Crystal-Field Paschen-Back Effect on Ruby in Ultrahigh Magnetic Fields

Zeeman spectra of the R lines of ruby (Cr$^{3}$: $\alpha$-Al$_{2}$O$_{3}$) were studied in ultrahigh magnetic fields up to 230~T by magneto-photoluminescence measurements. The observed Zeeman patterns exhibit nonlinear behaviors at above 100~T, evidencing the breakdown of the previously reported Paschen-Back effect for ${\bf B} \perp c$ geometry. We adopted the crystal-field multiplet theory including the cubic crystal field (${\mathcal H}_{\rm cubic}$), the trigonal crystal field (${\mathcal H}_{\rm trig}$), the spin-orbit interaction (${\mathcal H}_{\rm SO}$), and the Zeeman interaction (${\mathcal H}_{\rm Z}$). It is found that the nonlinear splitting of the R lines is owing to the hybridization between the $^{2}E$ and $^{2}T_{1}$ states, which leads to the quantization of these Zeeman levels with the orbital angular momentum. Our results suggest that the exquisite energy balance among ${\mathcal H}_{\rm cubic}$, ${\mathcal H}_{\rm trig}$, ${\mathcal H}_{\rm SO}$, and ${\mathcal H}_{\rm Z}$ realized in ruby offers a unique opportunity to observe the onset of the $crystal-field$ Paschen-Back effect toward the high-field extreme.

Zeeman effect is categorized into anomalous Zeeman (AZ) effect at the weak-field limit and normal Zeeman (NZ) effect at the high-field limit. When atoms are located in weak magnetic fields, the energy levels split nonlinearly due to the competition between the external magnetic field and the hyperfine or the spin-orbit interactions. Under high magnetic fields where the Zeeman energy far exceeds those interactions, on the other hand, the energy splitting is asymptotically quantized by µ B B, where µ B is the Bohr magneton and B is the magnetic field. Historically, the crossover from the AZ effect to the NZ effect, the so-called Paschen-Back (PB) effect [1], has been observed in various atoms [2][3][4][5][6][7][8] and molecules [9,10]. For example, the D lines of the sodium atom exhibit the hyperfine PB effect around 30 mT [4,5] and the spin-orbit PB effect around 50 T [6]. Through this process, the good quantum number changes from M F for the total angular momentum of an atom to M I and M J for that of a nucleus and electrons, respectively, and then M J are further decoupled to M L and M S for the orbital and spin angular momenta, respectively.
The PB effect can also be observed in solid states [11][12][13][14][15][16]. The most well-known example is the R lines of ruby [13][14][15][16]. Many spectroscopic works on ruby have been done for more than half century, motivated by the scientific interests [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] as well as its applications to a solid-state laser and pressure gauge [28][29][30][31][32][33]. The optical transitions of ruby stem from the Cr 3+ impurities in α-Al 2 O 3 , where Cr 3+ ions are subjected to the strong cubic crystal field of the octahedrally coordinated O 2− ions. Furthermore, the repulsion between neighboring cations causes the slight lattice distortion, lowering the symmetry to trigonal C 3 (In the following, we take the trigonal axis as c axis). As shown in Fig. 1(a), the absorption spectrum of ruby in visible region consists of two broad bands and three groups of sharp lines. In the notation of the cubic symmetry, the R lines correspond to an electric dipole transition from the ground state 4 A 2 to the first excited state 2 E, appearing at around 1.79 eV (694 nm). Due to the interplay of the trigonal crystal field and the spin-orbit interaction, the 2 E state further splits into two Kramers doublets E and 2A (R 1 and R 2 lines) with the energy gap of δ = 3.6 meV. The observed Zeeman patterns of the R lines up to 60 T agree well with the theory shown in Figs. 1(b) and 1(c) [15][16][17][18]. Here, eight optically allowed transitions for E ⊥ c are noted for each B c and B ⊥ c geometry. For B ⊥ c, the tendency of the PB effect is observed toward 60 T, where the Zeeman energy is large enough compared to the initial splitting (gµ B B ≈ 2δ, where g = 2) [15,16].
This PB effect in ruby can be understood as such, that the Cr 3+ spins initially oriented along the c axis are gradually quantized along the applied magnetic field B ⊥ c. Unlike  the case of isolated atoms, the Zeeman levels of the R lines become quantized only with M S in this PB region, while the orbital angular momentum of the 2 E state is still quenched in crystal. Hence, one can naively expect further quantization of these levels with M L toward the high-field limit.
