Class Physics English Volume

ν 0B Figure . Variation of stopping potential with frequency of the incident radiation for two metals Now a graph is drawn between frequency of incident radiation and the stopping potential for different metals (Figure . ). From this graph, it is found that stopping potential varies linearly with frequency.

Below a certain frequency called threshold frequency, no electrons are emitted; hence stopping potential is zero for that reason. But as the frequency is increased above threshold value, the stopping potential varies linearly with the frequency of incident light. . .

Laws of photoelectric effect The above detailed experimental investigations of photoelectric effect revealed the following results: i) For a given metallic surface, the emission of photoelectrons takes place only if the frequency of incident light is greater Unit dual nature of radiation and matter For the sake of simplicity, the following standard assumptions can be made when light is incident on the given material. a) Light is absorbed in the top atomic layer of the metal b) For a given element, each atom absorbs an equal amount of energy and this energy is proportional to its cross-sectional area A c) Each atom gives this energy to one of the electrons. (Given : The work function for cesium is . eV and the power absorbed per unit area is Wm which produces a measurable photocurrent in cesium.) Solution i) According to wave theory, the energy in a light wave is spread out uniformly and continuously over the wavefront.

The energy absorbed by each electron in time t is given by E = IAt With this energy absorbed, the most energetic electron is released with K max by overcoming the surface energy barrier or work function ϕ and this is expressed as K IAt max = − ϕ ( ) Thus, wave theory predicts that for a unit time, at low light intensities when IA < ϕ , no electrons are emitted. At higher intensities, when IA ≥ ϕ , electrons are emitted. This implies that higher the light intensity, greater will be K max . K max is dependent only on the intensity under given conditions – that is, by suitably increasing the intensity, one can produce But this does not happen.

The experiments show that maximum kinetic energy of the photoelectrons emitted does not depend on the intensity of the incident light. ii) According to wave theory, if a sufficiently intense beam of light is incident on the surface, electrons should be liberated from the surface of the target, however low the frequency of the radiation is. From the experiments, it is found that photoelectric emission is not possible below a certain minimum frequency of incident radition. Therefore, the wave theory fails to explain the existence of threshold frequency.

iii) Since the energy of light is spread across the entire wavefront, the electrons which receive energy from it are large in number. Each electron needs considerable amount of time (a few hours) to get energy sufficient to overcome the work function and to get liberated from the surface. But experiments show that photoelectric emission is almost instantaneous process (the time lag is less than – s after the surface is illuminated) which could not be explained by wave theory. Thus, the experimental observations of photoelectric emission could not be explained on the basis of the wave theory of light.

EXAMPLE . For the photoelectric emission from cesium, show that wave theory predicts that i) maximum kinetic energy of the photoelectrons ( K max ) depends on the intensity I of the incident light ii) K max does not depend on the frequency of the incident light and iii) the time interval between the incidence of light and the ejection of photoelectrons is very long. Unit dual nature of radiation and matter radiations emitted by a black body and the shape of its radiation curves. According to Planck, matter is composed of a large number of oscillating particles (atoms) which vibrate with different frequencies.

Each atomic oscillator - which vibrates with its characteristic frequency - emits or absorbs electromagnetic radiation of the same frequency. It also says that i) If an oscillator vibrates with frequency v , its energy can have only certain discrete values, given by the equation. E n = nhν n= , , .... ( .

) where h is a constant, called Planck’s constant. ii) The oscillators emit or absorb energy in small packets or quanta and the energy of each quantum is hν . This implies that the energy of the oscillator is quantized – that is, energy is not continuous as believed in the wave picture. This is called quantization of energy .

. . Particle nature of light: Einstein’s explanation Einstein extended Planck’s quantum concept to explain the photoelectric effect in . According to Einstein, the energy in light is not spread out over wavefronts but is concentrated in small packets or energy quanta.

