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Showing posts with label Bloch Theorem. Show all posts
Showing posts with label Bloch Theorem. Show all posts

PHYSICS DICTIONARY

Big Bang Theory

The Big Bang theory is an effort to explain what happened at the very beginning of our universe. According to the theory, our universe sprang into existence as "singularity" around 13.7 billion years ago. Our universe is thought to have begun as an infinitesimally small, infinitely hot, infinitely dense, something called as singularity. The Big Bang theory is the scientific theory that is most consistent with observations of the past and present states of the universe, and it is widely accepted within the scientific community. Georges Lemaître first proposed the Big Bang theory which he called as "hypothesis of the primeval atom". The framework for the Big Bang model relies on Albert Einstein's general relativity and on simplifying assumptions such as homogeneity and isotropy of space.

Bimetallic Strip

Two different metal strips of equal lengths placed on each other is called bimetallic strip. On heating, bimetal bends with material of greater linear expansion to convex side. On cooling, it bends with material of greater linear expansion on concave side.

Binary System

The pair of stars which orbit around each other is referred to as binary system. The center of mass of the binary system lies in between the two stars. The two stars rotate about this point.

Binding Energy per Nucleon

See average binding energy.

Binding Energy

The energy equivalent of mass defect is called binding energy. It is this energy which binds nucleons together. Hence it is the energy required for breaking a nucleus into free neutrons and protons.

Binoculars

Binoculars are a parallel combination of two telescopes for viewing an erect 3 dimensional image with both eyes. The image of same size can be viewed with both eyes which is uncomfortable for users to see with single eye. The distance between telescopes is adjustable.

Bioluminescence

It is the phenomenon of emission of light by living creatures because of chemical reaction.

Biophysics

Biophysics is an interdisciplinary science that deals with the application of physics to biological processes and phenomena.

Biot & Savart law

This law gives relation between magnetic field due to current carrying conductor and current flowing in the element. dB at a distance ‘r’ from a current element ‘dL’ carrying a current I is found to be proportional to I , to the length dL and inversely to the square of the distance r. The direction of magnetic field is perpendicular to line element dL as well as radius r.

Biprism

A triangular prism with vertex angle of nearly 180o used to obtain images of a single source in observing the interference light.

Birefringence

Some crystals have property of splitting incident light ray into two refracted rays. It is due to optical anisotropy in the binding forces between the atoms forming a crystal. They have two indices of refraction. This property is called birefringence.

Black Body

A body which completely absorbs radiation of all wavelengths incident on its surface and doesn’t reflect any part of it is called as black body.

(or)

It is a body which emits thermal radiations of all wavelengths when heated to high temperature.

Black Hole

A black hole is a region of space-time of extreme density with such strong gravitational attraction due to which nothing can escape, even light. When a star burns through the last of its fuel, it may find itself collapsing. For smaller stars, up to about three times the sun's mass, the new core will be a neutron star or a white dwarf. But when a larger star collapses, it continues to fall in on itself to create a stellar black hole.

Bloch Theorem

It is a mathematical theorem which gives us the form of electron wave function in a periodic potential. As per this theorem, electron in a one dimensional lattice behaves as plane wave.

Blue Moon

When two full moons occur in a single month, the second full moon is called a "Blue Moon".

BLOCH THEOREM

Bloch assumed that electrons move in a perfect periodic potential. He considered one dimensional array of lattice. The potential of electron at positive ion site is zero and is maximum in between. So long any line passing through the centers of positive ions, the potential variation must be as shown in below figure.

So Bloch gave a condition which is

𝚿(x+Na)=𝚿(x) .............................................................................................................(1)

It is considered as boundary condition.

Consider Schrodinger wave equation for one dimensional lattice.

(d²𝚿(x)/dx²) + (2m/ħ²)*[E-V(x)]*𝚿(x) = 0  .................................................................(2)

The Schrodinger equation for an electron in the potential at x+a is

[d²𝚿(x+a)/d(x+a)²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0  ..............................................(3)

Because of periodicity,

[d/d(x+a)] = d/dx  ; V(x+a) = V(x)

With  this, eqn (3) reduces to

[d²𝚿(x+a)/dx²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0  ...................................................(4)

This is Schrodinger  equation at x+a.

as 𝚿 at x+a is also obeying Schrodinger wave equation as 𝚿 at x there should exist a relation between 𝚿(x+a) & 𝚿(x).

Let    𝚿(x+a) = A𝚿(x)..................................................................................................(5)

𝚿(x+2a) = A²𝚿(x) [i.e. A𝚿(x+a) = A.A𝚿(x) = A²𝚿(x)]

𝚿(x+na) = Aⁿ𝚿(x)

from eqn(1),  Aⁿ =1 [i.e by using bloch condition]

Aⁿ =exp(2πij) [i.e. exp(2πij) =1 for j=01,2............]

or

A=exp(2πij/n)

Therefore,  𝚿(x+a) = exp(2πij/n)*𝚿(x) --------------------------------------------------(6) [ from eqn 5]

𝚿(x) can be written in terms of other function Uk(x )

𝚿(x) = exp(ikx)*Uk(x) where k=(2πj/n) ..................................................................(7)

From eqns (6) & (7),

exp[ik(x+a)]*Uk(x+a) = exp(2πij/n)*exp(ikx)*Uk(x)

exp[ika]*Uk(x+a) = exp(2πij/n)*Uk(x)

noting that  Ka = 2πj/n,

we can write that  Uk(x+a) = Uk(x) ..........................................................................(8)

Conclusion

Bloch Theorem is a mathematical theorem and it gives us the form of electron wave function in a periodic potential.

𝚿(x) = exp(ikx)*Uk(x) represents Plane Wave

Thus, electron in a one dimensional lattice behaves a a plane wave.It only gives Wave nature of electron.