E.3 Radioactive Decay

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2024/07/05 08:52

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E.3.1 Radioactive Decay

Radioactive decay and Rate of decay

Random	Spontaneous
We cannot predict which unstable nucleus in sample will decay or when there will be a decay	We cannot affect the rate of decay in a given sample any way

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Rate of decay is proportional to the number of particles left have not decayed

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Activity A : rate of radioactive decay

A = \frac{\Delta N}{\Delta t}

Particles and Their Symbols

Particle	Symbol
Nucleus	${}^{A}_{Z}\text{X}$
Proton	${}^{1}_{1}\text{P}$
Neutron	${}^{1}_{0}\text{n}$
Electron (beta minus particle)	${}^{0}_{-1}\text{e}$
Positron (beta plus particle)	${}^{0}_{+1}\text{e}$
Neutrino	${}^{0}_{0}{\nu}\, or\, {\nu}$
Anti-neutrino	${}^{0}_{0}\bar{\nu}\, or\, \bar{\nu}$
Alpha particle (helium nucleus)	${}^{4}_{2}{\alpha}$
Photon	${}^{0}_{0}{\gamma}\, or \,\gamma$

Types of decay:

Alpha decay

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An unstable nuclei emits an alpha particle (the same configuration as helium nucleus)

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Proton number and nucleon number must be conserved.

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For example,

{}^{238}_{92}\text{U} →{}^{4}_{2}\text{He} + {}^{234}_{90}\text{Th}\\ {}^{234}_{90}\text{Th}→{}^{4}_{2}\text{He} + {}^{230}_{88}\text{Ra}

Negative beta decay

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An electron and antineutrino are emitted from the parent particle

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Proton number remains the same, but the nucleon number increases by one

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For example,

{}^{131}_{53}\text{I}→{}^{0}_{-1}e + {}^{131}_{54}\text{Xe} + \bar{\nu}\\ {}^{234}_{90}\text{Th}→ {}^{0}_{-1}e + {}^{234}_{91}\text{Pa} + \bar{\nu}

Positron decay

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A proton and neutron are emitted from the parent particle

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Proton number remains the same, but the nucleon number reduces by one

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For example,

{}^{11}_{6}\text{C}→{}^{0}_{+1}e + {}^{11}_{5}\text{B} + \nu\\ {}^{21}_{11}\text{Na}→{}^{0}_{+1}e + {}^{21}_{10}\text{Ne} + \nu

Gamma ray emission

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Gamma ray is emitted when an electron moves to the ground state.

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This type of decay happens when the particle is in excited state

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For example,

{}^{60}_{27}\text{Co}→{}^{0}_{-1}e + {}^{60}_{28}\text{Ni}^{*} + \bar{\nu} + \gamma \\ {}^{60}_{28}\text{Ni}^{*}→ {}^{60}_{28}\text{Ni} + \gamma

Radioactive Decay Comparison

Type of decay	Alpha decay	Beta decay	Gamma Decay
Particles emitted	Alpha particle and a different nucleus	Electron, positron, neutrions, antineutrions and different nucleus	$\gamma$ and same nucleus
Penetrating power	medium	low	highest
Ionizing Power	highest	medium	low
Example	${}^{238}_{92}\text{U}$ → ${}^{234}_{90}\text{Th} + {}^{4}_{2}\alpha$	${}^{40}_{19}\text{K}$ → ${}^{40}_{20}\text{Ca} + {}^{0}_{-1}\text{e} + {}^{0}_{0}\bar{\nu}_e$ ${}^{22}_{11}\text{Na}$ → ${}^{22}_{10}\text{Ne} + {}^{0}_{+1}e + {}^{0}_{0}\nu_e$	${}^{60}_{28}\text{Ni}$ → ${}^{60}_{28}\text{Ni} + {}^{0}_{0}\gamma$

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All of radioactive decay follow the conservation of mass and conservation of charge before and after the decay

E.3.2 Half-life

Half-life

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Half-life is the time taken for number of particles of radioactive sample is halved

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Any given nucleus has a 50% chance of decaying within a time interval equal to the half life

E.3.2-1 Process of half-life with their probability

E.3.3 Nuclear Binding Energy

Zone of stability

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The nuclides within this zone are considered to be stable, and the nuclides outside this zone are unstable and spontaneously tending towards the stability zone.

