Isotope shift and search for metastable superheavy elements in astrophysical data

TL;DR

This paper investigates the spectral lines of heavy radioactive elements in astrophysical data, proposing they originate from decay of superheavy nuclei near the island of stability, and calculates isotope shifts to identify these isotopes.

Contribution

It introduces calculations of isotope shifts for superheavy elements to aid in identifying metastable nuclei in astrophysical spectra, linking laboratory data with cosmic observations.

Findings

Calculated isotope shifts for No, Lr, Nh, Fl, and Z=120.

Identification of spectral lines potentially from N=184 isotopes.

Support for the existence of metastable superheavy nuclei in space.

Abstract

Spectral lines belonging to the short-lifetime heavy radioactive elements up to Es ($Z$=99) have been found in the spectra of the Przybylski's star. We suggest that these unstable elements may be decay products of a "magic" metastable nucleus belonging to the the island of stability where the nuclei have a magic number of neutrons $N=184$. The laboratory-produced nuclei have a significantly smaller number of neutrons. To identify spectra of the $N=184$ isotopes of these nuclei and their neutron-reach superheavy decay products in astrophysical data we calculate the isotope shift which should be added to the laboratory - measured wavelenghs. The results for the isotopic shifts in the strongest optical electromagnetic transitions in No, Lr, Nh, Fl,and $Z$=120 elements are presented.

Tables

Table 1. Table 1: Isotope shift for strong electric dipole transitions from the ground state of some heavy elements. The shift is given by δ ​ E = a ​ ( A 1 1 / 3 − A 2 1 / 3 ) 𝛿 𝐸 𝑎 superscript subscript 𝐴 1 1 3 superscript subscript 𝐴 2 1 3 \delta E=a(A_{1}^{1/3}-A_{2}^{1/3}) , where A 1 subscript 𝐴 1 A_{1} and A 2 subscript 𝐴 2 A_{2} are atomic numbers of the two isotopes, the values of the parameter a 𝑎 a are presented in the last column. A 0 = Z + 184 subscript 𝐴 0 𝑍 184 A_{0}=Z+184 is the atomic number of the more stable isotope with the neutron number N = 184 𝑁 184 N=184 , A s subscript 𝐴 𝑠 A_{s} is the atomic number of the heaviest synthesised isotope. The ”Frequency” column presents experimental (E) or theoretical (T) values for the frequency of the transition found in literature. Theoretical uncertainty is presented in parentheses.

Atom				Transition			Frequency	Reference	$a$
$Z$	Symbol	$A_0$	$A_s$				(cm${}_{}{}^{-1})$		(cm${}_{}{}^{-1})$
102	No	286	259	$7s^2{}^1$S0	-	$7s7p^1$P${}_1{}^{}{}_{}{}^o$	29961.457${}_{-0.007}{}^{}{}_{}{}^{+0.041}$111For the ${}_{102}{}^{}{}_{}{}^{254}$No isotope.	E Laatiaoui et al. (2016)	34
103	Lr	287	266	$7s^{2}7p^2$P${}_{1/2}{}^{}{}_{}{}^o$	-	$7s^{2}8s^2$S1/2	20253(500)	T Dzuba et al. (2014)	-19
113	Nh	297	286	$7s^{2}7p^2$P${}_{1/2}{}^{}{}_{}{}^o$	-	$7s^{2}8s^2$S1/2	36041(440)	T Dinh and Dzuba (2016)	-17
114	Fl	298	292	$7p^2{}^1$S0	-	$7p8s^1$P${}_1{}^{}{}_{}{}^o$	43876(310)	T Dinh and Dzuba (2016)	-2.6
120	Ubn	304		$8s^2{}^1$S0	-	$8s8p^1$P${}_1{}^{}{}_{}{}^o$	27559(200)	T Ginges and Dzuba (2015)	148

