arXiv:1809.10349v1 [nucl-th] 27 Sep 2018 ain ou seilyo h ffc h pia-ae rad optical-laser the effect the a the inte on the increasing especially laser-relate under an focus in bound-states gations receiving from research is emission radiation topical charge electromagnetic active of an problem of the context the In proton radiation; electromagnetic decay; radiation alpha words: Key 03.50.de 03.65.Sq; 03.65.xp; 23.50.+z; 23.60.+e; PACS: spontaneous lcrmgei-aito ffc nalpha on effect Electromagnetic-radiation h aeo ttceeti edi lodiscussed. also is field a electric static slight disintegr a a the of of with case enhancement rate, The small a disintegration give the corrections to These co field conditions, second-order these electric brings In the radiation electromagnetic fields. nu the low the that on rather acting "sees" field clou nucleus electric electronic the mo the the alpha-particl that a appreciably given screens and in are emission Arguments estimated proton fields. is laser for in relevant nuclei be atomic may heavy which from emission charge eateto hoeia hsc,Isiueo tmcPhy Atomic of Institute Physics, Theoretical of Department h ffc fteeetoantcrdaino the on radiation electromagnetic the of effect The aueeBcaetM-,PBxM-5 Romania MG-35, POBox MG-6, Magurele-Bucharest .Mt.Ter hs 5 (2018) 155 1 Phys. Theor. Math. J. lh-atcedcyo h tmcnce,6-1]o nucl or nuclei,[6]-[11] atomic the of decay alpha-particle mi:[email protected] email: .Apostol M. Abstract decay 1 et oeinvesti- Some rest. ainmyhv on have may iation spontaneous rcin in rrections ti shown is it physics,[1]-[5] d nheavy in d to rate. ation msin laser emission; nisotropy. lu and cleus decay e to fthe of ction sics, del ear proton emission,[12, 13] but the area may be extended to or molecular or atomic clusters fragmentation.[14]-[17] The aim of the present paper is to estimate the effect of the adiabatically-applied electromagnetic radiation upon the rate of spontaneous nuclear alpha decay and proton emis- sion. Specifically, the paper is motivated by the interest in computing the rate of tunneling through a Coulomb potential barrier in the presence of elec- tric fields. It is claimed that the rate of alpha decay is practically not affected by electric fields,[8] or it is greatly enhanced by strong electric fields.[18] On the other side, the atomic cloud may screen appreciably the external electric fields, such that the may experience, in fact, rather low electric fields. It is this point, related to low electric fields, which may raise technical difficulties in estimating the small effect of these external fields upon the alpha decay. We adopt a nuclear model with Z protons and A Z neutrons, where A is the mass number of the nucleus, moving in the nuclear− mean field. The ex- periments proceed usually by placing a collection of heavy atoms in the focal region of a laser beam, and focusing radiation pulses upon that collection of atoms. We consider an optical-laser radiation with a typical frequency ω of the order 1015s−1 (corresponding to a period T 10−15s and a wavelength λ 0.8µm). We assume that the motion of the≃ charges under the action ≃ of the electromagnetic radiation remains non-relativistic, i.e. qA /mc2 1, 0 ≪ where q is the particle charge, m is the particle mass and A0 is the amplitude of the vector potential (c denotes the speed of light in vacuum). For protons in atomic nuclei (q =4.8 10−10esu, m 2 10−24g, c =3 1010cm/s) this condition yields a very high× electric field≃ E×= 3 1013V/cm× (1011 electro- 0 × static units), which corresponds to a maximum intensity of the laser beam in 2 24 2 the focal region of the order I = cE0 /8π = 10 w/cm . Typically, the dura- tion of the laser pulse is of the order of tens of radiation period (or longer), such that we may consider the action of the electromagnetic radiation much longer than the period of the radiation. The repetition rate of the laser pulses is usually much longer than the pulse duration. For simplification we consider linearly-polarized radiation plane waves; the calculations can be extended to a general polarization. The laser-beam shape or multi-mode operation have little relevance upon the results presented here. The electric fields are appreciably screened by the electronic cloud of the heavy atoms. The screening effects on the thermonuclear reactions, alpha decay and lifetimes have been considered previously.[10, 11, 19, 20] A conve-

