²²⁹Thorium-doped calcium fluoride for nuclear laser spectroscopy

The radioisotope ²²⁹thorium (lifetime 7932 years, [1]) is expected to have a long-lived, exceptionally low-energy, isomer state—probably the lowest of all nuclear levels. The currently most accepted value for its energy is 7.8 eV, obtained in an indirect gamma-spectroscopy measurement [2, 3], placing it into the vacuum ultraviolet (VUV) range (approximately 160 nm). The possibility of manipulating the quantum states of a nucleus with laser light has engaged researchers from various fields for more than 30 years. Possible applications range from a nuclear frequency standard [4–7] to gamma-ray lasers [8].

All of these exciting applications obviously require exact knowledge of the energy and lifetime of the isomer state, which currently remains elusive. The predicted isomer energy has undergone a series of changes over the years: from 3.5 eV [9] to 5.5 eV [10], and to 7.8 eV [2, 3], in the VUV range. Conclusions drawn from the experimental results and even the mere existence of the isomer state are still under discussion [11].

Predictions for the isomer lifetime are equally scattered: [12] yields a theoretical prediction of T_1/2 = (10.95 h)/ (0.025E³), where E is the isomer transition energy in eV, which leads to 55 min for E = 7.8 eV . Several experiments have been performed, proposing lifetimes between 2 min [13] and 6 h [14]. These results are heavily disputed [15].

A spectroscopy experiment to determine these values should therefore be able to cover a large range of wavelengths and be compatible with a maximum number of excitation sources and measurement schemes. Furthermore, the exact chemical composition of the thorium ions needs to be known and controlled, as it may have a crucial effect on the isomer lifetime and the nature of the de-excitation process [16].

It was first suggested in 2003 by Peik et al [4] that thorium ions may be implanted into UV transparent crystals to perform laser spectroscopy of the isomer state and eventually realize a nuclear clock. Due to shielding by the electron shell, the favourable properties of the nuclear transition (narrow line width) could be maintained, opening the possibility of building a solid-state optical clock [6, 7]. Due to the huge mass of the nucleus (compared to electrons), the nuclear transition would lie deep in the Lamb–Dicke regime, decoupling internal and external degrees of freedom to first order.

Although inhomogeneous effects [7] may ultimately hinder the design of high-performance solid-state optical clocks, such thorium-doped crystals can be a promising system for an initial 'coarse-grained' measurement of the isomer transition energy. Moreover, crystal effects (including temperature and pressure dependences) can be probed directly with a laser, transferring Mössbauer spectroscopy into the optical regime. Such investigations not only allow studying the host crystal properties but could give additional information about magnetic dipole and electric quadrupole momenta of the thorium isomer nucleus.

The host material for thorium doping obviously has to be transparent in the wavelength region in which the nuclear transition is to be investigated. Furthermore, it has to chemically accept the dopant in sufficient concentration. The Th-doped material must remain transparent, which means a sufficiently large band gap and no additional electronic levels emerging, which might quench the nuclear transition or inhibit excitation.

In our investigation, we have chosen calcium fluoride (CaF₂) single crystals as a host material for thorium doping. Other possible host systems, in particular LiCaAlF₆, have been considered by the group of Hudson [17]; an overview of crystal candidates can be found in [18]. CaF₂ is a standard material in UV optics due to its wide band gap. The literature values range from 11.6 eV [19] to 12.1 eV [20, 21] for a direct gap transition. The valence band of CaF₂ consists of 2p levels of the fluorine ions, while the bottom of the conduction band originates from the 4s and 3d orbitals of the calcium ions. CaF₂ presents a high radiation damage threshold, it can be grown to high quality using a multitude of established techniques such as Czochralski, Bridgman and micro-pulling; and cutting, polishing and coating techniques are readily available, even for large volume crystals.

The simplelattice structure of CaF₂ calls for ab initio calculations as performed in this work. Doping of CaF₂ has been studied extensively (both theoretically and experimentally) in the context of laser materials, focusing on trivalent dopants, exhibiting strong optical activity caused by electronic transitions. Due to the high electropositivity of thorium (1.3 Pauling), we expect to find the dopants in the Th⁴⁺ state, leading to a radon-like noble gas electron configuration. The ionic radii of Ca²⁺ and Th⁴⁺ are 1.26 Å and 1.19 Å, respectively [22]; their similarity should facilitate replacing a Ca²⁺ by Th⁴⁺ in the doping process.