Therefore, light (or any other electromagnetic waves) of frequency v from any source can be considered as a stream of quanta and the energy of each light quantum is given by E = hν . He also proposed that a quantum of light has linear momentum and the magnitude of that linear momentum is h c ν p = . The individual light quantum of definite energy and momentum can be associated with a photoelectric effect even if the frequency is less than the threshold frequency. So the concept of threshold frequency does not even exist in wave theory.

ii) According to wave theory, the intensity of a light wave is proportional to the square of the amplitude of the electric field ( ). E The amplitude of this electric field increases with increasing intensity and imparts an increasing acceleration and kinetic energy to an electron. Now I is replaced with a quantity proportional to E in equation ( ). This means that K max should not depend at all on the frequency of the classical light wave which again contradicts the experimental results.

iii) If an electron accumulates light energy just enough to overcome the work function, then it is ejected out of the atom with zero kinetic energy. Therefore, from equation ( ), IAt ϕ t IA r ϕ ϕ π By taking the atomic radius r = m and substituting the given values of I and ϕ , we can estimate the time interval as t = × × ≈ s days Thus, wave theory predicts that there is a large time gap between the incidence of light and the ejection of photoelectrons but the experiments show that photo emission is an instantaneous process. Concept of quantization of energy Max Planck proposed quantum concept in in order to explain the thermal Unit dual nature of radiation and matter Einstein’s explanation of photoelectric equation When a photon of energy hν is incident on a metal surface, it is completely absorbed by a single electron and the electron is ejected. In this process, a part of the photon energy is used in overcoming the potential barrier of the metal surface (photoelectric work function ϕ )and the remaining energy as the kinetic energy of the ejected electron.

From the law of conservation of energy, h ϕ ( . ) where m is the mass of the electron and v its velocity. This is shown in Figure . (a).

K = Metal Metal E = hv (a) (b) E = hv K max = hv – hv Figure . Emission of photoelectrons If we reduce the frequency of the incident light, the speed or kinetic energy of photo electrons is also reduced. At some frequency ν of incident radiation, the photo electrons are just ejected with almost zero kinetic energy (Figure . (b)).

Then the equation ( . ) becomes hv = ϕ where ν is the threshold frequency. By particle. The light quantum can behave as a particle and this is called photon.

Therefore, photon is nothing but particle manifestation of light. Characteristics of photons: According to particle nature of light, photons are the basic constituents of any radiation and possess the following characteristic properties: i) The photons of light of frequency ν and wavelength λ will have energy, given by E hv hc = l . ii) The energy of a photon is determined by the frequency of the radiation and not by its intensity and the intensity has no relation with the energy of the individual photons in the beam. iii) The photons travel with the speed of light and its momentum is given by p h hv c iv) Since photons are electrically neutral, they are unaffected by electric and magnetic fields.

v) When a photon interacts with matter (photon-electron collision), the total energy, total linear momentum and angular momentum are conserved. Since photon may be absorbed or a new photon may be produced in such interactions, the number of photons may not be conserved. According to quantum concept, intensity of light of given wavelength is defined as the number of energy quanta or photons incident per unit area per unit time, with each photon having same energy. Its unit is Wm – .

Note Unit dual nature of radiation and matter Slope = h O Cesium Potassium Sodium Calcium Kmax Frequency − .30eV − .14eV − .75eV − .20eV Figure . K max vs ν graph for different metals Millikan also calculated the value of Planck’s constant ( h = . × – Js) and work function of many metals (Cs, K, Na, Ca); these values are in agreement with the theoretical prediction. Explanation for the photoelectric effect: The experimentally observed facts of photoelectric effect can be explained with the help of Einstein’s photoelectric equation.

i) As each incident photon liberates one electron, then the increase of intensity of the light (the number of photons per unit area per unit time) increases the number of electrons emitted thereby increasing the photocurrent. The same has been experimentally observed. ii) From K max = hv – ϕ , it is evident that K max is proportional to the frequency of the incident light and is independent of intensity of the light. iii) As given in equation ( .