Equations

$$\nu =a(A_1^{1/3}-A_2^{1/3}).$$

$$\nu {(}^{286}\mathrm{No})=29961.457+$$

$$34(254^{1/3}-286^{1/3})=29952.8{\mathrm{cm}}^{-1},$$

$$\delta {ϵ}_s=8\sqrt{2}\xi K{ϵ}_s^{3/2}\frac{\gamma +1}{\Gamma (2\gamma +1{)}^2(2\gamma +1)}$$

$$\times (2Z{R}_N{)}^{2\gamma }\frac{\delta R_N}{R_N}\mathrm{Ry},$$

$$\mathrm{where}\xi =\frac{2{\gamma }^2(2-\gamma )(2\gamma +1)}{(\gamma +1)(2+\gamma )}.$$

$$\delta {ϵ}_p=\frac{1-\gamma }{1+\gamma }{\left(\frac{{ϵ}_p}{{ϵ}_s}\right)}^{3/2}\delta {ϵ}_s.$$

Full text

The existence of the hypothetical island of stability for superheavy elements (atoms with nuclear charge $Z\geq 104$) is an important problem of modern nuclear physics. All known nuclei with $Z>98$ have short lifetimes varying from fractions of seconds to hundreds of days. On the other hand, the theoretically estimated lifetime of the Fl nucleus with $Z=114$, $N=184$ is $10^{7}$ years  Oganessian et al. (2004); Hamilton et al. (2013). The nuclear shell model states that nuclei are most stable when both protons and neutrons fully occupy closed shells (double magic nuclei). For superheavy elements with spherical nuclei, the magic neutron number is believed to be 184, while magic proton numbers are $Z=114$, $Z=120$ and $Z=126$ (see, e.g. Oganessian et al. (2004); Hamilton et al. (2013)). (Magic numbers $Z=122$, $N=172,178,182,194$ are also mentioned in the literature Oganessian et al. (2004); Hamilton et al. (2013); Patra et al. (2000); Santhosh and Sukumaran (2017).) Therefore, the prospective candidates for the island of stability include the ${}^{298}_{114}$Fl, ${}^{304}_{120}$Ubn, ${}^{310}_{126}$Ubh and some other isotopes.

Superheavy elements are not found in nature but are produced in laboratories by colliding lighter atoms. All elements up to $Z=118$ have been synthesised so far  Hamilton et al. (2013). However, all superheavy elements synthesised in a laboratory are neutron-poor elements, with the number of neutrons being significantly smaller than required to make the most stable isotopes. For example, the heaviest isotope of Fl, produced in Dubna  Oganessian et al. (2004), ${}^{292}_{114}$Fl, is six neutrons short of the magic number $N=184$. This is a common problem because the Coulomb repulsion energy for protons increases as $Z^{2}$ and to compensate for this by the attractive strong interaction energy, the number of neutrons $N$ should increase with $Z$ faster than the number of protons. Therefore, the very large number of neutrons, $N=184$, needed for a more stable superheavy element, cannot be obtained by colliding any pair of lighter elements where the $N/Z$ ratio is smaller than that in the island of stabiltiy.

There is an alternative way of searching for the island of stability by looking for traces of superheavy elements in astrophysical data (see, e.g. Polukhina (2012)). For example, optical lines of many actinide atoms and ions up to einsteinium (Es, $Z=99$) have been found in the spectra of the Przybylski’s star Gopka et al. (2008). Since these lines belong to short-living isotopes the natural question is how the isotopes were produced. One possible answer is that they were produced via decay of heavier long-living elements.

Heavy elements are dispersed into the interstellar medium during supernova explosions. The neutron flux during the supernova explosion is very high, and this may lead to the production of the $N=184$ and other neutron reach isotopes. If they are close to the island of stability they may have sufficiently long lifetime to survive to present time and decay to isotopes of actinides and other elements. One needs to know frequencies of the strong electric dipole transitions for superheavy elements to search for them in astrophysical spectra. The heaviest element for which one such frequency is measured is No ($Z=102$) Laatiaoui et al. (2016). Work is underway for similar measurements in Lr ($Z=103$) Sato et al. (2015); Sato (2015); Block (2015). There are good prospects for further progress in this field.

