5.3 (2) & 5.4 Two-Hole and Particle-Hole Nuclei

In this posting, we treat the formalism of the two-hole and particle-hole nuclei.

 

Two-Hole Nuclei Formalism

Analogous to the two-particle formalism described in (https://djlab.tistory.com/3), we can write wave functions of two-hole nuclei as follows:

$$ | a^{-1} \, b^{-1} l J \, M \rangle = \mathcal{N}_{ab} [h_a^\dagger h_b^\dagger]_{JM} | {\rm HF} \rangle,$$

where state ${\rm HF}$ depends on our interest nuclei and the normalization factor is given as $\mathcal{J}_{ab} = \sqrt{1 +\delta_{ab} (-1)^J}/(1+\delta_{ab})$. 

 

By adopting this form, we can describe the state of one-proton hole and one-neutron hole nuclei as follows:

$$ |p^{-1} \, n^{-1}; J \, M \rangle = [h_p^\dagger h_n^\dagger]_{JM }|{\rm HF} \rangle. $$

Note that nuclei with two proton (neutron) holes are even-even nuclei, while nuclei with one proton hole and one neutron are odd-odd nuclei. 

 

Similar to the two-particle state, two-hole nuclei state has also following symmetry relation:

$$ \begin{eqnarray} && | b^{-1}\, a^{-1}; J \, M \rangle = (-1)^{j_a + j_b + J + 1} | a^{-1} \, b^{-1}; J \, M \rangle  \\        && | n^{-1}\,p^{-1}; j \, M \rangle =  (-1)^{j_n + j_p + J + 1} |p^{-1}\, n^{-1}l {J \, M} \rangle.  \end{eqnarray}  $$

 

As shown in the previous posting (https://djlab.tistory.com/3), $(-1)^{j_a + j_b + J}$ is derived from the symmetry properties of the Clebsch-Gordan coefficients, and the additional factor of $(-1)$ from the anti-commutation relation between two annihilation operators.

Example: A=38 Isobars ($^{38}_{18}{\rm Ar}_{20},\,     ^{38}_{19}{\rm K}_{19},\, {\rm and}  \ ^{38}_{20}{\rm Ca}_{18}.$)

With $|{\rm HF} \rangle$ as $|{\rm HF} = |{\rm HF}(Z=20)\rangle_\pi |  $, each isobar can be described as follows:

$$ \begin{eqnarray} &&|^{38}{\rm Ar}; 0^+, 2^+ \rangle = \frac{1}{\sqrt{2}} [h^\dagger_{\pi 0 d_{3/2}},  h^\dagger_{\pi 0 d_{3/2}}   ]_{0^+, 2^+} |{\rm HF} \rangle, \\[12pt] &&|^{38}{\rm K}; 0^+, 1^+, 2^+, 3^+ \rangle = [h^\dagger_{\pi 0 d_{3/2}},  h^\dagger_{\nu 0 d_{3/2}}   ]_{0^+, 1^+, 2^+, 3^+} |{\rm HF} \rangle, \\[12pt] &&|^{38}{\rm Ca}; 0^+, 2^+ \rangle = [h^\dagger_{\nu 0 d_{3/2}},  h^\dagger_{\nu 0 d_{3/2}}   ]_{0^+, 2^+ } |{\rm HF} \rangle. \\ \end{eqnarray}$$

 

Particle-Hole Nuclei

For the particle-hole nuclei, there are two types of excited states: 1) Charge-conserving type (proton (neutron) hole & proton (neutron) particle), and 2) Charge-changing type (proton (neutron) hole & neutron (proton) particle). The general formula of the particle-hole nuclei is given as

$$ |a \, b^{-1}; J\, M \rangle = [c_a^\dagger h_b^\dagger]_{JM} |{\rm HF} \rangle = [c_a^\dagger \tilde{c}_b]_{JM} | {\rm HF} \rangle, $$

which satisfies the orthonormality, i.e., $\rangle a\, b^{-1}; J\, M | c \, d^{-1}; J' \, M' \rangle = \delta_{ac} \delta_{bc} \delta_{JJ'} \delta_{MM'}.$

 

Then, for proton-particle and neutron-hole,

$$ | p \, n^{-1}; J \, M \rangle = [c_p^\dagger h_n^\dagger]_{JM} |{\rm HF} \rangle, $$

with $\langle p \, n^{-1}; J \, M | p' \, n'^{-1}; J\, M \rangle = \delta_{pp'} \delta_{nn'} \delta_{JJ'} \delta_{MM'}$.

