Precision atomic physics techniques for nuclear physics with radioactive beams

The Penning trap technique for mass determinations is one of the most powerful tools to reach the highest precision, and moreover, allows one to maintain control over precision and accuracy [2]. Penning traps were first conceived as storage devices for precision experiments on isolated single electrons. In fact, the Noble Prize for Physics was awarded in 1989 for 'the development of the ion trap technique' in part to Dehmelt and Paul. Dehmelt [9] invented and used the Penning trap system to successfully determine the g-factor of the electron. Paul [10] invented the so-called Paul trap, which plays an important role for mass spectrometry as devices to prepare the ions prior to the injection in the mass measurement Penning trap system. Nowadays Penning traps are used for mass measurements of stable and unstable atomic species, as well as molecules and antiparticles. For atomic mass measurements of stable species precisions of δm/m = 10⁻¹¹ [11] have been reached. Penning-trap mass measurement systems are currently in use (or planned) at practically all radioactive beam facilities around the world (see table 1).

Penning traps store charged particles by the combination of a strong homogeneous magnetic field and a weak static (typically) quadrupolar electric field. Figure 1 shows on the left the schematic configuration of a Penning trap system. The magnetic field points upwards and the electric field is generated by an applied voltage to the upper and lower end-cap electrodes, and one to the central ring electrode, leading to a potential difference of V₀. The typical electrode geometry is of hyperbolic nature, which allows one to achieve a harmonic potential. The trap-specific dimensions are r₀, the distance between trap center and ring electrode (in the minimum), and z₀ between trap center and end caps.

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For the mass measurements, the electrodes are segmented to allow for either a differential read out of induced image currents (used for Fourier-transform ion cyclotron resonance (FT-ICR)) or to apply radiofrequency (RF) excitations (for time-of-flight ion cyclotron resonance (TOF-ICR)). Both methods are explained below.

The Penning trap configuration leads to the storage of charged particles in three dimensions, and the motion of the particle is well understood and can be described classically [12] and quantum mechanically [13, 14]. An ion with a charge-to-mass ratio m/q is exposed to the electric and magnetic forces and the resulting net effect is a cyclotron motion, which is composed of three eigenmotions. These are the axial motion (up and down along the magnetic field axis) with frequency ω_z, and two rotational motions, the magnetron (ω₋) and modified cyclotron motion (ω₊). The latter are both perpendicular to the axial motion. Figure 1 shows on the right a calculated trajectory of an ion in a Penning trap and the deconvolution projection of the eigenmotions.

The cyclotron frequency ω_c, is proportional to the charge-to-mass ratio and the magnetic field strength B, and is given by

$$\begin{equation} \omega_{\mathrm{c}}= \frac{q}{m} B . \end{equation} \tag{ 1 }$$

Moreover, ω_c can be expressed as the square root of the quadratic sum of all eigenmotion frequencies as

$$\begin{equation} \omega_{\mathrm{c}}^{2}=\omega_{z}^{2}+\omega_{+}^{2}+\omega_{-}^{2}. \end{equation} \tag{ 2 }$$

This so-called Brown–Gabrielse-invariance theorem [15, 16] cancels field imperfections of the Penning trap to first order. Employing this relationship allows one to determine the mass of the stored particle via the determination of the cyclotron frequency ω_c. There are two main techniques applied in the field of radioactive beam Penning-trap mass measurements, the so-called time-of-flight ion cyclotron resonance (TOF-ICR) method and the Fourier-transform ion cyclotron resonance (FT-ICR) technique.

For the TOF-ICR [17] technique, the ions are stored in the Penning trap and (typically) excited by an RF quadrupole (RFQ) field. This RF field at frequency ν_RF is applied to a four-fold or eight-fold segmented ring electrode for a period T_RF. The RF excitation leads to a conversion from magnetron to cyclotron motion of the ions, which is accompanied by an increase in radial energy E_r. After the excitation, the ions are released from the trap and delivered to a detector system, typically a micro-channel plate or channeltron detector. The detector is installed outside the strong magnetic field region. The TOF from the trap to the detector is recorded. While traveling through the magnetic field gradient from high (on the order of 3–9 T) to the low magnetic-field region (typically 0.0001 T) radial energy E_r is converted to longitudinal energy E_z. Figure 2 on the left shows a schematic of the extraction from the trap to the detector on top of a graph that shows the magnetic field gradient and the corresponding radial energy as a function of distance from the trap. The minimal radial energy at the position of the detector indicates maximal longitudinal energy. By repeating this excitation, extraction, and TOF recording process for a set of frequencies around the expected cyclotron frequency ω_c a so-called TOF resonance spectrum is generated. The TOF is minimal for the maximal radial energy of the stored ions. This is reached when the excitation frequency ω_RF corresponds to the cyclotron frequency ω_c of the ions and the conversion from magnetron motion to cyclotron motion is maximal. For this excitation frequency the ions have maximal radial energy in the trap. A typical TOF resonance spectrum is shown in the right part of figure 2. Since the cyclotron frequency ω_c depends on the charge-to-mass ratio q/m of the ions as well as the magnetic field strength B, the derivation of the mass requires a determination of the charge state q as well as B. The charge state q is for most radioactive beam facilities (see [18]) either singly or doubly charged. Since the cyclotron frequency is directly proportional to q the difference from q = 1⁺ to 2⁺ leads to a doubling of the frequency and is easily detected. For facilities with highly charged ions (HCI) capabilities, like TITAN [19] at TRIUMF or the HITRAP facility [20] at GSI some charge selection technique is applied. HITRAP [21] employs a selecting dipol magnet and TITAN makes use of separation via TOF of a bunched beam with a Bradbury–Nilsen gate [22]. Both methods lead to unambiguous q identification. The magnetic field strength B is determined from calibration measurements using ions of very well-known mass such as ¹²C. To reach higher mass regions, one often uses carbon-clusters ¹²C_x. The mass of the ion of interest is then extracted from the frequency ratio of the two measurements, taking electron masses and binding energies into account.

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The method of FT-ICR mass spectrometry was originally developed in the 1970s for mass determinations of molecular species in the field of analytical chemistry [24] and has only recently been adapted for spectrometers for mass measurements of radioactive ions [25]. The schematic of the technique is shown in figure 3. A stored circulating ion in the segmented Penning trap electrode configuration induces image currents. Those are read out through a tuned tank circuit and a time dependent current spectrum can be recorded. This current signal is amplified using a low-noise amplifier and a mass spectrum is generated by employing a fast Fourier transform (FFT).

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The main advantage of the FT-ICR method over the otherwise very robust TOF-ICR approach is that a mass determination is in principle possible with a single ion. For the TOF-ICR a minimum of 200–300 ions are needed, however, with only one ion at a time in the trap. The downside of the FT-ICR is that the induced signal current is very small (a few fA). With that the electronic noise becomes a major obstacle for practical applications. It can be overcome by using dedicated high quality tuned tank circuits (typical quality factor Q ≈ 5000), reduction of the ambient Johnson-noise by going to cryogenic temperatures, or enhancing the induced signal by using higher charge states and/or longer accumulation times. Ideal applications for this method are mass measurements of super-heavy element (SHE) isotopes for nuclear structure studies as they are performed at SHIPTRAP [26, 27]. SHE are produced in minuscule quantities, and typically have half-lives of hundreds of milliseconds to a few seconds. A proof-of-principle experiment is planned at the TRIGA reactor facility TRIGA-TRAP at Mainz University [28] and applications for HITRAP with HCI are foreseen [29].

The specific requirements for the mass measurements of radioactive ions stem from the parameters of the ions themselves. For example the required sensitivity (depending on the production yield) and measurement speed (depending on the half-life) as well as the envisaged application which dictates the required precision. In order to be meaningful, all measurements should deliver reliable data, hence precise and accurate. The physics requirements can be categorized with the corresponding relative precision as follows:

• nuclear structure, δm/m ≈ 1 × 10⁻⁷,
• nuclear astrophysics, δm/m ≈ 1 × 10^−7/−8,
• test of fundamental symmetries, neutrino physics, δm/m ≈ 1 × 10^−8/−9.

Accuracy and precision.

Speed of the mass measurements.

The Penning-trap mass measurement method which is essentially exclusively applied to radionuclides is the TOF-ICR method (with the exception of the above mentioned proof-of-principle experiment using FT-ICR). As can be seen in equation (3), for the TOF-ICR method the achievable resolution (or mass uncertainty) is inversely proportional to the excitation time T_RF. Other factors are the charge state q, the magnetic field strength B, and a statistical factor N. Hence the required speed, or in other words the half-life of the radioactive isotope to be measured, dictates what duration can be used for the RF excitation T_RF and therefore governs the achievable mass uncertainty of the measurement

$$\begin{equation} \delta m \approx \frac{1}{\omega_{\mathrm{c}}} \propto \frac{1}{T_{\mathrm{RF}} \times q \times B \times \sqrt{N}}. \end{equation} \tag{ 3 }$$

Figure 4 shows the achievable precision in Penning-trap mass spectrometry as a function of the excitation time for the case of the short-lived radioactive isotope ⁷⁴Rb with a half-life of t_1/2 = 65 ms. Besides the actual time required for the RF excitation T_RF, the preparation of the ions prior to the excitation takes time.

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The initial condition of the ions before the RF fields are applied should be similar for all ions within one resonance spectrum. The preparation of these conditions has recently been dramatically improved by the invention of the so-called Lorentz-steerer [39], a device which introduces an electrical steering field within the magnetic field of the Penning trap. This preparation provides reliable injection parameters and initial conditions for the ions, while requiring virtually no time. Using such Lorentz-steerer devices, it was possible to reach isotopes as short-lived as ¹¹Li [40] with t_1/2 = 8.6 ms.

Improving the mass measurement precision.

The precision of the Penning-trap mass measurement method is given by equation (3). One parameter that cannot be manipulated is the half-life of the radioactive isotopes. In addition, the statistical factor N only scales with the square-root and it is hence not very beneficial, particularly for experiments at radioactive beam facilities. The other two factors are the charge state q and the magnetic field strength B.

