LimRank
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Datasets and Models for the LimRank paper.
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J = 2 state to the predominantly e 3 − wave functions) and are labeled for a given J by F 3 , F 2 , and F 1 starting from the highest to the lowest energy corresponding to J = N − 1, N, N + 1. From the mixing coefficients it is then possible to derive the lifetimes of the interacting levels. Unperturbed lifetimes of the A 1 1 and the e 3 − states are 10 ns and 5 μs, respectively. 58 The lifetimes of mixed states are given by 1 τ = 1 − (δ 2 + ε 2 ) 5 μs + δ 2 + ε 2 10 ns . The derived lifetimes are also given in Table III. The lifetimes for F 1 and F 3 levels are very similar, which is due to the strong spin-spin interaction between 1 (e 3 − ). The lifetimes of the F 2 levels is expected to be shorter (1.15 μs for J = 1 and
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two state, in other words, for any operator A, and any physical quantity is calculated by trace, Tr ρA = A 11 ρ 11 + A 22 ρ 22 . Here ρ 11 and ρ 22 are 1/2 for this case of π/2 pulse. There is one thing to be noted here. The excited state in this case is not exactly a simple state. Exactly it consists of combination of states with energy around E 2 . Especially the lifetime δt of the state around the hydrogen atom gives the range of energy contributing to this "state". This is determined by l/v g = lm/hk, where l is Bohr radius or the range of the electron localized in the ground state. We assumed here that the energy ambiguity coming from the finite duration of πpulse is negligible. Numerically, if E 2 is 100 eV, k is 4.9x10 10 /m and δt is 1.7x10 -17 s or 17 atoseconds. v g is 5x10 6 m/s and T is as large as 10 12 years for L of 137x10 8 light year. If we compromise L to be 1 light year, then T is rather short (about 60 years). Discussions The result illusturated by eq. (1) shows that the superposition of states is not lost. What is lost is the coherence, or, more precisely, the possibility of interference of two states. Therefore, decoherence is the loss of literal coherence and not the superposition. The loss of superposition is just apparent. When we observe the hydrogen atom, in other
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, n y = 1) will escape through the same leaky 1D level, n x,s , which remains the favoured decay channel as long as it is occupied. Therefore the total probability for one electron to escape from the occupied states with quantum number n x,s is the probability for a single 1D electron with energy E nx,s , multiplied by the number of electrons in occupied states with the same quantum number n x,s : q nx,s : and when the occupation q nx,s of the leaky level is constant, the linear variation of log 10 (τ ) reflects that of the lifetime of the leaky level. This is where the 2D nature of the quantum dot asserts its presence, even though the decay appears to proceed only in one dimension. In Fig. 7 we show the occupations of the two levels with the shortest lifetimes. One sees that when Q < 300 the occupation of the n x = 13 level stays practically constant and n x = 14 level remains empty. For higher values of Q both levels contribute significantly to the escape lifetime. In this situation: This is shown as the dotted curve in Fig. 6, and it accounts very well for the trend of the lifetimes predicted by the separable model. Our separable model favours the appearance of the linear decay sequences, because of the degeneracy in lifetime of states with the same n x,s . A non-separable model would lift that degeneracy and then the lifetime sequences should show a
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provide ballpark numbers of the lifetimes of different excited states. The lifetime of an electron in an S2 state is typically on the order of 10-15 second. The lifetime of an electron in an S1 state depends on the energy levels involved. For a $$\pi$$-$$\pi$$* system, the lifetimes range from 10-7 to 10-9 second. For a n-$$\pi$$* system, the lifetimes range from 10-5 to 10-7 second. Since $$\pi$$-$$\pi$$* molecules are more commonly studied by fluorescence spectroscopy, S1 lifetimes are typically on the order of 10-8 second. While this is a small number on an absolute scale of numbers, note that it is a large number compared to the lifetimes of the S2 state. The lifetime of a vibrational state is typically on the order of 10-12 second. Note that the lifetime of an electron in the S1 state is significantly longer than the lifetime of an electron in a vibrationally excited state of S1. That means that systems excited to vibrationally excited states of S1 rapidly lose heat (in 10-12 second) until reaching S1, where they then “pause” for 10-8 second. Transition 4 (Fluorescence) The transition labeled (4) in Figure $$\PageIndex{4}$$ denotes the loss of energy from S1 as radiation. This process is known as fluorescence. $\mathrm{S_1 = S_0 + h\nu}$ Therefore, molecular fluorescence is a term used to describe a singlet-to-singlet transition in a system where the chemical species was first excited by absorption of electromagnetic radiation. Note that the diagram in Figure $$\PageIndex{4}$$ does not show molecular fluorescence occurring from the S2 level. Fluorescence
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Q value, the 2 1∕2 level with a binding energy of = 3727 eV can also contribute to the transition. This level is located above the endpoint by 37 ± 190 eV. Unfortunately, the accuracy of measurement of the Q value does not allow to make an unambiguous conclusion about the position of the 2 1∕2 level relative to the endpoint. Here we consider two options: the subthreshold level for EC = 2690 eV and the resonance level for EC = 2730 eV. These Q values agree with each other within the experimental error, however, in relation to the energy spectrum near the endpoint the physical consequences turn out to be quite different. Calculations for EC = 2690 eV lead to a total halflife of 1∕2 = 3.8 × 10 4 years. Table 2 shows half-lives of the specific dominant channels. The experimental error in EC introduces about 10% uncertainty in these estimates. The possible contribution of the 2 1∕2 level to the total probability is small due to the smallness of the phase-space volume near the endpoint. For EC = 3730 eV and = 0 the partial half-life of the level can be estimated to be 1∕2 = 4.8 × 10 11 years. The forbidden transition is accompanied by a change in the nuclear spin by three units without a change in parity. This is the 2nd FU transition; its shape function has the form: for capture from the shells in units of years for the allowed 111 In(9∕2 + ) → 111 Cd(7∕2 +
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and factored out 1/(10 nm)2. Generally, the energy levels are not degenerate, i.e. all energies are different. However, some energy levels with different quantum numbers coincide, if the lengths along two or three directions are identical or if their ratios are integers. In our cubic QD case, all three lengths are identical. Consequently, we expect the following degeneracies: • $$E_{111}$$ = 0.43388 eV (ground state) • $$E_{112}$$ = $$E_{121}$$ = $$E_{211}$$ = 0.86776 eV = $$2 E_{111}$$ • $$E_{122}$$ = $$E_{212}$$ = $$E_{221}$$ = 1.30164 eV = $$3 E_{111}$$ • $$E_{113}$$ = $$E_{131}$$ = $$E_{311}$$ = 1.59090 eV = $$11/3 E_{111}$$ • $$E_{222}$$ = 1.73552 eV = $$4 E_{111}$$ • $$E_{123}$$ = $$E_{132}$$ = $$E_{213}$$ = $$E_{231}$$ = $$E_{312}$$ = $$E_{321}$$ = 2.02478 eV = $$14/3 E_{111}$$ • $$E_{333}$$ = 3.90493 eV = $$17/3 E_{111}$$ nextnano++ numerical results for a 10 nm cubic quantum dot with 0.50 nm grid spacing (The grid spacing is rather coarse but has the advantage that the calculation takes only a few seconds.): num_ev: eigenvalue [eV]: (0.50 nm grid) 1 0.432989 = E111 2 0.862425 (three-fold degenerate) E112/E121/E211 3 0.862425 (three-fold degenerate) E112/E121/E211 4 0.862425 (three-fold degenerate) E112/E121/E211 5 1.291860 (three-fold degenerate) E122/E212/E221 6 1.291860 (three-fold degenerate) E122/E212/E221 7 1.291860 (three-fold degenerate) E122/E212/E221 8 1.566392 (three-fold degenerate) E113/E131/E311 9 1.566392 (three-fold degenerate) E113/E131/E311 10 1.566392 (three-fold degenerate) E113/E131/E311 11 1.721296 = E222 12 1.995828 (six-fold degenerate) E123/E132/E213/E231/E312/E321 13 1.995828 (six-fold degenerate) E123/E132/E213/E231/E312/E321 14 1.995828 (six-fold degenerate) E123/E132/E213/E231/E312/E321 15 1.995828 (six-fold degenerate) E123/E132/E213/E231/E312/E321 16 1.995828 (six-fold degenerate) E123/E132/E213/E231/E312/E321 17 1.995828 (six-fold degenerate)
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state [38]. The highest-lying 5d 10 6s S 2 1/2 Pt − bound state, which has a different electron configuration from that of the ground state, was measured to lie 1.275 67(161) eV above the Pt − ground state [39]. Both excited states are expected to be long-lived since E1 transitions to the ground state are forbidden by the parity selection rule. Thøgersen et al. [38] used multi-configurational Dirac-Fock methods to calculate a spontaneous decay rate of Γ = 14 s −1 for the 5d 9 6s 2 D 2 3/2 → 5d 9 6s 2 D 2 5/2 transition in Pt − , corresponding to a lifetime of τ = 71 ms. It is interesting to note that a scaling of the τ = 15.1 ± 0.4 s experimental lifetime of the 3d 9 4s 2 D 2 3/2 level in Ni − [36] with the third power of the ratio between the corresponding transition energies in Ni − and Pt − suggests a lifetime of the 5d 9 6s 2 D 2 3/2 level in Pt − of τ = 53 ± 2 ms [40]. To the best of our knowledge, no calculation of the 5d 10 6s S 2 1/2 → 5d 9 6s 2 D 2 3/2,5/2 transition rates in Pt − , which are the main quantities of interest in the present study, has been reported in the literature. II. EXPERIMENTAL APPARATUS The present experiment was conducted at the cryogenic electrostatic ion beam storage ring DESIREE. A full description of this
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this state can only decay in a different way: by internal conversion, by emission of 2 photons or by the emission of an e+ e− -pair, if this last is energetically possible. Parity conservation does not permit internal conversion transitions between two levels with J = 0 and opposite parity. The lifetime of excited nuclear states typically varies between 10−9 s and −15 s, which corresponds to a state width of less than 1 eV. States which 10 can only decay by low energy and high multipolarity transitions have considerably longer lifetimes. They are called isomers and are designated by an “m” superscript on the symbol of the element. An extreme example is the second excited state of 110Ag, whose quantum numbers are J P = 6+ and excitation energy is 117.7 keV. It relaxes via an M4-transition into the first excited state 38 3 Nuclear Stability (1.3 keV; 2− ) since a decay directly into the ground state (1+ ) is even more improbable. The half life of 110Agm is extremely long (t1/2 = 235 d) [Le78]. Continuum states. Most nuclei have a binding energy per nucleon of about 8 MeV (Fig. 2.4). This is approximately the energy required to separate a single nucleon from the nucleus (separation energy). States with excitation energies above this value can therefore emit single nucleons. The emitted nucleons are primarily neutrons since they are not hindered by the Coulomb threshold. Such a strong interaction process is clearly preferred to γ-emission. The excitation spectrum above the threshold for particle emission
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of the coupling as we might expect. Indeed it turns out that g = g 1 + g 2 , where g 1 / h and g 2 /h are the escape rates introduced in the last Section. This comes out of a full quantum mechanical treatment, but we could rationalize it as a consequence of the "uncertainty principle" that requires the product of the lifetime ( = h/ g) of a state and its spread in energy ( g ) to equal h [3]. Another way to justify the broadening that accompanies the coupling is to note that the coupling to the surroundings makes energy levels acquire a finite lifetime, since an electron inserted into a state with energy E = e at time t = 0 will gradually escape from that state making its wavefunction look like This broadens its Fourier transform from a delta function at E = e to the Lorentzian function of width g = h/ t centered around E = e given in Eq.(4.2). There is thus a simple relationship between the lifetime of a state and its broadening: a lifetime of one picosecond (ps) corresponds to approximately 1.06e-22 joules or 0.7 meV. In general the escape of electrons from a level need not follow a simple exponential and the corresponding lineshape need not be Lorentzian. This is usually reflected in an energy-dependent broadening g(E) . The coupling to the contacts thus broadens a single discrete energy level into a continuous density of states given by Eq.(4.2) and we can
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# Transition between excited states 1. Nov 3, 2013 ### gildomar 1. The problem statement, all variables and given/known data An atom in an excited state has a lifetime of 1.2 x 10 -8 sec; in a second excited state the lifetime is 2.3 x 10 -8 sec. What is the uncertainty in energy for the photon emitted when an electron makes a transition between these two levels? 2. Relevant equations $\Delta$E$\Delta$t$\geq$$\frac{\hbar}{2}$ 3. The attempt at a solution So I just found the uncertainty in energies for the two excited states using the uncertainty principle, getting 2.74*10^-8 eV for the 1.2*10^-8 sec state, and 1.43*10^-8 eV for the 2.3*10^-8 sec state. And figured that the uncertainty in energy would just be the difference in the energies, giving 1.31*10^-8 eV. But the book gives an answer of 4.17*10^-8 eV, which I noticed is what you get if you add the energies instead. So is the book wrong, or is there some weird thing about the uncertainties combining such that I have to add them instead? 2. Nov 4, 2013 ### Basic_Physics $\Delta E \Delta t \geq \frac{\hbar}{2}$ that $\Delta E \geq \frac{\hbar}{2 \Delta t}$ ? 3. Nov 4, 2013 4. Nov 4, 2013 ### gildomar @Basic_Physics: Yes, that's how I got the energies. @gneill: That's a rule from statistics? Cause the book was a bit sparse on that point. 5. Nov 4, 2013 ### Staff: Mentor Yup. Consider that subtraction is just adding the negative of one of the values. The uncertainty in the negative value is
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radiative lifetimes of the excited states. Based on the values in Table II, we choose to consider two suitable initial excited electronic states: 5f 6d 2 at 30223 cm -1 with J = 15/2 and the 7s 2 7p at 31626 cm -1 with J = 1/2. The IC rates for these two configurations are presented in Tables III and IV. The excited state 5f 6d 2 has no electric-dipole decay channels; its calculated radiative lifetime is 0.4 s. IC from the Internal conversion rates for the state 5f 6d2 at 30223 cm −1 of 229m Th + as a function of the possible isomeric transition energy. For comparison, the half-life of the electronic excited state was calculated to be 0.4 s. 5f 6d 2 state becomes possible provided that the isomeric state energy would be higher than 9.0 eV, in which case the characteristic decay time becomes considerably less than the level lifetime. Provided that the isomeric energy indeed lies higher than 7.8 eV as speculated at present, this opportunity seems to be unique, as the other states at such high excitation energies typically decay very fast (see Table II, the case for the 7s 2 7p state at 31626 cm -1 ). On the other hand, in case the isomer energy is higher than 12 eV, the excitation method would no longer be applicable since the ground state 6d 2 7s electrons could undergo IC. We are therefore focusing on the range of 9.0 eV to 12.0 eV isomeric state energy. We note here
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# Transition between excited states ## Homework Statement An atom in an excited state has a lifetime of 1.2 x 10 -8 sec; in a second excited state the lifetime is 2.3 x 10 -8 sec. What is the uncertainty in energy for the photon emitted when an electron makes a transition between these two levels? ## Homework Equations $\Delta$E$\Delta$t$\geq$$\frac{\hbar}{2}$ ## The Attempt at a Solution So I just found the uncertainty in energies for the two excited states using the uncertainty principle, getting 2.74*10^-8 eV for the 1.2*10^-8 sec state, and 1.43*10^-8 eV for the 2.3*10^-8 sec state. And figured that the uncertainty in energy would just be the difference in the energies, giving 1.31*10^-8 eV. But the book gives an answer of 4.17*10^-8 eV, which I noticed is what you get if you add the energies instead. So is the book wrong, or is there some weird thing about the uncertainties combining such that I have to add them instead? Related Introductory Physics Homework Help News on Phys.org $\Delta E \Delta t \geq \frac{\hbar}{2}$ that $\Delta E \geq \frac{\hbar}{2 \Delta t}$ ? gneill Mentor @Basic_Physics: Yes, that's how I got the energies. @gneill: That's a rule from statistics? Cause the book was a bit sparse on that point. gneill Mentor @gneill: That's a rule from statistics? Cause the book was a bit sparse on that point. Yup. Consider that subtraction is just adding the negative of one of the values. The uncertainty in the negative value is the same as for the positive value.
