# The Role of Axions in the Formation of the Photoluminescence Spectrum in Dispersive Media

## Abstract

**:**

## 1. Introduction

## 2. The Bohr Frequency and Its Relation to the Lorentz Harmonic Oscillator Model

#### 2.1. Niels Bohr’s Postulates

_{1}to an energetically higher E

_{2}, which Bohr called excited. The transition between these states is characterized by the Bohr frequency—ν

_{21}[14], the value of which is determined from Equation (1)

_{21}= (E

_{2}− E

_{1})/h,

_{2}− E

_{1}) = ΔE is the energy of the electron transition between levels. Such relations can link many levels together in pairs, and the role of the lower state can be assigned not only to the lower level but also to the higher ones. The Equation (1) in this case should have the following form:

_{mp}= (E

_{m}− E

_{p})/h

_{mn}atom does not exceed 10–20 cm

^{−1}. When considering the PL process, in various environments with volumetric placement of multidimensional oscillators, and, consequently, having a large set of Bohr frequencies, the proposed approach turns out to be useful for practical use.

_{ij}is not observed at the output of DM.

#### 2.2. The Bohr Frequency and Its Relation to the Lorentz Harmonic Oscillator Model

_{ij}. E. Fermi pointed out that the refractive index n(ν) and the phase velocity V are not quantities that have a constant value in the medium under study [14]. For the classical harmonic Lorentz oscillator, according to [16] we have the Equation (2):

_{mp}= (E

_{m}− E

_{p})/h,

_{mp}is the natural frequency of the electron oscillations near the stable equilibrium position, E

_{m}is the value of the electron energy at the “m” level, and E

_{p}is the value of the electron energy at the “p” level [16]. Planck’s constant h = 6.6252 × 10

^{−27}erg × s.

_{mp}= (E

^{(m)}− E

^{(p)})/ħ

_{mp}is the Bohr transition frequency, E

^{(m)}is the energy of an electron occupying level “m”, and E

^{(p)}is the energy of an electron occupying level “p”. In this case, ħ = h/2 π = 1.0544 × 10

^{−27}erg × s.

_{mp}, then the photon velocity slows down. If the value of the refractive index n(ν) is greater than the Bohr frequency ν

_{mp}, then photons are reflected toward the pump light beam, which assists in the addition of photons, leading to the birth of an axion.

_{3}in a two-level medium can be calculated in accordance with the law of conservation of energy:

_{3}= 2ν − ν

_{21}

_{21}corresponds to the tabular value of the frequency of the interlevel transition.

#### 2.3. Starting the Analysis of the PL Spectra in the DM

_{21}, n(ν) ≅ 1;

(b) ν > ν

_{21}, n(ν) > 1;

(c) ν < ν

_{21}, n(ν) >1,

_{21}—the Bohr frequency resonant transition.

_{21}, the refractive index n(ν) of the medium is close to unity [15,16,17,18]. The reflection coefficient of such a medium increases as the excitation frequency ν approaches the Bohr frequency ν

_{21}, reaching a maximum value at ν = ν

_{21}[16]. A qualitative illustration of this circumstance is Wood’s experience in observing resonant radiation in the case of atomic sodium vapors placed in a cuvette illuminated by a sodium lamp. This experience practically illustrates the case of exposure to a two-level medium of photons whose frequency ν is equal to the Bohr frequency (ν = ν

_{21}).

_{21}+ hν = 2hν

_{21}, where ν

_{21}= (E

_{2}− E

_{1})/h is the transition frequency of the investigated duplex environment; the value of hν

_{21}—energy of an electron in the excited level (hν

_{12}= hν

_{21}).

_{21}, the feature limiting the propagation of photons is due to the fact that, according to the theory of dispersion, the refractive index n(ν) < 1. The propagation of photons of monochromatic radiation in this region of the spectrum, generally speaking, is impossible, because otherwise their velocity V = c/n(ν) would exceed the speed of light c, which contradicts existing concepts. If a high-power laser is used in the experiment, then due to three-photon electron Raman scattering [22] after leveling the populations of the levels of the transition under study, we will get n(ν) ≅ 1, which will allow part of the pumping to pass through such a medium. If the intensity distribution across the cross-section of a powerful beam is Gaussian (single-mode laser), then we have self-focusing at the output of such a medium [21].