The precision is directly proportional to the charge state, and hence breeding the charge to a higher level will boost the precision accordingly (neglecting statistical losses due to inefficiencies and decay). Charge breeding has been applied for mass measurements of stable species with SMILETRAP [41] and has now also been adapted to radioactive species using an electron beam ion trap (EBIT) [42] system at TITAN. Figure 5 shows the set-up of the TITAN facility at TRIUMF, where the originally singly charged radioactive isotopes from ISAC are cooled, bunched in a linear Paul trap, and charge bred using the EBIT, before injecting them into the Penning trap for the mass determination. A number of experiments with HCI were carried out [43, 44] and it was demonstrated that it is possible to increase the precision by up to two orders of magnitude at a constant excitation time, i.e. half-life of the isotope [45].

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The magnetic field strength also allows one to improve the precision, and nowadays Penning trap systems with magnetic field strengths of up to 9.6 T [46] are operational. This provides a gain in precision of about a factor of 2 over conventional systems with typical magnetic fields of 3–5 T.

Both approaches reach the gain by a boost of the cyclotron frequency and hence the excitation frequency ω_RF. This can also be reached by going to higher excitation modes. Typically quadrupole excitations are used, but recently it was demonstrated at the LEBIT facility at MSU/NSCL [47], the SHIPTRAP experiment at GSI [48], and ISOLTRAP [49] that octupole excitations are possible. Moreover, an improvement over conventional excitation and achievable precision of a factor 30 was achieved [50].

Another interesting new development is to enhance the reach of Penning traps at radioactive beam facilities by re-capturing decay products. This opens the door to mass measurements of species which would otherwise not be accessible due to limitations in the production mechanism, ionization, or extraction. This was recently demonstrated at the ISOLTRAP facility [51, 52] and is also planned for the TAMU-TRAP system [53] at Texas A&M University.

The complementary devices to Penning traps for high-precision mass spectrometry on radionuclides are the high-energy heavy-ion storage rings, which are designed to store charged particles at relativistic energies. Presently, there are only two storage ring facilities in the world performing mass measurements of exotic nuclei, the experimental storage ring (ESR) at GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt (Germany) [55] and the experimental cooler–storage ring (CSRe) at the Institute of Modern Physics of the Chinese Academy of Science in Lanzhou (China) [56]. Both rings are installed at facilities with in-flight production of exotic nuclei (for the different production mechanisms of radioactive ion beams see [18]). Similar to the Penning-trap technique, the mass-to-charge ratios (m/q) of the stored ions get related to their revolution frequencies ν according to [57, 58]

$$\begin{equation} \frac{\Delta \nu}{\nu}=-\frac{1}{\gamma_{\mathrm{t}}^2}\frac{\Delta (m/q)}{(m/q)}+\biggr(1-\frac{\gamma}{\gamma_{\mathrm{t}}^2}\biggl)\frac{\Delta v}{v}, \end{equation} \noindent \tag{ 4 }$$

with Δv/v being the velocity spread of the ions, γ the relativistic Lorentz factor and γ_t the transition point of the storage ring, which is constant for a given ion-optical setting of the ring.

In order to minimize the term containing the velocity spread and hence to get a direct relation between the revolution frequency and the mass-to-charge ratio of the stored ions, two complementary storage-ring mass spectrometry techniques have been developed: Schottky and Isochronous Mass Spectrometry, SMS and IMS, respectively [57]. While in SMS Δv/v → 0 is realized by stochastic and electron cooling of the stored ion beams, which unfortunately takes a few seconds and thus limits the accessible half-life of the exotic species, in IMS the ring is tuned to a specific ion-optical mode where γ_t = γ [59]. In this mode the different velocities of the stored ions are compensated by the corresponding orbit lengths, such that all ions of a given species have the same revolution frequency. Nuclides with a half-life as short as a few ten microseconds can be addressed with this technique since no cooling is required [60].

The measurement of the revolution frequency is in the case of SMS non-destructive by detecting the induced image currents of the circulating ions on a Schottky pick-up and performing the Fourier transform of the periodic signals at typically the 30th harmonic of the revolution frequency. In IMS operation a thin foil of a few μg/cm² is moved into the circulating beam. The penetrating ions produce secondary electrons, which are detected and allow for a measurement of the TOF and thus the revolution frequency of the stored particles. As becomes apparent from the physics applications and results that will be discussed below, both storage-ring mass spectrometry techniques provide single ion sensitivity and can be applied to measure masses of nuclides with production rates as low as one ion per day.

The isotope shift is the difference in the resonance frequency of an electronic transition between two isotopes. It is usually defined as

$$\begin{equation} \delta \nu _{\mathrm{IS}}^{AA'}=\nu ^{A'}-\nu ^{A}, \end{equation} \tag{ 5 }$$

i.e. the required change in frequency for tuning the light source that is in resonance with the isotope A at frequency ν^A to the resonance with the isotope of mass number A' at a frequency ν^A'. One of the isotopes within the chain—in most cases a stable isotope—is chosen as a reference. The contributions to the isotope shift can be separated into two terms

$$\begin{equation} \delta \nu _{\mathrm{IS}}^{AA'}=K_{\mathrm{MS}} \frac{ M_{A'}-M_{A}}{M_{A}M_{A'}}+F~\delta \left\langle r_{ \mathrm{c}}^{2}\right\rangle ^{AA'}. \end{equation} \tag{ 6 }$$

The first term is the mass shift that arises from the center-of-mass motion of the nucleus and the fact that this motion changes with the mass of the nucleus. It is obvious that a large relative change—as it is caused by adding a single neutron to a light nucleus—has a much larger impact on the shift than the addition of a neutron to a heavy nucleus. This is directly reflected in the 1/M_AM_A' dependence of the mass shift term. The proportionality constant K_MS is usually divided once more: the contribution expected for a two-body system

$$\begin{equation} K_{\mathrm{NMS}}=m_{e}\nu _{0} \end{equation} \tag{ 7 }$$

is called the normal mass shift (NMS) and always leads to a positive shift for the heavier isotope. The remaining part is caused by the correlations between the electrons and can have a positive or negative sign. It is called the specific mass shift (SMS) and is notoriously difficult to evaluate because of the electron correlation integrals. Both contributions have the same mass dependence and thus it is possible to write

$$\begin{equation} K_{\mathrm{MS}}=K_{\mathrm{NMS}}+K_{\mathrm{SMS}}. \end{equation} \tag{ 8 }$$

Reliable ab initio calculations with spectroscopic accuracy for the SMS are currently only available for systems with up to three electrons as will be discussed below. For systems with more electrons, large-scale many-body calculations must be performed, which are not as accurate. Alternatively, the separation of mass-shift and field-shift contributions to the isotope shift can be performed with a King-plot procedure if at least three stable isotopes exist of which nuclear charge radii were determined by other means, e.g. electron scattering or muonic atoms. The techniques involved here and the current status of such many-body isotope-shift calculations has been recently summarized in [61].

The second term in equation (6), the so-called field shift, arises from the change in the electron energy due to the finite size of the nucleus. For a radially symmetric charge distribution, the field outside the nucleus is equal to that of a point-like charge. But electrons that have a finite probability to be found inside the nucleus probe the deviations of the potential from the 1/r potential within the nuclear volume. It can be shown that the change in energy of an electronic state is proportional to the mean-square charge radius of the nucleus 〈r²_c〉  and to the probability density of the electron inside the nucleus |Ψ_e(0)|². Consequently, the field shift contribution to the total transition energy between two atomic states is proportional to the change in the probability density Δ|Ψ_e(0)|²_i→f between the lower state |i〉 and the upper state |f〉. This offers, in principle, the possibility to extract the absolute nuclear charge radius from the measurement of the total transition frequency if all other contributions are sufficiently well known. However, this is currently only possible for hydrogen [62] and hydrogenlike systems⁷. Already a second electron complicates the calculation of mass-independent quantum-electrodynamical (QED) corrections so much that the required accuracy cannot be achieved. With increasing number of electrons even the non-relativistic energies are not reliably calculable anymore. But the difference in the transition energy between two isotopes can be measured and is proportional to the change in the mean-square charge radius δ〈 r²_c〉^AA'. As a result of a non-relativistic calculation, assuming a constant probability density of the electron within the nuclear volume, one obtains

$$\begin{equation} \delta \nu _{\mathrm{FS}}^{AA'}=-\frac{Ze^{2}}{6\varepsilon _{0}} ~\Delta \left\vert \Psi _{\mathrm{e}}(0)\right\vert _{i\rightarrow f}^{2}~\delta \left\langle r_{\mathrm{c}}^{2}\right\rangle ^{AA^{\prime }}=F_{i\rightarrow f}~\delta \left\langle r_{\mathrm{c}}^{2}\right\rangle ^{AA'}. \end{equation} \tag{ 9 }$$

The index at the field shift constant F, specifying the electronic transition, is usually suppressed.

An estimation of the field shift dependence on the atomic number Z and the mass number A results in the rough approximation [6]

$$\begin{equation} \delta \nu _{\mathrm{FS}}^{AA'}\varpropto \frac{Z^{2}}{\sqrt[3]{A}}. \end{equation} \tag{ 10 }$$

In figure 6 the mass-shift (•) and field-shift (▴) contributions are plotted as a function of the atomic number. Both estimations are based on realistic numbers. For the mass shift, K_MS, the measured mass shift in the 2s → 2p resonance transition of Be⁺ ions (λ = 313 nm) was used and the mass shift for a similar transition in heavier elements was calculated according to the mass dependence in equation (6). The field shifts for the heavier elements are extrapolated based also on the field shift of beryllium using the simple relation given in equation (10). For the lightest elements (up to Z = 11, H–Na) data was taken from experiments. The scatter indicates the variation of sensitivity with the respective transition. Especially in the light elements there is little choice in which transition to take for a given charge state—particularly if the boundary conditions for the spectroscopy of short-lived isotopes and the availability of an appropriate laser system are considered. For He and Li the sensitivity is strongly reduced compared to hydrogen because the 2s electron does not probe the nuclear interior as strong as the 1s electron. Beryllium on the other hand uses also the 2s electron but now in a singly charged ion—hence the electron is much tighter bound. It is instructive to realize that the binding energy of an electron decreases with increasing nuclear charge radius. Assuming an increasing charge radius towards heavier isotopes and a transition with a reduction of the electron density at the nucleus, will result in opposite effects in the isotopes shift: the field shift decreases the transition frequency whereas the mass shift increases it. If both are approximately of the same size, the effect cancels and the isotope shift becomes very small. This is indeed what is observed around the elements Z ≈ 38 in accordance with the data in figure 6.

