Atomic Nucleus

Modern Physics of Class 12

ATOMIC NUCLEUS

The atomic nucleus consists of two types of elementary particles, viz. Protons and neutrons. These particles are called nucleons.

The proton (denoted by p) has a charge +e and a mass mp ≈ 1.6726 × 10-27 kg, which is approximately 1840 times larger than the electron mass. The proton is the nucleus of the simplest atom with Z = 1, viz. the hydrogen atom.

The neutron (denoted by n) is an electrically neutral particle (its charge is zero). The neutron mass mn ≈ 1.6749 × 10-27 kg. The fact that the neutron mass exceeds the proton mass by about 2.5 electron masses is of essential importance. It follows from this that the neutron in free state (outside the nucleus) is unstable (radioactive). During the time equal on the average to 12 min, the neutron spontaneously transforms to the proton by emitting an electron (e-) and a particle called the antineutrino (v). This process can be schematically written as follows:

n → p + e- + v.

The most important characteristics of the nucleus are the charge number
Z (coinciding with the atomic number of the element) and the mass number A. The charge number Z is equal to the number of protons in the nucleus, and hence it determines the nuclear charge equal to Ze. The mass number A is equal to the number of nucleons in the nucleus (i.e. to the total number of protons and neutrons). 

Nuclei are symbolically designated as ZXA

where X stands for the symbol of a chemical element. For example, the nucleus of the oxygen atom is symbolically written as Atomic Nucleus or 8O18.

Most of the chemical element have several types of atoms differing in the number of neutrons in their nuclei. These varieties are called isotopes. For example, oxygen has three stable isotopes: Atomic Nucleus, Atomic Nucleus and Atomic Nucleus. In addition to stable isotopes, there also exist unstable (radioactive) isotopes.

Atomic masses are specified in terms of the atomic mass unit or unified mass unit (u). The mass of a neutral atom of the carbon isotope 6C12 is defined to be exactly 12 u.

1u = 1.66056 × 10-27 kg = 931.5 MeV

Example 16.14

(a) Calculate the value of 1 u from Avogadro’s number.

(b) Determine the energy equivalent of 1u. 

Solution 

(a) One mole of C12 has a mass of 12 g and contains Avogadro’s number, NA, of atoms. 

By definition, each C12 has a mass of 12 u. 

Thus, 12 g corresponds to 12 NA u   which means 

1u = Atomic Nucleus

or 1u = 1.66056 × 10-27 kg 

(b) From Einstein relation  E = mc2

∴ E = (1.66056 × 10-27) (3 × 108)2 = 1.4924 × 10-10

Since 1eV = 1.6 × 10-19

∴ E = 931.5 MeV 

Hence 1u = 931.5 MeV

The shape of nucleus is approximately spherical and its radius is approximately related to the mass number by 

R ≈ 1.2 A1/3  fm

where 1 fermi (fm) = 10-15

Example 16.15 

Find the mass density of the oxygen nucleus 8O16.

Solution 

Volume V = Atomic Nucleus = 1.16 × 10-43 m3

Mass of oxygen atoms (A = 16) is approximately 16 u.

Therefore, density is ρ = m/v

or ρ = Atomic Nucleus = 2.3 × 1017 kg/m3

This is 1014 times the density of water.

Binding Energy 

The rest mass of the nucleus is smaller than the sum of the rest masses of nucleons constituting it. This is due to the fact that when nucleons combine to form a nucleus, the binding energy of nucleons is liberated. The binding energy is equal to the work that must be done to split the nucleus into particles constituting it.

The difference between the total mass of the nucleons and the mass of the nucleus is called the mass defect of the nucleus:

Δm = [Zmp + (A − Z)mn] − mnuc.

