The spectrum of electromagnetic waves emitted by an oscillator trapped in an external

potential well is studied. It is assumed that the natural frequency of the oscillator is much

greater than the frequency of oscillations in the potential well. We consider the quantum

model of emission with regard to the recoil e ect. The highest intensity of the absorption and

emission lines is observed on the natural frequency of the oscillator when the recoil energy is

equal to the energy of the quantum of low-frequency (LF) oscillations in the potential well.

A certain decrease in the amplitude of the emission and absorption lines is noted caused

by the oscillations of the potential well due to, for example, the presence of the phonon

spectrum. The relaxation times of the oscillator LF motion in the potential well, resulted

from the phonon emission, are estimated. It is concluded that these processes have no

in

uence on the observed features of the emission and absorption of high-frequency photons.

This model can be applied to description of the emission and absorption of gamma rays in

the crystal structures even in the presence of the phonon spectrum on condition that the

relaxation time of the LF movements in the potential well is greater than the lifetime of the

HF oscillator

Kirichok, A.V.

Kuklin, V.M.

Zagorodny, A.G.

English

arXiv

Електронного архіву Харківського національного університету імені В.Н.Каразіна (Electronic Archive V.N. Karazin Kharkiv National University)

ON THE NATURE OF THE MO¨SSBAUER EFFECTA.V. Kirichok,1, ∗ V.M. Kuklin,1 and A.G. Zagorodny21V.N. Karazin Kharkiv National University , Institute for High Technologies4 Svobody Sq., Kharkiv 61022, Ukraine2Bogolyubov Institute for Theoretical Physics14-b, Metrolohichna str., Kiev, 03680, UkraineThe spectrum of electromagnetic waves emitted by an oscillator trapped in an externalpotential well is studied. It is assumed that the natural frequency of the oscillator is muchgreater than the frequency of oscillations in the potential well. We consider the quantummodel of emission with regard to the recoil effect. The highest intensity of the absorption andemission lines is observed on the natural frequency of the oscillator when the recoil energy isequal to the energy of the quantum of low-frequency (LF) oscillations in the potential well.A certain decrease in the amplitude of the emission and absorption lines is noted causedby the oscillations of the potential well due to, for example, the presence of the phononspectrum. The relaxation times of the oscillator LF motion in the potential well, resultedfrom the phonon emission, are estimated. It is concluded that these processes have noinfluence on the observed features of the emission and absorption of high-frequency photons.This model can be applied to description of the emission and absorption of gamma rays inthe crystal structures even in the presence of the phonon spectrum on condition that therelaxation time of the LF movements in the potential well is greater than the lifetime of theHF oscillator.1. INTRODUCTIONThe scattering of high-energy photons by free electrons results in a decrease in energy of pho-tons due to the recoil effect (the Compton effect). This fact together with the phenomenon ofthe photoelectric effect confirmed the basic principles of quantum theory of radiation [1–3]. Theprocesses underlying the interaction of radiation and matter are characterized by an impressivevariety and form the basis for many physical research directions [4–6]. One of the problems thatarises when considering the processes of absorption and emission by a matter is the problem ofinteraction with the external radiation field of the oscillating particle trapped into the potential∗Electronic address: sandyrcs@gmail.comarXiv:1701.06223v1  [quant-ph]  22 Jan 20172well formed by the spatial structure of the medium. This problem requires the use of methods ofquantum electrodynamics for describing the behavior of an excited oscillator in a potential