 sq. units 50 45 40 ( ) arb units 35 30 25 20 15 10 5 0 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 Optic Mode Ratio Acoustic Mode Ratio

Two Phonon DOS. Approx. Area = 36 Sq. Units 350 300 250 ( ) arb units 200 150 100 50 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 Optic Mode Ratio Acoustic Mode Ratio

DOS graph of () vs for optic mode ratio K2/K1 = 0.598, constructed from surface of constant energy integral. :(0,0,0), L: ( L: ( / /a, a, / /a, a, / /a) a) , X:(0, 2 /a, 0) 80 75 70 ( ) arb units 65 60 55 50 45 40 35 30 25 20 15 10 5 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

DOS graph of () vs for optics ratio K2/K1 = 0.598, constructed from delta function with =0.003. q = (0,0,0), q = L( /a, /a, /a), q = U( /2a, 2 /a, /2a) 70 60 ( ) arb units 50 40 30 20 10 0 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200

Summary: 1. The one-phonon and two-phonon density of states exhibits features at the van Hove singularities at fixed frequencies, but the spectral features are not divergent due to the 3-D nature of the crystal. 2. For different q values, several van Hove singularities converge at one frequency and produce a large peak in the two-phonon DOS. 3. Next: search for q values at which the two- phonon density of states is enhanced and then examine whether the introduction of anharmonic interactions can bind the two- phonon excitations to produce a quantized ILM.

[1]. A. J. Sievers, S. Takeno, Phys. Rev. Lett. 61, 970 (1988). [2] W.Z. Wang, J.T. Gammel, A.R. Bishop and M.I. Salkola, Phys. Rev. Lett. 76, 3598-3601 (1996) [3] L. Proville, Phys. Rev. B, 71, 104306 (2005). [4] ME Manley AJ Sievers & JW Lynn, Phy. Rev B 79 134304 (2009) [5] D. Kanbur & P.S. Riseborough, Phil. Mag. Letts 94, 424 432 (2014)

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