Unit 12. Quantum and Nuclear Physics :The interaction of matter with radiation Notes

Understandings

➔ Photons

➔ The photoelectric effect

➔ Matter waves

➔ Pair production and pair annihilation

➔ Quantization of angular momentum in the Bohr model for hydrogen

➔ The wave function

➔ The uncertainty principle for energy and time, and position and momentum

➔ Tunnelling, potential barrier, and factors affecting tunnelling probability

Applications and skills

➔ Discussing the photoelectric effect experiment and explaining which features of the experiment cannot be explained by the classical wave theory of light

➔ Solving photoelectric problems both graphically and algebraically

➔ Discussing experimental evidence for matter waves, including an experiment in which the wave nature of electrons is evident

➔ Stating order of magnitude estimates from the uncertainty principle

Equations

➔ Planck relationship: E = hf

Einstein photoelectric equation: Emax = hf – Φ

➔ Bohr orbit energies: E = -13.6/n2 eV

➔  quantization of angular momentum: mvr = nh/2π

➔ Heisenberg relationships: position–momentum  ∆xph/4π

➔ Probability density P(r) = |Ψ|2 ΔV

➔energy–time:  ∆Eth/4π

12.1 The Interaction of Matter with Radiation

The photoelectric effect

​Definition: Phenomenon in which light (or other forms of electromagnetic radiation) incident on a metallic surface causes electron to be emitted from the surface.

Experiment: Evacuated tube, with a metallic photo-surface (P), in which light passes through a small opening and causes electrons to be ejected. These electrons are collected by a collecting surface (C).

  • Since the collecting plate is connected to the negative terminal of the power supply, it will repel normally repel electrons and only absorb the energetic ones.​

  • As the voltage is made more negative, there is a point at which the current ceases, called stopping voltage/potential (Vs).

Tip: The I-V graph will change if the frequency of the light is increased. Each photon will have more energy, and hence, the stopping potential will be greater. The saturation current will depend upon the intensity of the light, but in the case of two lights with the same intensity, the saturation current for the higher frequency will be less. This follows logically from the photon nature of light: Same current means the same amount of electrons (charge-carriers) per second, but higher frequency means that electrons have more energy. Less photons per second means that fewer electrons are emitted, and so, smaller saturation current.

Observations

  1. The intensity of the incident light neither affects the kinetic energy or the stopping voltage, solely the number of electrons emitted.

  2. The frequency of light influences the emitted electrons’ energy.

  3. Electrons are emitted without a time delay.

  4. There is a minimum/threshold frequency, fc, below which no electrons are emitted.

If light was only a wave…

  1. Intense beams, which have more energy, should cause the emissions of electrons with higher kinetic energy.

  2. Frequency should play a role in the energy of electrons

  3. Low intensity beams should cause a time delay, since energy would need to accumulate before the emission of an electron.

Einstein’s explanation​: Light consists of photons, which are quanta or bundles of energy and momentum.

  • Photon’s energy: E = hf = hc/λ

  • Planck’s constant = h = 6.63 x 10^-34

  • Photoelectric effect: Single photon of frequency f is absorbed by a single electron in the photo-surface, so the electron’s energy increases by hf. The electron will spend Ф Joules, called the work function, to free itself.

  • Electron’s kinetic energy (after emission): E = e Vs = hf – Ф

frequencyxmaxek.jpg

Matter (or “de Broglie”) Waves

As suggested by de Broglie, to any particle of momentum ​p, there corresponds a wave of wavelength given by the formula λ = h/p, something known as the duality of matter.

  • Electron diffraction: Electrons shot through or to a thin slice of crystal have a low probability of reaching a place where the path difference is not an integer number of wavelengths (constructive and destructive interference).

    • Electrons accelerated through a pd, they gain kinetic energy. Hence, we have eV = 1/2 mv = p²/2m.