well.When the energy of the single photon becomes comparable to the kinetic energy of the oscil-lator, the recoil effects should be taken into account. As follows from the energy and momentumconservation laws, the Lamb-Dicke parameter kb [7], which is proportional to both the wave num-ber of the resonant photon k and the spatial spread of the oscillator ground state wavefunctionb, becomes in this case rather large (as will be shown below kb ≈ 2) and extra spectrum linesappear both in the emission and absorption spectra, shifted on the value of the recoil energy Er/~.Similar effect was studied in [8]. The interaction between the external field and this extended set ofspectral lines can change the nature of absorption and emission processes of the oscillator trappedin the potential well.The purpose of this work is to analyze the interaction between a charged particle-oscillatortrapped in a potential well and external electromagnetic field. We found that the highest intensityof the absorption and emission lines is observed on the natural frequency of the oscillator ω0 whenthe recoil energy Er is equal to the energy of low-frequency quantum ~Ω (Ω is the frequency ofoscillations in the potential well). We also discuss the role of relaxation of low-frequency motion inthe potential well due to emission of sound waves and conditions under which the relaxation doesnot affect the considered emission and absorption of high-frequency quanta.2. THE EMISSION OF VIBRATING OSCILLATORFollowing the method described in [1], we consider the quantum model of the oscillator witha charge e, mass m and natural frequency ω0, which oscillates as a whole in the potential wellaligned along axis OZ. Let the oscillator emits the electromagnetic wave in the same direction~k = (0; 0; kz). The components of the oscillator’s velocity vector are:vx = vx0 cosω0t = aω0 cosω0t, vz = bΩ cos(Ωt). (1)The electromagnetic field can be found from~E = −1c~˙A, ~H = rot ~A. (2)The vector potential has the componentsAx = q(t)√2 cos(kz + δ), Ay = 0, Az = 0. (3)3Phase δ depends on the oscillator orientation. The form (3) choice is determined by normalizingcondition, so that the integral of the vector potential squared in the unit volume equals to unityand q(t) satisfies the equationq¨ + ω2q = 0. (4)The total field energy inside V -volume is equal toU =V4pic212(q˙2 + ω2q2). (5)Let define the effective mass of the oscillator asmeff = V/4pic2. (6)As follows from Eqs. (1) and (3) , the x-component of vector potential Ax in the location point ofthe oscillator is equalAx =√2q0 cosωt · cos (kb sin Ωt+ δ) (7)Suppose the particle occupies the lowest energy level in an external potential well and consider thecase, when it remains inside the well after the emission or absorption of a quantum. That meansthe recoil energy to be insufficient for the particle to leave the potential well it stays in.After absorbing a high-frequency quantum Eν = ~(ω0+Ω), the particle-oscillator gets the recoilmomentum mV and begins slow oscillations inside the potential well. The energy of the quantum,emitted by the particle-oscillator with natural frequency ω0 is equal Eν = ~(ω0 − Ω) due to therecoil effect.The conservation laws for the case of quantum absorption are:~(ω0 + Ω)/c = mV, (8)~Ω = mV 2/2. (9)It is noticeable that the oscillation energy of the particle in the potential well equals to ~Ω.That is why the exciting quantum energy has to exceed the oscillator‘s energy ~ω0 by this value.Assuming the condition~ω0  2mc2 (10)is fulfilled and using Eqs. (8)-(9), one can find the oscillation frequency inside the potential wellΩ ≈ ~ω20/2mc2. (11)4It is not difficult to show that Eq. (11) still stands and for the case of emission. Besides, it followsfrom Eqs. (1), (8), (9) thatV = bΩ = ~ω0/mc. (12)The Hamiltonian of the system including the oscillator and the field can be represented as [1]H =12m(~p− ec~A)(~p− ec~A)+ eΦ. (13)The interaction part may be written out asH ′ = −evxAx/c = −evx0cq0√2 cos δ exp (iωt+ ikb sin Ωt± Ωt) cosω0t ≈≈ −evx02cq0√2 