A recent experiment identified a weak spin-orbit-coupling in atomic Bk while a jj coupling scheme described atomic Cf. It concluded that these observations strengthen Cf as a transitional element in the actinide series. This paper uses Ramsauer-Townsend (R-T) minima and shape resonances (SRs) from low-energy electron elastic total cross sections (TCSs) calculated using the rigorous Regge pole method as novel confirmation of the observation. Regge pole-calculated TCSs for Bk and Cf atoms are characterized generally by ground, metastable and excited states negative-ion formation, R-T minima and SRs. Additionally, a polarization-induced metastable TCS with a deep R-T minimum near threshold characterizes the Bk TCSs, which flips over to a well pronounced SR appearing very close to threshold in the Cf TCSs. This demonstrates the sensitivity of R-T minima and SRs to the electronic structure of these atoms, thereby permitting their first time ever use as novel and rigorous validation of the experimental observation. .
Article
Rigorous probing of the electronic structure of Berkelium and Californium actinide atoms
https://doi.org/10.21203/rs.3.rs-1456195/v1
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posted 31 Mar, 2022
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generalized bound states
electron scattering
anionic binding energies
Regge poles
actinide atoms
Ramsauer-Townsend minima
The recent experiment [1] used a transmission electron microscopy to characterize the highly radioactive Bk and Cf atoms, using only a single nanogram and firmed that Cf is indeed a transitional element in the actinide series. This experimental breakthrough, including the recent first ever measurement of the electron affinity (EA) of the highly radioactive element At[2] and the recent measurements of the EAs of Th [3] and U[4,5] represent significant advances in the measurements of the challenging highly radioactive elements. And more such measurements of other radioactive atoms can be expected in the near future. Consequently, reliable theoretical predictions are essential for a fundamental understanding of the underlying physics. Already, the Regge pole method has accomplished a remarkable feat in determining the theoretically challenging to calculate EAs of the Au, Pt, and At atoms, etc. see Table 1 of [6] for comparisons, as well as the EAs of the fullerene molecules C20 through C92[7,8]. In this paper we present an entirely new approach to the validation of the experimental observation in [1], namely through the behavior of the Ramsauer-Townsend (R-T) minima and the shape resonances (SRs) in the metastable electron elastic total cross sections (TCSs) of atomic Bk and Cf.
Worth mentioning here and relevant to this paper as well is that the first break in periodicity in the actinide series was determined to occur between Pu and Am [9], see also Contemporary Chemistry of Berkelium and Californium[10]. The polarization-induced metastable TCS with the deep R-T minimum near threshold first appeared in the TCSs of the actinide atoms in the Pu atom [11]. It was attributed to the size effect and the first 6d-orbital collapse impacting significantly the polarization interaction. This near threshold R-T minimum first flipped over to a SR appearing very close to threshold in the metastable TCS of the Cf atom [12]. The flip over was attributed to the increase in the size of the actinide atom and the second 6d-orbital collapse, impacting significantly the polarization interaction. This provides a new mechanism of tuning R-T minima and SRs through the polarization interaction as well as a new understanding of the structural dynamics of the actinide atoms.
The paper is organized as follows: Section 2 presents the essentials of the Method of Calculation while Section 3 deals with the Results. Section 4 provides the Conclusion.
In this paper we have used the rigorous Regge pole method to calculate the electron elastic total cross sections (TCSs). Regge poles, singularities of the S-matrix, rigorously define resonances [13,14] and in the physical sheets of the complex plane they correspond to bound states [15]. In [16] it was confirmed that the Regge poles formed during low-energy electron elastic scattering become stable bound states. In the Regge pole, also known as the complex angular momentum (CAM), method the important and revealing energy-dependent Regge trajectories are also calculated. Their effective use in low-energy electron scattering has been demonstrated in for example [17,18].
The near-threshold electron–atom/fullerene collision TCS resulting in negative-ion formation as resonances is calculated using the Mulholland formula [19]. In the form below, the TCS fully embeds the essential electron-electron correlation effects [20, 21] (atomic units are used throughout):
In Eq. (1) S(λ) is the S-matrix, 
, m being the mass and E the impact energy, rn is the residue of the S-matrix at the nth pole, ln and I(E) contains the contributions from the integrals along the imaginary l-axis (l is the complex angular momentum); its contribution has been demonstrated to be negligible [17].
As in [22] here we consider the incident electron to interact with the complex heavy system without consideration of the complicated details of the electronic structure of the system itself. Therefore, within the Thomas-Fermi theory, Felfli et al [23] generated the robust Avdonina-Belov-Felfli (ABF) potential which embeds the vital core-polarization interaction
In Eq. (2) Z is the nuclear charge, α and β are variation parameters. For small r, the potential describes Coulomb attraction between an electron and a nucleus, U(r) ~ −Z/r, while at large distances it has the appropriate asymptotic behavior, viz. ~ −1/(αβr4) and accounts properly for the polarization interaction at low energies. For an electron, the source of the bound states giving rise to Regge trajectories is the attractive Coulomb well it experiences near the nucleus. The addition of the centrifugal term to the well 'squeezes' these states into the continuum [24]. Regge poles are generalized bound-states, namely solutions of the Schrödinger equation where the energy E is real, positive and the angular momentum λ is complex. The complex angular momentum methods have the advantage in that the calculations are based on a rigorous definition of resonances, viz. as singularities of the S-matrix, see also [24, 25]
The strength of this extensively studied potential [26, 27] lies in that it has five turning points and four poles connected by four cuts in the complex plane. The presence of the powers of Z as coefficients of r and r2 in Eq. (2) ensures that spherical and non-spherical atoms and fullerenes are correctly treated. Also appropriately treated are small and large systems. The effective potential 
is considered here as a continuous function of the variables r and complex λ. The details of the numerical evaluations of the TCSs have been described in [21] and references therein.