Multiplying the mass defect by the square of the velocity of light, we can find the binding energy of the nucleus:

BE = Δmc2 = [(Zmp + (A – Z)mn) – mnuc]c2     J (16.25)

If the masses are taken in atomic mass unit, the binding energy is given by 

BE = [ZmP + (A – Z)mn – mnuc]931.5  MeV (16.26)

Dividing the binding energy by the number A of nucleons in the nucleus, we obtain the binding energy per nucleon. Fig.(16.13) shows the dependence of the binding energy per nucleon BE/A on the mass number A of the nucleus. Nucleons in nuclei with mass numbers from 50 to 60 have the highest binding energy. The binding energy per nucleon for these nuclei amounts to 8.7 MeV and gradually decreases with increasing A. For the heaviest natural element − uranium − it amounts to 7.5 MeV. Figure shows that when a heavy nucleus (with A ≈ 240) splits into two nuclei with A = 120, the released energy is of the order of 1 MeV per nucleon i.e. 240 MeV per parent nucleus. It should be mentioned for comparison that when a carbon atom is oxidized (burnt) to CO2, the energy of the order of 5 eV is liberated, which is smaller than the energy released in fission of a uranium nucleus by a factor of 50 millions. 


Atomic Nucleus

It also follows from Fig(16.13) that the fusion (synthesis) of light nuclei into one should be accompanied by the liberation of a huge energy. For example, the fusion of two nuclei of heavy hydrogen  1H2 (this nucleus is called a deuteron) into a helium nucleus 2He4 would yield an energy equal to 24 MeV.

The forces binding nucleons in a nucleus manifest themselves at distances < 10-15 m.
In order to bring together two positively charged deuterons to such a distance, their Coulomb repulsion should be overcome. For this, the deuterons must have a kinetic energy equivalent to their mean energy of thermal motion at a temperature of the order of 109 K. For this reason, the fusion reaction of nuclei is also called a thermonuclear reaction. Actually, some thermonuclear reactions may occur at a temperature of the order of 107K. This is due to the fact that there is always a certain number of nuclei whose energy considerably exceeds the mean value.

Example  16.16

Find the binding energy of  ? Also find the binding energy per nucleon. 

Solution

One atom of 612C consists of 6 protons, 6 electrons and 6 neutrons. The mass of the uncombined protons and electrons is the same as that of six  atoms (if we ignore the very small binding energy of each proton-electron pair).

Mass of six  atoms = 6 × 1.0078 = 6.0468 u

Mass of six neutrons = 6 × 1.0087 = 6.0522 u 

Total mass of component particles = 12.0990 u

Mass of  612C atom = 12.00004

Mass defect = 0.0990 u 

Binding energy = (931)(0.099) = 92 MeV

Binding energy per nucleon = 92/12 = 7.66 MeV

Example 16.17

A neutron breaks into a proton and electron. Calculate the energy produced in this reaction in MeV. Mass of an electron = 9 × 10-31kg, Mass of proton = 1.6725 × 10-27kg, Mass of neutron 1.6747 × 10-27kg. Speed of light = 3 × 108 m/s. 

Solution

on11H1 + -1eo

Mass defect (Δm) = [Mass of neutron – (mass of proton + mass of electron)]

= [1.6747 × 10-27 – (1.6725 × 10-27 + 9 × 10-31)] 

= 0.0013 × 10-27 kg 

∴ Energy released   Q = Δm c2

Q = (0.0013 × 10-27) × (3 × 108)2 = 1.17 × 10-13 J

= 1.17 x 10-13 /1.6 x 10-19= 0.73 × 106eV = 0.73 MeV 

NUCLEAR  REACTION

A nuclear reaction in which a collision between particle a and nucleus X produces Y and particle b is represented as a + X → Y + b 

The reaction is sometimes expressed in the shorthand notation X(a, b)Y. 

Reactions are subjected to the restrictions imposed by the conservation of charge, energy, momentum and angular momentum. 

Energy of A Reaction

Atomic Nucleus

Atomic Nucleus

Initial energy:  Ei = m1c2 + m2c2 + K1 + K2

Final energy:    Ef = m3c2 + m4c2 + K3 + K4

Since Ei = Ef  (energy conservation) 

∴ [(m1 + m2) - (m3 + m4)]c2 = (K3 + K4) - (K1 + K2

The energy, that is released or absorbed in a nuclear reaction is called the Q - value or disintegration energy of the reaction.

Q = [(m1 + m2) - (m3 + m4)]c2  J (16.27 a)

or Q = [(m1 + m2) - m3 + m4)] 931.5 u (16.27 b)

If Q is positive, rest mass energy is converted to kinetic mass energy, radiation mass - energy or both, and the reaction is exoergic.

