Direct determination of the $^{138}$La $\beta$-decay $Q$ value using Penning trap mass spectrometry

TL;DR

This study precisely measured the $^{138}$La $eta$-decay $Q$ value using Penning trap mass spectrometry, confirming previous results and significantly reducing uncertainties, thereby improving theoretical decay models and experimental interpretations.

Contribution

The paper provides the most precise $^{138}$La $eta$-decay $Q$ value measurement to date, enhancing nuclear decay models and experimental data analysis.

Findings

Measured $Q$ value: 1052.42(41) keV

Improved precision by an order of magnitude

Confirmed previous $Q$ value with higher accuracy

Abstract

Background: The understanding and description of forbidden decays provides interesting challenges for nuclear theory. These calculations could help to test underlying nuclear models and interpret experimental data. Purpose: Compare a direct measurement of the $^{138}$La $\beta$-decay $Q$ value with the $\beta$-decay spectrum end-point energy measured by Quarati et al. using LaBr$_3$ detectors [Appl. Radiat. Isot. 108, 30 (2016)]. Use new precise measurements of the $^{138}$La $\beta$-decay and electron capture (EC) $Q$ values to improve theoretical calculations of the $\beta$-decay spectrum and EC probabilities. Method: High-precision Penning trap mass spectrometry was used to measure cyclotron frequency ratios of $^{138}$La, $^{138}$Ce and $^{138}$Ba ions from which $\beta$-decay and EC $Q$ values for $^{138}$La were obtained. Results: The $^{138}$La $\beta$-decay and EC $Q$ values were measured to be $Q$ = 1052.42(41) keV and $Q_{EC}$ = 1748.41(34) keV, improving the precision compared to the values obtained in the most recent atomic mass evaluation [Wang, et al., Chin. Phys. C 41, 030003 (2017)] by an order of magnitude. These results are used for improved calculations of the $^{138}$La $\beta$-decay shape factor and EC probabilities. New determinations for the $^{138}$Ce 2EC $Q$ value and the atomic masses of $^{138}$La, $^{138}$Ce, and $^{138}$Ba are also reported. Conclusion: The $^{138}$La $\beta$-decay $Q$ value measured by Quarati et al. is in excellent agreement with our new result, which is an order of magnitude more precise. Uncertainties in the shape factor calculations for $^{138}$La beta-decay using our new $Q$ value are reduced by an order of magnitude. Uncertainties in the EC probability ratios are also reduced and show improved agreement with experimental data.

Tables

Table 1. Table 1: Measured cyclotron frequency ratios for combinations of 138 La + , 138 Ba + , and 138 Ce + ions among themselves and against 136 Xe + . N is the number of individual ratio measurements contributing to the average, R ¯ ¯ 𝑅 \bar{R} . The uncertainties for R ¯ ¯ 𝑅 \bar{R} , shown in parentheses, have been inflated by the Birge Ratio, BR, when BR > > 1.

Num.	Ion Pair	N	BR	$\overline{R}$
(i)	${}_{}{}^{138}\text{La}_{}^{+}$/${}_{}{}^{138}\text{Ce}_{}^{+}$	33	1.2	$\mathrm{0.999\; 991\; 810\; 7}\left(37\right)$
(ii)	${}_{}{}^{138}\text{La}_{}^{+}$/${}_{}{}^{138}\text{Ba}_{}^{+}$	48	1.1	$\mathrm{0.999\; 986\; 387\; 2}\left(29\right)$
(iii)	${}_{}{}^{138}\text{Ce}_{}^{+}$/${}_{}{}^{138}\text{Ba}_{}^{+}$	32	1.0	$\mathrm{0.999\; 994\; 589\; 6}\left(56\right)$
(iv)	${}_{}{}^{138}\text{Ce}_{}^{+}$/${}_{}{}^{136}\text{Xe}_{}^{+}$	79	1.3	$\mathrm{0.985\; 506\; 162\; 7}\left(118\right)$
(v)	${}_{}{}^{138}\text{Ba}_{}^{+}$/${}_{}{}^{136}\text{Xe}_{}^{+}$	22	1.4	$\mathrm{0.985\; 511\; 499\; 9}\left(34\right)$

Table 2. Table 2: Q 𝑄 Q values for 138 La β 𝛽 \beta -decay or EC and 138 Ce 2EC calculated from cyclotron frequency ratios listed in Table I. The relevant ratios were measured directly and via an intermediary isotope (see text for details). The weighted average is listed along with the AME2016 value Wang  et al. ( 2017 ) and the difference Δ ​ Q = Q LEBIT − Q AME Δ 𝑄 subscript 𝑄 LEBIT subscript 𝑄 AME \Delta Q=Q_{\mathrm{LEBIT}}-Q_{\mathrm{AME}} .

