Suppose that the gas is in equilibrium with a radiation field B(?, T). In a steady state, the number atoms leaving the lower state must equal the number leaving the upper state. Combining the last two equations, The probability that an atom will be in a state with energy Eu is exp{- Eu/kT) . This is proved in any book on statistical mechanics, but it is beyond the scope of this chapter. The relative populations of the two states is Using equation (33), Therefore, combining equations (37) and (38), Since the constants and depend only on the structure of the atom, and not on temperature or other aspects of the environment,25 Then So, using the Plank black-body law [equation (2)], These coefficients are called the Einstein coefficients; they depend only on the nature of the atom and not on the temperature, the ambient radiant intensity, or any other aspect of the atom’s environment. Therefore, equations (42) and (44) are true in general. In particular, note that this argument shows that stimulated emission must exist; i.e., . This proved the existence of stimulated emission long before it had been observed in the laboratory; today, of course, it is the basis of the laser (Light Amplification by Stimulated Emission of Radiation) industry. Stimulated emission is necessary, as I have already mentioned, to keep the Second Law of thermodynamics from being violated, because without it, atomic absorption and emission processes could not be in balance with emission from a black body. If there is an enclosure with perfectly black walls with temperature T, then the radiation field inside the enclosure must be equal to the Planck function B(?, T). But now introduce a gas into that enclosure. Without stimulated emission, there would be a continuous transfer of energy either to or from the gas. This, as we know, is impossible.
Suppose that the gas is in equilibrium with a radiation field B(?, T). In a steady state, the number atoms leaving the lower state must equal the number leaving the upper state. Combining the last two equations, The probability that an atom will be in a state with energy Eu is exp{- Eu/kT) . This is proved in any book on statistical mechanics, but it is beyond the scope of this chapter. The relative populations of the two states is Using equation (33), Therefore, combining equations (37) and (38), Since the constants and depend only on the structure of the atom, and not on temperature or other aspects of the environment,25 Then So, using the Plank black-body law [equation (2)], These coefficients are called the Einstein coefficients; they depend only on the nature of the atom and not on the temperature, the ambient radiant intensity, or any other aspect of the atom’s environment. Therefore, equations (42) and (44) are true in general. In particular, note that this argument shows that stimulated emission must exist; i.e., . This proved the existence of stimulated emission long before it had been observed in the laboratory; today, of course, it is the basis of the laser (Light Amplification by Stimulated Emission of Radiation) industry. Stimulated emission is necessary, as I have already mentioned, to keep the Second Law of thermodynamics from being violated, because without it, atomic absorption and emission processes could not be in balance with emission from a black body. If there is an enclosure with perfectly black walls with temperature T, then the radiation field inside the enclosure must be equal to the Planck function B(?, T). But now introduce a gas into that enclosure. Without stimulated emission, there would be a continuous transfer of energy either to or from the gas. This, as we know, is impossible.
