FREE ELECTRON THEORY

1. FREE ELECTRON THEORY ARC

2. TOPICS TO BE COVERED

3. Chief Characteristics of Metals • Metal possesses high electrical and thermal conductivity • Metals obey Ohm’s law • Conductivity of metals decreases with rise of temperature • Metals obey Wiedemann-Franz law • Near absolute zero, certain metals exhibit superconductivity

4. He was working prior to the development of quantum mechanics, so he began with a classical model. Classical Free Electron Theory • Initially stated by Drude in 1900 CONCEPT • In Drude model, the valence electrons from each atom become detached and wander freely through the metal, while the metallic ions remain intact and play the role of the immobile positive particles. • Each electron behaves as a perfect gas molecule • Each electron is free to move through the entire volume of the metal • System of free electrons in a metal = free electron gas (1863-1906)

5. + + + + + + + + + + + + + + + + + + + + + + + + + Free Electron Model Schematic model of metallic crystal, such as Na, Li, K, etc. Equilibrium positions of the atomic cores are positioned on the crystal lattice and surrounded by a sea of conduction electrons.

6. Drude’s Assumptions • Matter consists of light negatively charged electrons which are mobile& heavy, positively charged ions. • 2. The only interactions are electron-ion collisions, which take place in a very short time t. • The neglect of the electron-electron interactions is THE INDEPENDENT ELECTRON APPROXIMATION. • The neglect of the electron-ion interactions is THE FREE ELECTRON APPROXIMATION • 3. Electron-ion collisions are assumed to dominate. These will abruptly alter the electron velocity & maintain thermal equilibrium. • 4. The mean time between collisions is 𝜏 • 5. Electrons emerge from each collision with their velocity changed. • Till the application of an external • electric field, the electrons move • about in a random manner

7. Drift Velocity In presence of applied electric field, electrons move in a specific direction. This directional motion of the free electrons is called DRIFT. Average velocity gained during this drift motion is called DRIFT VELOCITY. Steady state drift velocity produced for unit electric field is called MOBILITY (μ) Fig. Ref. Google

8. Relaxation Time (𝜏) When the applied electric field is switched off, the electrons again undergo collision. The electron gas resumes its equilibrium condition. Such a process which leads to the establishment of equilibrium in a system from which it was previously disturbed is called the relaxation process. The time taken for this process is RELAXATION TIME.

9. Mean free path (λ) It is the average distance travelled by the conduction electron between successive collisions with the lattice ions. Mean collision time () The average time taken by an electron between two successive collisions of an electron with lattice points during its motion. (averaging is done over a large number of collisions)

10. An electric field is applied. The equation of motion of free electron of mass is Integrating, we get If is the average time between collisions then the average drift velocity is Drift Velocity Expression

11. Ohm’s Law • Basic law concerning the flow of electricity. • Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. • Constant of proportionality, resistance, is introduced • In mathematical terms, V = I x R where V is voltage, I is current, and R is resistance • When an electric field, E is applied to a conductor, an electric current begins to flow, and the current density by Ohm’s law is • Materials that obey Ohm’s law are said to be ohmic

12. E Ohm’s Law Experimental observation: If J = current density for electric field E, then , where σ= conductivity

13. Amount of charge passing per unit time = - So, current density J = = = ……… (1) = I Distance covered = We know, ………… (2) From (1) and (2), σ This is the form of Ohm’s Law in terms of free electrons.

14. Wiedemann-Franz Law Wiedemann and Franz law states that the ratio of thermal and electrical conductivity of all metals is constant at a given temperature (for room temperature and above). Thermal conductivity = constant Electrical Conductivity Later it was found by L. Lorenz that this constant is proportional to the absolute temperature L = Lorentz Number

15. Drawbacks of Classical Free Electron Theory • Specific Heat: • Classical free electron theory all valence electrons in a metal can absorb thermal energy. So, molar electronic specific heat 1.5 times R, where R = universal gas constant. • This is about 100 times greater than experimentally predicted values. • Mean free path: • Experimental value of λ is much greater than the theoretical value • Temperature (T) Dependence of Electrical Conductivity(σ) • Classical free electron theory σ is inversely proportional to • experiments σ is inversely proportional to

16. Drawbacks (continued..) • Wiedemann- Franz law: • At low temperatures, K/σT is not a constant. But in classical free electron theory, it is a constant at all temperatures. • Paramagnetism of Metals: • Theoretical value of paramagnetic susceptibility is greater than the experimental value. Experimental fact that paramagnetism of metals is nearly independent of temperature could not be explained

17. Salient features of Quantum Free Electron Theory • Proposed by Sommerfeld in 1928 • Electrons obey the laws of quantum mechanics • Energy levels of electrons are quantized • Electrons occupy energy orbitals according to Pauli’s exclusion principle • Distribution of electrons in different energy levels are according to Fermi-Dirac statistics • Retained concept of free electrons moving in a uniform potential but prevented them from escaping the crystal by very high potential barriers at the surfaces

18. FERMI LEVEL : Highest energy level occupied by electrons at Absolute zero. All the energy states upto Fermi level are OCCUPIED and all energy levels above Fermi level are VACANT. FERMI ENERGY: Energy corresponding to Fermi Level. Constant for a particular system Probability of an electron occupying a particular energy level ‘E’ is given by Fermi-Dirac Distribution At T = 0 K and for E < E , f(E) = 1 for E > E , f(E) = 0 F F For lower energies, ftends to 1. For higher energies, ftends to 0. On increasing the temperature, electrons get excited to higher energy level. Distribution of electrons in different energy levels gets determined by Fermi-Dirac Distribution function.