In this Letter, we present the Zeeman spectra of the R lines of ruby in ultrahigh magnetic fields up to 230 T. For both B c and B ⊥ c geometries, the observed Zeeman patterns exhibit nonlinear splitting at above 100 T, already beyond the previous theory based on the effective Hamiltonian for the 2 E state [17,18]. The observed Zeeman patterns are analyzed by the standard crystal-field multiplet theory including the cubic crystal field, the trigonal crystal field, the spin-orbit interaction, and the Zeeman interaction [17][18][19][34][35][36][37]. The hybridizations between the 2 E and the upper excited states, which are responsible for the AZ effect at above 100 T, are discussed. We also refer to the Zeeman spectra at the highfield extreme and the possible crystal-field PB effect.
The magneto-photoluminescence (PL) measurements were carried out by using a horizontal single-turn-coil (HSTC) megagauss generator. Disk-shaped ruby samples with 2.5 mm diameter and 1.0 mm thickness (SHINKOSHA Co., Ltd.) were used. The concentration of the Cr 3+ ion was 0.70 wt%, which is low enough to neglect the effect of the exchange interaction. The magnetic fields were measured by a calibrated pickup-coil with the error of ±2 %. The 532 nm laser was used for the excitation 4 A 2 → 4 T 2 , and the PL spectra were measured by the high-speed streak camera. In our setup, the incident light is polarized as E ⊥ c for B c, while both E ⊥ c and E c components are mixed for B ⊥ c. The measurements were performed at around 200 K by using 4 He-flow-type cryostats. In this temperature range, electrons are thermally distributed within the 2 E levels and almost all of the optically allowed transitions are observed as relatively sharp peaks. Exceptionally, the measurements at above 190 T were performed without using the cryostat due to the limited space inside the STC. For the best signal-to-noise ratio, we performed several cycles of measurements with the different maximum field B max and integrated the spectra recorded near B max within 1 % for each pulse. Details of the setup and analysis are found in Supplemental Material [38].
Our main results are summarized in Fig. 2. The evolution of the R lines' Zeeman spectra are shown in Fig. 2(a). The peak positions as a function of magnetic field for B c and B ⊥ c are plotted in Figs. 2(b) and 2(c), respectively, which are extracted by multi-Lorentzian fits. The energy shift ∆E is measured from the center of the R 1 and R 2 lines at zero field. For B c, all of the eight optically allowed lines, corresponding to A-H in Fig. 1(b), are clearly observed. Remarkably, the Zeeman patterns deviate from the linear field-dependence to the lower energy side as the magnetic field increases, resulting in the shift of the average peak position as shown in the inset of Fig. 2(b). For B ⊥ c, three distinct peaks are observed at 91 T, which can be understood that six lines, corresponding to J-O in Fig. 1(c), merge into three lines in the previously reported PB region. Note that the rest two lines, I and P, are hardly observable due to the little transition probability under high magnetic fields [14]. Surprisingly, those three peaks split again into (at least) six peaks at above 100 T, contradicting the concept of the previously reported PB effect. The tendency of the peak shifts to the lower energy side is also seen as in the case of B c.
To the best of our knowledge, there is no prediction that the AZ effect emerge at around 100 T for the R lines of ruby. For explaining the Zeeman patterns of the R lines, a simple effective Hamiltonian for the 2 E state has been so far considered [14][15][16][17][18]. For B c, the Zeeman patterns of the R 1 and R 2 lines in the low-field region can be respec- Fig. 1(b)]. Here, g 0 = 1.98 [21] is the g-factor of the ground state, and g 1 = 0.23 and g 2 = 0.26 [16] are the g-shifts which are mainly caused by the third-order interactions between the 2 E and the upper excited states, 2 T 1 and 2 T 2 , through the trigonal crystal field and the orbital angular momentum along the c axis (L z ) [18]. For B ⊥ c, in contrast, the Zeeman patterns are described by the quadratic relations without lifting the degeneracies of the Kramers doublets as Fig. 1(c)]. As is evident from these formula, this model assumes that the center of the four energy levels remains constant even in the external magnetic field. This contradicts our experimental observations at above 100 T, indicating that the hybridization between the 2 E and the upper excited states need to be directly incorporated rather than perturbatively treated.
