SUBMITTED TO: Dr. RANJIT PRASAD BY: RAVI RANJAN,CSE019 MAYANK KUMAR,CSE049 ALOK HERMOM,CSE079 SAURAV KUMAR,CSE109

ON:JANUARY 10,2020

TOPIC: 1. THE DEPENDENCY OF THE ENERGY ON WAVE NUMBERS__1 2. THE DENSITY OF STATE CURVES___ 2 TABLE OF CONTENT 1.THE DEPENDENCY OF THE ENERGY ON WAVE NUMBERS__1

1.1 INTRODUCTION 1.2 ILLUSTRATION WITH DIAGRAM 1.3 DIFFERENT TYPES OF WAVE EQUATION 1.4 CONCLUSION

2. THE DENSITY OF STATE CURVES___2

2.1 INTRODUCTION 2.2 DENSITY OF CURVES 2.3 Fermi-Dirac Probability Functon 2.4 CONCLUSION The Dependence of Energy on Wavenumber

In the physical sciences, the wavenumber (also wave number or repetency) is the spatal frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit tme, wavenumber is the number of waves per unit distance. Diagram illustratng the relatonship

between the wavenumber and the other propertes of harmonic waves. In multdimensional systems, the wavenumber is the magnitude of the wave vector. The space of wave vectors is called reciprocal space. Wave numbers and wave vectors play an essental role in optcs and the physics of wave scatering, such as X-ray difracton, neutron difracton, electron difracton, and elementary partcle physics. For quantum mechanical waves, the wavenumber multplied by the reduced Planck's constant is the canonical momentum.

Wavenumber can be used to specify quanttes other than spatal frequency. In optcal spectroscopy, it is ofen used as a unit of temporal frequency assuming acertain speed of light.

Wavenumber, as used in spectroscopy and most chemistry felds, is defned as the number of wavelengths per unit distance, typically centmeters (cm−1):,

where λ is the wavelength. It is sometmes called the "spectroscopic wavenumber".

It equals the spatal frequency. where,

Rh is the Rydberg constant

n1 is the inital shell number of the electron n2 is the fnal shell number of the electron

In theoretcal physics, a wave number defned as the number of radians per unit distance, sometmes called "angular wavenumber", is more ofen used:

When wavenumber is represented by the symbol ν, a frequency is stll being represented, albeit indirectly. As described in the spectroscopy secton, this is done through the relatonship ,

Where vs is the frequency in hertz. This is done for convenience as frequencies tend to be very large.

It has dimensions of reciprocal length, so its SI unit is the reciprocal of meters (m−1). In spectroscopy it is usual to give wavenumbers in cgs unit (i.e., reciprocal centmeters; cm−1); in this context, the wavenumber was formerly called the kayser, afer Heinrich Kayser (some older scientfc papers used this unit, abbreviated as K, where 1 K = 1 cm−1).

The angular wavenumber may be expressed in radians per meter (rad⋅m−1),or as above, since the radian is dimensionless. For electromagnetc radiaton in vacuum,wavenumber is proportonal to frequencyand to photon energy. Because of this,wavenumbers are used as a unit of energy in spectroscopy.

A complex-valued wavenumber can be defned for a medium with complex- valued relatve permitvity , relatve permeability and refracton index n as:

where k0 is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses atenuaton per unit distance and is useful in the study of exponentally decaying evanescent felds

The Density of States

1. Density of States • Mathematcal functon that gives us the possible energy quantum state per unit volume per unit energy of the electrons inside the solid metal.

£(E)=8√2πm3/2E1/2 …equaton (1). H^3

2. Fermi-Dirac Probability Function

· Functon that describes the probability of the certain energy quantum state be- ing occupied by an electron in the metal solid at some temperature T. f(E)=1/. .. equaton(2) · We can combine equaton (1) and equa- ton (2) n(E) = £(E).f(E) =8√2πm3/2E1/2 . h3

This equaton is known as the density of occupied states. If we plot this equaton on the XY plane and integrate it with respect to energy, then it gives us the number of electrons per unit volume which occupies quantum states between some range of energies at some partcular temperature. · The shaded curve represents the density of occupied states, for a metal solid at some temperature T.

· The curve present the density of states at absolute zero. · That shaded curve tells us what hap- pens to the electrons in the metal solid as we increase the temperature. By in- creasing the temperature by approxi- mately 1000K , we increase the energy of electrons by approximately 0.1eV . Since this is very small amount only the elec- trons very close to the Fermi energy level can actually jump to higher and quantum states.

· Electrons with energy way below the Fermi level do not gain enough energy and remain in their quantum states.

· The free electron theory describes that the outer electrons of all the atoms in the metal roam freely about their entre metal solid, but very rarely do they actu- ally leave. Since electrons are trapped within the metal, we can imagine that the met- al is a fnal potental as well. That is why electrons have a potental energy of zero inside the well metal, but the potental energy to some high value at the edges of the metal.

· Recall that a partcle trapped in a po- tental well has quantzed energy given by the following equaton.

· Similarly, electrons trapped in metal solids have quantzed energy states.

· By the Pauli exclusion principle, all elec- trons must be given by a unique set of quantums and only two electrons can be found in any given electron orbital. Since there is a very large number of electrons in solid metals, we can presume that there must be a very large number of possible energy states. £(E)= 8√2πm3/2E1/2 h^3

· Mathematcal functon called the densi- ty state, gives the distributon of all pos- sible quantum states per unit volume per unit in exit that the electrons in the met- als can take. · We can integrate the functon between any two energy value to calculate the number of quantum states per unit vol- ume that exists between those energy values.