Beta Plus (β⁺) Decay

Beta Plus (β⁺) decay occurs in proton-rich nuclei. In this process, a proton is transformed into a neutron, and a positron (e⁺) along with an electron neutrino (ν_e) is emitted. The atomic number of the nucleus decreases by one, while the atomic mass number remains unchanged.

The general equation for a β⁺ decay reaction is:

In this equation:

• P represents the parent nucleus, which undergoes the β⁺ decay process.
• D represents the daughter nucleus formed after the decay.
• Z_P and Z_D are the atomic numbers of the parent and daughter nuclei, respectively.
• A_P and A_D are the mass numbers of the parent and daughter nuclei, respectively, and they remain the same.
• e⁺ is the emitted positron (the antimatter counterpart of an electron).
• ν_e is the emitted electron neutrino.

Example of Beta Plus Decay

A well-known example of β⁺ decay is the decay of fluorine-18 (¹⁸F), which is widely used in Positron Emission Tomography (PET) scans. Fluorine-18 undergoes β⁺ decay to form oxygen-18 (¹⁸O):

In this reaction:

• F¹⁸ is the parent nucleus (fluorine-18), which undergoes β⁺ decay.
• O¹⁸ is the daughter nucleus (oxygen-18), which is formed after the decay.
• A positron (e⁺) and an electron neutrino (ν_e) are emitted during the decay.

The half-life of fluorine-18 is approximately 110 minutes. This short half-life makes fluorine-18 ideal for use in PET scans, where it is often used in a compound called fluorodeoxyglucose (FDG). FDG is a glucose analogue that is taken up by active cells in the body, and because of its radioactive properties, it allows doctors to monitor the metabolic activity of tissues in real-time.

Energy Released in Beta Plus Decay

Every radioactive decay releases a certain amount of energy, known as the Q value of the reaction. The Q value can be calculated by the difference in the rest masses of the parent and daughter nuclei and the emitted positron.

The general formula for the Q value is given by:

Where:

• M(P) is the mass of the parent nucleus.
• M(D) is the mass of the daughter nucleus.
• M(e⁺) is the mass of the emitted positron.
• c is the speed of light in a vacuum.

If the Q value is positive (Q > 0), it indicates that the decay is energetically possible and exothermic (releases energy).

Example Calculation: Fluorine-18 Decay

To calculate the energy released during the β⁺ decay of fluorine-18, we need the masses of the parent nucleus, daughter nucleus, and emitted positron. For simplicity, let's use the following approximate values:

• Mass of fluorine-18 (M(P)) ≈ 17.999 amu
• Mass of oxygen-18 (M(D)) ≈ 17.999 amu
• Mass of positron (M(e⁺)) ≈ 0.000548 amu

Using the formula for the Q value:

The Q value for this reaction is approximately -0.511 MeV, which means the decay is not energetically favorable, and no energy is released in the process of β⁺ decay.

Importance of β⁺ Decay in Medicine

β⁺ decay plays a crucial role in medical imaging, especially in functional imaging techniques like Positron Emission Tomography (PET). When positrons are emitted, they quickly interact with electrons in the surrounding tissue, resulting in the annihilation of both particles and the production of two gamma photons. These photons are detected by the PET scanner and used to create highly detailed images of metabolic processes in the body.

PET scans are widely used for:

• Detecting cancer by observing areas of high metabolic activity.
• Evaluating brain activity, such as in the case of Alzheimer's disease or epilepsy.
• Studying heart function and blood flow.

This ability to track metabolic processes in real-time makes β⁺ decay and positron emitters invaluable in modern diagnostic medicine.