Photoelectric Effect.

The Black Body radiation spectrum forced upon us the idea of quantized oscillators
radiating EMW and a departure from the concepts of Classical Physics. Another phenomena was soon discovered which seemed to defy a classical explanation in terms of EMW. This was the photoelectric effect which demonstrated that incident radiation on the (metal) surface are able to eject electrons. The electrons are called photo electrons and may constitute a current called a photo current

In 1886 Hertz did his famous experiment on this effect



http://physics.info/photoelectric/circuit.html

* Image from Flickr "Robbo Coach's" photostream

Several interesting features may be noted

1. The current I is NOT ZERO till a "reverse voltage is applied". This shows that
the electrons are ejected with some kinetic energy. The current saturates at some value of the forward voltage and this is dependent on the intensity of the incident radiation.

2. The cut off voltage is independent of Intensity of the radiation but depends on the frequency.

3. Millikan showed that the stopping potential varied linearly with the frequency and there was a "cut off frequency" ν₀ beyond which no electrons are emitted.

*Graph

An explanation of this phenomena based on the Classical Wave picture of EMW fails
due to the following reasons

1. Kinetic energy of photo electrons in particular the maximum kinetic energy K_max
=e V₀ where V₀ is the stopping potential should depend on the intensity as Intensity is proportional to the Electric Field amplitude squared in an EMW and this is the field that energizes the electron in the wave picture. But experiment shows otherwise.

2.The effect should occur for any frequency provide the incident radiation is intense enough to energize the electron. Experiment shows a cut off frequency specific to the material below which no photo electrons are ejected.

3. Effect should show a time lag as the wave energy is diffused over a spatial extent. Experiment shows that the effect is instantaneous.


These discrepancies inspired Einstein to apply the idea of Quantization due to Planck to EM radiation. Einstein postulated that EM radiation is composed of a large number of discrete packets of localized energy which behave like particles. The particles were called photons and their energy was quantized as E=hν where ν was the frequency of the radiation.

Einstein postulated that in the photoelectric effect 1 photon was absorbed by 1 electron to be energized and ejected provided the energy was adequate for the electron to overcome the surface atomic attractions.

So the kinetic energy of the ejected electrons would be K=hν-W where W is the energy for overcoming the surface barriers.

The maximum kinetic energy K_max=hν -W₀ where the second term was called the work function of the material and referred to the minimum energy for electrons to overcome the surface atomic attractions.

The photon picture was able to explain the experimental results

1. Intensity of radiation was proportional to the number of photons or their density. However each electron could absorb only 1 single photon. Hence the maximum kinetic energy was independent of the intensity.

2. At threshold when an electron is Just ejected with zero kinetic energy ( corresponding to the stopping potential K_max=0 and hν₀=W₀ where ν₀ is the cut off frequency
and no emission was possible below this.

3. As the energy of radiation was highly localized in the photon picture the absorption is instantaneous and hence there was no time lag as observed experimentally but was predicted by the classical wave picture.


From the equation K_max=eV₀=hν -W₀
its is seen that the stopping potential V₀ varies linearly with the frequency and from the slope of the graph of the stopping potential vs frequency
we can fix the Planck's constant h knowing e and m for the electron, which comes out to be close to the measured value.

Black Body Radiation III

The Planck Distribution
------------------------

We saw that the Rayleigh Jeans law fails to explain the black body spectrum at higher frequencies. Planck in trying to explain this discrepancy noticed that the average total energy for the modes was kT at low frequencies but must fall to zero at high frequencies to describe the experimental curve. This required a high frequency cut off for the oscillators in violation of the equipartition theorem which assumes that the oscillators may have any energy from 0 to infinity with equal a priori probability resulting in an average total energy per mode independent of frequency.

Planck departed from this hypothesis and postulated that the oscillators have
discrete energies which are equally spaced that is E=0, ΔE. 2 ΔE..... and ΔE=hν was proportional to the frequency and the proportionality constant h was refered to as the Planck's constant.

As ΔE=hν the energy of the oscillators E=nhν with n=0,1,2,3.....
An average energy computed from this discrete values using the Boltzmann
distribution as in the Classical Equipartition Theorem ( using summations over discrete values instead of integrals) yielded the Plancks distribution formula for the average energy

E_av=[hν/ e^(hν/kT) -1]

It is obvious from the formula that the desired high frequency cut off for the oscillators is obtained as E goes to zero with large values of frequencies and E
goes to (kT) ( expanding the exponential to first order) for small values of frequency.

Multiplying this by the Jeans number of modes in an infinitessimal frequency interval we obtain the energy density of the cavity radiation to be

ρ(ν)dν=(8πν²/c³ [hν/ e^(hν/kT) -1] dν

This is the Planck Black body Spectrum and completely matches the experimental black body curve ( upto a proportionality constant).

http://www.youtube.com/watch?v=cW4vmr3hb2o&feature=related 

Black Body Radiation II

In the last 2 sessions we have established that the Cavity Radiation spectrum is identical to the black body radiation spectrum and this is expressed as

ρ(ν)dν=constant. R(ν)dν at some temperature T deg K at thermal equilibrium.

