# Precision half-life measurement of the
decay of ^{37}K

###### Abstract

The half-life of ^{37}K has been measured to be ,
a value nearly an order of magnitude more precise than the best previously
reported. The decay of ^{37}K occurs mainly via a
superallowed branch to the ground-state of its mirror, ^{37}Ar.
This transition has been used recently, together with similar transitions
from four other nuclei, as an alternative test of CVC and method for
determining , but the precision of its value was limited by
the relatively large half-life uncertainty. Our result corrects that
situation. Another motivation for improving the value was to determine
the standard-model prediction for the -decay correlation parameters,
which will be compared to those currently being measured by the
Trinat collaboration at Triumf. The new value, ,
is now limited in precision by the ground-state branching
ratio.

###### pacs:

21.10.Tg, 23.40.-s, 27.30.+t## I Introduction

Many precision measurements of the correlation parameters and values of -decaying nuclei have been used to help form our understanding of the fundamental symmetries of the weak interaction. Experiments of this kind continue to be performed to search for physics beyond the standard model. For example, the measured values for superallowed pure Fermi transitions, have been used to verify the conserved vector current (CVC) hypothesis to one part in ; to determine most precisely the value of , the up-down element of the Cabbibo-Kobayashi-Maskawa (CKM) matrix; and to set limits on scalar and right-handed currents HardyPRL:05 (); Hardy:05 (); Hardy:09 ().

Transitions between isospin doublets in mirror nuclei are
also useful weak-interaction probes because there are relatively few
corrections required to describe their decay. The most thoroughly studied
and theoretically cleanest example of such a transition is the decay of the
neutron mendenhallPRC2013 (); plasterPRC2012 (); PERKEO (), but a number of other
cases have attracted attention recently: the decays of ^{19}Ne,
^{21}Na, ^{29}P, ^{35}Ar and the subject of the present
work, ^{37}K. Naviliat-Cuncic and Severijns NavPRL09 () used the
values for these five transitions to determine , though less
precisely than has been achieved from the transitions.
With more precise measurements, however, the result for could
be improved enough to provide a valuable cross-check on the result from the
transitions. Our measurement is a first step on this
path.

For mixed Fermi/Gamow-Teller transitions, such as the decay of ^{37}K to
the ground state of ^{37}Ar, the measured value can be related to
via the corrected value, which incorporates calculated
corrections for radiative and isospin-symmetry-breaking effects, as
follows SevPRC08 ():

(1) |

where , the Fermi constant , and the vector coupling constant . The parameters , , and are the usual radiative and isospin-symmetry-breaking correction terms calculated for the vector current Hardy:09 (). The statistical rate functions for vector and axial-vector currents are denoted and . In these mirror nuclei their ratio generally is within a few percent of unity according to shell model calculations SevPRC08 ().

The Fermi matrix element in the limit of strict isospin-symmetry is

(2) |

for a transition from a nucleus having protons, neutrons, and isospin . It takes the value 1 for =1/2 mirror transitions.

Finally, the Gamow-Teller to Fermi mixing ratio is given by

(3) |

where is the Gamow-Teller matrix element, is the axial-vector coupling constant, and the correction terms with superscripts are equivalent to those with superscripts but must be evaluated for the axial-vector component. Since, in general, cannot be precisely calculated, must be taken from experiment. This also makes it unnecessary to calculate the axial-vector correction terms.

The decay of ^{37}K was one of the five cases used in
Ref. NavPRL09 () to determine . The relatively large uncertainty
of its contribution was overwhelmingly dominated by the precision of ,
which was obtained from a single measurement of the neutrino
asymmetry parameter, MelconianPLB2007 (). The
value is known much better, to SevPRC08 (), with the largest
contribution to its uncertainty being the lifetime of the decay. A survey
of mirror transitions and the theoretical corrections to their values
was made in Ref. SevPRC08 (), in which three ^{37}K half-life
measurements Sc58 (); Ka64 (); Az77 () were averaged to arrive at
. This uncertainty is much larger
than the and uncertainties on the other two
contributors to for ^{37}K: the branching ratio and value,
respectively.

The aim of the present work is to improve the lifetime of ^{37}K so that
it no longer dominates the uncertainty in the deduced value.
A more precise value on its own will not improve the value
of obtained from ^{37}K decay. For that purpose, an improved
measurement of via a correlation parameter such as or is
necessary.

