Chapter - 11
DUAL NATURE OF RADIATION AND MATTER
CATHODE RAYS
DISCHARGE TUBE EXPERIMENTS
When a very strong potential difference is applied across the two electrodes in a discharge tube and the pressure of the air is lowered gradually, then a stage is reached at which the current begins to flow through the air with cracking noise. The potential at which this happens is called sparking potential.
As pressure is lowered to 0.1 mm. Hg – cathode glow, Crooke’s dark space, negative glow, Faraday’s dark space and striations are observed.
At a pressure 0.01 mm. Hg entire tube is dark (Crooke’s dark space) except the glass wall behind anode. Colour is yellowish-green for soda glass and greyish-blue for lead glass.
The luminous streaks travelling from cathode to anode, below the pressure 0.01 mm. Hg, are called cathode rays.
PROPERTIES OF CATHODE RAY
They are emitted perpendicularly to cathode,
They travel in straight lines.
They carry energy.
They possess momentum.
They are deflected by electric and magnetic fields.
They produce fluorescence on ZnS plates.
They Ionise gases through which they pass.
They produce highly penetrating secondary radiation when suddenly stopped.
They affect photographic plates.
J.J. THOMSON’S E/M VALUE OF ELECTRON
This value of
is for electrons.
MILLIKAN’S OIL DROP EXPERIMENT FOR E/M
In fig.(a) we consider a single drop of mass m carrying a negative charge –q in the absence of electric field. Then
Fviscous 1 = mg [from Stoke’s law Fviscous = 6πηrv]
or 6π rv1 = mg …(1)
where r is coefficient of viscosity of air, r is radius of drop and v1 is the terminal velocity of drop.
In fig. (b) we consider a single drop of mass m, radius r carrying a negative charge –q in the presence of an electric field acting downward. Then by free body diagram (fig. (b)), we get
…(2)
where v2 is the terminal speed in this case. Then from equation (1), we have.
…(3)
and radius of drop from equation (1)
or …(4)
where ρ is the density of drop.
Millikan repeated these measurements on thousands of drops and he found that the charge q calculated for each drop was some integral multiple of an elementary charge e. (e = 1.6 × 10–19C).
Hence, q = ne, n = 0, ±1, ±2 ...(5)
This experiment gives the evidence of quantisation of charge.
EMISSION OF ELECTRON
Electrons from the metal surface are emitted by anyone of the following physical processes :
Thermionic emission : The emission of electrons by suitably heating the metal surface.
Field emission : The emission of electrons by applying a very strong field of the order of 108 Vm–1 to a metal.
Photoelectric emission : The emission of electrons when light of suitable frequency illuminates metal surface.
Photo-electric emission: Requires light energy
Field emission: Requires an electric field to release the electrons from the metal surface.
Thermionic emission: Requires thermal energy i.e. Heat.
PHOTOELECTRIC EFFECT (EINSTEIN’S PHOTOELECTRIC EQUATION)
In the 19th century, experiments showed that when light is incident on certain metallic surfaces, electrons are emitted from the surfaces. This phenomenon is known as the photoelectric effect & emitted electrons are called photoelectrons. The first discovery of this phenomenon was made by Hertz.
When light strikes the cathode C (metallic surface), photo electrons are ejected. Electrons are collected at anode A, constituting a current in the circuit. (Photoelectric effect)
The Concept of Photons
The photoelectric effect cannot be explained by considering light as a wave. However, this phenomenon can be explained by the particle nature of light, in which light can be visualised as a stream of particles of electromagnetic energy. These ‘particles’ of light are called photons. The energy held by a photon is related to the frequency of the light via Planck’s equation.
E = h𝜈 = hc/λ
Where,
E denotes the energy of the photon
h is Planck’s constant
𝜈 denotes the frequency of the light
c is the speed of light (in a vacuum)
λ is the wavelength of the light
According to Einstein, each photon of energy E is
E = hν
Where E = Energy of the photon in joule
h = Plank’s constant (6.626 × 10-34 J.s)
ν = Frequency of photon in Hz
Characteristics of the Photoelectric Effect
The threshold frequency varies with the material, it is different for different materials.
The photoelectric current is directly proportional to the light intensity.
The kinetic energy of the photoelectrons is directly proportional to the light frequency.
The stopping potential is directly proportional to the frequency, and the process is instantaneous.
Factors Affecting the Photoelectric Effect
With the help of this apparatus, we will now study the dependence of the photoelectric effect on the following factors:
The intensity of incident radiation.
A potential difference between the metal plate and collector.
Frequency of incident radiation.
Effects of Intensity of Incident Radiation on Photoelectric Effect
The potential difference between the metal plate, collector and frequency of incident light is kept constant, and the intensity of light is varied.