Einstein (1917) wrote the rates of absorption, stimulated emission and spontaneous emission between two levels as B12 u N1, B21 u N2 and A N2, and showed that if they balance in thermal equilibrium with Boltzmann level populations, the radiation u is Planck’s law and the ratio of A to B21 is 8πhν³/c³. This paper counts what happens when stimulated emission is removed from this ledger, and where in the coefficient 8π the number of polarizations enters. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──Einstein’s relations between the A and B coefficients (1917), Planck’s radiation law, Wien’s radiation law, that the occupation of radiation is 1/(exp(x) − 1), and that a population inversion can be expressed as a negative temperature are all known. No causal reason is given for why stimulated emission is needed──what is said is only the fact that removing stimulated emission from the ledger makes the balancing radiation Wien’s law rather than Planck’s. No claim is made about a world whose dimension is not 3──d − 1 times the area of the unit sphere is a value of the formula, and no such world is claimed to exist. Lasers are not treated──devices, linewidths, pumping and the degeneracies of real atoms are not included; 632.8 nm is used only as an example wavelength. The numerics of the rate equations are limited──A was set to 10⁶ per second, and only degeneracy ratios 1 and 3 and four values of x were integrated. Relation to earlier papers: the scope note of Paper 902 said that it removed the number of polarizations from the coefficient, leaving it out of the integrals that fit exponents and putting in 2 only when checking the 3-dimensional coefficient; this paper sits in that seat, puts the polarization count back into the coefficient of A over B, and counts that it does not enter the ratio of stimulated to spontaneous emission (no earlier occupant). Paper 902 showed that the 4π in the coefficient of black-body energy density is not a solid angle but the area of the unit sphere in k-space──here it is counted that multiplying that 4π by the 2 polarizations gives the 8π of A over B. Paper 265 showed that the 3 in the T³ of low-temperature heat capacity comes from the dimension of the density of states──here it is counted that the same density of states, together with the polarization count, enters the ratio of A to B. Paper 403 showed that negative temperatures are hotter than all positive ones──here it is counted that the population inversion in which stimulated emission beats absorption is a negative temperature. What is added is counting the ratio of Wien to Planck when stimulated emission is removed and finding the range within 1 percent, counting shell points and polarizations with integer points in a periodic box and splitting 8π into 2 and 4π, confirming by integrating the rate equations that populations settle to Boltzmann, counting the stimulated-to-spontaneous ratio in four settings, and confirming with lattice modes that the polarization count enters the ratio of A to B but not the stimulated-to-spontaneous ratio. First, and this is the core. Without stimulated emission, the balancing radiation is Wien’s law──its ratio to Planck’s law is 1 − exp(−x) with x = hν/kT: 0.095163 at x = 0.1, 0.632121 at x = 1 and 0.999955 at x = 10. It is within 1 percent only for x above 4.605170, which at 300 K means wavelengths shorter than 10.414 μm. The missing fraction exp(−x) equals the share of stimulated emission in all emission within Planck radiation (Section 4). Second, 8π is the product of the polarization count 2 and the area 4π of the unit sphere in k-space──counting integer points in a periodic box, the number in a shell is proportional to the area of the unit sphere (ratio 0.994578 for d = 3), and the number of directions perpendicular to the wave is d − 1. The product of 2 polarizations and 4π is 25.132741, equal to 8π (Section 2). Third, the ledger does settle to the Boltzmann distribution──integrating the rate equations in time inside Planck radiation, for degeneracy ratios 1 and 3 and x from 0.5 to 10, all 8 cases settled to level ratios within a relative difference of 7.077×10⁻¹⁴ of Boltzmann (Section 3). Fourth, the ratio of stimulated to spontaneous emission is the occupation 