New look at Geiger-Nuttall Law and α clustering of heavy nuclei


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Yi-Bin Qian and Zhong-Zhou Ren. New look at Geiger-Nuttall Law and α clustering of heavy nuclei[J]. Chinese Physics C.
Yi-Bin Qian and Zhong-Zhou Ren. New look at Geiger-Nuttall Law and α clustering of heavy nuclei[J]. Chinese Physics C. shu
Received: 2020-02-02
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New look at Geiger-Nuttall Law and α clustering of heavy nuclei

    Corresponding author: Yi-Bin Qian,
    Corresponding author: Zhong-Zhou Ren,
  • 1. Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China
  • 2. School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

Abstract: The Geiger-Nuttall (GN) law of α decay is commonly explained in terms of the quantum tunneling phenomenon. In this study, we show that such an explanation is actually not enough regarding the α particle clustering. Such an inference is drawn from the exploration on the involved coefficients of the GN law based on the conventional recognition of α decay, namely the formation of α cluster and its subsequential penetration. The specific roles of the two former processes, played in the GN law, are manifested themselves via the systematical analysis of the calculated and experimental α decay half-lives versus the decay energies across the Z=82 and N=126 shell closures. The α-cluster preformation probability is then found to behave as a GN-like pattern. This previously ignored point is explicitly demonstrated as the product of the interplay between the mean-field and pairing effect, which in turn reveals the structural influence on the formation of α cluster in a simple and clear way. Besides providing an effective way to evaluate the amount of surface α clustering in heavy nuclei, the present conjecture supports other theoretical treatments of the α preformation probability.


    • Dated back to the early stage of nuclear physics, Geiger and Nuttall discovered an empirical relationship between the traveling range of emitted $ \alpha $ particle from the parent nucleus and the period of transformation[1]. Accordingly, the partial half-life $ T_{1/2} $ of $ \alpha $ decay is related with the corresponding decay energy,

      $ \log_{10}T_{1/2}\; = \; A(Z)Q_{\alpha}^{-1/2}+B(Z), $


      where the coefficients A and B, interpreted as the function of atomic number Z, are determined by fitting the experimental data for each isotopic chain. This simple law has been confirmed since then and still in general holds for various isotopes, despite the increasing amount of $ \alpha $ decay data in heavy and superheavy nuclei[27]. Following this, a natural question is: what is the physical mechanics behind this simple and reliable relationship? Owing to the foundation of quantum mechanics, the $ \alpha $ decay process was clarified as the quantum tunneling phenomenon by Gamow[8] and independently by Condon and Gurney[9] in 1928, well explaining the linear dependence of $ Q_{\alpha}^{-1/2} $ in the above formula. This was a great milestone in modern physics and the first successful application of the quantum theory to nuclear physics.

      Nowadays, the original GN law has been developed and extended to make it universal or adjust the different external environments[1017]. In an earlier study, Buck et al. pointed out the picture that the emitted $ \alpha $ particle should orbit around the core nucleus with different global quantum numbers via the GN plots of isotopic chains[10]. The effects of these quantum numbers were embedded in a new version of GN law, successfully reproducing the $ \alpha $ decay half-lives of various isotopes across the $ N = 126 $ shell closure[12]. Based on the Thomas expression of $ \alpha $ emission, Qi et al. explored the validity of the GN law and identified its microscopic basis[13]. Very recently, the debate over the influence of the strong electromagnetic field on $ \alpha $ decay process is initiated from a proposed shifted GN law as well[14, 15]. As is well known, the $ \alpha $ decay is usually considered as a two-simultaneous-step process of the preformation of $ \alpha $ cluster at the nuclear surface and its penetration through the Coulomb barrier. Given the success of the GN law, it is expected that these two physical steps should respectively play an important role in this simple relationship (1). However, the GN plot has been always attributed to the penetration part since the Gamow scenario, while the $ \alpha $ preformation probability is supposed to vary smoothly in the off-shell region. Although the experimental $ \alpha $ decay half-lives can be well reproduced in this strategy, the systematical deviation, between the calculated and measured half-lives, still exists not to mention the situation across the shell closure. Hence there should be something missing, rooted in the preformation of $ \alpha $ cluster. The present study is exactly to tackle this subject in view of the GN law, which can not only lead to the novel understanding of an old rule, but also provide a new perspective of $ \alpha $ clustering in heavy nuclei.

