and Matter

and Matter particles. They were found to travel with speeds ranging from about . to . times the speed of light ( × m/s).

The presently accepted value of e/m is . × C/kg. Further, the value of e/m was found to be independent of the nature of the material/metal used as the cathode (emitter), or the gas introduced in the discharge tube. This observation suggested the universality of the cathode ray particles.

Around the same time, in , it was found that certain metals, when irradiated by ultraviolet light, emitted negatively charged particles having small speeds. Also, certain metals when heated to a high temperature were found to emit negatively charged particles. The value of e/m of these particles was found to be the same as that for cathode ray particles. These observations thus established that all these particles, although produced under different conditions, were identical in nature.

J. J. Thomson, in , named these particles as electrons, and suggested that they were fundamental, universal constituents of matter. For his epoch-making discovery of electron, through his theoretical and experimental investigations on conduction of electricity by gasses, he was awarded the Nobel Prize in Physics in .

In , the American physicist R. A. Millikan ( - ) performed the pioneering oil-drop experiment for the precise measurement of the charge on an electron. He found that the charge on an oil-droplet was always an integral multiple of an elementary charge, .

× – C. Millikan’s experiment established that electric charge is quantised. From the values of charge (e) and specific charge (e/m), the mass (m) of the electron could be determined. and Matter stopped.

These observations indicate that when ultraviolet radiations fall on the emitter plate C, electrons are ejected from it which are attracted towards the positive, collector plate A by the electric field. The electrons flow through the evacuated glass tube, resulting in the current flow. Thus, light falling on the surface of the emitter causes current in the external circuit. Hallwachs and Lenard studied how this photo current varied with collector plate potential, and with frequency and intensity of incident light.

Hallwachs, in , undertook the study further and connected a negatively charged zinc plate to an electroscope. He observed that the zinc plate lost its charge when it was illuminated by ultraviolet light. Further, the uncharged zinc plate became positively charged when it was irradiated by ultraviolet light. Positive charge on a positively charged zinc plate was found to be further enhanced when it was illuminated by ultraviolet light.

From these observations he concluded that negatively charged particles were emitted from the zinc plate under the action of ultraviolet light. After the discovery of the electron in , it became evident that the incident light causes electrons to be emitted from the emitter plate. Due to negative charge, the emitted electrons are pushed towards the collector plate by the electric field. Hallwachs and Lenard also observed that when ultraviolet light fell on the emitter plate, no electrons were emitted at all when the frequency of the incident light was smaller than a certain minimum value, called the threshold frequency.

This minimum frequency depends on the nature of the material of the emitter plate. It was found that certain metals like zinc, cadmium, magnesium, etc., responded only to ultraviolet light, having short wavelength, to cause electron emission from the surface. However, some alkali metals such as lithium, sodium, potassium, caesium and rubidium were sensitive even to visible light. All these photosensitive substances emit electrons when they are illuminated by light.

After the discovery of electrons, these electrons were termed as photoelectrons. The phenomenon is called photoelectric effect. and Matter polarity is reversed, the electrons are repelled and only the most energetic electrons are able to reach the collector A. The photocurrent is found to decrease rapidly until it drops to zero at a certain sharply defined, critical value of the negative potential V0 on the plate A.

For a particular frequency of incident radiation, the minimum negative (retarding) potential V0 given to the plate A for which the photocurrent stops or becomes zero is called the cut-off or stopping potential. The interpretation of the observation in terms of photoelectrons is straightforward. All the photoelectrons emitted from the metal do not have the same energy. Photoelectric current is zero when the stopping potential is sufficient to repel even the most energetic photoelectrons, with the maximum kinetic energy (Kmax), so that Kmax = e V0 ( .

) We can now repeat this experiment with incident radiation of the same frequency but of higher intensity I2 and I3 (I3 I2 I1). We note that the saturation currents are now found to be at higher values. This shows that more electrons are being emitted per second, proportional to the intensity of incident radiation. But the stopping potential remains the same as that for the incident radiation of intensity I1, as shown graphically in Fig.

. . Thus, for a given frequency of the incident radiation, the stopping potential is independent of its intensity. In other words, the maximum kinetic energy of photoelectrons depends on the light source and the emitter plate material, but is independent of intensity of incident radiation.

. . Effect of frequency of incident radiation on stopping potential We now study the relation between the frequency + of the incident radiation and the stopping potential V0. We suitably adjust the same intensity of light radiation at various frequencies and study the variation of photocurrent with collector plate potential.

The resulting variation is shown in Fig. . . We obtain different values of stopping potential but the same value of the saturation current for incident radiation of different frequencies.

The energy of the emitted electrons depends on the frequency of the incident radiations. The stopping potential is more negative for higher frequencies of incident radiation. Note from FIGURE . Variation of photocurrent with collector plate potential for different intensity of incident radiation.