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It shows the trend of the neutron-proton number ratio.

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Unstable nuclides that are neutron rich tend to have β decay.

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Unstable nuclides that are proton rich tend to have β+ decay.

E.3.3-1 Zone of stability

Unified atomic mass unit

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It is a unit of mass measured in atomic scale

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Symbol: u

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One-twelfth of the rest mass of the unbounded Carbon-12 atom in its nuclear and electronic ground state ( =1.661x10−27kg= 1.661 x 10^{-27} kg=1.661x10−27kg)

Mass and Energy

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Mass of a nucleus is less than the sum of mass of its constituent nucleons

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The difference in mass is called the mass defect

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The energy equivalent to the mass defect is called binding energy, which mathematically equal to :

\textit{Binding Energy} = \textit{mass defect} \times c^2\\ E = mc^2

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When a nucleus is broken up to its constituent nucleons, binding energy is supplied to the reaction

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Mass of products is greater than the mass of reactants

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Energy gained corresponds to the increase in mass of the products as compared to the reactants

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When constituent nucleons are assembled to form a nucleus, binding energy is released from the reaction process

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Mass of products is less than the mass of reactants in this case

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thus the energy lost corresponds to the loss in mass of the products as compared to the reactants

E.3.4 Fundamental Forces

Fundamental Forces

	Electromagnetic force	Weak force	Strong force	Gravitational force
Act on	Particles with electric charge	Protonos, neutrons, electrons, neutrinos, during beta decay	Attractive force between protons and neutrons inside nuclei	Attractive force between masses
Range	infinite	$10^{-18}m$	$10^{-15}m$	infinite
Relative strength	$\frac {1}{137}$	$10^{-6}$	$1$	$10^{-41}$

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As more protons are added to a nucleus, the tendency for the nucleus to break apart increases

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All the protons repel each other through the electromagnetic force, which has infinite range

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But the strong force has a short range, so any one proton only attracts its very immediate neighbors

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To keep the nucleus together, we need more neutrons that will contribute to nuclear binding through the strong force, but which will not be added to the repulsive forces

E.3.4-1 graph of neutron and proton number through periodic elements

E.3.5 The Law of Radioactive Decay

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As discussed in topic 7.1, radioactive decay is random and unpredictable.

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The probability that an individual nucleus will decay is given a time interval is the decay constant λ (time−1time^{-1}time−1).

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The activity of sample A is the number of nuclei decaying in a second (Bq).

The activity and the number of nuclei present (N) are negatively proportional since it is decaying:

A = -\lambda N\\ \frac{dN}{dt} = -\lambda N

The solution of this equation above is given by:

N = N_0 e^{-\lambda t}

Its relation to the activity can be expressed as the equations:

A = A_0 e^{-\lambda t}\\ A = \lambda N_0 e^{-\lambda t}

A_0

= the activity of a sample of radioactive material at time t=0

Half-Life

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Besides the whole number half-life we have learned in topic 7.1, we will now consider the half-life that is not a whole number.

E.3.5-1 Half-life decay of a number of nuclei

Since half-life

t_{1/2}

is the time taken for the number of radioactive nuclei to decay to half, the half-life of the nuclei can be found by taking the process below:

N = N_0 e^{-\lambda t}\\ \text{If} \quad t = T_{\frac{1}{2}}\\ N = \frac{N_0}{2}\\ \frac{N_0}{2} = N_0 e^{-\lambda T_{\frac{1}{2}}}\\ \therefore \frac{1}{2} = e^{-\lambda T_{\frac{1}{2}}}\\ \therefore \ln \frac{1}{2} = -\lambda T_{\frac{1}{2}}\\ \therefore -\lambda T_{\frac{1}{2}} = -\ln \frac{1}{2}\\ \quad \quad = \ln 2\\ \therefore T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}

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Some nuclides have long half-lives that are longer than the possible time interval of radioactive decay observations.

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A pure sample of the nuclide in a known chemical form needs to be separated, its mass measured, and then a count rate taken.

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The activity can be calculated by multiplying the count rate by the ratio:

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