7.8 - Types of Radioactive Decay

Particles Emitted in Nuclear Decay

Some decay processes throw out subatomic particles with distinctive properties.

Alpha particle. An alpha particle is a helium nucleus, made of two protons and two neutrons, written

\alpha

^4_2He^{2+}

. It has charge

+2e

, a mass of 4 u, and it’s relatively heavy and slow. It has low penetration (paper or skin stops it) but high ionizing power. In AP Physics 2, only He-4 nuclei count as alpha particles, not other helium isotopes.

Neutrinos and antineutrinos. Neutrinos (

\nu

) and antineutrinos (

\bar{\nu}

) have no electric charge and negligible (nearly zero) mass. They only interact with matter through the weak force and gravity, so they barely interact with ordinary matter at all. That means they can pass through huge amounts of material without touching anything, they’re extremely hard to detect, and trillions from the Sun stream through your body every second. They carry away energy and momentum in decays, which keeps the conservation laws satisfied even though they’re nearly invisible. AP Physics 2 doesn’t ask you to tell neutrino types apart or explain the weak force in detail.

Positrons. Positrons (antielectrons) have a charge opposite to the electron (

+e

instead of

-e

) and the same mass as an electron (

m_e = 9.11 \times 10^{-31}

kg), written

e^+

\beta^+

. A positron is the electron’s antiparticle, and when it meets an electron they annihilate:

e^+ + e^- \rightarrow 2\gamma

(two gamma rays). Positrons are produced in beta-plus decay and behave just like electrons except for the sign of their charge.

Types of Radioactive Decay

Nuclei can decay through alpha decay (

\alpha

), beta-minus decay (

\beta^-

), beta-plus decay (

\beta^+

), and gamma decay (

\gamma

Conservation Laws in Nuclear Decay

In every nuclear decay, these are conserved: nucleon number (total protons plus neutrons), lepton number (leptons minus antileptons), electric charge, energy (including mass-energy), and momentum. These rules decide which decays are allowed and which particles have to show up.

Lepton number is a bookkeeping rule for electrons and neutrinos. Electrons and neutrinos count as

+1

, and their antiparticles, positrons and antineutrinos, count as

-1

. Protons, neutrons, nuclei, and photons count as 0. The total has to be the same before and after.

Example: check every conservation law for the beta-minus decay of carbon-14 and the beta-plus decay of sodium-22. -

^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + e^- + \bar{\nu}

: nucleon number

14 = 14 + 0 + 0

, charge

+6 = +7 + (-1) + 0

, lepton number

0 = 0 + (+1) + (-1)

. -

^{22}_{11}\text{Na} \rightarrow {}^{22}_{10}\text{Ne} + e^+ + \nu

: nucleon number

22 = 22 + 0 + 0

, charge

+11 = +10 + (+1) + 0

, lepton number

0 = 0 + (-1) + (+1)

That’s why beta-minus decay has to come with an antineutrino and beta-plus decay with a neutrino: the new electron or positron changes the lepton count by 1, and the antineutrino or neutrino brings it back to 0.

Alpha Decay

Alpha decay is when a nucleus ejects an alpha particle:

^A_Z X \rightarrow {}^{A-4}_{Z-2}Y + {}^4_2He \qquad \text{or} \qquad {}^A_Z X \rightarrow {}^{A-4}_{Z-2}Y + \alpha

The mass number drops by 4 and the atomic number drops by 2, so the element moves two spots back on the periodic table. For example, uranium-238:

^{238}_{92}U \rightarrow {}^{234}_{90}Th + {}^4_2He

Uranium (Z=92) becomes thorium (Z=90). Alpha decay happens in heavy nuclei (usually Z > 82), reduces both protons and neutrons, produces relatively slow and easily stopped alpha particles, and gives the daughter nucleus a noticeable recoil. The released energy is shared between the alpha particle’s kinetic energy and that recoil, to conserve momentum.

Beta-Minus Decay

Beta-minus decay (

\beta^-

) is when a neutron turns into a proton, emitting an electron and an antineutrino:

^A_Z X \rightarrow {}^{A}_{Z+1}Y + {}^{0}_{-1}e + \bar{\nu}

or at the nucleon level:

n \rightarrow p + e^- + \bar{\nu}

The mass number stays the same and the atomic number goes up by 1, so the element moves one spot forward, with one neutron becoming a proton. For example, carbon-14:

^{14}_6C \rightarrow {}^{14}_7N + {}^{0}_{-1}e + \bar{\nu}

Carbon (Z=6) becomes nitrogen (Z=7). Beta-minus decay happens in neutron-rich nuclei, turns a neutron into a proton, and the emitted electron comes out with a continuous range of energies (because the antineutrino carries off a variable share). It’s more penetrating than alpha particles. The antineutrino is needed for energy conservation (the total energy has to balance) and momentum conservation (a third particle is required), which is why the observed electron energies fall short of expectations: the antineutrino carries the missing energy and momentum.