    • The present challenge is how to designate the role of the two above mentioned process acted in the GN law. Meanwhile, the fully microscopic description of the formation of $ \alpha $ cluster, in particular for a large range of nuclei, is still an open problem[18]. Hence a practical alternative is to present the systematics of calculated and experimental $ \alpha $ decay half-lives versus the $ Q_{\alpha}^{-1/2} $ in parallel, leading to the behavior of $ \alpha $ preformation factor indirectly. To this end, the available $ \alpha $-decaying nuclei are divided into three regions:

      (i) $ N\leqslant 126,\ Z\leqslant 82 $;

      (ii) $N\leqslant 126,\ Z > 82$;

      (iii) $ N>126,\ Z>82 $,

      guided by the present experimental facts. Note that the $ N>126,\ Z\leqslant 82 $ region is not considered here, because there is only one $ \alpha $ emitter up to now, namely $ ^{210} $Pb. After that, let us recall the relationship between the decay half-life and the decay width in logarithm scale,

      $ \begin{aligned}[b] \log_{10}T_{1/2} =& \log_{10}\frac{\hbar\ln2}{P_{\alpha}\Gamma}\\ =& -\log_{10}\Gamma-\log_{10}P_{\alpha}\\&+\log_{10}(\hbar\ln2). \end{aligned} $


      Obviously, the first two terms should take the responsibility for the $ Q^{-1/2}_{\alpha} $ dependence in the GN law of one isotopic chain. As mentioned above, the $ \Gamma $ term, or the related penetration probability, is indeed crucial for such an idea, based on the quantum tunneling interpretation. The $ \alpha $ preformation factor, especially in the logarithm scale of GN law, are usually believed to vary quite limitedly as compared to the decay width or the penetrability. The GN law was then well understood within the tunneling explanation, while its involved coefficients are generally constant for one certain isotopic chain. Of physical interest is the deviation from the this judgement especially when it comes to the extreme proton-neutron ratios or various isotopes across the shell closure[7, 19, 20]. The implementation of the present objective starts from the calculation of partial decay width based on the Thomas expression[21],

      $ \Gamma = \frac{\hbar^2k}{\mu}\frac{R^2 \phi_{n\ell j}(R)^2}{G_{\ell}(R)^2+F_{\ell}(R)^2}, $


      where the wave number k = $ \sqrt{2\mu Q_{\alpha}}/\hbar $, and $ \mu $ is the reduced mass $ A_{d}A_{\alpha}/(A_{d}+A_{\alpha}) $ in the unit of nucleon mass. The angular momentum $ \ell $, carried by the emitted $ \alpha $ particle, equals to zero for the present case as only ground-state ($ 0^{+} $) to ground-state transitions of even-even $ \alpha $ emitters are considered here. R is the $ \alpha $-core relative distance, while $ \phi_{n\ell j} $ and $ G_{\ell} $ ($ F_{\ell} $) are respectively the radial wave function of the $ \alpha $-core relative motion and regular (irregular) Coulomb function. When it comes to large enough R beyond the nuclear surface, the final $ \Gamma $ value is actually not dependent on the choice of R at all considering the asymptotic behavior of the former functions. In such an $ \alpha $ cluster model, the fundamental ingredient is the $ \alpha $-core relative motion, which is constrained by the large global quantum number G. The internal nodes n, in the above wave function, are connected with this quantum number by the Wildermuth condition[10],