), there must be minimum energy (equal to the work function of the metal) for incident photons to liberate electrons from the metal surface. Below this value of energy, emission of electrons is not possible. Correspondingly, there exists minimum frequency called threshold frequency below which there is no photoelectric emission. rewriting the equation ( .

), we get hv hv v ( . ) The equation ( . ) is known as Einstein’s photoelectric equation . If the electron does not lose energy by internal collisions, then it is emitted with maximum kinetic energy K max .

Then K max max = where v max is the maximum velocity of the electron ejected. The equation ( . ) is rearranged as follows: K max = hv – ϕ ( . ) Kmax hv hv v Frequency Figure .

K max vs ν graph A graph between maximum kinetic energy K max of the photoelectron and frequency ν of the incident light is a straight line as shown in Figure . . The slope of the line is h and its y-intercept is – ϕ . Einstein’s equation was experimentally verified by R.A.

Millikan. He drew K max versus ν graph for many metals (cesium, potassium, sodium and lithium) as shown in Figure . and found that the slope is independent of the metals. Unit dual nature of radiation and matter .

. Photo electric cells and their applications Photo cell Photo electric cell or photo cell is a device which converts light energy into electrical energy. It works on the principle of photo electric effect. When light is incident on the photosensitive materials, their electric properties will get affected, based on which photo cells are classified into three types.

They are i) Photo emissive cell: Its working depends on the electron emission from a metal cathode due to irradiation of light or other radiations. ii) Photo voltaic cell: Here sensitive element made of semiconductor is used which generates voltage proportional to the intensity of light or other radiations. iii) Photo conductive cell: In this, the resistance of the semiconductor changes in accordance with the radiant energy incident on it. In this section, we discuss about photo emissive cell and its applications.

Photo emissive cell Construction: It consists of an evacuated glass or quartz bulb in which two metallic electrodes – that is, a cathode and an anode are fixed as shown in Figure . . iv) According to quantum concept, the transfer of photon energy to the electrons is instantaneous so that there is no time lag between incidence of photons and ejection of electrons. Thus, the photoelectric effect is explained on the basis of quantum concept of light.

The nature of light: wave - particle duality We have learnt that wave nature of light explains phenomena such as interference, diffraction and polarization. Certain phenomena like black body radiation, photoelectric effect can be explained by assigning particle nature to light. Therefore, both theories have enough experimental evidences. In the past, many scientific theories have been either revised or discarded when they contradicted with new experimental results.

Here, two different theories are needed to answer the question: what is nature of light? It is therefore concluded that light possesses dual nature, that of both particle and wave. It behaves like a wave at some circumstances and it behaves like a particle at some other circumstances. In other words, light behaves as a wave during its propagation and behaves as a particle during its interaction with matter.

Both theories are necessary for complete description of physical phenomena. Hence, the wave nature and quantum nature complement each other. A reader may find it difficult to understand how light can be both a wave and a stream of particle. This is the case even for great scientist like Albert Einstein.

Einstein once wrote a letter to his friend Michel Besso in expressing his frustration: “All these fifty years of conscious brooding have brought me no closer to answer the question, ‘What are light quanta?’ Of course today everyone thinks he knows the answer, but he is deluding himself”. Unit dual nature of radiation and matter and switch off according to whether it is night or day use photocells. Photo cells are used for reproduction of sound in motion pictures and are used as timers to measure the speeds of athletes during a race. Photo cells of exposure meters in photography are used to measure the intensity of the given light and to calculate the exact time of exposure.

EXAMPLE . A radiation of wavelength nm is incident on a silver surface. Will photoelectrons be observed? [work function of silver = .

eV] Solution: Energy of the incident photon is E hv hc = λ (in joules) E hc = λ (in eV) Substituting the known values, we get E = × × E = . eV The work function of silver = . eV. Since the energy of the incident photon is less than the work function of silver, photoelectrons are not observed in this case.

