ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 9, с. 1326-1342
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
BETA DECAY AND OTHER PROCESSES IN STRONG ELECTROMAGNETIC FIELDS
©2011 E. Kh. Akhmedov*
Max-Planck-Institut fur Kernphysik, Heidelberg, Germany; Kurchatov Institute, Moscow, Russia
Received November 18, 2010
We consider effects of the fields of strong electromagnetic waves on various characteristics of quantum processes. After a qualitative discussion of the effects of external fields on the energy spectra and angular distributions of the final-state particles as well as on the total probabilities of the processes (such as decay rates and total cross sections), we present a simple method of calculating the total probabilities of processes with production of nonrelativistic charged particles. Using nuclear в decay as an example, we study the weak- and strong-field limits, as well as the field-induced в decay of nuclei stable in the absence of the external fields, both in the tunneling and multiphoton regimes. We also consider the possibility of accelerating forbidden nuclear в decays by lifting the forbiddeness due to the interaction of the parent or daughter nuclei with the field of a strong electromagnetic wave. It is shown that for currently attainable electromagnetic fields all effects on total в-decay rates are unobservably small.
1. INTRODUCTION
Study of quantum processes in intense electromagnetic fields is a very interesting subject. Strong external fields can help us to learn more about the properties of the involved particles and their interactions. Studying processes in strong fields may also have interesting implications for astrophysics and cosmology. Recently, there has been a renewed interest in this topic in connection with development of new powerful laser sources.
In this paper I will discuss effects of strong external electromagnetic fields on various characteristics of quantum processes. In Section 2 a rather general qualitative analysis of these effects is given, whereas Sections 3 and 4 are dedicated to a specific example — nuclear [ decay in the field of a strong electromagnetic wave.
My interest in this topic was raised in the early 1980s by A.M. Dykhne, who called my attention to a paper published in the Physical Review Letters [1]. It was claimed in that paper that under the influence of electromagnetic fields of existing at that time powerful lasers [ decay of tritium can be significantly accelerated. Simple estimates I made did not confirm this conclusion, but at the same time I could not pinpoint a mistake in the calculation done in [1]. The problem was that the calculation was very complicated and difficult to follow. It was based on the standard at that time procedure of infinite summation of partial probabilities corresponding to
E-mail: akhmedov@mpi-hd.mpg.de
absorption from the external wave (or emission into it) of all possible numbers of photons. This motivated me to look for a simpler way of calculation of the total probabilities of quantum processes in the fields of intense electromagnetic waves. My quest was strongly supported by Victor Khodel, a colleague of mine at the Kurchatov Institute, who used to say that "if there is a simple result, there must exist a simple way of obtaining it".
Eventually, I found a very simple way of calculating the total probabilities of quantum processes with emission of nonrelativistic charged particles. The method in particular allowed one to easily study all interesting limiting cases — the weak- and strong-field limits as well as the case of the field-induced decay of a particle (or a nucleus) that is stable in the absence of the external field, both in the tunneling and multiphoton limits. The results were published in [2, 3].
While I was working on this subject, several papers appeared [4—6], where the problem of [ decay in strong electromagnetic fields was re-investigated and it was shown that the results of [1] were erroneous. The analysis in [4, 6] was based on the same old summation technique, whereas the approach in Voloshin's paper [5] was close in spirit (though not identical) to the one I was developing. In Sections 3 and 4 I will discuss the method of [2, 3] using nuclear [ decay as an example, but it can actually be applied to a much wider class of problems.
Gratifyingly, in all considered limiting cases the results of direct calculations of the probability of
¡3 decay in the field of a strong electromagnetic wave [2, 3] agreed perfectly well with my previously made estimates, often even including the numerical coefficients. I learned a great deal about how to analyze physics problems qualitatively and how to make simple estimates from A.B. Migdal, both from my personal interactions with him and from his splendid book [7]. It is therefore a great pleasure and honor for me to write this article for the special issue of Yadernaya Fizika dedicated to the centennial anniversary of A.B. Migdal's birthday.