_{3}is recorded on the spectrograms [22,27].

## 3. Photoluminescence (PL)

_{pl}= 2hν − hν

_{ij},

_{pl}—the energy quanta (photons) PL, hν—the energy of the pumping quanta (photons)—the light radiation used to excite the PL, 2hν—the energy of “virtual” level, ν—the frequency of pumping radiation, and hν

_{ij}—the energy of electron, expended for nonradiative relaxation.

_{ij}is a combination of a large number of transition frequencies associated with nonradiative relaxation and transfer of thermal energy to the medium. Since the PL spectrum in DM is usually broadened, it is natural to assume that the PL process is accompanied by a multitude of electronic interlevel transitions responsible for nonradiative relaxation in the atoms of the element used for doping DM. Naturally, the law of conservation of energy is fulfilled for each frequency component of the broadened PL spectrum.

_{pl}—the quantum energy of the PL radiation,

_{pl}—the frequencies filling the broadened spectrum of the PL.

_{pl}and ν

_{ij}.

_{ij}− ν = ν − ν

_{pl}

_{pl}.

_{pl}− ν = ν − ν

_{ij}

_{pl}.

## 4. On the Results of Studies of PL and LG in Holmium-Doped Media

^{5}I (9 − 5

^{5}/

_{2}) −

^{4}I

^{0}(

^{15}/

_{2}−

^{9}/

_{2}) [44].

_{pl}.

_{pl}− ν = ν − ν

_{ij},

_{pl}is the frequency of the maximum peak on the spectrogam PL.

_{ij}is the frequency electronic transition between electronic levels (in case LG, for the holmium atom). Recall that the frequency ν

_{ij}corresponds to transitions between any pair of electronic levels of opposite parity.

_{3}, YAP), radiation in the region (~2 µm) was obtained by pumping, the wavelength of which is 791 nm.

_{lm}, and, in fact, responsible for the PL, and the energy for the table value of the wavelength 4939.01 Å [43] does not exceed 0.043 eV. Thus, in the case of glass fiber for the 755.09 nm pumping wavelength used in the experiment, the transition,

^{5}I (9 − 5

^{5}/

_{2}) −

^{4}I

^{0}(

^{15}/

_{2}−

^{9}/

_{2}), is responsible for the LG mode, corresponding to the frequency of 13243.35 cm

^{−1}.

^{5}I (9 − 5

^{5}/

_{2}) −

^{4}I

^{0}(

^{15}/

_{2}−

^{9}/

_{2}); its frequency—20246.97 cm

^{−1}.

_{3}(YAP) crystal, where a laser diode was used for pumping, the radiation wavelength of which is 794.8 nm. Numerical calculation shows in this case that the difference between the calculated value of the energy of the interlevel transition responsible for the PL and the tabular value does not exceed 0.027 eV.

_{pl}and, in fact, responsible for photoluminescence, and the energy of the table value of the wavelength 4939.01 Å (see [43]) does not exceed 0.043 eV.

^{5}I (9 − 5

^{5}/

_{2}) −

^{4}I

^{0}(

^{15}/

_{2}−

^{9}/

_{2}), corresponds to the LG mode, corresponding to the frequency of 13243.35 cm

^{−1}. In crystal structures, when pumping 493.901 nm, the transition

^{5}I (9 − 5

^{5}/

_{2}) −

^{4}I

^{0}(

^{15}/

_{2}−

^{9}/

_{2}) is responsible for LG, its frequency is 20,246.97 cm

^{−1}. Some results of the section are presented in [42].

## 5. Analysis of the Results, Observed in Bismuth-Doped Media

^{−1}.

^{2}D

_{3/2},

^{2}D

_{5/2},

^{4}P

_{5/2}, etc. These levels of bismuth correspond to the energy region of 0.5–3 eV.

_{ij}” of these transitions, then the conditions for anti-Stokes PL are met.

## 6. Axions in the Optical Range of the Spectrum and Their Lifetime

_{max}/2. Table 2 shows several examples illustrating the relationship between the width of the spectral band of the PL radiation and the lifetime of the axion. To estimate the lifetime of the axion, the ratio was used: 1 cm

^{−1}≈ 2.99793 × 10

^{10}s

^{−1}. The lifetime of the axion is significantly shorter than the lifetime of the excited energy levels, which, according to reference data [62], corresponds to a value of 10

^{−8}s.