Beta-Plus Decay

Beta-plus decay (

\beta^+

) is when a proton turns into a neutron, emitting a positron and a neutrino:

^A_Z X \rightarrow {}^{A}_{Z-1}Y + {}^{0}_{+1}e + \nu

or at the nucleon level:

p \rightarrow n + e^+ + \nu

The mass number stays the same and the atomic number drops by 1, so the element moves one spot back, with one proton becoming a neutron. For example, sodium-22:

^{22}_{11}Na \rightarrow {}^{22}_{10}Ne + {}^{0}_{+1}e + \nu

Sodium (Z=11) becomes neon (Z=10). Beta-plus decay happens in proton-rich nuclei. A free proton can’t do this on its own, since a proton is lighter than a neutron. Inside a nucleus it can, because the energy comes from the nucleus itself: the daughter nucleus is bound more tightly, so the parent has enough extra mass-energy to make the neutron and the positron. No outside energy is needed, and the decay happens spontaneously. The emitted positron eventually annihilates with an electron in matter, making gamma rays. The neutrino is needed for the same energy and momentum reasons as before, except beta-plus emits a neutrino rather than an antineutrino.

Gamma Decay

Gamma decay happens after a nucleus has already gone through alpha or beta decay and is left in an excited state. The excited nucleus drops to a lower energy state by emitting a photon:

^A_Z X^* \rightarrow {}^A_Z X + \gamma

where the asterisk marks an excited nuclear state. The mass number and atomic number both stay the same, so the element doesn’t change, only its internal energy state does. For example, after cobalt-60 beta decays:

^{60}_{27}Co \rightarrow {}^{60}_{28}Ni^* + e^- + \bar{\nu}

the excited nickel then gamma decays:

^{60}_{28}Ni^* \rightarrow {}^{60}_{28}Ni + \gamma

Gamma decay doesn’t change the element or nucleon number, emits a high-energy photon (a gamma ray), and usually follows alpha or beta decay rather than happening alone. Gamma rays are electromagnetic radiation like X-rays but higher energy, they’re very penetrating (you need thick lead or concrete to shield them), and they can come out in rapid succession as a cascade. The energy levels here are like atomic energy levels but in the nucleus, with much higher energies (MeV instead of eV), and the gamma ray carries off the excess.

Determining Decay Type

Which decay a nucleus undergoes is set by the specific isotope. A few factors decide the mode: the neutron-to-proton ratio (too many neutrons leans toward beta-minus, too many protons toward beta-plus), the nuclear size (very heavy nuclei with Z > 82 tend to alpha decay), the binding energy (the nucleus moves toward a more stable, higher-binding-energy configuration), and energy availability (beta-plus needs more energy than beta-minus). Some examples:

^{238}_{92}

U does alpha decay (heavy nucleus),

^{14}_6

C does beta-minus (too many neutrons),

^{22}_{11}

Na does beta-plus (too many protons), and

^{60}_{27}

Co does beta-minus followed by gamma decay. AP Physics 2 won’t make you memorize specific isotope decay modes, they’ll be given when needed.

Example (decay chain): uranium-238 (

^{238}_{92}

U) goes through a long chain of alpha and beta-minus decays and ends as lead-206 (

^{206}_{82}

Pb). How many of each decay happen? - Only alpha decay changes the mass number, by 4 each time:

(238 - 206)/4 = 8

alpha decays. - Eight alpha decays lower

Z

8 \times 2 = 16

, from 92 to 76. Lead has

Z = 82

, so

Z

has to go back up by 6. Each beta-minus decay raises

Z

by 1, so there are 6 beta-minus decays. - Answer: 8 alpha decays and 6 beta-minus decays. Check:

A = 238 - 8(4) = 206

and

Z = 92 - 8(2) + 6(1) = 82

Practice Problems

1. Complete this alpha decay:

^{226}_{88}Ra \rightarrow

+ {}^4_2He

2. Carbon-14 undergoes beta-minus decay. Write the full decay equation including the antineutrino.

3. Why is an antineutrino emitted in beta-minus decay instead of just an electron?

4. A nucleus undergoes beta-plus decay. If it started with 30 protons and 28 neutrons, how many protons and neutrons does the daughter have?

5. Which radiation (alpha, beta, or gamma) is (a) most ionizing, (b) most penetrating, (c) stopped by a sheet of paper?

6. Cobalt-60 does beta-minus decay to an excited nickel-60, which then gamma decays. Write both equations.

7. Why doesn’t gamma decay change the element?

8. A positron and an electron meet. What happens? Write the equation.

Solutions

1. In alpha decay, mass number drops by 4 and atomic number by 2:

^{226}_{88}Ra \rightarrow {}^{222}_{86}Rn + {}^4_2He

Check:

226 = 222 + 4

and

88 = 86 + 2

. Z=86 is radon, so the answer is

^{222}_{86}

Rn (radon-222).