    • λ = h/√(2meV)

  • Davisson-Germer experiment verifies de Broglie hypothesis (below).​

electronbeam.jpg

The Bohr Model

Model proposed by Niels Bohr to interpret the scattering of alpha particles, which states that electrons are found at orbitals: fixed multiples of angular momentum that can be represented as a wave function.

  • Electrons in any atom have a definite/discrete energy (which explains the emission and absorption spectra).

    • Energy levels = electron wave = standing wave, since there is no energy transfer in standing waves​

    • Hydrogen atom: Energy is given by E = -13.6/n², where n is the principal quantum number and represents the nth energy level.

 
modelsatom.jpg
  • Angular momentum (mvr): A vector product of the momentum of a particle and the radius of its orbit, of an electron in a stationary state is an integral value of h/2π. Hence, we have mvr = nh/2π.

  • Assumptions: 

    • Electrons in an atom exist in stationary states, without emitting any electromagnetic radiation.

    • Electrons may move from one stationary state to another by absorbing or emitting a quantum of electromagnetic radiation, with difference in energy between stationary states given by ∆E = hf.

  • Limitations: Bohr’s Model failed to explain…

    • Why some energy transitions are more likely to occur than others,

    • Predict behavior of other elements

    • Explain behaviors theoretically

Schrödinger’s equation (wave function)

Describes the quantum state of the particles, where the square of the amplitude of the wave function │Ψ│²  is proportional to the probability per unit volume of finding the particle at a distance r from the nucleus     ​ P(r) = │Ψ│²∆V or at (x,y), i.e.  P(x,y) = │Ψ│²∆V.

  • Copenhagen interpretation: For double-slit interference, the wave function is considered to be such that a single photon or electron passes through both slits and be everywhere on the screen until it is observed or measured.

    • Nothing is real unless it is observed. When observed, the wave function collapses.

Heisenberg’s uncertainty principle

It is impossible to simultaneously measure the position and momentum of a particle with indefinite precision. The same applies to energy and time.

  • Uncertainty in position and momentum: ∆x∆p ≥ h/4π

    • Example: ​Since we know the wavelength of the electron and momentum and wavelength are related by p = h/λ, ∆x is infinite.Single-slit diffraction: The uncertainty in position for beam going through a hole of diameter is approximately ∆x = b/2. When the opening is approximately of the same order as the de Broglie wavelength of the electrons, the wave will diffract. θ 

  • Uncertainty in energy and time: ∆E∆t ≥ h/4π (where E is half the difference between the excited state and the ground state).

    • Useful to estimate the lifetime of an electron in excited state.

  • Single-slit diffraction: The uncertainty in position for beam going through a hole of diameter is approximately ∆x = b/2. When the opening is approximately of the same order as the de Broglie wavelength of the electrons, the wave will diffract.

    • Formula: ∆x∆p = λp/2 = h/2.

  • Electron in a box: If an electron is confined to a region of length L where it can only move back and forth, the uncertainty in position is ∆x = L/2, and thus, ∆p = h/4π∆x = h/2πL.

    • Kinetic energy = ​p²/2m = h²/8π²mL²

Pair production and annihilation

  • Pair production: close to an atomic nucleus, where the electric field is very strong, a photon with minimum energy given by E = 2mc² can produce a particle and its anti-particle (e.g. e- and e+), where m is the rest mass.

    • The atomic nucleus helps conserving energy and momentum.​

    • Any excess energy (above 2mc²) will be converted into kinetic energy of the particles

incidentphoton.jpg
  • Pair annihilation: when a particle collides with its anti-particle, producing 2 photons.

    • When​ they move in the opposite directions, the total energy of the system is ET = 2(mc² + EK) and the photons will travel in opposite direction.

Quantum Tunneling

  • Tunneling: A particle can effectively “borrow” energy from its surroundings, pass through a barrier and pay the energy back.

  • The energy required to go through a potential barrier is due to the uncertainty principle less than eV.

 
hillquantumclassical.jpg
  • The wave function is continuous despite the fact that the particle requires more energy to “jump” the barrier, which is borrowed from surroundings

  • Energy level remains unchanged after barrier, but the amplitude decreases since it is proportional to P(r).