The cavity radiation is due to the energized atomic oscillators and at ε is a standing wave configuration ( assuming no dissipation at walls)

To calculate the energy density of the cavity radiation in an infinitesimal frequency interval dν we need

(i) The number of modes in that interval

(ii) The average total energy in each mode

so ρ_T(ν)= # of modes X average energy/volume

To count the modes for simplicity we assume a cubic cavity and we will ses that the final result is independent of the geometry of the cavity.

As a warm up exercise we first compute the modes for 1 dimension ( line cavity of length L). We solve the 1 d wave equation with the boundary condition of nodes at the walls at x=0 and x=L. The boundary conditions restricts the modes to be such that the wave vector k=(nπ/L).

from standard relations between frequency and wave vectors its easy to see that the number of modes in the frequency interval dν is dN=(2L/C) X dν

This is easily generalized to 3d with a cube of side L and
assuming independent propagation in (x, y,z) directions due to parallel walls and

n²=n²_x + n²_y
+ n²_z

Using the same construction as 1d we mark the n's on a 3 d lattice and we see that
the number of modes = volume in lattice space. To find the number of modes in the frequency interval dν it is required to find the volume between two spheres of radius N and N + dN in the first octant as all
n > 0.

N= (1/8)(4π/3)n³=(π/6)(2L/c)³ ν³

so that dN=N(ν)dν=(4π/c)³ L³ν²dν X 2

for the 2 states of polarization of EMW

so the density is dN/L³ = (8π/c³)ν² dν

This is to be multiplied by the average total energy of each mode which may have any energy continuously between 0 and infinity.

Since the standing waves are in Ε with each other the equipartition theorem may be applied to find the average total energy to be E=kT.
(potential and kinetic)

So ρ(ν)dν=kT (8π/c³)ν² dν

This is the famous Rayleigh Jeans law and shows a quadratic dependence on the frequency. It is quite obvious that the law predicts an indefinite increase in the
energy density with frequency. It describes the black body spectrum at low frequencies but breaks down at high frequencies ( ultraviolet direction). Hence this came to be known as the "Ultraviolet Catastrophe"

It actually marks the failure of classical physics comprising of Mechanics and
EM.

Black Body Radiation I

Mechanics: 

Heat and Thermodynamics (Macroscopic Large Scale Phenomena)

Kinetic Theory ( Statistical mechanics) ( Microscopic Small Scale Phenomena)

Average over microscopic properties give rise to macroscopic properties

Eg. Temperature is a measure of the average molecular kinetic energy. Pressure is average force on wall of container due to molecular collisons.

Electromagnetic Theory: Maxwell and the Wave Equation.

Radiation: Classically due to oscillating charge dipoles. Radiation from oscillators cause heat transfer through EM waves. Radiation transfer occur at all non zero Temperature T and is in the infra red part of the EM spectrum.

At thermal equilibrium the black body radiates and also absorbs at the same rate and the spectrum of radiation is

continuous and depends on the temperature T and the nature

of the material.

The power radiated by a black body is given by the Stefan-Boltzmann Law as

P=e σ A (T⁴ -T₀⁴)

where T₀ is the ambient temperature, e is the emissivity and σ is the Stefan constant. By Kirchoff's Law e=a at thermal equilibrium where a is the absorptivity of the body. For an ideal black body radiator e=a=1 and the spectrum is universal and dependent only on the temperature.

A graphite cavity with thick walls and a small hole may be approximated to
an ideal black body such that it is perfect absorber and absorbs all radiation incident on it. The cavity walls re radiate and at thermal equilibrium the cavity is full of radiation at te ambient temperature T. If this cavity is heated to T then the hole must radiate like a black body by Kirchoffs Law because a perfect absrober
is also a perfect emitter. So the radiation from the hole may be understood by analysing the cavity radiation.

The energy density ( energy/volume) of the cavity radiation ρ_T(ν) is proportional to R_T(ν) where R is the spectral radiancy (energy radiated/unit time/unit area) of the black body. A graph of the spectral radiancy was
obtained by Lummer and Pringsheim at various temperatures in deg K through experiemnets.

Spectral Radiancy Curves with Frequency

Image From: thermal-survey.co.uk via Google Images
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The characteristics of these curves are

1. Very little power radiated for low frequency P=0 for ν=0

2. Increases with increase in frequency and reaches a maxima for ν_max

3. Drops with further increase and asymptotes to zero for ν infinity.

4. The peak frequency ν_max shifts to higher frequecies for higher temperatures.

The total radiancy R_T=∫ R_T(ν) dν=σ T⁴

Wiens Law: W=T λ_max where W is the Wien's constant. This is
a theoretical fit to an experimental curve and the value of W is thus obtained.

Welcome to Quantum Physics ( PHY 204/SE 301) 2011 Semester II