Once again on the duration of nuclear gamma-ray-emission and gamma-ray-absorption processes
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CLEI Theory
Once Again on the Duration of Nuclear Gamma-Ray-Emission and Gamma-Ray-Absorption Processes A. V. Davydov* Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya 25, Moscow, 117218 Russia Received June 24, 2010
Abstract—In addition to previous indications of a long-term character of the processes of gamma-ray emission and absorption by nuclei, another two simple arguments in support of this picture are presented. It is shown that the Fourier integral for a short wave train is a frequency distribution whose width is many orders of magnitude larger than actual intrinsic widths of gamma lines. The uncertainty in the photon spatial position is found to be about τ c, which is the length of the wave train emitted by a nucleus over the average lifetime τ of this nucleus in an excited state. DOI: 10.1134/S1063778811010030
In his monograph [1], A.B. Migdal states that a gamma ray is emitted within a time of about λ ¯/c, where λ ¯ is the gamma-radiation wavelength divided by 2π. For gamma rays of energy about 100 keV, this time is about 6.6 × 10−21 s. There are, however, a number of experimental facts and theoretical conclusions that are not consistent with this statement. Earlier, the present author showed [2] that a quantum-mechanical calculation aimed at determining, on the basis of the assumption that the time of photon absorption by a nucleus is short, the mean lifetime of an excited state of a nucleus that ¨ occurred in this state upon the Mossbauer resonance absorption of gamma rays leads to a value that does not agree with its experimental counterpart. In [3], I indicated that the absence of a large broadening ¨ of a Mossbauer gamma line in experiments devoted to observing a gamma resonance in the long-lived isomer 109m Ag is incompatible with the assumption that the time of radiative nuclear processes is much shorter than the characteristic lifetime of a nucleus in an excited state. Another two simple arguments in support of the statement that photon absorption and emission by nuclei are long-term processes are presented in this article. Let us first consider the Fourier frequency spectrum of an extremely short wave train of a photon. If, in accordance with Migdal, we assume that a gamma ray is emitted within the time λ ¯/c, then the corresponding wave train must have a duration of about 1/6 of the period T ; that is, it must be proportional to *
sin ω0 t, where 0 < t < T /2π. We will now calculate the Fourier integral for a signal whose duration is equal to one period of oscillations. We have 1 f (ω) = 2π
T sin(ω0 t)eiωt dt
(1)
0
=
1 2π
T
eiω0 t − e−iω0 t iωt e dt 2i
0
T
1 ei(ω0 +ω)t − ei(ω0 −ω)t dt 4πi 0 ⎡ ⎤ ω ω exp 2πi exp 2πi − 1 − 1 ω0 ω0 1 ⎦ − =− ⎣ 4π ω + ω0 ω − ω0 exp 2πi ωω0 − 1 ω 0 . = 2π ω 2 − ω02 =
The squared modulus of this amplitude has the form 1 (2) |f (ω)|2 = 2 2 2π ω0 −2
ω ω 2 −1 1 − cos 2π . × ω0 ω0 The graph of this function is shown in the figure. The width of the resulting frequency spectrum (this width is readily transforma
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