Decay	Interm.	Q value (keV)		$\Delta Q$
		LEBIT	AME2016	(keV)
138La${\to }^{138}$Ce	Direct	1051.98(48)
	138Ba	1053.67(81)
($\beta$-)	Avg.	1052.42(41)	1051.7(4.0)	0.7(4.0)
138La${\to }^{138}$Ba	Direct	1748.67(37)
	138Ce	1746.98(86)
(EC)	Avg.	1748.41(34)	1742.5(3.2)	5.9(3.2)
138Ce${\to }^{138}$Ba	Direct	695.01(72)
	138La	695.68(1.58)
	136Xe	696.69(60)
(2EC)	Avg.	695.97(44)	690.7(4.9)	5.3(4.9)

Table 3. Table 3: Adjusted parameters of the experimental shape factor C exp ​ ( W ) = 1 + a ​ W + b ​ W 2 subscript 𝐶 exp 𝑊 1 𝑎 𝑊 𝑏 superscript 𝑊 2 C_{\text{exp}}(W)=1+aW+bW^{2} , to be applied on the theoretical shape factor Eqn. ( 6 ) to match the measured spectrum from Ref.  Quarati  et al. ( 2016 ) . The fitting procedure has been applied using the AME2016 Q 𝑄 Q value from Ref.  Wang  et al. ( 2017 ) and the LEBIT Q 𝑄 Q value from this work. Upper uncertainties are for E max + σ E max subscript 𝐸 max subscript 𝜎 subscript 𝐸 max E_{\text{max}}+\sigma_{E_{\text{max}}} and lower uncertainties for E max − σ E max subscript 𝐸 max subscript 𝜎 subscript 𝐸 max E_{\text{max}}-\sigma_{E_{\text{max}}} .

Parameter	AME2016		LEBIT
	value	uncertainty	value	uncertainty
a	$-$1.32	+0.07	$-$1.319	+0.006
		$-$0.07		$-$0.006
b	0.499	$-$0.043	0.4982	$-$0.0038
		$+$0.043		+0.0038
${\chi }^2$	9.0$\times$10-5		8.8$\times$10-5

Table 4. Table 4: Influence of the Q 𝑄 Q value on the theoretical predictions of the capture probability ratios for 138 La. Experimental values are from Ref.  Quarati  et al. ( 2016 ) . The AME2016 Q 𝑄 Q value is 1742(3) keV from Ref.  Wang  et al. ( 2017 ) and the LEBIT Q 𝑄 Q value is 1748.41(34) keV from this work.

EC Ratio	Experiment	AME2016	LEBIT
L/K	0.391(3)	0.403(8)	0.3913(26)
M/K	0.102(3)	0.0996(24)	0.0964(10)
M/L	0.261(9)	0.247(8)	0.2464(30)

Table 5. Table 5: Mass excesses, ME, for 138 Ba, 138 La, and 138 Ce obtained from the ratios listed in Table  1 . The results are compared to those listed in the AME2016 Wang  et al. ( 2017 ) . The column Δ Δ \Delta M is calculated as ME LEBIT – ME AME2016

Nuclide	Ref.	ME (keV/c2)		$\Delta$M
		LEBIT	AME2016	(keV/c2)
138Ba	136Xe	-88 262.13(0.44)	-88 261.64(0.32)	-0.49(0.54)
138La	138Ba	-86 513.44(0.57)	-86 519.2(3.2)	5.8(3.2)
	138Ba	-87 567.12(0.84)
	136Xe	-87 566.45(1.54)
	138La	-87 565.43(0.74)
138Ce	Avg.	-87 566.21(0.52)	-87 570.9(4.9)	4.7(4.9)

Equations

$$f_c=\frac{1}{2\pi }\frac{qB}{m},$$

$$f_{+}+f_{-}=f_c.$$

$$R=\frac{f_{c1}(t_2)}{f_{c2}(t_2)}=\frac{m_2}{m_1}.$$

$$Q=(M_p-M_d)c^2=(M_d-m_e)(1-\bar{R})c^2,$$

$$\frac{dN}{dW}\propto pW{q}^2{F}_0{L}_{0}C(W),$$

$$C(W)=q^4+\frac{10}{3}{\lambda }_2{q}^2{p}^2+{\lambda }_3{p}^4,$$

$$C_{{\kappa }_x}=\sum _{K,{\kappa }_{\nu }}[M_K({\kappa }_x,{\kappa }_{\nu })+S_{{\kappa }_x}{m}_K({\kappa }_x,{\kappa }_{\nu }){]}^2,$$

$$M_{int}=(M_{ref}-m_e)\frac{1}{\bar{R}}+m_e,$$

Full text

The understanding and description of forbidden decays provides interesting challenges for nuclear theory. These calculations could help to test underlying nuclear models and interpret experimental data.