cos δ∑mJm(kb) exp (i(ω − ω0m)t± Ωt) , (14)where vx = vx0 cosω0t. This means that the external field with frequency ω interacts with the setof oscillators, which frequencies are ω0m = ω0 +mΩ and the amplitude is proportional to Jm(kb),where Jm(x) is the Bessel function with argument kb ≈ ω0b/c/ In other words, the system possessesmenergy levels the transfer on each of them can be realized independently. The additional termin the exponent ±Ωt which doesn’t follow from the classic description, takes into account thefrequency variation when the HF quantum is emitted (upper sign) or absorbed (lower sign).Representing the system “oscillator in the potential well” as a set of oscillators with frequenciesω0m = ω0 + mΩ and imposing the condition of temporal synchronism one can find that for theexternal field with the frequencyω = (m∓ 1)Ω + ω0, (15)the interaction Hamiltonian (14) takes the formH ′ = −evx0cq0√2∑mJm(kb) cos δ. (16)The upper sign in Eq. (15) corresponds to the emission of the quantum with taking into accountthe recoil effect (the frequency of the external field at this is less than the proper frequency ofthe oscillator on the value of Ω/2pi, m = 0) . The lower sign corresponds to the absorption (thefrequency of the external field at this is greater than the proper frequency of the oscillator on thevalue Ω/2pi, m = 0). Note that the correction to the matrix element due to the recoil effect isproportional to the small parameter ~ω0/mc2 both for the individual processes of absorption andemission, as well as for the Compton scattering.5When the oscillator in rest, b = 0, the only term in the sum (16) is different from zero (m = 0):H ′ = −evx0cq0√2 cos δ. (17)Thus, the frequency of the absorbed radiation ω = ω0 +Ω differs from the frequency of the emittedradiation ω = ω0 −Ω on the value of 2Ω that corresponds to the value of the double recoil energy.For b 6= 0, the interaction Hamiltonian for the most interesting case of the absorption andemission on the proper frequency of the oscillator ω0 becomes:H ′ = −e · vxcq√2J±1(kb) cos δ. (18)The interaction Hamiltonian for emission on the frequency ω = ω0 − Ω and absorption on thefrequency ω = ω0 + Ω is similar to Eq. (18), where J±1(kb) should be replaced by J0(kb).Let estimate the value of the argument of Bessel functions. Since the frequency of LF oscillationsin the potential well is Ω = v/b and the energy of quantum ~Ω is equal to the recoil energy thenit follows from Eq.(12) [9]:kb ≈ ω0bc≈ 2. (19)The matrix element of the interaction Hamiltonian (18) can be written asHif = −ec√2ω0xcdqnn′J±1(kb) cos δ. (20)The subscripts c and d indicate two states of the emitted (n, n+ 1) or absorbed (n, n− 1) field.For the absorption case |qnn′ |2 = |qn,n−1|2 = n|q01|2 and for the emission case |qnn′ |2 = |qn+1,n|2 =(n+ 1)|q01|2.Taking into account the expression for effective mass (6), we can write the matrix element forthe spontaneous emission [1]|q01|2 = hc2V ω0, (21)and|Hif |2 = −2e2c2hc2Vω0(x2cd + y2cd)J21 (kb) cos2 δ n+ 1n , (22)where the upper value corresponds to emission and the lower value to absorption. The transitionprobability can be found by taking the product of Eq. (22) and 4pi2ρ(νcd)/h2, where ρ(νcd) is theoscillation density, with taking into account the averaging over initial phases 〈cos2 δ〉 = 1/2Pif =4pi2h2|Hif |2ρ = 8pie2hc3ω20(|xcd|2 + |ycd|2) J21 (kb) cos2 δ n+ 1n . (23)6Note that the probability of absorption on the frequency ω0 + Ω and the probability of emissionon the frequency ω0 − Ω can be obtained by replacing J21 (kb) by J20 (kb) in Eq. (23). MultiplyingEq.