2 nient means of treating the electron cloud in heavy atoms is the linearized Thomas-Fermi model.[21] According to this model, the radial electron distri- 1/3 bution is concentrated at distance R = aH /Z (screening distance), where 2 2 aH = ~ /mq is the and Z is the atomic number (Z 1); q and m denote the electron charge and mass, respectively. The atom≫ ic binding energy depends on R, and the atom exhibits an eigenmode re- lated to the change in R (a breathing-type mode), with an eigenfrequency 3 16 −1 ~ ω0 Z q / maH 4.5Z 10 s ( ω0 28Z(eV )). We recog- nize≃ in ω| the| plasma frequency≃ × 4πnq2/m of≃ a mean electron density 0 p n Z/R3 = Z2/a3 . It corresponds≃ to the atomic giant-dipole oscillations ≃ H discussed in Ref. [21]. In the presence of an external electric field E oriented along the z-direction the are displaced by u (with fixed nucleus), a displacement which produces an energy change z2u2/R2. By integrating ∼ over z, we get a factor 1/√3 in the eigenfrequency ω0, as expected. It follows that the displacement u obeys the equation of motion u¨ + Ω2u = qE/m, where Ω= ω /√3; the internal field is E = 4πnqu (polarization P = nqu 0 i − and the dipole moment p = Zqu). For E = E0 sin ωt the solution of this equation is u = u sin ωt, u = (qE /m)/(ω2 Ω2), and the internal field is 0 0 − 0 − E = Ω2E/(ω2 Ω2); the total elecric field acting upon the atomic nucleus is i − ω2 F = E + E = E sin ωt ; (1) i ω2 Ω2 0 − since ω Ω, we may use the approximation F (ω2/Ω2)E 10−3/Z2 (where ω≪= 1015s−1); we can see that the total field≃ − acting upon≃ the − nucleus is appreciably reduced by the electron screening. For Z = 50 this reduction factor is 4 10−7; the maximum field 3 1013V/cm is reduced to 107V/cm ( 104 electrostatic≃ × units). It follows that× we may limit ourselves here to low≃ fields acting upon the atomic nuclei. The case of strong fields have been analyzed in Refs. [8, 9, 18, 22, 23]. At the same time, an induced electric field generated by the dipolar eigenmodes occurs inside the atom, which oscillates with the higher eigenfrequency Ω. If the field is low, the bound-state charge oscillates, emits higher-order har- monics of electromagnetic radiation and tunneling may appear; in this lat- ter case, the charge accommodates itself in the field, in a long time, which amounts to an adiabatically-introduced interaction; this regime allows the usual, standard application of the tunneling approach. As we shall see be- low, the threshold field which separates the two regimes (low-field regime

3 2 from high-field regime) can be obtained from q E0/mω a = 1, where a is a distance of the order of the bound-state dimension| | (a = 10−13cm) (for protons, the threshold field is E 105V/cm (102 electrostatic units)). 0 ≃ Originally, the charge emission from bound states, like atom ionization, has been treated by using adiabatic hypothesis, either by time-dependent perturbation theory, or by imaginary-time tunneling, or other equivalent approaches.[24]-[32] Quasi-classical tunneling through the potential barrier generated by the field has been applied in classical works to static fields and the (in parabollic coordinates).[33]-[35] For alpha-particle de- cay or proton emission the situation is different. First, in spontaneous de- cay, the alpha particle (and, in general, the ejected charge) is preformed and, second, the tunneling through the Coulomb potential barrier must be included.[36]-[39] We analyze below the spontaneous charge emission, affected by the presence of an adiabatically-introduced electromagnetic radiation, in the presence of a Coulomb barrier; the problem may exhibit relevance for studies of alpha-particle decay or proton emission. The standard model of spontaneous alpha decay is based on Bohr’s con- cept of compound nuclei.[40] In an alpha-unstable nucleus the pre-formed alpha particle acquires a kinetic energy and penetrates (tunnels through) the Coulomb potential barrier. Consequently, the alpha-unstable nucleus is in fact in a "metastable state". In this simple model, the spontaneous alpha-particle decay and proton emission proceed by tunneling through the Coulomb potential barrier, as a result of many "attempts" the charge makes to penetrate the barrier. The (high) frequency of this process is of the or- der 1/ta, where ta corresponds, approximately, to the energy level spacing; in atomic nuclei this spacing, for the relevant energy levels, is of the order ∆ = 200keV , which gives t 3 10−21s;[40] also, the broadening of the E a ≃ × charge energy levels introduces an energy uncertainty (we leave aside the so- called tunneling through the internal potential barrier and the pre-formation factor of the alpha particle). The order of magnitude of the energy of the charge is a few MeV , which ensures a quasi-classical tunneling. The effect of the electromagnetic radiation upon the initial preparation of the charge for tunneling may be neglected. Similarly, we consider a sufficiently low electro- magnetic radiation, such that we may neglect its effects upon the mean-field potential. We limit ourselves to the effect of the electromagnetic interaction on the tunneling rate. Let us consider a charge q > 0 with mass m in the potential barrier V (r)