The electrode C, i.e., the collecting electrode, is made positive with respect to D (metal plate). For a fixed value of frequency and the potential between the metal plate and collector, the photoelectric current is noted in accordance with the intensity of incident radiation.
It shows that photoelectric current and intensity of incident radiation both are proportional to each other. The photoelectric current gives an account of the number of photoelectrons ejected per sec.
Effects of Potential Difference between the Metal Plate and the Collector on the Photoelectric Effect
The frequency of incident light and intensity is kept constant, and the potential difference between the plates is varied.
Keeping the intensity and frequency of light constant, the positive potential of C is increased gradually. Photoelectric current increases when there is a positive increase in the potential between the metal plate and the collector up to a characteristic value.
There is no change in photoelectric current when the potential is increased higher than the characteristic value for any increase in the accelerating voltage. This maximum value of the current is called saturation current.
Effect of Frequency on Photoelectric Effect
The intensity of light is kept constant, and the frequency of light is varied.
For a fixed intensity of incident light, variation in the frequency of incident light produces a linear variation of the cut-off potential/stopping potential of the metal. It is shown that the cut-off potential (Vc) is linearly proportional to the frequency of incident light.
The kinetic energy of the photoelectrons increases directly proportional to the frequency of incident light to completely stop the photoelectrons. We should reverse and increase the potential between the metal plate and collector in (negative value) so the emitted photoelectron can’t reach the collector.
Einstein’s theory of the photoelectric effect
According to Einstein’s theory of the photoelectric effect, when a photon collides inelastically with electrons, the photon is absorbed completely or partially by the electrons. So if an electron in a metal absorbs a photon of energy, it uses the energy in the following ways.
Some energy Φ0 is used in making the surface electron free from the metal. It is known as the work function of the material. Rest energy will appear as kinetic energy (K) of the emitted photoelectrons.
Features of Einstein’s Photoelectric Equation
The frequency of the incident light is directly proportional to the kinetic energy of the electrons, and the wavelengths of incident light are inversely proportional to the kinetic energy of the electrons.
If γ = γ th or λ =λ th then vmax = 0
γ < γ th or λ > λ th: There will be no emission of photoelectrons.
The intensity of the radiation or incident light refers to the number of photons in the light beam. More intensity means more photons and vice-versa. Intensity has nothing to do with the energy of the photon. Therefore, the intensity of the radiation is increased, and the rate of emission increases, but there will be no change in the kinetic energy of electrons. With an increasing number of emitted electrons, the value of the photoelectric current increases.
Fig. shows, when light strikes the cathode C(-), electrons are emitted & they are collected on anode A(+), due to potential difference provided by the battery and constitutes the current in the circuit (observed by Galvanometer G.)
A plot of photoelectric current versus the potential difference V between cathode & anode is shown in fig below.
Photoelectric current versus voltage for two light intensities.
At a voltage less than –V0 the current is zero.
It is clear from fig. that photoelectric current increases as we increase the intensity of light & obtain saturation value at larger value of potential difference V between cathode & anode. If V is negative then, photoelectrons are repelled by negative cathode and only those electrons reach anode, who have energy equal to or greater than eV. But if V is equal to V0, called stopping potential (i.e., cut off. potential), no electrons will reach the anode
i.e., Maximum kinetic energy of electron = eV0
or Kmax = eV0 ...(1)
where e is the charge of electrons (e = 1.6 × 10–19 coulomb).
But some features of the photoelectric effect cannot be explained by classical physics & the wave theory of light.
No photoelectrons are emitted, if the frequency of incident light is less than some cut-off frequency (i.e., threshold frequency) ν0. It is inconsistent with the wave theory of light, which predicts that photoelectric effect occurs at any frequency provided intensity of incident light is sufficiently high.
The maximum kinetic energy of the photoelectrons is independent of light intensity, but increases with increasing the frequency of incident light.
Electrons are emitted from surface almost instantaneously (less than 10–9 sec after the surface illumination), even at low intensity of incident light (classically we assume that the electrons would require some time to absorb the incident light before they acquire enough kinetic energy to escape from metal).
These above points were explained by Einstein in 1905 by treating the light as a stream of particles.
Taking Max Planck assumptions, Einstein postulated that a beam of light consists of small packets of energy called photons or quanta. The energy E of a photon is equal to a constant h times its frequency ν
i.e., ...(2)
where h is a universal constant called Planck’s constant & numerical value of h = 6.62607 × 10–34 J.s
When a photon arrives at the surface, it is absorbed by an electron. This energy transfer is an All-or-None process, in contrast to continuous transfer of energy in classical theory; the electrons get all photon’s energy or none at all. If this energy is greater than the work function (φ) of the metal (φ is the minimum energy required to remove the electron from the metal surface), the electron may escape from the surface. Greater intensity at a particular frequency means greater number of photons per second absorbed & consequently greater number of electrons emitted per second & so greater current.