2. In beta-minus decay a neutron becomes a proton, emitting an electron and antineutrino:

^{14}_6C \rightarrow {}^{14}_7N + {}^0_{-1}e + \bar{\nu}

Check: mass number

14 = 14 + 0

, charge

6 = 7 + (-1) + 0

, nucleons

14 = 14

. So

^{14}_6C \rightarrow {}^{14}_7N + e^- + \bar{\nu}

3. The antineutrino is there for conservation of energy and momentum. Experiments showed the electron didn’t always carry all the expected energy, some was missing, and momentum didn’t balance for a two-body decay (nucleus plus electron). Wolfgang Pauli proposed the neutrino (understood as an antineutrino for

\beta^-

) to cover the gap: the antineutrino carries a variable amount of energy, which explains the continuous spread of electron energies, and a three-body decay (nucleus, electron, antineutrino) lets momentum balance. Without it, energy and momentum wouldn’t be conserved.

4. The original nucleus has 30 protons and 28 neutrons (58 nucleons total). In beta-plus decay a proton becomes a neutron, so the daughter has

30 - 1 = 29

protons and

28 + 1 = 29

neutrons, still 58 nucleons.

5. (a) Most ionizing: alpha particles. They’re heavily charged (

+2e

) and relatively slow, so they interact strongly and create many ion pairs per unit distance, losing energy quickly.

(b) Most penetrating: gamma rays. They have no charge and no mass, travel at light speed, and interact weakly, so they pass through a lot of material and need thick lead or concrete to stop.

(c) Stopped by paper: alpha particles. Their large mass and charge mean strong interactions and a very short range, so paper, skin, or even a few centimeters of air stops them.

6. Beta-minus decay:

^{60}_{27}Co \rightarrow {}^{60}_{28}Ni^* + e^- + \bar{\nu}

where the asterisk means nickel-60 in an excited state. Then gamma decay:

^{60}_{28}Ni^* \rightarrow {}^{60}_{28}Ni + \gamma

Nickel-60 often emits two gamma rays in a cascade.

7. In gamma decay the nucleus just drops from an excited energy state to a lower one by emitting a photon. It’s like an electron falling to a lower energy level in an atom, except the nuclear levels are much higher (MeV instead of eV) and the photon is a gamma ray. Since no particles are gained or lost and no nucleons switch type (protons stay protons, neutrons stay neutrons), the mass number

A

and atomic number

Z

are both unchanged. Only the internal energy state changes, and since the element is set by the proton number, the element stays the same.

8. When a positron (antimatter) meets an electron (matter), they annihilate, turning all their mass into energy as photons. To conserve momentum, two gamma rays fly off in opposite directions:

e^+ + e^- \rightarrow 2\gamma

The energy produced is:

E = 2m_e c^2 = 2(9.11 \times 10^{-31})(3 \times 10^8)^2 = 1.64 \times 10^{-13} \text{ J} = 1.02 \text{ MeV}

so each gamma ray carries 0.511 MeV. They annihilate into two gamma rays:

e^+ + e^- \rightarrow 2\gamma

Common Mistakes

Separating alpha particles from helium. Alpha particles are helium nuclei ( $^4_2He^{2+}$ ), the same thing.

Wrong mass/charge changes. Alpha: $A-4$ , $Z-2$ . Beta-minus: same $A$ , $Z+1$ . Beta-plus: same $A$ , $Z-1$ . Gamma: no change in $A$ or $Z$ .

Leaving out neutrinos. Beta decays must include neutrinos or antineutrinos for the conservation laws. Don’t drop them.

Confusing neutrino and antineutrino. Beta-minus emits an antineutrino ( $\bar{\nu}$ ), beta-plus emits a neutrino ( $\nu$ ). They’re distinct.

Thinking gamma changes the element. Gamma decay only changes the energy state, not $Z$ or $A$ , so the element stays put.

Misstating the beta-minus mechanism. A neutron converts into a proton and emits an electron. The electron isn’t sitting “in” the nucleus beforehand, it’s created during the decay.

Mixing up electron and positron symbols. Electron: $e^-$ , $^0_{-1}e$ , or $\beta^-$ . Positron: $e^+$ , $^0_{+1}e$ , or $\beta^+$ .

Skipping the conservation check. Always confirm nucleon number and charge balance in decay equations.

Thinking positrons stick around. Positrons quickly annihilate with electrons in normal matter, making gamma rays.

Confusing penetration and ionization. Alpha is low penetration, high ionization. Gamma is high penetration, low ionization. They’re inversely related.

Wrong charge on alpha. An alpha particle is $+2e$ (two protons), not $+4$ or $+1$ .

Thinking neutrinos are easy to detect. They interact so weakly they’re extremely hard to catch, even though they’re everywhere.

Not realizing gamma follows other decays. Gamma decay usually happens after alpha or beta decay leaves the nucleus excited, rather than on its own.

Confusing mass number with atomic mass. Mass number $A$ is a whole-number count of nucleons. Atomic mass is an average that folds in isotope abundances and binding-energy effects.

Thinking beta particles all have the same energy. They come out with a continuous energy spectrum because the neutrino takes a variable share.

Not seeing why beta-plus needs energy. Turning a lighter proton into a heavier neutron takes energy, which comes from the change in nuclear binding energy.

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