  • In order to increase P(r), one may reduce:

    • The mass m of the particles ​

    • The width w of the barrier

    • The difference ∆E between the energy barrier and that of the particles

reducedprobnotreducedenergy.jpg
  • Responsible for the relatively low temperature fusion that occurs in the Sun and useful in scanning tunneling microscopes (STM).

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

  • Emitted perpendicularly to cathode,
  • Travel in straight lines
  • Carry energy
  • Possesses momentum
  • Deflected by electric and magnetic fields
  • Excite fluorescence
  • Ionise gas
  • Produce highly penetrating secondary radiation when suddenly stopped
  • Effect photographic plate

J.J. THOMSON’S E/M VALUE OF ELECTRON

This value of is for electron.

MILLIKAN’S OIL DROP METHOD 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 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 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 eqn (1), we have.
…(3)
and radius of drop from equation (1)
or …(4)
where ρ is 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 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.

PHOTOELECTRIC EFFECT (EINSTEIN’S PHOTOELECTRIC EQUATION)

In 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)
Fig. shows, when light strikes the cathode C, electrons are emitted & they are collected on anode A due to potential difference provided by 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 reaches 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 charge of electron (e = 1.6 × 10–19 coulomb).
But some features of 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 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 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 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 law of conservation of energy.
According to it
…(3)
or …(4)
or,
This is the 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 ultraviolet region of 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 (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 above failures of classical physics.
  • It is clear from eq. (3) that if energy of photon is less than the work function of 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 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).

VARIOUS GRAPHS RELATED TO PHOTOELECTRIC EFFECT

Photocurrent versus intensity of light graph
Photocurrent versus potential graph
Maximum kinetic energy versus potential graph
KEEP IN MEMORY
  1. 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.
  2. In photoelectric effect all photoelectrons do not have same kinetic energy. Their KE ranges from zero to Emax which depends on frequency of incident radiation and nature of cathode.
  3. The photoelectric effect takes place only when photons strike bound electrons because for free electrons energy and momentum conservations do not hold together.
  4. Cesium is the best photo sensitive material.
  5. Efficiency of photoemission
Therefore,
  1. Maximum velocity of emitted electrons
  1. Stopping potential

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 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 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 next topic that Compton effect confirm 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
de-Broglie wave is not an electromagnetic wave but the 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 wave 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 photon concept (i.e., particle nature of light) was discovered in 1923 by American Physicist, A.H. Compton. 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.
Diagram shows Compton scattering of an x-rays by free electron in a carbon target. The scattered x-rays has less energy than the incident x-rays. The excess energy is taken by recoiling electrons.
This effect cannot be explained by classical theory (by wave nature of light). According to classical model, when X-rays of frequency ν0 is incident on the material containing electrons, then electrons do oscillate & reradiate electromagnetic waves of same frequency ν0. Hence scattered X-rays has same frequency ν0 & same wavelength as that of incident X-rays.
Compton treated this processes as a collision between a photon & an electron. In this treatment, the photon is assumed as a particle of energy
E = hν 0 = 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
…(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 electron & 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 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.
KEEP IN MEMORY
  1. 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.
  2. 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 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 field.
  • 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 helps us to identify the element of target material.

 

When the atoms of the target material are bombard with high energy electrons (or hard X-rays), which possess enough energy to penetrate into the atom, knock out the electron of inner shell (say K shell, n = 1). When an electron is missing in the ‘K’ shell, an electron from next upper shell makes a quantum jump to fill the vacancy in ‘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 effective atomic number. Generally Ze is taken to be equal to Z – σ, where Z is 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 shell (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 series of spectral line.

 

The transitions of electrons from various outer shells to the innermost ‘K’ shell produces a group of X-rays lines called as 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’ shell 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 wavelength 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 wavelength are called soft X-rays.

MOSELEY’S LAW

Moseley used different elements as target in the X-ray tube. He found that Kα radiation of different elements were different
Mathematically,
where a and b depend on the particular line of the radiation
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 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.