Zoom In Zoom Out Reset image size

The GULP code (General Utility Lattice Program) is a general-purpose atomistic modelling code [23]. It uses effective interatomic potentials to calculate the lattice and defect properties of materials. The potential model used is the Buckingham form, supplemented by an electrostatic term:

$$\begin{equation*}V(r) = A \exp (-r/\rho ) - C r^{-6} + q_{1}q_{2}/r_{12} \end{equation*}$$

where A, ρ and C are parameters whose values are specified for each interaction between a pair of ions, and q₁ and q₂ are the charges on the interacting ions. The potential parameters are given in table 2.

In order to carry out defect calculations, the Mott–Littleton approximation is used, in which the region immediately surrounding the defect is modelled explicitly, and approximations are used for more distant regions of the lattice [24]. Region I is the region immediately surrounding the defect, and a consistent value of 10 Å was used in these calculations for its radius. Region IIA is the interface region, with a radius of 20 Å.

CaF₂ can be represented as a simple cubic lattice with space group Fm3m (225) (see figure 1). The Ca atoms occupy the 4a positions and the F atoms the 8c positions. Ca is surrounded by eight F atoms, whereas F sits in the centre of a Ca tetrahedron.

To calculate the electronic structure of Th-doped CaF₂ we assume a supercell based on the rhombohedral primitive cell. The impurity problem is simulated by assuming a periodically repeated cell containing 27 formula units (81 atoms). By replacing one Ca by Th we simulate an effective impurity concentration of 3.7% per Ca.

To calculate the electronic structure, we employ the Vienna ab initio simulation package (VASP 5.2/5.3) method, as devised by Kresse and Furthmüller [25]. Since for semiconductors and insulators there are well-known deficiencies of the local spin density approximation (LSDA) or the spin-polarized generalized gradient approximation (SGGA) in determining the proper gap size [26–28] and the position of the impurity states [29], we perform our calculations using the hybrid Hartree–Fock density functionals in the HSE06 version [30, 31]. To reduce the computation time, we relax the cell size and the ionic positions within a PBE [32] calculation. The geometries were relaxed until all force components were less than 0.01 eV  Å⁻¹. The relaxed lattice constant is only 0.88% larger than the experimental value, which gives confidence in the accuracy of the method. After convergence we keep these relaxed coordinates and change to the HSE functional. All results are obtained using the projector augmented plane wave method [33, 34] by explicitly treating 12 valence electrons for Th (6s², 6p⁶, 7s², 6d²), eight valence electrons for Ca (3p⁶, 4s²), seven valence electrons for F (2s², 2p⁵) and six valence electrons for O (2s², 2p⁴), respectively. In order to avoid Pulay stress and related problems, plane waves up to a cut-off energy of 500 eV were included in the basis set. The Brillouin zone integration was carried out over a 2 × 2 × 2 Γ-centred Monkhorst–Pack k-mesh with a Gaussian broadening of 0.02 eV. The total energy was converged to 1 × 10⁻⁶ eV .

These calculations were performed using the GULP code, as discussed in section 2.1.

3.1.1. Potentials and structural agreement

3.1.2. Intrinsic defects in CaF₂

3.1.3. Thorium doping calculations

When the CaF₂ structure is doped with Th⁴⁺, it is reasonable to assume that the Th⁴⁺ ion will substitute at the Ca²⁺ site in the structure (see figure 2). The calculated substitution energy is −43.44 eV. When the doping takes place, there is a charge imbalance in the lattice of +2, so charge compensation has to be involved. Two mechanisms were considered.

Zoom In Zoom Out Reset image size

3.1.4. Charge compensation by Ca²⁺ vacancy formation

The equation for the doping process may be written as

$$\begin{eqnarray*} &&{\rm ThF}_{4} + 2\ {\rm Ca}_{\rm Ca}\to {\rm Th}_{\rm Ca}^{\boldsymbol {\cdot \cdot }}+ {\rm V}_{\rm Ca}''+2\ {\rm CaF}_{2}. \end{eqnarray*}$$

Here, standard Kröger–Vink notation has been used for the defects [37].

The solution energy is then obtained from

$$\begin{eqnarray*} &&{E}_{\rm sol} = - {E}_{\rm latt} ({\rm ThF}_{4}) + {E} ({\rm Th}_{\rm Ca}^{\boldsymbol {\cdot \cdot }})+ {E} ({\rm V}_{\rm Ca}'') + 2\ {E}_{\rm latt} ({\rm CaF}_{2}). \end{eqnarray*}$$

Unbound and bound values have been obtained, with the bound values calculating the thorium doping and calcium vacancy formation together as a defect cluster (giving a formation energy of −22.74 eV). These are given in table 5.