Purpose

Compare a direct measurement of the 138La $\beta$-decay $Q$ value with the $\beta$-decay spectrum end-point energy measured by Quarati et al. using LaBr3 detectors [Appl. Radiat. Isot. 108, 30 (2016)]. Use new precise measurements of the 138La $\beta$-decay and electron capture (EC) $Q$ values to improve theoretical calculations of the $\beta$-decay spectrum and EC probabilities.

Method

High-precision Penning trap mass spectrometry was used to measure cyclotron frequency ratios of 138La, 138Ce and 138Ba ions from which $\beta$-decay and EC $Q$ values for 138La were obtained.

Results

The 138La $\beta$-decay and EC $Q$ values were measured to be $Q_{\beta}$ = 1052.42(41) keV and $Q_{EC}$ = 1748.41(34) keV, improving the precision compared to the values obtained in the most recent atomic mass evaluation [Wang, et al., Chin. Phys. C 41, 030003 (2017)] by an order of magnitude. These results are used for improved calculations of the 138La $\beta$-decay shape factor and EC probabilities. New determinations for the 138Ce 2EC $Q$ value and the atomic masses of 138La, 138Ce, and 138Ba are also reported.

Conclusion

The 138La $\beta$-decay $Q$ value measured by Quarati et al. is in excellent agreement with our new result, which is an order of magnitude more precise. Uncertainties in the shape factor calculations for 138La $\beta$-decay using our new $Q$ value are reduced by an order of magnitude. Uncertainties in the EC probability ratios are also reduced and show improved agreement with experimental data.

I Introduction

Historically, nuclear $\beta$-decay studies have played a crucial role in our understanding of nuclear and particle physics and in the development of the Standard Model. Presently, high-precision and low-background nuclear $\beta$-decay experiments are being used to test the assumptions of the Standard Model and to search for new physics e.g. Hardy and Towner (2015); Severijns et al. (2006). In addition to the exotic neutrinoless double $\beta$-decay process Avignone et al. (2008), interest in other rare weak decay processes such as ultra-low $Q$ value $\beta$-decays Mustonen and Suhonen (2010) and forbidden $\beta$-decays e.g. Mustonen et al. (2006); Haaranen et al. (2014); Gamage et al. (2016); Kandegedara et al. (2017), has grown in recent years. The need for more precise $\beta$-spectrum shape measurements and calculations for forbidden $\beta$-decays is becoming apparent in a number of applications Kostensalo and Suhonen (2018). For example, such input is necessary in the use of the proposed spectral shape method (SSM) to determine the effective value of the weak axial vector coupling constant, $g_{A}$ Haaranen et al. (2016), and for understanding antineutrino spectra in context of the reactor antineutrino anomaly Hayes et al. (2014); Sonzogni et al. (2015).

In this paper, we focus on the second forbidden unique decay of 138La. Naturally occurring 138La has a half-life of 1.03(1)$\times$1011 years, and can undergo both $\beta^{-}$-decay to the 2*+* state in 138Ba and electron capture (EC) to the 2*+* state in 138Ce. In addition, 138Ce is energetically unstable against double EC to the 138Ba ground state. However, this decay has not been observed Belli et al. (2010). A schematic of the decay scheme for this isobaric triplet system is shown in Fig. 1.

Evidence for the radioactive decay of 138La was first obtained in 1950 Pringle et al. (1950), just a few years after its discovery Inghram et al. (1947). Since then, a series of measurements were performed that provided an understanding of the 138La decay scheme and more precise determinations of the partial and total half-lives Pringle et al. (1951); Mulholland and Kohman (1952); Turchinetz and Pringle (1956); Watt and Glover (1962); Ruyter et al. (1966); Ellis and Hall (1972); Cesana and Terrani (1977); Taylor and Bauer (1979); Sato and Hirose (1981); Norman and Nelson (1983); Nir-El (1997). The long half-life has enabled the use of 138La for geochemical dating Bellot et al. (2015) and as a nuclear cosmochronometer Hayakawa et al. (2008).

