| Academic Open Internet Journal |
Volume 12, 2004 |
Nonlinear Impurity for a square and BCC Lattices
A. J. Sakaji, Physics Department, Ajman University, UAE,
e-mail: info@ajss.net
J. M. Khalifeh, Physics Department, University of Jordan,
Jordan, e-mail: jkalifa@ju.edu.jo
R. S. Hijjawi, Physics Department, Mutah University, Jordan,
e-mail: hijjawi@mutah.edu.jo.
Abstract:
Analytical and numerical analysis of nonlinear impurity for a square and BCC lattices, a closed forms of the bound state energy equation, bound state amplitude, and transmission coefficient are obtained , a numerical result for a critical nonlinearity threshold is calculated.
Keywords: Nonlinear Impurity, Lattice Green’s Function, Bound States, and Transmission.
1- Introduction
Nonlinear impurity appears in the problems considering strong electron-phonon interactions, where the local site energy at impurity site depends on the electronic probability at that site[1,2,3]. In systems where an electron ( or excitation) is propagating while strongly interacting with vibrational degrees of freedom[4,5,6].
The discrete nonlinear Schrodinger equation (DNLS) has the form[7]:
Where Cn is the probability amplitude of finding the electron on lattice at site n at time t, V is the nearest-neighbor hopping integrals, and χn is the nonlinearity parameter at site n proportional to the square of the electron-vibration coupling, and α is the nonlinear exponent.
We can write the solution of Eq.(1.1) by using Dysion equation in term of lattice Green’s function [6,7,8]as:
The bound states amplitudes can be obtained from the residues of the Gmn(z) at z=zb as:
And 1/γ is the bound state energy equation.
Using the Green’s function, we can write the equation for the transmission coefficient t of plane wave across the nonlinear impurity as [7]:
Solving this equation for t we get:
2- Square lattice
The Green’s function for a square lattice is[8]:
where
So that the bound state energy equation is
where z=E/4V, m=1/z2, and m=k2
of the elliptic integral modulus 
the bound state probability can be written as:
After some mathematical manipulation the transmission coefficient t , can be written as:
Figure (1-16) shows the shape of right hand side of bound state energy equation for a square lattice with several different nonlinear exponents α, for α=0 there is no critical nonlinearity threshold γc , but for α>0 there exist critical nonlinearity threshold γc , at γ= γc there is exactly one root (one bound state) which are tabulated in table 1 , and at γ> γc , there are two roots (two bound states), Figure 17 shows the transmission coefficient of plane wave across plane wave dimensionless energy (0-1) and nonlinear impurity strength(0-20) For square lattice.
Fig.1 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=0
Fig. 2 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=0.5
Fig. 3 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=1
Fig 4 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=1.5
Fig. 5 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=2
Fig. 6 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=2.5
Fig. 7 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=3
Fig.8 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=3.5
Fig. 9 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=4
Fig. 10 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=4.5
Fig. 11 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=5
Fig . 12 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=5.5
Fig. 13 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=6
Fig. 14 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=9
Fig. 15 Three dimensional graph of the bound state energy equation as a function of nonlinear energy strength(0-9), and energy m(0-1) outside the band
Fig. 16 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent (α=0,0.5,1,2,3,4,5,6,7,8,9)
Fig. 17 Transmission coefficient of plane wave across plane wave dimensionless energy (0-1) and nonlinear impurity strength (0-20) For square lattice
|
Nonlinear exponent α |
Critical nonlinearity threshold γc |
Bound states |
|
1 |
1.031475 |
0.868039 |
|
1.5 |
1.214759 |
0.774103 |
|
2 |
1.365495 |
0.698324 |
|
2.5 |
1.496997 |
0.636681 |
|
3 |
1.615404 |
0.58561 |
|
3.5 |
1.724135 |
0.542557 |
|
4 |
1.825334 |
0.505721 |
|
4.5 |
1.920444 |
0.473804 |
|
5 |
2.010491 |
0.445853 |
|
5.5 |
2.096238 |
0.421148 |
|
6 |
2.178274 |
0.399139 |
|
6.5 |
2.257061 |
0.379395 |
|
7 |
2.332976 |
0.361574 |
|
7.5 |
2.406316 |
0.345402 |
|
8 |
2.477332 |
0.330654 |
|
8.5 |
2.546253 |
0.317147 |
|
9 |
2.613252 |
0.304725 |
Table 1 showing, the critical nonlinearity and its bound state energy for several nonlinear exponents for square lattice
3- Body Centered Cubic (BCC) Lattice
The Green’s function for a BCC lattice is[9]:
where
and
and K[k], E[k] are the complete elliptic integrals of first and second kind.
So that the bound state energy equation is
The bound state probability can be written as:
After some mathematical manipulation the transmission coefficient t , can be written as:
Figures (18-30) show the shape of right hand side of bound state energy equation for BCC lattice with several different nonlinear exponents α, for α=0 there is no critical nonlinearity threshold γc , but for α>0 there exist critical nonlinearity threshold γc , at γ= γc there is exactly one root (one bound state) which are tabulated in table 2 , and at γ> γc , there are two roots (two bound states), Figure 31 shows the transmission coefficient of plane wave across plane wave dimensionless energy (0-1) and nonlinear impurity strength(0-20) For BCC lattice.
Fig. 18 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=0
Fig. 19 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=0.5
Fig. 20 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=1
Fig. 21 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=1.5
Fig. 22 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=2
Fig. 23 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=2.5
Fig. 24 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=3
Fig. 25 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=3.5
Fig. 26 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=4
Fig. 27 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=4.5
Fig. 28 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent α=5
Fig. 29 Real Part of the right-hand side of the bound state energy equation for nonlinear exponent (α=0,0.5,1,2,3)
Fig. 30 Three dimensional graph of the bound state energy equation as a function of nonlinear energy exponent (0-5), and energy (0-1) outside the band
Fig. 31 Transmission coefficient of plane wave across plane wave dimensionless energy (0-1) and nonlinear impurity strength(0-20) For BCC lattice.
|
Nonlinear exponent α |
Critical nonlinearity threshold γc |
Bound states |
|
0.5 |
3.89813 |
0.979986 |
|
1.0 |
7.38842 |
0.963939 |
|
1.5 |
13.653 |
0.952762 |
|
2.0 |
24.9587 |
0.9446 |
|
2.5 |
45.3652 |
0.938391 |
|
3 |
82.1716 |
0.933515 |
|
3.5 |
148.505 |
0.929578 |
|
4.0 |
267.968 |
0.926338 |
|
4.5 |
482.99 |
0.923629 |
|
5 |
869.821 |
0.921322 |
Table 2 showing , the critical nonlinearity and its bound state energy for several nonlinear exponents for BCC lattice
References
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[8] A. Sakaji ,” Green’s Function for a Point Defect”, M. Sc. Thesis , University of Jordan (1994)
[9] A. Sakaji, R. Hijjawi, N. Shawagfeh, J. M. Khalifeh, Inter. J. of Theo. Phys., 41 (5): 973, May (2002).
Technical College - Bourgas,