To obtain maximum kinetic energy
for an emitted electron, applying the law of conservation of energy.
According to it
...(3)
or ...(4)
or,
This is Einstein's photoelectric equation.
where V0 = cut-off potential
νmax = maximum velocity obtained by photoelectrons
ν = frequency of incident light i.e., photon
ν0 = cut off frequency or threshold frequency.
ν0 is different for different metallic surfaces. For most metals the threshold frequency is in the ultraviolet region of the spectrum. (Corresponding to λ between 200 & 300 nm), but for potassium & cesium oxides, it is in the visible spectrum (λ between 400 & 700 nm).
Work Function or Threshold Energy (Φ)
The minimal energy of thermodynamic work that is needed to remove an electron from a conductor to a point in the vacuum immediately outside the surface of the conductor is known as work function or threshold energy.
The minimum amount of energy required to remove an electron from the metal is called the threshold energy (denoted by the symbol Φ). For a photon to possess energy equal to the threshold energy, its frequency must be equal to the threshold frequency (which is the minimum frequency of light required for the photoelectric effect to occur). The threshold frequency is usually denoted by the symbol 𝜈th, and the associated wavelength (called the threshold wavelength) is denoted by the symbol λth. The relationship between the threshold energy and the threshold frequency can be expressed as follows.
Φ = h𝜈th = hc/λth
The work function is the characteristic of a given metal. If E = energy of an incident photon, then
If E < Φ, no photoelectric effect will take place.
If E = Φ, just a photoelectric effect will take place, but the kinetic energy of ejected photoelectron will be zero
If E > photoelectron will be zero
If E > Φ, the photoelectric effect will take place along with the possession of the kinetic energy by the ejected electron.
Work Function (φ) of Some Elements Given in Brackets :
Al (4.3eV) , Ni (5.1 eV), C (5.0 eV), Si (4.8 eV)
Cu (4.7 eV), Ag (4.3 eV), Au (5.1eV) Na (2.7 eV)
where 1 eV = 1.602 × 10–19 joule.
Within the framework of photon theory of light (Quantum theory of light) we can explain the above failures of classical physics.
It is clear from eq. (3) that if the energy of a photon is less than the work function of a metallic surface, the electrons will never be ejected from the surface regardless of intensity of incident light.
Kmax is independent of intensity of incident light, but it depends on the frequency of incident light i.e., Kmax ∝ ν (frequency of light).
Electrons are emitted almost instantaneously consistent with the particle view of light in which incident energy is concentrated in small packets (called photons) rather than over a large area (as in wave theory).
Light Intensity
Since it is a well-known fact that the photoelectric current depends on the number of electrons escaping the metal surface in one second.
This indicates that the photoelectric current is in direct relation to the intensity of the light.
So, the graph between photoelectric current and light intensity will be a straight line.
Potential
Stopping potential is the minimum provided on the metal plate that stops the photoelectric current.
The minimum negative potential is the retarding potential as it is retarding the photoelectric current.
The Kinetic Energy of the photoelectrons doesn’t depend on the intensity of the incident light.
Effect of frequency on the stopping potential
The frequency of incident light is directly proportional to the maximum kinetic energy of the electrons.
This indicates that there will be a greater requirement of retarding potential to stop the electrons from emitting out of the metal plate.
Two important graphs to represent the same are given below:
VARIOUS GRAPHS RELATED TO PHOTOELECTRIC EFFECT
Photoelectric current vs Retarding potential for different voltages
Photoelectric current vs Retarding potential for different intensities
Electron current vs Light Intensity
Stopping potential vs Frequency
Electron current vs Light frequency
Electron kinetic energy vs Light frequency
Electron current vs Light Intensity
Stopping potential vs Frequency
Electron kinetic energy vs Light frequency
Photoelectric current vs Retarding potential for different voltages
Photoelectric current vs Retarding potential for different intensities
Electron current vs Light frequency
Problems on the Photoelectric Effect
1. In a photoelectric effect experiment, the threshold wavelength of incident light is 260 nm and E (in eV) = 1237/λ (nm). Find the maximum kinetic energy of emitted electrons.
Solution:
Kmax = hc/λ – hc/λ0 = hc × [(λ0 – λ)/λλ0]
⇒ Kmax = (1237) × [(380 – 260)/380×260] = 1.5 eV
Therefore, the maximum kinetic energy of emitted electrons in the photoelectric effect is 1.5 eV.
