Lifetimes and g-factors of the HFS states in H-like and Li-like bismuth

For that purpose, the laser is fixed to the resonance wavelength and the rate of fluorescence photons is measured as a function of time after the laser pulses. While the time interval of 33.3 ms between individual laser pulses is sufficiently long for the excited HFS state in H-like bismuth to decay completely, the longer lifetime of the upper HFS state in Li-like Bi of about 118 ms at β = 0.71 requires a shutter system to periodically block the laser for several lifetimes, allowing the observation of a sufficiently large part of the decay curve to obtain a good fit of the lifetime in the laboratory frame τ_lab.

For the analysis the acquired signals are accumulated in timing bins with a width of 25 μs for H-like ions and 5 ms for Li-like ions. For each of these bins, histograms of the number of counts as a function of time synchronized to the revolution period of the ions are extracted (see figure 2, left plot). In such a display, signals from the two ion bunches appear as peaks with one-sided tails resulting from an afterglow of excited residual gas molecules after passage of the bunches. The bin width of these histograms is given by the time resolution of the time-to-digital converter which operates with a 300 MHz clock. The left-hand side of figure 2 shows in blue the accumulated counts for Li-like ions during the first 5 ms after excitation by the laser on resonance and subsequent closure of the shutter system. The signal peak (left) clearly exhibits a significantly higher number of events caused by fluorescence from the decay of the excited ions together with background processes, while the reference peak (right) contains background events only. Several lifetimes after shutter closure, the population of the upper HFS state has almost completely returned to the ground state and signal and reference peaks now both exhibit roughly the same number of background counts (red histogram, obtained by summing up the individual 5 ms histograms in the time interval between 600 ms and 900 ms and dividing by their number). The right-hand side in figure 2 displays the difference of the blue and red histograms that corresponds to photons from the laser-excited HFS states only. This plot allows the determination of the center and width of the fluorescence signal contribution to the overall event display.

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To extract lifetime curves of the HFS transitions from the bunch histograms obtained for each time bin, the most robust method turned out to be a summation of the signal peak in a ±nσ window around the center of the signal count distribution determined as described above [16] (for the choice of the actual width of the summation window, see section 3.4). No background subtraction is performed in the process as especially for the parts of the lifetime curves where most HFS states have already decayed, one deals with the subtraction of two similarly large numbers that only leads to an increase in the statistical uncertainty of the extracted signal counts. Instead the background counts are summed together with the HFS signals and lead to a constant offset in the extracted lifetime curves that is treated as a fit parameter in the subsequent analysis. Figure 3 displays an example for such a lifetime curve obtained with Li-like bismuth ions. The number of counts is plotted against the time after closure of the shutter required due to the long decay time in the case of Li-like bismuth. In the first 1000 ms one observes the expected exponential decay of the signal rate together with a constant offset from background events (approximately 4370 in this case). Following that, the shutter is opened for 500 ms such that the upper HFS state can be populated again by the laser. The displayed data is accumulated over many shutter cycles to obtain sufficiently high statistics. The shutter cycle is synchronized to the timing of the laser pulses and the pulse structure of the laser with 33.3 ms period can therefore be observed in the rising branch of the data when the shutter is open.

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While fitting lifetime data from the measurements with H-like bismuth ions is straightforward as one is dealing with a simple exponential decay plus background, more effort is required in the Li-like case, where one would also like to extract the information contained in the rising branch of the curve, during which the upper HFS state is gradually repopulated by a number of consecutive resonant laser pulses. The fit model for the latter case has to take into account the processes of excitation, stimulated emission and spontaneous emission. The rate equation for the population of the upper HFS state N₂(t) reads:

$$\begin{eqnarray}&&\dot{{N}_{2}}(t)={N}_{1}(t)\,{B}_{12}\,u-{N}_{2}(t)\,{B}_{21}\,u-{N}_{2}(t)\,{A}_{21},\end{eqnarray} \tag{ 1 }$$

with the Einstein coefficients

$$\begin{eqnarray}&&{A}_{21}=\displaystyle \frac{1}{\tau }\hspace{10mm}{B}_{21}={A}_{21}\displaystyle \frac{{\lambda }^{3}}{8\pi h}\hspace{10mm}{g}_{1}\,{B}_{12}={g}_{2}\,{B}_{21}.\end{eqnarray} \tag{ 2 }$$

