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## Radioactivity and binding energy of nuclei, Physics tutorial

:Properties of the nucleusThe main properties of any nucleus comprise mass, mass defect, binding radius, binding energy, energy per nucleon, and angular momentum. Orbital radii (or radius of the orbit) in Bohr model is: ε

_{0}n^{2}h^{2}/πme^{2}and orbital speed in Bohr's model is: V_{n }= e^{2}/ε_{0}2nh where all symbols maintained usual meanings as has been stated.:Binding EnergyBinding Energy is energy required to add to nucleus to separate it in individual protons and neutrons (or nucleons). I.e. Binding energy EB = [(Mass of protons + Neutron) - Mass of

^{N+Z}_{Z}X]c2 or Binding energy (E_{B}= ZM_{H}+ NM_{n}-^{A}_{Z}M)c^{2}where M is the mass defect and c is speed of light.:Binding energy per nucleonBinding energy per nucleon is binding energy divided by total number of protons and neutrons.

I.e binding energy per nucleon = Binding energy/total number of protons and neutrons

Total mass of separated neutrons and protons is greater than mass of nucleus

:Mass defectMass defect is difference in mass between total mass of individual protons and neutrons and mass of nucleus. Mass and energy are related by E = mc

^{2}where E = mass, c = speed of light = 3 X 10^{8}m/s. Unit of mass is atomic mass unit (amu).1 amu = 1.66 X 10

^{-27}kg. Unit of energy in modern physics is Megaelectonvolt (MeV). 1 MeV = electronic charge X 10^{6 }Joules.1.6 x 10

^{-19}x 10^{6}=1.6 X 10

^{-13}J1 amu = 1.65 X 10

^{-27}X ( 3 X 10^{8})^{2}= 1.49 X 10

^{-10}J.= 931 MeV

:Nuclear StabilityStability of atom depends on both number of protons and neutrons in the atom. For stable nuclides the given points emerge.

i) Lightest nuclides have almost equal number of protons and protons.

ii) Heaviest nuclides need more neutrons than protons, heaviest having about 50% more.

Most nuclides have both the even number of protons and neutron. This signifies that and alpha particle that has two protons and two neutrons (

^{4}_{2}He or^{4}_{2}α) very stable and so also oxygen (^{16}_{8}O) , Silicon(^{28}_{14}Si),Iron(^{56}_{28}Fe) etc.For unstable nuclides, given can be noted.

i) They continue to disintegrate until new stable nuclides is created.

ii) The unstable nuclide nuclear can experience β

^{-1}decay so as to give the increase of proton number (in which neutron changes to the proton and electron). It's neutron - to - proton ratio is thus decreased.iii) Unstable nuclide can suffer a decay so that its proton number decreases and its neutron - to - proton ratio increases. In heavy nuclides, it happens by emission of alpha particles (i.e

^{4}_{2}He or^{4}_{2}α)Radioactivity::Radioactive decay lawRadioactivity is spontaneous emission of α, β, and γ by unstable nuclides to become stable.

Radioactive decay law: Law defines that rate of disintegration of the given nuclide at anytime t is directly proportional to number of nuclei N of nuclide present at that time'.

Mathematically, law is

-dN/dt α N

Negative (-ve) sign signifies that N decreases (or is decaying) as time t increases.

DN/dt = -λN.............................Eq.1

Where λ is radioactive decay constant and is proportionality constant. Unit of λ is disintegration per second.

Integrating Eq.1 above

dN/N = -λdt

∫

_{No}^{N}dN/N = -λ∫_{0}^{t}dtWhere N

_{0}= number of undecayed nuclei at time t = 0, N = number of undecayed nuclei at present time t.[ln N]

_{N0}^{N}= -λtln N - lnN

_{0}= -λtln(N/N

_{0}) = -λtN/N

_{0 }= e^{-λt}N = N

_{0}e^{-λt}...........................................Eq.2Eq.2 is known as decay law that defines that radioactive substance decays exponentially with time.

:Half - lifeHalf - life is time taken for radioactive nuclei to decay by half its original (or initial) quantity. Half - life is denoted by T

_{1/2}.From Eq.2, if N = N

_{0}/2 then t = T_{1/2}and Eq.2 becomes:N

_{0}/2 = N_{0}e^{-λT1/2}1/2 = e

^{-λT1/2}Take log of both sides

ln (1/2) = -λT

_{1/2}ln 1- ln 2 = --λT

_{1/2}0 - ln 2 = -λT

_{1/2}-ln 2 = -λT

_{1/2}T

_{1/2 }= ln2/λT

_{1/2}= 0.693.................................Eq.3:Radioactive seriesRadioactive Series: Radioactive series is the sequence of nuclides, each of which transforms by radioactive disintegration in the next, until the stable nuclide is attained. It is also called as decay series or disintegration series.

Accelerators and detectors:Accelerators are machines made to accelerate charge particles like protons to high speed by means of potential differences of hundreds of thousands of volts for nuclear physicists' experiments.

Kinds of accelerators:1) Van de Graff generator

2) Linear accelerator

3) Cyclotron

4) Synchrotron

:DetectorsDetectors are sensing devices that sense presence of charges or changes in something like radiation or pressure.

In the nuclear radiation detector energy is transform from radiation to atoms of detector and may cause:

i) Ionization of the gas in ionization chamber e.g. Geiger Muller tube, cloud or bubble chamber.

ii) Exposure of a photographic emulsion

iii) Fluorescence of the phosphor as in the scintillation counter.

iv) Mobile charge carriers in the semiconductor solid state detector.

Radiation is therefore detected by effect it produces.

Kinds of detectors:1) Ionization Chamber

2) Geiger Muller tube

3) Cloud Chamber

4) Bubble Chamber

5) Scintillation counter

6) Solid state detector.

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