Recently, the development of LaBr3 and LaCl3 scintillation detectors has enabled new measurements of the 138La $\beta$-decay and EC x-ray spectra McIntyre et al. (2011); Giaz et al. (2015); Quarati et al. (2012, 2016). From these measurements, more precise determinations of the relative EC probabilities and the $\beta$-decay spectrum shape can be made and compared with theoretical calculations. An experimental quantity that enters into these calculations is the $Q$ value for the decay, corresponding to the energy equivalent of the mass difference between the parent and daughter atoms, taking into account the energy of the daughter nuclear state. Before the 138La $\beta$-decay spectrum measurement by Quarati et al. Quarati et al. (2016), the uncertainties in the relevant $Q$ values were limited by the uncertainties in the masses of 138La and 138Ce, as given in the 2012 atomic mass evaluation (AME2012) Audi et al. (2012). The determination of the 138La $\beta$-decay spectrum end-point energy in Ref. Quarati et al. (2016) reduced the uncertainty in the 138La $\beta$-decay and EC $Q$ values to 4.0 and 3.2 keV Audi et al. (2012), respectively. In this paper, we present for the first time direct determinations of the 138La $\beta$-decay and EC $Q$ values using Penning trap mass spectrometry. We use these new $Q$ values to calculate EC ratios and $\beta$-spectrum shape factor coefficients. We also provide updated atomic masses for 138Ba, 138La, and 138Ce and for the 138Ce 2EC $Q$ value.

II Experimental Description

The 138La $\beta$-decay and EC $Q$ value measurements and absolute mass measurements were performed at the Low Energy Beam and Ion Trap (LEBIT) Penning trap mass spectrometry facility at the National Superconducting Cyclotron Laboratory (NSCL), a schematic of which is shown in Fig. 2. LEBIT was designed for online measurements of rare isotopes from the Coupled Cyclotron Facility, but also houses two offline sources—a laser ablation source (LAS) Izzo et al. (2016) and a plasma ion source—that can be used for the production of stable and long-lived isotopes. These offline sources provide reference ions during rare isotope measurements, but also provide access to a wide range of isotopes that have been used for studies related to neutrinoless double $\beta$-decay Redshaw. et al. (2012); Lincoln et al. (2013); Bustabad et al. (2013a, b); Gulyuz et al. (2015); Eibach et al. (2016), highly forbidden $\beta$-decays Gamage et al. (2016); Kandegedara et al. (2017), and ultra-low $Q$ value $\beta$-decays Sandler et al. (2019).

The LAS, described in detail in Izzo et al. (2016), uses a pulsed Nd:YAG laser to ablate material from a solid target. For this experiment, the LAS was fitted with 25 mm $\times$ 12.5 mm $\times$ 1 mm thick Ba, La, and Ce sheets of natural isotopic abundance esp . Two targets were placed on either side of the holder at one time and a stepper motor was used to alternate between the two sides. The high temperatures produced by the laser pulse results in the evaporation of surface material and the emission of positive ions and electrons to produce a high-temperature plasma. In addition to surface ionization, electron impact ionization of the ablated material, as well as other mechanisms contribute to the total ion production, see, e.g. Sin1990 for a complete description. After production, ions are accelerated to an energy of 5 keV and focused into a 90 degree quadrupole bender that steers them into the main beamline.

The plasma ion source is a DCIS-100 Colutron hot cathode discharge source bea . It consists of a tungsten filament within an alumina chamber. The chamber is filled with helium gas mixed with a small amount of xenon gas. As current is run through the filament it produces a discharge, creating a plasma within the gas-filled chamber. The ions are extracted through a radiofrequency quadrupole (RFQ) mass filter to suppress the helium ions, after which the xenon ions are focused into the other side of the 90 degree quadrupole bender and steered into the main beamline.

After entering the main beamline, ions are injected into an RFQ cooler and buncher Schwarz et al. (2016). Helium buffer gas is used to thermalize the ions, which are then released in packets of 100 ns duration to be accelerated to 2 keV and transported into the 9.4 T magnet containing the LEBIT Penning trap. At the entrance of the magnetic field is a fast electrostatic kicker, which only allows ions of the chosen $A/q$ to pass based on their time-of-flight. A series of electrodes decelerates the remaining ions to be captured in the Penning trap.

The Penning trap itself consists of a hyperbolic ring electrode, two hyperbolic endcap electrodes, and two correction ring and correction tube electrodes that sit within a uniform magnetic field produced by a 9.4 T superconducting solenoidal magnet. The ring electrode of the Penning trap is segmented so that dipole and RFQ fields can be applied to address the radial modes of the ions’ motion. Ions are confined radially in the trap via their cyclotron motion in the magnetic field that, without the presence of the electric field, has the frequency

[TABLE]

where $B$ is the magnetic field strength and $m/q$ is the mass-to-charge ratio of the ion.

