Table of Contents
INTRODUCTION OF QUANTUM THEORY OF RADIATION
Photon: We know that photoelectric action cannot be explained by wave theory. For this reason, in 1905, scientist Einstein expanded Planck’s quantum theory and introduced the concept of photon particles and explained the photoelectric effect. This particle theory of radiation is now known as quantum theory of radiation. The key to this theory is this: “Electromagnetic radiation is not a wave, but a stream of particles called photons.”
Properties of Photon:
- Every photon particle is neutral
- Each photon particle travels at the speed of light (c = 3 X 10⁸ m/s), in no case does any decrease in the photon’s velocity increase.
- The amount of energy carried by a photon is E = h𝛎; where 𝛎 = frequency of radiation, h = Planck’s constant. As the number of photon particles in the photon stream increases, so does the amount of energy carried, thereby increasing the brightness of the radiation.
- The photons or rest mass is zero.
- According to relativistic theory, if a particle has rest mass m0 and momentum p, the energy of the particle, E = √p²c²+m0²c⁴, is m0 = 0 for the photon; Hence, E = pc or, p = E/c = h𝛎/c. That is, the photon is a massless particle but has definite momentum.
Planck’s Constant: Planck’s constant is a universal constant. It is denoted by h.
SI unit of h = (unit of E / unit of 𝛎) = J/s-1 = J·s.
In the CGS method unit of h is erg·s.
This unit is identical to the unit of angular momentum. So, h is actually a measure of angular momentum.
Value of h = 6.625 X 10-34 J·s = 6.625 X 10-27 erg·s = 4.14 X 10-15 eV·s.
RELATION BETWEEN THE WAVELENGTH OF RADIATION AND THE PHOTON ENERGY
Energy of a Photon, E = h𝛎 = hc/λ;
1Å = 10-10 m and 1eV = 1.6 X 10-19 J
So, expressing E in eV units and λ in Å units,
λ Å = λ X 10-10 m, E eV = E X (1.6 X 10-19) J
So, E X (1.6 X 10-19) = (6.625 X 10-34) X (3 X 108) / (λ X 10-10)
Or, E = (6.625 X 10-34 X 3 X 108) / (1.6 X 10-19 X λ X 10-10) = 12422/λ
Generally, the number is taken as 12400. In that case it can be written,
E (in eV unit) = 12400/ λ (in Å unit)
EINSTEIN'S PHOTOELECTRIC EQUATION
Einstein developed an equation that describes the effects of photoelectric effects based on Planck’s quantum theory. This equation is known as Einstein’s Photoelectric Equation. Einstein states that light emits quantum energy together with a potential quota of photons and energy. In this case, each photon strikes an electron inside the atom, giving the electron energy in the process. The photoelectric equation takes into account Planck’s quantum theory of radiation, which ultimately emerges at the surface at the kinetic energy of the electron’s emission from the atom.
Postulates of Einstein
- A beam of light is incident on a metal surface as a stream of photon particles. Energy of each particle for light of frequency 𝛎, E = h𝛎 (h = Planks Constant)
- The incident photon collides with the electrons of the metal. This collision can have two outcomes: 1. Either the photon will be reflected with the entire hv energy, 2. or the entire h𝛎 energy will be transferred to the electron.
Clearly, Einstein fully utilized the quantum theory of radiation in his analysis of photoelectric action.
When all of the energy h𝛎 of the incident photon is transferred to the electrons in the metal, it is spent in two ways:
- A part is used to remove the electron from the metal. Its minimum value is equal to the work-function W0 of the metal surface. However, the interaction of positive and negative charges between metals required more energy than W0 to emit most electrons.
- The remainder is converted into kinetic energy of the emitted electrons; These moving electrons are photoelectrons, which can cause photoelectric current. If the energy absorbed by the electron to leave the metal surface is minimum i.e. W0, then the ejected electron attains the maximum kinetic energy (Emax).