1/(exp(x) − 1), and it moves by 45.453 orders of magnitude across four settings──it was 0.006884 at 500 nm on the solar surface (5772 K), 2.203651×10⁻⁴² at 500 nm at room temperature and 6.250486×10³ at 1 GHz at room temperature. The ratio is 1 at a wavelength of 69.1905 μm at 300 K (Section 5). Fifth, stimulated emission beats absorption only at negative temperature──at 632.8 nm, a state with twice as many atoms in the upper level as in the lower is T = -32802.090 K; at positive temperature the lower level holds more and absorption wins (Section 6). Sixth, the separator is whether the number of polarizations enters the ratio of A to B or the ratio of stimulated to spontaneous emission──the coefficient of A over B in d dimensions is d − 1 times the area of the unit sphere (2π, 8π, 6π², 105.275780) and carries the polarization count. The ratio of stimulated to spontaneous emission in radiation built from lattice-counted modes was the same with d − 1 polarizations as with 1 (Section 7). Without stimulated emission, radiation follows Wien’s law, and only the 8π of A over B carries the polarization count. Removing stimulated emission from Einstein’s ledger makes the ratio of the balancing radiation to Planck’s law 1 − exp(−x), which is 0.095163 at x = 0.1 and within 1 percent only for x above 4.605170──the ledger with stimulated emission settled to Boltzmann in all 8 cases of integrating the rate equations, within a relative difference of 7.077×10⁻¹⁴.8π is the product of the polarization count 2 and the area 4π of the unit sphere in k-space, and in d dimensions it becomes d − 1 times the area of the unit sphere──the ratio of stimulated to spontaneous emission is the occupation, 0.006884 at 500 nm on the Sun and 2.203651×10⁻⁴² at 500 nm at room temperature, and in lattice-built radiation it did not change between d − 1 polarizations and 1. A population inversion is a negative temperature: at 632.8 nm, twice as many upper as lower gives -32802.090 K. The separator is whether the number of polarizations enters the ratio of A to B or the ratio of stimulated to spontaneous emission. Placed among the earlier papers──Paper 902 read the 4π of the coefficient as the area of the unit sphere in k-space and left the polarization count out of the coefficient. Paper 265 showed the dimension of the density of states, and Paper 403 that negative temperatures are hotter than positive ones. Here the polarization count that was left out was put back into the ratio of A to B, and it was counted that it does not remain in the stimulated-to-spontaneous ratio. To be honest──no causal reason is given for why stimulated emission is needed, and neither worlds whose dimension is not 3 nor laser devices are treated. The rate equations were integrated only for limited A, degeneracies and x. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: stimulated emission, spontaneous emission, Einstein coefficients, Wien's law, number of polarizations, negative temperature. ----- アインシュタイン(1917)は、二つの準位のあいだの吸収・誘導放出・自発放出の速さを B12 u N1・B21 u N2・A N2 と置き、熱平衡で釣り合って準位の数がボルツマン分布になるなら、放射 u はプランクの式で、A と B21 の比は 8πhν³/c³ になることを示した。本稿は、この帳簿から誘導放出を外したとき何が起きるか、そして係数の 8π のどこに偏光の数が入るかを数える。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──アインシュタインの A・B 係数の関係(1917)、プランクの放射則、ウィーンの放射の式、放射の占有数が 1/(exp(x) − 1) であること、反転分布を負の温度で表せることは、いずれも既知である。誘導放出が要る理由を因果で言わない──言うのは、帳簿から誘導放出を外すと釣り合う放射がプランクではなくウィーンの式になる、という事実だけである。次元が 3 でない世界を主張しない──(d−1) 倍の単位球の面積は式の値で、そのような世界が在るとは言わない。レーザーを扱わない──装置、線幅、ポンプ、実在の原子の縮退の値は入れない。632.8 nm は波長の例として使っただけである。速さの式の数値を限る──A を 10⁶ 毎秒に置き、縮退の比 1 と 3、x の 4 点だけを積分した。既刊との関係:論文902 の射程注記は、偏光の数を係数から外し、指数を当てる積分では入れず、3 次元の係数を確かめるときだけ 2 を入れたと書いた──本稿はその席に座り、偏光の数を A と B の比の係数に戻して、それが誘導放出と自発放出の比には入らないことを数える(先客なし)。論文902 は、黒体の放射密度の係数の 4π が立体角ではなく k 空間の単位球の面積であることを示した──ここでは、その 4π に偏光の 2 を掛けたものが A と B の比の 8π であることを数える。論文265 は、低温比熱の T³ の 3 が状態密度の次元から来ることを示した──ここでは、同じ状態密度が偏光の数と合わさって A と B の比に入ることを数える。論文403 は、負の温度が正の温度のすべてより熱いことを示した──ここでは、誘導放出が吸収に勝つ反転分布が負の温度であることを数える。