MECHANISM OF "GSI OSCILLATIONS" IN ELECTRON CAPTURE BY HIGHLY CHARGED HYDROGEN-LIKE ATOMIC IONS
V. P. Krainov*
Moscow Institute of Physics and Technology 141700, Dolgoprudny, Moscow Region, Russia
Received January 11, 2012
We suggest a qualitative explanation of oscillations in electron capture decays of hydrogen-like 14uPr and 14JPm ions observed recently in an ion experimental storage ring (ESR) of Gesellschaft fur Schwerionenforschung (GSI) mbH, Darmstadt, Germany. This explanation is based on the electron multiphoton Rabi oscillations between two Zeeman states of the hyperfine ground level with the total angular momentum F = 1/2. The Zeeman splitting is produced by a constant magnetic field in the ESR. Transitions between these states are produced by the second, sufficiently strong alternating magnetic field that approximates realistic fields in the GSI ESR. The Zeeman splitting amounts to only about 1CT'J eV. This allows explaining the observed quantum beats with the period 7 s.
1. INTRODUCTION
The authors of Rof. fl] reported recently 011 time-modulated weak decays observed in the orbital electron capture of hydrogen-like 140Pr58+ and 142Pni6o+ ions (these ions have odd odd nuclei) coasting in the ion experimental storage ring (ESR) in Gesellschaft fur Schwerionenforschung (GSI) mbH, Darmstadt, Germany. Using a nondestructive single-ion time-resolved Schottky mass spectrometry, they found that an exponential decay is modulated in time with a modulation period of about 7 seconds for both ions. The authors of Ref. fl] attributed this observation to a coherent superposition of finite -mass eigenstates of the electron neutrinos from the weak decay into a two-body final state. This idea was developed in Ref. [2], where time modulation was explained in terms of the interference of two massive-neutrino mass eigenstates. But it was concluded in Ref. [3] that the decay rate measured at GSI cannot oscillate using approach in Ref. [2] if only standard physics of the weak interaction is involved.
Further, another explanation was proposed in [4]. It is based 011 a mechanism related to Rabi multiphoton oscillations between atomic hyperfine levels that can offer such a period. The Rabi multiphoton frequency-was derived using the multiphoton perturbation theory.
E-mail: vpkrainov'fi'mail.ru
Fig. 1. Hyperfine splitting of the ground Is state for odd-odd nuclei 140 Pr.™ and 142Pm(ii
Only a weak perturbing oscillating (magnetic) field was considered in [4]. Here, we extend this approach to a strong perturbing field based 011 our previous adiabatic approximations for a two-level system [5,6]. The nuclear spin of odd odd nuclei 14OPr50 and 142Pm6i is equal to one. The energy of the hydrogen-like l,s state with the nuclear charge Z = 59 is equal to
E = me2 \/l — (Za)'2 — me2 « ^52 keV.
This level is splits (Fig. 1) into two hyperfine levels with the total angular momentum F = 1/2 (lower) and F = 3/2 (higher). The electron capture decay from the F = 3/2 state is forbidden because the fully ionized daughter nucleus has the spin 1 = 0. 50% of
In the GSI storage ring, ultrarelativistic atomic ions circulate in the plane perpendicular to the strong magnetic field about H0 = 0.1 T. This is an average value of the field because the actual value varies. The radius of the circle in the ring is r = 15 in. The Zeenian splitting of the lower state F = 1/2 is hcoo — 2/ieiJo — _ j 210~5 eV, where yUg IS 9.11 electron Bohr magneton. Hence, tJo ~ 10lol/s. The lower state is (1/2,-1/2) and the upper state is (1/2,+1/2). The second oscillating magnetic field in the ESR produces transitions between these states (the focusing of the ion beam occurs due to several quadrupole magnets). The oscillating magnetic field hsuiuit is directed in the plane perpendicular to the constant magnetic field. The field has the frequency of the order of ui ~ Nc/r ~ 10B l/'s. Here, N ~ 100 is the possible number of magnetic devices on the ring. Hence, ujo/uj ~ 10.
Fig. 2. The number N of electron capture decays of hydrogen-like 14uPr.-)11 ions per second as a function of the time t after the injection into the experimental storage ring [1]. The inset shows the fast fourier transform of these data
the decay of the gfoPr50 nucleus is the Ganiov Teller 1+ 0+ /?+-decay:
140 81
Pr,
■59
140 82
Co,
'.58
The other 50% of the decay of the gfoPr50 nucleus is an electron capture:
140 81
Pr,
■59
140 82
Co,
'.58
The relativistic hyperfine splitting of the l.s1(/2 state with the charge Z for nuclear spin 1 is [7]
AE =
2 Z3
[i Ry'"
y/1 - (Za)'2 [2^1 - (Za)2 - l] MP(2
where //.¡y is the nuclear magneton and //. = +0.88//¡y is the magnetic moment of the odd odd nucleus with nuclear spin 1. In the case of the gfoPr50 nucleus, this splitting is AE = 0.4 eV. The Ml spontaneous gamma-decay 3/2+ —¥ 1/2+ occurs during the short time
9 H3 e2 (trie2 \ 3 _
4 m> 1 In- \ AE J
= 0.03 s,
and we should therefore discuss electron capture from the lower hyperfine state F = 1/2.