4 in the presence of an electromagnetic radiation with the vector potential A = A cos(ωt kr), where A is the amplitude of the vector potential, ω 0 − 0 is the radiation frequency and k is the radiation wavevector (ω = ck); the electromagnetic field is transverse, i.e. kA = 0. Since the phase velocity of the non-relativistic charge is much smaller than the speed of light c in vacuum, we may neglect the spatial phase kr in comparison with the tem- poral phase ωt; consequently, the vector potential may be approximated by A A0 cos ωt. This approximation amounts to neglecting the effects of the magnetic≃ field. It is assumed that this potential is introduced adiabatically. The charge is immersed in the radiation field, such that we may start with the standard non-relativistic hamiltonian 1 q 2 H = p A + V (r) , (2) 2m − c   where the momentum p includes the electromagnetic momentum qA/c beside the mechanical momentum mv, where v is the velocity of the particle. We consider the Schrodinger equation ∂ψ i~ = Hψ ; (3) ∂t since the interaction is time-dependent we need the time evolution of the wavefunction. Consequently, in equation (3) we perform the well-known Kramers-Henneberger transform[41]-[44] (with a vanishing electromagnetic interaction for t ) → −∞ ψ = eiSϕ , (4) q2A2 S = q A p sin ωt 0 (2ωt + sin 2ωt); ~mcω 0 − 8~mc2ω the Schrodinger equation becomes

~ ∂ϕ 1 2 i ∂t = 2m p ϕ + V (r)ϕ , (5) V (r)= e−iSV (r)eiS = V (r eqA sin ωt/mcω); − 0 it is conveniente to introduce the electric field E = E0 sin ωt, E0 = ωA0/c; we get q q2A2 S = E p sin ωt 0 (2ωt + sin 2ωt) (6) ~mω2 0 − 8~mc2ω 5 and V (r)= V (r qE/mω2) . (7) − We can see that for high-intensitye fields the potential (including the mean- field potential) is rapidly vanishing along the field direction. Here we assume 2 that the field intensity is low; specifically we assume qE0/mω a, where a is the dimension of the region the charge moves in (the atomic≪nucleus); for protons this inequality means E 3 104V/cm (102 electrostatic units), 0 ≪ × as stated above. The preformed alpha particle (or emitted proton) may tunnel through the potential barrier given by equation (7); the "attempt" frequency to penetrate the barrier and the energy uncertainty are practically not affected by the low-intensity field. We adopt a model of nuclear decay by assuming a Coulumb potential barrier V (r) Zq2/r (with the center-of-mass of the original nucleus placed at the ≃ origin); in the absence of the field the tunneling proceeds from r1 = a to 2 r2 = Zq / r, where r is the radial energy of the charge; it is convenient to introduce theE parameterE ξ = qE /mω2a 1, which includes the effect of the 0 ≪ field. In the presence of the field these limits become

r = a qE/mω2 (8) 1 − and r = r , where a = ar/r. We expand r in powers of ξ and get 2 2 e 1 1 e r = a 1 ξ sin ωt cos θ + ξe2 sin2 ωt sin2 θ + ... , (9) 1 − · 2 ·   where θ ise the angle the radius vector r makes with the electric field E0. To continue, we assume that the free charge attempting to penetrate the 2 potential barrier has momentum pn and kinetic energy n = pn/2m, where n is a generic notation for its state; we may leave aside theE orbital motion and denote by prn the radial momentum and by rn the radial energy. Let pr and 2 E r = pr/2m be the highest radial momentum and, respectively, the highest radialE energy; they correspond to the total momentum p and, respectiveley, total energy = p2/2m (in general, a degeneration may exist). This charge E may tunnel through the potential barrier V (r) from r1 to r2. The relevant factors in the wavefunction ψ given by equation (4) are

e iqE(t) i r2 e e cos θ·(p2−p1)+ ~ Re dr·pr(r) e ~mω2 r1 , (10)