 

KEEP IN MEMORY
  1. 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.
  2. Photoelectric effect doesn’t take place below the threshold frequency for the photo metal used.
  3. 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).
  4. The quantity is called Compton wavelength.
  5. The maximum wavelength change possible in Compton effect is 0.05Å.
  6. Compton effect can’t be observed for visible light rays.
  7. In Compton effect, the direction of recoil electron is given by
  8. The kinetic energy of recoil electron is given by
  1. de-Broglie wavelength of a particle of K.E., Ek is given by
  2. de-Broglie wavelength for a charged particle with charge q and accelerated through a potential difference V is given by
  3. 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 new dimension to X-rays crystallography. Various diffraction patterns are used to determine 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 body (cells are ionized and molecules are broken). So X-rays damages the malignant growths like cancer and tumours which are dangerous to life, when is 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 fracture in bones can be easily located. Similarly intestine and digestive system abnormalities are also detected by X-rays.

ATOMS

THOMSON’S ATOMIC MODEL

This model suggests an atom to be a tiny sphere of radius, containing the positive charge. The atom is electrically neutral. It contains an equal negative charge in the form of electrons, which are embedded randomly in this sphere, like seeds in a watermelon.
This model failed to explain
  • large scattering angle of α-particle
  • origin of spectral lines observed in the spectrum of hydrogen atom

ALPHA-PARTICLE SCATTERING AND RUTHERFORD’S NUCLEAR MODEL OF ATOM

In Rutherford α- particle scattering experiment a very fine beam of α-particle passes through a small hole in the lead screen. This well collimated beam is then allowed to fall on a thin gold foil. While passing through the gold foil, α-particles are scattered through different angles. A zinc sulphide screen is placed out the other side of the gold foil, this screen is movable, so as to receive the α-particles, scattered from the gold foil at angles varying from 0 to 180°. When a α-particle strikes the screen, it produces a flash of light.
FINDINGS
  • Most of the α-particles went straight through the gold foil and produced flashes on the screen as if there were nothing inside gold foil. This suggests that the most part of the atom is empty.
  • Few particles collided with the atoms of the foil which have scattered or deflected through considerable large angles. Very few particles even turned back towards source itself.
CONCLUSIONS
  • The entire positive charge and almost whole mass of the atom is concentrated in small centre called a nucleus.
  • The electrons revolving round the nucleus could not deflected the path of α-particles. This suggests that electrons are very light.
In 1911 Rutherford, proposed a new type of model of the atom. According to this model, the positive charge of the atom, instead of being uniformly distributed throughout a sphere of atomic dimension is concentrated in a very small volume at its centre. This central core, called nucleus, is surrounded by clouds of electrons makes the entire atom electrically neutral.
According to Rutherford scattering formula, the number of
α-particles scattered at angle θ by a target,
N ∝ cosec4 (θ/2)
Impact parameter
Distance of closest approach
 
RESULT OF RUTHERFORD SCATTERING EXPERIMENT
Nucleus is central, massive, positively charged core, its size of the order of 10–15 m, number of electrons surrounding nucleus is such that atom is electrically neutral.
Unit for nuclear dimension measurement : 1 fermi = 10–15m.