2. In a photoelectric experiment, the wavelength of the light incident on metal is changed from 300 nm to 400 nm and (hc/e = 1240 nm-V). Find the decrease in the stopping potential.
Solution:
hc/λ1 = ϕ + eV1 . . . . (i)
hc/λ2 = ϕ + eV2 . . . . (ii)
Equation (i) – (ii)
hc(1/λ1 – 1/λ2) = e × (V1 – V2)
⇒V1 – V2 = (hc/e) × [(λ2 – λ1)/(λ1 λ2)]
= (1240 nm V) × 100nm/(300nm × 400nm)
=12.4/12 ≈ 1V
Therefore, the decrease in the stopping potential during the photoelectric experiment is 1V.
3. When ultraviolet light with a wavelength of 230 nm shines on a particular metal plate, electrons are emitted from plate 1, crossing the gap to plate 2 and causing a current to flow through the wire connecting the two plates. The battery voltage is gradually increased until the current in the ammeter drops to zero, at which point the battery voltage is 1.30 V.
a) What is the energy of the photons in the beam of light in eV?
b) What is the maximum kinetic energy of the emitted electrons in eV?
Solution:
Assuming that the wavelength corresponds to the wavelength in the vacuum.
f = 1.25 × 1015 Hz
The energy of photon E = hf
E = (4.136 × 10-15)( 1.25 × 1015)
Note: Planck’s constant in eV s = 4.136 × 10-15 eV s
E = 5.17 eV.
b) The maximum kinetic energy related to the emitted electron is stopping potential. In this case, the stopping potential is 1.30V. So the maximum kinetic energy of the electrons is 1.30V.
4. Two photons, each of energy 2.5eV are simultaneously incident on the metal surface. If the work function of the metal is 4.5eV, then from the surface of metal. How many electrons will be emitted?
Solution: Not even a single electron will be emitted. Since the energy of each photon (2.5 eV) is lesser than the work function (4.5 eV), there will not be any emission of electrons. The photons will be absorbed and will excite the electrons. But the electrons will still remain bound to the metal.
5. In which type of electron emission, the electrons use high speed electrons to emit from the surface?
Solution: The minimum energy required by an electron to just escape (i.e. with zero velocity) from the metal's surface is called the Work function (W0) of the metal.
6. A radiation of wavelength 300 nm is incident on a silver surface. Will photoelectrons be observed?
Solution: Energy of the incident photon is
E = hv = hc/λ (in joules)
E = hc/λe (in eV)
Substituting the known values, we get
E = 6.626×10−34 × 3×108 / 300×10−9 ×1.6 × 10−19
E = 4.14 eV
Since the energy of the incident photon is less than the work function of silver, photoelectrons are not observed in this case.
Important Points to Remember
If we consider the light with any given frequency, the photoelectric current is generally directly proportional to the intensity of light. However, the frequency should be above the threshold frequency in such a case.
Below threshold frequency, the emission of photoelectrons completely stops despite the high intensity of incident light.
A photoelectron’s maximum kinetic energy increases with an increase in the frequency of incident light. In this case, the frequency should exceed the threshold limit. Maximum kinetic energy is not affected by the intensity of light.
Stopping potential is the negative potential of the opposite electrode when the photo-electric current falls to zero.
The threshold frequency is described as the frequency when the photoelectric current stops below a particular frequency of incident light.
The photoelectric effect establishes the quantum nature of radiation. This has been taken into account to be proof in favour of the particle nature of light.
POINTS TO REMEMBER
Mass spectrograph is an apparatus used to determine the mass or the specific charge (e/m) of positive ions. Examples are (a) Thomson mass spectrograph (b) Bain bridge mass spectrograph (c) Aston mass spectrograph (d) Dempster mass spectrograph etc.
In the photoelectric effect all photoelectrons do not have the same kinetic energy. Their KE ranges from zero to Emax which depends on frequency of incident radiation and nature of cathode.
The photoelectric effect takes place only when photons strike bound electrons because for free electrons energy and momentum conservation do not hold together.
Cesium is the best photo sensitive material.
Efficiency of photoemission
Therefore,
Maximum velocity of emitted electrons
Stopping potential
Properties of the Photon
For a photon, all the quantum numbers are zero.
A photon does not have any mass or charge, and they are not reflected in a magnetic and electric field.
The photon moves at the speed of light in empty space.
During the interaction of matter with radiation, radiation behaves as it is made up of small particles called photons.
Photons are virtual particles. The photon energy is directly proportional to its frequency and inversely proportional to its wavelength.
The momentum and energy of the photons are related, as given below
E = p.c where
p = Magnitude of the momentum
c = Speed of light
DE-BROGLIE EQUATION (DUAL NATURE OF MATTER)
In 1924, Louis de Broglie, wrote a doctoral dissertation in which he proposed that since photons have wave and particle characteristics, perhaps all forms of matter have wave as well as particle properties.