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For heavier elements, the radial change in the electron density inside the nucleus and relativistic effects cannot be neglected anymore. The nuclear factor has then to be replaced by a power series of radial moments

$$\begin{equation} \lambda ^{AA'}=\delta \left\langle r_{\mathrm{c}}^{2}\right\rangle ^{AA'}+\frac{G(Z)}{F(Z)}\delta \left\langle r_{\mathrm{c} }^{4}\right\rangle ^{AA'}+\frac{H(Z)}{F(Z)}\delta \left\langle r_{ \mathrm{c}}^{6}\right\rangle ^{AA'}+\cdots, \end{equation} \tag{ 11 }$$

where the fractional coefficients are the Seltzer coefficients tabulated in [67]. The non-relativistic change in the probability density of the electron is corrected by a factor that is obtained from a realistic solution of the Dirac equation for a finite-size nucleus with radius of liquid-drop dimension $R_{0}=r_{0}\sqrt[3]{A}$. The effect has been included for illustration in figure 6 (□) based on the table provided in [68]. It is obvious that the relativistic contributions significantly contribute for heavy elements.

The magnetic interaction of an atomic nucleus with spin I ≠ 0 and the electron shell can be approximated in first order as that of a point magnetic dipole interacting with the magnetic field that is produced by the electrons at the center of the nucleus B_e(0). It leads to a coupling of the total angular momentum J of the electron shell and the nuclear spin I to the atomic total angular momentum F = J + I. According to the angular momentum coupling rules, F can take any value between I + J and |I − J| and the energy of the corresponding level is proportional to the scalar product I·J, which can be written

$$\begin{eqnarray*} &&\Delta E=a\,I\cdot J=a\textstyle\frac{1}{2}\left( F^{2}-I^{2}-J^{2}\right) \end{eqnarray*}$$

and leads to a hyperfine splitting energy

$$\begin{equation} \Delta \nu _{\mathrm{mag}}=\frac{A}{2}C=\frac{A}{2}\left[ F\left( F+1\right) -J\left( J+1\right) -I\left( I+1\right) \right] \end{equation} \tag{ 12 }$$

with

$$\begin{equation} A=\frac{\mu _{I}B_{\mathrm{e}}(0)}{hIJ}. \end{equation} \tag{ 13 }$$

The magnetic interaction is dominated by the contact term, i.e. the interaction of the electron's intrinsic magnetic moment with the magnetic moment of the nucleus. Since this term contributes only if there is a non-vanishing electron-spin density inside the nucleus, it is largest for s-electrons.

The ratio of the A factors of different levels is – besides the 'trivial' dependence on the quantum numbers I and J—governed by the ratio of the magnetic field at the nucleus. If we look at the A factors of different isotopes along a chain, it is obvious that—in first order—the ratios should be determined by the ratios of the magnetic moments. We obtain for the ratio of the magnetic dipole hyperfine structure constant in two electronic levels i and f for two isotopes 1 and 2 the relation

$$\begin{equation} \frac{A_{1}^{f}}{A_{1}^{i}}=\frac{A_{2}^{f}}{A_{2}^{i}} \end{equation} \tag{ 14 }$$

which is often used as a constraint in fitting spectra with low statistics. Similarly, the unknown nuclear moment of one isotope can be connected to the known moment of a reference isotope via

$$\begin{eqnarray*} &&\mu =\frac{A}{A_{\mathrm{Ref}}}\frac{I}{I_{\mathrm{Ref}}}\mu _{\mathrm{Ref}}. \end{eqnarray*}$$

This relation neglects finite-size contributions of the nucleus, which arise from the change of the electron wavefunction due to the spatial extension, the distribution of the nuclear charge (Breit–Rosenthal effect, BR) and the distribution of the nuclear magnetic moment inside the nucleus (Bohr–Weisskopf effect, BW). They are usually accounted for by two factors

$$\begin{equation} A=A_{\mathrm{point}}~\left( 1+\epsilon _{\mathrm{BW}}\right) \left( 1+\epsilon _{\mathrm{BR}}\right) . \end{equation} \tag{ 15 }$$

These modifications from the point-like nucleus give rise to the so-called hyperfine anomaly in the ratio of the A factors between two isotopes

$$\begin{equation} \frac{A_{1}}{A_{2}}\approx \frac{g_{I}(1)}{g_{I}(2)}\left( 1+\,^{1}\Delta ^{2}\right), \end{equation} \tag{ 16 }$$

where g_I is the nuclear gyromagnetic ratio μ_I = g_Iμ_NI and ¹Δ² the differential hyperfine anomaly. In extreme cases, ¹Δ² can be of the order of a few percent but in most cases it is on the 10⁻⁴ level or even smaller. A recent compilation can be found in [69]. The effect is usually too small to be extracted from optical spectra but microwave spectroscopy is a useful tool for its determination [70].

A non-spherical nucleus with spin posseses also higher electromagnetic multipole moments up to the order 2I. Besides the monopole term—accounted for by the Coulomb term—the quadrupole moment is the next possible higher order of an electric moment. It arises from the energy of the orientation of the nuclear charge distribution in the inhomogeneous electric field of the electron shell and is given by

$$\begin{equation} \Delta \nu _{\mathrm{el}}=B\,\frac{\frac{3}{4}C\left( C+1\right) -I\left( I+1\right) J\left( J+1\right) }{2I\left( 2I-1\right) J\left( 2J-1\right) }, \end{equation} \tag{ 17 }$$

where

$$\begin{equation} B=\frac{eQ_{\mathrm{s}}}{h}\left. \frac{\partial ^{2}V}{\partial z^{2}}\right\vert _{r=0}, \end{equation} \noindent \tag{ 18 }$$

with Q_s being the spectroscopic quadrupole moment of the nucleus, and V_zz(0) the electric field gradient at the nucleus. The electric hyperfine structure appears only for nuclei and electronic states with I > 1/2 and J > 1/2, respectively. It shifts the magnetic hyperfine levels in a characteristic way, depending on the prolate or oblate form of the nucleus.

The observation of the atomic hyperfine structure allows us to determine also the spin of a nucleus if it is unknown. As long as I < J, the spin can directly and unambiguously be determined from the number of observed hyperfine components. Otherwise, the relative distances between the hyperfine components as well as the relative intensities are signatures of the spin. The theoretical line strength S_FF' of a hyperfine transition between the hyperfine levels F and F' arising from the fine structure levels with total angular momentum J and J' are related to the line strength in the underlying fine structure transition S_JJ' by

$$\begin{equation} S_{FF'}=\left( 2F+1\right) \left( 2F'+1\right) \left\{ \begin{array}{@{}ccc@{}} F & F' & 1 \\ J' & J & I \end{array} \right\} ^{2}~S_{JJ'}, \end{equation} \tag{ 19 }$$

where {··· } denotes the 6-j coefficient that can be taken from standard textbooks on angular momentum theory. However, these intensities must be handled with care since they are only correct for the excitation with unpolarized light and isotropic detection efficiency. Otherwise the detection geometry and the intensity distribution must be taken into account. Moreover, optical pumping can significantly alter the intensity ratios and lineshapes. In such cases, detailed studies of the lineshape have to be carried out.

Figure 7 shows a nuclear chart with all radioactive isotopes that have been studied with laser spectroscopy at on-line facilities—and in a few cases off-line—to extract properties of the nuclear ground state or sufficiently long-lived isomers. In recent years the region of the lightest nuclei, the region of the island of inversion around N = 20 and the region of the refractive elements above Sr, where a sudden-onset of deformation at N = 60 is observed, were studied intensively. Furthermore, the filling of the neutron pfg-shell in isotopes above the doubly-magic ⁵⁶Ni have lately been of much interest and was studied in copper and gallium. Further work in progress includes the region at the N = 28 shell closure above potassium and calcium and the N = 50,82 shell closures for nuclei in the tin region. Some of these developments have been enabled by new experimental techniques as will be discussed in the following.

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The majority of laser spectroscopic data at on-line facilities has been obtained using two techniques: collinear laser spectroscopy (CLS) [71, 72] and resonance ionization (mass) spectroscopy (RIS/RIMS) [73, 74] in various modifications. Both techniques are rather complementary to each other: generally CLS provides higher resolution but less sensitivity, while RIS is usually performed with pulsed lasers and is therefore limited in resolution but can be extremely sensitive. However, in special cases the sensitivity of collinear spectroscopy can be strongly increased by particle detection and RIS/RIMS can provide high-resolution if cw lasers are applied. Also mixed forms of CLS/RIS have already been demonstrated. Examples will be given below. Besides these techniques, dedicated methods were developed for the lightest isotopes in order to meet the challenges for a charge radius determination with extremely small field shifts and low production rates. This will be one focus of this issue.

Collinear laser spectroscopy.

