Photoelectric effect notes PDF

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Notes on the photoelectric effect from Serway Moses Modern Physics...

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Photoelectric Effect and Photon Hypothesis Note: Reference in Modern Physics: Section-3.4 In these notes, I discuss the following topics: • Photoelectric effect • Failure of wave theory to explain features of Photoelectric effect • Distinction between Planck’s and Einstein’s picture of quantum of light • Explanation of photoelectric effect through photon hypothesis

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Photoelectric Effect

Heinrich Hertz showed that clean metal surfaces emit charges when exposed to ultraviolet light. Later it was shown that these emitted charges are electrons and are named photoelectrons. Lenard studied photoelectric effect with sources of different frequencies and also different intensities. He observed four important facts: • Photoelectrons are emitted only when the frequency of the incident light is above critical frequency, for a given metal. • For a given frequency, the number of photoelectrons emitted, or equivalently the photocurrent, is larger for larger intensity. • The stopping potential, that is the potential needed to stop the photocurrent, is independent of the intensity for a given frequency. It also larger for larger frequency. • There is no time delay in the emission of photoelectrons. As soon as the light source is turned on, the emission of photoelectrons starts. Some of the results of Lenard are illustrated in the figure below.

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Failure of wave theory to explain features of photoelectric effect

Lenards observations are inexplicable in wave theory of light. In the wave theory, the intensity of light is a measure of the energy carried by the wave. The larger the intensity, the more energy the wave carries and it should be able to transfer that energy even to a tightly bound electron and free it. The energy absorbed by an atom is the product of the intensity (whose dimension is Watt/area), area of absorption (which is the area of atom which is exposed to the light) and time of exposure. Intensity of a light wave is proportional to the square of the electric field of the light wave, according to Maxwell’s theory. The frequency of the light does not enter anywhere in the expression for the energy of the light wave. Thus the wave theory of light can not explain the frequency dependence of the emission of photoelectrons nor can it explain the frequency dependence of the stopping potential. Let us also take a look at the time interval between the beginning of absorption of radiation and emission of photoelectrons. By energy conservation, it is easy to see that the electron is emitted only when the energy absorbed 2

by the electron is greater than the work-function (which is the binding energy of the electron in the metal). Sodium has a work-function of 2.28 eV and it is observed that a light of intensity of 1.0×10−7 mW/cm2 produces measurable photocurrent in sodium. We assume the radius of sodium atom is about an Angstrom (= 10−8 cm). We calculate the time interval of absorption as t=

2.28 eV × 1.6 × 10−16 mJ/eV ≈ 107 sec ≈ 100 days. 3.14 × 10−16 cm2 × 10−7 mW/cm2

(1)

According to the wave theory, photoelectrons should appear only after a time delay of more than 100 days after the light is turned on. According to the experiment, there is no measureable time delay between turning on the light and emission of photoelectrons. Thus we see that the wave theory of light is unable to explain at three of the four features of photoelectric effect, establised by Lenard.

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Distinction between Planck’s and Einstein’s picture of quantum of light

Planck, in explaining the blackbody radiation, said that the energy of the oscillators in the blackbody is quantized and hence the energy of the emitted radiation is quantized. In Plancks picture, radiation of frequency ν can exist as a quantum of energy hν or 2hν or 3hν or any nhν, where n is a positive integer. If an energy of 2hν is emitted, the following question arises: Is it emitted as one quantum of 2hν? OR Are two quanta of energy hν emitted? The derivation of Planck’s Blackbody radiation formula makes no distinction between the above two cases. We get the same answer in both cases. Einstein pictured the radiation in the blackbody to be an Ideal gas of light quanta. Suppose you have a monoatomic gas of mass M = N × m. Such mass can be a result many possibilities. Two such possibilities are: There are N molecules, each of mass m OR there are N/2 molecules, each of mass 2m. From your study of the thermodynamic properties of ideal gases, you know that the two cases behave very differently. If both gases are at temperature T , then by equipartition theorem, a molecule of each gas will have kinetic energy 3kT /2. Hence the internal energy of the gas in the first case is 3N kT /2 whereas the internal energy of the gas in the second case is 3

