6.1 - Radioactive Decay

Radioactive Decay

Radioactive decay occurs when an unstable nucleus emits radioactive radiation and becomes more stable.
Decay is random and spontaneous: temperature, pressure and other physical factors do not influence it.
The three types of radiation are alpha particles, beta particles and gamma rays.
In nuclear notation $_{Z}^{A} X$ , $A$ is the nucleon number and $Z$ is the proton number. The original nucleus is the parent nucleus; the nucleus produced is the daughter nucleus.

An alpha particle, $_{2}^{4} He$ , is a helium nucleus containing two protons and two neutrons.
An unstable nucleus emits an alpha particle and forms a more stable nucleus of a new element.
The daughter’s proton number decreases by 2 and nucleon number decreases by 4.

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

_{84}^{210} Po \to_{82}^{206} Pb +_{2}^{4} He

_{0}^{1} n \to_{1}^{1} p +_{- 1}^{0} e

The proton remains in the nucleus; the electron leaves with high kinetic energy.
The daughter’s proton number increases by 1; its nucleon number does not change.

_{Z}^{A} X \to_{Z + 1}^{A} Y +_{- 1}^{0} e

_{11}^{24} Na \to_{12}^{24} Mg +_{- 1}^{0} e

Gamma rays are high-frequency electromagnetic radiation with no mass and no charge.
An excited nucleus emits excess energy as gamma radiation and becomes less energetic and more stable.
Its proton number and nucleon number remain unchanged.

_{Z}^{A} X \to_{Z}^{A} X + γ

Example: an excited cobalt-60 nucleus releases gamma radiation without changing its nuclear composition.

_{27}^{60} Co \to_{27}^{60} Co + γ

One decay can emit more than one type of radiation. Uranium-234 emits an alpha particle and gamma radiation:

_{92}^{234} U \to_{90}^{230} Th +_{2}^{4} He + γ

Half-life, $T_{\frac{1}{2}}$ , is the time taken for half the initial radioactive nuclei in a sample to decay.
After one half-life, half the original parent nuclei remain undecayed. As parent nuclei decay, daughter nuclei form.
A daughter nucleus may also be unstable. A decay series continues through successive decays until a stable nucleus forms.
The uranium-238 series ends at stable lead-206. Each unstable member has its own half-life.

Parent nucleus → daughter nucleus	Radiation emitted; parent half-life
$_{92}^{238} U \to_{90}^{234} Th$	$α$ ; $4.5 \times 1 0^{9}$ years
$_{90}^{234} Th \to_{91}^{234} Pa$	$β$ ; 24.1 days
$_{91}^{234} Pa \to_{92}^{234} U$	$β$ ; 1.8 minutes
$_{92}^{234} U \to_{90}^{230} Th$	$α$ ; $2.5 \times 1 0^{5}$ years
$_{90}^{230} Th \to_{88}^{226} Ra$	$α$ ; $7.5 \times 1 0^{4}$ years
$_{88}^{226} Ra \to_{86}^{222} Rn$	$α$ ; 1 622 years
$_{86}^{222} Rn \to_{84}^{218} Po$	$α$ ; 3.83 days
$_{84}^{218} Po \to_{82}^{214} Pb$	$α$ ; 3.05 minutes
$_{82}^{214} Pb \to_{83}^{214} Bi$	$β, γ$ ; 26.8 minutes
$_{83}^{214} Bi \to_{84}^{214} Po$	$β, γ$ ; 19.7 minutes
$_{84}^{214} Po \to_{82}^{210} Pb$	$α$ ; $1.64 \times 1 0^{- 4}$ seconds
$_{82}^{210} Pb \to_{83}^{210} Bi$	$β$ ; 22.26 years
$_{83}^{210} Bi \to_{84}^{210} Po$	$β$ ; 5.0 days
$_{84}^{210} Po \to_{82}^{206} Pb$	$α$ ; 138.4 days

Radioisotopes are isotopes with unstable nuclei that can emit radioactive radiation.
Radioactive substances are stored in thick, sealed lead containers for safety.

Uranium-238 becomes trapped when rocks form and slowly decays through its series to lead-206.
Measure the lead-206 : uranium-238 ratio in a rock sample to estimate its age. A higher ratio indicates an older rock.
This geological dating method also helps estimate the age of the Earth.
Uranium-238’s long half-life makes its decay rate low and the dating process suitable for very old rocks.

Activity is the number of decays per second. It is directly proportional to the number of radioactive nuclei present at that time.
A decay curve plots the number of undecayed nuclei, $N$ , against time, $t$ . $N$ decreases as time increases.
To read the half-life from a curve:
1. Identify the initial number, $N_{0}$ .
2. Find $N_{0} /2$ on the vertical axis and read its corresponding time on the curve.
3. The interval for a further decrease from $N_{0} /2$ to $N_{0} /4$ is another half-life.
For samples with the same initial number of nuclei, a shorter half-life means a higher decay rate.
In a dice model, each die represents a nucleus. Dice showing 3 represent decayed nuclei; their number in each throw represents activity. Remaining dice represent undecayed nuclei. Choosing 1 instead of 3 gives the same decay probability because every face is equally likely.

n = \frac{t}{T _{\frac{1}{2}}}, N = (\frac{1}{2})^{n} N_{0}

$n$ is the number of half-lives; the repeated-halving series uses positive integer values of $n$ .
The number of decayed parent nuclei is $N_{0} - N$ .
Activity follows the same halving relationship:

A = (\frac{1}{2})^{n} A_{0}

Activity after a given time: initial activity $480 s^{- 1}$ , half-life 6 minutes, elapsed time 30 minutes.

n = \frac{30}{6} = 5, A = (\frac{1}{2})^{5} (480) = 15 s^{- 1}

Time to reach 25% activity: with a half-life of 10 days, the sequence $100% \to 50% \to 25%$ takes two half-lives.

t = 2 (10) = 20 days

Age of volcanic rock: argon-40 : potassium-40 is 3 : 1, with no argon initially trapped. Potassium-40 has a half-life of 1 250 million years.
After one half-life, parent and daughter nuclei have a 1 : 1 ratio. After two, potassium-40 remaining is $N_{0} /4$ and argon-40 formed is $3 N_{0} /4$ , giving 3 : 1.

Rock age = 2 (1250) = 2500 million years