The trap electrodes produce a quadratic electrostatic potential that confines ions axially. The electric field also has the effect of reducing the frequency of the cyclotron motion of an ion and introducing an additional radial motion, the magnetron mode. As such, an ion in the Penning trap has three normal modes of motion: the axial, reduced cyclotron, and magnetron modes, with eigenfrequencies $f_{z}$, $f_{+}$ and $f_{-}$, respectively Brown and Gabrielse (1986). For an ideal Penning trap, the frequencies of the radial modes are related to the true cyclotron frequency of Eqn. (1) Gabrielse (2008, 2009) via

[TABLE]

Before entering the trap, ions are deflected off-axis by a Lorentz steerer and captured in a magnetron orbit of well-defined radius, typically $\sim$0.5 mm Ringle et al. (2007). A dipole RF pulse of 20 ms duration at the reduced cyclotron frequency of any previously identified contaminant ions is then applied to drive the contaminant ions into the trap walls Bollen et al. (1996). Next, the cyclotron frequency of the ion of interest is measured using the time-of-flight ion cyclotron resonance (TOF-ICR) technique Gräff et al. (1980). In this technique, an RFQ pulse of appropriate amplitude and duration is applied at the frequency $f_{RF}\approx f_{+}+f_{-}$. This pulse couples the reduced cyclotron and magnetron modes, which converts magnetron motion into cyclotron motion and increases the radial energy of the ions. The ions are then released from the trap and their time-of-flight to a microchannel plate (MCP) detector is recorded, which depends on the ions’ initial radial energy. The measurement cycle is repeated over a range of values of $f_{RF}$ close to $f_{+}+f_{-}$ and a time-of-flight resonance curve such as the example shown in Fig. 3 is obtained. The minimum in time-of-flight corresponds to maximum radial energy, which results from a full conversion of magnetron to cyclotron motion by an RF pulse with $f_{RF}=f_{+}+f_{-}=f_{c}$. Hence, $f_{c}$ is obtained from a fit of the theoretical lineshape König et al. (1995) to the data, as shown in Fig. 3.

Our data taking procedure involved alternating between cyclotron frequency measurements on two ion species to account for temporal magnetic field variations. We measured $f_{c1}$ of ion 1 at time $t_{1}$, $f_{c2}$ of ion 2 at time $t_{2}$, and $f_{c1}$ of ion 1 again at time $t_{3}$. We then linearly interpolated the two $f_{c1}$ measurements to find the cyclotron frequency of ion 1 at time $t_{2}$. From this, we found the cyclotron frequency ratio, using the equation

[TABLE]

We repeated this series of measurements twenty to fifty times and found the average cyclotron frequency ratio $\bar{R}$, as seen in Fig. 4. The Birge ratio Birge (1932) for each series was calculated and, when the Birge Ratio was greater than 1, the uncertainty of $\bar{R}$ was inflated by the Birge ratio to account for possible underestimation of systematic uncertainty.

III Results and Discussion

The cyclotron frequency ratios that we measured in this work, corresponding to inverse mass ratios of singly charged 138La, 138Ce, 138Ba, and 136Xe ions, are given in Table 1.

III.1 138La and 138Ce $Q$ value determinations

The $\beta$-decay and EC $Q$ values are defined as the energy equivalent of the mass difference between parent and daughter atoms, $M_{p}$ and $M_{d}$, respectively. From this definition and Eqn. (3), the $Q$ value for each decay can be obtained from the cyclotron frequency ratio measurement via

[TABLE]

where $m_{e}$ is the mass of the electron and $c$ is the speed of light. Here we have ignored the ionization energies, which are nearly two orders of magnitude smaller than our statistical uncertainties and therefore do not affect our final results. The $Q$ values calculated using the cyclotron frequency ratios listed in Table 1 are given in Table 2. For each $Q$ value determination, we measured the relevant ratio in Eqn. (4) directly, e.g. ratio (i), 138La+/138Ce+, is used to obtain $Q_{\beta}(^{138}$La$)$. However, we can also obtain the same ratio independently from the data in Table 1 from a ratio of ratios, e.g. (ii)/(iii) also gives 138La+/138Ce+ where the intermediary nuclide is 138Ba. For each $Q$ value we list all such results and take the weighted average.

