Nuclei

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Formula Sheet

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- All the formula of the chapter in one go

Nuclear Composition and Atomic Mass
Z - atomic number = number of protons
N - neutron number = number of neutrons
A - mass number = Z + N = total number of protons and neutrons

Nuclear Notation:

_{Z}^{A} X

where X is the chemical Symbol of the element

1 Atomic mass unit (1u)=

\frac{m a s s o f 1 ^{12} C a t o m}{1 2} = 1.660539 \times 1 0^{- 27} k g

Mass of proton,

m_{p} = 1.00727 u = 1.67262 \times 1 0^{- 27} k g

Mass of neutron,

m_{n} = 1.00866 u = 1.6749 \times 1 0^{- 27} k g

Size of Nucleus
Radius of nucleus:

R = R_{o} A^{1 / 3}, w h e r e A = M a s s N u m b e r, R_{o} = 1.2 f m = 1.2 \times 1 0^{- 15} m

Volume of a nucleus:

V = \frac{4}{3} π R^{3} = \frac{4}{3} π R_{o}^{3} A

Density of Nucleus(ignoring mass difference between proton and neutron:

ρ = \frac{M a s s o f n u c l e u s}{V o l u m e o f N u c l e u s} = \frac{A m}{\frac{4}{3} π R _{o}^{2} A} = 2.7 \times 1 0^{17} k g / m^{3}

Mass Defect and Binding Energy

Mass defect:

M=Mass of Nucleus

m_{p}

=Mass of proton

m_{n}

=Mass of neutron

Δ M = [Z m_{p} + (A - Z) m_{n}] - M

Binding energy:

E_{b} = Δ M c^{2} = {[Z m_{p} + (A - Z) m_{n}] - M} c^{2}

1 u = 931.5 M e V / c^{2}

Binding energy per nucleon:

E_{b n} = \frac{E _{b}}{A}

A<30:

E_{b n}

small; unstable

A>170:

E_{b n}

small; unstable

Radioactivity

Radioactive decay law:

(\frac{d N}{d t}) \propto - N

N (t) = N_{o} e^{- λ t}

N(t): Number of active nuclei at time t

N_{o}

: Number of active nuclei at t=0

λ

: decay constant

Activity of a radioactive substance:

The decay rate R of a radioactive substance is

R = - \frac{d N}{d t}

R = λ N_{o} e^{- λ t} = R_{o} e^{- λ t}

SI unit: becquerel (bq) ; Other units: curie(ci), rutherford(rd)

1 b q = 1 d e c a y / s

1 c i = 3.7 \times 1 0^{10} B q

1 r d = 1 0^{6} B q

Half life:

T_{1 / 2} = \frac{l n 2}{λ} = \frac{0 . 6 9 3}{λ}

Mean Lifetime:

τ = \frac{T o t a l l i f e t i m e o f N _{o} n u c l e i}{N _{o}} = \frac{\int _{o}^{\infty} t [ R d t ]}{N _{o}} = \frac{1}{λ}

After n half lives:

a. number of nuclei left=

N_{o} (\frac{1}{2})^{n}

b. fraction of nuclei left=

(\frac{1}{2})^{n}

c. percentage of nuclei left=

100 (\frac{1}{2})^{n}

Number of nuclei decayed after time t:

= N_{o} - N = N_{o} (1 - e^{- λ t})

Probability of nucleus for survival at time t:

P_{s u r v i v a l} = \frac{N}{N _{o}} = e^{- λ t}

Probability of Nucleus for disintegration at time t:

P_{d i s i n t e g r a t i o n} = 1 - P_{s u r v i v a l} = 1 - \frac{N}{N _{o}} = 1 - e^{- λ t}

Decay Schemes

Alpha decay

_{Z}^{A} X \to_{Z - 2}^{A - 4} Y +_{2}^{4} H e

Q value of alpha decay

mass defect=

Δ m = m_{X} - m_{Y} - m_{α}

Q = Δ m c^{2} = (m_{X} - m_{Y} - m_{α}) c^{2}

Beta Decay:

β^{-} d e c a y :_{Z}^{A} X \to_{Z + 1}^{A} X + e^{-} + \overline{ν}

n \to p + e^{-} + \overline{ν}

Example:

_{6}^{14} C \to_{7}^{14} N + e^{-} + \overline{ν}

β^{+} d e c a y :_{Z}^{A} X \to_{Z - 1}^{A} X + e^{+} + ν

p \to n + e^{+} + ν

Example:

_{11}^{22} N a \to_{10}^{22} X + e^{+} + ν