FIGURE . Variation of photoelectric current with collector plate potential for different frequencies of incident radiation. Fig. .

that the stopping potentials are in the order V03 > V02 > V01 if the frequencies are in the order + > + > + . This implies that greater the frequency of incident light, greater is the maximum kinetic energy of the photoelectrons. Consequently, we need greater retarding potential to stop them completely. If we plot a graph between the frequency of incident radiation and the corresponding stopping potential for different metals we get a straight line, as shown in Fig.

. . The graph shows that (i) the stopping potential V0 varies linearly with the frequency of incident radiation for a given photosensitive material. (ii) there exists a certain minimum cut-off frequency + for which the stopping potential is zero.

These observations have two implications: (i) The maximum kinetic energy of the photoelectrons varies linearly with the frequency of incident radiation, but is independent of its intensity. (ii) For a frequency + of incident radiation, lower than the cut-off frequency + , no photoelectric emission is possible even if the intensity is large. This minimum, cut-off frequency + , is called the threshold frequency. It is different for different metals.

Different photosensitive materials respond differently to light. Selenium is more sensitive than zinc or copper. The same photosensitive substance gives different response to light of different wavelengths. For example, ultraviolet light gives rise to photoelectric effect in copper while green or red light does not.

Note that in all the above experiments, it is found that, if frequency of the incident radiation exceeds the threshold frequency, the photoelectric emission starts instantaneously without any apparent time lag, even if the incident radiation is very dim. It is now known that emission starts in a time of the order of – s or less. We now summarise the experimental features and observations described in this section. (i) For a given photosensitive material and frequency of incident radiation (above the threshold frequency), the photoelectric current is directly proportional to the intensity of incident light (Fig.

. ). (ii) For a given photosensitive material and frequency of incident radiation, saturation current is found to be proportional to the intensity of incident radiation whereas the stopping potential is independent of its intensity (Fig. .

). (iii) For a given photosensitive material, there exists a certain minimum cut-off frequency of the incident radiation, called the threshold frequency, below which no emission of photoelectrons takes place, no matter how intense the incident light is. Above the threshold frequency, the stopping potential or equivalently the maximum kinetic FIGURE . Variation of stopping potential V0 with frequency + of incident radiation for a given photosensitive material.

and Matter energy of the emitted photoelectrons increases linearly with the frequency of the incident radiation, but is independent of its intensity (Fig. . ). (iv) The photoelectric emission is an instantaneous process without any apparent time lag ( – 9s or less), even when the incident radiation is made exceedingly dim.

and Matter In Einstein’s picture, the basic elementary process involved in photoelectric effect is the absorption of a light quantum by an electron. This process is instantaneous. Thus, whatever may be the intensity i.e., the number of quanta of radiation per unit area per unit time, photoelectric emission is instantaneous. Low intensity does not mean delay in emission, since the basic elementary process is the same.

Intensity only determines how many electrons are able to participate in the elementary process (absorption of a light quantum by a single electron) and, therefore, the photoelectric current. Using Eq. ( . ), the photoelectric equation, Eq.

( . ), can be written as e V0 = h +– - ; for or V0 = h e e ( . ) This is an important result. It predicts that the V0 versus + curve is a straight line with slope = (h/e), independent of the nature of the material.

During - , Millikan performed a series of experiments on photoelectric effect, aimed at disproving Einstein’s photoelectric equation. He measured the slope of the straight line obtained for sodium, similar to that shown in Fig. . .

Using the known value of e, he determined the value of Planck’s constant h. This value was close to the value of Planck’s contant (= . × –34J s) determined in an entirely different context. In this way, in , Millikan proved the validity of Einstein’s photoelectric equation, instead of disproving it.

The successful explanation of photoelectric effect using the hypothesis of light quanta and the experimental determination of values of h and - , in agreement with values obtained from other experiments, led to the acceptance of Einstein’s picture of photoelectric effect. Millikan verified photoelectric equation with great precision, for a number of alkali metals over a wide range of radiation frequencies. and Matter EXAMPLE . eV0 = h+ – - = hc – - or, # = hc/(eV0 +- ) ( .

Js) ( m/s) ( .60eV .14eV) . J m ( .74eV) . J m nm . .

J Example . The wavelength of light in the visible region is about nm for violet colour, about nm (average wavelength) for yellow- green colour and about nm for red colour. (a) What are the energies of photons in (eV) at the (i) violet end, (ii) average wavelength, yellow-green colour, and (iii) red end of the visible spectrum? (Take h = .

× – J s and eV = . × –19J.) (b) From which of the photosensitive materials with work functions listed in Table . and using the results of (i), (ii) and (iii) of (a), can you build a photoelectric device that operates with visible light? Solution (a) Energy of the incident photon, E = h+ = hc/# E = ( .