EXAMPLE . When light of wavelength 2200Å falls on Cu, photo electrons are emitted from it. Find (i) the threshold wavelength and (ii) the stopping potential. Given: the work function for Cu is ϕ = .

eV. The cathode C is semi-cylindrical in shape and is coated with a photo sensitive material. The anode A is a thin rod or wire kept along the axis of the semi-cylindrical cathode. A potential difference is applied between the anode and the cathode through a galvanometer G .

_ + G C Radiation Figure . Construction of photo cell Working: When cathode is irradiated with suitable radiation, electrons are emitted from it. These electrons are attracted by anode and hence a current is produced which is measured by the galvanometer. For a given cathode, the magnitude of the current depends on i) the intensity of incident radiation and ii) the potential difference between anode and cathode.

Applications of photo cells: Photo cells have many applications, especially as switches and sensors. Automatic lights that turn on when it gets dark use photocells, and street lights that switch on Unit dual nature of radiation and matter ii) The number of photons reaching the surface per second is E p = × × photons / sec The rate of emission of photoelectrons is = ( n p = . photoelectrons / sec ´ EXAMPLE . Light of wavelength nm is directed at a metal electrode.

To find the energy of electrons ejected, an opposing potential difference is established between it and another electrode. The current of photoelectrons from one to the other is stopped completely when the potential difference is . V. Determine i) the work function of the metal and ii) the maximum wavelength of light that can eject electrons from this metal.

Solution i) The work function is given by ϕ = hv – K max λ hc eV since K max = eV × ×      −   J = eV ii) The threshold wavelength is Å λ ϕ × × hc Solution i) The threshold wavelength is given by λ ϕ . × × hc = Å ii) Energy of the photon of wavelength Å is E hc × × λ J eV We know that kinetic energy of fastest photo electron is K max = hv – ϕ = . – . = eV From equation ( .

), K eV max = V K = × max Therefore, stopping potential = V EXAMPLE . The work function of potassium is . eV. UV light of wavelength Å and intensity Wm – is incident on the potassium surface.

i) Determine the maximum kinetic energy of the photo electrons ii) If % of incident photons produce photo electrons, how many electrons are emitted per second if the area of the potassium surface is cm ? Solution i) The energy of the incident photon is E hc × × λ E = J eV Maximum KE of the photoelectrons is K max = hv – ϕ = . – . = .

eV Unit dual nature of radiation and matter . . De Broglie wave length: The momentum of photon of frequency ν is given by p hv c h c λ λ since The wavelength of a photon in terms of its momentum is λ = h p ( . ) According to de Broglie, the above equation is completely a general one and this is applicable to material particles as well.

Therefore, for a particle of mass m travelling with speed v , the wavelength is given by λ = h h p ( . ) This wavelength of the matter waves is known as de Broglie wavelength . This equation relates the wave character (the wave length λ ) and the particle character (the momentum p ) through Planck’s constant. .

. De Broglie wave length of electrons: Let an electron of mass m be accelerated through a potential difference of V volt. The kinetic energy acquired by the electron is given by eV v = Therefore, the speed υ of the electron is v = eV ( . ) Hence, the de Broglie wavelength of the matter waves associated with electron is .

MATTER WAVES . . Introduction - Wave nature of particles So far, we learnt that the characteristics of particles and waves are different. A wave is specified by its frequency, wavelength, wave velocity, amplitude and intensity.

It spreads out and occupies a relatively large region of space. A particle specified by its mass, velocity, momentum and energy occupies a definite position in space and is very small in size. Classical physics treated particles and waves as distinct entities. But quantum theory suggested dual character for radiations – that is, radiation behaves as a wave at times and as a particle at other times.