(23) by ~ω0 one can obtain the intensity of radiation along axis OZ and integrating over theangle θ = ~k ∧O~Z – the total intensity over all directions.It is easy to verify that since J21 (kb) J20 (kb) the intensity of absorption and emission spectrallines on the proper frequency of the oscillator ω0 exceeds the intensity on the frequency ω0±Ω byan order. Note that the width of the potential well should be greater than b and its depth shouldexceed the recoil energy. The proposed quantum-mechanical description may be more correctlyclarify the mechanism of emission and absorption of γ- quanta without recoil [6]. In addition, it canbe shown that taking into account fast oscillations of the potential well as whole with a frequencyωS and amplitude bS , which satisfy the conditions b2Sω2S > b2Ω2 and b2S  b2, results in decreaseof the vector potential Ax and interaction energy H′, defined by Eqs. (12) and (16), by the factorexp(−W/2) = (1− b2s/b2) and reduces the transition probability (23) in exp(−W ) times.In a similar way, the presence of a relatively high frequency phonon spectrum in the environmentwill effect on the probability of the transition. For the case of a rather broad spectrum, this influencecan be described by a factor exp{−W} = (1−b−2∑i=1 b2i ) under the conditions b2Ω2 ∑i=1 b2iω2iand b2 >∑i=1 b2i .3. THE RELAXATION OF LOW-FREQUENCY OSCILLATIONSWhen the above-discussed system is placed in solid medium, it is necessary to consider a pos-sibility of emission of a low-frequency phonon Ω = ω0(~ω0/2mc2). The relatively low velocityacquired by an oscillator due to recoil is often significantly less than the phase velocity of phononsvs  c(~ω0/2mc2). (24)This makes impossible the direct transfer of kinetic energy to phonon. This is evidenced by theinability to fulfill the requirements of conservation of energy and momentum. For example, it canbe shown for the momentum that~ω0/c ~Ω/vs = ~ω0(~ω02mc2)/vs =~ω0c~ω02mc2cvs. (25)However, if the oscillator is trapped to the potential well, its movement becomes irregular and itbecomes capable to emit phonons. At this, the spatial period of low-frequency oscillations in the7potential wellb =VΩ=~ω0mc2mc2~ω20=cω0(26)is much less than the wavelength of phonon oscillations, which frequency is equal to the frequencyof oscillations in the potential wellksb =Ωvscω0=~ω02mc2cvs 1. (27)Let estimate the lifetime of the low-frequency oscillator. If the lifetime occurs much greater thanthe period of oscillations, the above consideration of the isolated system ”oscillator in potentialwell” remains applicable for the case when this system is placed in a medium. In other words, therelaxation processes caused by the interaction with the phonon spectrum can be neglected.In one-dimensional case, the perturbation of the medium density generated by low-frequencymotion of the oscillator can be written asρext(t, x) = mδ[x− V cos(Ωt)] (28)Its Fourier image is equalρ(ω, k) =ω2v2sk2 − ω2m4pikb2[δ(ω − Ω)− δ(ω + Ω)] (29)The inverse transform givesρ(x, t) = −mbΩ28v2s[sin(Ωt− k0z) + sin(Ωt+ k0z)]. (30)Since the perturbation of the medium speed u = vsρ/ρ0, the ratio of emission intensity to theenergy of the quantum is equalI~Ω=116mρ0λsΩ. (31)The small parameter m/ρ0λs here is equal to the ratio of oscillator’s mass to the total mass ofsimilar particles located on the wavelength of sound wave. The relaxation time of low-frequencyoscillationsτLF ≈ 8ρ0λs/pimΩ (32)occurs, as was supposed earlier, much greater than the period of low-frequency oscillations in thepotential well. Moreover, if τLF far exceeds the lifetime of high-frequency quantum, the relaxation8process does not affect the character of emission and absorption of photons considered above. Note,that in the three-dimensional case the characteristic relaxation time of low-frequency motionτLF ≈ 3(ρ0λ3s/m)(ω0/pi2Ω20) (33)is proportional to a large parameter ρ0λ3s/m. This parameter is equal to the ratio of total massof atoms within a cube with edge λ to the mass of single atom. The factor (ω0/Ω) is large too.In this case the lifetime of high-frequency oscillator may occur less than the relaxation time oflow-frequency motion caused by emission of sound.4. CONCLUSIONWe considered the quantum-mechanical model of electromagnetic emission by one-dimensionaloscillator, trapped in the external potential well. On the assumption of the energy of the emittedquantum is much less that the rest energy of the oscillator (~ω0/mc2  1), the perturbation