1 Introduction

The positive muon (µ+) is an elementary particle classified as a lepton. Using accelerators such as cyclotrons and synchrotrons is appropriate for providing beams of muons of sufficient intensity within a reasonable amount of time. In the presence of most organic compounds, insulators, or semiconductors, the positive muon can capture an electron and become a muonium (Mu = [μ+e−]) that can be regarded as a light isotope of hydrogen. The H-atom surrogate Mu can add to the unsaturated units in the organic molecule and provide the corresponding paramagnetic species. In most cases, the muon hyperfine coupling (hfc) constants of the muoniation products from the organic compounds are considerably smaller than muonium itself (4463.3 MHz) [1]. The muon hfc constants have been quite informative in characterizing the structures of the muoniated radicals formed from organic compounds such as benzenes, alkenes, alkynes, and aldehydes/ketones [2,3,4].

In addition to the ordinary organic molecules, the progress in main group chemistry has realized the heavier congeners including alkenes and benzenes as well as imines and carbenes. West and Percival have analyzed the paramagnetic radicals formed from the heavier group 14 congeners of unsaturated organic molecules [5, 6]. Subsequently, the phosphorus congeners of alkenes became targets of muon spin rotation/resonance/relaxation (µSR) spectroscopy, and paramagnetic muonium adducts of acyclic phosphaalkenes [7, 8] and phosphasilenes [9] were characterized.

This paper presents an overview of the µSR studies on phosphorus heterocyclic compounds featuring the most fundamental cyclic biradical-type structure and fused polycyclic aromatic ring systems [10]. Fundamentals of µSR for characterizing reactions of muonium in organic molecules are described in the literature [1,2,3,4, 10].

2 Radical reactions of 1,3-diphosphacyclobutane-2,4-diyls with singlet biradical properties

A cyclobutane-1,3-diyl is the most fundamental cyclic organic biradical species and is normally observable only under cryogenic conditions. The exchange of the skeletal elements in the cyclobutane-1,3-diyl with the hetero-elements is an effective approach to reduce the open-shell character and instability. 1,3-Diphosphacyclobutane-2,4-diyl bearing sterically demanding substituents (Fig. 1a) have been useful for developing air-tolerant derivatives, which have enabled finding low-voltage organic semiconductors [11,12,13] and H2/HF capturing materials [14, 15]. The presence of biradical electrons should produce these functional aspects of 1,3-diphosphacyclobutane-2,4-diyls and can be assessed by chemical processes using radical additions. However, almost all the chemical radical additions failed. So far, only the O2 adduct was characterized but incompletely (Fig. 1b, Mes* = 2,4,6-(Me3C)3C6H2). These results have stimulated the employment of µSR for elucidating radical reactions of 1,3-diphosphacyclobutane-2,4-diyl via muonium of a light isotope of hydrogen.

Fig. 1
figure 1

a Formula of air-tolerant 1,3-diphosphacyclobutane-2,4-diyl. Me = methyl (CH3). b Tentatively characterized O2 adduct of 1,3-diphosphacyclobutane-2,4-diyl. Mes* = 2,4,6-(Me3C)3C6H2. The regioselectivity would be promoted by the steric effects

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1,3-Diphosphacyclobutane-2,4-diyl 1 in Fig. 2a is an extremely air-stable crystalline compound and has been used to fabricate a low-voltage p-type organic field-effect transistor (OFET) [11, 13]. An anisotropic solid sample of 1 was investigated by µLCR, and the spectrum shown in Fig. 3 was obtained. The Δ1 (ΔM =  ± 1) resonance signal \({B}_\mathrm{res}^{{\Delta }_{1}}\) was observed at 0.18–0.20 T. Applying Eq. 1, the muon hfc constant (Aµ) was determined to be 46.2 MHz [16].

$${B}_\mathrm{res}^{{\Delta }_{1}}= \frac{1}{2}{A}_{\mu }\left[\frac{1}{{\gamma }_{\mu }}-\frac{1}{{\gamma }_{e}}\right]$$
(1)

where γμ and γe refers to the gyromagnetic ratios of the muon and electron, respectively.