The experimental data in Ref. fl] are shown in Fig. 2. The number of electron capture decays of hydrogen-like 140Pr ions per second is given as a function of time. The case of the 142Pm6i ion is similar.
2. ADIABATIC APPROXIMATION FOR A TWO-LEVEL SYSTEM IN A STRONG OSCILLATING FIELD
The goal of this section is to investigate the interaction of a strong classical magnetic field with a two-level system in the case of a multiphoton resonance [5]. The basic assumption is the smallness of the frequency of the field in comparison with the separation between levels (in atomic units), i.e., u; -C uio- Of course, the well-known results of time-dependent perturbation theory are not applicable for strong alternating fields. We use the adiabatic approximation to calculate the rate for transition from the lower state (+1/2) into the upper state (— 1 /2). As is well known, it is mathematically equivalent to the WKB approximation for the problem of above-barrier reflection. We direct the constant magnetic field producing the Zeenian splitting along the i axis. We also first direct the oscillating magnetic field hs'm(uit) along the x axis.
We seek eigenstates of the adiabatic Schrodinger equation
H(№(t) = E(№(t),
H(t) = fic(T~H0 + ii.fj h sinU/).
where ar and <r- are Pauli matrices. The wave function is presented in the form of a superposition of the unperturbed lower and upper states
=
Wo obtain the following equations for u\ and и2:
E(t)ui = E(t)u2 =
huln
huln
Ui + fichsm(u)t)a2, «2 + //c/isin(u>i)ui.
(1)
From system (1), we find that the energy eigenvalues are given by
Eh2(t) = yl-rshru/)
where we introduce the dimensionless quantity
4 =
2 fi.ch h
huin
#0
(2)
(3)
Similarly to the problem of above-barrier reflection, we are only interested in the complex turning points t^ that lie in the upper half-plane. They are determined from the condition ) = ¿^(ifch whence
ti< =
kn
и)
iarcshi
u>
'/
к = 0, ±1, ±2,... (4)
These points are the fundamental branch points for E1}2(t).
We first consider the point to- According to [8,9], we obtain the following result for the contribution to the transition probability introduced by the point t0 (k = 0):
«>12 = exp I -21m j [E2(t) - Et(t)] dt > . (5)
Here, ti is an arbitrary point on the real time axis. Evaluating the integral, we find
R = exp ■
«>12 = R UJ0
uVl + Ч2
D
(6)
Г
where D(x) is the complete elliptic integral of the third kind. It follows that the quantity wi2 is exponentially small in terms of the adiabatic parameter uio/ui 1. Referring to this formula, we also emphasize that the unit pro-exponential factor is exact.
Taking the turning points with k ^ 0 into account (they all lie at the same distance from the real time axis) allows passing from absolute probabilities to probabilities per unit time (rates). According to the principle of superposition of quantum mechanics [10], the total amplitude A12 for a transition into the upper state is a sum of the amplitudes a/, associated with the individual turning points ifc. The resonance condition
is that these amplitudes be added together coherently. The phase factor cxp(iS) appears in the amplitude a/, in connection with the transition from a given turning point to the next. Here, the quantity
S = —
u>
1 + q2 sirr if dip
(7)
represents the accumulation of the classical action between neighboring turning points. The minus sign appears in front of the exponential in the phase factor because the WKB wave contains the factor [£7(#)]1/2 ; it changes sign during the transition from the point t^ to the point ffc+i. It follows from Eq. (7) that
s = — \/Y
u)
■ q2 E
•r
(8)
where E(x) is the complete elliptic integral of the second kind. The quantity S is a semiclassical phase (classical action) taken 011 the temporal interval [0, Tr/ul].
Summing the amplitudes from the AT turning points and taking the absolute value squared of ,4 12) W6 find
sin [N(S Л12 —'-'-'12"
тг)/2]
sin2 [(5 — тг)/2] The resonance condition has the form
(5 •— тг)/2 = mn + 7, 7 —^ 0,
(9)
(10)
where m is an integer. The time interval T is related to AT by the formula T = ttAt/ui. Finally, we express the transition rate as
W12 =
\A
12
2t^2
T
x S
UJo_ nul
ж
J \j 1 + q2 sin2 if dip — Kuj
(H)
where K = 2m + 1. Therefore, the transition only occurs in odd harmonics. The ¿-function in Eq. (11) expresses the energy conservation law, and its argument contains the magnitude of the energy shift due to the external field (the Stark effect).
We next proceed to an analysis of the transition rate W'i-2- If Kq2 -C 1, we use Eq. (11) to obtain the transition rate in the perturbation theory:
W12 =
2t^2
2 A"
6( U!Q — Kul)
(12)
We see that the perturbation theory validity criterion with regard to the transition rate is more stringent than
with regard to the Stark shift. On the other hand, we write the well-known expression for the transition rate obtained by direct application of tini
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