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5d 9 6s 2 D IV. SUMMARY AND CONCLUSIONS In conclusion, the lifetimes of the two excited states of the Pt − anion are reported. The highest-lying bound excited state in Pt − , 5d 10 6s S 2 1/2 , was found to have an intrinsic lifetime of 2.54 ± 0.10 s, while only a range of 50-200 ms could be estimated for the intrinsic lifetime of the 5d 9 6s 2 D 2 3/2 excited fine-structure level. This lifetime range is consistent with available theoretical predictions for the D 2 3/2 state lifetime [38,40]. Based on estimates of the ratio of E2 decay rates from the S 2 1/2 state, the estimated lifetime of the S 2 1/2 state via the 5d 10 6s S 2 1/2 → 5d 9 6s 2 D 2 3/2 E2 transition is on the order of 100 days. Therefore, the measured lifetime of the 5d 10 6s S 2 1/2 excited state directly yields the 5d 10 6s S 2 1/2 →5d 9 6s 2 D 2 5/2 transition rate. To date, this is the first measurement of a lifetime of a bound excited state of an atomic negative ion with a different electron configuration than that of the ground state. The present results may thus be used to benchmark ab initio calculations that include electron correlation effects, which can be expected to differ significantly between bound anionic states of different electron configurations. This is in contrast to the situation involving transitions between fine-structure levels of the same
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This leads to the resonance condition in the form This determines the complex eigenvalues of k 1 = ζ n /L and then the complex eigenenergies E = 2 k 1 2 /(2m 1 ) = 2 ζ n 2 /(2mL 2 ). (The origin of the energy is set at the Fermi energy of the background Al 0.4 Ga 0.6 As. The bottom of the quantum well is at −300meV.) Figure 3 The resonance lifetime is given by then the eigenfunction decays as We plotted the resonance lifetime of each resonant state in Fig. 3(b). The lifetime becomes longer in some regions. The most prominent ones are listed in Tab. I. (For references, we also list the bound states in Tab. II. Note again that the bottom of the quantum well is at −300meV.) For the fourthgeneration Cantor set of length L = 405nm, the longest lifetime was 0.2ns found at 60meV. A shorter one of 0.02ns was found at a higher energy of 100meV. For the fifth-generation Cantor set of length L = 1215nm, the longest lifetime was 2.8µs found at 44meV. A shorter one of 0.02µs was found at a higher energy of 102meV. We can see that the lifetime grows very rapidly as we go to higher generations. Figure 4 shows the amplitude of the wave function of a scattering state that has the wave number equal to the real part of the eigen-wave-number of the resonant state with the longest lifetime in each generation (marked by the symbol * in Tab.
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fs at 0.1 eV above E F and 20 fs at 2 eV as well as different lifetimes for excitation of sp-and d-band electrons. [14] Another interesting feature to be studied by 2PPE are image potential states, which were first predicted in 1978 and which are well described by theory. [15] They arise from an electron being trapped in front of the surface in the potential well generated by its own attractive image force and the band gap. They form Rydberg-like series of states with a main quantum number n and l = 0. Lifetimes around 30 fs were measured on Cu(100) for the n = 1 state [16] and it can be deduced from theory that the lifetime will scale with n 3 . Höfer et al. managed to prepare image potential states with n = 4…6 and measured lifetimes between 0.63 and 2.0 ps respectively. For coherent excitation of several states centered around ñ = 7 they observed quantum beating with a period of 800 fs resulting from the electron wave packet oscillating in front of the surface. [17] More recently the availability of wellcharacterized ultra-short attosecond (1 as = 10 -18 s) laser pulses opened the door to time-resolved experiments of bound electrons. [18,19] Photoemission is usually described as a three-step process: excitation, transport and escape of the photoelectron through the surface. [20] In a proof-of-principle experiment, Cavalieri et al. delivered the first direct as-time-resolved measurement of electron transport in solid matter. [21] A ~300 as XUV pulse was focused on a W(110)
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the lifetime width is linear in energy E measured from E F , i.e. FWHM w = α|E − E F |. The coefficient α, which phenomenologically represents the intensity of the lifetime of the photo-hole with increasing binding energy, is a parameter which is determined so that the measured spectra are well reproduced. For both compounds we have taken α = 0.24 and 0.40 for the photoemission and inverse-photoemission spectra, respectively. The spectra have been normalized to their total area. As shown in Fig. 5 (a), the agreement in the wide energy range between theory and experiment is quite satisfactory as has been already reported 12,13 . The main features observed around E = 2-3 eV, −(1-2) eV, and −(3-6) eV in the BIS and PES spectra are well reproduced in the theoretical simulation. As shown in the figure, the energy difference between the main peak in the He II spectra and that in the BIS spectra equals ∼ 4.1 eV for both compounds, while the counterpart in the theory is ∼ 3.8 eV for Mo 6 Se 7.5 and ∼ 3.6 eV for Sn 1.2 Mo 6 Se 7.5 . Although the theoretical values are slightly smaller than the experimental ones, the values of the energy splittings, which directly reflects the electronic structure of the Mo 6 Se 8 cluster, fall in the same range for both theory and experiment. In the theoretical studies, it is known that a molecular-cluster approach is a good first approximation 9 , which is due to the localized
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\cite{parthey}. They have gotten an accuracy of about $4,2 \times 10^{-14}$ eV (2,466,061,413,187,035(10)Hz, an accuracy of 4 parts in $10^{15}$). The calculation of the energy of the 2S state is a very boring job. However, in order to make a rough estimation, we do not need to find the exact energy of the 2S state, since the contribution of the lowest order to the correction of the energy of the 2S state must be of ${\cal O}(\alpha^{4})$, because the 1S and 2S ordinary states have the same symmetry. Therefore the energy difference between the 1S and 2S states for minimal length correction must lead to a result of ${\cal O}(\alpha^{4})$, since the 1S and 2S levels differ only in the $n$ quantum number. If we attribute this error entirely to the minimal length corrections and assume that the effects of the minimal length can not yet be seen experimentally, from (\ref{edml}), we find \begin{equation} \Delta X_{i}^{min} \leq 10^{-17} m. \end{equation} As it was expected the result is identical to one obtained by Brau \cite{brau}. \section{Conclusion} \label{Concl} \hspace{.5cm} The aim of this work was to calculate, in a relativistic approach, the energy of the ground state of the hydrogen atom in a minimal length scenario. The minimal length has been introduced in the theory through the generalization of the Heisenberg's algebra chosen by Kempf and in the special case $\beta^{\prime} = 2\beta$, see Eqs. (\ref{rc1}), (\ref{rc2}) and (\ref{rc3}). In order to avoid the problem of substituting $\hat{X}_{i}$ for derivatives of $\hat{p}_{i}$ in the Coulomb potential ($\frac{1}{r}$) we
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states given by ρ(E) = 1/(E(v f +1 ) − E(v f )). On figure 5, we observe the continuity of the results around the dissociation limit. The lifetime, as well as the relative contribution from bound and continuum states, are presented in table IV. We observe that the lifetime increases with the vibrational number v. We also see that the contribution of the continuum to the lifetime is very small (5%) for v i = 0, but increases with the value of v i up to 30% for v i = 5. The contribution of the higher lying Π states will probably reduce the lifetime by a few percent, but our calculations provide an upper bound of 150 s on the lifetime of the v = 0 level of the a 3 Σ + state in the absence of collisional decay. IV. CONCLUSIONS We have determined the radiative lifetime of the a 3 Σ + state of HeH + using ab initio methods. The decay of this state onto the ground X 1 Σ + state is spin-forbidden but can occur through spin-orbit coupling. We took into account FIG. 4. Adiabatic potential energy curves of the first triplet a 3 Σ + state (above) and of the ground X 1 Σ + state (below) of HeH + and position of the bound vibrational levels supported by these states. the interaction of the a 3 Σ + and X 1 Σ + states with the first five 1 Π and three 3 Π states of HeH
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∼ m H ∼ 10 14 GeV. The lifetime for the H quantum is then F8 For spin-zero φ, one can make the lifetime much larger by applying (1 + bγ 5 ) to ψ ντ in the mixing interaction (after eq. (10)), and letting b approach unity. This can be a reason for the projection, implying parity nonconservation. This is the time t H used in eq. (9) for the end of the first time interval in which matter dominates, and the beginning of the second time interval in which radiation dominates. For example, considering only photons, an energy density ρ H M ∼ m 4 H ∼ 10 56 GeV 4 , assumed to be present F9 at about t H ∼ = 10 −34 sec, and the decay H → 2γ, leads to a radiation energy density which evolves to a present-day value of A relevant point about the above ρ H M (t H ) (as well as about other possible F10 (cold) matter energy densities such as ρ L,L M (t H ), ρ b M (t H )), is that the values lie well below the maximum total energy density at the end of inflation which has been estimated [12] from the upper-limit value for the Hubble parameter at the end of inflation. This energy density is [12] Another point is that massive φ quanta, and possible L, L quanta, have virtually no interaction contact with surrounding thermal conditions in the model of eq. (1), from which the potential in
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the state. The cross indicates the energy and lifetime that minimize χ 2 while the white line is the 1-σ contour line. hindered single-particle E2 transition for the decay to the ground state. For example, the E2 decay of a state at 673 keV with a reduced transition probability of 1 W.u. would result in a level lifetime of 476 ps. The mean decay position was thus located 107 mm after the reaction vertex, leading to a low energy tail in the Doppler corrected γ-ray energy spectrum as observed in Fig. 2. A combined fit of both transition energy and level lifetime results in best fit values of E = 673(17) keV and τ = 1130 +520 −330 ps. The parameters for the double exponential background included in the fit were constrained from fits to spectra of neighboring nuclei for analogue reactions. Alternatively, the spectrum can be fitted with two or more transitions with negligible lifetimes. However, a coincidence can be ruled out and the occurrence of two low-lying states populated in the reaction is at variance with the expectation of the ground state configuration of 56 Ca and theoretical calculations discussed below. It should be noted that the extracted intensity of the 673(17) keV transition depends strongly on the lifetime assumed in the simulation of the response function. For the extraction of the exclusive cross sections shown in Table 1 this correlation was taken into account. States in 57 Ca were populated in the proton removal from 58 Sc. In the same setting as employed
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18 C and 20 C respectively. Figure 8 shows the calculated exact eigenvalue of the pairing Hamiltonian for 18 C and 20 C. Experimentally, only one excited state in 18 C is known. It is a (2 + ) state at 1620 keV from the (0 + ) ground state. Considering what we learn in the previous spectra one may place some confidence on the theoretical estimation for the levels C. Results: Complex Energy Representation The first step in the determination of the complex representation is to find the resonant partial waves. This is done by evaluating the outgoing solutions (Gamow states) of the Schrodinger equation [25,31] of the mean field Hamiltonian defined in Sec. III A 1. Then, Table VII compares the characteristic time with the half-life of the states ε 0d 3/2 and ε 0f 7/2 . The half-life of the state 0d 3/2 is around nine times bigger than the characteristic time. The 0f 7/2 state seems to be a wide resonance, but the comparison with the characteristic time shows that its half-life is a bit bigger than τ c . The effect of the resonant continuum was already investigated in ref. [9]. In order to investigate the effect of the non resonant continuum on the many-body correlations we compare in fig. 9 the ground state energy of the nucleus 22 is not very strong [9]. As the interaction increases the continuum starts to be important. The curve labeled as "Continuum Representation" gives the ground state energy when the resonant and non resonant
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sharply. Of course, ensuring the condition ∆ = 0 demands a much better knowledge of E is than is available today ((3.5 ± 1) eV [68]). For the range 2 ≤ E is ≤ 5 eV theoretical calculations give T1 2 10 min [52], whereas the authors of [68] claim that T1 2 45 hr for E is = 3.5 eV (M1 transition). Taking into account the ±1 eV uncertainty in the excited state energy, T1 2 could be as long as ≈ 120 h or as short as ≈ 20 h. However, one has to realize that the half-life time of an anomalously low-lying nuclear level (E is ≈ several eV) is a very subtle quantity. It is influenced by the interaction not only with the electronic shell, but, due to the extremely small value of the isomeric state energy, it will be affected also by the physical and chemical properties of the sample containing the atoms of 229 T h. Experimental and theoretical studies of such a phenomena may open new interesting directions into atomic and condensed matter physics. We now turn to the question of the laser-assisted deexcitation of the isomeric 229 T h state to its ground state using the resonant, discrete IC with bound electrons, applying a laser field with the appropriate frequency. As we saw above and in Section 5, this leads to a drastic acceleration of the nuclear isomeric decay [52,27]. Very recently, the probability for this process was recalculated [69]. As atomic levels, in contrast of [52], the
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Direct measurement of the 3 d 2 D 3 / 2 to 3 d 2 D 5 / 2 lifetime ratio in a single trapped 40 Ca + We present for the first time a direct measurement of the lifetime ratio between the 3 d 2 D 3 / 2 and 3 d 2 D 5 / 2 metastable states in a single trapped 40Ca+. A high-efficiency quantum state detection technique is adopted to monitor the quantum jumps, and a high precision and synchronized measurement sequence is used for laser control to study the rule of spontaneous decay. Our method shows that the lifetime ratio is a constant and is irrelevant to the dwell time; it is only determined by the spontaneous decay probabilities of the two metastable states at one random decay time, independent of the lifetimes of the two metastable states. Systematic errors such as collisions with background gases, heating effects, impurity components, the shelving and pumping rates and the photon counts are analyzed, and the lifetime ratio between the 3 d 2 D 3 / 2 and 3 d 2 D 5 / 2 states is directly measured to be 1.0257(43) with an uncertainty of 0.42%. Our result is in good agreement with the most precise many-body atomic structure calculations. Our method can be used to obtain the lifetime of a state which is usually difficult to measure and can be used to determine the magnetic dipole transition matrix elements in heavy ions such as Ba+ and Ra+, and can also be universally
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In some instances there are indications that several transitions are not resolved. Even though the values plotted in Fig. 10 display a lot of scatter, the overall trend indicates a monotonically increasing width of the final state lifetime broadening, but not necessarily a quadratic dependency as formulated earlier. There is no obvious dependence on crystal orientation, which is not unexpected. Very low energy excited electrons within a few eV above E F display a very sharp drop in lifetime with increasing excitation energy and correspondingly a lifetime width which increases much stronger than linear (47)(48)(49). On the other hand, for the energy range from 20 eV to above 100 eV the lifetime τ increases much less rapidly. Goldmann et al. (46) empirically determined a linear rise approximated as τ -1 = 0.13 (E -E F ). This empirical relation is shown as a yellow solid line in Fig. 10. Also included in Fig. 10 is the lieftime as predicted by a free electron gas model (blue line) using the electron density of Cu as calculated by Echenique et al. (42). We consider it appropriate to view the lifetime width data in Fig. 10 as an upper bound of the actual lifetime or self energy in the final state of the photoemission process. Other factors such as phonon scattering or unresolved multiple transitions might lead to a measured width which is larger than the lifetime width. Therefore it might be appropriate to concentrate on the sharpest transitions observed, rather than all transitions. This is suggested by the
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The energy difference between equidistant interaction partners decreases with increasing distance. For the case of Ne 2 pairs these energy differences between the peaks stemming from different interatomic distances are given by: 3 Å, 4 Å 1.20 eV, 4 Å, 5 Å 0.68 eV and 5 Å, 6 Å 0.48 eV. This means that the spectrum is better resolved for smaller distances and that small distance changes like vibrations will mainly affect this lower energy part of the spectrum. the dimer to confirm our findings, that the shoulder stems from ICD with interaction partners of the next shell [15]. In the neon dimer several vibrational states of the ionized initial state are involved in the decay [7]. It has furthermore been shown, that the experimentally determined lifetime of t = 150 50 fs NeNe [46] is only in agreement with those theoretical lifetime calculations that explicitly include nuclear dynamics of the intermediate state [47]. However, the early lifetime measurements in neon clusters with a mean cluster size of < N >=900 atoms show a lifetime of 30 fs for surface atoms and 6 fs for bulk atoms. This lifetime decrease is caused by the possibility to decay with several decay partners. Nuclear dynamics occur in the range of tens of femtoseconds in the dimer. Additionally, the driving force for a bond contraction after the initial ionization is higher in dimers than in clusters, where one bond contraction usually leads to several bond elongations. Therefore, we consider the influence of nuclear dynamics on
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ground state arising from the 5s 2 5d 2 D term with the higher energies (71,406 and 72,048 cm −1 ) are the more favorable transitions with shorter lifetimes, while the ones from the 5s5p 2 2 D term with the lower energies (58,844 and 59,463 cm −1 ) are much longer lived. Previously this was known only theoretically. Our MCDHF calculations support these facts as well.