In CLS the laser beam is superimposed with a beam of fast ions or atoms as shown in figure 8 and the resonance fluorescence is detected with a photomultiplier perpendicularly to the flight direction. Since the atoms are propagating parallel or antiparallel to the laser beam with velocity β = υ/c, the resonance frequency ν₀ of the atom is shifted in the laboratory system by the relativistic Doppler effect according to

$$\begin{equation} \nu _{\mathrm{p,a}}=\nu _{0}\gamma (1\pm \beta ), \end{equation} \tag{ 20 }$$

with the time dilation factor γ = (1 − β²)^−1/2. CLS utilizes the fact that the acceleration of an ion ensemble with a static electric potential compresses the longitudinal velocity distribution. Thus, the Doppler width is considerably reduced [71, 75]. In a simplified way, the kinematic Doppler compression can be deduced using the differential variation of energy with velocity dE = mυ dυ. The energy uncertainty δE in the source, caused by the Doppler distribution δυ_D of the ions, stays constant under electrostatic acceleration with a potential U. From δE = δ(mυ²/2) = mυ δυ = const., it follows that any increase in velocity must be compensated by a corresponding reduction of δυ_D

$$\begin{equation} \delta \upsilon =\frac{\delta E}{m\upsilon }=\frac{\delta E}{\sqrt{2meU}}. \end{equation} \tag{ 21 }$$

Since the remaining Doppler width Δν_D = ν₀(δυ/c) for a resonance transition at frequency ν₀ is proportional to δυ_D it follows that

$$\begin{equation} \Delta \nu _{\mathrm{D}}=\nu _{0}\frac{\delta E}{\sqrt{2eUmc^{2}}}. \end{equation} \tag{ 22 }$$

Hence the Doppler width decreases proportionally to $1/\sqrt{U}$ and beam energies of about 50 keV are usually sufficient to reduce the Doppler width to some 10 MHz and therefore into the order of the natural line width for allowed dipole transitions. CLS has been applied on-line first at the TRIGA reactor in Mainz [76] and soon thereafter at ISOLDE [5, 77].

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The collinear geometry allows for sufficiently long interaction times and high resolution. The velocity dependence of the laser frequency in the rest frame of the ion according to equation (20) can be utilized to keep the laser frequency in the laboratory system fixed and to tune instead the resonance frequency of the ion across the laser frequency by changing the ion velocity. This Doppler-tuning is usually realized by applying an additional post-acceleration potential to the optical detection region (ODR) in which the resonance fluorescence is detected. This has the additional advantage that optical pumping into dark states cannot occur before the ions enter the detection region.

Spectroscopy of fast neutral atoms can also be carried out if a charge exchange cell (CEC), filled with a thin vapor of an alkaline element, is mounted in front of the detection region. The cross sections for charge exchange collisions of a beam with ions M⁺ on, e.g. Cs vapor

$$\begin{equation} \mathrm{M}^{+}+\mathrm{Cs}\longrightarrow \mathrm{M}+\mathrm{Cs}^{+}+\Delta E \end{equation} \tag{ 23 }$$

is large (≈10⁻¹⁴ cm²) at typical beam energies and leads preferably to the charge exchange into states with ΔE close to 0 (resonant charge exchange). This provides also the opportunity to populate metastable higher lying levels if the electron donator is properly chosen. Doppler-tuning must in this case be realized by applying the post-acceleration voltage to the CEC and the ODR should be mounted as close as possible behind the CEC. Technical details and a comparison of two CECs—a horizontal and a vertical design as they are in use at COLLAPS and at TRIUMF, respectively—were recently presented in [78].

While the direct optical detection is the most straightforward detection scheme, its sensitivity is limited by the small solid angle, the quantum efficiency of the detectors, background from scattered laser light and several other factors. During the last decades CLS was therefore combined with various other detection techniques with the aim to increase sensitivity and accuracy of the technique [3, 4, 79]. Some of the recent developments will be discussed below.

Resonance ionization spectroscopy.

The isotopic chain of helium has 4 bound nuclei, the stable ^3,4He, the two-neutron halo ⁶He, and the four-neutron halo ⁸He. Table 2 shows the half-life and the mass uncertainty of these isotopes. The mass measurement requirements for the short-lived isotopes included fast preparation and measurements cycle while maintaining accuracy and precision on the sub-keV level. The measurements were carried out at the TITAN facility [19, 54] at the TRIUMF-ISAC radioactive beam facility in Vancouver, Canada. For helium, the isotopes were produced using the 500 MeV proton cyclotron. A silicon-carbide target coupled to a forced electron beam ion arc discharge source [127] was bombarded with 70 μA of protons. The singly charged ions were extracted from the target, mass separated using the ISAC dipole isotopes separator with a typical resolution of R ≈ 3000, and delivered as a continuous beam to the TITAN experiment. The TITAN experiment (see figure 5) consists of three ion traps in series, a linear Paul trap [22, 128], for cooling and bunching, an EBIT [129], for charge breeding (not used for these experiments), and a mass measurement Penning trap apparatus [38, 130]. For the mass measurements the TOF-ICR method is applied, and a fast excitation cycle in combination with the Lorentz-steerer for injection into the Penning trap and preparation of initial condition was employed. Figure 10 (right) shows the mass of the ⁸He isotopes as measured with TITAN [131] in comparison to previous indirect mass determinations. The resonance spectra are shown on the left. The uncertainty for ⁸He was improved by more than an order of magnitude and a final uncertainty of δm = 110 eV was reached [132]. For ⁶He the new mass measurements led to a deviation from the AME2003 [133] of 4σ and an uncertainty of δm = 54 eV was achieved [132], corresponding to a 14-fold improvement over previous measurements.

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The isotope shift of the helium isotopes was measured on trapped and cooled helium atoms in a magneto-optical trap (MOT) [134, 135]. A schematic of the setup is shown in figure 11. The halo nuclei ⁶He and ⁸He were produced in the stripping reaction ⁷Li + ¹²C → ⁶He + ¹³N at the Argonne Tandem Linear Accelerator System (ATLAS) and ¹³C + ¹²C → ⁶He, ⁸He + X at GANIL. Laser spectroscopy of helium atoms is only possible from an excited metastable atomic state since excitation from the ground state would require laser light at 59 nm. Thus, the ortho-helium 1s2s ³S₁ metastable state has been populated in a gas discharge cell (efficiency ≈10⁻⁵) and laser cooling and spectroscopy was performed on the 1s2s ³S₁ → 1s2p ³P₂ (1083 nm) and 1s2s ³S₁ → 1s3p ³P_J (389 nm) transitions, respectively. A simplified experimental setup is shown in figure 11. Excited helium atoms leaving the RF discharge cell are first transversely cooled before they enter a Zeeman slower in which they are slowed down by repetitive absorption of counter propagating photons until they are sufficiently slow to be captured in the shallow potential of a MOT located at the end of the Zeeman slower. Even though the total efficiency of the apparatus is only on the order of 10⁻⁷, it was still possible to perform laser spectroscopy with production rates as low as 10^{5 8}He atoms per second. The capture rate of only 30 atoms per hour is compensated by the large amount of photons that can be scattered by a single atom even within the short half-life of ⁸He (t_1/2 = 119 ms). The atom trap is operated in the capture modus, i.e. only the cooling lasers are on until an atom is trapped. Such an event is observed by a significant increase of scattered photons on the photodetector. Then, the MOT operation is switched to the spectroscopy modus in which cooling and spectroscopy is alternately performed: the spectroscopy laser is switched on for typically 2 μs and this period is followed by 8 μs of laser cooling. The laser beam intensities have to be carefully balanced in order to reduce any systematic shift as much as possible and the spectroscopy was performed on all three fine-structure transitions in ⁶He and on the transitions to the ³P_J=1,2 states in ⁸He. The results for the field shift are consistent within their uncertainties.

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The results of the isotope shift measurements presented in [134, 135] were reevaluated in [132] based on the new mass measurements of ^6,8He at TITAN. Charge radii are shown in figure 12. The charge radius increases from ⁴He to ⁶He but decreases then again toward ⁸He. The Gamov shell model has recently been used to analyze the differences in charge radii [136]. It was concluded that the change in the charge radius between ⁴He and ^6,8He is dominated by the center-of-mass motion of the α-core, Smaller, but not negligible, are contributions from the spin–orbit term of the halo neutrons and a swelling of the ⁴He core due to core polarization. The estimated size of the latter effect, taken from Green's function Monte-Carlo calculations, amounts to about 5% of the ⁴He point-proton size [106]. The decrease from ⁶He to ⁸He is accordingly caused by the difference in the recoil term and the spin–orbit contribution, which is about twice as large and negative for ⁸He compared to ⁶He.

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In the case of the lithium isotopes, spectroscopy was performed using Doppler-free two-photon excitation on neutral lithium atoms, followed by resonance ionization and mass separation [126, 137–139]. The technique was developed to achieve an accuracy of about 100 kHz in the isotope shift for the low production rate (≈10 000 ions s⁻¹) and the short lifetime (t_1/2 = 8.75 (14) ms) of the isotope ¹¹Li. As it is shown on the left in figure 13, the beam of lithium ions, produced either at the GSI on-line mass separator beamline in Darmstadt (⁶⁻⁹Li) or at the ISAC mass separator facility at TRIUMF (⁶⁻¹¹Li), is first stopped in a thin graphite foil (about 300 nm thickness, ≈80 μg cm⁻²). The foil is heated by a CO₂ laser to about 2000 °C, such that the implanted and neutralized atoms quickly diffuse out of the foil and drift into the laser interaction region which is placed a few mm in front of the foil inside the ionization region of a quadrupole mass spectrometer. The neutral atoms are first excited by two photons at 735 nm from the 2s ²S_1/2 ground state into the 3s ²S_1/2 level. In this transition the atom absorbs two photons simultaneously. In order to obtain the two counterpropagating beams and the relatively high intensity that is required for this type of nonlinear spectroscopy, the 735 nm laser-beam intensity is enhanced in a resonator by reflecting the photons back and forth between two highly reflective mirrors. The (non-relativistic) Doppler shift

$$\begin{equation} \Delta \nu _{\mathrm{D}} = \nu _{\mathrm{L}} \left( 1 + \frac{\upsilon _{\Vert }}{c} \right) \end{equation} \tag{ 24 }$$

of the laser frequency ν_L in the rest frame of an atom with velocity component υ_∥ along the wave vector $\vec{k}$ of the laser beam has equal amplitudes but opposite signs for two counterpropagating beams with identical frequency. The sum frequency of two photons is then independent of the atom's velocity if one photon is taken from each beam. This results in a (first-order) Doppler-free signal to which all atoms of the thermal ensemble contribute. After excitation to the 3s ²S_1/2 level the atoms will relax via the 2p level into the 2s ground-state. A second laser is used to transfer the population of the 2p level resonantly into the 3d state out of which the atoms can undergo non-resonant ionization by absorbing another photon either from the 735 or the 610 nm beam. The power of the 610 nm laser beam is also enhanced in the same cavity as the 735 nm laser beam. This excitation scheme leads to very selective and efficient ionization only of those atoms which have previously been excited in the 2s → 3s transition. The ionized atoms are extracted from the ionization volume with the ion optics of the quadrupole mass spectrometer and mass separated before being detected with a continuous dynode electron multiplier. A typical spectrum of ¹¹Li is shown on the right in figure 13. It shows the two hyperfine structure components from which the center of gravity can be obtained. Hyperfine-induced fine-structure mixing could in principle affect the center of gravity but is expected to be far smaller than the accuracy of these measurements. The charge radii within the lithium chain, as shown in figure 12, exhibit a decrease of the radius from ⁶Li to ⁹Li, followed by a significant increase to ¹¹Li. This behavior is caused by the strong clusterization of the light nuclei [140]. The increasing charge radius for the halo isotopes is again dominated by the center-of-mass motion of the core nucleus, caused by the halo neutrons. Core polarization can also contribute to the larger charge radius, but its contribution is expected to be small. The amount of core polarization cannot be extracted from laser spectroscopic investigations alone, but by combining measurements of matter radii, charge radii and the low-lying dipole strength B(E1) distribution as it is discussed e.g. in [140, 141]. However, the quantification is hampered by other contributions like, e.g. the spin–orbit contribution that must be determined theoretically.

