To determine which option represents beta decay, we need to understand the nature of beta decay. In beta decay, a neutron is transformed into a proton or vice versa, with the emission of a beta particle, which can either be an electron (`β⁻` decay) or a positron (`β⁺` decay). The essential points of beta decay are:
1. The mass number (A) remains unchanged.
2. The atomic number (Z) changes by ±1.
Let's examine each option in this context:
### Option A
[tex]\[ {}_{64}^{148} \mathrm{Gd} \rightarrow {}_{62}^{144} \mathrm{Sm} + {}_{2}^{4} \mathrm{He} \][/tex]
- Initial: Gadolinium-148 ([tex]\(Z = 64\)[/tex], [tex]\(A = 148\)[/tex])
- Final: Samarium-144 ([tex]\(Z = 62\)[/tex], [tex]\(A = 144\)[/tex]) and Helium-4 ([tex]\(Z = 2\)[/tex], [tex]\(A = 4\)[/tex])
This transformation involves a change in both the atomic number and the mass number, and emits an alpha particle ([tex]\( {}_{2}^{4} \mathrm{He} \)[/tex]). This is alpha decay, not beta decay.
### Option B
[tex]\[ {}_{64}^{160} \mathrm{Gd} \rightarrow {}_{65}^{160} \mathrm{Tb} + {}_{-1}^{0} e \][/tex]
- Initial: Gadolinium-160 ([tex]\(Z = 64\)[/tex], [tex]\(A = 160\)[/tex])
- Final: Terbium-160 ([tex]\(Z = 65\)[/tex], [tex]\(A = 160\)[/tex]) and an electron ([tex]\( {}_{-1}^{0} e \)[/tex])
Here, only the atomic number changes from 64 to 65 (increased by 1) while the mass number remains the same (160). This emission of an electron (beta particle) and the change in atomic number by +1 indicate that this is [tex]\(\beta^-\)[/tex] decay.
### Option C
[tex]\[ {}_{63}^{150} \mathrm{Eu} + {}_{-1}^{0} e \rightarrow {}_{62}^{150} \mathrm{Sm} \][/tex]
- Initial: Europium-150 ([tex]\(Z = 63\)[/tex], [tex]\(A = 150\)[/tex]) and an electron ([tex]\( {}_{-1}^{0} e \)[/tex])
- Final: Samarium-150 ([tex]\(Z = 62\)[/tex], [tex]\(A = 150\)[/tex])
This reaction involves the absorption of an electron and a decrease in the atomic number by 1, which describes electron capture (not beta decay).
### Option D
[tex]\[ {}_{43}^{90} \mathrm{Tc} \rightarrow {}_{43}^{90} \mathrm{Tc} + \gamma \][/tex]
- Initial: Technetium-90 ([tex]\(Z = 43\)[/tex], [tex]\(A = 90\)[/tex])
- Final: Technetium-90 ([tex]\(Z = 43\)[/tex], [tex]\(A = 90\)[/tex]) and a gamma photon ([tex]\( \gamma \)[/tex])
This reaction implies the emission of a gamma photon without a change in the atomic number or the mass number, indicating it is a gamma decay, not beta decay.
### Conclusion
Therefore, the transformation that represents beta decay is found in Option B, as it fits the characteristic criteria of beta decay where the atomic number changes by 1 while the mass number remains constant.
Thus, the option representing beta decay is:
[tex]\[ {}_{64}^{160} \mathrm{Gd} \rightarrow {}_{65}^{160} \mathrm{Tb} + {}_{-1}^{0} e \][/tex]
This corresponds to option B.