The coefficient for spontaneous emission A₂₁ is the inverse of the mean lifetime τ. g₁ and g₂ are the number of magnetic sub-states of the ground state (F = 4) and the excited state (F = 5) of the given HFS transition. N₁(t), N₂(t) are the occupation numbers of the two states, which yield the total number of stored ions N via $N={N}_{1}(t)+{N}_{2}(t)$. As the typical storage time of the ions in the ring was about 20 min, the time dependence of the total ion number over the timescale of a single 1.5 s shutter cycle can be neglected. λ denotes the wavelength of the transition and u the time averaged spectral energy density of the laser. Equation (1) can be re-written as follows:

$$\begin{eqnarray}&&\dot{{N}_{2}}(t)=\left\{\displaystyle \frac{{g}_{2}}{{g}_{1}}N-\left(\displaystyle \frac{{g}_{2}}{{g}_{1}}+1\right){N}_{2}(t)\right\}\,\displaystyle \frac{{\lambda }^{3}u}{8\pi h}\,\displaystyle \frac{1}{\tau }-{N}_{2}(t)\,\displaystyle \frac{1}{\tau }.\end{eqnarray} \tag{ 3 }$$

With the abbreviation $\phi := \tfrac{{\lambda }^{3}u}{8\pi h}$ equation (3) simplifies to

$$\begin{eqnarray}\begin{array}{rcl}\dot{{N}_{2}}(t) & = & \mathop{\underbrace{\displaystyle \frac{{g}_{2}}{{g}_{1}}N\displaystyle \frac{\phi }{\tau }}}\limits_{:= a}-\mathop{\underbrace{\left(\displaystyle \frac{\left(\tfrac{{g}_{2}}{{g}_{1}}+1\right)\phi +1}{\tau }\right)}}\limits_{:= b}\\ {N}_{2}(t) & = & a-{{bN}}_{2}(t).\end{array}\end{eqnarray} \tag{ 4 }$$

The differential equation (4) can be solved using the ansatz ${N}_{2}(t)=\tfrac{a}{b}+c\,{{e}}^{-bt}$, where c is a constant, introduced by the boundary conditions at t = 0 and is therefore given by $c={N}_{2}(0)-\tfrac{a}{b}$. Together it yields

$$\begin{eqnarray}&&{N}_{2}(t)=\displaystyle \frac{a}{b}+\left({N}_{2}(0)-\displaystyle \frac{a}{b}\right)\exp \left(-b\,t\right).\end{eqnarray} \tag{ 5 }$$

Stimulated emission as well as excitation do not take place during the full measurement interval since the laser is blocked for a defined time. Therefore a distinction is needed, with the two cases

• (i)  
  $t\leqslant {t}_{c}$: The laser is blocked, therefore only spontaneous emission takes place
• (ii)  
  t > t_c: The ions are excited by the the laser and all three processes are taken into account

The corresponding fit function is given by

$$\begin{eqnarray}&&\begin{array}{l}f(t)\,=\\ \left\{\begin{array}{ll}s\cdot {N}_{2}(0)\exp \left(-\displaystyle \frac{t}{\tau }\right)+d & \mathrm{for}\,\,t\leqslant {t}_{c}\\ s\cdot \left[\displaystyle \frac{a}{b}+\left({N}_{2}({t}_{c})-\displaystyle \frac{a}{b}\right)\exp \left(-b\,(t-{t}_{c})\right)\right]+d & \mathrm{for}\,\,t\gt {t}_{c},\end{array}\right.\end{array}\,\end{eqnarray} \tag{ 6 }$$

where d accounts for a constant background and s is a scaling factor representing further experimental parameters like measurement time and observed solid angle. Equation (6) holds for excitation with a continuous laser beam and does not yet take into account the pulse structure of the laser. Therefore a further modification of the model has been implemented in the fitting code, taking into account that during the rising branch of the lifetime curve only each 6th or 7th bin actually contains a laser shot.

The main part of the LIBELLE beam time in 2014 was spent on the precise determination of the wavelengths of the HFS transitions in ${}^{209}{\mathrm{Bi}}^{82+}$ and ${}^{209}{\mathrm{Bi}}^{80+}$. For that purpose the laser wavelength was scanned repeatedly across the respective transitions and a number of dedicated systematics measurements, among others investigating the influence of the electron cooler current or the buncher amplitude on the observed transition wavelength, were performed [4, 5]. In addition to this, some runs were recorded in which the laser was fixed to the transition wavelength of either ${}^{209}{\mathrm{Bi}}^{82+}$ or ${}^{209}{\mathrm{Bi}}^{80+}$ (in the latter case with the optical shutter system in operation) to obtain high statistics data on the lifetime of the states.