      $ G\; = \; 2n+\ell\; = \; \sum\limits_{i = 1}^{4}g_{i}, $


      presenting the effect of the Pauli principle. $ g_i $ is here the oscillator quantum number of nucleons to form the $ \alpha $ particle, and its value is taken as: $ g_i = $ 4 for nuclei with $ 50\leqslant Z,N\leqslant 82 $, $ g_i = $ 5 for nuclei with $ 82< Z,N\leqslant 126 $, and $ g_i = $ 6 for nuclei above the $ N = $ 126 neutron shell closure. In this sense, all the nucleons in the $ \alpha $ cluster should occupy the states above the Fermi surface of the residual daughter nucleus, and the cluster-daughter scheme is then somewhat constructed on the building blocks of the shell model. The $ \alpha $ decay width $ \Gamma $ can be then obtained by following the above procedure after the $ \alpha $-core potential is constructed via the double-folding integral (see details in Refs.[2224]). The $ \log_{10}\dfrac{\hbar\ln2}{\Gamma} $ in the right side of Eq. (2) is here written as $ \log_{10}T_{1/2}^{calc} $. Once the term at left side of Eq. (2) is chosen as the experimental value, namely $ \log_{10}T_{1/2}^{expt} $, the $ P_{\alpha} $ term can be extracted as the deviation between the $ \log_{10}T_{1/2}^{expt} $ and $ \log_{10}T_{1/2}^{calc} $, namely

      $ \log_{10}P_{\alpha} = \log_{10}T_{1/2}^{calc}-\log_{10}T_{1/2}^{expt}. $

    • To see the contribution of penetration part in the GN law, we calculate the $ \alpha $ decay width of various even-even isotopes with $ N\leqslant 126 $. Indeed, these calculated $ \alpha $ decay half-lives follow the GN law with different coefficients A and B fitted for each isotopic chain as shown in Fig. 1, including elements from hafnium to thorium. On the other hand, when it comes to the next major shell above $ N = 126 $, the global quantum number G will change from 20 to 22. This can result in the deviation away from the initial linear dependence of $ \log_{10}T_{1/2}^{calc} $ on $ Q_{\alpha}^{-1/2} $ to a certain extent, which is of the essence in explaining the different GN plots for $ N<126 $ and $ N>126 $[10]. To present this key factor of the $ \alpha $-cluster model, the calculated $ \alpha $ decay half-lives are plotted as a function of $ Q_{\alpha}^{-1/2} $ for the whole Rn and Ra isotopic chains. As expected, two linear relations can be clearly distinguished for the regions (ii) and (iii) as shown in Fig. 2, while the deviation between them is quite limited. However, the GN plots of experimental $ \alpha $ decay data are believed to quite obviously change especially when crossing the $ N = 126 $ shell closure. Hence one may doubt whether it is enough that the well-known GN law is purely understood as the tunneling phenomenon.

      Figure 1.  The logarithm of calculated $ \alpha$ decay half-lives versus the $ Q_{\alpha}^{-1/2}$ for the even-even Hf-Th isotopes, in which the preformation factor is excluded, namely $ P_{\alpha} = 1$. Note that the straight line is to guide the GN law for each isotopic chain.

      Figure 2.  Same as Fig. 1 but only for the Rn and Ra nuclei including the regions with $ N>126$ (square) and $ N\leq 126$ (circle). Note that the two straight lines correspond to these two regions with different G values.