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larger (more than 100 times) than the ones of the E2 transition. The radiative lifetimes of the levels 4d 8 ( 3 F )4f 4 I 15/2 and 4d 8 ( 3 F )4f 4 I 13/2 are extremely large (more than 10 seconds) due to their position in the energy spectra. Although these two levels are relatively very low in the spectra, the level 4d 8 ( 3 F )4f 4 I 15/2 has three open radiative decay channels through rather weak transitions (one M1 and two E2). The level 4d 8 ( 3 F )4f 4 I 13/2 has only one decay channel through the E2 transition. Since the total number of possible M1 and E2 transitions depends on a particular level location, for the high-lying levels with J = 13/2, as well as for other high-lying metastable levels, the M1 transitions primarily determine the calculated level radiative lifetime as their probability values are significantly larger compared to those of the E2 transitions. The lifetimes of the levels with J = 11/2 and J = 9/2 are also determined by the transitions to the levels of the same parity, but in this case, the M2 and E3 transitions to the ground configuration of the opposite parity are allowed. Since there are only two levels in the ground configuration, the number of M2 and E3 transitions is strictly defined. Only one E3 transition is possible from each level with J = 11/2; two E3 transitions and one M2 transition is allowed from each level with
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≈ 10 78 , the electron number N e ≈ 10 81 , the photon number N tγ ≈ 10 88 , and the phonon number N tp ≈ 10 75 are conserved good quantum numbers as described above. The matter current at the Planck scale, the (V -A) current and the electromagnetic current at the electroweak scale, the baryon current and the proton current at the strong scale, and the lepton current at the present scale might be conserved currents at different energy scales. The proton number conservation is the consequence of the U (1) f gauge theory just as the electron number conservation is the consequence of the U (1) e gauge theory. In gravitational interactions, the predicted typical lifetime for a particle with the mass 1 GeV is τ p = 1/Γ p ≃ 1/G 2 N m 5 ≈ 10 50 years using the analogy of the lifetime of the muon τ µ = 192π 3 /G 2 F m 5 µ in weak interactions. Therefore, in the proton decay p → π 0 + e + at the energy E << M P l , the proton would have much longer lifetime than 10 32 years predicted by GUT [14] if the decay process is gravitational. In fact, the lower bound for the proton lifetime is 10 32 years at the moment. If the electric charge is completely conserved, the electron can not decay. The present lower bounds for the electron lifetime are bigger than 10 21 years for the electron
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= -1.5 eV, E4 = - 0.85 eV Now, from the levels given above, E2 - E4 = 2.55 eV Therefore, quantum numbers of the two levels involved in the emission of these photons are 4 and 2. (c) Change in angular momentum in transition from 4 to 2 will be Here, angular momentum (L) is given by $\dfrac{nh}{2\pi}$, where, n = quantum number and h = planck’s constant, Therefore, $\Delta L=L_2-L_4 = \dfrac{2h}{2\pi}-\dfrac{4h}{2\pi}$ Hence, Change in angular momentum will be $-\dfrac{h}{\pi}$ Key Point: Kinetic energy of the fastest moving electron is maximum kinetic energy only because kinetic energy depends on mass and velocity only. 2. The mean lives of an unstable nucleus in two different decay processes are 1620 yr and 405 yr, respectively. Find out the time during which three-fourth of a sample will decay. Sol: Given, Mean life of Process- 1 (t1) = 1620 yr, Mean life of Process- 2 (t2) = 405 yr, Let decay constants of process-1 and process- 2 be λ1 and λ2, respectively. Then, $\lambda_1 = \dfrac{1}{t_1}$ $\lambda_2 = \dfrac{1}{t_2}$ If the effective decay constant is λ, then λN = λ1N + λ2 λ = λ1 + λ2 $\lambda = \dfrac{1}{t_1}+\dfrac{1}{t_2}$ $\lambda = \dfrac{1}{1620}+\dfrac{1}{405}$ year-1 $\lambda = \dfrac{1}{324}$ year-1 Now, when three- fourth of the sample will decay. The sample remaining will be one- fourth of the total sample. If the initial sample was N0, then final sample will have N0/4, So, $\dfrac{N_0}{4} = N_0e^{-\lambda t}$ Applying logarithms on both the sides, we get, $-\lambda t = \ln\lgroup\dfrac{1}{4}\rgroup$ =
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M2 and E3 transitions affect the lifetime of 4d 8 ( 1 D)4f 2 H 9/2 , but no so significantly as the lifetimes of previously discussed levels due to the amount of the M1 and E2 transitions allowed from this level. The radiative lifetimes of other levels with J = 9/2, located higher in the energy spectra than the discussed ones, are determined mainly by the numerous M1 transitions. The E2 transitions are also allowed, but they are much weaker than the M1 transitions. So here the influence of the E3 and M2 transitions is not so outstanding. Although these transitions cannot be discarded at all, when the nuclear charge Z is between 60 and 70, these transitions decrease the calculated lifetime value up to 4 times. Usually, the decrease drops very quickly. For heavy ions (Z > 80), the lifetime values change only by few percent. Summary The configurations 4p 6 4d 9 , 4p 5 4d 10 , 4p 6 4d 8 5s, 4p 6 4d 7 5s 2 , 4p 6 4d 8 5p and 4p 6 4d 8 4f of the Rh-like ions are investigated using the quasirelativistic approach with a small CI1 expansion for the range from Z = 48 up to Z = 92. Calculation results demonstrate that the ions with Z < 60 do not have metastable levels in the excited configurations, except for two levels of the the configuration 4p 6 4d 8 4f with the largest total angular momentum, J = 15/2. The levels with J
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for this factor is the renormalisation of the density of emitted photon states. For a CaF 2 crystal, n = 1.55 at wavelength λ = 159 nm (corresponding E = 7.8 eV), which gives a reduction factor of 3.7 for the lifetime. A more significant lifetime reduction can be connected with electronic bridge and bound internal conversion effects, i.e. isomer state decay can be amplified by interaction between the nucleus and the electron shell. There are a number of theoretical calculations of electronic bridge processes. In an isolated neutral Thorium atom the isomer state lifetime is about 10 −5 s for E = 7.6 eV and about 4.5 min for E = 3.5 eV [20]. In the Th 3+ ion for E = 7.6 eV the lifetime remains the same if the valence electron is in the ground state, and decreases 20 times if the valence electron is in the metastable 7s state [21]. Similar calculations for the Th 4+ ion which is most likely to occur in the solid-state approach have yet to be performed. In the Th + ion the lifetime decreases 10 2 − 10 3 times for E = 3.5 and E = 5.5 eV [22], there is not sufficient data about the Th + ion spectrum to calculate the lifetime at E = 7.8 eV. Experimental attempts to measure the 229m Th lifetime were performed in [23,24]. These experiments are based on the theoretical prediction [26] that the α decay rate for 229m Th is 2 to 4 times higher than
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to the differences between the initial and final rovibronic states are in the range from 10 meV to 50 meV and are typical for the case under consideration. The mentioned electronic non-adiabatic couplings allow the calculation of the vibrational coupling element between selected states according to Eq. 8. Assuming that the energy of the detached electron is in the range given above, and the maximal lifetime of the electron emission reaction is less than 10 s, which corresponds to the duration of the experiment reported by Anderson et al. 13 , then the mean value of this coupling element between all possible rovibronic states of Ag − 2 and Ag 2 for J = 0 is 1.4·10 −9 a.u. Averaging is performed by all possible pairs of vibrational levels, for which energy gaps are in the range related to the kinetic energy of the ejected electrons. As expected, the value of the vibrational coupling element is very small. In turn, the mean lifetime of the spontaneous electron emission (Ag − 2 → Ag 2 + e − ) as estimated by Eq. (9), is around 3 seconds, and the shortest calculated lifetime is 262 ms. All this confirms the experimental finding that both processes (i.e. spontaneous electron emission and rotational predissociation) compete with each other in the same timescale and their duration can be quite long, in specific cases up to a few seconds. IV. CONCLUSIONS In reference to the recent experiment, we set ourselves the goal of creating a theoretical description of the two-channel decay
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energy (<300 keV) electrons. At higher energies (>300 keV), the measured lifetimes become smaller than model lifetimes during low AE activity (green solid line), falling in closer agreement with model lifetimes for moderate activity (100 < AE < 300 nT). This behavior is probably partly due to the upper limit of <20 days imposed on measured lifetimes by the method of empirical lifetime determination, in the presence of low measured fluxes at high E . Possible physical causes of discrepancies between estimated and measured lifetimes will be discussed in the next section. As shown in Figure 5a the model lifetimes at GEO follow approximately the same analytical scaling with energy given by Equation 2. Here we refine and generalize Equation 2, deriving analytical lifetime fits to the full model lifetimes as a function of energy and L-shell in the range of 30 ≤ E ≤ 2,000 keV and 3 ≤ L ≤ 6.5, respectively. We use the analytical formula in Equation 2 to find numerically the polynomial function Bw(L, E) that provides the best agreement at all (E, L) with model lifetimes. Table 1 shows the average Bw that provides the best agreement between analytical and model lifetimes as a function of E and L for quiet (AE < 100 nT), moderate (100 ≤ AE ≤ 300 nT), and active (AE > 300 nT) geomagnetic conditions. In general, for quiet and moderate geomagnetic conditions the average Bw does not change significantly with E. Therefore, for quiet and moderate geomagnetic conditions we simply drive polynomial fits for
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having ℓ ′ = ℓ ± 1 (which themselves mix with states ℓ ′ ± 1). However, m ℓ (or, in atoms with large fine structure splittings like Rb, m j ) remains a good quantum number. This mixing obviously has the potential to impact our lifetime measurements, given the generally much longer lifetimes of the np states than of the ns and nd states (whose lifetimes are similar) at comparable n eff . To estimate the extent to which our measurements are an average over Stark-mixed nℓ states, we have calculated the amount of mixing that could result from an electric field using a program that diagonalizes the Stark energy matrix [51]. For a given electric field, the mixing is stronger as n increases, so we calculated the mixing at the highest n eff for any state whose lifetime was measured in our experiments (44d 5/2 , corresponding to n eff ≈ 43). Additionally, we assumed an electric field of 2 V/cm, i.e., twice the field that was present in our experiments. We found that in this regime, the admixture of states with a different orbital angular momentum quantum number ℓ was negligible. For instance, the 44d 5/2 |m j | = 1/2 state is 1.57% (probability) ℓ = 1, 97.10% ℓ = 2, and 1.29% ℓ = 3, while the 46s 1/2 |m j | = 1/2 state is 98.95% ℓ = 0, and 1.05% ℓ = 1. We looked at all the possible |m j | eigenstates in the 46s 1/2 , 45p
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lifetime τ is bounded by (roughly 95% confidence level) 0.66 × 10 6 s eV 3 /ω 3 ≤ τ ≤ 2.2 × 10 6 s eV 3 /ω 3 . In Fig. 5, the bound is plotted as a function of isomeric transition energy (blue dash-dotted lines). For completeness, the energy ranges from 2.5 eV, which includes the now-rejected value of the transition energy from Ref. [20] with one standard deviation towards lower energies, up to 10.5 eV, which is expected to be the largest possible value for the transition energy based on Sec. V. Further, the energy range for the currently accepted value of the transition energy of Ref. [41] including two standard deviations is highlighted (see Fig. 5, blue dotted lines). The intersection of these two bounds, each at roughly the 95% confidence level, is marked as as blue shaded area and gives the primary region of interest. The energy and lifetime of the isomeric transition should be found at roughly the 90% confidence level in this region. It is very likely that this region is somewhat conservative in its upper lifetime bound for two reasons. First, reduced transition probabilities calculated from the Alaga rules are typically smaller than actual values when the spin of the nucleus increases during the transition, as is the case here [67]. Second, the measurements of the B(M 1; 9/2 + 5/2[633](97.14 keV) → 7/2 + 3/2[631](71.83 keV)) reduced transition probability rely on calculated values of the internal conversion coefficient, and there is evidence [68] that these calculated
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will be about the same for all of them. | S( k ) | 2 ≈A e −B/ E . Here B=( π 2m /ℏ )( Z−2 )2 e 2 is not an adjustable parameter: and plotting ln | S( k ) | 2 against 1/ E for Polonium212 (which emits α ’s with energy 8.95MeV, and lasts 3× 10 −7 seconds) Thorium232 (4.05MeV α ’s, 1.4× 10 10 years), and several intermediate lifetime nuclei gives a straight line plot with the correct slope within a few percent! These elements can all be understood in terms of essentially the same barrier being tunneled through at the different heights corresponding to the α energy. The treatment here is a slightly simplified version of the WKB method, to be discussed in detail later. Further refinements make little difference to the final result in this case. Source: many of the topics covered in this lecture are elementary, and treated in any quantum textbook. For some of them I’ve followed (more or less) the excellent book by French and Taylor, An Introduction to Quantum Physics, Norton, 1978.
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2S state in atomic hydrogenlike systems is 8.229 Z 6 s −1 (inverse seconds). At Z = 1, this is equivalent to the "famous" value of 1.3 Hz which is nowadays most frequently quoted in the literature. The lifetime of a hydrogenic 2S level is thus 0.1215 Z −6 s. This latter fact has been verified experimentally for ionized helium [45,46,47]. We now briefly recall the expression for the two-photon decay involving two emitted photons with polarization vectors ε 1 and ε 2 , in a two-photon transition from an initial state |φ i to a final state |φ f . The two-photon decay width Γ is given by [see, for example, Eq. (3) of Ref. [31]] where ω 2 = ω max − ω 1 and ω max = E − E ′ is the maximum energy that any of the two photons may have. The Einstein summation convention is used throughout this article. Note the identity [48,49] which is valid at exact resonance ω 1 + ω 2 = E i − E f . This identity permits a reformulation of the problem in the velocity-gauge as opposed to the length-gauge form. In a number of cases, the formulation of a quantum electrodynamic bound-state problem may be simplified drastically when employing the concepts of an effective low-energy field theory known as nonrelativistic quantum electrodynamics [50]. The basic idea consists in a correspondence between fully relativistic quantum electrodynamics and effective low-energy couplings between the electron and radiation field, which still lead to ultraviolet divergent expressions.