× –34J s) ( × m/s)/# – . J m (i) For violet light, # = nm (lower wavelength end) Incident photon energy, E1 = – – . J m × = . × –19J – – .

J . × J/eV = . eV (ii) For yellow-green light, # = nm (average wavelength) Incident photon energy, E2 = – – . J m × = .

× – J = . eV (iii) For red light, # = nm (higher wavelength end) Incident photon energy, E3 = – – . J m × = . × – J = .

eV (b) For a photoelectric device to operate, we require incident light energy E to be equal to or greater than the work function - of the material. Thus, the photoelectric device will operate with violet light (with E = . eV) photosensitive material Na (with - = . eV), K (with - = .

eV) and Cs (with - = . eV). It will also operate with yellow-green light (with E = . eV) for Cs (with - = .

eV) only. However, it will not operate with red light (with E = . eV) for any of these photosensitive materials. EXAMPLE .

and Matter wavelength extends all over space. By Born’s probability interpretation this means that the electron is not localised in any finite region of space. That is, its position uncertainty is infinite (x ), which is consistent with the uncertainty principle. In general, the matter wave associated with the electron is not extended all over space.

It is a wave packet extending over some finite region of space. In that case x is not infinite but has some finite value depending on the extension of the wave packet. Also, you must appreciate that a wave packet of finite extension does not have a single wavelength. It is built up of wavelengths spread around some central wavelength.

By de Broglie’s relation, then, the momentum of the electron will also have a spread – an uncertainty p. This is as expected from the uncertainty principle. It can be shown that the wave packet description together with de Broglie relation and Born’s probability interpretation reproduce the Heisenberg’s uncertainty principle exactly. In Chapter , the de Broglie relation will be seen to justify Bohr’s postulate on quantisation of angular momentum of electron in an atom.

Figure . shows a schematic diagram of (a) a localised wave packet, and (b) an extended wave with fixed wavelength. Example . What is the de Broglie wavelength associated with (a) an electron moving with a speed of .

m/s, and (b) a ball of mass g travelling at . m/s? Solution (a) For the electron: Mass m = . – kg, speed v = .

m/s. Then, momentum p = m v = . – (kg) . (m/s) p = .

– kg m/s de Broglie wavelength, # = h/p = . . – – Js kg m/s #= . nm (b) For the ball: Mass m ’ = .

kg, speed v ’ = . m/s. Then momentum p’ = m’ v’ = . (kg) .

(m/s) p ’= . kg m/s de Broglie wavelength #’ = h/p’ FIGURE . (a) The wave packet description of an electron. The wave packet corresponds to a spread of wavelength around some central wavelength (and hence by de Broglie relation, a spread in momentum).

Consequently, it is associated with an uncertainty in position (x) and an uncertainty in momentum (p). (b) The matter wave corresponding to a definite momentum of an electron extends all over space. In this case, p = and x EXAMPLE . EXAMPLE .

EXAMPLE . ± . . Js kg m/s #’= .

– m The de Broglie wavelength of electron is comparable with X-ray wavelengths. However, for the ball it is about – times the size of the proton, quite beyond experimental measurement. Example . An electron, an (-particle, and a proton have the same kinetic energy.

Which of these particles has the shortest de Broglie wavelength? Solution For a particle, de Broglie wavelength, # = h/p Kinetic energy, K = p2/2m Then, / h mK For the same kinetic energy K, the de Broglie wavelength associated with the particle is inversely proportional to the square root of their masses. A proton 1H is times massive than an electron and an (-particle 2He four times that of a proton. Hence, ( – particle has the shortest de Broglie wavelength.

EXAMPLE . and Matter h h p mv Mass, m = h/#v For an electron, mass me = h/#e ve Now, we have v/ve = and #/#e = . × – Then, mass of the particle, m = me e e v v m = ( . × – kg) × ( / ) × ( / .

× – ) m = . × – kg. Thus, the particle, with this mass could be a proton or a neutron. Example .

What is the de Broglie wavelength associated with an electron, accelerated through a potential differnece of volts? Solution Accelerating potential V = V. The de Broglie wavelength # is # = h /p . V nm # .

nm = . nm The de Broglie wavelength associated with an electron in this case is of the order of X-ray wavelengths. and Matter SUMMARY . The minimum energy needed by an electron to come out from a metal surface is called the work function of the metal.

Energy (greater than the work function - ) required for electron emission from the metal surface can be supplied by suitably heating or applying strong electric field or irradiating it by light of suitable frequency. . Photoelectric effect is the phenomenon of emission of electrons by metals when illuminated by light of suitable frequency. Certain metals respond to ultraviolet light while others are sensitive even to the visible light.

Photoelectric effect involves conversion of light energy into electrical energy. It follows the law of conservation of energy. The photoelectric emission is an instantaneous process and possesses certain special features. .