From this wave – particle duality of radiation, the concept of wave nature of matter arises which we will see in this section. De Broglie wave: The wave–particle duality of radiation was extended to matter by a French physicist Louis de Broglie (pronounced as de Broy) in . Greatly influenced by the symmetry in nature, de Broglie suggested that if radiation like light can act as particles at times, then material particles like electrons can also act as waves at times. According to de Broglie hypothesis, all material particles like electrons, protons, neutrons in motion are associated with waves.

These waves are called de Broglie waves or matter waves. Unit dual nature of radiation and matter H.T. L.T. F Ni crystal Scattered beam Incident beam Electron detector Aluminium cylinder Tin aluminium diaphragms Electron gun θ Figure .

Experimental set up of Davisson – Germer experiment The electrons scattered by Ni atoms in different directions are received by the electron detector which measures the intensity of scattered electron beam. The detector is capable of rotation in the plane of the paper so that the angle θ between the incident beam and the scattered beam can be changed at our will. The intensity of the scattered electron beam is measured as a function of the angle θ . ° ° ° ° ° Intensity of diffracted electron beam V = V θ Figure .

Variation of intensity of diffracted electron beam with the angle θ λ = h h emV Substituting the known values in the above equation, we get λ = V V m (or) Å λ = V For example, if an electron is accelerated through a potential difference of 100V, then its de Broglie wavelength is . Å. Since the kinetic energy of the electron, K = eV , then the de Broglie wavelength associated with electron can be also written as λ = h mK ( . ) .

. Davisson – Germer experiment Louis de Broglie hypothesis of matter waves was experimentally confirmed by Clinton Davisson and Lester Germer in . They demonstrated that electron beams are diffracted when they fall on crystalline solids. Since crystal can act as a three-dimensional diffraction grating for matter waves, the electron waves incident on crystals are diffracted off in certain specific directions.

Figure . shows a schematic representation of the apparatus for the experiment. The filament F is heated by a low tension (L.T.) battery. Electrons are emitted from the hot filament by thermionic emission.

They are then accelerated due to the potential difference between the filament and the anode aluminium cylinder by a high tension (H.T.) battery. Electron beam is collimated by using two thin aluminium diaphragms and is allowed to strike a single crystal of Nickel. ( . ) Unit dual nature of radiation and matter Figure .

shows the variation of intensity of the scattered electrons with the angle θ for the accelerating voltage of V . For a given accelerating voltage V , the It is to be noted that electrons are not the only particles with which wave nature can be demonstrated. The waves are associated with particles like neutrons and alpha particles also when they are in motion. They undergo diffraction when they are scattered by suitable crystals.

Neutron diffraction studies are highly useful for investigating crystal structures. Note Diffraction is one of the properties of waves. Whenever waves are incident on an obstacle, they bend around the edges of the obstacle. This bending of waves is called diffraction.

The amount of bending depends on the wavelength of the waves. We have learnt in unit that as the wavelength of light is very small, diffraction effects of light are very small. In order to study diffraction of light, diffraction gratings are used. Since x-rays and de Broglie waves of electrons have wavelengths (in the order of – m) much shorter than that of the light wave, diffraction grating cannot be used in x-ray diffraction studies.

In a crystal, the spacing between atomic planes is comparable to the wavelength of x-rays and de Broglie waves of electrons. Hence, in x-ray diffraction studies, the crystals are used which serve as three-dimensional grating. Note scattered wave shows a peak or maximum at an angle of ° to the incident electron beam. This peak in intensity is attributed to the constructive interference of electrons diffracted from various atomic layers of the target material.

From the known value of interplanar spacing of Nickel, the wavelength of the electron wave was experimentally calculated as .65Å. The wavelength can also be calculated from de Broglie relation for V = V from equation ( . ). λ λ V Å Å Å This value agrees very well with the experimentally observed wavelength of .65Å.

Thus this experiment directly verifies de Broglie’s hypothesis of the wave nature of moving particles. . . Electron Microscope Principle This is the direct application of wave nature of particles.