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10 –6 eV . size 12{ΔE =" 5" "." "3 " times " 10" rSup { size 8{"–25"} } " J " cdot { {"1 eV"} over {1 "." "6 " times " 10" rSup { size 8{"–19"} } " J"} } =" 3" "." "3 " times " 10" rSup { size 8{"–6"} } " eV" "." } {}$ 29.51 Discussion The lifetime of $10−10s10−10s size 12{"10" rSup { size 8{ - "10"} } s} {}$ is typical of excited states in atoms—on human time scales, they quickly emit their stored energy. An uncertainty in energy of only a few millionths of an eV results. This uncertainty is small compared with typical excitation energies in atoms, which are on the order of 1 eV. So here the uncertainty principle limits the accuracy with which we can measure the lifetime and energy of such states, but not very significantly. The uncertainty principle for energy and time can be of great significance if the lifetime of a system is very short. Then $ΔtΔt size 12{Δt} {}$ is very small, and $ΔEΔE size 12{ΔE} {}$ is consequently very large. Some nuclei and exotic particles have extremely short lifetimes (as small as $10−25s10−25s size 12{"10" rSup { size 8{ - "25"} } s} {}$), causing uncertainties in energy as great as many GeV ($109eV109eV size 12{"10" rSup { size 8{9} } "eV"} {}$). Stored energy appears as increased rest mass, and so this means that there is significant uncertainty in the rest mass of short-lived particles. When measured repeatedly,
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with the recent width of meV derived from measured cross sections in recent collision experiments (Lee et al 1996). The shortness of the time for which exponential decay (ED) holds and the fact that the survival probability, P(t), is still significant at the beginning of the NED, does not allow the rigorous justification of the definition of the lifetime from , or the equivalence of this with the observed energy width. Thus, we propose that a mean life, , should be obtained from Calculation produces s and s. For the Ca state, whose bound - free interaction is smooth and nearly constant from zero to about 5.5 eV, NED appears after 17 lifetimes. The lifetime of Ca is deduced from the exponential decay (ED) part of P(t) to be s. From our examination of the case of the level by a number of methods based on the use of state-specific wavefunctions, we conclude that for metastable states whose lifetimes are in the range - s, the ab initio calculation of P(t) is, at present, prohibited by the huge requirements for computer time. Finally, having computed the amplitude , we obtain numerically the energy distribution function, , of the two autoionizing states. In the case of Ca it is a perfect Lorentzian.
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200 eV higher in energy, which makes mixing negligible. 40 To our knowledge, the present spectrum is the first Br KLL Auger spectrum reported in the literature since the work of Erman and coworkers published in 1965. 6 In their work, the 1s À1 core hole in a 79 Br atom was formed by an electron-capture decay of 79 Kr and the experiment had a resolution of 10 eV, which is rather modest by modern standards. The Br KLL Auger transitions were fitted in order to extract the lifetime broadenings of the Br 2p À2 , 2s À1 2p À1 and 2s À2 double core-hole (DCH) states. In the analysis it was assumed that all DCH states of the Br 2p À2 configurations show the same lifetime broadening. For the 2s À1 2p À1 configurations the present calculations suggest different lifetimes for the individual states. In particular, for the 2s À1 2p À1 ( 3 P) states similar lifetime broadenings are calculated, while for the 2s À1 2p À1 ( 1 P) state a significantly larger value was obtained. Following these results we assumed in the fit analysis the same lifetime broadening for the 2s À1 2p À1 ( 3 P) states and a different one for the 2s À1 2p À1 ( 1 P) state. The upper panel of Fig. 4 shows the Br 1s À1 -2p À2 Auger transitions. All the four diagram lines Br 1s À1 -2p À2 ( 1 S 0 , 1 D 2 , 3 P 0 , 3 P
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long lifetime of 4 ps for 760 nm (1.63 eV) shown in Fig. 7(b) can be explained by assuming an excited state "Int" lying 1.63 eV above the "def" state. Three electronic levels, "def", "Int", and 3eg can explain the observed asymmetrical delay-time dependence. the valence band top when the bandgap energy is 3.2 eV. Then, electrons even in the valence band can be excited to the eg state by four photons of 800 nm. The total photon energy of 4.65 eV for three photons of 800 nm is not large enough to excite valence electrons to the eg state. Therefore, in the case of NPs showing a three-photon slope, another excited state is required below the eg state, as shown by "Int" in Fig. 8. Otherwise, we need to assume that the states observed by Zhang et al. 19 with a lifetime of 80 ps were excited to the eg state. The state observed by Zhang et al. 19 can be the same as the "def" state from which electrons were resonantly excited to the eg state by a 3.66 eV photon in Argondizzo et al. 32 In our experiment, the "def" state in the bandgap might have been fully filled because pulses from the 80 MHz laser irradiated the sample every 12. 5 ns. The energy between "def" and the eg state is 3.66 eV and three photons of 800 nm are enough to populate the eg state. In the case of thermionic emission in Au NPs observed in our previous study, 38 the
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+ 1 and 8 + 1 states are always larger than 6 + 2 and 8 + 2 states, confirming the isomeric nature for the first yarst states of these spins, despite the change of configuration. The 6 + 2 state decays with the M 1 + E2 transition, and the M 1 transition shows a large impact by reducing the half-life from ns to ps. Similarly, the 8 + 2 state is solely dominated by the M 1 transition, thus giving a smaller half-life than the 8 + 1 state. Our calculation supports the experimental counterpart of the B(E2) and half-lives except for 142 Nd and 218 U, whose half-lives show large discrepancies with the experimental values. For 142 Nd, the energy levels are reproduced well, but the B(E2) transition value is very small (0.0003 W.u.). This could be the reason for the large magnitude of the calculated half-life. In the case of 218 U, as there is no experimental information available for the electromagnetic transition of 8 + , we can not say due to which part discrepancy is arising. We have calculated M 1 transition for the 8 + 2 state of 218 U, which is not measured experimentally. In comparison with the isomeric nature of 8 + 1 state in 216 Th, πh 9/2 f 7/2 configuration has been proposed in [50] for the 218 U. Following the behavior of 6 + states in the Sn-region and our shell-model results, we can propose the configuration of 8 + 2 state for 218
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two modes estimated above is of order 10 −1 , then the lifetime τ S is of order 10 −37 s (this is an "initial" time interval after hypothetical inflation). The Hubble expansion time scale corresponding to an energy scale of ∼ 10 16 GeV is ∼ 10 −38 s. The S decay thus occurs in a non-equilibrium situation; back-reactions do not vitiate the primary ν-ν asymmetry. From an estimate of the absorptive part of the diagram in Fig. (1b), we calculate this number asymmetry in N ν,ν (for one type of neutrino), relative to photon number N γ , The first factor in (18) arises explicitly from the CP-violating interference between the amplitudes from Figs. (1a,b). The numerical estimate of this factor is for , and m b ∼ 2.7 × 10 −6 eV as estimated above. Two aspects of this factor are noteworthy (1) There is no enhancement because of the long range of the final-state interaction from exchange of a very low-mass b quantum F 8 . ( 2) The small energy scale F b does not appear explicitly. This is because it does not appear in the scalar interaction term in (14), form ν ≪ (g ν F b ) (i. e. sin α ν ∼ = 1). The last factor in (18) is (N S /N γ ), the ratio of the original number of decaying S to the number of (eventual) photons (∼ 10 88 ). We have used ∼ 10 −2 for this ratio, because this corresponds to an
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and H 2 18 O energy level with quantum numbers K a , K c , ν 1 , ν 2 and ν 3 . On their own, the rigorous quantum numbers from DVR3D do not provide enough information to match with the empirical energy levels from MARVEL. Hence, energy level differences must be used together with this information to match with MARVEL states. This approach makes the labeling of closely lying states not straightforward. For H 2 18 O we supplemented the MARVEL data with new pseudo-experimental levels taken from Polyansky et al. [53]. Each energy level calculated with DVR3D is unique, but can naturally occur more than once as both an upper or lower state within a transition. It is therefore important that the same energy level from DVR3D is not assigned differently in various transitions. We approached this matching problem in stages, where the maximum energy difference interval for matching, i.e |E calc − E meas |, is increased in values of 0.22 cm −1 , 0.40 cm −1 , 0.50 cm −1 , 0.70 cm −1 , 1.00 cm −1 , 1.30 cm −1 and 1.50 cm −1 . Increasing beyond this final value would dramatically increase the possibility of mis-labeling. Using the uniqueness, once a calculated level from DVR3D is matched to its corresponding level in MARVEL, only that particular calculated level may carry that MARVEL label from then on. Also, once a MARVEL level is matched to a level in our line list, it does not carry onto the next
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lifetimes of J = 13/2 levels The configuration 4p 6 4d 8 4f has five levels with J = 13/2. The radiative lifetimes of the lowest level 4d 8 ( 3 F )4f 4 I 13/2 are presented in Fig. 5. The lifetimes of this level increase with the Z increasing, because the location of this level in the energy spectrum is going down, i.e. the energy of this level relative to the ground level decreases unlike in the case of the lowest level with J = 15/2, where the energy difference increases when the Z increases. The 4d 8 ( 3 F )4f 4 I 13/2 level has three decay channels open when Z = 60 and Z = 62. When Z ≥ 64, only one E2 transition to the 4d 8 ( 3 F )4f 4 I 9/2 level is allowed. This transition determines the lifetimes of the level 4d 8 ( 3 F )4f 4 I 13/2 up to Z = 80. For the higher Z, the decay channel to the 4d 8 ( 3 F )4f 4 I 9/2 is also closed. For the ions with Z ≥ 82, the 4d 8 ( 3 F )4f 4 I 13/2 level does not have a radiative decay channel, because the radiative transitions of all calculated multipole orders to the levels, which are lower than the investigated one, are forbidden by the selection rules for J. The lifetime of this level calculated for the tungsten ion is 3.8 · 10 15 s −1 (see
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|1E 2g x2−y2 > is E 2E1u − E 1E2g = 5.75 eV, with corresponding period T 2E1u,1E2g = 719 as. Likewise, the energy gap between states |2E 1u y > and |1A 1g > is E 2E1u − E 1A1g = 14.23 eV, with period T 2E1u,1A1g = 291 as. These two periods are short enough to satisfy condition (47) for two laser pulses with durations τ 7 = 2.5 fs and τ 8 = 7.5 fs, respectively. Their durations add up to t f = τ 7 + τ 8 = 10 fs, in accord with the constraint for the conservation of nuclear symmetry [7,8]. The smaller value of τ 7 compared to τ 8 is suggested by condition (47), due to the smaller value of T 2E1u,1A1g compared to T 2E1u,1E2g . After the sequence of two laser pulses 7 and 8 with the same polarizations e' 1 and again e' 1 , for times t ≥ t f = τ 7 + τ 8 = 10 fs, the wavefunction (51) evolves in field-free environments, with coefficients specified in Equation (26). The corresponding swapping part of the one-electron densities ρ e '1&e'1 (r,t) − ρ 1A1g (r) is illustrated in Figure 10a by snapshots analogous to those shown in Figures 6-8. Corresponding inspection and analyses of these snapshots reveal that ρ e '1&e'1 (r,t) displays charge migration with conservation of the symmetry elements of the subgroup D 2h,1 = {E,C 2 ,C 2 ' 1 ,C 2 " 1 ,i,σ h , σ
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to 20, the solid line results from equation (1) using LS-coupling according to Laughlin [11]. The dashed line shows the present modification of Laughlin's calculations taking into account the splitting between the 3 P 0 and 3 P 1 energy levels [12]. For details see text. increases the predicted lifetimes by factors ranging from 1.5 to 2.5. For Xe 50+ this leads to a lifetime prediction of 78 instead of 30 seconds. More reliable theoretical descriptions of E1M1 transitions in heavy Be-like systems, which take into account relativistic effects, are not available. For 2E1 two-photon transitions in helium-like ions from the excited 2 1 S 0 state to the 1 1 S 0 ground state the comparison of non-relativistic [14] and relativistic [15] treatments shows that for xenon relativistic 2E1 lifetimes are about 30% larger as non-relativistic 2E1 lifetimes [15]. To perform lifetime measurements at ion-storage rings the transition lifetime τ has to be longer than the time needed for ion beam preparation which is usually of the order of seconds. Moreover, τ should not significantly exceed the ion-beam storage lifetime t store . The latter is affected by vacuum conditions including residual gas composition, by intra-beam scattering, and by additional ion loss processes due to collisions with the electron beam of the electron cooler. Some of the corresponding loss rates depend strongly on the ion velocity. In case of the heavyion storage ring ESR at GSI, which can store heavy ions with energies up to about 400 MeV/u, t store can be up to several
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t τ , we expect that it decays as e −t/τ 1 − t/τ , where τ is the lifetime given by Fermi's golden rule 1/τ ∼ ρ(E 0 )W 2 , where ρ(E 0 ) is the density of states (per site) of the atomic band at energy E 0 . The lifetime should therefore scale as 1/W 2 . This is indeed what we observe: for example, for E 0 3.087, we find τ ∼ 3.5/W 2 , see Fig. 17. At longer time, the evolution is more complicated. Time evolution of the probability P (t) = | ψ(0)|ψ(t) | 2 for three different disorder strengths: W = 0.1 (blue), 0.5 (yellow) and 1 (green). The initial bound state |ψ(0) is an eigenstate at W = 0 with energy E0 3.087 for Nx = 30 andŪ = 2. From P (t) 1 − t/τ (dashed lines), the lifetime τ is (a) 365, (b) 16 and (c) 4, which agrees with τ ∼ 3.5/W 2 .
