Quenching of magnetic form factors of $s\text{−}d$ shell nuclei ${}^{17}\mathrm{O},{}^{25}\mathrm{Mg},{}^{27}\mathrm{Al},{}^{29}\mathrm{Si}$, and ${}^{31}\mathrm{P}$ within the relativistic mean field model

The magnetic form factors of the odd-$As\text{−}d$ shell nuclei ${}^{17}\mathrm{O},{}^{25}\mathrm{Mg},{}^{27}\mathrm{Al},{}^{29}\mathrm{Si}$, and ${}^{31}\mathrm{P}$ are studied within the relativistic frame with single-nucleon wave functions generated using the relativistic mean field model. Single valence-nucleon contributions to the nuclear magnetic form factors are calculated, and two sets of quenching ratios of the nuclear magnetic form factors relative to the single valence-nucleon contributions are extracted for each nucleus. It is found that the single valence-nucleon contributions can generally give a good approximate description for the shapes of the nuclear magnetic form factors, including the positions of the minima and maxima, and after the quenching ratios are applied the experimental data can be very well reproduced. The quenching ratios are compared with those obtained within some other nuclear models, and the physical implications for each nucleus are discussed. Calculations also show that no reasonable quenching ratios could be found for ${}^{27}\mathrm{Al}$ when ${\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}$ is chosen as the $M1$ multipole quenching ratio. This implies that the common way of choosing ${\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}$ as the $M1$ multipole quenching ratio for the odd-$A$ nuclei is not always valid.

Figure 1

The magnetic form factors of ${}^{17}\mathrm{O}$. The solid curve shows the result of the pure contribution of the single $1d_{5/2}$ neutron; the dashed curve denotes the result including correction with ${\alpha }_1=0.887,{\alpha }_3=0.514$, and ${\alpha }_5=0.891$; the dotted curve denotes that including correction with ${\alpha }_1$ fixed to ${\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}=0.990,{\alpha }_3=0.475$, and ${\alpha }_5=0.880$. The filled dots represent the experimental data from Refs. [1, 7, 17].

Figure 2

The magnetic form factors of ${}^{25}\mathrm{Mg}$. The solid curve corresponds to the contribution of the single $1d_{5/2}$ neutron; the dashed curve corresponds to the result including correction with ${\alpha }_1=0.333,{\alpha }_3=0.205$, and ${\alpha }_5=0.510$, and the dotted curve corresponds to that including correction with ${\alpha }_1={\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}=0.447,{\alpha }_3=0.124$, and ${\alpha }_5=0.499$. The filled dots represent the experimental data from Refs. [18, 19].

Figure 3

The magnetic form factors of ${}^{27}\mathrm{Al}$. The solid curve corresponds to the contribution of the single $1d_{5/2}$ proton; the dashed curve corresponds to the result including correction with ${\alpha }_1=0.716,{\alpha }_3=0.435$, and ${\alpha }_5=0.606$, and the dotted curve corresponds to the result including correction with ${\alpha }_1={\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}=1.304,{\alpha }_3=1.90\times {10}^{-4}$, and ${\alpha }_5=0.575$. The filled dots represent the experimental data from Refs. [1, 23].

Figure 4

The magnetic form factors of ${}^{31}\mathrm{P}$. The solid curve denotes the pure contribution of the single $2s_{1/2}$ proton; the dashed curve shows the result including correction with ${\alpha }_1=0.526$, and the dotted curve shows the result including correction with ${\alpha }_1$ fixed to ${\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}=0.405$. The filled dots represent the experimental data from Refs. [1, 22].

Figure 5

The magnetic form factors of ${}^{29}\mathrm{Si}$. The solid curve corresponds to the pure contribution of the single $2s_{1/2}$ neutron; the dashed curve corresponds to the result including correction with ${\alpha }_1=0.532$, and the dotted curve corresponds to the result including correction with ${\alpha }_1={\mu }_{\mathrm{expt}.}/{\mu }_{s.n.}=0.290$. The filled dots show the experimental data from Refs. [1, 22].