III.1.1 138La $\beta$-decay $Q$ value

One of the main motivations of this work was to perform a precise measurement of the 138La $\beta$-decay $Q$ value using Penning trap mass spectrometry to compare with the result of Quarati et al. Quarati et al. (2012) obtained from a measurement of the end-point energy of the 138La $\beta$-decay spectrum using LaBr3 detectors. A comparison of these results can be seen in Fig. 5 along with results from the AME2012 and AME2016 Audi et al. (2012); Wang et al. (2017) (we note that the AME2016 analysis includes the Quarati et al. result). For this comparison, we compute the $\beta$-decay spectrum end-point energy, corresponding to the $Q$ value defined in Eqn. (4) with the energy of the 138Ce(2+, 788.74 keV) daughter state subtracted. The Quarati et al. result of 264.0(4.3) keV is in excellent agreement with our new value of 263.68(41) keV, which is an order of magnitude more precise.

III.1.2 138La EC $Q$ value determination

Our direct measurement of the 138La EC $Q$ value shows a 5.9(3.2) keV shift with respect to the AME2016 value and a reduction in uncertainty of almost an order of magnitude. Our direct mass determinations of 138Ce and 138Ba, described in section III.3, indicate that this disagreement is due to a shift in the mass of 138Ce compared to the AME2016 value. Since the mass of 138La is directly linked to the mass of 138Ce in the AME2016 through the 138La $\beta$-decay $Q$ value measurement of Quarati, et al., Quarati et al. (2016) the 138La mass is also shifted with respect to the AME2016 value. Our new measurement enables more precise calculations of the 138La relative EC probabilities, as described in section III.2.

III.1.3 138Ce 2EC $Q$ value

Finally, in Table 2, we list three independent results for the 138Ce $Q_{2EC}$-value along with their weighted average. The first result is a direct measurement of the $Q$ value obtained from ratio (iii) in Table 1, using Eqn. (4). The second and third results are from the ratio of ratios of (ii)/(i) and (iv)/(v), respectively using 138La and 136Xe as an intermediary. These results and their weighted average are plotted in Fig. 6 along with the value obtained from the AME2016. Our three independent measurements of the 138Ce $Q_{2EC}$-value are in good agreement with each other, but the average shows a 5.3(4.9) keV discrepancy with respect to the AME2016 value. Again, our direct mass determinations of 138Ce and 138Ba, described in section III.3, indicate that this disagreement is due to a shift in the mass of 138Ce compared to the AME2016 value.

III.2 138La $\beta$-spectrum shape factor and $EC$ ratio calculations

It has been well known for a long time that the mass region around 138La cannot be depicted by a naive shell model Helton et al. (1973) and that the collective structure of the nuclear states is critical to reproduce low energy data  Suhonen (1993). In this context, precise measurements are of high importance to test and constrain nuclear models. In this section, we study the influence of a precise knowledge of $Q$ values on the theoretical predictions. We first look at the electron energy spectrum from the $\beta$-decay to 138Ce and then at the capture probabilities from the EC decay to 138Ba.

III.2.1 138La $\beta$-spectrum shape factor

The $\beta$-decay spectrum can be described, following the formalism of Behrens and Büring Behrens and Bühring (1982), as

[TABLE]

where $W$ is the total electron energy, $p$ its momentum and $q$ the antineutrino energy. The Fermi function $F_{0}L_{0}$ is defined from the Coulomb amplitudes of the relativistic electron wave functions which are solutions of the Dirac equation for a static Coulomb potential from a uniformly charged sphere. The theoretical shape factor $C(W)$ couples the nuclear structure of the nuclei involved in the decay with the lepton dynamics. Describing the weak interaction as a current-current interaction, a multipole expansion can be performed for each current —the hadron current and the lepton current. Keeping only the main terms, the nuclear component can be factored out of the theoretical shape factor for allowed and forbidden unique transitions. In the present work, we have calculated the second forbidden unique transition from the ground-state of 138La to the first excited state of 138Ce, for which one has:

[TABLE]

where the $\lambda_{k}$ parameters are ratios of Coulomb amplitudes of the electron wave functions.

This treatment of the shape factor usually gives good agreement with measurements Mougeot (2015). However, 138La exhibits a specific nuclear structure which leads to an accidental cancellation of the nuclear matrix elements. The leading multipole orders are not sufficient anymore to describe the transition and higher orders have to be included. This mechanism hinders the transition and drastically increases the half-life. As shown in Fig. 7, it also modifies the shape of the $\beta$ spectrum, our calculation (green) being far from the measured spectrum (black) from Ref. Quarati et al. (2016). Therefore, we have performed fits to these data to determine an experimental shape factor $C_{\text{exp}}$ defined as the distortion to be applied on the theoretical shape factor to get the measured spectrum. A minimum of two parameters was necessary to fit the data, with the form $C_{\text{exp}}(W)=1+aW+bW^{2}$. For these fits we used an end-point energy, $E_{\text{max}}$, of either 263.3(4.0) keV obtained from the AME2016 Wang et al. (2017), or 263.68(41) keV found in this work. Uncertainty limits on the parameters were determined by refitting the data with $E_{\text{max}}\rightarrow E_{\text{max}}\pm\sigma_{E_{\text{max}}}$. The resulting parameters and corresponding uncertainties are shown in Table 3 and are illustrated in Fig. 7. As can be seen, the results are very consistent and the new $Q$ value reduces uncertainties in the shape factor fit parameters by a factor of $\sim 11$, putting a stronger constraint on the precision of future predictions of the nuclear matrix elements.