3.3.1. Li-like bismuth

After the first half of the beam time it was discovered that one of the drift-electrodes inside the electron cooler was not properly connected leading to fluctuations in the revolution frequency of the ion bunches that were then compensated by adapting the electron cooler voltage. After the electrode had been properly grounded, these fluctuations disappeared. At the same time however, the experimental background in the fluorescence detection increased, possibly due to a slight change in the beam properties following the repair action. We therefore collect the lifetime measurements for the HFS transition in Li-like bismuth in two datasets, one before and one after the maintenance of the electrode. Figure 4 displays the corresponding datasets together with results from the fit. The reduced χ²-values of both fits are consistent with one indicating that the model provides an adequate description of the data. The residuals do not show any non-statistical fluctuations and follow a Gaussian distribution around zero. The extracted lifetimes of both datasets are consistent with each other and with fit results obtained from the first 1000 ms of the data, where only the exponential decay is observed (see table 1).

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Table 1.  Fit results for Li-like bismuth lifetimes as observed in the laboratory frame before and after the maintenance. Lifetimes were extracted using the model described in section 3.2, using only the exponentially decaying part of the lifetime curves (0–1000 ms) or the complete datasets (0–1500 ms). The uncertainties are purely statistical.

Fit interval	0–1000 ms	0–1500 ms
Before maintenance	(118.3 ± 2.1) ms	(118.2 ± 1.7) ms
After maintenance	(118.5 ± 3.2) ms	(117.8 ± 2.3) ms
Weighted mean	(118.4 ± 1.8) ms	(118.1 ± 1.4) ms

3.3.2. H-like bismuth

For H-like bismuth the relevant measurements were performed after the repair and we are therefore dealing with one dataset only. Figure 5 displays the accumulated data together with the corresponding fit result delivering a value for the lifetime of the upper HFS state in ${}^{209}{\mathrm{Bi}}^{82+}$of (566.7 ± 2.2) μs in the laboratory frame.

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A number of measurements were performed to investigate possible systematic effects on the measured wavelengths of the HFS transitions in ${}^{209}{\mathrm{Bi}}^{82+}$and ${}^{209}{\mathrm{Bi}}^{80+}$ [4, 7]. By selecting from these runs only those periods when the laser was on resonance, we can also perform checks of systematic effects on the lifetime of the HFS states, albeit with very restricted statistics. Figure 6 displays the results of these investigations for the lifetime of the upper HFS state in Li-like bismuth. To investigate possible effects of space charge of the electron beam generated by the cooler or of the ion beam itself, lifetimes have been extracted for different values of the electron cooler current and the ion current (figures 6(a) and (b)). Additionally a series of measurements were taken with different amplitudes of the RF buncher voltage that is used to concentrate the ions stored inside the ESR into two bunches (figure 6(c)). The green bands illustrate 1σ uncertainty intervals of linear fits applied to the corresponding displayed data points. Within the uncertainties the slopes of all three fits are compatible with zero and, therefore, none of the described measurements displayed a systematic change of the lifetime with the respective experimental parameters. The same findings apply to the systematic tests of the lifetime of the upper HFS state in ${}^{209}{\mathrm{Bi}}^{82+}$.

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A systematic uncertainty arising in the analysis of the experimental data is given by the chosen width of the analysis window used to sum up the counts in the signal peaks of the individual bunch histograms (see figure 2). Choosing a too small window leads to shifts in the observed lifetimes as the widths of the signal peaks has been found to slightly fluctuate over the decay period [16]. Choosing a too large window on the other hand results in an increased fit uncertainty and a lesser quality of the fit demonstrated by a reduced χ²-value that starts to significantly deviate from one. The effect of a systematic variation of the width of the analysis window from ±0.5σ to ±4σ is displayed in figure 6(d), for the Li-like dataset before the cooler maintenance. It can be seen that for small window widths the lifetime of the state is underestimated but stabilizes when moving to larger widths. The optimum value chosen for the analysis is ±2.5σ, where the lifetime τ has already reached the plateau, the uncertainty dτ of the fit value is comparably small and the reduced χ² is compatible with one. To estimate the systematic uncertainty connected to this analysis procedure, we calculate the differences between the lifetimes derived with ±2σ, ±2.5σ and ±3σ width of the analysis window. The weighted mean of these differences is then taken as systematic uncertainty of the analysis of the given dataset. The systematic uncertainty for the lifetime of the upper HFS state in ${}^{209}{\mathrm{Bi}}^{82+}$was calculated in the same way.