BOHR’S ATOMIC (HYDROGEN ATOM) MODEL

In 1913 Bohr gave his atomic theory primarily to explain, the spectra of hydrogen and hydrogen-like atoms. His theory, contained a combination of views from Plank’s quantum theory, Einstein’s photon concept and Rutherford model of atom. The Bohr theory can explain, the atomic spectra of hydrogen atom and hydrogen-like ions such as He+, Li2+, Be3+…(one electron ions). But his theory failed to explain, the spectra of more complex atom and ions.
BASIC POSTULATES OF BOHR’S MODEL
  1. The electron moves in circular orbits around the nucleus under the influence of coulombic force of attraction between the electron and the positively charged nucleus (as shown in figure below).
Bohr’s model of hydrogen atom
  1. The electron rotates about the nucleus in certain stationary circular orbits, for which the angular momentum of electron about the nucleus is an integral multiple of, where h is plank’s constant
i.e., Angular Momentum, …(1)
(where n = 1, 2, 3……… principal quantum number)
  1. When the electron is in one of its stationary orbits, it does not radiate energy, hence the atom is stable. These stationary orbits are called allowed orbits.
  2. The atom radiates energy when the electron “jumps” from one allowed stationery state to another. The frequency of radiation follows the condition
hν = Ei Ef …(2)
Where Ei and Ef are total energies of initial and final stationary states. This difference in energy (Ei -Ef) between two allowed stationary states is radiated/absorbed in the form of a packet of electromagnetic energy (hν – one photon of frequency ν) called a photon.
Now we calculate the allowed energies of hydrogen atom,
For moving an electron in a circular orbit the required centripetal force is provided by the coulomb force of attraction which acts between nucleus [Ze+, here Z = 1 (atomic number) for hydrogen atom] & electron (e),
i.e., …(3)
where is electrostatic constant & εo is permittivity of free space.
Eliminating v from eqn. (1) and (3) we obtain radius of nth orbit
(where n = 1, 2, 3 …..) …(4)
Equation (4) gives the radii of various orbits (have discrete values).
The smallest radius (also called Bohr radius) corresponds to n = 1 is
…(5)
r = 0.529 n2 Å for hydrogen atom and
r = 0.529 × for hydrogen like ions.
From equation (4) & (1) we obtain,
Velocity of electron in nth state
or (for hydrogen atom ) …(6)
for hydrogen like ions
The total energy of electron is given by
E = K.E. + P.E. = Kinetic energy + Potential energy
…(7)
(Allowed energy state)
After substituting numerical values in eqn.(7), we obtain
(for hydrogen atom) …(8)
(for hydrogen like ions)
The lowest energy state, or ground state, corresponds to n = 1 is
The next state corresponds to n = 2 i.e., first excited state has an energy, E = –3.4 eV

LIMITATIONS OF BOHR’S MODEL

  • It could not explain the spectra of atoms containing more than one electron.
  • There was no theoretical basis for selecting mvr to be an integral multiple of .
  • It involved the orbit concept which could not be checked experimentally.
  • It could not explain Zeeman & Stark effect and fine lines of spectra.
  • It was against de-Broglie concept and uncertainty principle.
KEEP IN MEMORY
  1. Total energy of electron = – Kinetic energy
  1. The reference level for potential energy has been taken as infinity
  2. The energy gap between two successive levels decreases as the value of n increases
  3. The radius difference between the successive orbit (or shells) increases as the value of n increases
  4. The velocity of electrons around the nucleus goes on decreasing as n increases
  5. The time period of the electron in an orbit
  6. Maximum number of spectral lines that can be emitted when an electron jumps from nth orbit is

ENERGY LEVELS AND THE LINES SPECTRA OF HYDROGEN ATOM

An energy level diagram of the hydrogen atom is shown in figure. The upper most level corresponding to n→, represents the state for which the electron is completely removed from the atom.
Some transitions for Lyman, Balmer & Paschen series are shown. The quantum numbers are at left & energies of levels are at right.
E = 0 for r = (Since n = )
If the electron jumps from allowed state ni to allowed state nf, then frequency of emitted photon is given by
…(1)
and the wavelength of emitted photon is
for hydrogen atom …(2)
and ( for H-like atoms)
where R = 1.096776 × 107m–1 is known as Rydberg constant. By using this expression we can calculate the wavelengths for various series (Lyman, Balmer…) in hydrogen spectrum, i.e.
  1. Lyman series ni = 1 & nf = 2, 3, 4……………
  2. Balmer series ni = 2, & nf = 3, 4, 5……………
  3. Paschen series ni = 3 & nf = 4, 5, 6…………..
  4. Brackett series ni = 4 & nf = 5, 6, 7……………
  5. P fund series ni = 5 & nf = 6, 7, 8……………
First three series of hydrogen atom are shown in figure.
But in practice, the value of Rydberg constant varies between and R
This is because in above calculations we assumed that electron revolves around a massive fixed nucleus of mass M. But in reality, the electron and nucleus each revolve round their common center of mass i.e., the motion of nucleus cannot be ignored. The correction for nuclear motion amounts to replacing electronic mass m by reduced mass μ which is defined as
…(3)
So total energy by taking this correction is
…(4)
If we are dealing with hydrogen like ions such as – He+, Li2+, Be3+, Be4+ (one electron ions), each can be considered as a system of two charges, the electron of mass m & charge –e & nucleus of mass M and charge +Ze, where Z is atomic number. The radii of circular orbits for these one electron ions can be written as
(n = 1, 2, 3…………) …(5)
and the allowed energies are given by
(n = 1, 2, 3………) …(6)

