Alpha, beta and gamma decay Knowledge-Based Study

Study notes

1. Nuclear Stability and Spontaneous Decay

   Atomic nuclei consist of protons and neutrons, collectively termed nucleons. The stability of a nucleus depends primarily on the balance between two fundamental forces: the repulsive electrostatic force acting between positively charged protons, and the attractive strong nuclear force acting over very short distances among all nucleons.

   When the neutron-to-proton ratio ($N/Z$) lies outside the zone of stability, or when a nucleus contains an excessive number of nucleons (typically $Z > 82$), the nucleus is unstable. To attain a lower, more stable energy configuration, an unstable parent nucleus undergoes spontaneous radioactive decay, emitting particles, electromagnetic radiation, or both, thereby transforming into a daughter nucleus.

   Mechanics of Alpha Decay

   Alpha decay typically occurs in very heavy, proton-rich nuclei, such as uranium, thorium, and radium. In this process, the parent nucleus expels an alpha particle ($\alpha$), which is identical to a helium-4 nucleus ($^{4}_{2}\text{He}$).

   An alpha particle consists of two protons and two neutrons, carrying a $+2$ charge and an atomic mass of $4\text{ u}$. The general nuclear equation for alpha decay is formulated as:

   $$^{A}_{Z}\text{X} ightarrow \,^{A-4}_{Z-2}\text{Y} + \,^{4}_{2}\text{He}$$

   • The mass number ($A$) decreases by 4.
   • The atomic number ($Z$) decreases by 2, changing the elemental identity of the atom.

   For example, the alpha decay of uranium-238 produces thorium-234:

   $$^{238}_{92}\text{U} ightarrow \,^{234}_{90}\text{Th} + \,^{4}_{2}\text{He}$$

   In all balanced nuclear reactions, both total electric charge (sum of $Z$) and total nucleon count (sum of $A$) are strictly conserved.

   Beta-Minus and Beta-Plus Emission

   Beta decay occurs when a nucleus has an unfavorable neutron-to-proton ratio and stabilizes via the weak nuclear force. There are two primary forms of beta decay:

   1. Beta-Minus Decay ($\beta^{-}$)

   Beta-minus decay occurs in neutron-rich nuclei. A neutron converts into a proton, expelling a high-speed electron (beta-minus particle, $\beta^{-}$ or $^{\phantom{-}0}_{-1}\text{e}$) and an electron antineutrino (ar{\nu}_e):

   ^{1}_{0}\text{n} ightarrow \,^{1}_{1}\text{p} + \,^{\phantom{-}0}_{-1}\text{e} + ar{\nu}_e

   The general transformation is:

   ^{A}_{Z}\text{X} ightarrow \,^{\phantom{-}A}_{Z+1}\text{Y} + \,^{\phantom{-}0}_{-1}\text{e} + ar{\nu}_e

   • The mass number ($A$) remains unchanged.
   • The atomic number ($Z$) increases by 1.

   Example: Carbon-14 decaying into nitrogen-14:

   ^{14}_{6}\text{C} ightarrow \,^{14}_{7}\text{N} + \,^{\phantom{-}0}_{-1}\text{e} + ar{\nu}_e

   2. Beta-Plus Decay ($\beta^{+}$ / Positron Emission)

   In proton-rich nuclei, a proton converts into a neutron, releasing a positron ($\beta^{+}$ or $^{\phantom{+}0}_{+1}\text{e}$) and an electron neutrino ($u_e$):

   $$^{A}_{Z}\text{X} ightarrow \,^{\phantom{-}A}_{Z-1}\text{Y} + \,^{\phantom{+}0}_{+1}\text{e} + u_e$$

   De-excitation of Excited Nuclei

   Unlike alpha and beta emissions, gamma decay ($\gamma$) involves neither a change in nucleon count nor the conversion of nucleons. Gamma rays are high-energy photons belonging to the shortest wavelength, highest frequency end of the electromagnetic spectrum.

   Following an alpha or beta transition, the newly formed daughter nucleus is frequently left in an excited, metastable nuclear state, denoted by an asterisk ($^{*}$). The nucleus releases this excess energy by emitting one or more gamma photons:

   $$^{A}_{Z}\text{X}^* ightarrow \,^{A}_{Z}\text{X} + \gamma$$

   • The mass number ($A$) remains unchanged.
   • The atomic number ($Z$) remains unchanged.
   • The chemical identity of the isotope is fully preserved.

   An important example is technetium-99m de-exciting to its ground state:

   $$^{99\text{m}}_{43}\text{Tc} ightarrow \,^{99}_{43}\text{Tc} + \gamma$$

   Comparative Physical Properties

   The physical hazards and detection methods for radioactive emissions depend directly on their ionizing power and penetrating power, which have an inverse relationship:

   • Alpha particles ($\alpha$): Possess the highest ionizing power due to their relatively large mass and $+2$ charge. Consequently, they lose kinetic energy rapidly through collisions and have the lowest penetrating power. They are stopped by a standard sheet of paper or the outer layer of human skin.
   • Beta particles ($\beta$): Possess intermediate ionizing power and intermediate penetrating power. Being much smaller and lighter than alpha particles, they travel several meters in air and require a few millimeters of aluminum or plastic to be absorbed.
   • Gamma rays ($\gamma$): Possess the lowest ionizing power per interaction because they are uncharged, massless photons. However, they have the highest penetrating power, passing easily through soft matter and air. Significant attenuation requires dense, thick shielding such as several centimeters of lead or meters of concrete.

   Summary of Decay Characteristics

   • $\alpha$ decay: Emits $^{4}_{2}\text{He}$; $A ightarrow A - 4$; $Z ightarrow Z - 2$. High ionization, stopped by paper.
   • $\beta^{-}$ decay: Emits $^{\phantom{-}0}_{-1}\text{e}$; $A ightarrow A$; $Z ightarrow Z + 1$. Moderate ionization, stopped by aluminum.
   • $\gamma$ emission: Emits energy quanta ($\gamma$); $A$ and $Z$ remain constant. Low ionization, attenuated by thick lead.

   Key Verification Problem

   Consider the nuclide radium-226 undergoing alpha decay:

   $$^{226}_{88}\text{Ra} ightarrow \,^{A}_{Z}\text{Rn} + \,^{4}_{2}\text{He}$$

   • Conservation of mass number gives: $226 = A + 4 \implies A = 222$
   • Conservation of charge gives: $88 = Z + 2 \implies Z = 86$

   The resulting daughter product is radon-222 ($^{222}_{86}\text{Rn}$).