3.1.5. Charge compensation by $\text {F}_{\text {c}}^-$ interstitial formation

The equation for the doping process may be written as

$$\begin{eqnarray*} &&{\rm ThF}_{4} + {\rm Ca}_{\rm Ca} \to {\rm Th}_{\rm Ca}^{\boldsymbol {\cdot \cdot }} + 2{\rm F}_{\rm i}' + {\rm CaF}_{2}. \end{eqnarray*}$$

The solution energy is then obtained from

$$\begin{eqnarray*} &&{E}_{\rm sol} = -{E}_{\rm latt} ({\rm ThF}_{\rm 4}) +{E} ({\rm Th}_{\rm Ca}^{\boldsymbol {\cdot \cdot }}) + 2{E} ({\rm F}_{\rm i}') + {E}_{\rm latt} ({\rm CaF}_{2}). \end{eqnarray*}$$

Since two interstitials are formed, their respective location needs to be considered when bound solution energies are calculated. Three possible configurations have been analysed: (i) linear, (ii) 90^° and (iii) 125.3^°, as illustrated in figures 2(b)–(d).

Table 5.  Solution energies for thorium doping. (Note: ThF₄ lattice energy =−79.98 eV.)

Configuration	Solution energy per defect (eV)
Vacancy compensation, unbound	3.42
Vacancy compensation, bound	1.67
Interstitial compensation, unbound	2.90
Bound interstitial—180°	0.47
Bound interstitial—90°	0.32
Bound interstitial—125.3°	0.67

The clear conclusion is that interstitial compensation is most favourable in this case, and further, that the configuration where the two F interstitials are at 90^° is of lowest energy. Note that the solution energies calculated with the GULP code allow us to compare different stoichiometries.

CaF₂ is a wide gap insulator; its electronic structure is well described within the applied ab initio method. For pure CaF₂ we calculate an effective gap of about 9 eV (see figure 3 and table 6), which is in fair comparison with the experimental value of 12 eV. It is also known that for these wide gap insulators advanced post-DFT functionals such as HSE06 tend to underestimate the gap size. However, the positions of the impurity bands are scaled down accordingly, which makes it straightforward to compare our calculated results with experimental values. The valence band consists of completely filled 2p states of the F⁻ ions with a small admixture of Ca²⁺ states. The conduction band contains the empty s and d states of Ca²⁺, again with a small admixture of F states. The density of states (DOS) of undoped CaF₂ is shown in figure 3.

Zoom In Zoom Out Reset image size

3.2.1. Charge compensation by $\text {F}_{\text {c}}^{-}$ interstitial formation

Substituting one Ca by Th means that the tetravalent Th⁴⁺ provides two additional electrons which would occupy states in the conduction band, leading to a metallic state. To remain an insulator, these two electrons need to be bound to additional atoms that provide charge compensation. Our GULP study showed that an F Frenkel defect is energetically most favourable, which means that the presence of two additional ${\rm F}_{\rm c}^{-}$ ions will provide the necessary charge compensation; see table 5. Here, we simulate three different configurations for these two ${\rm F}_{\rm c}^{-}$ ions, forming 180^°, 90^° and 125.3^° angles with respect to the Th impurity. While a simple electrostatic argument would point to the 180^° configuration being most favourable, our calculation finds the 90^° geometry as a ground state, which confirms the GULP results.

The insertion of Th⁴⁺ and the two additional ${\rm F}_{\rm c}^{-}$ ions leads to relaxations of the crystalline lattice, which are shown in figure 4. The main effect is a movement of Th⁴⁺ and the two ${\rm F}_{\rm c}^{-}$ ions towards each other. The presence of the negatively charged ${\rm F}_{\rm c}^{-}$ charge compensation ions exerts a repulsive force on the neighbouring F lattice ions, pushing them outwards.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The charge density in figure 5 shows in the left panel a section through the plane spanned by the Th and the two ${\rm F}_{\rm c}^{-}$ ions (110) in the 90^° configuration. The ${\rm F}_{\rm c}^{-}$ ions move slightly towards the Th, which is also reflected by the increase in charge density between Th and ${\rm F}_{\rm c}^-$. The right panel is a section in the $(01\bar {1})$ plane, and it clearly shows the outward movement of the next-neighbour lattice F ions.