## 7. Summing Up

^{−}

^{10}s). Stokes’s law states that PL light has a longer wavelength compared to the light used for excitation. According to Lommel, the PL spectrum as a whole and its maximum are always shifted in comparison with the excitation spectrum and its maximum towards long waves.

## Funding

## Data Availability Statement

## Conflicts of Interests

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**Table 1.**Juxtaposition tabular value of the transition wavelength with the result of calculating the wavelength of the transition.

1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|

Link number in the list literatures | Wavelength corresponding to the maximal valued ordinal curve of PL (LG): λ_{pf} (nm) | Wavelength of excitation source PL: −λ (nm) | The result of calculating the wavelength of the transition: −λ_{lm} (nm) | Tabular value of the transition wavelength: −λ _{t} (nm) |

which is associated with non-radiative relaxation. | ||||

[45] | ~1130 nm | 1058 nm | 994.6 nm | 982.8 nm |

[46,47] | ~720 nm | 514 nm | 399.6 nm | 388.6 nm |

[48,49] | ~750 nm | 500 nm | 374.99 nm | 359.6 nm |

[48,49] | ~1140 nm | 500 nm | 320.2 nm | 323.9 nm |

[48,50] | ~1300 nm | 800 nm | 577.7 nm | 574.2 nm |

[48,51] | ~1315 nm | 808 nm | 583.1 nm | 574.2 nm |

[48,52,53] | ~1310 nm | 808 nm | 584.1 nm | 574.2 nm |

[48,54] | ~1150 nm | 980 nm | 853.8 nm | 854.4 nm |

[48,55] | ~1210 nm | 405 nm | 243.2 nm | 243.3 nm |

[56] | ~1260 nm | 798 nm | 584 nm | 574 nm |

[56] | ~1153.5 nm | 502 nm | 314.9 nm | 306 nm |

[56] | ~1153,5 nm | 525 nm | 339.8 nm | 339.7 nm |

[56] | ~1085.4 nm | 680 nm | 472 nm | 472.2 nm |

[56] | ~1171.6 nm | 738 nm | 540 nm | 555.2 nm |

[56] | ~1260 nm | 798 nm | 584 nm | 527.4 nm |

References | Alloying Material; in Parentheses—Environment of the Test Sample | The Wavelength of the Pump Radiation, nm | Frequency Corresponding to the Maximum Value of Intensity PL, cm^{−1} | Width of the PL Spectrum at Half-Length, ∆W, cm^{−1} | Axion Lifetime τ, s |
---|---|---|---|---|---|

[35,36] | Silicon (ethanol) | 488 | ~5480 | ~1160 | ~3.48 × 10−13 |

[63] | Bismuth-Bi (glasscorderite)) | 514 | ~8547 | ~420 (~700) ~1120 | 2.93 × 10−13 |

[64] | Bismuth-Bi (aluminosil-rolled glass T = 77 K | 1075 | ~8880 | ~1213 | 3.64 × 10−13 |

[65] | Bismuth-Bi (phosphorus-silicate glass) | 1240 | ~7463 | ~340 | ~1.02 × 10−13 |

[66] | Bismuth-Bi T = 1.4 K | 375 | ~6803 | ~340 | 1.02 × 10−13 |

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**MDPI and ACS Style**

Ogluzdin, V.E.
The Role of Axions in the Formation of the Photoluminescence Spectrum in Dispersive Media. *Foundations* **2022**, *2*, 184-198.
https://doi.org/10.3390/foundations2010011

**AMA Style**

Ogluzdin VE.
The Role of Axions in the Formation of the Photoluminescence Spectrum in Dispersive Media. *Foundations*. 2022; 2(1):184-198.
https://doi.org/10.3390/foundations2010011

**Chicago/Turabian Style**

Ogluzdin, Valeriy Evgenjevich.
2022. "The Role of Axions in the Formation of the Photoluminescence Spectrum in Dispersive Media" *Foundations* 2, no. 1: 184-198.
https://doi.org/10.3390/foundations2010011