Fig. 2
figure 2

a Addition of muonium to 1 affording 1Mu. b A dominant zwitterionic canonical formula of 1

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Fig. 3
figure 3

µLCR spectra of 1 and Mes*H [1,3,5-(Me3C)3C6H3] showing the muonium adduct 1Mu (orange circles) and the tri-tert-butylcyclohexadienyl radical Mes*H-Mu (green squares). Spectrometer: Helios (TRIUMF)

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Figure 3 accompanies the µLCR spectrum of 1,3,5-tri-tert-butylbenzene (Mes*H). The Δ1 signal of the corresponding muonium adduct Mes*H-Mu (Aµ = 468 MHz) does not appear in the field of 0–0.5 T but at 1.71 T [16]. Therefore, the aromatic rings in 1 did not accept muonium leading to the cyclohexadienyl radicals (Aµ = ca. 450 ~ 520 MHz). DFT calculations of the possible additions of muonium to 1 were used to assign the regioselective muoniation affording 1Mu under the kinetic and electrostatic controls promoted by the steric hindrance and the substituent effect. In the DFT calculations, hydrogen was used instead of muonium, and accordingly, the muon hfc constants were calculated by multiplying the relative gyromagnetic ratio of proton (γμ/ γp = 3.1833). In addition, the effects of the increased zero-point energy caused by the light mass of muon were estimated by using a truncated molecule. The anharmonic potential curve was obtained by altering the P–H bond distance in the H adduct of 1,3-diphosphacyclobutane-2,4-diyl manually, and the increase of the zero-point energy was determined by calculating the vibrational frequencies. As a result, the 4% elongation of the P–Mu bond compared with the P–H distance would be caused by the increased zero-point energy [16]. The calculated muon hfc value of 1Mu including the empirical structure correction was 44.3 MHz at the UB3LYP/6-311G(d,p)//UM06-2X/6-31G(d) level. The regioselective addition of relatively nucleophilic muonium of [17, 18] was compatible with the dominant canonical formula in Fig. 2b.

Because 1 allowed the addition of muonium affording 1Mu in the crystalline state, we next examined µSR experiments with a similar derivative 2. However, 2 did not afford the expected muonium adduct (Fig. 4). In contrast to the monoclinic crystal structure of 1, the triclinic crystal structure of 2 would prevent the construction of effective pathways of the carriers [13]. We speculated that the lack of semiconductor functionality might correlate with the low reactivity with muonium.

Fig. 4
figure 4

1,3-Diphosphacyclobutane-2,4-diyl 2 is similar to 1 but does not show the addition of muonium

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Besides the electron-donating derivatives such as 1, we prepared 3 bearing the strongly electron-withdrawing nitrogen heterocycle. In contrast to 1, 3 showed different regioselectivity of muoniation to the 1,3-diphosphacyclobutane-2,4-diyl unit (Fig. 5) [11, 19]. A Fourier power TF-µSR spectrum of 3 indicated two muon hfc parameters from the combinations of vR1 and vR2 and vR1' and vR2', respectively, and the diamagnetic peak was observed at 27.2 MHz (B = 0.20 T) (Fig. 6a). The paramagnetic signals in Fig. 6a indicated an Aµ value of 6.7–7.2 MHz, which was compatible with the muoniation reaction of 3 affording the thermodynamically preferable paramagnetic species trans-3Mu of the larger muon hfc parameter. It was difficult to characterize the smaller muon hfc parameter from the vR1' and vR2' signals due to the large diamagnetic signal in the power spectrum. An alternative TF-µSR spectrum of 3 composed of Fourier amplitude and frequency indicated the small Aµ values of ca. 4.5 MHz (Fig. 6b), which could also be shown by using musrfit [20]. The ultraviolet–visible (UV–Vis) absorption spectrum and the redox potentials of 3 indicated a smaller energetic difference ("band gap") between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) ΔEH-L of 1.38 eV compared with ΔEH-L of 1 (1.56 eV), indicating a smaller energetic difference between the singlet ground state and triplet state [11]. Therefore, 3 would have a larger open-shell character compared with 1, and the increased radical character around the skeletal carbon centre in 3 would facilitate capturing muonium [19]. The aromatic nitrogen heterocycle in 3 was not affected by muonium, because it would have interfered with the aromatic character.