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for almost all strong E1 transitions, our radiative rates are accurate to better than 10%. However, for the weaker transitions the accuracy is comparatively poorer. Lifetimes The lifetime τ for a level j is defined as follows: Since this is a measurable parameter, it provides a check on the accuracy of the calculations. Therefore, in Table 1 we have also listed our calculated lifetimes, which include the contributions from four types of transitions, i.e. E1, E2, M1, and M2. To our knowledge, no calculations or measurements are available for lifetimes in Kr XXXV with which to compare. However, we hope the present results will be useful for future comparisons and may encourage experimentalists to measure lifetimes, particularly for the level 1s2s 1 S 0 which has comparatively a larger value. 5 Collision strengths Collision strengths (Ω) are related to the more commonly known parameter collision cross section (σ ij , πa 0 2 ) by the following relationship: where k 2 i is the incident energy of the electron and ω i is the statistical weight of the initial state. Results for collisional data are preferred in the form of Ω because it is a symmetric and dimensionless quantity. For the computation of collision strengths Ω, we have employed the Dirac atomic R-matrix code (darc), which includes the relativistic effects in a systematic way, in both the target description and the scattering model. It is based on the jj coupling scheme, and uses the Dirac-Coulomb Hamiltonian in the R-matrix approach. The R-matrix radius adopted for Kr
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of 0.06-0.1 eV for V Si −As 3 . Similarly to the case of V Si we do not find for V Si − As and V Si − As 2 an energetically favored trapped positron state but just a metastable configuration in disagreement with the experimental trend. The trend in our trapping energies is exactly the opposite; the trapping energy increases with the increasing number of As atoms (n = 0, ..., 3). This is because the large As ions around the vacancy do not relax inwards as strongly as the neighboring Si ions so that the energy stored in the ionic lattice in the positron trapping process is smaller for the As decorated vacancy than for the clean vacancy. The fact that a bound positron state can be found at V Ge or at V As in GaAs when freezing the ions at ideal positions means that the bound positron states are very close to appear, and an improvement in the theoretical description could lead to bound positron states also for optimized ion positions. In experimental works 48,49,50 positron lifetime components between 279 and 292 ps are assigned to V Ge . The measured lifetimes can be contrasted to the measured bulk lifetime of 228 ps (Ref. 48). These vacancy lifetimes are already quite close to the theoretical estimate of 316 ps for an ideal divacancy in Ge (the corresponding bulk lifetime is 229 ps). 16 For an ideal neutral monovacancy we get the lifetime of 246 ps which is only 33 ps longer
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but we can estimate the order of magnitude in this preliminary assessment, which is considered a 2S → 2P transition: The number of excited atoms or the number of events will be: RN t = 4π 3 χ 2 e 2 I A ′ a 2 0 N t = 1.93 * 10 8 χ 2 N (t/second) (M/eV) (13) where N is the number of populated 2S states and t is the integration time. For a case that N ∼ 10 −6 mole, χ ∼ 10 −15 and M ∼ 10 −5 eV, we have the number of events is 11.6 per second. These excited 2P atoms will decay rapidly into 1S atoms, and the emitted Lyman-α photons can be registered as the number of events. Because the electric dipole transition of 2S → 1S is forbidden, the 2S state is semistable with a lifetime of about 1/7s, which is much larger than the lifetime, 2 * 10 −11 s, of the 2P states. The 2S semistable states can be populated with electrons [46]. Let us assume 10 −5 mole/sec of the 2S state are excited, which takes order of 1W power, then the populated 2S states are about 10 −6 mole at any given time. The set-up of the proposed experiment can be very similar to the experiments measuring the Lamb shift or the 1S-2S transition frequencies of atoms [47]. A major difference between the experiments is that for the existing experiments, microwaves are used to stimulate transitions between the 2S and the 2P
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All one sees are some broadened, weak peaks and gaps superimposed on a broad, continuous density of states. We now analyze the nature of the electronic states for these configurations. We start with the case which has only disorder. In Fig. 6 we plot |G R (k min , k max ; E)| 2 as a function of the energy E, for different values of δ (computation details were given in Sec. IV B 3). As already discussed, extended states are indicated by large values of this quantity, as well as a strong (roughly 1/δ 2 ) dependence on the value of the small parameter δ. Figure 6 reveals that as δ is reduced, resonant behavior appears in a narrow energy interval E = 0.02 − 0.36 meV. Panel (a) shows that results corresponding to δ = 10 −7 eV and δ = 10 −8 eV indeed differ by roughly 2 orders of magnitude, with δ = 10 −8 eV showing sharper resonance peaks. The difference between results for δ = 10 −8 eV and δ = 10 −9 eV shown in panel (b), is no longer so definite. The reason is simply that for such small δ, the denominator in the Green's function expression is usually limited by |E − E n,α,σ | and not by δ [see Eq. (12)], and the dependence on δ is minimal. Only if E is such that |E − E n,α,σ | < δ can we expect to see a δ dependence, and indeed this is observed at some
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remain separated. Figure 12 shows the 6 1 Σ + g radiative lifetimes obtained from the full (Eq. 4) calculation for J = 1 and J = 31, plotted as a function of their vibrational level. The large difference of approximately a factor of three between the J = 1 and J = 31 lifetimes for v = 40 and for v = 100 occurs due to a "switch" of the wavefunction, which is predominantly outer well for the long lifetime value, v = 40, J = 31 and v = 100, J = 1, and is predominantly inner well for the shorter lifetime value, v = 40, J = 1 and v = 100, J = 31. The results of these calculated radiative lifetimes are presented in Table I. V. RESULTS AND DISCUSSIONS A time-resolved double resonance spectroscopy technique has been applied to the sodium diatomic molecules in a heat-pipe and the radiative lifetime of the 6 1 Σ + g (7, 31) state was measured using a time-correlated photon counting technique. Also, lifetime calculations in various ro-vibrational levels of the 6 1 Σ + g state using bound-bound and bound-free transitions were performed. The radiative lifetimes are measured and the effect of argon gas collisions is eliminated with the Stern-Volmer extrapolation, explained in section III. The measured radiative lifetime (extracted from the zero-pressure of the extrapolation) of the 6 1 Σ + g (7, 31) state is found to be 39.56 (±2.23) ns and our calculations showed this value to be 42.8 ns. Experimental
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Across all AC sizes (Figure \ref{ac_zz_rel}(d)), S$_3$ lifetimes are never more than 2 ps and higher-energy state lifetimes are within 10-50 fs ($E/E_G >$ 1.5, where $E_G$ is the first excitation energy). In contrast, the S$_3$ lifetime is two orders of magnitude larger for ZZ150 ($\tau^{ZZ150}_{S_3} =$ 19.9 ps) and further lengthened by decreasing ZZ GQD size ($\tau^{ZZ54}_{S_3} =$ 39.1 ps, Figure \ref{ac_zz_rel}(c)). Increasing ZZ GQD size leads to a sharp decrease in S$_3$ lifetime to 3.1 ps for ZZ216 (Figure \ref{ac_zz_rel}(e)). \begin{figure}[h] \includegraphics[scale=1]{Fig3_ac_zz_rel_func.eps} \caption{Summary of relaxation dynamics for GQDs with armchair (AC) and zigzag (ZZ) termination edges. All phonon-induced relaxation occurs after a 1.6 $E/E_G$ photoexcitation, where $E_G$ is the energy of the lowest excitation. Colors from blue to red indicate population changes from 0 to 1, where 1 represents the entire population in one state at a given energy and time. Figures a) through c) plot the energy of the excited carrier as a function of time for AC114, ZZ150, and ZZ54, respectively, and list lifetimes ($\tau$) of important states. In all cases, S$_3$ corresponds to the absorption peak close to the electronic band gap and S$_1$ is an excitonic state. d) and e) plot carrier lifetimes ($\tau$) as a function of $E/E_G$ and size for AC and ZZ GQDs, respectively.} \label{ac_zz_rel} \end{figure} An intriguing effect of size on lifetimes occurs for high-energy states in ZZ GQDs (Figure \ref{ac_zz_rel}(e)). Wider energy spacing at $E/E_G >$ 1.7 (Figure S1) leads to increased lifetimes as a function of size, reaching picosecond lifetimes in ZZ216 (Figure \ref{ac_zz_rel}(e)).
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= 0f(z) SI I SOj valid solutionf() + f(-t) 2f' (2t) = 2 Q2 Consider continuously differentiable functions f R R such that for every & > 0, the following relation holds: ay Ta? JIf f dr dy dz = (f(a) + sin(a) - 1) () 122 + y D(a} where D(a) = {(1,y,2) : 22+y + 22 < &',lyl < 43} Which of the following is true? f(0) + f'(0) = 1 ... ##### QUESTION 2pointsSave AnswerA particular atomic state has energy of 2.20 eV above the ground state, and has a lifetime of 2.9 ns What is the fractional uncertainty of the wavelength of light emitted when it decays to the ground state? 5.2x10-86.Ox10-84.9x10-8 7.6x10-84.5x10-8 QUESTION 2 points Save Answer A particular atomic state has energy of 2.20 eV above the ground state, and has a lifetime of 2.9 ns What is the fractional uncertainty of the wavelength of light emitted when it decays to the ground state? 5.2x10-8 6.Ox10-8 4.9x10-8 7.6x10-8 4.5x10-8... ##### . Certain material is at 15 ° C. If its temperature increases twice, it will be... . Certain material is at 15 ° C. If its temperature increases twice, it will be at What is heat?... ##### Give the two substitution products (skeletal isomers) when the following compound is added to a solution... Give the two substitution products (skeletal isomers) when the following compound is added to a solution of sodium acetate in acetic acid.... ##### Perform the indicated operations. The voltage across a certain inductor is $V=\left(8.66 \underline{/ 90.0^{\circ}}\right)\left(50.0
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π = 9/2 − assignment is a better solution from the angular correlation fits. It also presents the longest lifetime, a newly measured 1170(300) fs, with respect to the other levels discussed in this section. An enhanced B(E2) value of 27 +15 −9 W.u. for the 689.6 keV E2 transition to the 5/2 − one-phonon state suggests this state as a member of the negative-parity IS coupling excitations. This large B(E2) value is also predicted by our SM calculations, together with the isoscalar character of the transition to the one-phonon states. − 2122.8 * 100 † Data taken from Ref. [38]. q Lifetime taken from Ref. [46]. These large B(E2) values support the IS character of the state. Decay to the ground state has not been observed. The strong E2 transitions to the one-phonon states confirm the IS character. 6. 1588.1 5/2 − IS state A lower limit for the lifetime of >1260 fs has been determined for this level, giving upper limits for the B(E2) values to the 3/2 − and 5/2 − one-phonon states of <8 and <13 W.u., respectively. Although the experimental data are not conclusive, our SM calculations predict relatively strong transitions and isoscalar character for the transitions to the one-phonon states, which support the IS character. Although the identification of M S states have already been discussed in [8], we include them for completeness. The previously proposed (5/2 − ) level at 1779.7 keV yields a new 1092-keV branch to the first 3/2 − 1 excited state that has been revealed
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supports an infinite number of image states exibiting the well-known Rydberg series form: E n =−͑13.6/ 16n 2 ͓͒͑⑀ −1͒ / ͑⑀ +1͔͒ eV, where n =1,2,3,... is the principal quantum number. Since E n ϳ 1 / n 2 , the states with higher n have weaker binding energies. This series converges to the vacuum level. The electrons are confined along the surface normal but can move freely parallel to the surface leading to discrete parabolic energy bands. Recently, Höfer et al. 2 applied two-photon photoemission techniques 3 to populate the coherent wave packets of image states close to Cu͑100͒ and Cu͑111͒ surfaces. The states observed in these experiments had n Յ 6 and binding energies of 15-40 meV. These surface states collapsed onto the Cu surface with lifetimes of a few femtoseconds. The states with larger n have longer lifetimes. 4 For example, for the Cu͑100͒ surface, the lifetimes of electrons excited to the image-potential states for n =1,2,3 are 1 Ϸ 40 fs, 2 Ϸ 110 fs, and 3 Ϸ 300 fs, respectively. 2 In this paper, we calculate the image potentials that exist in front of layered two-dimensional ͑2D͒ electron systems ͑ESs͒, which may be comprised of a 2D electron gas ͑EG͒ or a graphene layer. Of course, an electron near the surface would experience the combined effect of the Coulomb-type attractive image potential and the repulsive surface-barrier potential. 4 We show that the potential created by the polarization which a charge induces at the surface of a layered 2DES is qualitatively
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possible to obtain precise values of either E1 matrix elements or transition strengths from the measured lifetimes due to association of many transition probabilities with these quantities. However, the 4P states are the first two excited states which decay to the ground state only via one allowed transition each. There are precise measurements of lifetimes of these states are available in K atom. The lifetime of the 4P 1/2 state is reported to be 26.69(5) ns [44]. By combining this result with the experimental value of wavelength λ = 7701.1 A[28] of the 4P 1/2 → 4S transition, we find the E1 matrix element of the 4P 1/2 → 4S transition to be 4.110(5) a.u. against our calculated result 4.131(20) a.u. Similarly, the lifetime of the 4P 3/2 state is measured to be 26.34 (5) ns [44]. This state has an allowed transition channel to the ground state and it can also decay to the first excited 4P 1/2 state via both the electric quadrupole (E2) and magnetic dipole (M1) channels. It is found from our analysis that the transition probabilities of an electron due to the above forbidden channels from the 4P 3/2 state are very small and can be neglected within our estimated uncertainties. Therefore combining the measured lifetime of the 4P 3/2 state with the experimental value of the 4P 3/2 → 4S transition wavelength λ = 7667.0Å, we [35]; d [36]; e [37]; f [38]; g [39]; h [40]; i [41]; j [42]; k [43] obtain E1 matrix element of the 4P
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the 2 → 1 transition would result in the emission of −8 eV − (−12 eV) = 4 eV. Therefore, if the atom is initially in the _n_ = 3 state, it could emit photons of energy 3 eV, 4 eV, or 7 eV. 7. **D** The energy of the _n_ th level is given by the equation _E_ _n_ = _E_ 1/ _n_ 2, where _E_ 1 is the ground-state energy. Therefore, the energy of level _n_ = 2 (the next level above the ground state) is _E_ 2 = (−54.4 eV)/4 = −13.6 eV. Therefore, the difference in energy between the _n_ = 1 and _n_ = 2 levels is ∆ _E_ = −13.6 eV − (−54.4 eV) = 40.8 eV. 8. **C** The energy of a photon is given by the equation _E_ = _hf_ , or equivalently by _E_ = _hc_ /λ. Therefore, _E_ is inversely proportional to λ. If λ decreases by a factor of 2, then _E_ will increase by a factor of 2. 9. **A** The equation that relates the mass difference _m_ and the disintegration energy _Q_ is Einstein's mass–energy equivalence formula, _Q = mc_ 2. Because _c_ 2 is a constant, we see that _Q_ is proportional to _m_. Therefore, if _m_ decreases by a factor of 4, then so will _Q_. 10. **B** In order to balance the mass number (the superscripts), we must have 2 + 63 = 64 + _A_ , so _A_ = 1. In order to balance the charge (the subscripts), we
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at the Heidelberg EBIT consisted in the lifetime measurement of the first-excited energy level of boronlike Ar XIV. The forbidden transition at 441.26 nm arising from this level was used to monitor its depopulation (cf. Fig. 3). The high light collection efficiency of our setup allowed us to observe evidence of a small population of low-energy electrons trapped in the drift tube region of the trap, which are responsible for reexcitation and quenching of the low-lying metastable levels. The sensitivity of the decay curves to trap parameters makes it possible to use such electrons to probe ion losses from the trap and to correct this systematic contribution from the apparent lifetimes. In this way, an accuracy level of 0.22% is achieved at the determination of the lifetime of the Ar 13+ 2s 2 2p 1 2 P 3/2 level. The result is (9.597 ± 0.021) ms, in agreement with a recent measurement at the LLNL EBIT [3], but almost an order of magnitude more precise, allowing to distinguish between different theoretical models. This measurement sets a new standard regarding lifetime measurements in highly charged ions and offers the possibility of observing the influence of quantum electrodynamic (QED) contributions to the lifetimes of excited-states. Laser spectroscopy of hydrogenic ions at the Heidelberg EBIT In order to improve the resolution achievable in the study of transitions in the visible range, an experimental setup for high resolution laser spectroscopy of highly charged ions has also been installed. Heavy ions with few (or even only one electron) are ideal systems
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a shift in the B 2 Σ + potential minimum marked by the hollow dot in Fig. 7(a). To minimise the effect of errors in r e , the calculated B 2 Σ + state is shifted by this value prior to determination of the decay channels. All the tabulated theoretical values are performed following this transformation and the equivalent shift for the A 2 Π 1/2 state. The available decay paths and their relative strengths are shown in Fig. 8. The calculated lifetime of the B 2 Σ + 1/2 state of 120.3 ns is in good agreement with the lifetime of the J = 11/2 level measured 19 by Kelly and co-workers. Unlike B 2 Σ + 1/2 , there is no lifetime measurement currently available for A 2 Π 1/2 state. The calculated v = 0 lifetime (Fig. 7(c)) is 136.2 ns, slightly longer than the corresponding vibrational level in B 2 Σ + . The difference is consistent both with the change in the ω 3 term in the lifetime formula 36 on moving from the B 2 Σ + to the lower lying A 2 Π state and that the A -X transition dipole moments is slightly larger across the ground state v = 0 wavefunction. In addition, comparing a DUO calculation that includes the E 2 Π state with another that does not reveals that mixing with the E 2 Π 1/2 state at shorter range lowered the lifetime of the A 2 Π 1/2 state by around 5 ns.