This is called the dual nature of matter. According to which a matter particle moving with a velocity v can be treated as a wave of wavelength λ. This λ is called de-Broglie wavelength & it is defined as :
...(1)
where m is the mass of a matter particle & these waves are called matter waves.
Further with the analogy of photon, the frequency of matter waves is
...(2)
The dual nature of matter is quite apparent in these two equations (equations (1) & (2)). i.e., each equation contains both particle concepts (mv & E) & wave concepts (λ & ν). It is clear from the next topic that the Compton effect confirms the validity of p = h/λ for photons, and the photoelectric effect confirms the validity and E = hν for photons.
de-Broglie wavelength associated with electron accelerated under a potential difference V volt is given by
The de-Broglie wave is not an electromagnetic wave but a matter wave.
WAVELENGTH OF MATTER WAVES ASSOCIATED WITH ACCELERATED CHARGED PARTICLES
If V is the accelerating voltage applied then :
For the charged particle
Energy E = qV ;
Velocity
Momentum
Wavelength
For electron λe = Å
For proton λp = Å
For alpha particle λα = Å
For deuteron λd = Å
For neutral particles (neutron, atom or molecule)
If E is the energy of the particle, then,
If T is the temperature, then,
DAVISSON-GERMER EXPERIMENT
Idea of de-Broglie waves was tested beautifully in 1926 in an experiment performed by C. Davisson (1881-1958) and L.H. Germer (1896-1971). They directed a beam of electrons at a crystal and observed that the electrons scattered in various directions for a given crystal orientation.
In this experiment the pattern formed by the electrons reflected from the crystal lattice of aluminium is almost identical to that produced by X-rays. This strongly suggests that the electrons have a wavelength λ associated with them and that the Bragg condition for X-ray diffraction holds true for electron also :
BRAGG’S EQUATION
nλ = D sin θ or nλ = 2d sin φ.
Diffraction maximum of electrons accelerated with 54 volt is obtained at θ = 50º for the Nickel crystal.
EXPLANATION OF BOHR’S QUANTUM CONDITION
According to Bohr’s quantum conditions :
Angular momentum,
Matter waves associated with the electrons moving in an orbit are stationary waves.
For the production of stationary waves in the orbit the circumference of the orbit should be integral multiple of wavelength of waves associated with the electron,
i.e., 2πrn = nλ, where
∴
COMPTON EFFECT
Further experimental proof for the photon concept (i.e., particle nature of light) was discovered 100 years ago in 1923 by American Physicist, A.H. Compton.
Compton effect can be observed when there is an increase in the wavelength caused by the dispersion of x-rays and gamma rays on the material.
The Compton Effect
The Compton effect is the term used for an unusual result observed when X-rays are scattered on some materials. By classical theory, when an electromagnetic wave is scattered off atoms, the wavelength of the scattered radiation is expected to be the same as the wavelength of the incident radiation.But, observations show that when X-rays are scattered off some materials, such as graphite, the scattered X-rays have different wavelengths from the wavelength of the incident X-rays. This classically unexplainable phenomenon was studied experimentally by Arthur H. Compton and his collaborators, and Compton gave its explanation in 1923.
To explain the shift in wavelengths measured in the experiment, Compton used Einstein’s idea of light as a particle. The Compton effect,with so much significance in the history of physics as it shows that electromagnetic radiation cannot be explained as a purely wave phenomenon. The explanation of the Compton effect could give a convincing argument to the physics community that electromagnetic waves can indeed behave like a stream of photons, which placed the concept of a photon on firm ground.
According to which, when a monochromatic beam of X-rays (wavelength λ0) strikes the electron in a carbon target, two types of X-rays are scattered. The first type of scattered wave has same wavelength (λ0) as the incoming X-rays, while second type has a longer wavelength (λ) than incident rays (First type of X-rays are called unmodified x-rays, while second type of X-rays are called modified X-rays.)
This change in wavelength i.e. Δλ = λ – λ0 is called Compton shift & this effect is called Compton effect.
Because of the relation between energy and wavelength, the scattered photons have a longer wavelength that also depends on the size of the angle through which the X-rays were diverted. The increase in wavelength, or Compton shift, does not depend on the wavelength of the incident photon. The wavelength of the photon after (λ′) and before (λ) the scattering event differs by
λ′ − λ = (h/mc)(1 − cos θ).
Here h is Planck’s constant, m is the rest mass of the electron, c is the speed of light, and θ is the angle through which the photon is scattered.