加えたのは、誘導放出を外したときのウィーンとプランクの比を数え、1 パーセント以内に入る範囲を出したこと、周期箱の整数点で殻の数と偏光の数を数え、8π を 2 と 4π に分けたこと、速さの式を積分してボルツマン分布に落ち着くことを確かめたこと、誘導/自発の比を四つの場面で数えたこと、偏光の数が A と B の比には入り、誘導/自発の比には入らないことを格子のモードで確かめたことである。 第一に、これが本稿の芯である。誘導放出を外すと、釣り合う放射はウィーンの式になる──プランクとの比は 1 − exp(−x)(x=hν/kT)で、x=0.1 で 0.095163、x=1 で 0.632121、x=10 で 0.999955 だった。1 パーセント以内に入るのは x が 4.605170 より大きいとき、300 K なら波長 10.414 μm より短いときだけである。欠ける割合 exp(−x) は、プランクの放射の中で誘導放出が放出の全体に占める取り分と同じだった(第4節)。 第二に、8π は偏光の数 2 と k 空間の単位球の面積 4π の積である──周期箱の整数点を数えると、殻の点の数は単位球の面積に比例し(d=3 で比 0.994578)、波の向きに垂直な向きの数は d−1 だった。偏光 2 と 4π の積は 25.132741 で 8π と一致した(第2節)。 第三に、帳簿は実際にボルツマン分布へ落ち着く──プランクの放射の中で速さの式を時間で積分すると、縮退の比 1 と 3、x=0.5〜10 の 8 通りとも、準位の数の比がボルツマン分布と相対差 7.077×10⁻¹⁴ 以内に落ち着いた(第3節)。 第四に、誘導放出と自発放出の比は占有数 1/(exp(x) − 1) で、四つの場面のあいだで 45.453 桁動く──太陽の表面(5772 K)の 500 nm で 0.006884、室温の 500 nm で 2.203651×10⁻⁴²、室温の 1 GHz で 6.250486×10³ だった。比が 1 になる波長は 300 K で 69.1905 μm である(第5節)。 第五に、誘導放出が吸収に勝つのは、温度が負のときだけである──632.8 nm で上の準位の数が下の 2 倍になる状態は T=-32802.090 K で、正の温度では下の準位が多く吸収が勝つ(第6節)。 第六に、分離子は、偏光の数が A と B の比に入るか、誘導放出と自発放出の比に入るかである──A と B の比の係数は d 次元で (d−1) 倍の単位球の面積(2π・8π・6π²・105.275780)で偏光の数を運ぶ。一方、格子で数えたモードで組んだ放射の中の誘導/自発の比は、偏光を d−1 本にしても 1 本にしても同じだった(第7節)。 誘導放出を抜くと放射はウィーンの式になり、A と B の比の 8π だけが偏光の数を運ぶ。アインシュタインの帳簿から誘導放出を外すと、釣り合う放射のプランクとの比は 1 − exp(−x) で、x=0.1 で 0.095163、1 パーセント以内に入るのは x が 4.605170 より大きいときだけだった──誘導放出を入れた帳簿は、速さの式の積分で 8 通りともボルツマン分布へ相対差 7.077×10⁻¹⁴ 以内
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
The phenomena which are principally responsible for the interaction of radiation with matter in everyday life are absorption and spontaneous emis sion. Light falling on matter is absorbed, leaving atoms in excited states which spontaneously emit radiation with a spread of frequency inversely proportional to the decay time. The situation is modified when the atomic decay times are sufficiently long and the radiation sufficiently intense, for then radiation may fall on excited atoms, resulting in stimulated emission rather than absorption. We saw in Chap. 2 that this third process was required to obtain the Planck blackbody radiation law in Einstein’s derivation. It is stimulated emission which enables an atomic medium to amplify incident radiation, and therefore is an essential ingredient in light amplifiers and oscillators. In fact, the word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
The interaction of radiation and matter is discussed on the basis of Einstein's theory. The distribution of energy in an enclosure containing quantized particles is indeterminate unless there is thermal or other non-radiative interaction between the particles; the latter process is therefore essential to the production of the black-body distribution. Fluorescence cannot lower the entropy of the system.Exciting power for fluorescence.---No sharp distinction can be drawn between thermal emission and fluorescence. If ${\ensuremath{\psi}}_{\mathrm{yx}}$ is the fluorescence of frequency ${\ensuremath{\nu}}_{y}$ excited by radiation of frequency ${\ensuremath{\nu}}_{x}$ and if ${\ensuremath{\alpha}}_{x}$, ${\ensuremath{\alpha}}_{y}$, ux, uy are the corresponding absorption coefficients and black-body densities, respectively, then on certain assumptions ${\ensuremath{\psi}}_{\mathrm{yx}}=\frac{k{\ensuremath{\alpha}}_{x}{\ensuremath{\alpha}}_{y}{u}_{y}}{{\ensuremath{\nu}}_{x}}$, k being independent of ${\ensuremath{\nu}}_{x}$ and ${\ensuremath{\nu}}_{y}$. Much more general assumptions lead to: $\frac{{\ensuremath{\psi}}_{\mathrm{yx}}}{{\ensuremath{\psi}}_{\mathrm{xy}}}=\frac{{\ensuremath{\nu}}_{y}{u}_{y}}{{\ensuremath{\nu}}_{x}{u}_{x}}$. These equations ought to apply to gas fluorescence, and it is quite possible that the latter may hold generally for liquids and solids and the former in those cases where the shape of the fluorescence band does not vary with the exciting frequency. The equations agree with the few data in existence.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.