Overall we arrive at the following lifetimes for the upper HFS states in ${}^{209}{\mathrm{Bi}}^{82+}$ and ${}^{209}{\mathrm{Bi}}^{80+}$ in the laboratory frame:

$$\begin{eqnarray}&&{\tau }_{\mathrm{lab}}{(}^{209}{\mathrm{Bi}}^{82+})=(566.7\pm {2.2}_{\mathrm{stat}.}\pm {1.3}_{\mathrm{sys}.})\,\mu {\rm{s}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\tau }_{\mathrm{lab}}{(}^{209}{\mathrm{Bi}}^{80+})=(118.1\pm {1.4}_{\mathrm{stat}.}\pm {1.3}_{\mathrm{sys}.})\,\mathrm{ms}.\end{eqnarray} \tag{ 8 }$$

To transform the measured lifetimes from the laboratory frame to the rest frame of the ions we need to know the velocities of the ions. In the experiment the ion velocity is determined by the velocity of the electrons in the cooler, which itself can be calculated from the acceleration voltage applied to the cooler and space-charge corrections taking into account a reduction of the potential felt by the electrons due to shielding effects in the electron beam. To determine the latter, several measurements were conducted with H-like ion beams during which the electron cooler current was lowered stepwise and the change in velocity of the ion beam was then corrected for by adapting the electron acceleration voltage until the original revolution frequency of the ions (and thus their velocity in the ring) was restored. From a linear fit to the obtained data points, the space-charge correction was found to be (−0.162  ±  0.009) V mA⁻¹ [7]. The average voltage during the lifetime measurements extracted from the datasets and corrected for the space-charge effect is for the case of H-like bismuth found to be ${U}_{\mathrm{eff}}{(}^{209}{\mathrm{Bi}}^{82+})=(-213878\pm 9)$ V and for Li-like bismuth ${U}_{\mathrm{eff}}{(}^{209}{\mathrm{Bi}}^{80+})=(-213871\pm 10)$ V (the latter value has been obtained from the datasets after the cooler maintenance). Besides the uncertainties of the space-charge correction the quoted error bars include the standard deviation of the high voltage measurements from the calculated mean (1.3 V for ${}^{209}{\mathrm{Bi}}^{82+}$ and 2.1 V for ${}^{209}{\mathrm{Bi}}^{80+}$), the uncertainty of the scale factor of the PTB HVDC voltage divider (13 ppm [17]) and uncertainties due to the difference in contact potentials of cathode and drift tubes of the electron cooler ($\leqslant 3$ V). For a conservative estimate of the overall uncertainty on the effective cooler voltage U_eff these contributions were added linearly.

We can then calculate the Lorentz factor using

$$\begin{eqnarray}&&\gamma =1+\displaystyle \frac{e\,{U}_{\mathrm{eff}}}{{m}_{e}\,{c}^{2}}\end{eqnarray} \tag{ 9 }$$

with the electron charge e and electron rest mass m_e to obtain $\gamma {(}^{209}{\mathrm{Bi}}^{82+})=1.418548(18)$ and $\gamma {(}^{209}{\mathrm{Bi}}^{80+})\,\,=1.418534(20)$. With these results the measured lifetimes can be transfered into the laboratory frame using $\tau ={\tau }_{\mathrm{lab}}/\gamma$ yielding the results listed in table 2. The values are in excellent agreement with the theoretical predictions in both cases. Our measurement of the lifetime of the HFS state in ${}^{209}{\mathrm{Bi}}^{82+}$ agrees within the uncertainties with the previous measurement by Winter et al [11, 12] while excluding the older value of Klaft et al [18].

Table 2.  Results for the lifetimes of the upper HFS state in ${}^{209}{\mathrm{Bi}}^{82+}$ and ${}^{209}{\mathrm{Bi}}^{80+}$ compared to existing experimental and theoretical data.

	H-like bismuth ${}^{209}{\mathrm{Bi}}^{82+}$	Li-like bismuth ${}^{209}{\mathrm{Bi}}^{80+}$
This work	$(399.5\pm {1.6}_{\mathrm{stat}.}\pm {0.9}_{\mathrm{sys}.})\,\mu {\rm{s}}$	$(83.3\pm {1.0}_{\mathrm{stat}.}\pm {0.9}_{\mathrm{sys}.})\,\mathrm{ms}$
Winter et al (exp.) [11, 12]	(397.5 ± 1.5stat.)μs
Klaft et al (exp.) [18]	(351 ± 16stat.)μs
Shabaev (theo.) [19]	(399.01 ± 0.19)μs
Shabaev et al (theo.) [14]		(82.0 ± 1.4) ms
Calculated from theoretical		(82.85 ± 0.61) ms
g-factor [20] and wavelength [21]