      To illuminate the above ambiguity, we systematically analyze the GN relationship of $ \log_{10}T_{1/2}^{expt} $ values with the decay energy term $ Q_{\alpha}^{-1/2} $ for available even-even $ \alpha $ emitters in the listed three regions. Accordingly, the coefficients A and B are determined for one isotopic chain in one certain region. The same procedure is also made for the $ T_{1/2}^{calc} $ case, and all these coefficients A and B are presented as a function of the atomic number Z of parent nuclei in Fig. 3. For convenience, the comparative analyses of experimental data $ T_{1/2}^{expt} $ and calculated results $ T_{1/2}^{calc} $ are respectively symbolized as case I and case II. A clear point is that different sets of coefficients (divided by the vertical dotted line), in the GN plot, are requested for regions (i) (with $ Z\leqslant 82 $) and (ii) (with $ Z>82 $), whatever the case I or II is concerned. This is expected due to the shell effect on the $ \alpha $ decay, which is not a new story. However, the unexpected and impressive issue is that there is an unambiguous discrepancy between the coefficients A (and B) of case I and those of case II, no matter which region is concerned. We may therefore conclude that the tunneling, corresponding to case II, is definitely not the whole picture of the $ \alpha $ decay process. The only way to fill this gap should be dependent on the $ \alpha $ preformation probability. By combing with Eq. (5), the $ \alpha $ preformation factor $ P_{\alpha} $ is supposed to follow in an exponential law with the $ Q_{\alpha}^{-1/2} $, i.e.

      Figure 3.  Coefficients A and B of even-even nuclei involved in the GN law versus the atomic number Z for regions (i) plus (ii) at the left panel and region (iii) at the right panel. The square indicates the computational case II, and the experimental case I is denoted by the circle. The dashed lines are to guide the eye for presenting the different dependence in different regions.

      $ \log_{10}P_{\alpha} = {\cal{A}}(Z)Q_{\alpha}^{-1/2}+{\cal{B}}(Z). $


      Furthermore, by a careful analysis on the specific values of A and B for both cases I and II, one can combine two panels of this figure to denote two issues about the preformation probability of emitted $ \alpha $ particle as below:

      i). The two terms in the right side of the above $ P_{\alpha} $ expression appear to compete with each other considering their signs and absolute values. What are their microscopic basements?

      ii). The slope of linear relationship $ {\cal{A}} $ or $ {\cal{B}} $ appears to be different for the three focused regions. Moreover, the values of $ {\cal{A}} $ and $ {\cal{B}} $ in region (ii) are obviously larger than those of two other regions.

      To answer these questions and pursue a deep understanding of clustering at the surface of heavy nuclei, we recall the formation amplitude in the conventional formalism, namely[18, 25]

      $ F(R) = \int d\hat{R}d\xi_{\alpha}d\xi_{d}[\Psi_{d}(\xi_{d})\phi_{\alpha}(\xi_{\alpha})Y_{l}(\hat{R})]^{*}\Psi_{p}, $


      where $ \xi_{d} $ and $ \xi_{\alpha} $ are respectively the internal degrees of freedom for the daughter nucleus and the $ \alpha $ particle. The $ \Psi_{d}(\xi_{d}) $ and $ \Psi_{p}(\xi_{d},\xi_{\alpha},\hat{R}) $ present the wave functions of the daughter and parent nuclei. The intrinsic $ \alpha $ particle wave function behaves in a $ n = l = 0 $ harmonic oscillator eigenstate[26]

      $ \phi_{\alpha}(\xi_{\alpha}) = \sqrt{\frac{1}{8}}\left(\frac{\nu_{\alpha}}{\pi}\right)^{9/4}\exp[-\nu_{\alpha}(r_{nn}^2+r_{pp}^2+r_{np}^2)/4]S_{\alpha} $


      with the relative distances $ r_{nn} $ ($ r_{pp} $) of neutron-neutron (proton-proton) pairs plus the the distance $ r_{np} $ between the mass centers of the nn and pp pairs. The quantity $ \nu_{\alpha} $ is the $ \alpha $ particle harmonic oscillation parameter, and $ S_{\alpha} $ is its spin-isospin function. Very recently, given the extreme case induced by four non-interacting protons and neutrons, the particle decay unit (p.d.u.) was defined to set a simple averaged single-particle limit for the $ \alpha $ particle formation amplitude[27],