So, h𝛎 = W0 + Emax
or, Emax = h𝛎 – W0 …………(i)
If, Mass of electron = m & maximum velocity of Photoelectron = Vmax
Emax = 1/2mv²max
So, from equation no (i),
1/2mv²max = h𝛎 – W0 ……….(ii)
Again, for light of frequency v, if the value of the evanescent potential is V0, we know that Emax = eV0 (e = charge of an Electron). Hence, from equation (i) no.,
eV0 = h𝛎 – W0 ………(iii)
Equations (i), (ii), (iii) above are practically identical. Either of these is called Einstein’s photoelectric equation.
In most cases the photon collides with the electron but no energy is transferred, i.e. the photon returns with its full hv energy. Because of this, the probability of photoelectron emission from the metal surface is low, so the photocurrent value is almost never very high.
EXPLANATION OF PHOTOELECTRIC EFFECT BY QUANTUM THEORY OF RADIATION
Einstein’s photoelectric equation is based on the quantum theory of radiation. This equation correctly explains the following aspects of photoelectric action, which cannot be derived from the wave theory of light or radiation.
(I) Maximum Kinetic Energy of Photoelectrons: The specific metal surface work-function (W0) is constant; again, the frequency of monochromatic light, 𝛎 = constant. Hence from Einstein’s equation Emax = h𝛎 – W0, Emax = constant. That is, no matter how much the intensity is increased keeping the wavelength and frequency of the incident light constant, the photoelectrons can never attain a kinetic energy greater than Emax.
(II) Starting Frequency: W0 = constant of specific metal surface. Now if the incident light frequency 𝛎 is reduced, the equation Emax = h𝛎 – W0 shows that the value of Emax will eventually decrease to zero. In that case if frequency = V0,
0 = h𝛎0 – W0 or, h𝛎0 = W0
or, 𝛎0 = W0/h ……….(i)
The kinetic energy of a photoelectron can never be negative. Therefore, if the frequency of light is less than V0, no photocurrent is emitted; that is, this 𝛎0 is the starting frequency. Putting h𝛎0 = W0 into Einstein’s equation Emax = h𝛎 – W0 gives,
Emax = h𝛎 – h𝛎0
or, Emax = h (𝛎- 𝛎0) ………(ii)
This equation (ii) is also an alternative form of Einstein’s photoelectric equation.
(III) Instantaneous Behavior of Photoelectric Action: Direct elastic collisions of electrons with photons of energy hv transfer energy. Therefore, there is no time lag between the incident light beam and the emission of photoelectrons.
(IV) Dependence of Luminous Flux on Intensity of Incident Light: Increasing the incident light intensity while keeping the frequency constant increases the number of incident photons on the metal sheet. Increasing the number of incident photons also increases the number of collisions between photons and electrons in the metal, resulting in more electrons being ejected and increasing the photocurrent. This is consistent with the experimental results.
Expressing Einstein's Equation Graphically
From Einstein’s photoelectric equation, eV0 = h𝛎 – W0 or, V0 = h𝛎/e – W0/e. Plotting the induced potential V0 against the frequency 𝛎 gives a straight line of form y = mx + c. If the electron charge e is known, the Planck constant h from the slope h/e of this straight line and the work function W0 of the metal surface from the intercept -W0/e on the vertical axis can be determined. Also, the intercept of the horizontal axis is the starting frequency 𝛎0.
Note that the slope h/e of the straight line is the same for all metals, but the intercepts of the horizontal and vertical axes, 𝛎0 and -W0/e respectively, are different for different metals.
CONCLUSIONS
In summary, radiation’s role in both quantum and kinetic energy emission may be well explained by Einstein’s photoelectric equation in quantum theory of radiation. Additionally, the investigation has determined that, in terms of electron ejection during the impact, the photoelectric equation has conserved energy. In contrast, it can be argued that the photoelectric equation primarily relies on photon radiation to maintain kinetic energy, which ultimately increases its success value in real-time physics. In terms of effectively resolving the complexity of quantum energy emission, the equation also works. In summary, this formula effectively captures the collision between electrons and photons inside quantum energy.