10 Photo Electric Effect

N. Panchapakesan and J.D. Anand

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Learning Outcomes

  • Learn about the photo electric effect where an electron is knocked out of an atom by a photon. The frequency of the photon (not intensity of light) has to have more than a certain value for the process to work.
  • Learn how the photo-electric differential and total cross-section can be obtained from the transition probability by taking the initial state as a bound state and treating the wave function of the final ejected electron state  as a plane wave.
  • Learn that using the plane wave instead of an electron wave function distorted by the coulomb field of ionized atom is called Born approximation.
  • Learn to obtain the total cross section of the inverse recombination process from the photoelectric cross section by using the principle of detailed balance.
  • Learn  how this plays an important role in cooling of radiation in the early universe.

   1. Introduction

 

We calculated the decay life time of an excited state of hydrogen atom in an earlier module. In this module we shall calculate the production cross section for photo electric effect. First the differential cross section along solid angle d and then the total cross section. In photo-electric effect a light quanta ejects electron out of the atom. The energy of the electron emitted in the process is such that one can treat electron as non-relativistic particle and so we shall treat electrons of the atoms by means of Schrodinger equation. The radiation field shall be described by the quantized vector potential as in earlier modules.

 

In photo-electric effect we  treat the initial state A of the electron as bound state with energy  ?? = −?,whereI is the binding energy, and the final state B as that of a free electron. We evaluate the cross-section of the photo-electric effect using the results from time dependent perturbation theory. In section 2 the derivation of the photo-electric effect is carried out by assuming the initial state of the electron to be the ground state and the final state of the final electron is taken as plane wave. The plane wave approximation is called Born approximation. The total cross-section of photo-electric effect is obtained by treating the velocity of the final electron to be large, though it is still much less than velocity of light. We have computed the total cross-section for the case of 1KeV energy of the incident photon. The error due to the Born approximation is also calculated which is about 11 % in the cross section. Finally the radiative recombination process is calculated assuming the principle of detailed balance.

 

2. Photo – electric effect in the Born Approximation.

 

In module M8 we had studied the radiative transitions with emission and absorption of a photon between atomic levels of the hydrogen atom. The photoelectric effect differs from such a photon absorption process only in that the final state is that of a free electron, whose energy levels belong to a continuous spectrum.

 

Let the electron in the initial state be at a level ?? = −? where I is the ionization potential of the atom and the photon has a definite momentum ? ⃗ and polarization ? ?. In the final state, the electron has momentum ? and energy ??. Here ? takes continuous values. From equation (3.14-15) from the unit M7 we have (??? = 1, and one electron only)

 

 

⃗

 

This is the photo-electric differential cross-section when the photon momentum is in the zdirection and the polarization is along x-axis. ? is the velocity of the electron with momentum ? making an angle ? with the photon momentum ? ⃗ . If the velocity of the final electron is small we can neglect it in the denominator of (17).

 

It can be seen from eq. (3.15) that most of the photo electrons are emitted in the direction of polarization of the incident light quantum ( = π/2 , ?=0 ). To get the total cross section we have to integrate over d?, the solid angle element. Solid angle is defined area of the small area divided by ?2 .

 

A small elementary area has sides rd? along k p plane and r sin ? d? along xy plane (see figure above eq. (3.12)) . Dividing the area ?2 sin d? d by ?2, d? = sin d? d . To evaluate total cross section we have to multiply eq. (3.15) by sin d? d? and integrate over ? and ?. The integrals are ∫ ?? ???3 ? =

Here ???? varies as ?−7⁄2, with photon energy ?. If we take ? = 1 Kev, ? = 511 Kev then ???? comes out to be ≃ 10−22 Cm2 . For K-shell electrons the effect is large. For Z higher ,than 1,we have to multiply by 2 for the 2 electrons in K-shell.