6 where p (r) = 2m [ V (r)], p = p (r ) = 2m [ V (r )]; it is r E − 1,2 r 1,2 E − 1,2 easy to see that p2 =0. It follows that the tunneling probability (transmission p −γ p coefficient) is given by w = e , where e e γ = Aξ sin ωt cos θ + B, − · (11) 2a|p1| qE0 2 r2 A = , ξ = 2 , B = e dr p (r) ~ mω a ~ r1 r e | | R and p1 = 2m [V (r1) ], pr(r) = 2m [V (r) ] (the condition V (r1) > ensures| | the existence−E of the| bound| state). We expand−E the coefficient A in E p p powers of ξ and takee the average with respect to time; we get e

Zq2 2m γ = ξ2 cos2 θ + B... ; (12) − 2~ Zq2/a s −E the same procedure applied to the coefficient B leads to

2 2 B = γ aξ 2m(Zq2/a )+ aξ 2m (3Zq2/2a ) cos2 θ , 0 − 2~ −E 2~ Zq2/a−E −E p q (13) where γ0 corresponds to the absence of the radiation; finally, we get

aξ2 Zq2/2a γ = γ 2m(Zq2/a ) 1 −E cos2 θ . (14) 0 − 2~ −E − Zq2/a   p −E We can see that the effect of the radiation is to increase the rate of charge emission by a factor proportional to the square of the electric field (ξ2) and to introduce a slight anisotropy. It is worth noting that the radiation field contributes not only to the tunneling factor, as expressed by the coefficient B, but it is present also in the coefficient A, via the time-dependence of the wavefunction provided by the Kramers-Henneberger transform. We can define a total disintegration probability

aξ2 Zq2/2a w 1+ 2m(Zq2/a ) 1 −E w0 (15) tot ≃ 2~ −E − 3(Zq2/a ) tot    p −E 0 −γ0 by integrating over angle θ, where wtot = e . The disintegration rate per unit time is (1/τ)wtot, where τ is related to the time ta estimated above and the time introduced by the energy uncertainty.[40]

