# PHYSICS FOR 6- ATOMIC PHYSICS- QUANTUM

PHYSICS FOR 6- ATOMIC PHYSICS- QUANTUM

UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU: 0787237719

## PHYSICS FOR 6- ATOMIC PHYSICS- QUANTUM

PLANCK’S QUANTUM THEORY OF BLACK BODY RADIATION

The findings in the black body radiation led Max plank in 1901 to postulate that radiant energy is quantized i.e. it is radiated in form of energy packets.

BASIC POSTULATES OF THE PLANK’S THEORY

1.      Any radiation is associated with energy.

2.      Radiant energy is emitted or absorbed in small packets known as quanta.

3.      The energy associated with a quantum is proportional to the frequency f  of the radiation

4.      The energy is absorbed or emitted only in whole number of quanta

A blackbody is a substance that absorbs all light fall on it and does not reflect any light.

It is not easy to get a black body however a sealed metal box with a very small hole on it is very close to a black body.

From law of physics it follows that a good absorber of radiation also is a good radiator.  A black body is supposed to be the best radiator.

When a black body is heated it emits light. The colour of light emitted changes from red  to yellow then to white as the temperature is increased.

The change in colour with temperature shows that the frequency changes with temperature.

This in contradiction with the classical wave theory since in the classical wave theory energy is uniformly distributed over the wave form when heating the black body the colour of radiation should stay the same

Only the intensity is supposed to increase with temperature.

DISTRIBUTION OF ENERGY IN THE SPECTRUM OF A BLACK BODY

Lamer and Pringshein investigated the distribution of energy amongst the different wavelength of a thermal spectrum of a back body radiation

The results obtained by Lamer and Pringshein are shown in figure below

Results:

1.   At a given temperature the energy is not uniformly distributed in the radiation spectrum of a hot body

2.   At a given temperature the intensity of radiations increases with increased in wavelength  and at a particular  wavelength λ its value is maximum with further increase in wavelength   the intensity of heat  radiations decreased

3.    With increase in temperature wave length  increases, wavelength emission of energy takes place.

The points on the dotted line represent wavelength at various temperatures

4.     For all  wavelength an increase in temperature causes an increase in the energy emission The area  under each curve represents the total energy  emitted for the complete spectrum at a  particular temperature

5.    This area increases in temperature of the body. It is found that the area is directly proportional to fourth power of the temperature of the body

This represents Stefan’s Boltzmann’s law.

Plank’s constant

Plank’s constant is a fundamental constant equal to the ratio of the quantum energy to the frequency of the radiation.

ELECTRON EMISSION

This is the liberation of electron from the surface of a substance.

For electron emission metals are used because they have many free electrons.

If a piece of metal is investigated at room temperature the random motion of free electrons is as shown in figure below

However these electrons are free only to the extent that they may transfer from one atom to another within the metal but they cannot leave the metal surface to provide electron emission.

It is because the free electron that start at the surface of metal find behind them positive nuclei pulling them back and none  pulling forward.

This at the surface of a metal a free electron encounters force that prevents it to leave the metal

In other words the metallic surface  offers a barrier to free electrons and is known as surface barriers

However if sufficient external energy is given to the free electron its kinetic energy is increased and thus electron will cross over the surface barrier to leave the metal.

WORK FUNCTION OF THE METAL

This is the additional energy required by an electron to overcome the surface barrier of the metal.

Or

This is the minimum energy required by an electron to just escape from the metal surface.

The work function depends on

i)  Nature of metal

ii) Conditions of the metal surface

It is measured by a smaller unit of energy called electron volt. (eV) because this is the conventional unit of energy i.e. joule is very large for computations in atomic and nuclear physics.

Electron volt.

One electron volt is the amount of energy acquired by an electron when it is accelerated through a potential difference of IV

Since potential difference V

Work done = QV

For an electron

The electron volt is the kinetic energy gained by an electron being accelerated by a potential difference of one volt.

The work function of pure metal varies roughly from V to V as shown in table below

 Metal Work Function Wo (eV) Cs 2.14 K 2.30 Na 2.75 Ca 3.20 Mo 4.17 Pb 4.25 Al 4.28 Hg 4.49 Cu 4.65 Ag 4.70 Ni 5.15 Pt 5.65

It is clear from the table above that the work function of platinum is the highest while it is lowest for Cesium.

It is desirable that metal used for electron emission should have low work function so that a small amount of energy is required to cause emission of electrons

PHOTOELECTRIC EFFECT

Photoelctric effect is the phenomenon of emission of electron from a metallic surface when radiation of suitable frequency falls on it or is the phenomenon where electromagnetic radiation of certain frequency when incident on certain material liberates electron from the surface of the material.

Photo emission

Photo emission is the emission of electron by electromagnetic radiation.

Photo electrons

These are emitted or rejected electrons from the surface of the cathode.

Photo electric effect is a general phenomenon exhibited by all substances but is most easily observed with metals.

When radiation of suitable frequency the threshold frequency is incident on a metallic surface electrons are emitted from the metal surface.