## Ii Experiment

A high purity radioactive beam is essential for performing a
precision half-life measurement. We have measured the half-life of ^{37}K
at the Cyclotron Institute of Texas A&M University using a
continuous-gas-flow proportional counter and a fast
tape-transport system. We produced ^{37}K via an
inverse kinematics reaction (fusion-evaporation reaction) by accelerating a primary beam of ^{38}Ar
to in the K500 superconducting cyclotron, and
impinging it on a -atm H gas target cooled to liquid
nitrogen temperature. The forward focused reaction products were analyzed
using the Momentum Achromat Recoil Separator
(MARS) mars (); tamu-beams+MARS (). Selection and focusing of the desired
ions in MARS was done according to their value. At the focal plane of
MARS, a 16-strip position-sensitive detector of thickness
with an active area of could be inserted to
detect the secondary reaction products, which were identified according to
their position and energy loss in the strip detector. The position of the
reaction products on the strip detector was related with its ratio.
This detector was initially inserted to help us tune the spectrometer
and was then removed before the measurement began. During the measurement,
it was reinserted once a day to ensure that no changes had occurred.
In this way we determined that the 22.7- ^{37}K
beam passing through the spectrometer extraction slits was
pure, the remaining contaminants being ^{35}Ar, ^{34}Cl
and ^{33}Cl.

This separated beam then exited the vacuum system
through a thick Kapton foil. Immediately following this
foil was a -thick BC-404 plastic scintillator, which counted
the number of ions. The beam then passed through a stack of Al degraders,
and finally was implanted into the 76--thick aluminized Mylar
of the fast-tape transport system. The degrader thickness was adjusted so
as to place the implanted ^{37}K at a depth mid-way through the tape. At
this setting (m of Al), the combination of electromagnetic separation
in MARS and range selection in the degraders resulted in the ^{37}K beam
reaching the tape being pure.

For part of the measurement we used two different thicknesses of degrader:
m, which resulted in the ^{37}K being placed at the front of
the tape; and m, which put it at the back of the tape. The former
led to a higher proportion of impurities; the latter to a much lower
proportion. In both cases the intensity of ^{37}K was reduced. In the
best case, with the ^{37}K deposited near the back of the tape, the
sample purity was .

A sample of ^{37}K was collected by implanting the beam into a section
of aluminized Mylar tape for a time interval of . The beam
was then turned off and the tape-transport system moved the sample in
to a well-shielded location, stopping it in the center of
a proportional gas counter. There, the particles from the
decay were counted for half-lives (). This
cycle was repeated continuously, with each sample being collected on
a fresh region of tape, until the desired statistics had been accumulated.

The gas counter and the data acquisition system used in the present measurement were similar to those originally described in Ref. koslowskyNIMA1997 (). The current system has also been described in detail in reports of previous half-life measurements of superallowed emitters performed at Texas A&M University (see, for example, iacobPRC2006 (); iacobPRC2010 ()). Briefly, the preamplified signals from the gas counter are passed to a fast-filter amplifier with a high gain (). At this high gain many of the pulses would saturate the amplifier so, to ensure that the amplifier recovers quickly, large pulses are clipped with a Schottky diode inserted after the first stage of amplification. The amplified and clipped pulses are then sent to a discriminator with very low threshold ( mV).

Since deadtime is a serious concern for half-life measurements, the discriminator signals were split and sent to two fixed-width nonretriggering and nonextending gate generators, which established different dominant deadtimes in the two separate streams, both of which were multiscaled into 500-channel time spectra.

The total measurement was split into 13 runs with different experimental
conditions: three discriminator threshold settings (150, 200 and 250 mV),
two combinations of dominant deadtimes (4 and 6 s; 3 and 8 s),
three detector bias voltages within the detector’s plateau region (2600, 2650
and 2700 V) and three degrader thicknesses (13, 51 and 70 m). The
initial activity of ^{37}K for each cycle was in the range of
. All runs were composed of cycles, which
yielded a total of counts in each. In addition, a background
measurement was performed for which all conditions were the same as for
normal data taking except that the tape motion was disabled. The background
rate was found to be four orders of magnitude lower than the initial count
rate for each collected sample.

## Iii Data Analysis

The selection of data for final analysis was made according to three criteria. First, cycles with too few counts – less than 5000 – were rejected. Second, the number of ’s recorded in each cycle was compared to the corresponding number of heavy ions recorded in the scintillator at the exit of MARS. If their ratio was abnormally low, it meant that the tape did not stop with the activity positioned correctly within the proportional gas counter. In that case, the cycle was rejected. Finally, a cycle was rejected if a half-life fit of the recorded data resulted in a very poor reduced , which corresponds to a goodness-of-fit statistic less than .