Accordingly, we adopted a general multiplet Hamiltonian comprised of all the 120 bases in the 3d 3 state. In this Hamiltonian, the trigonal crystal field (H trig ), the spin-orbit interaction (H SO ), and the Zeeman interaction (H Z ) were involved in together with the cubic crystal field (H cubic ). The bases are expressed as |(αS Γ)M S γ , where α is the electronic configuration, M S the spin quantum number in the spin-S state, and γ the orbital function in the cubic irreducible representation Γ. We take M S and γ quantized along the trigonal c axis (u ± for E, a ± and a 0 for T 1 , and x ± and x 0 for T 2 ) [17,18]. In the following, the notation (αS Γ) is omitted if it is evident from the context. Several empirical parameters were introduced in the multiplet Hamiltonian for numerical diagonalization: the cubic crystal-field strength 10Dq, Racah parameters B and C [34], the trigonal crystal fields K and K ′ defined as K ≡ (t 2 )+ 1 2 x + | H trig |(t 2 )+ 1 2 x + and K ′ ≡ −(1/ √ 2) (t 2 )+ 1 2 x + | H trig |(e)+ 1 2 u + , the spin-orbit interactions ζ and ζ ′ defined as ζ ≡ −2 (t 2 )+ 1 2 x + | H SO |(t 2 )+ 1 2 x + and ζ ′ ≡ − √ 2 (t 2 )+ 1 2 x + | H SO |(e)+ 1 2 u + , and the orbital reduction factors k and k ′ defined as k ≡ − (t 2 )+ 1 2 x + | L z |(t 2 )+ 1 2 x + and k ′ ≡ −(1/ √ 2) (t 2 )+ 1 2 x + | L z |(e)+ 1 2 u + , which reflect the Cr-O bond covalency. Note that K, K ′ , ζ, ζ ′ , k, and k ′ are the matrix elements between one-electron states, and K = K ′ , ζ = ζ ′ , and k = k ′ = 1 hold for the free ion. Similar theoretical approach focusing on the low-field limit was attempted previously, but the discrepancies between experimental and theoretical values of δ and g-factor are relatively large [19,20]. Hence, although no perfect quantitative match seems to be achieved with any set of parameters, reexamination of the appropriate values was required in this work. We chose the parameters in H cubic as 10Dq = 2.320 eV, B = 0.071 eV, and C = 0.429 eV following the latest analysis in Ref. [27], which succeeded in reproducing the spectrum of ruby at zero field in a wide energy range including the UV region. Then, we searched for the combination of the values of K, K ′ , ζ, ζ ′ , k, and k ′ which simultaneously satisfy (i) the initial splitting δ = 3.6 meV, (ii) the g-values of the R 1 and R 2 lines for B c in the low-field limit, g 0 + 2g 1 = 2.44 and g 0 − 2g 2 = 1.46 [16], and (iii) our new experimental results in the high-field region. Here, we imposed additional constraints of K < 0, ζ > 0, 0 < K ′ /K < 1, 0.8 < ζ ′ /ζ < 1, and k ′ /k = ζ ′ /ζ.
The calculated energy diagrams of the four Zeeman levels are shown up to 250 T in Figs. 3(a) and 3(b). The parameters are chosen as K = −0.037 eV, K ′ = −0.013 eV, ζ = 0.025 eV, k = 0.72, and k ′ /k = ζ ′ /ζ = 0.86. Here, K and K ′ are a bit smaller compared to those in the previous works, whereas ζ and ζ ′ the opposite [19,20,27]. It can be seen that all of the four curves for each geometry show upwardly convex magnetic field dependence at above 100 T. These trends can be clearly captured as the decrease in the field-derivatives of the energy g ′ r ≡ 2 µ B dE r dB (r = 1±, 2±) as shown in Figs. 3(c) and 3(d). Furthermore, for B ⊥ c, the splitting of each Kramers doublet starts to be clearly seen at around 100 T and becomes larger toward higher fields as shown in Fig. 3(b). The ground state 4 A 2 is found to show the nearly linear Zeeman splitting up to 250 T (not shown). From the above, the theoretical Zeeman patterns of the R lines are obtained as shown in Figs. 2(b) and 2(c), which agree well with the experimental peak plots for both geometries. This indicates that the semiempirical crystal-field multiplet Hamiltonian taking all the bases in the 3d 3 state can account for the Zeeman spectra of the R lines of ruby even in the megagauss region.
As shown in Fig. 2(c), our calculation predicts that each of the three merged peaks at around 100 T splits into four individual peaks beyond the previously reported PB region for B ⊥ c. All of them are optically allowed, suggesting that the observed PL spectra at above 100 T are composed of 12 peaks, while not all lines are well resolved due to their overlapping. In this work, those spectra are tentatively fitted by six peaks with relatively large errors because the peak assignments with the 12-peak fit are challenging within our experimental accuracy. In Supplemental Material, we give a detailed analysis of the experimental and theoretical peak intensities [38].