 

The Error Due to Born Approimation

 

In the above derivation of photo-electric cross-section we have treated the wavefunction of the electron as plane wave which is possible only in the Born approximation. Let us now estimate the error due to it. As you know for Born approximation to be valid the parameter ?⁄? , where ? is the magnitude of the velocity should be small. Now

    4. Radiative Recombination Process and Early Universe

 

The time reversed process to the photoelectric effect is the radiative recombination process of an electron with an ion of the atom (proton for z=1) to form a neutral atom with emission of photon.. The cross-section ???? for this process can be found from that for the photo-electric effect (??ℎ) by using the principle of detailed balance. The main point is that transition probability is proportional to the matrix element squared, which is same for both processes. To allow for different phase space factors in ?? we multiply by 2 ?2⁄?2 where k is the momentum of photon and p that of electron. The cross-sections for the processes ? → ? and ? → ? (with two particles in each of the states ? and ?) are related by

????2??→? = ????2??→? − − − − − − − − − −(3.19)

where ?? and ?? are the momenta of the relative motion of the particles and ?? , ?? the spin statistical weights of the states ? and ?. Since ? = 2 for the photon (which has two possible directions of polarizations we find for the Hydrogen atom ground state with ?? =2 for photon , and ?? = 1 for electron

 

???? = ??ℎ2 ?2⁄?2 − − − − − − − −(3.20)

 

Here p = |?? | is the momentum of the incident electron and k that of the emitted photon.

 

Early Universe

 

The existence and relationship of the time reversed reaction, like above , plays a crucial role in the progress towards equilibrium of a thermodynamic system. In early universe, in cosmology, the cooling down of the universe below a temperature corresponding to the binding energy of hydrogen atom reduces the number of photons above the binding energy of hydrogen. This stops the ejection of photo electrons. This leads to formation of neutral hydrogen atom and stoppage of interaction between photons and protons. However protons and electrons continue to combine to form neutral hydrogen atoms though at a reduced rate. The equilibrium between ionization and recombination breaks down at this temperature. (Interestingly this is really combination for the first time in the universe and not recombination, as there never was neutral hydrogen earlier to this time.) After this time, photons (radiation) and protons (matter) cool independently. The cooled photons form the cosmic microwave radiation at 2.7 K at present.

  1. Summary

The photoelectric effect is obtained in the Born approximation from the absorption of a photon by the nonrelativistic electron of the atom. It is seen that the total cross-section varies as ??−7⁄2 where ?? is the
energy of the incident photon. Also for Hydrogen like atom where ? ≠ 1, the total cross-section is proportional to the fifth power of z. The total cross-section of the recombination process is also obtained.

you can view video on Photo Electric Effect

Learn More

 

Books

  1. Advanced Quantum Mechanics by J.J. Sakurai, Pearson Education Inc. (1998).
  2. Quantum Mechanics, Vol.3 by L.D. Landau and Lifshitz, Pergamon Press.
  3. Quantum Electrodynamics by V.B. Berestetskii, E.M. Lifshitz and Pitoevskii, Pergamon Press 1982.
  4. A Text Book of Quantum Field Theory by Amitabha Lahiri, Palash B. Pal, Narosa Publishing House, 2000, 2005
  5. An Introduction to Quantum Field Theory by M.E.Peskin and D.V. Schroeder, Westview Press, 1995, Indian Edition, Levant Books, Kolkata, 2005.
  6. Quantum Field Theory, A Modern Introduction, by Michio Kaku. OUP 1993.

Web sites

 

WWW. Santarosa.edu/~ya taiiya/4D/PhotoElecEffectPhy4D.ppt   (Very elementary)

 

https://en.wikipedia.org/wiki/Photoelectric_effect     (Descriptive)

 

Place in History of Science

 

For the present generation it will be almost impossible to understand the strong opposition to the idea of a quantum, especially the photon. Though Einstein gave the (correct ) explanation of photoelectric effect based on the idea of photon in 1905, it took another twenty years before the idea was accepted by the scientific community and Einstein given the Nobel prize for it in 1928.

 

See the book for example by Abraham Pais,    “Subtle is the Lord” , OUP, 1982.