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GENERAL: See also Table 8.3 [Table of Energy Levels] (in PDF or PS). Theory: See (1955HE1E, 1956KU1A, 1956PE1A, 1957BI1C, 1957FR1B, 1958WI1E). Recent Q-values are 93.7 ± 0.9 keV (1957CO59: 9Be(p, d)8Be), 90 ± 5 keV (1955TR03: 11B(p, α)8Be): the weighted mean of all measurements is 94.1 ± 0.7 keV (1957VA11). The width of the ground state is 4.5 ± 3 eV (1956RU41: 15% of Wigner limit), ≤ 3.5 eV (1956HE57). The second value leads to τm ≥ 2 × 10-16 sec. (Combination of these values places the mean life in the range τm = 2 to 4.5 × 10-16 sec.) An upper limit to the mean life is 6 × 10-15 sec (1955TR03). See also (1955AJ61). Absolute differential cross sections are reported for Eα = 0.15 to 3.0 MeV (1956HE57), Eα = 3.0 to 5.9 MeV (1956RU41), Eα = 12.9 to 21.6 MeV (1953ST52), Eα = 12.3 to 22.9 MeV (1956NI20), Eα = 20 and 20.4 MeV (1951BR92, 1951MA1B), Eα = 30 MeV (1951GR45, 1952GR1A), Eα = 38.5 MeV (1957BU13), and Eα = 44.7 MeV (1957CO63). See also (1958CH35). Phase shifts summarizing the work of (1956HE57), (1956RU41) and (1956NI20) are presented in (1956RU41) and (1958NI05). These three sets of data appear to join smoothly, but do not appear to fit well with the data of (1953ST52). For Eα < 3 MeV, only the s-wave phase shift is important. A careful survey in the range 146 - 202 keV reveals no effect of the ground state and places an upper limit of Γ ≤ 3.5 eV on
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energy E(v, J) lower than U J (R J ) has a finite lifetime before it will be decomposed due to a quantum tunneling effect. These states are called quasibound states and formally belong to the continuum. What is important is that during their lifetimes they can be regarded as bound states. When the energy E(v, J) exceeds the barrier maximum U J (R J ) then any bound state is not possible. Following Way and Stwalley 39 , we introduce a critical value of the rotational quantum number J c which obeys the two following inequalities: and In other words, for a given v, the state with the energy E(v, J c ) is the last of the quasibound states series supported by the barrier, and the state with the energy E(v, J c + 1) already belongs to the continuum. By solving Eq. 8 we obtain E(v, J c ) and estimate by extrapolation E(v, J c + 1). Respectively, the differences E(v, J c ) − E(0, 0) and E(v, J c + 1) − E(0, 0) may refer to the last observed and the first unobserved rotational predissociation experimental result. Molecular dynamics The time-dependent approach which is mathematically equivalent to the time-independent one can be regarded as a complimentary tool and is often used in studying photodissociation processes. Here, it serves as an alternative and quite illustrative method for testing results of our structural calculations. We start our consideration from the time-dependent Schrödinger equation written in the following form where Φ(R,t) is
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presented in Wong et al. (2017), see Fig. 5. Due to the change in the model and consequently the wavefunctions, the Einstein-A coefficients between the states of X 2 Π as well as the corresponding lifetimes have also changed. The energies of the lower rovibronic states (v ≤ 29 and J ≤ 99.5) of X 2 Π were replaced with the NOname ones which were calculated by Wong et al. (2017) using the programs SPFIT and SPCAT (Pickett 1991), based on the work of Müller et al. (2015). These states are labeled with EH to indicate that they were calculated from effective Hamiltonian, which replaces the label e (i.e. empirical) used in NOname for these EH levels. The energies of the other states of X 2 Π were shifted from the results of DUO, using the same strategy as Wong et al. (2017), to avoid energy jumps above v = 29 or J = 99.5. These shifted states were labeled with Sh while they were labeled with c (i.e. calculated) in NOname. We also replaced the DUO energies of the A 2 Σ + , B 2 Π and C 2 Π states with MARVEL energies where available, these states are labeled Ma in the .states file. The jumps between MARVEL and DUO energies in the excited electronic states are negligible. Therefore, we did not shift the other calculated energies of the A 2 Σ + , B 2 Π or C 2 Π states and labeled these states with Ca. For this line list we
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instance for small screenings ξ = 0.5 ) the energy E 10 goes to zero, while E 1,±1 -to minus infinity. This means that at strong disorder and finite screening, our 2D excitons are more stable for higher values of orbital quantum number m. Our numerical analysis shows that this effect realizes for higher excited states also. Of course, for higher n this effect is even more diversified as now we have all m < n at our disposal. The latter feature is also important for the proper functioning of different optoelectronic devices, based on disordered 2D structures. It can also be shown, that energy level crossing (having again much more crossing points than for n = 1 ) occurs also for higher n > 1. www.nature.com/scientificreports/ One more interesting feature is seen in Fig. 4b. Namely, while E 10 grows monotonously at screening radius ξ increase, the energy E 1,±1 (ξ ) is notably nonmonotonous having minimum around ξ = 0.5 , which depends on Lévy index µ . The higher is µ deviation from 2, defining the degree of disorder, the deeper is minimum. In other words, we see that the branch E 1,±1 has lower energy than that of exciton in the ordered 2D model without screening. It can be shown that this feature occurs also for E 2,±2 , E 3,±3 and possibly higher excited states. This shows that the synergy between disorder and (nonlocal 27 ) screening in two dimensions minimizes the energy (as compared to the ordered unscreened case) for
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+ ions, where a decoherence-free entangled state of two ions in the electronic 2 D 5/2 manifold was created to suppress ambient noise 4,5 . The relatively short 1.2 s radiative lifetime of the 2 D 5/2 state in Ca + and the requirement for high fidelity quantum gates limit the scalability and, ultimately, the sensitivity of LV tests with this scheme 5 . The highly relativistic 2 F 7/2 state in the Yb + ion is an order of magnitude more sensitive to LV than the 2 D 5/2 state in Ca + 7,8 and its radiative lifetime was measured to be about 1.6 years 26 . These beneficial properties were recently exploited in a 45-day comparison of two separate state-of-the-art single-ion optical 171 Yb + clocks, both with a systematic uncertainty at the 10 −18 level, resulting in a more than ten-fold improvement of the LV bounds 6 . In this work we present new bounds on LV in the electron-photon sector using a simple, robust and scalable measurement scheme in a single trapped ion experiment. We combine the high susceptibility of the 2 F 7/2 state to LV with an improved radio-frequency (rf) spin-echoed Ramsey sequence 8,27 to make a direct energy comparison between the nearly orthogonal atomic orbitals of the F -state Zeeman manifold within a single trapped 172 Yb + ion. In this manner, we fully exploit the most sensitive stretched m = ±7/2 states 8 and eliminate the requirement of optical clock operation or high-fidelity quantum gates. With the applied
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horizon of our universe. Besides, for a free particle with mass m 0 , its energy also assumes discrete values E n = n 2 h 2 32m 0 R 2 U . For instance, the minimum energy is E 1 ≈ 10 −72 eV for free electrons, which is much smaller than the minimum energy of photons 9 . It is interesting to see whether this tiny discreteness of energy makes the collapse dynamics more abrupt. Suppose the energy uncertainty of a quantum state is ∆E ≈ 1eV , and its energy ranges between the minimum energy E 1 and 1eV . Then we can get the maximum energy level l max ≈ 1eV 10 −33 eV ≈ 10 16 . The probability of most energy eigenstates in the superposition will be about P ≈ 10 −16 . During each discrete instant t P , the probability increase of the energy branch with tiny collapse is ∆P ≈ ∆E E P (1 − P) ≈ 10 −28 . This indicates that the probability change during each discrete instant is still very tiny. Only when the energy uncertainty is larger than 10 23 eV or 10 −5 E P , will the probability change during each discrete instant be sharp. Therefore, the collapse evolution is still very smooth for the quantum states with energy uncertainty much smaller than the Planck energy. On the consistency of the model and experiments In this section, we will analyze whether the discrete model of energy-conserved wavefunction collapse is consistent with
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the energy of the 2 + state in 136 Gd to be 165 keV. Figure 9 shows the expected half-life as a function of Q p . It is expected that for lifetimes longer than the limit marked by the grey line, beta-decay will dominate [46]. C. Theoretical Uncertainties It should be emphasized that our method contains no adjustable parameters; there are a few parameters which are set by experiment. These include Q p and the placement of the lowest few levels in the ground-state band of the daughter nucleus. Since the higher levels are energetically forbidden, even if they are needed in the calculation to ensure proper convergence, the half-lives and branching ratios are fairly insensitive to their placement. We shall now discuss the sensitivity of the calculated halflives and branching ratios to various quantities used in the calculations. For concreteness, we will focus on the [411] 3 2 level in 131 Eu. All other levels studied show similar sensitivities. The largest effect on the lifetime comes from the Q p value. The Q p value for 131 Eu is currently taken as 950 (7) keV [7]. The uncertainty of 7 keV leads to an uncertainty in the calculated lifetime of −7.5/ + 9.8 ms. This is a difference of roughly −22/ + 30%. Since a change in the Q p value also affects the energies of excited states, the change in branching ratio is much smaller. orbital, the effect is ±1.3 %. On the other hand, the placement of the 2 + level has
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−4 (eV) 2 , the condition (34) (or (32)), with n = 2, leads to ∆ξ (2) reach as deep as ∼ 10 −24 . Alternatively, one may wish to relinquish this high sensitivity of ∆ξ (n) for the sake of probing deviations from E 2 = p 2 + m 2 that are suppressed by m P at a much higher order. As an extreme example, one can imagine a model in which quantum gravity induced deviations become significant only at the seventh order of suppression by the Planck energy. Then, with σ-setting as high as 10 +8 , condition (34), with n = 7, dictates the values of E ∼ 10 21 eV and L ∼ 10 9 light-years, giving ∆m 2 reach ∼ 10 −10 (eV) 2 and ∆ξ (7) reach ∼ 10 −3 . Thus, flavor oscillating neutrinos with energy in the range of ZeVprovided they have originated at a cosmological distance of some 10 9 light-years-are capable of probing the effects of quantum gravity that may be as minute as, say, septically suppressed by the Planck energy. V. PROSPECTS FOR OBSERVATIONS IN THE MEDIAN FUTURE It is clear from the above discussion that, at least in principle, ultra-high-energy neutrinos of cosmic origin can serve as a remarkably sensitive probe of the Planck regime. In fact, within the next decade, thanks to the surge of progressively larger neutrino detectors, even in practice this intriguing possibility may become reality. To be sure, as yet no extraterrestrial neutrino of energy greater than a few
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alleged lifetime of 10 ns is extracted from τ = Q/ω, which is the classical ring-down time of a classical oscillator with frequency ω/2π and quality factor Q. This is not the implied lifetime T 1 of a quantum system with discrete energy levels. Furthermore, T 1 of different energy levels in an atom is different for different energy levels 3 , spanning a broad range of timescales, from tens of nanoseconds to seconds or minutes (metastable states). In a transition with multiple quanta the decay time observed in a statistical measurement will depend on the distribution of T 1 for all energy levels. Fundamentally, the decay of an atom could be observed in any timescale, since the exponential decay form extends all the way to infinity. Only the number of counts at longer timescales becomes exponentially small. Finally, in the paper we never mentioned anything with regard to single quantum accuracy. 3) "The magnetomotive response of the suspected 1.48 GHz mode is anomalous. Fig#3b shows the amplitude versus magnetic field where the authors claim that it demonstrates the expected quadratic dependence. The authors fail to point out and explain that the quadratic fit is not symmetric about the origin as expected and widely observed..." The fit to the data in Fig. 3b is indeed a quadratic fit (quadratic defined as y = a 0 + a 1 B + a 2 B 2 ), as we clearly mention in the caption. This is different from a fit to B 2 . The difference is the
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38 . Here, the center of gravity of the experimental scaled lifetime (≈ 25 fs eV 2 ) is close to the GW +T curve for energies higher than ∼ 0.7 eV. Below this value, towards E F , the experimental lifetime goes down faster than the GW + T one. However, the experimentally observed less rapid increase in the lifetime towards E F in comparison to the Fermi-Liquid model in this energy region is qualitatively well reproduced by the theoretical results, which follow the slope of the data reasonably well. Some discrepancy for energies smaller than 1.0 eV can be caused by both the approximations made for the self-energy (e.g., electron-phonon coupling can be important here) and secondary processes which effect the experimental observed relaxation times. 37 Note that this discrepancy can be eliminated by adding to ImΣ GW +T a weak energy dependent term of ∼ 5 meV. B. Rhodium The calculated band structure and DOS of Rh are shown in Fig. 7. Compared to Mo, the Rh d-band is more narrow (∼ 7 eV). Almost all the d states are occupied except for a small fraction which constitutes the sharp peak at 0.46 eV and pronounced edge of the d-band at 0.9 eV above E F . At higher energies unoccupied bands show comparable contributions of sp and d states. Below E F , the sp-states contribution to the bands is negligible in comparison with the d states. Another peak comparable with the previous one is situated at −2.58 eV. In Fig. 8,
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b 3 Σ + , v = 5 system. To verify the fraction of triplet character, f b5 (F, n, ±), of the hyperfine levels, lifetime measurements are performed. The lifetime τ of a certain hyperfine level characterized by the quantum number F, the counting integer n and the parity, is given by and largely differs for the different hyperfine levels. The lifetime of the A 1 Π, v = 0 state has been measured to be τ A0 = 1.90(3) ns. 7 The amount of singlet-triplet mixing in this v = 0 state is very small and its lifetime is the radiative lifetimes of the unperturbed vibrational ground state. The found value agrees very well with the theoretically predicted value of 1.89 ns 13 . These calculations predict that the lifetime of the bare A 1 Π, v states increases with v, and that τ A6 = 2.05 ns. The lifetime of the b 3 Σ + , v = 0 state is found to be τ b0 = 190(2) ns. 8 The lifetimes of the v = 1 and 2 states are mea- sured in the context of this study using time-delayed ionization to be τ b1 = 189(4) ns and τ b2 = 191(8) ns. For these three states, the effect on the lifetime due to the perturbation with the A 1 Π, v + 1 state can be neglected. In the absence of the perturbation with the A 1 Π, v = 6 state, the b 3 Σ + , v =
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from Table 4. Lifetimes for the calculated states are displayed in Fig. 5. For the (1 − , 0) state, the lifetime is infinite within the present description. The large difference between the lifetimes of the (2 + , 0) state and, for example, the (0 + , 1) state is essentially due to the factor (E i − E f ) 5 in expression (7) of the E2 transition probability. Indeed the vibrational energy difference in the transition from the (0 + , 1) state is about 10 times larges than the rotational energy difference in the transition from the (2 + , 0) state which leads to an order of magnitude of 10 5 for the ratio of the lifetimes. Except for the ground-state rotational band, where the lifetimes decrease rather fast from the very high values obtained at low orbital momenta (about 3300 years for L = 2), the lifetimes do not depend much on L. They are rather constant up to about L = 15. Then they slowly decrease till about L = 30 before starting increasing again slowly. The decrease is less than a factor of five for v = 1, three for v = 2 and two for v = 3. For v = 1, the lifetimes are of the order of twenty-two days at small L and of four days near L = 30. When v increases from 1 to 3, they decrease progressively for L ≤ 18 and increase progressively for L > 18. Conclusion In this paper,
