# Wave and Particle Theory of Light applied to the Photoelectric Effect

### Introduction

It is generally argued that the photoelectric effect provides evidence for the particle nature of light as the photoelectrons are almost released instantaneously when the light hits the detector. This argument is based on the classical wave theory of light and the assumption that a light wave has an energy flux c.E2/4π where E is the electric field amplitude of the wave and c the speed of light. Using this expression, the energy absorbed by an atom would indeed be so small that it would take about one second for sunlight to release any photoelectrons, contrary to what is observed and hence suggesting that the wave model of light could not explain the photoeffect. However, this conclusion is based on the flawed assumption that the above expression can be assigned to the 'energy flux' of the electromagnetic wave and in fact that such a physical quantity exists in the first place for electromagnetic waves. The point is that this expression is a result from classical electrostatics describing the potential energy of a capacitor (i.e. the work that is needed to charge the capacitor). However, even here it is wrong to interpret this as energy stored in the electric field, because kinetic and potential energies of the particles fully describe the system and there is no room for a separate field energy, neither in electrostatics nor in electrodynamics. Additionally of course, there is no explanation as to how this assumed electromagnetic wave 'energy' should be converted to the kinetic energy of the released photoelectrons (see also my page regarding Energy and Momentum Conservation in this context).
In the following it is shown that a consistent semi-classical treatment of the interaction process between an atomic electron and an electromagnetic wave does indeed yield ionization times that may be short enough to explain the observations. On the other hand, it is shown that the particle model would in fact not enable any photoelectric effect at all as the collisional energy transfer would be much too small.

### Wave Model

The interaction of light with atoms can generally be described as a forced oscillator (in the case of photoionization as a Pseudo-oscillator; see http://www.plasmaphysics.org.uk/#pseudosc). If the oscillation is in phase with the driving field of the electromagnetic wave throughout, this is exactly equivalent to the acceleration of an electron by a constant electric field E (where E is the amplitude of the wave). The electron will therefore be accelerated to the velocity v within a time

Tc=v/a  =  v/(eE/m)   ,

where e is the elementary charge.
Since v is related to the energy ε by

v= (2ε/m)   (m=electron mass)   ,

and ε has to be taken as identical to hν (h=Planck-constant, ν=wave-frequency), this yields

Tc= (2hν.m) /(eE) .

After evaluating the constants this gives the numerical expression

Tc= 7.10-18.ν /E     [sec]     (in Gausssian cgs-units i.e. ν [Hz], E [statvolt/cm](=3.104V/m) ) .

For sunlight (assuming E=10-2 statvolt/cm), this amounts to about 10-8 sec, i.e. the almost instantaneous release of electrons in the photoeffect can well be explained within the wave theory of light if a proper interaction model is used (of course, the wave frequency ν has to be high enough here to enable photoionization in the first place)(note: the assumption E=10-2 statvolt/cm should be merely considered to be exemplary here as the electric field strength E of the radiation field is in fact unknown; it has been derived here by equating the well known 'energy' flux of the sun with c.E2/4π , which however, as pointed out in the first paragraph, is theoretically flawed and is ambiguous anyway due other physical parameters affecting the measured intensity as addressed on this page).

Note: if Tc is larger than the coherence time of the electromagnetic wave or the time between particle collisions disturbing the ionization process, then the above argument has to be modified as the acceleration of the electron can not be uniform due to phase jumps occurring. This is done on my site plasmaphysics.org.uk under Photoionization Theory for Coherent and Incoherent Light .

### Particle Model

In contrast, the particle theory of light is completely inconsistent and would in fact not enable photoionization at all:
Assuming that a photon with energy ε has a mass

μ= ε/c2   ,

the energy transfer Δε to the electron with mass m is determined by the equation for momentum conservation

μc  =   mv  =  m.(2.Δε/m)  ,

and therefore

Δε  =  μ2c2/2m  =  μ/2m .ε   .

With μ corresponding to ultraviolet radiation, this means that only about a fraction 10-5 of the initial photon energy can be transferred to the electron. This amount is much too small for photoionization to occur (which would require a factor of the order of 1). The energy transferred to the atomic nucleus would be even 3-4 orders of magnitude smaller due to its higher mass (that is if there would be in the first place an interaction force by means of which the photon could transfer its kinetic energy).

A further proof for the incorrectness of the particle model for light is the experimental fact that photoelectrons are not primarily released in the direction of propagation of light (as one should expect it for particles) but perpendicular to this in the direction of the electric field vector (see reference) .

### Conclusions

It is obvious that the picture of a photon as a particle with mass and momentum is completely wrong. In general, momentum can only be transferred from one particle to another by means of some static interaction force. However, no such interaction force (like the Coulomb- force for electron- impact ionization) is known for photons. The flaw with the particle picture of light (and Quantum Field Theory in general) is that the principles of Classical Mechanics are generalized to non-mechanical physical phenomena , i.e. it is assumed that a Hamiltonian (or Lagrangian) function exists for the corresponding systems (and with it the laws of momentum and energy conservation). The above analysis shows that such a 'mechanization' of light is not only theoretically unfounded but also clearly contradicts experimental results.

The discrete nature of the photoelectric effect is clearly due to the properties of the material (i.e. the existence of individual electrons and the quantization of atomic energy levels) rather than that of the light which can consistently be described as an electromagnetic wave field with a given frequency spectrum and coherency. The usual description of light intensities in terms of photons is therefore in general ambiguous due to the number of factors affecting the photoionization process (see http://www.plasmaphysics.org.uk/research/#A5).

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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