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Due to the short half-life of ¹¹Li, a mass measurement in a Penning trap is also extremely challenging. It was also carried out at TITAN, using a measurement cycle of 50 Hz, allowing a mass measurement precision of δm = 690 eV [40]. It improved the uncertainty δm(AME2003) = 19 keV of the AME2003 [133] by almost a factor 30. The measurement practically eliminated the uncertainty arising from the mass for the derivation of the charge radius [140]. ¹¹Li is now the shortest-lived isotope for which a mass measurement in a Penning trap was performed, with almost an order of magnitude shorter half-life than the previous record, set by ISOLTRAP for ⁷⁴Rb [142]. Measurements of the lighter lithium isotopes were also carried out in order to extract the two-neutron separation energies along the lithium chain and to verify the masses used for the mass-shift calculations required to determine their charge radii [40, 130, 143].

Beryllium mass measurements were carried out for ⁹⁻¹²Be [144, 145]. The results for the short-lived isotopes are included in figure 14. The reached precision was sufficient for the mass-shift calculations to determine the charge radius from isotope shift measurements. The mass measurement of the two-neutron halo candidate ¹⁴Be remains a challenge for the future. With a half-life of t_1/2 = 4.35 (17) ms it would establish a new Penning-trap mass measurement record.

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Laser spectroscopy on beryllium had to be performed on the three-electron system Be⁺ because sufficiently accurate calculations of the mass shift term are still not available for four-electron systems. Singly charged beryllium ions offer an alkaline excitation spectrum with the resonance transition being in the near ultraviolet at 313 nm. Due to the strong binding, the valence electrons wavefunction has a relatively high probability density at the nuclear site, providing a sensitivity to changes in the mean-square charge radius that is approximately ten times larger than in the case of helium or lithium [114, 125]. Consequently, an accuracy of about 1 MHz in the determination of the isotope shift is sufficient to extract the nuclear charge radius with a relative precision of about 1%. This was achieved by performing collinear laser spectroscopy quasi-simultaneously in collinear and anticollinear geometry [146, 147]. In the usual approach of CLS—performing fluorescence spectroscopy just collinearly or anticollinearly—the available relative precision of 10⁻⁴ in the voltage determination of the 50 kV acceleration potential, leads to an uncertainty of about 40 MHz for the isotope shift between ¹²Be and the stable reference isotope ⁹Be. This is disastrous for a determination at the level of accuracy that is required to extract the small field shift. To become independent from the exact knowledge of the ion velocity, the absolute transition frequencies required for excitation in collinear and anticollinear geometry ν_c,a according to equation (20) are determined simultaneously. Their product can then be easily evaluated

$$\begin{equation} \nu _{\mathrm{a}}\,\nu _{\mathrm{c}}=\nu _{0}^{2}\,\gamma ^{2}\,(1+\beta )\,(1-\beta )=\nu _{0}^{2} \end{equation} \tag{ 25 }$$

and provides the rest-frame transition frequency ν₀ independently of the acceleration voltage. This collinear–anticollinear approach was previously used for precision tests of QED on helium-like ions of stable isotopes from helium to fluorine [148–150] but was never applied to radioactive isotopes. The prize one has to pay for this advantage is the determination of the absolute laser frequencies with accuracy better than 10⁻⁹, whereas standard CLS requires only that the laser frequency is sufficiently stable and known on a 10⁻⁵ level. The new approach was therefore strongly facilitated by the invention of the frequency comb [151], a device that allows to directly link RFs with optical frequencies and has been demonstrated to be able to determine transition frequencies with accuracy at the 10⁻¹⁷ level [152].

At ISOLDE the approach was realized using two dye lasers at 628 and 624 nm, respectively [146]. One laser was locked to well-known reference transitions in molecular iodine and the other one directly to a frequency comb. Both laser beams were frequency doubled and sent into the COLLAPS laser spectroscopy beamline in collinear (312 nm) and anticollinear (314 nm) configuration. Using fast shutters, spectra in both directions were taken alternately and integrated until a sufficiently large signal-to-noise ratio was observed. A typical spectrum of ¹²Be, the isotope with the shortest half-life (20 ms) and production yield (≈8000 ions s⁻¹) is shown in figure 15. For this isotope it was necessary to implement additionally an ion-photon coincidence detection in order to reduce the background from scattered laser light [147]. The data points are the experimentally observed count rates normalized by the number of ions as a function of the Doppler-tuned laser frequency in the rest-frame of the ion. The solid line is a fit to the spectrum using a Voigt profile. Beryllium isotopes with an odd mass number exhibit hyperfine structure that was analyzed by fitting the complete spectra with similar profiles for each peak and adjusting the hyperfine parameters, the center-of-gravity, peak intensities, and an offset as well as the Doppler width for all Voigt profiles. In these cases it was required to include a small secondary peak that arises 4 V below the main peak. This is caused by non-elastic collisions of the ions in which they are excited into the 2p excited state during the encounter of the residual gas atom. The energy for the transition is taken from the ions kinetic energy and subsequently irradiated in the form of a 4 eV photon.

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The charge radii obtained in these experiments are included in figure 12. The trend is quite similar to that observed for the lithium isotopes and the underlying physics as well. The cluster basis in this chain is α + ³He for ⁷Be and α + α + xn for ^9–12Be. The center-of-mass motion in the one-neutron halo isotope ¹¹Be is again the dominant contribution to the increase of the charge radius. Since ¹²Be is not believed to be a halo isotope and has a significantly smaller matter radius than ¹¹Be it is (at first glance) surprising that the measured charge radius of ¹²Be is larger than that of ¹¹Be. The observation can be explained theoretically by a strong admixture of (sd)² to the ground-state wavefunction of ¹²Be [147]. It has been shown in fermionic molecular dynamics calculations that this actually increases the α–α distance in the underlying ¹⁰Be cluster structure, such that the ¹²Be charge radius is again comparable to that of ⁹Be.

The proton-rich isotopes of the neon isotopic chain presented a difficult challenge for mass measurements, and even though laser spectroscopy had been done at ISOLDE in the late 1990s, the charge radius extraction was hindered by a mass shift determination. This was recently accomplished using results from high precision experimental mass measurements performed at ISOLTRAP [156].

The mass measurements were carried out using the ISOLDE facility for the production of the neon isotopes, where 1.4 GeV protons impinge on CaO or MgO targets to produce the isotopes. A plasma source was used to ionize the isotopes, which were then mass selected and delivered as a pulsed 60 keV beam to the ISOLTRAP setup [153]. ISOLTRAP consists of four main devices in series, a linear RFQ cooler and buncher [154], a recently added multi-reflection TOF separator [155], a buffer-gas filled isobar separator trap, and a mass measurement Penning trap. The measurement cycle was adjusted for this measurement such that the short half-life of ¹⁷Ne with t_1/2 = 109.2(6) ms could be accommodated within a 400 ms measurement cycle from proton pulse to ion detection. The uncertainty was reduced by a factor 50 compared to AME2003 and reached a level of δm = 570 eV for ¹⁷Ne. Figure 16 shows a resonance spectrum of the measurement [156] from ISOLTRAP. These mass measurements provided the required precision for the extraction of the charge radius.

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The isotope ¹⁷Ne is the only proton-halo candidate that has so far been studied using laser spectroscopy. CLS was also applied in this case, and state-selective ionization was used for efficient detection of the induced resonant transition [156, 157]. The neon ions produced at ISOLDE were ionized in a plasma ion source and sent to the COLLAPS apparatus. Here, resonant charge exchange into the [2p⁵(²P^o_3/2)3s]₂ metastable level was induced by passing the ion beam through the CEC operated with sodium. Subsequent laser excitation into the metastable [2p⁵(²P^o_3/2)3p]₂ state leads to the decay into the [2p⁵(²P^o_3/2)3s]₁ level which is quickly relaxing into the 2s² 2p⁶ ground state of the neon atom. On resonance, the populated metastable level will be quickly depopulated. Electrons bound in the 2s² 2p⁶ ground state have ionization energies of about 20 eV and are much less susceptible to collisional ionization than the metastable state with a binding energy of only about 5 eV. Hence, the ratio of neutralized to ionized atoms after transmission through a second gas cell is used to detect the transition as it is shown in figure 17. This detection technique was also applied for the isotopes of other noble gases (Ar, Kr, Xe, Rn) [158–162]. A speciality is again the voltage calibration for the neon measurements. Since the ionization occurs in a plasma ion source, the starting potential is not as well defined as for a surface ion source. Therefore a peculiarity of the neon excitation scheme was exploited: the transition frequencies for the [2p⁵(²P^o_3/2)3s]₂ → [2p⁵(²P^o_1/2)3p]₂ and the [2p⁵(²P^o_3/2)3s]₂ → [2p⁵(²P^o_1/2)3p]₁ transitions coincide in the laboratory frame if the first transition is excited anticollinearly and the second one collinearly at a beam energy of 61 758.77 eV. Hence, a single laser beam can be used to excite both transitions if it is retroreflected at the end of the beamline. This approach was used in the neon measurements for the beam energy determination [163].