      $ F_{\alpha;pdu} = \frac{\sqrt{8}\nu_{\alpha}^{-3/4}\pi^{-7/4}}{R^3}, $


      when regarding the ground-state to ground-state (g.s.) transitions of even-even nuclei. The collectivity involved in the $ \alpha $ decay process is then revealed via the ratio between the extracted $ \alpha $ particle formation amplitude and the corresponding p.d.u. value. Inspired by this, we assume the $ \alpha $ preformation probability can be written as

      $ P_{\alpha} = P_{\alpha}^{c}P_{\alpha}^{0}, $


      where the $ P_{\alpha}^{0} $ denotes the contribution of "mean-filed"-like part in terms of uncorrelated nucleons, corresponding to the $ |F_{\alpha;pdu}|^{2} $ value. The former $ P_{\alpha}^{c} $ part corresponds to the enhancement factor due to the collectivity degree of freedom. On the other hand, as shown above, the $ F_{\alpha;pdu} $ actually relates with the nuclear volume, i.e., the mass number of parent nuclei. With these in mind, the $ {\cal{B}}(Z) $ term in Eq. (6) is expected to present the effect of single particle pattern, while the $ {\cal{A}}(Z)Q_{\alpha}^{-1/2} $ should be derived from the collective behavior of corresponding parent nuclei. The latter $ Q_{\alpha}^{-1/2} $ dependence of $ P_{\alpha} $ can be unravelled in the shell-model context. It is believed that the clustering of $ \alpha $ particles at the nuclear surface is governed by the pairing force acting among the involved neutrons and protons. For two particles in a non-degenerate system with a constant pairing strength $ {\cal{G}} $, the corresponding two-particle wave function amplitudes $ X_{i} $ should be related with the $ {\cal{G}} $ value. On one hand, the $ \alpha $ (four-particle) formation amplitude is connected with the $ X_{i} $ values. This can be also confirmed in the multistep shell-model method (MSM), where the MSM basis is consisted of the tensorial product of the two-particle correlated states[28]. On the other hand, the g.s. energy of even-even nuclei can be presented as the function of the pairing strength $ {\cal{G}} $, which determines the $ \alpha $ decay energy via the difference between g.s. energies of parent and daughter nuclei. As a result, it can be expected that there should be a connection between $ Q_{\alpha} $ and the final $ \alpha $ preformation probability, namely $ P_{\alpha}^c\sim X_{i}\sim {\cal{G}}\sim Q_{\alpha} $. In this sense, the pairing correlations can be understood as the origin of the term $ {\cal{A}}(Z)Q_{\alpha}^{-1/2} $ in the $ P_{\alpha} $ expression (6).

      In a nutshell, the interplay between the mean-filed and pairing effect leads to the final clustering of two neutrons and two protons at the nuclear surface, performing a simple but meaningful relationship similar to the GN law. After the above discussion on issue i), the second issue can be explained as follows. The valence neutrons and protons are supposed to locate in the same major shell in the region (ii) with $ 82<Z,N<126 $, leading to larger overlaps of single particle radial wave functions and relative stronger pairing correlation. Hence the absolute values of coefficients $ {\cal{A}} $ and $ {\cal{B}} $ in the region (ii) are relatively larger than those in two other regions. Moreover, despite the discrepancy, the comparable competition between these two terms in Eq. (6) causes the smooth varying trend of $ P_{\alpha} $ off the shell closure. This is the deep reason why the Geiger-Nuttal law can be successfully explained as the quantum tunneling phenomenon while excluding the contribution of $ \alpha $ preformation process. Let us mention here that the four-nucleon correlations (quartetting) may play an important role in the collective dynamics of clustering[2931]. Considering this, we plot the coefficients $ {\cal{A}} $ and $ {\cal{B}} $ in Eq. (6), generated by the difference between the systematics of experimental data and that of the calculated results based on Eq. (5), versus the number of quartets $ N_{q} $ in Fig. 4. The $ N_{q} $ value is determined by the number of proton pairs above the closet proton magic number $ Z_{mag} $, namely $ N_{q} = (Z-Z_{mag})/2 $, as suggested in Ref.[31]. As one can see from this figure, the involved coefficients in the $ P_{\alpha} $ systematics vary more obviously with the quartet numbers in the region (ii) as compared to those in the region (i). These $ \alpha $ decaying nuclei in the region (ii) are more close to the doubly-magic nuclei, implying that the large quartetting effect is present in emitters near the shell closure. This point is consistent with other studies[30, 31], and the quartetting effect is supposed to diminish greatly due to the Pauli principle when more valence nucleons are added.