The threshold frequency is different for metals.

Certain alkali, metals e.g. sodium potassium, calcium show photo electric effect when visible light falls on them.

However, metal like zinc, calcium, magnesium etc show photo electric effect to ultra violet light.

Threshold Frequency

The threshold frequency is the minimum frequency of the incident radiation which is just sufficient to eject photo electron from surface of a metal Or is the minimum frequency of radiation below which no photo electron emission occurs.

It is denoted by

Illuminating a metal surface with light of frequency less than  will not cause ejection of photo electrons, no matter now great is   the intensity of radiation.

But illumination with a frequency greater than    causes emission   of photo electrons even if the radiation intensity is very small

Threshold wavelength

The threshold wavelength is the maximum wavelength of the incident radiation at which photo electric emission occurs.

It is the wavelength corresponding to threshold frequency

If the wavelength of the incident radiation is greater than threshold wavelength then there will be no photo electric emission.

Photoelectric current is the photo-electrons emitted per second

EXPERIMENTAL STUDY OF PHOTO ELECTRIC EFFECT

Figure below shows the experimental set up for studying the photoelectric effect.

The arrangement consists of an evacuated glass or quartz tube inclusive a photosensitive cathode C and metallic A.

A transparent window W is sealed onto the glass tube which can be covered with different filters to obtain the desired frequency.

The anode and cathode  are connected to a battery through a potential divided by which potential difference between anode and cathode can be changed.

The reversing switch RS tends to make anode positive or negative with  respect  to cathode.

The P.D between anode and cathode is measured by the voltmeter V while photoelectric current is indicated   by the micro ammeter.

1.
1.Effect of intensity of light on photo electric current

The anode A is maintained at positive potential with  respect  to cathode C and a radiation of suitable frequency (above threshold frequency) is incident on cathode C.

As a result photo electric current is set up.

Keeping the frequency of incident radiation and accelerating potential fixed the intensity of the incident radiation is changed in steps.

For each value of intensity of radiations the corresponding value of photo electron current is noted.

If we plot a graph between  intensity of radiation and photoelectric current it is found to be a straight line passing through the origin O as shown in figure below

This shows that photoelectric current is directly proportional to the intensity of incident radiation

The intensity of radiation can be changed by changing the distance between cathode C and the source of radiation.

Effect of potential of anode with  respect  to cathode on photoelectric current

We keep the anode at some positive accelerating potential with  respect  to cathode C and illuminate the cathode with radiation of fixed frequency f above threshold frequency and fixed intensity I

If we increase the positive potential on anode gradually, it is found that photo electric current also increases a stage comes when the photo electric current becomes maximum.

If we increase the positive potential on anode further the photo electric current does not increase.

This maximum value of photo electric current is called saturation current and corresponds to the photo electrons emitted by the cathode reach the anode A

Now saturation current is of higher value as shown in figure below

This is expected because the greater the intensity of incident radiation the greater is the photo electric current

Stopping potential

Stopping potential is the minimum retarding potential at which photoelectric current becomes zero Or  is the potential difference when no electrons are able to reach the anode.

It is also known as stopping voltage or cut-off potential.

Stopping potential is a measure of the maximum kinetic energy of the photo electrons.

Since potential difference V

For stopping Voltage Vo

For an electron

At V0, even the photo electrons having maximum kinetic energy K.E max (i.e. fastest photo electrons) cannot reach the anode A.

Therefore, the stopping potential V0 is a measure of the maximum kinetic energy K.E max of the photo electrons.

eV0 is the work done by the retarding force to stop the photo electron with maximum kinetic energy and is therefore equal to K.Emax.

At V0, it is found that the photoelectric current cannot be obtained even if we increase the intensity of radiation. It is same for different intensities I1, I2 and I3 of incident radiation.

3.   Effect of frequency of incident radiation on stopping potential.

We now study the relation between the frequency f of the incident radiation and the stopping potential V0.

For this purpose, we take the radiations of different frequencies but of the same intensity.

For one frequency say f1, of the incident radiation, we plot the graph between photoelectric current and potential of anode A with  respect  to cathode C at a constant intensity of incident radiation

Keeping the intensity of incident radiation the same, we repeat the experiment for frequency f2 of the incident radiation.

The following is the resulting graph

Observations from the graphs

1.      The value of stopping potential is different from radiation of different  frequencies

2.      The value of stopping potential is move in low higher frequency. This implies that the value of maximum kinetic energy depend on the frequency of incident radiation.

The greater the frequency of incident radiation, the greater is the kinetic energy of emitted photo electrons.

3.       The value of saturation current depends on the intensity of incident radiation but is independent of the frequency of incident radiation.

If we draw a graph between the frequency of incident radiation (f) and the stopping potential (V0) at constant intensity of radiation, it will be a straight line AB as shown in figure below

From the graph

At fo, stopping potential V0 = 0. It means that at fo, the photo electric current is just zero (i.e. photo electrons and emitted with zero velocity) and there is no retarding potential.