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The charge radii obtained from that analysis are plotted together with the neighboring isotopes of Na and Mg in figure 18. The FMD approach has been used to describe the neon and the magnesium isotopes and the results reproduce the trend of the charge radii very well. For neon, especially the strong increase in charge radius from ¹⁸Ne to ¹⁷Ne is remarkable. This reflects a structural change in the isotopic chain and is correctly reproduced in the FMD calculation from which an amount of approximately 40% s² contribution is extracted. This clearly indicates at least the onset of a proton halo [156].

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After the discovery of the halo isotopes, the question arose whether the large radii are due to a strong nuclear deformation or the long tails of a halo state. Particularly the nuclear moments of ¹¹Li were considered to be a key property of this nucleus. While magnetic moments and quadrupole moments can usually be extracted from optical spectra as discussed above, the quadrupole interaction in very light elements can hardly be resolved. Furthermore, nuclear magnetic resonance (NMR) is often able to provide more accurate values. NMR requires a beam of spin-oriented (polarized) nuclei that can be produced by optical pumping with σ^±-light in collinear geometry. The NMR technique can then be applied for radioactive beams if the isotopes are afterwards stopped inside a suitable crystal and the β-asymmetry is detected [3]. Since detection of charged particles is much more efficient than photon detection, the β-asymmetry detection can be applied together with CLS to enhance the sensitivity. Hence, it seems only natural to combine these two techniques. In addition, the combined measurement of the hyperfine structure from which μ can be extracted and of the g-factor allows one to directly measure the nuclear spin as it will be discussed below for the case of Mg. Another technique with high sensitivity and accuracy is the RF-optical double resonance. This method has been applied to the radioactive beryllium isotopes.

NMR with β-asymmetry detection (β-NMR).

At ISOLDE, a first measurement of the nuclear magnetic dipole moment and the spin of ¹¹Li was performed already in 1987 [115], followed by a measurement of the electric quadrupole moment in 1992 [116]. Later, the nuclear magnetic moment of ¹¹Be has also been measured with this technique [164]. The experimental setup for these measurements is presented in figure 19: the laser light is circularly polarized (σ⁺, σ⁻) and is used to optically pump the atoms into the m_F -state with the maximum (absolute) value of m_F . Then both, the electronic and the nuclear moments are maximally aligned along the quantization axis provided by the direction of a weak magnetic guiding field produced by a set of coils along the beamline. In order to use β-asymmetry detection, the optically pumped and oriented ions or atoms must be implanted into a crystal inside a strong magnetic field. Then, the β-asymmetry is measured as the difference of β-particles emitted parallel and antiparallel to the magnetic field normalized to their total number

$$\begin{equation} a=\frac{N^{\uparrow }-N^{\downarrow }}{N^{\uparrow }+N^{\downarrow }}. \end{equation} \tag{ 26 }$$

Recording a as a function of laser frequency (or Doppler tuning voltage) allows to record the hyperfine structure of the atom with higher sensitivity than with pure optical detection. The total size of the asymmetry a depends on the nuclear transition but also on the host crystal in which the ion is implanted, its structure and its purity, and often this has to be clarified in systematic pilot investigations.

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Once the laser resonance has been found that provides the maximum asymmetry, NMR can be used to measure the nuclear g factor or the quadrupole moment with high accuracy. For a measurement of the g factor a crystal with a cubic lattice structure is chosen since there is no electric field gradient at the regular lattice positions. Consequently, the measurement becomes independent from a possibly existing quadrupole moment and the magnetic substates m_I of the nucleus remain degenerate. If an external magnetic field is applied, this degeneracy is lifted and the additional energy of the substates is proportional to m_Iμ_IB. The laser-induced polarization—i.e. the predominant population of the m_I = ± I substates for σ^± pumping—can then be destroyed by inducing transitions between neighboring magnetic substates with the Larmor frequency

$$\begin{equation} \omega _{\mathrm{L}}=\frac{g_{I}\mu _{N}I}{\hbar }B_{0} \end{equation} \tag{ 27 }$$

which is directly connected to the nuclear g-factor provided that the magnetic field is known. The latter is usually calibrated using a second species with a well-known nuclear magnetic moment like, e.g. ⁸Li.

Contrary, if the quadrupole moment is the property of interest, a crystal with a non-cubic lattice structure has to be used, where an electric field gradient is present at the regular lattice position. If an implanted ion inhabits such a regular position, the quadrupole interaction leads to a non-equal splitting between the different magnetic substates. In the NMR spectrum, the corresponding transitions occur at equidistant RFs. In the easiest case (axially symmetric electric field gradient along the z-axis of the principle axis system), the frequency difference Δν between neighboring lines is directly related to the quadrupole interaction constant and given by

$$\begin{eqnarray} \Delta \nu_{Q} &=& \nu(m\leftrightarrow m-1)-\nu(m-1 \leftrightarrow m-2)\nonumber\\ &=& \frac{3eQ_{\mathrm{s}}V_{zz}/h}{2I(2I-1)}. \end{eqnarray} \tag{ 28 }$$

Already the early results for the spin and the magnetic moment of ¹¹Li supported the halo picture and excluded a strong deformation: the spin was determined to be I = 3/2 and the magnetic moment of μ = 3.6673(23) μ_N [115] to be very close to the Schmidt-value for a spherical 1p_3/2 proton. Finally, the result of the measurements of the spectroscopic quadrupole moments of ⁹Li and ¹¹Li | Q_s(¹¹Li)/Q_s(⁹Li)| =1.14(16) excluded again a large deformation as the source of the large matter radius and also implied that the ⁹Li core is essentially unchanged by the presence of the halo neutrons [116]. The accuracy of the ¹¹Li moments has been increased by an order of magnitude since then, resulting in |Q_s(¹¹Li)/Q_s(⁹Li)| = 1.088(15). The currently most accurate values for the nuclear moments of ¹¹Li are μ = 3.6712(3) μ_N and Q_s(¹¹Li) = −33.3(5) mb compared to Q_s(⁹Li) = −30.6(2) mb [119, 120]. An even more accurate measurement of the quadrupole moment of ⁹Li relative to ⁸Li has been reported from TRIUMF [165]: using β-detected nuclear quadrupole resonance it was measured to be |Q_s(⁹Li)/Q_s(⁸Li)| =0.96675(9). The ion beam of ⁹Li was neutralized in a CEC, optically pumped, reionized in a helium gas jet and implemented in a SrTiO₃ crystal. Contrary to the measurements at ISOLDE, the magnetic field at the implantation site in SrTiO₃ is actively compensated in order to obtain a pure quadrupole resonance spectrum. The same technique will soon be applied for a measurement of the quadrupole moment of ¹¹Li.

The small increase of Q_s from ⁹Li to ¹¹Li of 8.8(1.5)% can in principle arise either from a ⁹Li core polarization or from the change of the charge distribution of the core due to the recoil effect caused by a d component in the halo-neutron wavefunction. In [166] a pure three-body wavefunction was designed that was optimized to reproduce experimentally known properties of the halo isotope consistently. The model wavefunction indeed reproduces the quadrupole moment, magnetic moment and the nuclear charge radius as well as the binding energy in excellent agreement with experiment. Also all other properties are within the measurement uncertainties. For microscopic theories this halo nucleus is still a challenge but the set of high-precision data from atomic physics low-energy experiments that has now been established along the complete chain of isotopes provides excellent benchmarks for ab initio, cluster, and shell-model theories [140].

RF-optical double resonance in an ion trap.

The nuclear moments of beryllium isotopes are also the subject of an experiment at the SLOWRI facility at RIKEN. Here, radioactive beryllium isotopes are produced using projectile fragmentation of a 100 MeV u^{−1 13}C beam. The fragments are stopped in a gas cell, extracted as 2 eV low-energy beam and transferred into a cryogenic (10 K) Paul trap. The central region of the segmented Paul trap is shown on the left in figure 20 [167]. During the ion accumulation time, a small amount of cold He gas is introduced for buffer gas cooling. Laser Doppler-cooling on the 2s²S_1/2 → 2p ²P_3/2 transition is performed using a red-detuned laser introduced at a small angle relative to the trap axis in order to cool all degrees of freedom simultaneously and a final temperature of the ions of the order of 10 mK is reached. Hence, the energy of the ions is reduced by about 15 orders of magnitude [168]. In order to obtain a closed two-level system on the transition, a magnetic field is employed, to lift the degeneracy of the m_F levels. During the cooling process the ions are optically polarized and pumped with σ^± light into one of the 2s_1/2(F = 2, m_F  = ± 2) → 2p_3/2(F = 3, m_F  = ± 3) closed two-level transitions. The fluorescence is observed with a two-dimensional (2D) photon-counting detector. The separation between the F = 2, m_F  = ± 2 and the F = 2,m_F  = ± 1 states is then measured by introducing an RF via the microwave antenna that couples the respective substates and therefore removes population from the laser transition. Thus, the observed amount of fluorescence decreases and a dip in the RF spectrum is observed as it is shown on the right in figure 20 [167]. Measuring the RF transition frequencies for the two transitions at two magnetic field strengths and fitting the Breit–Rabi formula to the results allows the extraction of the A factors of ⁷Be and—assuming no significant hyperfine anomaly—the determination of the magnetic moment of this isotope to μ(⁷Be) = 1.399 28(2) μ_N. The A factor in the ²S_1/2 ground state of ¹¹Be has also been measured with high precision [169] but has not been published yet. In order to extract the hyperfine anomaly, an independent measurement of the magnetic moment with higher accuracy than the previous CLS measurement in [164] is required.

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The trapped ions were also used for isotope shift measurements, applying an optical-optical double-resonance technique [170]. Final results have not been published so far, but reasonable agreement with the collinear data has been obtained in the preliminary data analysis [169].