7 The exponent γ0, corresponding to the absence of the radiation, is Zq2 γ = 2m/ arccos a/Zq2 a/Zq2 1 a/Zq2 ; (16) 0 ~ E E − E −E p  p p p  since Zq2/a (for protons q2/a =2.5MeV ) , we may use the approximate formulae ≫E πZq2 γ 2m/ (17) 0 ≃ 2~ E and p 5aξ2 w 1+ 2mZq2/a w0 . (18) tot ≃ 12~ tot   As it is well know the interplay betweenp the very large values of 1/τ and the very small values of e−γ0 , makes the disintegration rate to be very sensitive to the energy values, and to vary over a wide range.[40] The result can be cast in the form of the Geiger-Nuttall law, which, in the absence of the 0 radiation, can be written as ln(wtot/τ) = a0Z/√ + b0, a0 and b0 being well-known constants;[40] the only effect of− the radiationE is to modify the 2 2 constant b0 into b = b0 + (5aξ /12~) 2mZq /a. The correction to b0 can also be written as (5ξ2/12)[(Zq2/a)/(~2/2ma2)]1/2 for ξ 1. The maximum p value of this correction is of the order of the unity; it follo≪ws that the decay rate is enhanced by the radiation by a factor of the order ξ2 1. ≪ After the emission of the charge, the mean-field potential suffers a recon- figuration (re-arrangement) process and the potential V (r) is modified; this is the well-known process of "core shake-up" (or "core excitation"); a new bound state is formed and a new transformation process may begin for the modified potential V (r). The tunneling probability w given above is a trans- mission coefficient (we can check that w < 1); with probability 1 w the − charge is reflected from the potential barrier; in these conditions the bound state is "shaken-up" and the charge resumes its motion, or its preformation process, untill it tunnels, or is rescattered back to the core; the latter is the well-known recollision process.[8],[45]-[49] The case of a static field requires a special discussion. Within the present formalism a static electric field E can be obtained from a vector poten- tial A = cEt; the position vector in the mean-field potential is shifted to r r +−ζ, where ζ = qEt2/2m; the special discussion is necessary be- → cause the parameter ζ is unbounded in time. The distance a is covered in −13 time t0 = 2ma/qE; for proton, a is of the order a = 10 cm and the p 8 4 2 threshold field is E = E0 = 3 10 V/cm (10 electrostatic units) given −15 × above; we get t0 10 s. This is a very long duration, in comparison with ≃ −21 the relevant nuclear times, in particular the attempt time τ (ta 10 s estimated above). In general, the condition of adiabatic interaction≃ reads t ~/∆ , where ∆ is the mean separation of the energy levels; it im- 0 ≪ E E plies qEa (∆ )2/(~2/ma2), which allows for high static fields. In these conditions≪ the protonsE accommodate themselves to the electric field, which is absorbed into slightly modified energy levels; this change, which can be estimated by perturbation theory, is irrelevant for our discussion, since the field strength is small. However, it has an important consequence in that the electric field, once taken in the energy levels, is not available anymore for the Kramers-Henneberger transform given by equation (4); therefore, the present time-dependent formalism cannot be applied. Instead of using the hamiltonian given by equation (2), we start with the (equivalent) dipole hamiltonian which includes the interaction term qEr. Consequently, the − potential barrier V (r) Zq2/r is changed into ≃ Zq2 Zq2 Er2 V (r)= qEr = 1 cos θ . (19) r − r − Zq   We compute the tunneling rate by using this potential barrier. In view of the small value of the correction parameter proportional to E in equation (19), we may expand the momentum p = 2m [ V (r)] in powers of r E − this parameter and replace the powers of r2 by their mean values over the 2 p tunneling range from r1 = a to r2 = Zq / ; since r2 a, we get the small 2 E ≫ parameter α = Er2/Zq = Eqr2/ 1 in equation (19). For Z = 100 and = 1MeV this parameter is α =E 10 ≪−4E, which is much smaller than unity Efor any usual static field. Integrating over angles and assuming αγ 1, 0 ≪ where γ0 is given by equation (17), we get finally α2γ2 w 1+ 0 w0 . (20) tot ≃ 108 tot   We can see that the correction brought by a static electric field to the decay rate is extremely small, as expected. Finally, it is worth discussing the case of intermediate fields, i.e. field 2 strengths which satisfy the inequality qE0/mω > a (ξ > 1) (in our case, fields from 3 104V/cm to 107V/cm).[50] In this case the adiabatic hypoth- esis cannot be× used anymore, and the initial conditions for introducing the

9 interaction are important. The corresponding Kramers-Henneberger trans- form diminishes appreciably the potential barrier and the charge is set free in a short time, which is the reciprocal of the decay rate; this rate may exhibit oscillations as a function of the field strength. In conclusion, we may say that in low-intensity electromagnetic radiation the bound-states charges accommodate themselves in the field, which amounts to an adiabatically-introduced interaction, as it is well known. In these con- ditions, besides oscillating and emitting higher harmonics, the charge may tunnel out from the bound state. This is the standard ionization process, which was widely investigated for atom ionization. In spontaneous alpha de- cay or proton emission the situation is different, because of the preformation stage and the tunneling through the Coulomb potential barrier. We have analyzed above the disintegration rate for the charge emission from atomic nuclei in the case of the adiabatic introduction of electromagnetic interac- tion, with application to nuclear alpha-particle decay and proton emission. Under these circumstances, it has been shown in this paper that the tunnel- ing rate (through Coulomb potential) is slightly enhanced by the presence of the radiation, by corrections whose leading contributions are of second-order in the electric field, with a slight anisotropy. Similar results are presented in this paper for static fields. Acknowledgements. The author is indebted to S. Misicu and the members of the Laboratory of Theoretical Physics at Magurele-Bucharest for many fruitful discussions. This work has been supported by the Scientific Research Agency of the Romanian Government through Grants 04-ELI / 2016 (Pro- gram 5/5.1/ELI-RO), PN 16 42 01 01 / 2016 and PN (ELI) 16 42 01 05 / 2016.

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