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an order of magnitude larger than the empirical value E A 1 emp ≈ 6 × 10 −4 extracted from the 5700 a half life of 14 C. We also find that the matrix element E A 1 depends on the energy of the first excited 1 + state in 14 N. For the three different cutoffs Λ χ = 450, 500, 550 MeV this excited 1 + state is at 5.69, 4.41, 3.35 MeV, respectively (compared to 3.95 MeV from experiment). As the value of E A 1 decreases strongly with decreasing excitation energy, a correct description of this state is important for the half-life in 14 C. Summary. -We studied β − decays of 14 C, and 22,24 O. Due to 2BCs we found a quenching factor q 2 ≈ 0.84−0.92 from the difference in total β decay strengths S − − S + when compared to the Ikeda sum rule value 3(N − Z). To carry out this study, we optimized interactions from χEFT at NNLO to scattering observables for chiral cutoffs Λ χ = 450, 500, 550 MeV. We developed a novel coupled-cluster technique for the computation of spectra in the daughter nuclei and made several predictions and spin assignments in the exotic neutron-rich isotopes of fluorine. We find that 3NFs increase the level density in the daughter nuclei and thereby improve the comparison to data. The anomalously long half life for the β − decay of 14 C depends in a complicated way on 3NFs and 2BCs. While the former increase
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measured. In similarity with the other $N=89$ isotones, this first excited state can be considered part of the g.s. band built on the $\nu5/2[523]$ configuration. \textbf{The 175.3-keV level \soutthick{$(1/2^-_1)$} $(3/2^-_1)$:} The measured lifetime for this state is surprisingly short. Assuming a pure $E2$ character for the $1/2^-_1 \rightarrow 5/2^(-)_1$ transition (the only branch observed decaying from this state), yields a B(E2) value of 400(20)~W.u. This value is significantly larger than the RUL of BE(2)$<300$~W.u. and is a strong indication that this transition may have a different multipolarity. In order to bring this value in line with the measured B(E2) in neighbouring nuclei, the lifetime would have to be 5 to 10 times longer, which seems very unlikely. Careful checks were made in the analysis to confirm accurate timing for low-energy transitions. A more likely explanation is that the spin assignment of this parent state is incorrect. The $(1/2^-)$ assignment comes from angular correlations reported in Ref.~\cite{Robertson1986}. The authors claimed to have observed isotropic ($A_{22}=0$ and $A_{44}=0$, where $A_{ii}$ are the normalized coefficients of the Legendre polynomials) angular distributions when the 175~keV transition was used as the gate, a strong suggestion that this level has spin 1/2. However, taking the uncertainties the authors reported as a sensitivity of $|A_{22}|, |A_{44}|<0.05$, a large number of spin combinations and mixing ratios ($\delta_1$ and $\delta_2$) can fit their measured distributions. As a relevant example for this discussion, angular correlations for a $5/2 \rightarrow 3/2 \rightarrow 5/2$ cascade are almost flat for $\delta_1=\delta_2=0$, with $A_{22}=0.01$ and $A_{44}=0$, or $-0.05<A_{22}<+0.05$ for a
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1, n α is the initial occupancy of the shell α, and the emitted electron has energy ε. A configuration interaction calculation using the atomic code AMBiT [39] indicates initial shell occupancies for the uranium ground state of 7s 1.79 6d 1.09 3/2 6d 0.12 5/2 5f 2.73 5/2 5f 0.27 7/2 . With these values of n α , we calculate the electronic factor from (2) The internal conversion is dominated by the contribution of core 6p shells with emission of a d-wave electron [40], unlike in thorium where internal conversion mainly comes from from the 7s shell [41]. Using the measured internal conversion lifetime of 26 min we obtain B(E3, m → g) = 0.036 W.u., consistent with previous calculations [40]. At this point it is worthwhile to make a brief aside and calculate the natural linewidth of the transition. Using the standard formula [29,38] we obtain The natural lifetime is therefore much larger than the half-life of the 235 U nucleus, and is even longer than the lifetime of the Universe. However, this longevity is only realised in special systems, such as a bare uranium nucleus: electronic bridge interactions will generally dominate. Indeed, the mere presence of atomic electrons may induce virtual internal conversion rates several orders of magnitude larger than suggested by (3) [22,23]. III. ELECTRONIC SPECTRUM In order to overcome the smallness of (3), and enable laser spectroscopy of this nucleus, we seek an electronic structure which maximises the electronic bridge mechanism. In this work we concentrate on U 7+ ,
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Q, and filling the levels as follows: a) First we list the (E nx , τ nx ), in order of increasing n x (and therefore of increasing energy and decreasing lifetime.) This list is truncated at an n x = n x,max whose lifetime is less than 0.01 sec. b) Next we form a list of 2D levels (n x , n y ) by choosing those for which The levels in this list are occupied in order of increasing energy and according to Fermi statistics, see eqs. A5 A6. We choose the dot Fermi level so that the number of electrons is the desired Q. It is supposed that, for the long lifetimes observed in the experiment, the electrons have time to lose energy by phonon collisions and occupy the quasibound states of lowest energy. Then, as described in Appendix A, we determine the lifetime for one electron to escape from the dot. This involves a weighted average of the level lifetimes, according to the occupancy of each level at the experimental temperature, T ′ = 100 mK. To produce a sequence of decays for comparison to experiment we proceed as follows: i) we start with a dot containing a number of electrons, Q 0 , chosen large enough so that the lifetime for one electron to escape is smaller than those observed in experiment. ii) We redetermine the barrier and dot configuration for Q = Q 0 − 1 electrons, as described in the above paragraph and determine again the corresponding lifetime for
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3, 4, 5 are shown, n r = 2 having the longest lifetime and n r = 5 having the shortest. The dashed line shows the ratio t 1 /(ν 1 U ) which should be less than unity for the expression to be valid. As a comparison, the resulting lifetime obtained in the wideband limit Eq. (20) is also shown as the dash-dotted line. The most interesting part of the result shown in Fig. (15) is the sudden decay of the lifetime. This is, as was the case in the wide-band limit, a result of the diverging density of states ρ(ǫ 2 (k)) near the band edge. For lattice potentials deeper than v 0x ≈ 20, there is no phase space (no available final energy levels for the excited particle), available for the first order decay. To find out the lifetime for larger values of v 0x , second order perturbation theory is needed. Consider again the same initial state |i as above. Adhering to energy conservation arguments, there are three different, mutually orthogonal, final states reachable through a second order process, |i , |i . The corresponding decay rates w 1,2,3 are obtained from the text-book relation Numerical evaluations of the decay rates reveal that the dominant contribution to the total decay rate w tot = w 1 + w 2 + w 3 comes from w 1 . The reason for this is easily understood; the contribution from w 2 is small due to destructive interference of time reversed processes while the smallness
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over {1 "." "6 " times " 10" rSup { size 8{"–19"} } " J"} } =" 3" "." "3 " times " 10" rSup { size 8{"–6"} } " eV" "." } {} (12) Discussion The lifetime of 1010s1010s size 12{"10" rSup { size 8{ - "10"} } s} {} is typical of excited states in atoms—on human time scales, they quickly emit their stored energy. An uncertainty in energy of only a few millionths of an eV results. This uncertainty is small compared with typical excitation energies in atoms, which are on the order of 1 eV. So here the uncertainty principle limits the accuracy with which we can measure the lifetime and energy of such states, but not very significantly. The uncertainty principle for energy and time can be of great significance if the lifetime of a system is very short. Then ΔtΔt size 12{Δt} {} is very small, and ΔEΔE size 12{ΔE} {} is consequently very large. Some nuclei and exotic particles have extremely short lifetimes (as small as 1025s1025s size 12{"10" rSup { size 8{ - "25"} } s} {}), causing uncertainties in energy as great as many GeV (109eV109eV size 12{"10" rSup { size 8{9} } "eV"} {}). Stored energy appears as increased rest mass, and so this means that there is significant uncertainty in the rest mass of short-lived particles. When measured repeatedly, a spread of masses or decay energies are obtained. The spread is ΔEΔE size 12{ΔE} {}. You might ask whether this uncertainty in energy could be
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significant. The larger the Yukawa coupling y 1 and y 2 , the larger the difference. In these cases, both λ(t) and λ(t) have no minimum for energy scale below the Planck scale. The energy scale of bounce, Λ B , is chosen as the Planck scale for these two cases. We note that the positive sign of λ − λ shown in Fig 4 means that the lifetime calculated using λ in these plots is longer than that computed using λ. In Table. IV, we list more numerical results for the SM and for some benchmark points in the SDFDM model. As a comparison, we also list results just using λ(t). We can see that usingλ(t) and Z 2 in the effective action leads to some differences in the probability of false vacuum decay. For SM case, we can see that the lifetime of EW vacuum computed using effective action is shorter than that just using λ(t) althoughλ(t) and λ(t) are very close at high energy scale. This is caused mainly by the presence of Z 2 in the effective action. In the SM, (Z 2 ) 2 term in Eq. (34) is about 0.9 which makes S cl smaller and leads to a larger decay rate. For SDFDM model, we find that in the region that y 1 and y 2 are less than about 0.3, the situation is similar to the SM case. When y 1 and y 2 are larger, the Z 2 factor becomes greater than 1. In this case the
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1 / 40 =8.99E9 ( N-m 2 / C 2 ) The energy increment required to create such a pair includes the mass energy, the potential energy, and the kinetic energy of revolution: E = 2mc 2e 2 / 40r + ½ m 2 r 2 = 2mc 2 -½ e 2 / 40r Inspect these two equations for an e+e-mass of 1.02 MeV and a dipole frequency range from the mid IR to the upper UV, 200 -3000 nm. The resulting radius of gyration, (r is not the Fresnel surface coordinate here), ranges over a fraction (0.11 to 0.68) of r0, where r0 is the classical electron radius 2.82E-15 m. (Physically, r0 is that separation where the potential energy of an electron exactly balances its mass energy.) The energy increment is a deficit which must be borrowed from Heisenberg uncertainty at the expense of a very short lifetime of the pair. The lifetimes are estimated from ℏ⁄ and range from 33E-20 to 5.4E-20 seconds. This is a very small fraction of the period of the dipole radiation, which is of the order of 1E-14 seconds. If electron-positron pairs play a role in the propagation of Huygens wavelets they must do so with very short in-phase kicks to the dipole field, each of which adds a new wavelet with retarded and advanced potentials. This may be expected, since the Aharonov Bohm effect tells us that the electromagnetic potential of a wave, including the phase, is physically real and will persist though the field disappears. Emergent pairs
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lifetimes have been calculated to lie in the range of a couple of seconds (Table 2), which should be much easier to measure than the hour-range lifetimes of the 3s 2 3p 3 2 D o 3/2,5/2 levels. Figure 5 reveals that the scatter of the predictions is sizeable in this case, too. The latest data point is from a data base without a creation date of this entry. The data entries can be identified from Table 2. The color bars indicate a visual mean value of the more recent results and a ±10% interval for each of the levels. Previous Experiments on Other Ions There are no experimental data on the two level lifetimes of primary interest in S II yet. The E1-forbidden transition rates scale steeply with the transition energy ((∆E) 3 or (∆E) 5 for M1 and E2 transitions, respectively), and that energy interval in turn scales steeply with the nuclear charge Z (fine structure intervals grow with Z 4 ). The level lifetime drops correspondingly steeply with increasing Z, and measurements with established techniques have been feasible so far only for a range of multiply charged ions. The lowest-Z measurements on ions of the P I isoelectronic sequence have been reported for Z = 25 (Mn), and the highest-Z ones for Z = 36 (Kr). Figure 6 shows comparisons of these lifetime data [15][16][17][18][19] obtained by work with a variety of ion traps (electrostatic (Kingdon) trap, heavy-ion storage ring, electron beam ion trap) with various predictions. To accommodate all experimental and computational
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0 set by the long range asymptotic behaviour of the van der Waals interaction (see Fig. 1). This trend is confirmed by our direct calculations of the molecular life- (9) and (12), respectively. The magnitude of the bare state lifetime τres is indicated by the dotted line. For the purpose of comparison, we have slightly corrected the axis of the magnetic field strength in such a way that the theoretical resonance position recovers the measured value of B0 = 15.5041 mT [5]. time shown in Fig. 2. The exact lifetime can be obtained from the energy dependence of the inelastic scattering cross section between any pair of open decay channels. We have chosen transitions between the open d wave channels associated with the internal atomic quantum numbers (2, −2; 2, −1) and (2, −2; 2, 0). Fitting the standard Breit-Wigner formula to the resonance peak at the binding energy E b (B) [2,5] of the metastable Feshbach molecules then determines the decay rate γ(B) = 1/τ (B) through its width. The squares in Fig. 2 indicate lifetime measurements by Thompson et al. [1] in which the dimer molecules were produced in a dilute vapour of 85 Rb atoms at a temperature of 30 nK using a Feshbach resonance crossing technique (see, e.g., Refs. [6,7]). Our theoretical predictions agree with the experimental lifetimes in both their systematic trends and magnitudes. Figure 2 shows the large differences in the order of magnitude of τ (B) from roughly 30 ms to only 100 µs in the experimentally accessible [1]
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Relativistic configuration interaction calculations of lifetimes of Si− 3p3 bound excited states Relativistic configuration interaction lifetimes of Si− 3p3 2D3/2, 2P1/2 and 2D5/2 states are calculated (using M1 and E2 length gauge transition probabilities) to be 162 s, 23.6 s and 27.3 h, respectively. Detailed analysis of individual configurational contributions to transition probabilities are used to improve the bases of the 4S3/2 and 2D3/2 calculations resulting in excellent E2 gauge agreement and a final M1 transition probability that is nearly three orders of magnitude larger than its initial Dirac–Fock value. Relatively poor E2 gauge agreement for transitions between the two J = 3/2 states and 2P1/2 are found to have negligible impact on the 2P1/2 lifetime, as length gauge transition probabilities remain stable in the final stages of the calculation (changing only by a few per cent while the velocity value shifts by 30% or more).
