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  • Published on: 15th October 2019
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Lattice vibration

What is the lattice vibration ?!! it is mean that the atoms in the crystal can oscillate and it is not motionless or rigid body , and the atoms vibrates in groups in  a synchronized and harmonic way .

Imagine that the atoms at their equilibrium case , and for some reason there is a force acted upon them , so what will happen in this case of course  the atoms will resist to return back to their equilibrium  sites or position ,because of that there is a lattice vibrations , and we should not ignore the motion of the entire lattice and take it into consideration .

elements and components  are made of atoms , all solids with the exception of amorphous solids and glasses have periodic arrays of atoms , atoms and ions are bounded each other with considerable and huge interatomic forces to form a crystal lattice  , with different types of bounds  , (atoms oscillate in a solid about their equilibrium positions , in a crystal lattice these positions form a regular lattice , because atoms are bounded to  the neighboring atoms ,vibration of neighbors are dependent of each other ) , we was assumed that the crystal lattice is completely a rigid body which is not accurate , in a regular lattice with harmonic  forces between atoms the normal modes of vibration are lattice waves , they are a progressive waves which mean that they are elastic waves at low frequencies  in the corresponding anisotropic  continuum ,  The spectrum of lattice waves ranges from these low frequencies to frequencies of the order of 1013 Hz, and sometimes even higher. The wavelengths at these highest frequencies are of the order of interatomic spacing ,  the vibration of lattice effects directly on the properties of the materials and it is important for materials .

The phenomena of lattice vibrations depend on the properties of the medium  , which is reflect the interaction of micro-particles with each other .

Most of the thermal energy resides in the waves of highest frequency  at room temperature and above , because of the short wavelength, the motion of neighboring atoms is essentially uncorrelated ,we can consider the vibration  for many purpose  like the independent vibration of the atoms , each moving about its average position in three dimensions with average vibrational energy of 3kT, where k is the Boltzmann constant and T the absolute temperature , however we should study the properties of these waves to know more about heat transport ,  electrical resistance of the materials and so on .

Lattice Vibrations is one of the main types of internal motion of a solid , the particles ( atoms or molecules ) are oscillate about equilibrium point or positions , when there is an external force or influences that varies all the time acted upon the crystal lattice , lattice vibrate in the form of standing or traveling sound waves arise , even when there is no external force acted upon the lattice steady state of vibrations is established in a crystal that is in thermal equilibrium with its environment , it is like a fixed distribution of  the atoms and molecules that is established in a gas according to their rate of translator  motion .

What can effect on the vibration of lattice ?

the symmetry of the crystal, number of atoms in its unit cell , the type of chemical bond , and the type and concentration of the crystal defects, all these condition or cases have a big role in the nature of lattice vibrations , also the temperature , when the temperature become higher the atoms will move actually it is  oscillating it will have an kinetic energy , but even up to the melting point at which the solid become a liquid it is still much smaller than the lattice constant , the force that keep the atoms in equilibrium positions are proportional to their relative displacements , it looks like that they are interconnected to each other by springs. Crystal represented as a set of particles which that connected by ideally elastic forces is called a harmonic approximation .

if there is  a crystal consisting of N unit cells of n atoms each there exist 3nN ��� 6 types of the simplest vibrations in the form of standing waves, which are called normal (or natural) oscillations, or modes. Their number is equal to the number of degrees of freedom of the set of particles in the crystal, less the three degrees of freedom that correspond to the translator motion and the three that correspond to the rotary motion of the crystal as a whole. The number 6 may be disregarded, since 3nN is a quantity of the order of 1022-1023 per cubic centimeter of the crystal.

 A simple harmonic oscillators mean that all particles of the crystal oscillate about their own equilibrium positions with the same constant frequency �� , all possible normal vibrations may be present simultaneously in a crystal, and each occurs as if the others were not present. Any motion of atoms in a crystal that does not disrupt its microstructure may be represented as a superposition of the normal vibrations of the crystal .

 we have to study the lattice vibration to know and understand many phenomena in the natural  we can explain sound velocity, thermal properties  such as Heat capacity, Specific heat, Thermal expansion and  Thermal conductivity (for insulators and semiconductors), elastic properties such as Hardness and Temperature dependence of elastic constants  and optical properties such as Infrared absorption of materials .

heat transport ,  Heat can be transported through a crystal either by the conduction electrons (metals) or by lattice vibrations (all solids) , it is transport as a waves of lattice vibrations in a rigorously harmonic crystal, since the phonons are perfect solutions . we can note that that a metal with a perfect rigid lattice (without defects or vibrations) would exhibit an in���nite electric conductance.

The phonon :

 Actually the energy of lattice vibrations between the lattice spacing  is  quantized and  The quants of lattice vibrations are called phonons .

So it is a quantum vibration condition occurring in solid crystalline  atoms lattice , It is important to study the phonon because it is an important topic  in the solid physics and it is play a big an essential role in the physics of the  solid materials and it contributing of  determine some characteristics or properties of the steel body such as thermal conductivity and electrical conductivity , all parts of the crystal lattice vibrate with the same frequency , and long ��� wave phonon can consider the main causative  of the phenomenon of the emergence  sound in the solid body , phonons are also forming the main heat transfer mechanism within objects that are classified as an insulator , the phonon in Phonetics is equal to the photon in optics , Photons , of the correct frequency are visible light, Phonons , of the correct frequency are sound   ( the long wavelength property of phonon also gives attributes to sounds in solids ), Phonons are also a quantum mechanical version of a special type of vibrational motion. This type of vibrational motion is called a normal mode (elementary blocks of lattice vibration),which is a pattern of motion in which all parts of the system move in a sinusoidal fashion, with the same frequency , a standing wave is an example of normal mode  , which is a continuous form of normal mode and all space elements, (x, y and Z coordinates) are oscillating at the same frequencies, and phase but different amplitude.

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