WAVELENGTH LIMITS IN VARIOUS SPECTRAL SERIES OF HYDROGEN ATOM

  1. For Lyman series (lies in ultraviolet region)
Here
  1. For Balmer series (lies in visible region)
Here
  1. For Paschen series (lies in infrared region)
Here
  1. For Brackett series (lies in infrared region)
Here
  1. For p-fund series (lies in infrared region)

Here
KEEP IN MEMORY
  1. The first line of Lyman series is when electron jumps from 2 → 1, It is also called α–line
The second line of lyman series is when electron jumps from 3 → 1, It is also called β–line
The limiting line of lyman series is when electron jumps from ∞ → 1
  1. Energy of electrons in different orbits in an atom varies inversely with the square of the number of orbits. So, energy of electrons increases (decreases in negative) as the orbit becomes higher.
  2. If energy of a particular orbit is E for H-atom then its value for a H-like atom with atomic number Z is given by E’ = E × Z2.
  3. If the radius of a particular orbit of H-atom is R then its value for a H-like atom is given by
  1. If velocity of an electron in a particular orbit of H-atom be v then its value for H-like atom is given by
v’= v × Z.
  1. If kinetic energy and potential energy of an electron in a particular orbit of H-atom be T and V respectively then their corresponding values for H-like atom are given by
    T’ = T × Z2 and V’ = V× Z2.
COMMON DEFAULT
🗶 Incorrect. Bohr’s formula for spectral lines does not differentiate between isotopes. For example the first line of Lyman series in hydrogen and deuterium will have same wavelength because
Correct. The value of R will be different for hydrogen and deuterium and therefore λ will be different for the two cases. In fact

Wavefunction

wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively).The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.

Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory. Rather, it is represented as a wave; here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. Panels (C–F) show four different standing-wave solutions of the Schrödinger equation. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves.

Schrödinger’s equation

The Schrodinger equation plays the role of Newton’s laws and conservation of energy in classical mechanics – i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrodinger equation will predict the distribution of results.

The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to generate the evolution of the wavefunction in time and space. The Schrodinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated.

Tunnelling, potential barrier and factors affecting tunnelling probability

  • Imagine throwing a ball at a wall and having it disappear the instant before making contact and appearing on the other side. The wall remains intact and the ball did not break through it. Believe it or not, there is a finite (if extremely small) probability that this even would occur. This phenomenon is called quantum tunnelling.

300px-tunneleffektkling1

  • The position of a particle is described as a wave function (see previous section).
  • From the graph above, the observable particle is most likely to be at the position where its wave function has the largest amplitude. However, although the amplitude of the wave function will decay exponentially, since the wave function does not reach an amplitude of zero, the wave function can exit the barrier. Once the wave function exits the barrier, its amplitude no longer decays. This means that a particle has a certain probability of bouncing off a barrier and a certain probability of passing through the other side.
FactorEffect towards tunnelling probability
Increase barrier lengthDecrease
Increase particle massDecrease

This explains how tunnelling is frequent in nanoscale but negligible at the macroscopic level.

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