 

For neutron-particle and proton-hole,

$$ | n \, p^{-1}; J \, M \rangle = [c_n^\dagger, h_p^\dagger]_{JM} | {\rm HF} \rangle = [c^\dagger_n \tilde{c}_p]_{JM} | {\rm HF} \rangle, $$

with $\langle n \, p^{-1}; J \, M | n' \, p'^{-1}; J \, M \rangle = \delta_{pp'} \delta_{nn'} \delta_{JJ'} \delta_{MM'}.$

 

These wavefunctions also have symmetric properties:

$$ \begin{eqnarray} &&|b^{-1}\,a; J \, M \rangle = (-1)^{j_a + j_b + J + 1} | a \, b^{-1}; J \, M \rangle \\[12pt] &&|n^{-1}\,p; J \, M \rangle = (-1)^{j_p + j_n + J + 1} | p \, n^{-1}; J \, M \rangle \\[12pt] &&|p^{-1}\,n; J \, M \rangle = (-1)^{j_p + j_n + J + 1} | n \, p^{-1}; J \, M \rangle \end{eqnarray} $$

 

Example: A=4 Isobars ($^{4}_{1}{\rm H}_{3},\,     ^{4}_{2}{\rm He}_{2},\, {\rm and}  \ ^{4}_{3}{\rm Li}_{1}.$)

With $|{\rm HF} \rangle = | {\rm HF} (Z=2) \rangle_\pi | {\rm HF} (Z=2) \rangle_{\nu}$, 

$$\begin{eqnarray} &&|^4{\rm H}; 1^-, 2^- \rangle = [c_{\nu 0 p_{3/2}}^\dagger h_{\pi 0 s_{1/2}}^\dagger   ]_{1^-, 2^-} |{\rm HF} \rangle \\[12pt]  &&|^4{\rm Li}; 1^-, 2^- \rangle =  [c^\dagger_{\pi 0 p_{3/2}} , h^\dagger_{\nu 0 s_{1/2}}  ]_{1^-, 2^-} |{\rm HF} \rangle. \end{eqnarray} $$

 

 

For higher-lying excited states,

$$\begin{eqnarray} &&|^4{\rm H}; 0^-, 1^- \rangle = [c_{\nu 0 p_{1/2}}^\dagger h_{\pi 0 s_{1/2}}^\dagger   ]_{0^-, 1^-} |{\rm HF} \rangle \\[12pt]  &&|^4{\rm Li}; 0^-, 1^- \rangle =  [c^\dagger_{\pi 0 p_{1/2}} , h^\dagger_{\nu 0 s_{1/2}}  ]_{0^-, 1^-} |{\rm HF} \rangle. \end{eqnarray} $$

For $^{4}{\rm He}$, it can be described by the linear combination of the basic excitation states:

$$ | ^4{\rm He}; 1^- , 1^-, 2^-, 2^- \rangle = \frac{1}{\sqrt{2}} \left( [c_{\pi 0 p_{3/2}}^\dagger h_{\pi 0 s_{1/2}}^\dagger ]_{1^-, 2^-} |{\rm HF} \rangle \pm [c_{\nu 0 p_{3/2}}^\dagger h_{\pi 0 s_{1/2}}^\dagger ]_{1^-, 2^-} |{\rm HF} \rangle \right). $$

However, even a simplified model based on the mean-field approach cannot sufficiently explain the energy split observed in the experimental data. Careful choice of appropriate residual interaction is crucial, which will be treated in Chapter 8.