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=== Nuclear reaction ===
Rarely used are direct nuclear reactions in which nuclei are converted into PAC probes by bombardment by high-energy elementary particles or protons. This causes major radiation damage, which must be healed. This method is used with PAD, which belongs to the PAC methods.
== Theory ==
=== General theory ===
[[File:PAC-Spectroscopy-cascade.png|thumb|right|General γ-γ-cascade with life-time <math>\tau_N</math> of the intermediate state.]]
In the general theory for a transition <math>M_i\rightarrow M_f</math> is given:
:<math>W(M_i\rightarrow M_f)=\left|\sum_M\langle
M_f|\mathcal{H}_2|M\rangle\langle M|\mathcal{H}_1|M_i\rangle\right|^2
</math>
:<math>W(\vec{k}_1,\vec{k}_2)=\sum_{M_i,M_f,\sigma_1,\sigma_2}\left|\sum_M\langle
M_f|\mathcal{H}_2|M\rangle\langle M|\mathcal{H}_1|M_i\rangle\right|^2
</math>
:<math>W(\vec{k}_1,\vec{k}_2)=W(\Theta)=\sum_{k_{gerade}}^{k_{max}}A_k(1)A_k(2)P_k(\cos{\Theta})
</math>
:<math>0\leq k\leq</math> Minimum von <math>(2I,l_1+l_1',l_2+l_2')</math>
[[File:PAC-Spectrosocpy-detector-angles.png|thumb|right|Detector angles]]
:<math>W(\Theta,t)=\sum_{k=2,4}A_{kk}P_k(\cos{\Theta})</math>
:<math>|M_a\rangle\rightarrow\Lambda(t)|M_a=\sum_{M_b}|M_b\rangle\langle M_b|\Lambda(t)
|M_a\rangle </math>
:<math>W(\vec{k}_1,\vec{k}_2,t)=\sum_{M_i,M_f,\sigma_1,\sigma_2}\left|\sum_{M_a}\langle
M_f|\mathcal{H}_2\Lambda(t)|M_a\rangle\langle
M_a|\mathcal{H}_1|M_i\rangle\right|^2=\langle\rho(\vec{k}_2)\rangle_t
</math>
:<math>W(\vec{k}_1,\vec{k}_2,t)=\sum_{k_1,k_2,N_1,N_2} A_{k_1}(1)A_{k_2}(2)\frac{1}{\sqrt{(2k_1+1)(2k_2+1)}}\times
Y_{k_1}^{N_1}(\Theta_1,\Phi_1)\cdot
Y_{k_2}^{N_2}(\Theta_2,\Phi_2)G_{k_1k_2}^{N_1N_2}(t)
</math>
with:
:<math>
G_{k_1k_2}^{N_1N_2}=\sum_{M_a,M_b} (-1)^{2I+M_a+M_b}\sqrt{(
2k_1+1)(2k_2+)}\times\langle
M_b|\Lambda(t)|M_a\rangle\langle
M_b'|\Lambda(t)|M_a'\rangle^{*}\times\begin{pmatrix}
I&I& k_1\\
M_a'&-M_a&N_1
\end{pmatrix}
\begin{pmatrix}
I&I&k_2\\
M_b'&-M_b& N_2
\end{pmatrix}
</math>
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