III.2.2 138La $EC$ ratio calculations

We have performed the calculation of the second forbidden unique electron capture transition from the ground-state of 138La to the first excited state of 138Ba. The modeling used has already been described in Ref. Mougeot (2018) and takes into account overlap, exchange, shake-up and shake-off, and hole effects. However, radiative corrections based on Coulomb-free theory Bambynek et al. (1977) have also been considered in the present work. In addition, the relativistic atomic wave functions were determined using the precise atomic orbital energies from Refs. Kotochigova et al. (1997a, b) which include the effect of electron correlations. The resulting EC probability ratios for $K$, $L$, and $M$ shells are shown in Table 4. The calculations were performed using $Q_{EC}$ = 1742(3) keV from the AME2016 Wang et al. (2017) and $Q_{EC}$ = 1748.41(34) keV obtained in this work and are compared with the precise measurements from Ref. Quarati et al. (2016). A reduction in the uncertainties of the calculated values by factors of 2.4 to 3 is achieved with the new $Q$ value. It is noteworthy that a change of the $Q$ value by less than 0.4% leads to a perfect agreement of the predicted $L/K$ value with the measured one. The differences between predictions and measurements for the $M/K$ and $M/L$ values can be explained by the low energies of the $M$ subshells, which make both their high-precision calculation and measurement very difficult.

The calculations shown in Table 4 have been performed following the usual approximation of a constant nuclear component, identical for each subshell, which cancels when looking at capture probability ratios. This assumption is considered to be correct for both allowed and forbidden unique transitions Bambynek et al. (1977). However, in order to investigate the sensitivity of our theoretical predictions to the inclusion of the nuclear component, besides that reported in Table 4, we have performed additional calculations of the capture probability ratios. We have followed the formalism of Behrens and Büring Behrens and Bühring (1982) in which, as for $\beta$-decays, the coupling of the nuclear and lepton components is given for each subshell through a double multipole expansion by:

[TABLE]

where $\kappa_{x}$ and $\kappa_{\nu}$ are quantum numbers of the electron and neutrino respectively, and $S_{\kappa_{x}}$ is the sign of $\kappa_{x}$. The $M_{K}$ and $m_{K}$ quantities include nuclear and lepton matrix elements. They have been determined in impulse approximation considering the single decay of a $1g_{7/2}$ proton in 138La to a $3s_{1/2}$ neutron in 138Ba. A non-relativistic harmonic oscillator modeling has been considered for the large component of the relativistic nucleon wave functions, and the small component has been estimated following the method given in Ref. Behrens and Bühring (1982). With the $Q$ value from this work, we found a significant change in the $L/K$ ratio by taking into account the nuclear component—$L/K_{nuc}=0.3827(26)$—while the other two capture probabilities remain consistent—$M/K_{nuc}=0.0962(10)$, and $M/L_{nuc}=0.2514(31)$. One can clearly see that a high-precision determination of the $Q$ value allows for testing of the accuracy of the nuclear model, eventually providing nuclear structure information. A more realistic treatment would necessitate taking into account nucleus deformation and configuration mixing.

III.3 138La, 138Ce, 138Ba atomic mass determinations

The absolute masses of 138La, 138Ce and 138Ba were obtained from our cyclotron frequency ratio measurements listed in Table 1 and the relation

[TABLE]

where $M_{int}$ and $M_{ref}$ are the atomic masses of the nuclide of interest and reference nuclide, respectively. Ratio (v) in Table 1, 138Ba+/136Xe+, provided a direct link to obtain the mass of 138Ba using 136Xe as a reference, which has been measured to a precision of 0.007 keV using the Florida State University Penning trap Redshaw. et al. (2007). We then used 138Ba as a secondary mass reference along with ratios (ii) and (iii) from Table 1 to obtain atomic masses for 138La and 138Ce respectively. Ratio (iv) in Table 1, 138Ce+/136Xe+, provided an independent check for the mass of 138Ce. The two results for 138Ce are in good agreement, although the second is a factor of two less precise. This was due to the fact that after operating the LAS with barium, it became contaminated and a background of 138Ba+ was produced along with 138Ce+. Finally, 138La was used as a secondary mass reference along with ratio (i) in Table 1 to calculate a third atomic mass. The three values of 138Ce are in good agreement and were used to calculate an average value for the atomic mass. The resulting masses excesses for 138Ba, 138La, and 138Ce are listed in Table 5 and plotted in Fig. 8.