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states have lifetimes that scale as 1/β 2 , where β is the size of the asymmetric perturbation. To get lifetimes in SI units, the conversion is (1 unit of time in atomic units = 2.42 * 10 −17 sec). In an experiment, this lattice could experience small noisy perturbations and still have impressively long-lived states in the continuum. 2 As can be seen in Fig. 2, in the energy interval π 2 2 ≤E≤ 4π 2 2 at the Γ point, there are three propagating modes, corresponding to modes with wavevectors In this energy interval there are four localized reaction region eigenstates. These states are shown in Fig. 10. As one might expect from their structure, they are long-lived quasibound states in the continuum (similar states have been called QBICs [39]), and they come in odd-even pairs. If we include the evanescent modes k 0 −2 = k 0 2 = i √ 4π 2 − 2E, then the "S-matrix" in Eq. (13) is a 10×10 matrix, which contains in it a 6×6 unitary scattering matrix that has the form where, for example,R BB is the 3×3 matrix of reflection amplitudes between the three modes ν = 0, ±1 for particle waves entering the lattice from below. It can be writtenR where m and p refer to the channels ν = −1 and ν = +1, respectively. Here R BB mm is the reflection amplitude for a wave entering the lattice from below in channel ν = −1 and then reflecting back into channel ν
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of the 5 − state. However, the contribution of the configuration involving ν1h 9 2 in both the ground state and the 5 − state is calculated to be less than 1%. In the present work, the transition probability for the above mentioned 5 − level to the 2 − ground state was calculated by considering the resultant 33 keV γ transition as either of E4 or M3 in nature. The said transition probabilities come out to be 51.4 µ 2 N f m 3 for the M3 transition and 8545 e 2 f m 8 for the E4 transition respectively. These values correspond to 1.3×10 4 years (M3) and 2.5×10 5 years (E4) of lifetimes respectively for γ decay from the predicted 5 − level. These values corresponding to the 150 keV energy for the isomer yield lifetime of 5.7 days (M3) and 31 days (E4) respecively. However, while considering an energy value of 454 keV for the isomer the above lifetimes go down to 3.5 m (M3) and 2.1 m (E4) respectively. The Weisskopf estimates for these respective M3 and E4 transition rates were also calculated and comes out about 1315 µ 2 N f m 3 (M3) and 39500 e 2 f m 8 (E4) respectively. These, in turn, corresponds to the γ decay half lives as 5.0×10 2 years (M3) and 5.4×10 4 years (E4) respectively. Hence, both from the half life values obtained from shell model as well as from single particle estimates, it can be concluded that the lowest 5
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He+ 2p state lifetime by a quenching-asymmetry measurement. An interference asymmetry in the angular distribution of the Ly-α quenching radiation emitted by He + ions in the metastable 2s 1/2 state is measured to high precision to obtain the lifetime of the 2p 1/2 state. The derived lifetime of (0.997 17±0.00075) ×10 -10 s is the mostaccurate available for a fundamental atomic system. A detailed discussion of systematic corrections is included
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than 5 keV. Some discrepancies can be explained at low energies due to the low efficiency of the small E1 events. Distributions of ∆T are shown in Fig. 5 (d) together with results from exponential fits for only large ∆T events. This is because the two-pulse selection strongly suppress small ∆T events. Only events with ∆T>600 ns are fitted for the E2<13 keV distributions. Because twopulse discrimination is more efficient for large E2 events, E2>13 keV events with ∆T>350 ns are fitted. The fitted lifetimes are 299±11 ns and 115±25 ns for the E2 = 6.28 keV and E2 = 20.19 keV states, respectively, where only statistical uncertainties are considered. B. three-pulse events If we consider the level scheme of 228 Ac as Fig. 1, the 20.19 keV state transits to the 6.67 keV state before decaying to the ground state. Because of O(100 ns) lifetime of the 20.19 keV state transition, the 20.19 keV to 6.67 keV transition must have the same lifetime. However, the number of two-pulse events with E2>13 keV is only a few percent of those with E2<13 keV. Considering the relative intensities of the 228 Ra β-decay shown in Fig. 1, the observed isomeric transition of the 20.19 keV state is only O(1%) of the total β-decay to the 20.19 keV state. This may indicate that the 6.67 keV state is also an isomeric state. In this case, three distinct emissions have to occur, but at a rate that cannot be seen in the current analysis. Only O(1%) of these events
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Be-like ion, and E = E γ 1 + E γ 2 is the 3 P 0 → 1 S 0 transition energy shared by the two photons γ 1 and γ 2 given in eV. The theoretical E1M1-lifetimes obtained from equation 1 by using theoretical excitation energies E [12] are shown by the (red) solid line in figure 2. For Xe 50+ one obtains a lifetime τ E1M 1 of ≈ 30 s. The predictions by Laughlin were made within a non-relativistic framework using LS-coupling which is not appropriate for heavy ions. In particular, equation 1 was derived under the assumption that the 2s2p 3 P 0 and the 2s2p 3 P 1 state are degenerate levels. Clearly, this assumption is inappropriate for Xe 50+ where the splitting (E 01 ) is ≈ 23 eV [12] which is ∼ 20% of the 2s2p 3 P 0 excitation energy (≈ 104 eV) [12] (figure 1). Based on the formulas that were provided by Laughlin [11] we derived a more general formula for τ E1M 1 which takes into account the splitting E 01 , i.e. It can be easily seen, that for E 01 → 0 equation 2 approaches equation 1. With the 2s2p 3 P 0 and 2s2p 3 P 1 level energies from the relativistic many-body-perturbation calculations of Safronova et al [12] (color online) Non-relativistic lifetime predictions for 1s 2 2s2p 3 P 0 → 1s 2 2s 2 1 S 0 E1M1 two-photon transitions. Open circles represent calculations by Schmieder [10] for Z=12
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element term in our case. However, we do not see the QPI pattern to be present apart from these two energy scales, unless a large lifetime broadening is used in the calculation to smear out the quasiparticle states. (b1)-(b2) Corresponding experimental QPI data at these two energy scales for electron doped pnictide at x = 0.06 taken from Ref. 25. Experimental data show all four q vectors depicted in Fig. 1 at two energies, although their relative intensities are representative of our calculations in (a1) and (a2), respectively, due to low experimental resolution. As our phenomenological calculation do not capture the actual intensity of QPI maps, we normalize all QPI maps to their maximum. (c1)-(c2) Experimental data 40 for a 111 compound LiFeAs at two bias energies ΩQP I =-1.9 meV in (c1), and ΩQP I =-11.7 meV in (c2). Despise the differences in the FS topology for this 111 and our 122 system, we see that both results are in qualitative agreement. The experimental value of ∆ for LiFeAs is 10 meV, which makes the result (c2) slightly above the SC region. 40 Once magnetic field is applied, the relative intensity of q 1 with respect to that of q 3,4 evolves with the strength of the field, it has been observed in Fe(Se,Te) compounds, in accord with our calculations. [17][18][19] The proximity of q 4 to the reciprocal vector (2π, 0) and its equivalent directions, which has been observed both in pnictide 25 as well as in iron-chalcogenide 17 , has been ar- in
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0.14 T/s. The long dotted lines indicate the minima of P th whereas the short dotted lines indicate the minima of P −10,10 . by The probability for a spin to pass into the excited level m can be estimated by τ −1 m e −E10,m/kBT , where E 10,m is the energy gap between the levels 10 and m, and τ m is the lifetime of the excited level m. One gets: (4.10) Note that this estimation neglects higher excited levels with |m| < 8. 11 Fig. 4.19 displays the measured P th for the three isotopic Fe 8 samples. For 0.4 K < T < 1 K we fitted Eq. 4.10 to the data leaving only the level lifetimes τ 9 and τ 8 as adjustable parameters. All other parameters are calculated using the parameters in Sect. 4.1.3.2. We obtain τ 9 = 1.0, 0.5, and 0.3 × 10 −6 s, and τ 8 = 0.7, 0.5, and 0.4 × 10 −7 s for D Fe 8 , st Fe 8 , and 57 Fe 8 , respectively. These results evidence that only the first excited level have to be considered for 0.4 K < T < 0.7 K. Indeed, the second term of the summation in Eq. 4.10 is negligible in this temperature interval. It is interesting to note that this finding is in contrast to hysteresis loop measurements on Mn 12 [120,167] which suggested an abrupt transition between thermal assisted and pure quantum tunneling [168]. Furthermore, our result shows clearly the influence
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in Table 1 we obtain the lowest order values for multiphonon relaxation of 8, 7, 6, and 5 for lead The measured lifetime for the 4 I 13/2 upper laser state of erbium decreases, whereas lifetime for the 5 D 0 state of europium increases with increasing phonon energy in glass matrices. Moreover, the change of the 4 I 13/2 measured lifetime by a factor of 8 when the phonon energy changes from 775 cm −1 to 1320 cm −1 is very high as compared to about two-fold increase of the 5 D 0 lifetime. This marked dissimilarity stems from the competition between radiative and multiphonon relaxation processes that remove the excitation of metastable levels of rare earth ions. The multiphonon relaxation consisting of simultaneous emission of the highest energy phonons in the lowest order process to cover the energy separation between a luminescent level and the next lower-lying energy level is consistent with the "energy gap law". Considering the 4 I 13/2 metastable level located at about 6500 cm −1 above the next lower energy level 4 I 15/2 and phonon energies gathered in Table 1 we obtain the lowest order values for multiphonon relaxation of 8, 7, 6, and 5 for lead germanate, lead silicate, lead phosphate and lead borate glasses, respectively. We attribute observed dissimilarity of the 4 I 13/2 lifetime values to the contribution of multiphonon relaxation rates, especially in lead borate and lead phosphate glasses where the orders of the process are relatively low. The results are completely different for the
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total energy of the two-particle quantum system, obtaining Let us also remark that the total BH energy E = − M 2 is negative and different from the BH inert mass M. Thus, by using Eq. (51), one gets from Eq. (127) Remarkably, Eq. (129) is exactly the same Eq. (53). Thus, one can use again Eq. (51) in order to re-obtain Eq. (54). In complete consistence with Section 2, Eq. (127) is interpreted in terms of a particle, the "electron", quantized on a circle of length which is equal to Eq. (66) which was derived in Section 2 through a full quantum analysis. The semi-classical physical interpretation of the QNMs in Eq. (125) is of a collection of damped harmonic degrees of freedom, which releases an intuitive physical picture of a BH as a whole [32]. The larger is |ω n | , the shorter is the lifetime, as one expectes from physical intuition [32]. Thus, the increasing of absorptions "triggers" shorter and shorter lived modes, while subsequent emissions of Hawking quanta "trigger" longer and longer lived modes. It has been shown that different approaches give the same results concerning the BH mass and energy spectra. It also means that one gets different pictures concerning the SBH. In the classical framework of Einstein's general relativity, a BH is a "dead object" with inert mass. It is a definitive prison where anything that enters cannot escape. This means that it can only become more massive and bigger with time. In the semi-classical approximation, Hawking has shown,
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most important transition determining the lifetime is ν2p 1/2 → π2p 3/2 . In the presence of pairing correlations, the energy of the corresponding 2qp excitation is increased and the lifetime becomes longer for increasing values of f iv . This transition remains the dominant one also in the case of 80-84 Ni, but is less influenced by pairing correlations due to the shift of the chemical potential to the shell above. As a consequence, the lifetime is insensitive to the value of f iv . Turning now to Sn isotopes, the situation for 136-140 Sn is qualitatively similar to the case of ing pairing on tends to reduce the lifetime; further increasing the value f iv the competing effect on quasiparticle energies prevails and the lifetime increases, as was already noticed in Fig. 2(c). The QRPA+QPVC results are shown in Fig. 2(b) and (d), again for different values of the isovector pairing strength. With the inclusion of QPVC, the half-lives are reduced systematically. The experimental half-lives of Ni isotopes are well reproduced, while the half-lives of Sn isotopes are still overestimated, although much less than before. We remind that no isoscalar pairing has been introduced so far. At the QPVC level, we find that the effect of isovector pairing is greatly reduced; we will discuss the underlying reason making reference to Fig. 4, in which 70 Ni has been picked up as an example. In Fig. 4, the GT strength distributions (with respect to the daughter nucleus), the cumulative sums of the GT strengths, and
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29\,eV respectively. Therefore the expected energy of the isomer transition (7.8\,eV) is above the ionization energy of the neutral Thorium atom.}. In the \emph{Th$^{3+}$ ion} for $E=7.6$\,eV the expected lifetime is the same as in a bare nucleus described above, if the valence electron is in the ground $5f_{5/2}$ state. It decreases by a factor of 20 if the valence electron is in the metastable $7s$ state~\cite{Porsev10_3}. In the \emph{Th$^{+}$ ion} the lifetime decreases $10^2-10^3$ times for $E=3.5$ and $E=5.5$\,eV~\cite{Porsev10}, there is not sufficient data about the Th$^{+}$ ion electronic structure to calculate the lifetime at $E=7.8$\,eV. Similar calculations for the \emph{Th$^{4+}$ ion}, as it will occur in the solid-state approach presented here, have yet to be performed. There were several experimental attempts to determine the $^{229m}$Th lifetime without knowledge of the exact isomer energy. Inamura et al. measured a temporal variation of the $\alpha$ decay activity in a sample of $^{229}$Th after excitation in a hollow-cathode electric discharge~\cite{Inamura09}. They propose a value of $T_{1/2}=2\pm 1$\,min for the half-life of the isomeric state which corresponds to a total excited state decay rate of $\gamma=6\,^{+6}_{-2}\cdot 10^{-3}$\,s$^{-1}$. Kikunaga et al. studied the $\alpha$ spectrum of chemically ``fresh" $^{229}$Th produced in the $\alpha$ decay of $^{233}$U~\cite{Kikunaga09}. The authors obtain $T_{1/2}<2$\,h with $3\sigma$ confidence. The latter experiment was not suited to measure half-lives on the order of minutes because of the long chemical preparation process. The analysis of both these experiments is significantly complicated by the presence of various chemical compositions of Thorium, mainly ThO$^+$, Th and Th$^+$ in~\cite{Inamura09} and
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and so it has a lifetime of 2.2 × 10 6 seconds, or ∼25 days, which is several orders of magnitude longer than any of the other levels. IV. SUMMARY AND CONCLUSION We have calculated energies, g-factors, and lifetimes of several low-lying atomic levels in Lr + . A striking agreement between the calculated FSCC and CI+MBPT energies is achieved. Similar calculations for the lighter homologue Lu + support the high accuracy of both approaches. In view of the prospects opened up by the forthcoming experiments, we identified two strong ground-state transitions in Lr + , leading to 7s7p 3 P 1 and 7s7p 1 P 1 states at 31540 cm −1 and 47295 cm −1 , respectively, that should in principle be amenable for experimental verification. In this case and for practical reasons, however, level searches are likely to focus on the 3 P 1 state with a convenient transition wavelength and of smallest uncertainty. AB is grateful for the support of the UNSW Gordon God-frey fellowship.
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the fitting procedure. In general, the agreement with the data is better for the states with even J values, while the calculated theoretical energy levels with odd J values appear systematically above the experimental excitation energies. We carry out a more detailed comparison for both the energies and E2 properties in the region where shape coexistence shows up most clearly (180 ≤ A ≤ 188) in the later part of this section. A more stringent test than a good reproduction of the energy systematics comes from calculating those observables that probe the corresponding wave functions and a comparison with the data. Recent experimental efforts have given rise to absolute B(E2) values along the yrast band, deduced from lifetime measurements [40,41], as well as through Coulomb excitation at ISOLDE-CERN [54,146], the latter also giving first results for the quadrupole moments of the 2 + 1 and 2 + 2 states. The systematics for a number of important E2 transitions and the electric quadrupole moments are presented in figures 5, 6, and 7, respectively. Because these figures highlight only the specific set of important B(E2) values, we detail the comparison between the present theoretical study and the existing data (which is most documented in the 180 ≤ A ≤ 190 region of shape coexistence) in tables III, IV, and V in which we compare the experimental absolute B(E2) values as well as the relative B(E2) values, respectively, with the corresponding IBM-CM theoretical results. In Fig. 5 (panel (a)) one observes a rather constant value for the B(E2; 2
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of the remaining atoms, N e , by optically pumping 5s5p 3 P 0 , 3 P 2 → 5s5p 3 P 1 with light resonant on the 5s5p 3 P 0 , 3 P 2 ↔ 5s6s 3 S 1 transitions at 679 nm and 707 nm. Atoms then rapidly decay to the ground state, via the 21 µs lived 5s5p 3 P 1 [23], where they are subsequently imaged with 461 nm light. We note that this readout method counts not only atoms in |e, m F = 1/2 , but all atoms in the metastable 5s5p 3 P 0 , 3 P 2 manifold in the quantity N e . The decay of the excited state population ρ ee = N e /(N g + N e ) is then fit to extract a 1/e lifetime. These lifetimes are measured for various lattice depths, V z , is the lattice photon recoil energy, h the Planck constant, and m the atomic mass. Fig. 1(a) shows the trap depth dependence of the extracted lifetimes. We find the measured lifetimes to be significantly shorter than the predicted τ 0 = 145(40) s natural lifetime [24], yet largely consistent with numerical simulations with no free parameters (shaded red region) in which two-photon Raman transitions, stimulated by the lattice light, distribute atoms amongst the 5s5p 3 P J manifold where the atoms can then spontaneously decay to the ground state from 5s5p 3 P 1 . The vacuum limited lifetime of atoms prepared in |g is independently
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[43]. When m ≥ ω, the time-delay factor Ω = 0, the quantum system will be locked up in the initial state. This is due to the fact that the entanglement generation and degradation is via the the excitation and de-excitation of the two-level systems, which are possible only when m < ω. This is the selection rule for quantum systems coupled with a massive scalar field [43]. Therefore, the initial entanglement can be maintained as long as we want by manipulating the energy level spacing of the two-level systems with respect to the mass of the field. can be found from the general expression of concurrence (A7) or negativity (A8) that the necessary and sufficient condition that the system get disentangled within a finite time is 4eg < (a − s) 2 < 4e, and the lifetime of entanglement τ is τ = 1 ΩΓ 0 ln 2e 2(a + e) 2 + 2(e + s) 2 − 4e + a + 2e + s Here, τ 0 is the lifetime in the massless case. This indicates that the lifetime of the entanglement for quantum systems coupled with massive scalar field can be much longer than those coupled with massless scalar field when Ω is close to 0. Long-distance generation of entanglement It is well-known that, for quantum systems coupled with massless fields, the two separable subsystems can get entangled only when the separation is smaller than or of the order of the transition wavelength [20]. However, it is clear from Eqs. (25) and (26)
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a nuclear clock of extremely high stability, due to an expected high resilience against external influences and a radiative lifetime in the range of minutes to hours [7][8][9][10]. It is generally assumed that direct nuclear laser excitation of 229m Th requires a considerably improved knowledge of the isomeric transition energy (see e.g. Ref. [11] and references therein). The reasons are that, first, the present uncertainty in the knowledge of the isomeric energy value is still rather large. The currently best energy value of 7.8±0.5 eV leads to an energy range of at least 1 eV (corresponding to 2.4·10 14 Hz) to be scanned when searching for the isomeric excitation. Second, the radiative lifetime of 229m Th was theoretically predicted to be in the range of hours [12][13][14], leading to long required detection times when searching for a radiative decay channel. This has led to the assumption that the required time for laserbased scanning of the large energy range of 1 eV would be prohibitively long. In order to shorten the required scanning times, there are worldwide efforts ongoing to decrease the uncertainty of the transition energy, which would bring the isomeric state into realistic reach of direct laser excitation (see e.g. Refs. [15][16][17][18]). A recent review is found in Ref. [11]. Here we propose a different approach, which allows for a direct laser excitation of 229m Th without the requirement of an improved knowledge of the transition energy. As the required laser technology is already available, this proposal paves the way for direct nuclear laser spectroscopy
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