3N kT /4. It was well known that the entropy of a gas is an extensive quantity and Boltzmann derived an expression for the entropy of an ideal gas, showing it to be proportional to the number of molecules in the gas. Einstein used the same thermodynamic arguments and argued that the behaviour of one quantum of energy 2hν is very different from the behaviour of two quanta of energy hν. He also argued that the data indicate that radiation of frequency ν MUST exist in the form of quanta of energy hν. It can not be in quanta of 2hν or 3hν etc. Einstein labelled these radiation quanta as photons.

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Explanation of photoelectric effect through photon hypothesis

The figure below gives a possible picture you can use to think of light with photon hypothesis. We should picture each photon as a bundle of energy. In the case of macroscopic phenomena, we must consider an Avogadro number of such photons and in such cases the collective behaviour of the photons can be described by an electromagnetic wave, which is also shown in the figure. Now let us consider how the photon hypothesis can explain different features of photoelectric effect. We assume that a given electron can only absorb only one photon at a time. This assumption is justified by later developments in quantum mechanics. They showed that the probability of interaction between an electron and a photon is very small (less than 1%). Therefore we neglect the probability of an electron absorbing two photons at the same time. Why can’t we assume that an electron absorbs a photon, goes to an excited state and then absorbs another photon? For that to happen, a

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stable excited state must exist. In metals, no such state exists. If an electron absorbs a photon with energy less than its work function, its energy increases but such an electron comes back to its ground state within a nanosecond, as shown by quantum mechanics. Therefore, we are justified in neglecting the processes where one electron can absorb two photons within a short time interval. • Existence of Critical Frequency: Electron can escape the metal only if the energy of the photon is greater than its binding energy (or the work function of the metal). If φ is the work function of the metal, the critical frequency is given by the expression νcrit =

φ . h

(2)

• Larger Photocurrent for Larger Intensity: This can be explained by classical picture also. In photon picture, the larger intensity is related to larger number of photons. If the number photons falling on the metal is larger, then larger number of electrons are released. Thus, the number of photoelectrons released is proportional to the intensity. But photocurrent is a product of the number of photoelectrons and their speed. Therefore we find that the photocurrent is not proportional to the intensity but is larger for larger intensity. • Dependence of Stopping Potential on Frequency: Suppose the photon has frequency ν > νcrit. Its energy is greater than the binding energy of the electron. The difference (Ephoton − Ebinding ) becomes the kinetic energy of the electron. Sometimes, the electron can lose some kinetic energy in coming out of the metal. So the above difference is the maximum possible kinetic energy of the electron Kmax . A fraction of photoelectrons do have this value of kinetic energy and they form a part of the photocurrent. The stopping potential is the value of potential, applied to oppose the motion of the electrons, such that the photocurrent just becomes zero. To completely stop the photocurrent, the voltage opposing the movement of electrons should be large enough so that even the maximum kinetic energy will be fully converted to potential energy. If Vs is the stopping potential and e is charge of the electron, then eVs = Kmax = hν − φ. (3) 5

This equation predicts a linear relation between the stopping potential and frequency. It was verified by Millikan, who plotted eVs vs ν for a given metal. By repeating the experiment for different metals, he verified that the above equation holds for all of them with the same slope but with different values of intercepts, as illustrated in the figure above. By measuring the slopes these graphs, Millikan also obtained an accurate value of the Planck’s constant h. • No time-lag between absorption of photons and emission of photoelectrons: The absorption of photon (energy bundle of light) by an electron can be treated as an instantaneous process. Quantum Mechanical calculations later showed the time scale of this interaction will be less than a nanosecond. Hence there is no observable time-lag between absorption of radiation and emission of photocurrent.

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