This shell closure is in-depth discussed in [187]. Mass measurements provide excellent insight into the complex nuclear interaction as all effective forces are reflected in the nuclear binding energy. Hence, by providing precise and accurate binding energies (or total ground state properties) it is possible to refine and ultimately improve theoretical predictions. Neutron-rich isotopes are particularity suited for such tests, as can for example be seen from the studies of halo nuclei. Another example, where new phenomena appear in very neutron-rich isotopes, is the chain of oxygen isotopes (see [188–190]). Predictions for changes or evolution of the magic numbers of the nuclear shell model now also exists for isotopes of calcium (see [110]). In order to test such predictions, ground state mass measurements are well suited, but reliable data are often difficult to obtain due to the low production yields and short half-lives, for example t_1/2(⁵⁴Ca) = 86(7) ms. Recently mass measurements of the Ca-chain and nearby K-chain were carried out. In a first campaign ⁴⁷⁻⁵⁰K and ^49,50Ca were measured with the TITAN facility. Surprisingly large deviations from literature were found ranging from 4 to 10 standard deviations [191]. A second campaign made use of a UC_x target and even more exotic nuclei were accessible. The measurements of ⁵¹K and ⁵²Ca revealed again large deviations from previous AME2003 [133] values and a strong change in the general trend of the two-neutron separation energies S_2n appeared. The results together with AME2003 data are shown in figure 23. The observed separation energies and the general trend along the isotopic chain are in good agreement with theory if three-nucleon (3N) interactions are taken into account. This has been shown, for example, using a coupled cluster method with phenomenological 3N forces [192] and using chiral effective field theory [193].

Laser spectroscopy in this region was recently performed at JYFL with investigations on Sc [194], Ti [195], and Mn [197]. Laser measurements have also been completed for K and Ca isotopes at COLLAPS and are currently being analyzed.

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With ISCOOL at ISOLDE the isotopic chains of copper and gallium have been studied [198–205]. Further information was obtained from in-source spectroscopy of ^77,78Cu, where magnetic moments and spins were measured [206]. The main objective of these works was the investigation of the behavior of the proton πp_3/2 and πf_5/2 single-particle levels, while the νg_9/2 orbital is filled with neutrons. The monopole interaction between protons and neutrons in these states changes their relative energies and leads to a lowering of the πf_5/2 level while the energy of the πp_3/2 state is increased when neutrons are filled in the νg_9/2 shell. This behavior is understood in terms of the tensor force [207, 208] which is repulsive when protons and neutrons both occupy orbitals that are either j_> = ℓ + 1/2 or j_< = ℓ − 1/2 single-particle states (see also [196]). Contrary, the tensor force is attractive when neutrons and protons occupy states of opposite coupling between orbital angular momentum and spin. In copper and gallium this attraction lowers the energy of the πf_5/2 (j_<) state when filling neutrons in the νg_9/2 shell and it was expected that this interaction would lead to an inversion of the p_3/2 and the f_5/2 single-particle states somewhere between ⁷³Cu and ⁷⁹Cu. While the odd copper isotopes up to A = 73 exhibit an I = 3/2 ground-state spin as expected from the simple shell model, the isotope ⁷⁵Cu was found to have an I = 5/2 ground state. The spin has been tentatively assigned already after first low-resolution in-source measurements at ISOLDE and was then confirmed by CLS, both reported in [198]. In the gallium isotopic chain the equivalent transition from an I = 3/2⁻ to an I = 5/2⁻ ground state occurs between ⁷⁹Ga and ⁸¹Ga. The ground state of ⁷³Ga, previously assigned to be 3/2⁻ was surprisingly proven to be a I = 1/2⁻ state [199]. This was concluded from the number of observed resonances in both investigated transitions: the 4p ²P_3/2 → 5s ²S_1/2 exhibits only three instead of six lines and the 4p ²P_1/2 → 5s ²S_1/2 three instead of four resonances. Additionally, the peak intensity ratio agreed to that expected for a I = 1/2 ground state. The corresponding 1/2⁻ excited state in the neighboring isotopes shows a strong variation of energy along the chain. In shell-model calculations using the effective jj44b [209] and the JUN45 [210] interactions, it dives down in energy, until a minimum is reached around ^73,75Ga but in the calculations it does not become the ground state. The 1/2⁻ state seems to be of strongly mixed character and its behavior cannot easily be understood in terms of the tensor force. Analysis of the magnetic and quadrupole moments of the gallium isotopes revealed further details of the ground-state composition, e.g. the sign inversion of the spectroscopic quadrupole moment around ⁷³Ga indicates a transition from a dominant π(p³_3/2) state (Q_s > 0) to a π(p¹_3/2) configuration (Q_s < 0) with two protons excited to higher levels [4]. The high-precision mass data required for the isotope-shift analysis were obtained with the Penning-trap mass spectrometer ISOLTRAP [211–213].

It should also be noted that the moments of various copper isotopes were addressed by RIS with limited resolution. In-source spectroscopy was applied at ISOLDE in order to study the moments of ⁶⁸Cu^g,m, ⁷⁰Cu^g,m1,m2 [214], and ^58,59Cu [215], while a higher resolution for these isotopes and sensitivity even for ⁵⁷Cu was obtained using gas-cell RIS at LISOL [216, 217]. ⁵⁷Cu was considered a key isotope since an earlier β-NMR measurement reported a magnetic moment that was considerably smaller than expected from shell-model calculations [218] and its explanation required strong excitation across the closed N = 28 shell. However, this was disproven by the laser spectroscopic measurements which were substantially in agreement with shell-model calculations [216, 217].

At JYFL the RFQ cooler–buncher technique was applied for the first time for studies of the isotope shift and hyperfine structure of ¹⁷⁵Hf [182]. It was then used to systematically study the region of deformation at N = 60 above Sr A sudden onset of deformation above N = 59 was observed at ISOLDE for Rb [219] and Sr [220] but not in Kr [160]. Above Sr a region of refractory elements starts, which cannot be studied at ISOL facilities. The IGISOL at JYFL can produce these isotopes but only in small amounts. The bunching technique was therefore the breakthrough for laser spectroscopy with IGISOL. During the last decade the chains of Y [221–223], Zr [224–226], Nb [227] and Mo [228] were studied. An additional technique facilitated the spectroscopy of Y, Zr and Nb: optical pumping in the cooler–buncher was applied to prepare the ions in states that are more favorable for laser spectroscopy than the respective ground state. For example, the 5s^2 1S₀ ground state of Y⁺ has a resonance transition into a 4d5p ¹P₁ state. This J = 0 → J = 1 transition is not well suited for nuclear spin determinations. The 4d5s³D₂ → 4d5p³P₁ transition is more advantageous but the population of the metastable ³D₂ state is small. Therefore, a pulsed laser was applied inside the RFQ cooler–buncher to excite ions from the ground into the 4d5p ¹P₁ level, which has a ≈30% branching ratio into the metastable 4d5s³D₂ state. The signal was significantly enhanced by the optical pumping as it is shown in figure 24 [223].

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Optical pumping in the RFQ for state preparation before CLS has further possible applications. It can be used to pump the population into states that have transitions easier to excite with cw laser light. While pulsed lasers—as they are used for the state preparation—can be easily frequency-doubled, tripled and even quadrupled, these processes require much more effort with cw lasers as they are used for the high-precision spectroscopy. Metastable ionic states might in other cases offer stronger transitions, better knowledge of the atomic factors for mass and field shifts, larger hyperfine splitting, or the possibility to distinguish the fluorescence light from the laser excitation. Depopulation of the metastable state after cw excitation might also be a possible detection scheme.

The results of the investigations in this region at JYFL are shown in figure 25 in combination with previous results from ISOLDE. For Y and Zr the deformation is strong and similar to that observed in Sr. In the chain of Nb the transition already weakens and for Mo it is practically absent. It is interesting to see that this behavior seems to be also reflected in mass measurements on these isotopes, which were mainly performed at Jyväskylä and partially with ISOLTRAP as it is shown in the upper part in figure 25, taken from [229].

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The region of the heaviest stable elements has early been studied by optical spectroscopy and later by laser spectroscopy. One of the highlights was the discovery of the shape coexistence of mercury which leads to a large odd–even staggering for the isotopes lighter than ¹⁸⁴Hg [230, 231]. Many of the isotopic chains around mercury were investigated using resonance ionization laser spectroscopy (RIS). The observation of shape coexistence in ¹⁸⁶Pb [232] revived the field. Recent studies at ISOLDE were performed, e.g. on lead [233, 234], the light even isotopes of Po from ¹⁹²Po up to ²¹⁰Po and on the neutron-rich isotopes ^216,218Po at ISOLDE [235]. Compared to the neighboring elements, Po showed a large deviation from sphericity already at less exotic neutron numbers.

SHEs are chemical elements beyond the actinides, with atomic numbers Z > 103. Even though there were some speculations about the existence of very heavy elements already in the 1950s [236], the year 1966 is usually regarded as the birth of SHE research. Three theoretical papers were published in this year that discussed the existence of an 'Island of Stability' due to shell-effects that stabilize the nuclei of these elements against spontaneous fission [237–239]. Since then it was shown that the shell effects can indeed provide significant stability giving some of the very heavy isotopes half-lives in the range of seconds to hours. During the first period of SHE research, the term SHE was solely coined for the expected island of stability around Z = 114. However, it is now widely used for all those nuclei, which are stabilized by nuclear-shell effects, starting around rutherfordium (Z = 104). To date the periodic table has been extended up to element 118 [240]. New elements are typically discovered and identified in fusion-evaporation experiments and subsequent decay identification of the daughter chains. The production of all elements from 113 up to 118 has been claimed in the meanwhile, but according to the last report from the IUPAC/IUPAP Joint Working Party on Discovery of Elements (JWP) [241] only the papers on element 114 and 116 meet the strong criteria of the working group to really announce the discovery an element. These elements have in the meanwhile been named Flerovium (Fl) and Livermorium (Lv) (for more details see [242]).