If Q is negative, the reaction is endoergic. The minimum amount of energy that a bombarding particle must have in order to initiate an endoergic reaction, is called Threshold Energy Eth .

Eth = -QAtomic Nucleus (16.28)

where m1 = mass of the projectile

m2 = mass of the target.

Example 16.18

Neon - 23  beta decays in the following way : 

Atomic Nucleus

Find the minimum and maximum  kinetic energy that the beta particle Atomic Nucleus can have.  The atomic masses of  23Ne  and 23Na are 22.9945 u and 22.9898 u, respectively.

Solution

Reactant Products

Atomic Nucleus 22.9945 - 10me Atomic Nucleus 22.9898 - 11 me

-me

 

Total 22.9945 – 10 me Total 22.9898 – 10 me

 

Mass defect = 22.9945 - 22.9898 = 0.0047 u 

Q = (0.0047)(931) = 4.4 MeV 

The β - particle and neutrino share this energy. Hence the energy of the β-particle can range from 0 to 4.4 MeV.


Example 16.19

How much energy must a bombarding proton possess to cause the reaction.

Atomic Nucleus

Solution

Since the mass of an atom include the masses of the atomic electrons, the appropriate number of electron masses must be subtracted from the given values.

Reactants Products

Atomic Nucleus 7.01600 - 3 me Atomic Nucleus 7.01693 - 4me

Atomic Nucleus 1.0783 - 1 me Atomic Nucleus 1.0866

  

Total 8.02383 - 4me Total 8.02559 - 4me 

The Q-value of the reaction

Q = -0.00176 u = −1.65 MeV

The energy is supplied as kinetic energy of the bombarding proton. The incident proton must have more than this energy because the system must possess some kinetic energy even after the reaction, so that momentum is conserved.

With momentum conservation taken into account, the minimum kinetic energy that the incident particle can be found with the formula.

Eth = -Atomic Nucleus MeV

Example 16.20

In a nuclear reactor, fission is produced in 1 g for U235(235.0439 u) in 24 hours by a slow neutron (1.0087 u). Assume that 35Kr92 (91.8973 u) and 56Ba141 (140.9139 amu) are produced in all reactions and no energy is lost.

(a) Write the complete reaction 

(b) Calculate the total energy produced in kilowatt hour. Given 1 u = 931 MeV.

Solution

The nuclear fission reaction is 92U235 + on156Ba141 + 36Kr92 + 3 on1

Mass defect Δm = [(mu + mn) – (mBa + mKr + 3mn)]

Δm = 256.0526 – 235.8373 = 0.2153 u 

Energy released  = 0.2153 × 931 = 200 MeV

Number of atoms in 1 g = 6.02 x 1023/235= 2.56 × 1021

Energy released in fission of 1 g of U235 is Q = 200 × 2.56 × 1021 = 5.12 × 1023 MeV 

= (5.12 × 1023) × (1.6 × 10-13) = 8.2 × 1010

=  8.2 x 10 10/3.6 x 106 = 2.28 × 104 kWh 

Example 16.21

It is proposed to use the nuclear fusion reaction:

1H2 + 1H2 = 2He4

in a nuclear reactor of 200 MW rating. If the energy from above reaction is used with a 25% efficiency in the reactor, how many grams of deuterium will be needed per day. (The masses of 1H2 and 2He4 are 2.0141 and 4.0026 u respectively). 

Solution

Energy released in the nuclear fusion is 

Q = Δmc2 = Δm(931)MeV 

⇒ Q = (2 × 2.0141 – 4.0026) × 931MeV = 23.834 MeV = 23.834 × 106eV 

Since efficiency of reactor is 25% 

So effective energy used = 25 / 100 × 23.834 × 106 × 1.6 × 10-19 J = 9.534 × 10-13

Since the two deuterium nucleus are involved in a fusion reaction, therefore, energy released per deuterium is 9.534 x 10 -13/2

For 200MWpower per day

number of deuterium nuclei required = Atomic Nucleus = 3.624 × 1025

Since 2g of deuterium constitute 6 × 1023 nuclei, therefore amount of deuterium required is 

g = Atomic Nucleus = 120.83 g/day

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