There are many high-quality atomic spectra calculations for superheavy elements (see, e.g. Eliav et al. (1996); Yu et al. (2008); Borschevsky et al. (2015); Eliav et al. (2015); Borschevsky et al. (2013); Dzuba et al. (2014); Ginges and Dzuba (2015); Dinh and Dzuba (2016)). However, the accuracy of the calculations is not sufficiently high to reliably identify spectral lines in astrophysical data. A possible solution involves the following three-stage process:

Measure the frequencies of strong electric dipole transitions in a laboratory-produced superheavy element. This will be a neutron-poor isotope. 2. 2.

To find the frequencies for the more stable neutron-rich isotope, the isotopic shifts are calculated and added to the laboratory frequencies. 3. 3.

The results are used to search for the spectral lines from the more stable neutron-rich isotope in astrophysical data.

One needs to know the isotope shifts for superheavy elements to follow this path. The heaviest elements for which isotope shift experimental data are available are Pu, Am, Cm Blaise and Wyart (1992) and No Laatiaoui et al. (2016). These data can be used for searching heavier isotopes of the elements. However, no experimental data are available for the superheavy elements in the vicinity of the island of stability, such as 114Fl, 120Ubn, etc.

In the present work we calculate the isotope shift for some elements within nuclear charge range $102\leq Z\leq 120$, including candidates for the island of stability, Fl and Ubn. The isotope shift in superheavy elements is strongly dominated by the field (volume) shift Sobel’man et al. (1972). Therefore, we ignore the mass shift and calculate only the energy shift due to the change of the nuclear charge radius. We assume a Fermi distribution for the nuclear charge with the radius given by $R_{N}=1.1A^{1/3}$ fm with a skin thickness of 2.3 fm. Here $A$ is the number of nucleons in the nucleus.

We use a combination of the configuration interaction (CI) method with many-body perturbation theory (MBPT) to perform the calculations (the CI+MBPT method Dzuba et al. (1996); Dzuba and Johnson (1998); Dzuba (2005)). The field shift is obtained by repeating calculations with different values of nuclear radius and taking differences between the results. We consider only the strongest optical transitions from the ground state, which are the $s-p$ electric dipole transitions to states of opposite parity having the same total electron spin (the change of the spin leads to a partial suppression of the $E1$ transition probability since the electric dipole operator conserves the spin).

Since $R_{N}\propto A^{1/3}$ it is convenient to approximate the calculated isotope shift $\nu$ by an analytic formula:

[TABLE]

The coefficient $a$ in (1) is found by fitting to the results of the CI+MBPT numerical calculations. The values of $a$ for No, Lr, Nh, Fl and Ubn are presented in Table 1.

For No there is an additional source of the information about the isotope shift. We may use the experimental value of the isotope shift: $\nu(^{252}{\rm No})-\nu(^{254}{\rm No})=0.32$ cm*-1* Block (2015). Then Eq. (1) provides the value $a=19$ cm*-1* for No. This is about 1.8 times smaller than the calculated value $a=34$ cm*-1* (see Tavle 1). The difference is partly due to an uncertainty of the atomic calculations and partly due to an uncertainty in the nuclear radius change. The formula $R_{N}\propto A^{1/3}$ gives an average trend in the dependence of $R_{N}$ on $A$ while the actual change of the nuclear radius depends on what orbitals are occupied by the additional neutrons. Thus, we may trust the dependence of $R_{N}\propto A^{1/3}$ only for a significant change in $A$ when the shell model fluctuations are relatively suppressed in comparison with the average trend. The 254No isotope of nobelium is the heaviest element for which experimental frequency is available. The isotope with the magic neutron number $N=184$ is 286No. The difference in neutron numbers for these two isotopes is large, $\Delta N=286-254=32$. Therefore, the $R_{N}\propto A^{1/3}$ trend should hold to a high precision. This means that prediction transition frequency for the 286No isotope using theoretical isotope shift might be more accurate than using experimental shift. The value for the frequency is

[TABLE]

if the theoretical value $a=34$ cm*-1* is used (and $\nu(^{286}{\rm No})=29956.6\ {\rm cm}^{-1}$ if the experimental value $a=19$ cm*-1* is used).

It is important to have an independent way of estimating the isotope shift. We use analytic solutions for the isotope shift problem in the single-electron approximation. The accuracy of this approach is significantly lower than the accuracy of numerical many-body calculations. However, it helps to avoid mistakes and it can be used to extrapolate the isotope shift from lighter atoms (where experimental data are available) to heavier elements with a similar electron structure. Such an extrapolation makes sense since the relative value of the many-body corrections to the single-electron formula is approximately the same in atoms with similar structure of external shells.