Ideally, one can identify the proton number of the product, and then link the subsequent decay chain(s) to well known nuclei, such that no unambiguous event is falsely counted. However, the production cross-sections are extremely low, and typical production rates are around a few atoms per hour to few atoms per days, going down to even a few atoms per month. An additional potent path forward toward a 'clean' identification as well as to carry out systematic studies, is via direct mass measurements of the isotopes that act as the anker points of the decay chains and to map out where the shell closures occur. This would be of particular interest for the chains of hot fusion, where the decay chains can yet not be linked to known isotopes, since the α-decay chains end in nuclei that are not investigated so far and decay by spontaneous fission.

The approach of direct Penning-trap mass measurements has been realized with the SHIPTRAP system at GSI Darmstadt. SHIP is a velocity filter system for fusion evaporation reaction products, and is used for the generation and identification for SHEs. This in-flight production and separation system is coupled via a gas-filled stopper cell to a mass measurement Penning trap (SHIPTRAP). Here, binding energies of isotopes of trans-uranium elements were obtained from direct mass measurements for the first time [243]. Masses of the ^252−254No isotopes have been measured [243] and confirmed in combination with measurements of ²⁵⁵No and the lawrencium isotopes ^255,256Lr [27] the deformed neutron-shell closure at N = 152. With these mass measurements it has now been possible to map out the shell gap at N = 152 and therefore providing information where it will be most promising to investigate SHE and find the predicted island of stability. With production cross sections as low as tens of nano-barns the SHIPTRAP measurements constitute the Penning-trap mass measurement of isotopes with the lowest cross section performed so far.

The small production rates are also an obstacle for laser spectroscopic studies in the region of SHEs. Moreover, laser spectroscopy must first be applied in order to identify atomic transitions that can then be used for hyperfine structure investigations, because nothing is known about the atomic structure for elements with Z > 100. This requires to scan huge frequency spans since the predictive power of corresponding atomic structure calculations is still limited and the results have rather large uncertainties. These boundary conditions make laser spectroscopic studies of SHEs extremely challenging. Gas-cell spectroscopy with pulsed lasers was shown to be the technique of choice to study these nuclei [244, 245]. The heaviest element studied so far by laser spectroscopy is the element fermium (Z = 100). Off-line studies with the 20.5 h isotope ²⁵⁵Fm with low-resolution RIS in a buffer-gas cell resulted in the observation of two transitions that can in the future be used for higher resolution spectroscopic studies [245]. The resonances were observed by extracting the ions formed by the laser pulse from the gas cell through a quadrupole ion guide system, subsequent mass separation in a quadrupole mass spectrometer and ion detection with a channeltron detector. The technique has been further developed towards on-line application at the GSI SHIP installation for first spectroscopy of the element nobelium (Z = 102). The RADRIS (radiation detected resonance ionization spectroscopy) method is based on stopping the separated fusion-evaporation product in a buffer gas cell at pressures of about 100 mbar. The fraction of the ions that are not neutralized can be guided by suitable electric fields to a filament, where they are collected. The remaining neutralized atoms in the gas can be laser ionized and the laser-produced ions are guided to a particle detector where the ion is detected by its α-decay. Alternatively, in the so-called ICARE technique (Ion Collection and Atom Re-Evaporation), the catcher filament is heated by a short current pulse and the collected atoms are evaporated. Spectroscopy is then performed by a laser pulse triggered shortly after the evaporation and detection after resonance ionization as before. The technique has been demonstrated on-line with ¹⁵⁵Yb, a chemical homologue of nobelium [246]. Recently, in-gas jet, high-repetition, high-resolution laser RIS of SHEs, also stopped previously in a buffer gas cell, has been proposed (www.kuleuven.be/eu/erc/2011/vanduppen.html). Besides the possibility to study nuclear structure by hyperfine interactions, the identification of atomic transitions as a first step is a very interesting subject in itself due to the strongly relativistic effects in the electron shell.

The most promising attempt to get the absolute mass value of the electron-antineutrino ν_e is the investigation of the β-decay spectrum of tritium

$$\begin{equation} ^3\mbox{H} \rightarrow \, ^3\mbox{He}^+ + e^- + \bar{\nu}_e. \end{equation} \tag{ 32 }$$

The KArlsruhe TRItium Neutrino experiment (KATRIN) [288] aims for a direct measurement of the antielectron neutrino rest mass or at least to improve its present upper limit of 2.0 eV (95% c.l.) [289] to about 0.2 eV (90% c.l.) by investigating the β-decay of tritium gas molecules. The excess energy Q is shared between the kinetic energy of the electron, the recoil of the nucleus and the total energy of the antineutrino. The shape of the energy spectrum for the decay electrons would be modified near the endpoint Q₀ in case of a finite electron antineutrino rest mass. Q₀ is determined by the mass difference of the mother and the daughter nucleus, i.e. Q₀ = m(³H) − m(³He), minus the electronic excitation energy of the final state of ³He. Though electron retardation spectrometers, like KATRIN, can determine this quantity in addition to the neutrino mass from the analysis of the electron spectral shape, independent measurements help to rule out possible systematic errors in the neutrino mass measurement. Furthermore, as a future perspective, a significantly improved uncertainty of Q₀ compared to the currently best value from SMILETRAP (Stockholm, Sweden) of Q₀ = 18 589.8(1.2) eV [290] might allow for an absolute calibration and thus for an improved limit on the neutrino mass from KATRIN [288]. An attempt to achieve δQ₀ < 100 meV is presently pushed forward at the Max–Planck-Institute für Kernphysik (MPIK Heidelberg, Germany) within the THe-TRAP project [291] as well as at the Florida State University by E Myers [292].

A comparable limit of 0.2 eV (90% c.l.) for the ν_e mass is aimed for in the MARE project [293], which studies similarly to KATRIN the β⁻-decay of ¹⁸⁷Re. Here, the Q₀ value determined by the mass difference between ¹⁸⁷Re and ¹⁸⁷Os needs to be known with an unprecedented relative uncertainty of a few parts in 10¹². The high-precision cryogenic multi-Penning-trap mass spectrometer PENTATRAP [294, 295], also at MPIK Heidelberg, aims for such a precision employing HCI.

Another interesting case for precision mass measurements related to neutrino mass studies is the Q-value of the β-decay of ¹¹⁵In to the first excited level of ¹¹⁵Sn. This decay has been observed first in [296] (see also [297]) and the most precise Q-value of 155(24) eV was extracted from precision Penning-trap mass measurements at the Florida State University [298].

In the last few years tremendous efforts have been put into experiments to clarify the ambiguity whether the neutrino is a Dirac or a Majorana particle. The observation of a neutrinoless double β⁻-decay (0νβ⁻β⁻) or a neutrinoless double electron-capture decay (0νECEC) would give an unambiguous answer since these decays can only exist if the neutrino is of Majorana type, i.e. its own antiparticle (ν_e ≡ ν_e). This would violate the conservation of the total lepton number and manifest a clear sign for physics beyond the SM.

There are presently several experiments in operation or under preparation aiming for the direct observation of the neutrinoless double-beta decay (0νββ) [299] for the clarification of the nature of the neutrino–antineutrino duality. Investigations concerning neutrinoless double-electron capture (0νECEC) are still on the theory level [300–302] since they are even more challenging as will be discussed below.

A survey of all masses of known nuclides in the nuclear chart reveals that only 35 of them can undergo a 0νβ⁻β⁻ decay due to energetic reasons. But only eleven of them are of practical use for the search of such a decay in large scale detectors due to their sufficiently large Q-values [303]. The experimental signature would be the observation of an increased event-rate at the Q₀-value of the decay in a sum-energy spectrum of the two emitted electrons. Due to the huge background from the orders-of-magnitude stronger decay under the emission of two neutrinos (2νβ⁻β⁻), the knowledge of the Q₀-value with an uncertainty of less than 1 keV is of utmost importance in order to search for 0νβ⁻β⁻ events at the right energy. To date, as shown in figure 28, the Q₀-values of eight out of the eleven most interesting double β⁻-decay transitions have been determined by high-precision Penning-trap mass spectrometry with uncertainties as low as a few 100 eV. Among them the four transitions under investigation in large-scale experiments, i.e. the double β⁻-decays of ⁷⁶Ge [304, 305], ¹⁰⁰Mo [306], ¹³⁰Te [307, 308], and ¹³⁶Xe [309, 310].

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An experiment for the observation of the 0νECEC process is not yet planned, although it would allow for the same physics conclusions as the neutrinoless double β⁻-decay experiments. The reason lies in the expected four to six orders of magnitude longer half-lives and thus in the amount of material needed to search for such a virtually unobservable process. But as pointed out already in 1983 by Bernabeu et al [312], the process can be resonantly enhanced by a favorable nuclear and atomic structure: an energy degeneracy between the initial and the final state of the transition, i.e. when the Q-value is close to the energy of an excited state in the daughter atom, the decay probability can be substantially increased. The determination of such resonance conditions for specific transitions [301]—in order to find suitable candidates for 0νECEC searches—called for high-precision Penning-trap mass measurements. In total 15 Q-value measurements of double-electron capture transitions, as indicated in figure 28, have been performed since the first measurement by JYFLTRAP on ¹¹²Sn in 2009 [313], all with uncertainties well below 1 keV. However, so far only one system, namely ¹⁵²Gd–¹⁵²Sm, meets the criteria of a full energy degeneracy, which results in a half-life estimate of 10²⁶ years when normalizing to an effective neutrino mass of 1 eV [314]. For a detailed overview on all so-far investigated double-electron capture transitions by Penning-trap mass spectrometry we refer to the recent review by Eliseev et al [302]. Among those one other system is worth mentioning due to its uniqueness: ¹⁵⁶Dy–¹⁵⁶Gd. Here, a multi-resonance phenomenon has been discovered with four resonantly enhanced transitions to nuclear excited states in the daughter nuclide [315]. Still, the estimated half-life of ¹⁵⁶Dy exceeds that of ¹⁵²Gd by two orders of magnitude and thus excludes it as a suitable candidate for the search of 0νECEC-decays.