Diagram shows Compton scattering of x-rays by free electrons in a carbon target. The scattered x-rays have less energy than the incident x-rays. The excess energy is taken by recoiling electrons.
This effect cannot be explained by classical theory (by the wave nature of light). According to the classical model, when X-rays of frequency ν0 are incident on the material containing electrons, then electrons do oscillate & reradiate electromagnetic waves of same frequency ν0. Hence scattered X-rays have the same frequency ν0 & same wavelength as that of incident X-rays.
Compton treated this process as a collision between a photon & an electron. In this treatment, the photon is assumed as a particle of energy
E = hν0 = hc/λ0 ...(1)
Further, the rest mass of photon is zero (because photon travels with the speed of light) hence the momentum of photon can be written as
Pf =h/λ ...(2)
To derive the Compton shift. Δλ, we apply both conservation of energy & momentum.
CONSERVATION OF ENERGY
...(3)
Where hc/λ is energy of scattered X-rays, Ke is kinetic energy of recoiling electrons & hc/λ0 is the energy of incoming X-rays. Since the electron may travel with the speed of light, so we must use relativistic expression of Ke in equation (3), and we obtain
...(4)
where m is rest mass of electron and mc2 is the rest mass energy of the electron
where
CONSERVATION OF MOMENTUM
x-component ...(5)
y-component ...(6)
where pe = γmv is the relativistic expression for momentum of a recoiling electron.
By eliminating v & φ from equation (4) to (6), we obtain
...(7)
or ...(8)
It is clear from expression (7) that Compton shift Δλ depends on scattering angle θ & not on the wavelength.
Difference Between Compton Effect and Photoelectric Effect
Solved Example:
Find the wavelength of the scattered X-ray, for the given scattering angle, θ=30°.
Solution:
We use Equation Δλ=λc(1−cosθ). Then we add this shift to the incident wavelength to obtain the scattered wavelength. The largest Compton shift occurs at the angle θ when 1−cosθ has the largest value, which is for the angle θ=180°.
Solution
The shift at θ=30° is
Δλ =λc(1− Cos 30°)
=0.134λc
=(0.134)(2.43)pm
=0.32
The scattered wavelength is given by:
𝛌′=𝛌+Δ𝛌
=(71+0.325)pm
=71.325pm.
The largest shift is
(Δλ)max
=λc(1−Cos 180°)
=2(2.43pm)=4.86pm
Problem For Practice
An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a 60° angle. What is the smallest shift that can be expected in this experiment?
POINTS TO REMEMBER
The wave nature of light shows up in the phenomena of interference, diffraction and polarisation whereas photoelectric effect and Compton effect shows particle nature of light.
To find the wavelength of the scattered X-ray, first we must find the Compton shift for the given scattering angle, θ=30°. We use Equation Δλ=λc(1−cosθ). Then we add this shift to the incident wavelength to obtain the scattered wavelength. The largest Compton shift occurs at the angle θ when 1−cosθ has the largest value, which is for the angle θ=180°.
The maximum kinetic energy of the photoelectrons varies linearly with the frequency of incident radiation but is independent of its intensity.
X-RAYS
The X-rays were discovered by Prof. Roentgen, a German scientist in 1885. He was awarded the Nobel Prize for this discovery in 1901. X-rays are electromagnetic waves.
The modern apparatus for the production of X-rays was developed by Dr. Coolidge in 1913.
X-rays are produced when fast moving electrons are suddenly stopped on a metal of high atomic number.
PROPERTIES OF X-RAYS
They are not deflected by electric or magnetic fields.
They travel with the speed of light.
There is no charge on X-rays.
X-rays show both particle and wave nature.
They are invisible.
CONTINUOUS AND CHARACTERISTIC X-RAYS
Experimental observation and studies of spectra of X-rays reveal that X-rays are of two types and so are their respective spectras. Characteristic X-rays and Continuous X-rays.
CHARACTERISTIC X-RAYS
The spectra of this group consists of several radiations with specific sharp wavelengths and frequency similar to the spectrum (line) of atoms like hydrogen. The wavelengths of this group show characteristic discrete radiations emitted by the atoms of the target material. The characteristic X-rays spectra help us to identify the element of target material.
When the atoms of the target material are bombarded with high energy electrons (or hard X-rays), which possess enough energy to penetrate into the atom, knock out the electron of the inner shell (say K shell, n = 1). When an electron is missing in the ‘K’ shell, an electron from the next upper shell makes a quantum jump to fill the vacancy in the ‘K’ shell. In the transition process the electron radiates energy whose frequency lies in the X-rays region. The frequency of emitted radiation (i.e. of photon) is given by
where R is constant and Ze is an effective atomic number. Generally Ze is taken to be equal to Z – σ, where Z is the proton number or atomic number of the element and σ is called the screening constant. Due to the presence of the other electrons. The charge of the nucleus as seen by the electron will be different in different shells.