In figures 1 and 2 we present the Regge pole-calculated electron elastic TCSs for the Bk and Cf actinide atoms, respectively. As seen the TCSs are characterized generally by dramatically sharp resonances, representing ground, metastable and excited states negative-ion formation, shape resonances (SRs) (broad peaks) and Ramsauer-Townsend(R-T) minima. Also, the highest excited states TCSs (green curves) exhibit fullerene molecular behavior [7]. The energy positions of the sharp resonances, well delineated correspond to the binding energies (BEs) of the formed negative ions during the electron collisions with the Bk and Cf atoms. Each figure consists of ground (red curve), metastable (blue and orange curves) and excited (brown and green curves) states TCSs. At first glance these TCSs appear a little complicated. However, they can be readily understood if each curve is discussed separately, see [6, 11] for example. The R-T minima manifest the effects of the polarization interaction [28], while the SRs convey the trapping effect of the centrifugal potential. The underlying physics has already been discussed in [6, 11]; it will not be repeated here.
For the purpose of this paper we focus mainly on the polarization-induced TCSs (orange curves) and the ground state TCSs in both Figs. 1 and 2. For a better understanding and appreciation of the results, it is appropriate to place in perspective the polarization-induced TCSs that are characterized by a deep R-T minimum near threshold in the TCSs of Bk and a pronounced SR very close to threshold in the Cf TCSs. The polarization-induced TCS with the deep R-T minimum near threshold first appeared in the actinides TCSs through the atomic Pu TCSs [11]. It was attributed to the size effect and the first 6d-orbital collapse impacting the polarization interaction significantly. The first 6d-collapse occurred in the transition Np[Rn]7s25f46d to Pu[Rn]7s25f6. This caused the ground state anionic BE of the Np atom to increase from 3.06eV to 3.25 eV in Pu. Also the anionic BE of the first metastable state increased from 1.47eV in Np to 1.57eV in Pu, see also Table 1 of Ref. [11]. It is the increase in the ground state energy space that facilitated the first appearance of the polarization-induced metastable TCS with the deep R-T minimum near threshold to appear in the Pu TCSs. This R-T minimum in the Pu TCSs continued through the Bk TCSs [6]. Indeed, the increase in the number of polarization-induced TCSs with size has already been clearly demonstrated in the fullerene molecular TCSs, see Fig. 1 of [11] as an example.
The second 6d-orbital collapse occurs in the transition Cm[Rn]7s25f76d to Bk[Rn]7s25f9. To facilitate the discussion we have included for convenience Table 1. Most of the data have been taken from [6, 11]. Table 1 shows that the ground state anionic BE increased significantly from 3.32eV in Cm to 3.55eV in Bk, thereby widening the energy space for the flip over to take place. Subsequently, the ground state anionic BE dropped from 3.55eV in Bk to 3.32eV in Cf. Similarly, for the first metastable states the anionic BEs increased from 1.57eV in Cm to 1.73eV in Bk, while it decreased to 1.70eV in Cf.
The size effect and the 6d-orbital collapse impact significantly the Cf TCSs. The additional energy space created by the ground state anionic BE when it increased from 3.32eV in Cm to 3.55eV in Bk, facilitated the flipping over of the R-T minimum from the Bk TCSs to the SR in the Cf TCSs. After the completion of the flipping of the R-T minimum, the ground state anionic BE of Cf stabilized at its optimal value of 3.32eV. Similarly, the increased energy space in Bk impacts the metastable state anionic BE of Cf.
Table 1: Negative ion binding energies (BEs), in eV extracted from the TCSs for the actinide atoms Np through No are taken mostly from [6, 11]. GRS, MS-n and EX-n (n=1, 2) represent respectively ground, metastable and excited anionic states. The experimental EAs, EXPT, denoted by N/A are unavailable.
|
Z Atom |
BEs GRS |
EAs EXPT |
BEs MS-1 |
BEs MS-2 |
BEs EX-2 |
BEs EX-1 |
|
93 Np |
3.06 |
N/A |
1.47 |
- |
0.521 |
0.248 |
|
94 Pu |
3.25 |
N/A |
1.57 |
1.22 |
0.527 |
0.225 |
|
95 Am |
3.25 |
N/A |
1.58 |
0.968 |
0.619 |
0.243 |
|
96 Cm |
3.32 |
N/A |
1.57 |
1.10 |
0.519 |
0.258 |
|
97 Bk |
3.55 |
N/A |
1.73 |
0.997 |
0.505 |
0.267 |
|
98 Cf |
3.32 |
N/A |
1.70 |
0.955 |
0.577 |
0.272 |
|
99 Es |
3.42 |
N/A |
1.66 |
0.948 |
0.642 |
0.272 |
|
100 Fm |
3.47 |
N/A |
1.79 |
1.02 |
0.623 |
0.268 |
The flipping over of the near threshold deep Ramsauer-Townsend minimum from the polarization-induced metastable Bk TCS to a pronounced shape resonance very close to threshold in the Cf metastable TCS manifests the sensitivity of the electronic structure of both Bk and Cf to the polarization interaction. This sensitivity allows the first time ever use of the Ramsauer-Townsend minimum and the shape resonance as rigorous confirmation of Cf as a transitional element in the actinide series, consistent with the recent experimental observation [1]. Indeed, the rigorous Regge pole method requires no assistance whatsoever from either experiment or any other theory for the remarkable feat, namely of probing reliably the electronic structure of these complicated actinide atoms.
Acknowledgements
Research was supported by the U.S. DOE, Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Energy Sciences, Office of Energy Research. The computing facilities of National Energy Research Scientific Computing Center, also funded by U.S. DOE are greatly appreciated.
Conflict of interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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