Fig. 5
figure 5

The addition of muonium to 3 afforded two geometrical isomers trans-3Mu and cis-3Mu. The calculated muon hfc parameters were obtained at the UPBE-MBD@rsSC/TZ2P-J//UωB97XD/6-311G(d,p) level

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Fig. 6
figure 6

a A Fourier power TF-µSR spectrum obtained from a powder sample of 3 at 298 K and a field of 0.20 T. A strong diamagnetic resonance of muon (vD) appears at 27.2 MHz, and one pair of paramagnetic signals vR1 and vR2 indicated trans-3Mu. Another pair of paramagnetic signals vR1' and vR2' indicates the smaller muon hfc constant. b A Fourier amplitude TF-µSR spectrum obtained from a powder sample of 3 at 298 K and a field of 0.20 T. The smaller muon hfc constant of ca. 4.5 MHz is compatible with cis-3Mu. Spectrometer: LAMPF (TRIUMF)

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3 Radical reactions of a peri-trifluoromethylated 9-phosphaanthracene

Anthracene is a typical condensed aromatic hydrocarbon. Its radical reactivity is of interest from the views of bioactivity, combustion science, and environmental chemistry. Exchange of the skeletal sp2 carbon atom with the heavier p-block elements has been performed because the small bandgap of the heavier unsaturated bonds would realize novel functional characters by the heavier π-conjugated heterocyclic systems. 9-Phosphaanthracene is a heavier congener of anthracene. Whereas most of the 1,8-non-substituted 9-phosphaanthracenes are labile, the use of the trifluoromethyl (CF3) groups as the peri (1,8-positions) substituents is a promising method to synthesize air-tolerant 10-aryl-9-phosphaanthracenes [21].

In this µSR study, 10-mesityl-1,8-bis(trifluoromethyl)-9-phosphaanthracene (4, mesityl = 2,4,6-(CH3)3C6H2) was employed (Fig. 7). Figure 8a shows a TF-µSR spectrum of a THF solution of 4 at 298 K [22]. The transverse magnetic field of 1.45 T corresponds to the diamagnetic signal (vD) of 196 MHz. A paramagnetic species of Aµ = 222 MHz was observed, indicating the regioselective addition of muonium. Taking the high reactivity and instability of the P = C double bonds in 4 into account, muonium should be added to the sp2-type phosphorus in 4 providing the corresponding paramagnetic species (see Fig. 7). Subsequently, µLCR experiments with the solution sample of 4 were conducted. According to Eq. 2, where γk refers to the gyromagnetic ratio nucleus of I = 1/2 spin, the nucleus hyperfine constant (Ak) of a half-integer nuclear spin (I = 1/2) can be determined by using the Δ0 (ΔM = 0) avoided level-crossing resonance magnetic field and muon hyperfine constant (Aµ).

Fig. 7
figure 7

Structure of a peri-trifluoromethylated 9-phosphaanthracene (4). The most preferable mode of muonium addition to the 9-phosphaanthracene skeleton in 4 should afford 4Mu

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Fig. 8
figure 8

a TF-µSR spectrum of 4 (THF solution, 0.16 M) at 298 K and field of 1.45 T. b A µLCR spectrum of a 0.16 M solution of 4 in THF at 298 K. The differential-type line shape is due to the field modulation. Spectrometer: Helios (TRIUMF)

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$${B}_{{\text{res}}}^{{\Delta }_{0}}= \frac{1}{2}\left[\frac{{A}_{\mu }-{A}_{k}}{{\gamma }_{\mu }-{\gamma }_{k}}-\frac{{A}_{\mu }+{A}_{k}}{{\gamma }_{e}}\right]$$
(2)

A µLCR spectrum obtained from a THF solution of 4 at 298 K is shown in Fig. 8. The regioselective addition of muonium at the sp2-phosphorus in 4 means that 31P is the only spin-active I = 1/2 nucleus with a large enough abundance to generate the spectrum. From the avoided level-crossing resonance at 0.52 T and the Aµ value obtained from the TF-µSR spectrum in Fig. 8a we assigned the hyperfine coupling constant of the 31P nucleus [Ak(31P)] as 98 MHz. In combination with the TF-µSR experiment, the µLCR data strongly suggested that the muonium added to the phosphorus atom in 4 leading to the paramagnetic fused heterocyclic radical 4Mu.