V0 = 0

This limiting frequency fo is called threshold frequency for the cathode material.

It is a minimum frequency of the incident radiation which is just sufficient to eject photo electrons (i.e. with zero velocity) from the surface of a metal.

Stopping potential is directly proportional to the frequency of incident radiation.

V0 α f

The greater the frequency of incident radiation, the higher is the stopping potential and vice versa.

Experiments show that photo electric emission is an instantaneous process.

As soon as light of suitable frequency (equal to or greater than fo) is incident on the surface of the metal, photo electrons are emitted from the metal surface.

The time delay is less than 10-9 second

### PHYSICS FOR 6- ATOMIC PHYSICS- QUANTUM

LAWS OF PHOTOELECTRIC EMISSION

The above experimental study of photoelectric effect leads to the following laws of photoelectric emission.

i.          For a given metal, there exists a certain minimum frequency of incident radiation below which no emission of photo-electrons takes place. This cut off frequency is called threshold frequency fo.

ii.         For a given metal and frequency of incident radiation (>fo) the photo electric current is directly proportional to the intensity of incident radiation.

iii.        Above the fo, the maximum kinetic energy of the emitted photo-electron is independent of the intensity of the incident radiation but depends only upon the frequency of the incident radiation.

iv.        The photoelectric emission is an instantaneous process.

The above laws of photoelectric emission cannot be explained on the basis of light or radiation. This gave death blow to the wave theory of light or radiation.

FAILURE OF WAVE THEORY/CLASSICAL PHYSICS TO EXPLAIN PHOTOELECTRIC EFFECT

The wave theory of radiation  failed to explain photoelectric effect. This will become clear from the following discussion.

I. According to wave theory of radiation the greater the intensity of the wave the greater  the energy of the wave.

So wave theory does explain why the number of emitted photoelectrons increase as the intensity of radiation is increased.

But it fails to explain the experimentally observed fact that the velocity or kinetic energy of the emitted photoelectron is independent of the intensity of incident radiation.

According to the wave theory, an increasing in the intensity of radiation should increase the kinetic energy of the emitted photoelectrons but it is contrary to the experimentally observed fact.

II.  According to wave theory, intensity of radiation is independent of it is frequency it depends upon the amplitude of electric field vector.

Therefore, an increase in the frequency of radiation should not affect the velocity or kinetic energy of the emitted electrons.

But it is observed experimentally that if the frequency of the incident radiation is increased, the kinetic energy of the emitted electrons also increases.

III. According to the wave theory, electrons should always be emitted from a metal by radiation of any frequency if the incident been is strong enough.

However experiments show that no matter how great is the intensity of the incident radiation; no electrons are emitted from the metallic surface if the frequency of radiation is less than a particular value i.e.  threshold frequency.

IV.  According to the wave theory the energy of radiation is spread continuously over the wave fronts of the radiation.

Therefore, a single electron in the metal will intercept only a small fraction of the wave’s energy.

Consequently considerable time should be needed for an electron to absorb enough energy from the wave to escape the metal surface.
But experiment show that electron are emitted as soon as radiation of suitable frequency falls on the metallic surface.

In other words photoelectric emission is instantaneous there is no delay.

The above discussion is a convincing proof of the inability of the wave theory to explain the photoelectric effect.

EINSTEIN QUANTUM THEORY OF LIGHT

Einstein explained photoelectric effect on the basis of Planck’s quantum theory.

According to Einstein light radiation consist of tiny packets of energy called quanta.

Photon

Photon is the single quantum of light radiation which travels with the speed of light.

The energy of a photon is given by E.

E=hf

where

h – Plank’s constant.

Further, Einstein assumed that one photon of suitable frequency (=fo or >fo) can eject only one photoelectron from the metal surface.

He suggested that the energy of a single photon cannot be shared among the free electron in the metal.

Only one electron can absorb the energy of a single photon.

EINSTEIN’S PHOTOELECTRIC EQUATION

According to Einstein, one photoelectron is emitted from a metal surface if one photon of suitable frequency is incident on the metal.

Suppose a photon of suitable frequency f (< than the fo for the metal) is incident on the metal.

The energy hf of the photon is spent in two ways

I.            A part of photon energy is used in liberating the least tight bound electron from the metal surface which is equal to the work function W0 of the metal.

II.             The rest of the energy of photon appears as maximum kinetic energy of the emitted electron.

Einstein summarized this idea in what is called the Einstein photoelectric equation.

Photon = work function + maximum kinetic energy

The above equation is known as Einstein’s photoelectric equation

If the frequency of the incident radiation is fo then the emitted photoelectron will have zero velocity.

Maximum kinetic Energy of emitted photoelectrons is

If  f < fo, then from above equation K.Emax is negative which is impossible therefore, photoelectrons emission cannot occur if the frequency of incident radiation is less than fo.

If f > fo, then equation above K.Emax α f. This means that max kinetic energy of photoelectrons depends only on the frequency (f) of the incident radiation.