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VIBRATIONAL DYNAMICS OF ATOMS IN A ONE-DIMENSIONAL
CRYSTAL LATTICE
Muminov Islomjon Arabboyevich
Fergana State University, Doctor of philosophy in physics and mathematics (PhD) [email protected] Axmedov Aloviddin Nomonjon o'g'li Academic Lyceum of Fergana Polytechnic Institute, teacher of physics.
A crystal is composed of a periodic arrangement of atoms and molecules that vibrate around their equilibrium positions due to thermal motion at finite temperatures. When atoms are displaced from these positions, they experience a restoring force that follows Hooke's law and is proportional to the displacement. This study analyzes the vibrational dynamics of a one-dimensional lattice, where the atoms are modeled as masses connected by springs with a force constant. The motion of atoms in this system is governed by the equations of motion, and the resulting lattice vibrations are examined in the context of phase velocity, sound velocity, and Brillouin zones. The analysis is further extended to a lattice consisting of two different kinds of atoms, leading to the formation of distinct acoustic and optical modes. The dispersion relations and the role of wave vectors defined by cyclic boundary conditions are explored, with emphasis on the periodic nature of the w-q curve and the significance of the first Brillouin zone.
Keywords: Crystal lattice, lattice vibrations, Hooke's law, one-dimensional lattice, phase velocity, sound velocity, Brillouin zone, dispersion relation, wave vector, cyclic boundary conditions, acoustic mode, optical mode.
INTRODUCTION
A crystal consists of a periodic arrangement of atoms and molecules. However, the atoms or molecules of a crystal vibrate around the equilibrium positions at finite temperatures because of their thermal motion. When a atom is displaced from its equilibrium position, the atom is subject to a restorating force depending on the displacement. The restorating force follows Hooke's law and is proportional to the displacement. When each atom vibrates randomly, each atom suffers random forces and its vibration is immediately damped. On the other hand, if each atom vibrates with a small relative displacement from the neighboring atoms, then the vibration will
ABSTRACT
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continue with its small vibration energy. For simplicity we consider the one-dimensional lattice shown in Fig. 1, where the mass of the atoms is M and the distance between the nearest neighbor atoms (lattice constant) at equilibrium is a. We assume that the interatomic forces act on neighboring atoms only and that the atoms are connected to each other by a spring constant (force constant) k0. Defining the displacement of the atoms from their equilibrium positions by ; 1 ■"; i ! I J i ! I 2 , ,un, un+1, the equation of motion of nth atom
(1)
As stated above, when the displacement of each atom is independent, an atom will be subject to a strong force from the neighboring atoms and the
n-2 n-1 n n +1 n + 2
i i i i i i i i i i i i i i i i i i
it i i i i i i i i i i i i i i i i
u
n-1
u
n+l
u.
+2
Fig. 1. Displacement of atoms from their equilibrium positions (lattice vibration) displacement is damped. Therefore the lowest excitation energy of the lattice vibration corresponds to a wavelike displacement, keeping the displacement of neighboring atoms almost in phase, when we see neighboring atoms. In such a case we may write the solution of (1) as
u^ = /lexp[i(qii.a — wt)] where q = (o/vs = 2n/A is the wave vector, is the angular frequency, vs is the velocity of sound, and A is wavelength. Inserting (2) into (1), we obtain
Moo2 = -k0(eiqa + e~iqa - 2) = 4Ji0sin2 (y) From this we have following relation:
When the wavelength is much longer than the lattice constant (qa «1), the phase velocity vs is given by
(2)
(3)
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METHODS
This result indicates that a lattice vibration with long wavelength has a constant phase velocity vs = J kQ/Ma, which corresponds to the sound velocity. Figure 2 shows the calculated curve from (4), where we see that the co — q curve is a periodic function with a period of 2n/a. The whole dispersion curve is the repetition or displacement of the curve in the period — n/a < q < n/a, and thus this feature enables us to discuss lattice vibrations in that period. The region of the period is called the first Brillouin zone, and the second Brillouin zone is shown in Fig. 2. The second, third and other Brillouin zones are equivalent to the first Brillouin zone, which is shown in Sect. 1.4 in detail in connection with the reduced zone scheme. Where there are N atoms, N degrees of freedom of motion exist. Adopting cyclic boundary conditions we can define wave vectors by q = 2nn/(Na) , where n = (—N/2 + 1) - (+N/2), giving rise to N values of the wave vectors. The above defined wave vectors are obtained from the cyclic boundary condition such that the displacement un of (2) is equivalent for n = 0 and n = N.