The ordinary DFT structure optimization of 4Mu at the UωB97XD/6-311G(d,p) level, where Mu was calculated as H atom in optimizing the structures, provided the structure showing remarkable envelope- or saddle-type conformation of the 6-membered phosphorus heterocycle (Fig. 9a). The non-planar tricyclic skeleton is promoted by both the P-Mu unit and the flanking trifluoromethyl groups. However, the calculated muon hfc value of the normally optimized structure was considerably smaller (107 MHz) than the experimentally determined value, suggesting that the light mass of Mu accompanying larger zero-point energy avoids the formation of the most stable conformer. Because the conventional corrections including the elongation of the X–Mu bond (X = C or other main group elements) and scanning the angles around the X–Mu moiety for wagging and rocking were unsuccessful, many trials were conducted. As a result, it was possible to conclude that the revised structure shown in Fig. 9b had the muon hfc of 242 MHz which was comparable with the experimental result in Fig. 8a. The tricyclic skeleton of 4Mu in Fig. 9b is flat, and the sum of the internal bond angles of the P-containing heterocycle is up to 723.6°. Whereas the calculated 31P hfc constant for Fig. 9b (74 MHz) deviated from the experimentally determined 31P hfc parameter (98 MHz), decreasing the pyramidalization around the phosphorus atom could produce a calculated 31P hfc value of 96 MHz. The successfully characterized effect of muon of the light mass causing the remarkable distortion of the molecular skeleton in 4Mu would be promoted by the flexible fused heavier heterocyclic system and the fluorine-containing substituents [22].

Fig. 9
figure 9

a A normally optimized structure of 4Mu. b A revised structure of 4Mu according to the muon isotope effect

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In addition, µLCR measurements were applied to a solid sample of 4. Figure 10 displays a µLCR spectrum of 6 in the solid state. The weak Δ0 resonance at 0.51 T was compatible with the hfc constant of the 31P nucleus in 4Mu. The Δ1 resonance at 0.93 T means Aµ of 254 MHz of 4Mu, which was consistent with the TF-µSR spectrum of the solid sample [10]. Correspondingly, the 31P hfc [Ak(31P)] would be 132 MHz in the solid state, and the deviation from that of the solution was up to 34 MHz. The observation of both Δ0 and Δ1 of 4Mu should correlate with the solid-state properties of 4 [23]. Meanwhile, another Δ1 resonance at 1.78 T could be a small amount of the paramagnetic species via muoniation to the meso-substituted mesityl (= 2,4,6-trimethylphenyl) group in 4 leading to 4-aromMu, because the muon hfc of 486 MHz is typical of a cyclohexadienyl radical derived from a benzene derivative [10].

Fig. 10
figure 10

A μLCR spectrum of a powder sample of 4 at 298 K. The Δ0 resonance at 0.51 and the Δ1 resonance at 0.93 T are compatible with 4Mu. The Δ1 resonance at 1.78 T would correlate with minor muoniation to the meso-mesityl group in 4 generating 4-aromMu. Spectrometer: Helios (TRIUMF)

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4 Conclusions and future perspectives

This paper has summarized the μSR studies for elucidating radical reactions of the unsaturated phosphorus heterocycles featuring the cyclic singlet biradical character and the acene-type fused ring system. The use of muonium as a light hydrogen surrogate was advantageous to monitor the radical addition processes which can not be observed with more conventional means. Two modes of muonium addition were characterized by employing both electron-sufficient and -deficient 1,3-diphosphacyclobutane-2,4-diyls 1 and 3, and the particular paramagnetic 4-membered phosphorus heterocycles were characterized experimentally. The muoniation processes could correlate with the semiconductor functionality and the open-shell character as well as the zwitterionic canonical structure. The peri-trifluoromethylated 9-phosphaanthracene 4 displayed the predominant addition of muonium to the sp2-type phosphorus, and the CF3 groups were responsible for promoting the unprecedented isotope effect leading to the planar tricyclic molecular unit. Besides confirming the most favorable muoniation process with both Δ0 and Δ1 resonances, the solid-state µLCR experiment revealed the addition of muonium to the meso-aromatic group in 4 leading to a cyclohexadienyl radical.

Muons from accelerators are useful for understanding organic radical reactions in detail. As overviewed in this paper, the light hydrogen surrogate gave valuable information about the unsaturated phosphorus heterocycles concerning the radical reactions. Uncovered physical phenomena will exist and be fruitful in exploring novel paramagnetic molecular functionality produced by the chemistry of phosphorus heterocycles including the largely π-extended molecular systems [24].