Next, we discuss the wave functions of these four levels in magnetic fields. The wave functions are expressed by a linear combination of the 120 bases in the 3d 3 state, |ϕ r = k c k |(αS Γ)M S γ , where c k is a complex number coefficient. Our calculation reveals that |ϕ 1± (|ϕ 2± ) at zero field mainly consists of one base (t 3 2 2 E)± 1 2 u ∓ (t 3 2 2 E)± 1 2 u ± with |c k | 2 = 0.947 (0.941) [39]. When a magnetic field is applied, the second excited states 2 T 1 start to hybridize with the 2 E states in the first order via the orbital term of H Z . Note that the 2 T 1 states are composed of three Kramers doublets, all of which are ∆ = 60 ∼ 90 meV away from the 2 E states [40]. The field dependences of |c k | 2 for the 2 E and 2 T 1 states up to 250 T are summarized in Fig. 4, where only the crucial components are shown. For B c, as shown in Figs. 4(a)-4(d), the main component of the 2 E states decreases in each of the four wave functions, associated with the increase in the contribution of the 2 T 1 states with the same M L and M S . Meanwhile, their field-dependences in |ϕ 2− and |ϕ 1+ are different from those in |ϕ 2+ and |ϕ 1− . This is responsible for the difference in the field dependence of g ′ , exhibiting convex upward and downward behaviors, respectively [see Fig. 3(c)]. In contrast, for B ⊥ c, both the summation of the + 1 2 u + and − 1 2 u − components and that of the + 1 2 u − and − 1 2 u + components approach 0.5 toward high magnetic fields as shown in Figs. 4(e)-4(h). These features signal the tendency of the quantization of the spin along the field direction, i.e. the previously reported PB effect [14]. Besides, as shown in Figs. 4(i)-4(l), we see that the ± 1 2 a 0 components significantly increase only in |ϕ 2− and |ϕ 1+ . They are related to the lowest energy level of the three Kramers doublets of the 2 T 1 states. Therefore, the hybridization of those components is responsible for the shift of E 2− and E 1+ to lower energy than E 2+ and E 1− , respectively [see Fig. 3(b)].
These detailed theoretical analyses enable us to get more insights about the Zeeman patterns in ultrahigh magnetic fields. As previously reported, the Zeeman patterns of the R lines exhibit (almost) linear behaviors around 60 T. This is because of the special energy relations among the initial level splittings and the Zeeman energy, δ < H Z ≪ ∆, where the hybridization between the first and second excited states 2 E and 2 T 1 is negligible. However, their interactions are already important at above 100 T, resulting in the nonlinear behaviors that are experimentally observable. Indeed, our calculation suggests that 2 ∼ 12 % of the 2 T 1 components are contributing to the R lines at 250 T as is seen form Figs. 4(a)-4(d) and Figs. 4(i)-4(l). In the field region of 10 3 T, where ∆ < H Z ≪ 10Dq is achieved, the 2 E and 2 T 1 states are completely mixed. Importantly, both of them do not directly interact with the higher excited states 4 T 2 , 2 T 2 , and 4 T 1 shown in Fig. 1(a) via the orbital term of H Z . Thus, the Zeeman patterns of these 2 E and 2 T 1 states would approach the linear behaviors µ B (kM L + 2M S )B with k = 0.72, M L = 2, 1, 0, −1, −2, and M S = 1/2, −1/2 (For detailed energy diagrams, see Supplemental Material [38]). It is noteworthy that on further increasing the magnetic field much higher than 10 4 T, where H Z ≫ 10Dq is achieved, the Zeeman patterns of all the120 levels would be finally quantized with M L + 2M S = 6, 5, 4, · · · , −5, −6 [41].
The Zeeman patterns on ruby discussed above can be regarded as a kind of the PB effect. However, to the best of our knowledge, such PB effect has not be proposed for describing the crossover from the AZ effect to the NZ effect under the crystal-field splitting, possibly due to the lack of proper situation. The energy scale of the crystal field as well as the complexity of the level splittings or hybridizations make it challenging to clearly observe the crystal-field PB effect. Note that the magnetic field accessible by the current technology is at most 10 2 ∼ 10 3 T [42,43]. Therefore, our experimental observation of the onset of the crystal-field PB effect at above 100 T owes to the accidental crystal-field splitting manner in ruby, i.e. the exceptionally small energy gap ∆ between the 2 E and 2 T 1 states of the 3d 3 multiplets.
In conclusion, the AZ effect was observed for the R lines of ruby at above 100 T. The crystal-field multiplet theory with several empirical parameters successfully reproduces the experimental Zeeman patterns up to 230 T, proving that the hybridization with the second excited states are responsible for their nonlinear behaviors. Notably, the observed Zeeman patterns for B ⊥ c signal the crossover from one to another PB effect, characterized by the reconstruction of the good quantum number from only M S to both M L and M S . The present work offers a new concept of the PB effect, which is distinct from the conventional PB effects describing atomic energy levels, and it could be universally encountered in crystal if the conditions are met.
This work was partly supported by the JSPS KAKENHI Grants-In-Aid for Scientific Research (No. 18H01163, No. 19K23421, and No. 20J10988) and carried out in the science camp course for undergraduate students at ISSP. M.G. was supported by the JSPS through a Grant-in-Aid for JSPS Fellows. The authors appreciate for joining the science camp course to T. Anan, Y. Ebihara, K. Kawauchi, and R. Nagashima. M.G. and T.N. also appreciate for fruitful discussions to M. Hagiwara and S. Takeyama.