2. QUALITATIVE CONSIDERATIONS
Consider quantum processes such as
A + B — A + B (scattering),
A + B — C + D + ... (reactions),
A — B + C + ... (decay). How can external electromagnetic fields influence these processes? They can:
1) modify the differential characteristics of the process (i.e. the energy spectra and angular distributions of final-state particles);
2) affect the total probabilities of the processes, such as total cross sections and decay rates;
3) finally, new channels of reactions or decays, which were not available in the absence of the external fields, may open up.
Let us discuss qualitatively all these possibilities in turn.
We will now make several assumptions that will be used throughout this paper. First, it turns out that the smaller the characteristic energies of the charged particles, the stronger the effects of external electromagnetic fields on the processes in which they are involved. For this reason we shall consider processes with nonrelativistic charged particles. We will assume that the system is subjected to the field of a monochromatic electromagnetic wave of frequency u and electric field strength E, and that the field is a low-frequency one:
hw < £q,
(1)
be considered as a time-dependent uniform field E(t). Such a field can be conveniently described either in the Coulomb gauge
A^(t, x) = (0, A(t)), E(t) = -
1 OA (t) c dt
(2)
where e0 is the characteristic energy of the process. The low-frequency limit very often also means that the wavelength of the photons of the external field c/u is large compared to the characteristic length of the process lx (such as, e.g., the atomic size in photo-ionization processes or the nuclear radius in nuclear 3 decay): ulx/c < 1. The assumption that all the participating charged particles are nonrelativistic allows us to concentrate only on effects of the electric component of the external field, whereas the condition ulx/c < 1 implies that we can adopt the dipole approximation, in which the external electric field can
with the coordinate-independent vector-potential A(t), or in the scalar gauge
A»(t, x) = ($(t, x), 0), (3)
0(t, x) = -E(t)x, E(t) =
In different situations different gauges turn out to be most convenient for calculations; we will use the Coulomb gauge in Section 3 and the scalar gauge in Section 4.
2.1. Differential Characteristics
Strong external fields can modify energy spectra and angular distributions of particles produced in scattering, reaction, or decay process. This happens due to the absorption by a charged particle of photons of the external field (or stimulated emission of photons into it).
Free on-shell particles cannot absorb or emit photons due to energy—momentum conservation. However, a particle that undergoes a scattering which changes its momentum, or is produced in some process (such as an electron production in 3 decay or emission of an electron from an atom due to photo-ionization or electron-impact ionization) can exchange photons with the external field.
Let as a result of some process a nonrelativistic particle of charge e and mass m be produced, and let its kinetic energy be e < e0, where eo is the energy release in the process, i.e. the maximum kinetic energy available to the particle under consideration. The particle can receive some energy from the external field or give to the field a fraction of its energy. Let us estimate the corresponding energy Ae. The exchange of the energy between the particle and the field effectively takes place during a characteristic time of order of the period of oscillations of the external field: ichar ~ 1/u (the contribution of an integer number of full field periods T = 2n/u averages to zero). Therefore the momentum that the particle can obtain from the field is of order
Ak = eEotchar '
eEo
w
(4)
where E0 is the amplitude of the electric field strength. For particles with a characteristic energy e0 (i.e. with the characteristic momentum = ^2meo) we have
Ak _ eEo _ ^ ^
ko ^2meouo
This parameter characterizes, in particular, the modification of the angular distribution of the produced charged particle. If £ is not too large, during the characteristic time tchar the particle moves over the distance I ~ t>oiChar = sj2eo/mtchar• The energy obtained by the particle from the external field is just the work of the field on the particle over the distance l, which gives
Ae
£o
eE01 eE0
£0
£0
'2£o m to
= 2£. (6)
This result is only valid assuming that £ < 1;for £ » 1 one has to take into account that the velocity of the particle increases with time and is no longer equal to its original velocity v0. In this case
l — votchar +
eEo t
char
(7)
m 2
where eE0/m is the particle's acceleration in the external field. This yields
Ae eE0l
£o
(8)
AT A£
N0 ~ — ~ .
hu hu
(9)
In the considered low-frequency limit (1) it can be very large even for not too strong fields, when £ is relatively small.
The above estimates should be taken with some caution, though. The paramete
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