Zoom In Zoom Out Reset image size

Figure 6 shows the density of states for the Th-doped and 90^° charge-compensated case. The electronic states of the two additional ${\rm F}_{\rm c}^-$ ions are located at the bottom of the gap, forming a narrow impurity peak. Th, which replaces a Ca site, is in a 4+ state, so the empty Th states form the conduction band together with the unoccupied Ca states. The unoccupied Th 5f states form a narrow band at the top of the insulating gap, reducing it to about 9 eV. Although we observe a reduction of the band gap upon Th doping (5.5% compared to the undoped case), the band gap remains large enough to be transparent for the 7.8 eV optical excitation. Both other ${\rm F}_{\rm c}^-$ configurations (180^°, 125.3^°) exhibit an almost identical density of states; the small energetic differences do not appear.

3.2.2. Charge compensation by Ca²⁺ vacancy formation

A different route of charge compensation is the creation of a Ca²⁺ vacancy, as discussed in 3.1.4. The respective density of states looks almost identical to the ${\rm F}_{\rm c}^-$ compensated case, but without the additional ${\rm F}_{\rm c}^-$ states at the bottom of the gap. Our GULP analysis (see table 5) gives a defect formation energy of 1.67 eV, which is more than five times larger than the energy involved in the creation of two ${\rm F}_{\rm c}^-$ interstitial ions. We therefore consider this configuration as unlikely.

Zoom In Zoom Out Reset image size

3.2.3. Oxygen impurities

Since in the crystal production process oxygen could be easily present as an impurity, we also studied compensation mechanisms with oxygen involved. These is one ${\rm O}_{\rm c}^{2-}$ at an interstitial position, there are two O⁻ on two F positions and there is a mixture of an O⁻ on an F position and an ${\rm F}_{\rm c}^-$ interstitial (see figure 8). In all cases we find oxygen states appearing inside the gap in the density of states, as is shown in figure 9. The presence of O reduces the gap to about 6 eV (see table 6, cases 6–8), making it unfavourable for the desired optical excitation. Hence, oxygen impurities have to be avoided in the crystal fabrication process.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 6 shows the main results for the eight different cases studied. Since within VASP a comparison of the total energies makes sense only for the three stoichiometrically identical ${\rm F}_{\rm c}^-$ charge compensations (rows 2–4), for all other cases the total energies are omitted. The coordinates are given for the unrelaxed positions with respect to Th in multiples of the lattice constant. It should be noted that due to the relaxation of the atomic positions the true coordinates and angles differ slightly.

In the first row we give the band gap for the undoped CaF₂. The next three rows are the different configurations for the two additional fluorine atoms at interstitial positions. The 90^° configuration yields the lowest total energy, which is set to zero. The two other configurations are higher in energy. The next three rows give the results for a Ca vacancy (row 5) and two types of O impurities at different positions, O at an interstitial site (row 6) and two O replacing two F atoms (row 7). Row 8 gives the result for a mixed configuration with one O at an F site and an additional ${\rm F}_{\rm c}^-$ interstitial.

Table 6.  Calculated band gaps, total energies and electric field gradients for the different configurations investigated in the fully relaxed structure.

Case #	Charge compensation		Angle (deg)	Relaxed energy (eV cell)	Band gap (eV)
1	Undoped CaF2				9.07
2	${\rm F}\ \{\tfrac {1}{2}00\}$	${\rm F}\ \{0\tfrac {1}{2}0\}$	90	0	8.59
3	${\rm F}\ \{\tfrac {1}{2}00\}$	${\rm F}\ \{-\tfrac {1}{2}00\}$	180	0.42	8.07
4	${\rm F} \{\tfrac {1}{2}00\}$	${\rm F}\{-\tfrac {1}{2} -\tfrac {1}{2} -\tfrac {1}{2}\}$	125.3	0.52	8.00
5	${\rm Ca}\ \{\tfrac {1}{2}\tfrac {1}{2}0\}$				8.33
6	${\rm O}\ \{\tfrac {1}{2}00\}$				6.25
7	${\rm O}\ \{\tfrac {1}{4}\tfrac {1}{4}\tfrac {1}{4}\}$	${\rm O}\ \{-\tfrac {1}{4}\tfrac {1}{4}-\tfrac {1}{4}\}$	109.5		6.68
8	${\rm O}\ \{\tfrac {1}{4}\tfrac {1}{4}\tfrac {1}{4}\}$	${\rm F}\ \{\tfrac {1}{2}00\}$	54.7		7.07