Next, we consider a lattice consisting of two kinds of atoms with masses M1 and M2. The atomic distance of the atoms of mass M1 is a and the same is true for the atoms M2. When the nearest-neighbor interaction is assumed, the equations of motion are written as in the case of the one-dimensional lattice stated above as
Fig. 2. Dispersion curve of one-dimensional lattice vibration for the atoms Mt and M2. We assume once again wave-type solutions of the same type as before. In addition we may expect another type of solution where atoms of even order and odd order are displaced in the reverse direction to each other, and thus atoms of even
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order form a wave and odd order atoms form another wave. To satisfy these vibration types we express the displacement as
Inserting these equations into (6), we obtain following relations:
(MiCO2 - 2k0)A± + {2k0cos(qa/2)}A2 = 0| (M2co2 - 2k0)A2 + (2k0cos(qa/2)}A1 = oi We easily find that the condition A1 = A2 = 0 correspond to all atoms at a standstill, and therefore we have to find solutions such that both A1 and A2 are not zero simultaneously. This condition is satisfied by requiring the determinant of the simultaneous equations with respect to A1 and A2 in (8) to be zero, giving rise to
M±a>2 — 2k0) 2k0cos(qa/2) k0cos(qa/2) (M2co2 — 2k0) This equation is regarded as a quadratic equation with respect to co2, and the solutions co2 and co2 are given by
(8)
(9)
It is evident from (10b) and (10b) that a>_and approach zero and a constant not equal to zero as q -» 0. Taking account of the fact that the
Fig. 3. Angular frequency versus wave vector relations for the two vibration modes of a lattice consisting of two kinds of atoms angular frequency is positive, and are plotted as a function of the wave vector q for several values of a = M1/M2 as a parameter in Fig. 3, where we define 1 /Mr = 1/M± + 1/M2. Let us consider the two branches oj_ and co+ inthe long wavelength limit. For q = 0, (8 i, [ 10 ) and (10b) lead us to the following results:
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l2 LVll
RESULTS
When we put Mx = M2, the branch gives the same result as a lattice consisting of one kind of atom, corresponding to a sound wave, and thus called the acoustic branch, the acoustic mode of vibrations or the acoustic phonon. Since the acoustic branch satisfies the condition (A1 = A2), the neighboring atoms move in the same direction and thus the relative displacement is zero in the long wavelength limit. On the other hand, the co+ branch satisfies A1/A2 = -M2/M1 and thus different atoms are displaced in the opposite directions to each other, resulting in zero of the center of mass motion. The w+branch exhibits a relative displacement between the different atoms and induces an electric field when the atoms are ionic. The induced electric field interacts strongly with the external electromagnetic field and absorbs the external waves. This often occurs in the infrared region, leading to it to be called the optical branch, optical mode of lattice vibrations or the optical phonon. Figure 6.4 shows a schematic illustration of the atomic displacement for (a) the acoustic mode (acoustic phonon) and (b) the optical mode (optical phonon), where we see the difference in the displacement between the two modes of lattice vibrations.
(b) Optical mode
Fig. 4. Schematic illustration of two types of lattice vibrations: (a) acoustic mode (acoustic phonon) and (b) optical mode (optical phonon)
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The acoustic mode in the limit of qa « 1 gives the following relation:
and thus the sound velocity (phase velocity) is given by
which corresponds to (5). DISCUSSION
In ionic crystals unusual reflectivity has been observed in the infrared wavelength region of 20 — 100/im. For example, the reflectivity of NaCl exhibits maxima around 40/mi and 60/vm, where the angular frequencies of light for the corresponding wavelengths are w = 2nc/A ~ 4 x 1013 s_1 and their wave vectors are q = 2k/A ~ 1Ü3 cm-1, which are much smaller than the wave vector at the Brillouin zone edge nf a ^ 10s cm-1. From the energy and momentum conservation rules, such high energy excitation in the small wave vector region is easily found to correspond to the optical phonon branch co+.
In general, as stated previously, the optical mode of lattice vibration appears in a crystal with two or more atoms in a unit cell. Since there exist one longitudinal and two transverse acoustic modes, a crystal with s atoms in a unit cell gives rise to 3is - : j optical modes of lattice vibration.
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