Our result for the mass of 138Ba is in good agreement with the AME2016 value, which was determined from ($n,\gamma$) measurements along the barium isotope chain, a 134Cs $\rightarrow$ 134Ba $\beta$-decay measurement, a 133Cs($n,\gamma$)134Cs measurement, and a Penning trap measurement of 136Ba*+/136Xe+* Kolhinen et al. (2011). These measurements anchor 138Ba to 133Cs Bradley et al. (1999); Mount et al. (2010) and 136Xe Redshaw. et al. (2007), which have been precisely measured with Penning traps and can be considered secondary mass standards.

The determination of the masses of 138La and 138Ce in the AME is more convoluted. The mass of 138Ce is determined almost entirely from the Quarati et al. $\beta$-decay end-point energy measurement and the mass of 138La. The mass of 138La on the other hand is partially obtained from a 138La($d,p$)139La reaction measurement, and a 139Ba $\rightarrow$ 139La $\beta$-decay measurement that link it to the barium isotopes and ultimately 133Cs and 136Xe, as discussed above. It is also partially determined from a network of neutron capture, $\beta$-decay and $\alpha$-decay measurements that link the lanthanides up to 163Dy and 163Ho for which precise Penning trap measurements have been performed Eliseev et al. (2015). Our results, listed in Table 5 and displayed in Fig. 8, indicate a discrepancy in the AME2016 mass values for both 138La and 138Ce of about 5 keV/c2.

As a check of possible systematics we performed a measurement of the mass ratios of 134Xe+/136Xe+ and 136Ba+/136Xe+, with the results $\bar{R}$ = 0.985 270 617 0(22) and 0.999 980 585 7(23) respectively. The ratios differ from those calculated using the AME2016 mass values for 134,136Xe and 136Ba, and $m_{e}$ = 5.485 799 090 70(16) $\times$ 10*-4* u Mohr et al. (2016) by only $-$0.8(2.2) and 0.1(3.3) $\times$ 10*-9*, respectively. This is well within acceptable deviation and is considered consistent with the AME.

IV Conclusion

Using Penning trap mass spectrometry, we have measured the $Q_{\beta}$-value of 138La to be 1052.42(41) keV and the $Q_{EC}$-value of 138La to be 1748.41(34) keV. Both measurements reduce the uncertainties compared to previous values by an order of magnitude. The determination of the 138La $\beta$-decay $Q$ value from a measurement of the end-point energy of the $\beta$-spectrum obtained with LaBr3 detectors by Quarati, et al. Quarati et al. (2016) is in excellent agreement with our new, more precise result.

We have used our new $Q_{\beta}$ value in theoretical fits to the data of Ref. Quarati et al. (2016) and extracted new values for the experimental shape factor parameters with uncertainties that are reduced by about an order of magnitude compared to those obtained using the $Q$ value from the AME2016. We have used our new $Q_{EC}$ value in theoretical calculations of the EC probabilities that we compare with the experimental EC ratio results of Ref. Quarati et al. (2016). Our new $Q$ value reduces the uncertainties in the calculated ratios by factors of up to 3 compared calculations using the $Q$ value from AME2016, and, for the case of the L/K ratio significantly improves the agreement between experiment and theory.

Finally, we also present the first direct mass measurements of 138La, 138Ce, and 138Ba. Our result for 138Ba is in good agreement with the AME2016 value with a similar level of precision. Our results for 138La and 138Ce show an $\approx$5 keV/c2 shift with respect to AME2016 and reduce the uncertainties by factors of 6 and 9, respectively.

Acknowledgments

This research was supported by Michigan State University and the Facility for Rare Isotope Beams and the National Science Foundation under Contracts No. PHY-1102511 and No. PHY1307233. This material is based upon work supported by the US Department of Energy, Office of Science, Office of Nuclear Physics under Award No. DE-SC0015927. The work leading to this publication has also been supported by a DAAD P.R.I.M.E. fellowship with funding from the German Federal Ministry of Education and Research and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007/2013) under REA Grant Agreement No. 605728.

The theoretical work was performed as part of the EMPIR Projects 15SIB10 MetroBeta and 17FUN02 MetroMMC. These two projects have received funding from the EMPIR programme co-financed by the participating states and from the European Union’s Horizon 2020 research and innovation programme.