Knocking out e- of K shell by incident electron ei emission of X-ray photon (Kα- series)
Another vacancy is now created in the ‘L’ shell which is again filled up by another electron jump from one of the upper shells (M) which results in the emission of another photon, but of different X-rays frequency. This transition continues till outer shells are reached. Thus resulting in the emission of a series of spectral lines.
The transitions of electrons from various outer shells to the innermost ‘K’ shell produces a group of X-rays lines called K-series. These radiations are most energetic and most penetrating. K-series is further divided into …. depending upon the outer shell from which the transition is made.
The jump of electrons from outer shells to ‘L’ shells results in L-Series X-rays and so on.
CONTINUOUS X-RAYS
In addition to characteristic X-rays tubes emit a continuous spectrum also. The characteristic line spectra is superimposed on a continuous X-rays spectra of varying intensities. The wavelengths of the continuous X-rays spectra are independent of material. One important feature of continuous X-rays is that they end abruptly at a certain lower wavelength for a given voltage. If an electron beam of energy eV (electron volts) is incident on the target material; the electrons are suddenly stopped. If the whole of the energy is converted to continuous radiation, then λmin (corresponding to energy maximum) = hc/Ve where V is the voltage applied.
The classical theory of electromagnetism states that the suddenly accelerated or decelerated electrons emit radiations of electromagnetic nature called as bremsstrahlung (braking radiation) and wavelength of such radiation is continuous because the loss in energy is statistical. At the peak, the probability of maximum number of electrons producing radiation.
The wavelength of X-rays emitted is minimum corresponding to the electron which hits the target with maximum speed. This electron is completely stopped and will emit the photon of highest energy.
As the electrons lose energy by collision, longer wavelengths are produced the shape of the curve is statistical.
WAVELENGTH OF X-RAYS (DUANE HUNT LAW)
When an electron is accelerated through a potential difference V then the energy acquired by electron
When these high energy electrons fall on target T of high atomic number, then X-rays are produced, whose wavelength is given by
The energy of X-rays of wavelength λ is
The shortest wavelength of X-rays emitted is
i.e.
It is called Duane Hunt law.
TYPES OF X-RAYS
Hard X-rays : The X-rays of high frequency or low wavelength are said to be hard X-rays. They have higher penetrating power.
Soft X-rays : The X-rays of longer wavelengths are called soft X-rays.
MOSELEY’S LAW
Moseley used different elements as targets in the X-ray tube. He found that Kα radiation of different elements were different.This law came to existence because when Henry Moseley was studying graphs, he found a strange relationship between the lines and the atomic number. This law also helped with organising the elements on the periodic table based on atomic numbers rather than atomic mass.
The statement of his law is,
“The square root of the frequency of the x-ray emitted by an atom is proportional to its atomic number”.
Mathematically,
V = a . (Z – b)²
Where,
V is the frequency of the x-ray emitted line, and
a and b depend on the particular line of the radiation.
Moseley Periodic Law
Here, we will measure the x-ray spectra of a number of elements and also identify several unknown elements by looking at their characteristics, viz: X-ray spectra.
Moseley’s law was discovered and published by an English Physicist named Henry Moseley. This law is an empirical law that concerns the characteristics of X-rays emitted by atoms.
We also call these constants proportionality and screening or shielding constants.
Equation (1) is Moseley’s X-ray Characteristic formula and here the two physical constants ‘a’ and ‘b’ are independent constants of an element; however, these two depend on the X-ray series.
The relation between a and b is determined by experiments using Henry Moseley’s law and the graph for this relationship is as follows:
The line intersecting in the graph at the Z-axis shows that Z = b, where b is 1 for K series elements and 7.4 for elements in L series.
For Kα, and b = 1
where R = Rydberg constant and c = speed of light
In general the wavelength of K - lines are given as
where n = 2, 3, .....
ABSORPTION OF X-RAYS
X-rays are absorbed by the materials according to the relation I = I0 e–μx, where μ is the absorption coefficient and x is the thickness of the material. Here I is the intensity after penetrating the material through distance x and I0 is the initial intensity of the X-rays.
The coefficient of absorption (μ) of the material is given by
Where, x1/2 is the distance after traversing which the intensity of X-rays is reduced to half.Absorption coefficient depends on the nature of material and wavelength of X-rays i.e. μ = cZ4 λ3.
It means that (a) μz4 (b) μλ3 (c) μν–3.
FLUORESCENCE
Certain substances (like quinine sulphate, fluorescein, barium platinocyanide, uranium oxide etc.), when illuminated with light of high frequency (ultraviolet, violet, etc.) emit light of lower frequency. The phenomenon is called fluorescence.
