Atoms and radiation: the Einstein picture

Albert Einstein Max Planck Charles Townes Theodor Maiman Nobel prize in Nobel prize in Physics Nobel prize in Physics 1921 1918 1954 The inventor of “especially for his discovery “for his discovery “for the maser- the LASER of the law of the of energy quanta” principle” photo-electric The ammonia maser effect”

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs The photo-electric effect in atoms

Electrons are released only if the ionization threshold is passed e- (independent of intensity) IP

Electron kinetic energy: 1 E(e− ) = mv2 = hν − E hν 2 IP

Note: in solids IP = “work function” Eg=0

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Radiation in a two-level system

E2, n2, g2

E2 − E1 = hν Cuν A Buν

E1, n1, g1

uν the incident radiation field (mode density)

Cuν 1. Absorption process A 2. Emission process

Buν 3. process

A, B, C are the Einstein coefficients

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs E2, n2, g2

Rate equation model Cuν A Buν

E1, n1, g1

Population of states: Statistical Physics: thermal excitation ν dn 2 = Cu n − An − Bu n n(T )= exp(− E / kT ) dt 1 2 ν 2

Impose an equilibrium condition Relative population: dn n exp(− E / kt) hν 2 = 0 2 2 ⎡ ⎤ = ()= exp⎢ ⎥ dt n1 exp − E1 / kt ⎣kT ⎦ In steady state: Atomic two-level system in equilibrium n A + Bu 1 = ν with radiation field: n2 Cuν A u = ν C exp[]hν / kT − B

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs E2, n2, g2

Cuν A Buν Compare with Planck’s model E1, n1, g1

A black body is in equilibrium Einstein scheme withπ a νradiation field: 3 Atomic two-level system in equilibrium 8 h 1 with radiation field: uν = c3 exp[]hν / kT −1 A u = ν C exp[]hν / kT − B

A two-level atom in equilibrium with a radiation field: C = B A 8πhν 3 = B c3

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs E2, n2, g2 Results from the Einstein model Cuν A Buν

E1, n1, g1

A two-level atom in equilibrium with a radiation field:

C = B The B constant related to the Transition dipole moment Stimulated emission is equally strong (the QM radiation strength) as absorption The A-constant follows from From a quantum treatment it follows A 8πhν 3 π 2 = e 2 3 B = μ B c ε 2 ij 3 0h

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs E2, n2, g2 Decay of an excited state A

E1, n1, g1 In the absence of a radiation field:

uν = 0

Rate equation reduces: dn 2 = −An dt 2 With a boundary condition;

n2 (0) = N numbers in 106 s-1 Solution: −At −t /τ n2 (t) = Ne = Ne Note for multiple decay channels Lifetime of an excited state 1 1 τ = τ = ∑ Ai A i

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Properties of a two-level system

Under all circumstances, i.e. for arbitrary radiation fields u ν ν Cu < A + Buν

Emission is stronger than absorption Optical pumping can never result in a population inversion So if we start at

n1(0) = N

(all population in the ground state) A two-level LASER is impossible

It is not possible to reach:

n2 > n1

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Level schemes for

Three-level LASER Four-level LASER

E 3 E4 fast decay fast decay E 3 E3 pump hν pump ν lasing h lasing

E 1 E2

E1

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs The He-Ne Laser The CO2 Laser

Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs