Nuclear Physics

RANI CHANNAMMA UNIVERSITY BELAGAVI

B.Sc VI Semester

PHYSICS

UNIT-III

Nuclear Physics

Paper-I

Rani Channamma University

BSc VI Semester Physics Paper –I UNIT-III NUCLEAR PHYSICS

Alpha –rays: Theory of a decay, Range, Ionization, specific ionization andGeiger-Nuttal relation.

Beta – decay: Continuous beta spectrum, and Neutrino Hypothesis.

Nuclear Models: Liquid drop model- Explanation of semi empirical massformula, Explanation of nuclear fission on the basis of liquid drop model,Shell model (qualitative) and Magic numbers.

Nuclear Instruments: GM counter, Scintillation counter, Linear acceleratorand Cyclotron.Problems

Size of the atom and its constituents

Atom and its constituentsNucleus is the central part of the atom

where its diameter is of the order of 10-14 m

Nucleus = Protons + Neutrons

Protons are positively charged and neutronsare neutral particles

Number of protons = Z = Atomic number

Number of nucleons = A= (Protons + Neutrons)= Mass Number

Number of neutrons = A-Z

Rutherford first suggested theexistence of atomic nucleus when hewas postulating atomic model

Properties of the nucleus

Mass of the nucleus M =Zmp+(A-Z)mn-ΔMMass of the proton, mp = 1.67261×10-17 kgMass of the neutron,mn = 1.67492×10-17 kgΔM = Mass defect = Sum of all protons and neutrons – actual mass of nucleus

Charge on the nucleus is given by +Ze where e is the charge on each protone = +1.6 × 10-19 C

Radius of the nucleus R=R0A1/3 where R0=1.4×10-15 m and A=mass number

Alpha particles are doubly ionized Heatoms

Beta particles are fast moving electrons

Gamma rays are photons of wavelength10-11 m to 10-13 m

α –Decay

Most of the radioactive nuclei with atomic number greater than 82 spontaneouslydisintegrate. The process in which an α particle spontaneously ejected fromdisintegrating nucleus by loosing two protons and two neutrons is called α-decay.

This reduced mass corresponds to coulomb energy which is carried away by the α–particle.

Reduction in the nuclear mass = Mass of parent nucleus – Mass of daughter nucleus

Properties of the alpha (α) particles

1. They are positively charged particles. They are represented by 2He4

2. They are deflected by electric and magnetic fields.

3. Velocity of the alpha particle depend upon the nature of emitting radioactive element.

4. They produce heating effect when stopped

5. They affect photographic plates

6. They produce fluorescence in zinc sulphide, barium platinocyanide, etc.

7. They cause intense ionization when passes through gases. Their ionizing capacity is higher

than beta (β) and gamma (γ) rays.

8. They have lowest penetrating power as compared to beta and gamma rays when they pass

through matter.

Range of an α particle (R)

The range of the α particle is defined as the distance travelled through air at NTP before it loses its ionizing power.

Range R of an α particle depends on 1. Initial velocity of α particle2. The nature of radioactive element emitting α particle3. The pressure of the gas through which they pass

Range, R =a V03 where a = 9.6× 10-24

But V0= E0

2.08 ×10−𝟏𝟒

𝟏 𝟐Therefore R= 𝐛𝑬𝟎

𝟑 𝟐 where b= 3.18 ×10−𝟑

If Rs and R are the ranges of α particle in solid and in air respectively then

Rs = 𝟎.𝟑𝟏𝟐𝑹𝑨 𝟏 𝟐

ρwhere ρ is the density of the solid.

Specific Ionization at atmospheric pressure is the number of ion pairs per cm of the path produced by a radioactive element.

Geiger-Nuttal Relation(Connects Range and Energy with Disintegration Constant)

The emitted α particles do not carry same energy and hence their ranges are different.

Geiger and Nuttal observed that nuclides with shorter half-life emit α particles of higher energy and vice-versa

The Relation between disintegration constant (λ) and range (R) is log λ = A log R +B Where A and B are constants

The Relation between disintegration constant (λ)

and energy(E) is log λ = 𝟑

𝟐A log 𝑬𝟎 + B

Where A and B are constants

Gamow’s Theory of α Decay

α particles and the nucleus of the radioactive elements are bothpositively charged entities. Hence they repel each other.

Suppose if they are held at a certain distance then the potential energyof the α particle is given by

V(r)=𝟐𝒆 𝒁−𝟐 𝒆

𝟒𝝅𝜺𝟎𝒓For U238, Z=92 then V(r) =26 MeV

Thus according to this classical theory, 26 MeV of energy is required forα particle to escape from the nucleus.

But the emitted α particle by the U238 is only 4 MeV. Hence it isimpossible for an α particle to escape the nucleus as per the classicaltreatment.

Gamow’s Theory of α Decay

Gamow’s Theory of α Decay

Gamow theory successfully explained the escape of an α particle through the wavemechanical treatment and this process is known as tunneling effect.

Using WKB approximation the probability of escape of an α particle is obtained interms of

𝒍𝒐𝒈𝒆 λ = 𝒍𝒐𝒈𝒆𝑣

2𝑟0+ 𝒍𝒐𝒈𝒆P But 𝒍𝒐𝒈𝒆P = 2.97𝒁 𝟏 𝟐𝒓𝟎

𝟏 𝟐 − 𝟑. 𝟗𝟓𝒁𝑬− 𝟏𝟐

When the 𝒍𝒐𝒈𝒆 is removed we obtain

𝒍𝒐𝒈𝟏𝟎 λ = 𝒍𝒐𝒈𝟏𝟎𝑣

2𝑟0+ 1.29𝒁 𝟏 𝟐𝒓𝟎

𝟏 𝟐 − 𝟏. 𝟕𝟐𝑬− 𝟏𝟐 (Theoretical form of the Geiger-

Nuttal relation)

v is velocity of α particle moving inside the potential wellE is the kinetic energy of the α particle.

β Decay - Properties

1. β particles are fast moving electrons 2. They are deflected by electric and magnetic fields. 3. Their velocity ranges from 1% to 99% of velocity of light4. They affect photographic plates but the effect is much higher than α particles.5. They produce fluorescence in barium platinocyanide, calcium tungstate, etc.6. They produce ionization in air and it is 1/100 times less effective than α particles.7. They are more penetrating than α particles and can pass through 1 mm of Al sheet.8. The range of β particles is greater than the α particles.

Continuous Spectrum of β Particles

The kinetic energy of the emitted β particles is calculatedfrom the observations made from magnetic spectrometer.

Number of β particles per unit time are counted using theGM counter.

The above two measurements are plotted against eachother. The graph revealed two observations

1. The β particles have continuous distribution ofenergies.

2. Only a small number of electrons have this maximumkinetic energy, represented as KEmax on the graph;most of the electrons emitted have kinetic energieslower than that predicted value.

Conservation of both linear momentum and angular momentum are violated

Neutrino Hypothesis The existence of additional particle is predicted by Pauli in 1930 to conserve the

momentum and to carry the missing energy.

Later Enrico Fermi developed theory and called the missing particle as neutrino.This particle has (a) zero charge (b) intrinsic spin of ½ (c) zero rest mass

Thus missing neutrino finally balanced the charge and angular momentum.

With the introduction of the neutrino the correct form of the β decay is shown below.

𝒁𝑨𝑿 𝒁+𝟏

𝑨𝑿 + 𝒆− + ν

𝒁𝑨𝑿 𝒁−𝟏

𝑨𝑿 + 𝒆+ + ν

Since β particles and neutrino simultaneously emitted the kinetic energy is sharedamong them.

Nuclear ModelsThe phenomenon such as stability, spin, magnetic moment etc., are explained on the

basis of nuclear models. Two most successful models are

1. Liquid drop model 2. Shell model

Liquid Drop Model –Assumptions

1. The nuclei of all elements are considered to be behave like a liquid drop ofincompressible liquid of very high density

2. Drop of a liquid is spherical due to surface tension. Similarly in an equilibriumstate the nuclei of atoms remain spherically symmetric under the action of strongattractive nuclear forces.

3. The density of a nucleus is independent of its size just like the liquid drop.

4. The motion of the nucleons inside the nucleus is similar to the molecules of theliquid within the spherical drop.

5. The binding energy per nucleon of a nucleus is constant just like the latent heat ofvaporization of a liquid.

Weizsacker derived the formula for mass M and binding energy EB of a nucleus whichis given by M =Zmp+(A-Z)mn-𝑬𝑩

where 𝐸𝐵 is binding energy 𝑬𝑩 = 𝑬𝒗 − 𝑬𝒔 − 𝑬𝒂 − 𝑬𝒄 ± 𝜹(𝑨, 𝒁)

1. Volume Energy 𝐸𝑣 𝛼 𝐴 then 𝑬𝒗 = 𝒂𝒗𝑨 where 𝑎𝑣 is constant

2. Surface Energy 𝐸𝑠 𝛼 𝐴 2 3 then 𝑬𝒔 = −𝒂𝒔𝑨

𝟐 𝟑 where 𝑎𝑠 is constant

3. Asymmetry Energy 𝐸𝑎 𝛼(𝐴−2𝑍)

𝐴then 𝑬𝒂 = 𝒂𝒂

(𝑨−𝟐𝒁)

𝑨where 𝑎𝑎 is constant

4. Coulomb Energy 𝐸𝑐 𝛼𝑍(𝑍−1)

𝐴 1 3then 𝑬𝒄 = −𝒂𝒄

𝑍(𝑍−1)

𝑨 𝟏 𝟑where 𝑎𝑐 is constant

5. Pairing Energy 𝜹 𝑨, 𝒁 This term 𝑒𝑥𝑝𝑙𝑎𝑖𝑛 𝑡ℎ𝑒 𝑠𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦 of the nucleus 𝑤ℎ𝑒𝑡ℎ𝑒𝑟𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑟𝑜𝑡𝑜𝑛𝑠 𝑎𝑛𝑑 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑟𝑒 𝑒𝑣𝑒𝑛/odd in number. If both or eventhen nucleus is more stable and vice versa.

Weizsacker Semiempirical Mass Formula

Now substitute all the termsin the equation of bindingenergy 𝐸𝐵 then Weizsackermass formula becomes

Weizsacker Semiempirical Mass Formula

Explanation of Nuclear Fission on the basis Liquid Drop Model

In 1939 Bohr and Wheeler gave the theoretical treatment of nuclear fission on the basis of liquid drop model. The nucleus of an atom is considered to be a charged liquid drop

In an equilibrium state the nucleus remains spherically symmetric.

The nucleons on the surface develop surface tension effects similar to the liquid drop.

When heavy nucleus (U235)is excited through the capture of a thermal neutron, the shape of thenucleus distorts from spherical shape to ellipsoidal shape.

If the excitation energy is high then nucleus may attain dumb-bell shape.

If the energy given is above threshold energy then nucleus separates into two nearly equal massescausing nuclear fission.

If the given energy to the nucleus is low then fission does not occur. In such cases excited nucleus emitseither a 𝛾 ray or a neutron and returns to a stable state.

Merits of Liquid Drop Model1. Succeeded in explaining nuclear reactions and nuclear fission2. Succeeded in the accurate calculation of atomic masses and binding energies.

2. The Shell Model

According to this model two protons with opposite spin and two neutrons with opposite spin are accommodated in a shell. Such a system is more tightly bound than other shells.

Magic Numbers

In 1917 Harkins found that nuclei with even number of protons or neutrons are more stable than those which have odd numbers.

In 1934 Elasser observed that the nuclei with protons or neutron numbers 2, 8, 20, 50, 82 and 126 are extraordinarily stable than the other nuclei. These numbers are called magic nuclei.

If both proton numbers and neutron numbers are magic numbers then the nucleus is called doubly magic.

The proton or neutron numbers 14, 28, 40 show less stability than that of the magic nuclei and are called semi magic nuclei.

2. The Shell Model (Independent Particle Model)

According to this model protons and neutrons are grouped in shells.

Assumptions

1. Nucleons move in a nucleus independently in an average potential.

2. Pair of nucleus have zero spin and zero magnetic moment.

3. The paired nucleons from the inert core do not contribute to the properties of the

nucleus.

4. The nucleon are characterized by the unpaired nucleon.

5. The average potential field determined by all the nucleons except one nucleon which

determines the quantum states of the individual nucleons

Nuclear Instruments1. Geiger Muller (GM) Counter 2. Scintillation Counter3. Linear Accelerator 4. Cyclotron

1. Geiger Muller (GM) Counter

Construction

C is a metal chamber containing air/gas at apressure of about 10 cm of Hg.

W is fine tungsten wire stretched along the axisof the tube.

EE is the end points of the tungsten wire madefrom ebonite.

R is the resistance which connects tungsten wireto a positive terminal of a battery.

Working of GM Counter

When an ionizing particle (Ex. Α particles) enters the counter, ionization takesplace and few ions are produced.

Once the applied potential difference is strong an avalanche of electrons movestowards the central wire.

The critical potential is lowered causing a sudden discharge through R.

Vacuum tube circuits amplifies the potential difference developed across R and asingle particle can be registered.

The sudden pulse of discharge sweeps away the ions from the chamber and thecounter is ready to register the arrival of the next particle.

The voltage characteristics of GM counter

The zero count rate at the threshold voltage meansGM tube does not work.

As the applied voltage increases count rate reachesplateau where counting rate is independent of appliedpotential difference.

Continuous discharge takes place in the GM tubewhen the applied voltage is further increased. Thisregion must be avoided for the experiment.

The efficiency of the counter is defined as the ratio ofthe observed counts/second to the number ofionizing particles entering the counter per second.

Counting efficiency is defined as the ability of its counting.

Counting Efficiency = ε = 1- 𝒆𝒔𝒍𝒑 where s =Specific ionization at 1 atm pressure, l =Pathlength of the ionization particle in the counter, p = Pressure in atmospheres.

Scintillation CounterPrinciple When a charged particles falls on a fluorescent material flashes of light areproduced and these flashes are converted into electrical signal.Construction

Construction And Working

The various parts of the scintillation counter are as shown in the diagram. Phosphors such as inorganicsalts, alkali halides (like sodium iodide activated with thallium), crystalline organic materials, solutionsof organic compounds, etc. are used.

Photomultiplier tube (PMT) device is one which converts scintillations (flashes of light) into a amplifiedelectrical pulses.

Light from the phosphors strikes cathode which is made of Antimony and Cesium and ejects electronsthrough photoelectric effect. The PMT consists of several electrodes called dynodes which multiply theincident photoelectrons due to the increased applied potential to them.

When a, b and g particles falls on phosphor, the atoms inside it are excited. These excited atoms givesout photons in blue and ultra violet region during their deexcitation.

These emitted photons strikes the photocathode inside the photomultiplier tube causing the ejection ofphotoelectrons. The number of these photoelectrons gets multiplied each time when they fall on adynode. Finally millions of electrons are emitted due to the dynodes in the photomultiplier tube.However, scintillation counter measures all the charged particles but a particles causes more ionizationthan b and g because of their inherent properties. However a particles can be stopped by using 0.01 cmAl sheet to obtain contributions only from b particles and g rays.

Linear Accelerator It consists of a series of co-axial hallow metal cylinders T1, T2, T3, T4, etc,.

They are arranged linearly in a glass vacuum chamber. The alternate cylinders are connectedtogether like the odd numbered to one terminal and even numbered cylinders to one terminal of RFgenerator.

Thus in one half cycle if tubes T1 and T3 are negative T2 and T4 will be positive. The ions areaccelerated only in the gap between the tubes where they are acted upon by the electric field presentin the gas.

The ions travel with constant velocity in the field free space inside the drift tubes. Positive ions enteralong the axis of the accelerator from an ion source through an aperture A.

Figure: Linear Accelerator

Then velocity v1 of the ion on reaching the drift tube is given by𝟏

𝟐𝒎𝒗𝟏

𝟐 = 𝑽𝒆 or 𝒗𝟏 =𝟐𝑽𝒆

𝒎

Energy of the ionIf n be the number of gaps through the ion travels in the accelerator and vn be the nal velocity acquiredby the ion then

Velocity through the nth tube= 𝒏𝟐𝑽𝒆

𝒎

Kinetic energy acquired by the ion =𝟏

𝟐𝒎𝒗𝟏

𝟐 = n 𝑽𝒆

The final energy of the ions depends upon (i) the total number of gaps, (ii) the energy gained in eachgap

The Limitations of Accelerators

1. Large accelerator length and difficulty in maintaining its vacuum chamber

2. The ion current available is in the form of short interval pulses.

Cyclotron

The cyclotron consists of two hollow semicircularmetal boxes D1 and D2.

A source of ions is located near the mid point of thegap between D1 and D2.

D1 and D2 are insulated from each other and areenclosed in another vacuum chamber.

D1 and D2 are connected to a powerful radiofrequency oscillator.

The whole apparatus is placed between the polepieces of a strong electromagnet.

Theory Under the action of applied magnetic field the ions travel from the center of the chamber in a circular path of radius r is given byBev=

𝒎𝒗𝟐

𝒓…..(1) where B is flux density of the magnetic field.

𝒓 =𝒎𝒗

𝑩𝒆........ (2) or

𝒗

𝒓=𝑩𝒆

𝒎= 𝒘 … . . (3) where w is angular velocity of the ion in its circular path.

Now the time taken by the ion to travel the semicircular path is

T = 𝝅

𝒘=

𝝅

𝑩𝒆 m=𝝅𝒎

𝑩𝒆……(4)

From equation (4) it is clear that the time taken by the ion to describe semicircle is independent of both the radius of the path (r) and velocity (v) of the ion. This ion thus spirals round in circles of increasing radius and acquires high energy. The ion will finally come out of the D1 and D2 in the direction indicated through the window.

Energy of the ion The equation for the motion of the ion in a magnetic field is

𝑩𝒆𝒗𝒎𝒂𝒙 =𝒎𝒗𝒎𝒂𝒙

𝟐

𝒓𝒎𝒂𝒙

Where 𝒓𝒎𝒂𝒙 is the radius of the outermost orbit and v𝒎𝒂𝒙 is the maximum velocity of the ion in the finalorbit.

𝒗𝒎𝒂𝒙 = 𝑩𝒆

𝒎𝒓𝒎𝒂𝒙......... (5)

The energy of the ion

E=𝟏

𝟐𝒎𝒗𝒎𝒂𝒙

𝟐 =𝑩

2

𝒆𝟐

𝒎𝒓𝒎𝒂𝒙𝟐 …………… (6)

From the equation (4) the period (T) of the ion to describe the semicircular path is

𝝅𝒎

𝑩𝒆=𝑇

2T=

𝟐𝝅𝒎

𝑩𝒆

The frequency of the ion is f=1𝑇 =𝑩𝒆

𝟐𝝅𝒎…….. (7)

Therefore energy of the ion is given by

E=2π2𝒓𝒎𝒂𝒙𝟐 𝑓2𝑚……… (8)

The particles ejected out of the cyclotron not continuously but as pulsed streams.

Limitations of the cyclotronThe energy to which particles can be accelerated in a cyclotron are limited by the relativistic increase of mass with velocity.

Therefore frequency of the ion is

f =𝑩𝒆

𝟐𝝅𝒎=

𝑩𝒆

𝟐𝝅𝒎𝟎

𝟏−𝒗𝟐

𝒄𝟐

f =𝑩𝒆

𝟐𝝅𝒎=𝑩𝒆 𝟏− 𝒗

𝟐

𝒄𝟐

𝟐𝝅𝒎𝟎

The frequency of the rotation of the ion decreases with increase in velocity. Then ions lag behind theapplied potential and finally they are not accelerated further. Therefore the energy of the ions producedby the cyclotron is limited. This limitation can be overcome in the following two ways.

(1) Field Variation Here the frequency of the ion can be kept constant by increasing the magnetic field B.Suh machines are known as Synchrotron

(2) Frequency modulation The magnetic field is kept constant and frequency of the applied electric field is varied Such a devie is called a synchrocyclotron.

References and image Credit: Google and other books

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