When quinine sulphate is illuminated with ultraviolet or violet light it gives out blue light. The fluorescence of barium sulphate as well as uranium oxide gives out green light when illuminated with ultraviolet or violet light.
The house hold tubes are painted from inside with magnesium tungstate or zinc-beryllium silicate. They are fluorescent materials. The ultraviolet light generated inside the tube falls on the walls, where magnesium tungstate gives blue light and zinc beryllium silicate gives yellow orange light. The mixture of the two produces white light. If the inner side of the tube is painted with cadmium borate it gives fluorescence of pink light and when painted with zinc silicate, it gives fluorescence of green light.
The fluorescence occurs as long as the material is illuminated.
PHOSPHORESCENCE
Fluorescent materials emit light only so long as light is incident on them. There are certain substances which continue emitting light for some time after the light incident on them is stopped. This phenomenon is called phosphorescence. For example, if we make blue light incident on a zinc-sulphide (ZnS) screen, then it produces phosphorescence of green colour. Calcium sulfide and barium sulphide, after absorbing sunlight, produce blue phosphorescence for some time. Time of phosphorescence is different for different materials.
REMEMBER
The stopping potential (and hence the maximum kinetic energy of emitted electrons) is independent of the intensity of light but that the saturation current (and hence the number of emitted photoelectrons) is proportional to the intensity.
Photoelectric effect doesn't take place below the threshold frequency for the photo metal used.
In Compton effect, the change in wavelength is independent of incident photon as well as of the nature of the scatterer, but depends only on the angle of scattering (θ).
The quantity is called Compton wavelength.
The maximum wavelength change possible in Compton effect is 0.05Å.
Compton effect can't be observed for visible light rays.
In Compton effect, the direction of recoil electron is given by
The kinetic energy of recoil electron is given by
de-Broglie wavelength of a particle of K.E., Ek is given by
de-Broglie wavelength for a charged particle with charge q and accelerated through a potential difference V is given by
de-Broglie wavelength of a material particle at temperature T is given by
, where k is Boltzmann’s constant.
APPLICATION OF X-RAYS
Following are some important and useful applications of X-rays.
Scientific applications : The diffraction of X-rays at crystals opened a new dimension to X-rays crystallography. Various diffraction patterns are used to determine the internal structure of crystals. The spacing and dispositions of atoms of a crystal can be precisely determined by using Bragg’s law : nλ = 2d sin θ.
Industrial applications : Since X-rays can penetrate through various materials, they are used in industry to detect defects in metallic structures in big machines, railway tracks and bridges. X-rays are used to analyse the composition of alloys and pearls.
In radiotherapy : X-rays can cause damage to the tissues of the body (cells are ionised and molecules are broken). So X-rays damage malignant growths like cancer and tumours which are dangerous to life, when used in proper and controlled intensities.
In medicine and surgery : X-rays are absorbed more in heavy elements than the lighter ones. Since bones (containing calcium and phosphorus) absorb more X-rays than the surrounding tissues (containing light elements like H, C, O), their shadow is casted on the photographic plate. So the cracks or fractures in bones can be easily located. Similarly intestine and digestive system abnormalities are also detected by X-rays.
Some Q&A:
Q1 What is an X-ray?
An X-ray is a penetrating type of electromagnetic radiation. Almost every x-ray has a wavelength from 10 picometers to 10 nanometers.
Q2 What is a gamma-ray?
A gamma-ray is another penetrating type of electromagnetic radiation generated from the radioactive decay of atomic nuclei. Gamma-ray is made of the shortest wavelength (among the important electromagnetic waves).
Q3 What is meant by radiation?
Radiation is the transmission or emission of energy in the form of particles or waves through a medium or vacuum.
Q4 What is meant by the photoelectric effect?
The photoelectric effect is a process in which electrons are emitted from a metal’s surface when light strikes it.
Q5 Who discovered the Compton effect?
Arthur Compton discovered the Compton effect.
Q6 What is meant by the Compton effect?
The Compton effect is the enlargement of wavelengths of X-rays and other EM waves that have been scattered by electrons. It is a fundamental way in which radiating energy is absorbed by matter.
Q7 What is wavelength?
A wavelength is a length between successive troughs or crests.
Q8 Which symbol is used to represent the Compton wavelength?
Compton wavelength is represented by the letter λ (Lambda).
Q9 What is a photon?
A photon is a fundamental particle that is the force carrier of electromagnetic force. It is massless and travels at the speed of light 299792458 m/s (in vacuum).
Q10 Who explained the photoelectric effect?
Albert Einstein explained the photoelectric effect.
