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The Dependency of the Energy on Wave Numbers__1 2 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 FuncSon 2.4 CONCLUSION The Dependence of Energy on Wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the spa8al 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 8me, wavenumber is the number of waves per unit distance. Diagram illustra8ng the rela8onship between the wavenumber and the other proper8es of harmonic waves. In mul8dimensional 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 essen8al role in op8cs and the physics of wave scaAering, such as X-ray diffrac8on, neutron diffrac8on, electron diffrac8on, and elementary par8cle physics. For quantum mechanical waves, the wavenumber mul8plied by the reduced Planck's constant is the canonical momentum. Wavenumber can be used to specify quan88es other than spa8al frequency. In op8cal spectroscopy, it is oIen used as a unit of temporal frequency assuming acertain speed of light. Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically cen8meters (cm−1):, where λ is the wavelength. It is some8mes called the "spectroscopic wavenumber". It equals the spa8al frequency. where, Rh is the Rydberg constant n1 is the ini8al shell number of the electron n2 is the final shell number of the electron In theore8cal physics, a wave number defined as the number of radians per unit distance, some8mes called "angular wavenumber", is more oIen used: When wavenumber is represented by the symbol ν, a frequency is s8ll being represented, albeit indirectly. As described in the spectroscopy sec8on, this is done through the rela8onship , 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 cen8meters; cm−1); in this context, the wavenumber was formerly called the kayser, aIer Heinrich Kayser (some older scien8fic 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 electromagne8c radia8on in vacuum,wavenumber is propor8onal to frequencyand to photon energy. Because of this,wavenumbers are used as a unit of energy in spectroscopy. A complex-valued wavenumber can be defined for a medium with complex- valued rela8ve permi\vity , rela8ve permeability and refrac8on index n as: where k0 is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses aAenua8on per unit distance and is useful in the study of exponen8ally decaying evanescent fields The Density of States 1. Density of States • MathemaScal funcSon 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 …equaSon (1). H^3 2. Fermi-Dirac Probability Function · Func8on 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/. .. equa8on(2) · We can combine equa8on (1) and equa- 8on (2) n(E) = £(E).f(E) =8√2πm3/2E1/2 . h3 This equa8on is known as the density of occupied states. If we plot this equa8on 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 par8cular 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 en8re 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 final poten8al as well. That is why electrons have a poten8al energy of zero inside the well metal, but the poten8al energy to some high value at the edges of the metal. · Recall that a par8cle trapped in a po- ten8al well has quan8zed energy given by the following equa8on. · Similarly, electrons trapped in metal solids have quan8zed 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 · Mathema8cal func8on called the densi- ty state, gives the distribu8on 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 func8on between any two energy value to calculate the number of quantum states per unit vol- ume that exists between those energy values. .
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