19. Fermi Distribution Function at Different Temperatures For temperatures greater than zero, Fermi function plot begins to fall close to E and at E = E , f(E) = F F

20. FERMI VELOCITY= velocity associated with Fermi Energy i.e. velocity of electrons occupying Fermi Level FOR SODIUM, = 3.2 X 1.6 X J So, FERMI VELOCITY FERMI TEMPERATURE = Temperature associated with Fermi energy FERMI TEMPERATURE So, Ref: Eisberg, R. and Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. New York: Wiley, 1985.

21. DENSITY OF ENERGY STATES • In a macroscopically small energy interval, there are many discrete energy levels. • Difference between neighbouring energy levels is as small as eV CONCEPT OF DENSITY OF ENERGY STATES simplifies OUR CALCULATION. It helps in finding the number of energy states with a specific energy. Density of states (DOS) of a system describes the number of available states in a unit volume per unit energy range. In a system, if N(E) = number of electrons with energy E, g(E) = number of energy states with energy E, f(E) = probability of an electron to occupy energy state E, then N(E) dE = g(E)dEf(E)

22. We consider a free electron of mass ‘m’ trapped inside a cubical metal block of side length, ‘a’. According to quantum mechanics, energy of the free electron, ----- (1) where h = Planck’s constant where + Let us consider a space of points represented by coordinate system along the three mutually perpendicular directions. Let each point with integer values of the coordinates represent an energy state. Let n be the radius vector from origin (0,0,0) to a point represented by (). So, + -------- (2) All points on the surface of the sphere of radius ‘n’ will have the same energy. As per quantum condition, values of are restricted to be positive. Only in one octant of the sphere, each point corresponds to only positive values of .

23. SPACE OF POINTS

24. If n = radius of sphere whose octant encloses all the points upto an energy ‘E’, then Number of allowed energy values upto an energy E = number of points in the octant of sphere of radius ‘n’ We consider another sphere of radius n+dn whose octant encloses all points upto an energy ‘E+dE’, then Number of allowed energy values upto an energy E + dE = number of points in the octant of sphere of radius ‘n + dn’ So, number of allowed energy states in energy range dE = number of points in the space between two octant shells of radii n and n+dn = (volume of space between two octant shells of radii n and n+dn) X (number of points / unit volume) = π Since are all integers, A unit volume of plot consists of Just one point ---------------- (3)

25. If g(E) = number of energy states per unit energy range, then number of energy states in the energy interval dE = g(E)dE So, g(E) dE = ------------ (4) From (1), = ---------- (5) ---------- (6) Differentiating (5) , we get So, ----------- (7)

26. Using (4), (6) and (7), we get ---------- (8) Each energy value is applicable to two energy states, one for an electron with spin-up, and the other for an electron with spin down (Pauli’s exclusion principle). So, the number of allowed energy states in the energy interval dE = = Hence, the number of energy states present in unit volume having energy values lying between E and E + dE (DOS) is given by = [

27. Density of energy states for a free electron gas General Expression for = TASK: Find out the number of electrons present per unit volume of a cubical metal block at absolute zero temperature

28. Thermionic Emission The emission of electrons from a metal under the effect of thermal energy is called THERMIONIC EMISSION. Emitted electrons are called THERMIONS. Electrons are free to move inside the metal Electrons cannot come out of the metal surface on its own as high potential barrier is present at the surface but when the temperature of the metal is sufficiently high, electrons gain sufficient energy to overcome the barrier and ESCAPE from the metal surface Free electron theory assumes that the potential within the metal is constant. The minimum energy to be supplied to the electron for its emission from the metal is termed as WORK FUNCTION (Ф) of the metal

29. Richardson’s Equation If W = minimum energy of the electron for its emission from the surface, E = Fermi energy of the metal, then, Ф = W – E = work function of the metal. No. of energy states / unit volume in energy range E to E + dE, F F D(E)dE= We know, So, density of energy states per unit volume in momentum range p to p+dp, ----------- (1) We construct a plot in ‘momentum space’ such that each point represents a particular combination of momenta components of an electron along x-, y- and z-directions. So, + Volume element in momentum space, -------- (2)

30. From (1) and (2), the density of states in momentum space, = --------- (3) Hence, no. of electrons/unit volume having momenta in the range and + and = Now, we consider the metal plate to be in Y-Z plane. Electrons will be emitted in a direction perpendicular to Y-Z plane i.e. along x-axis. Only those electrons will be emitted whose energy, E > W. . So, 1 in denominator can be neglected.

31. Standard Integral Form So, current density,

32. J J [since J [since ] J