The energy shift of an $s$ state due to change of the nuclear radius is given by  Sobel’man et al. (1972); Smorodinsky (1947)

[TABLE]

Here $\gamma=\sqrt{1-(\alpha Z)^{2}}$, $\Gamma$ is the gamma-function, $R_{N}$ is nuclear radius, $\epsilon_{s}$ is the ionization energy of the $s$ orbital in atomic units (2 Ry), Ry=109737 cm*-1*. The parameter $\xi$ describes the non-perturbative relativistic correction which is important at $Z>110$ (when $Z$ approaches 137, the finite nuclear size effect is not a small perturbation because of the collapse of the point-like nucleus spectrum for $Z>137$). The analytic result in Eq. (3) with the factor $K=1$ was obtained in Ref. Sobel’man et al. (1972); Smorodinsky (1947) for the unrealistic charge distribution where all the charge is located on the surface of the sphere of the radius $R_{N}$. A more realistic case with the electric charge homogeneously distributed inside the sphere may be obtained by introducing an approximate correction factor $K=3/(2\gamma+3)$ to the isotope shift in Eq. (3) (this factor follows from the perturbative treatment of the isotope shift for the cases of the surface and volume charge distributions - see Racah-Rosental-Breit formulae for the isotope shift presented in the book  Sobel’man et al. (1972)). With this factor included the analytic results are close to the experimental values of the isotope shifts (see below).

The formula for the isotopic shift for $p$ wave is not presented in the text books. Indeed, in the non-relativistic approximation, the single-particle isotope shift for $p$ orbitals is zero since the $p$ orbitals vanish at the origin $r=0$. However, the relativistic $p_{1/2}$ wave density near the nucleus is proportional to the $s$ wave density. The relativistic calculation of the proportionality factor gives the following isotopic shift for $p_{1/2}$ states in heavy atoms:

[TABLE]

Thus, in the superheavy atoms, where $(1-\gamma)\sim 1$, the $p_{1/2}$ shift is not suppressed significantly.

Formulae (3,4,5) give $a=160$ cm*-1* for Ubn in good agreement with the calculated value $a=148$ cm*-1*(see Table 1). The difference can be attributed to the many-body effects neglected in (3,4,5). The same formulae give $a=9.3$ cm*-1* for Ra, which is a lighter analogue of Ebn. On the other hand, the fitting of the experimental data Dammalapati et al. (2016) gives $a=12$ cm*-1*.

Note the very rapid increase of the isotope shift when the nuclear charge is approaching 120. Such an increase is clearly seen in both the analytic formulae and the numerical many-body calculations. This is the result of the relativistic effects mentioned above.

Another feature of the isotope shift is its smaller value in $7p_{1/2}$ - $8s$ transitions in Lr, Nh and Fl. This is the result of the cancellation of the shifts of the $7p_{1/2}$ and $8s$ states. For example, for Nh the formulae give nearly the same shifts of the energies of the lower $7p_{1/2}$ and upper $8s$ states: $\delta\epsilon(7p_{1/2})=1.11$ cm*-1*, while $\delta\epsilon(8s_{1/2})=0.97$ cm*-1*. When such a cancellation occurs, the accuracy of the analytic formulae is low: the formulae give $a=-4.5$ cm*-1* for Nh in apparent disagreement with the result of the many body calculations $a=-17$ cm*-1* (see Table 1).

Thus, the formulae (3,4,5) give a reasonable accuracy when an $s$ state is lower than a $p$ state and indicate cancellation when a $p_{1/2}$ state is lower than an $s$ state.

We hope that this work provides a motivation for a further progress in the measurements of the transition frequencies for superheavy elements, calculations of the isotope shifts and search for the corresponding transitions in astrophysical spectra.

We are very grateful to J.-T. Hu and G. Gr$\ddot{\rm u}$ning for the information about measured heavy element spectra. This work is supported by the Australian Research Council and Gutenberg Fellowship.