      Figure 4.  Extracted coefficients $ {\cal{A}}$ and $ {\cal{B}}$ of the $ P_{\alpha}$ formula Eq. (6), based on the Eq. (5), versus the quartet number for $ \alpha$ emitters in regions (ii) $ Z>82$ and (i) $ Z\leq 82$, respectively.

      Last but not least, we would like to discuss the consistency of the present conjecture about $ P_{\alpha} $ with other theoretical studies. Recently, the cluster formation model (CFM) was proposed to evaluate the $ \alpha $ preformation probability in an effective way[32, 33]. Within the CFM, the $ \alpha $ formation amplitude is connected with the formation energy and the decay energy, which is supported by the above conclusion. Given the two-particle formation treatment in the BCS approach, the similarity between the neutron/proton pairing gaps and the $ \alpha $ preformation amplitude is also presented[4], supporting the present GN-like law of $ P_{\alpha} $ to some extent. Dated back to the earlier microscopic calculation on the absolute $ \alpha $ decay width in the shell model plus the cluster component[34], the g.s. energy of $ ^{212} $Po, with respect to the four-nucleon threshold, was found to be very sensitive to the shell-model basis states including the cluster states. This g.s. energy, after dividing the $ \alpha $ particle energy, equals to the $ \alpha $ decay energy, while the $ \alpha $ formation is regulated by the components of the state vector in the standard cluster basis. Consequently, it is expected that the final $ \alpha $ preformation probability is related with the decay energy accompanied by the Z dependence. Moreover, the penetration process is well addressed no matter the microscopic or the semi-classical (like the WKB approximation) way is concerned, which is actually the basement for various phenomenological formulas of $ \alpha $ decay half-lives after considering some extra factors for different types of parent nuclei[11, 16, 17]. However, through the present analysis, one can find that the missing point is the similar behavior of $ \alpha $ preformation factor, as demonstrated here, which is the other key ingredient for the success of these empirical evaluations.

    • In conclusion, the surface $ \alpha $-clustering is an important and profound phenomenon for both the cluster structure and decay properties of heavy neutron-deficient nuclei. We derive an effective relationship for the $ \alpha $-cluster preformation probability by digging out the famous Geiger-Nuttall law of $ \alpha $ decay, to shed some new light on this topic. By comparably analyzing the involved coefficients of GN law from the systematics of experimental and calculated $ \alpha $ decay half-lives, the amount of surface $ \alpha $ clustering is supposed to follow a GN-like pattern. This conjecture is requested by the full description of the well-known GN law rather than the simple recognition in terms of tunneling. Within the shell-model context, the behavior of $ \alpha $ preformation factor is determined by the mean-field and pairing effects, which are explicitly manifested themselves via the two Z dependent terms of the GN-like formula of $ P_{\alpha} $. Furthermore, the balance between these two ingredients produces the relative smooth value of $ \alpha $ preformation probability, leading to the success of most of the effective approaches on $ \alpha $ decay. Besides, the present study is consistent with other available theoretical results on $ \alpha $ clustering of heavy nuclei. Encouraged by this, we hope that the present derivation can offer an novel window to understand the $ \alpha $ correlations in probing the fundamental nuclear interactions.

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