ofthe Hamiltonian matrix elements caused by the recoil effect can be neglected, that simplifies theircalculation.It is shown that the intensity of the absorption and emission lines at the natural frequency ω0 ofthe oscillator significantly exceeds the intensity of other spectral lines, particularly at frequenciesω0 ± Ω. This is caused by the equality of the oscillation energy in the potential well to the recoilenergy, and the fact that the amplitude of the oscillation in the potential well in this case iscomparable to the wavelength of the radiation.It is noted a slight decrease in the amplitude of the emission and absorption lines due to theoscillations of the potential well as whole, due to the presence of the phonon spectrum. It isestimated the relaxation time of low-frequency movements in the potential well caused by thephonon radiation to the environment. It is concluded that these processes have no effect on theobserved features of the processes of emission and absorption of high-frequency photons.Such a quantum-mechanical description may be more correctly explains the phenomenon ofemission and absorption of electromagnetic field quanta without recoil by the matter [6]. Really, theemission of the oscillator with natural frequency ω0 produces the equidistant spectrum ω0+nΩ thatcan be considered as result of emission of a set of oscillators with oscillation amplitude proportionalto J2n(kb). Note that the existence of such an equidistant spectrum of the oscillator captured inthe potential well was also discussed in [8] under slightly different conditions. Moreover, sincethe amplitude of low-frequency oscillations in the potential well b is comparable with the emission9wavelength λ = 2pi/k, the emission and absorption on the proper frequency ω0 dominate in thiscase. The nature of the high-frequency oscillator is not critical for the manifestation of the above-described properties of the system.Note, that the application of this model to description of emission and absorption of γ-quantain crystal structures is possible even in presence of the phonon spectrum, when the relaxation timeof low-frequency motions in the potential well significantly exceeds the period of oscillations inthe potential well and the lifetime of high-frequency oscillator. For example, the absorption ofγ-quanta with energy 14,4 kEv and 23,8 kEv by nuclei 57Fe and 119Sn correspondingly results inaccordance with formula ω0b/c ≈ 2 to the oscillations of atoms in the potential well, formed by thecrystal, with amplitude excursion 0, 55 · 10−8 cm and 0, 33 · 10−8cm. The relaxation time of thisoscillatory motion is about 0.1 sec for 57Fe and about 0.01 sec for 119Sn, that is much more thanlife-time of excitation state of these nuclei. The absorption and emission on the proper frequencyof the atomic nucleus ω0 is dictated basically by the presence of the sufficient number of atoms,which oscillate in the potential wells formed by the crystal [9].5. REFERENCES[1] M. Born, Atomic physics (London, Blackie & Son, 1969).[2] D. Iwanenko and A. Sokolow, Klassische Feldtheorie (Akademie-Verlag, Berlin, 1953).[3] A. Akhiezer and V. Berestetskii, Quantum Electrodynamics (Interscience, New York, 1965).[4] A. Abramov, Y. A. Kazanskii, and E. Matusevich, Atomizdat, Moscow pp. 463–469 (1977).[5] V. Vinetsky and K. GA, Radiation Physics of Semiconductors (Naukova dumka, Kiev, 1979).[6] G. K. Wertheim, Mo¨ssbauer effect: principles and applications (Academic Press, 2013).[7] J. Javanainen and S. Stenholm, Applied physics 24, 151 (1981).[8] Q. Li, D. Xu, C. Cai, and C. Sun, Scientific reports 3 (2013).[9] O. Kuklina and V. Kuklin, The Journal of Kharkiv National University, physical series: Nuclei, Particles,Fields pp. 20–28 (2009).

On the nature of the Mossbauer effect

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Електронного архіву Харківського національного університету імені В.Н.Каразіна (Electronic Archive V.N. Karazin Kharkiv National University)

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