We analyzed the decay data from accepted cycles using two different methods: one based on summing all accepted cycles in a run (“summed” analysis) and the other on performing a simultaneous fit on all the cycles of a run (“global” analysis). All fits employed a fortran code based on the Marquardt algorithm for minimization. The program assumes Poisson rather than Gaussian statistics in the data.

The fit function contained a constant background plus four exponentials
corresponding to the decays of ^{37}K and the main contaminants
^{35}Ar, ^{34}Cl and ^{33}Cl. The half-lives of the
contaminants were fixed at their well-known measured values:
for ^{35}Ar SevPRC08 (),
for ^{34}Cl Hardy:09 () and
for ^{33}Cl SevPRC08 (). The intensity of
the contaminants could not be fit separately because of the high correlations
between them, so we fixed the relative normalization of the contaminants
at the values measured in the MARS focal-plane detector (corrected for losses
in the degraders), and allowed only the total amount of contaminant activity
to vary relative to ^{37}K in the fitting procedure.

In the “summed” method of analysis, the first step was to correct the measured decay spectrum in each cycle channel-by-channel for deadtime, based on the measured rate in each channel. Next, the cycles of a given run were summed into two decay curves, one for each imposed dominant deadtime. Finally, these spectra were fitted as already described. Figure 1 shows a typical deadtime-corrected decay curve for a single run analyzed using the summed analysis technique. Note that the fitted background level in the figure agrees well with the value obtained in our dedicated background run.

In the “global” analysis, the fitting function itself is adjusted to account
for deadtime effects. In this method, many separate decay spectra within a
single run were fit simultaneously, all with the same lifetime for ^{37}K but
with separate contamination levels and backgrounds allowed for each
spectrum. Runs had between 120 and 750 cycles; so, to create a more manageable
number of spectra and also to improve the statistics in the individual spectra,
we summed the cycles together in groups and for each group took the effective
count rate to be the average of the rates in its component cycles. We then
performed a global fit on the resulting summed spectra. The ^{37}K half-life
extracted turned out to be insensitive to the number of cycles summed together
for each spectrum. Figure 2 shows a simultaneous
fit using the global-analysis method; it is of the same run for which the
summed analysis is illustrated in Fig 1. The extracted
values for the half-life obtained from the two analysis methods are well
within uncertainties, as one can see from the results shown in
Figs. 1 and 2.

Method | half-life [s] |
---|---|

Summed analysis | |

Global analysis | |

Average ^{37}K half-life |

^{37}K obtained from two different analysis techniques. All of the uncertainties shown are purely statistical. The adopted half-life is taken to be the unweighted average of the two methods.

### iii.1 Systematic Uncertainties

The changing of parameters from run to run allowed us to test for potential systematic effects that could contribute to the uncertainty in the final results. In Fig. 3 we plot the half-life obtained for 11 different run conditions using both analysis techniques. Since two of the 13 runs had the same conditions as another run, each has been averaged with its twin and plotted as a single half-life. Each plotted result is labeled by the four measurement parameters that pertained to that result: detector bias, discriminator setting, degrader thickness and imposed deadtime. As already mentioned, each run produced two data streams, each with a different imposed deadtime. In all cases, the half-lives obtained from the two streams agreed very closely. Since both contain the same primary data, only the result from one of the data streams is presented in Fig. 3. No systematic dependence on measurement parameters is evident.

In Table 1 we present the half-lives, averaged over all runs, obtained from the two different analysis techniques, “summed” and “global”. We take our final result to be the unweighted average of these two half-lives, with the statistical uncertainty on the average conservatively chosen to be the larger of the two individual values.

Source | Uncertainty | |||
---|---|---|---|---|

Statistical | ||||

Systematics: | ||||

Contaminants | ||||

Analysis method | ||||

Deadtime | ||||

Fitting range | ||||

Bias voltage | ||||

Discriminator threshold | ||||

Total systematic | ||||

Total uncertainty |

^{37}K lifetime measurement. All of the values listed are in milliseconds.

Although we regularly monitored impurities with the detector at the focal
plane of MARS and accounted for the observed impurities in our fitting
procedures, we nevertheless made an additional check at the analysis stage
for unidentified short-live impurities. We did this by artificially removing
up to the first (60 channels) of the counting period in
steps of . At each step the remaining channels in the reduced
data set were fitted and a half-life extracted. The results are plotted in
Fig. 4, from which it is clear from that the derived
half-life is stable against these changes. This test also demonstrates our
system’s independence of counting rate. We examined the sensitivity of our
fit results to all the potential sources of systematic errors described above.
The outcome is the error budget given in Table 2. We
found the greatest variation in the extracted lifetime when we fixed the value
of the total impurity level to the amount observed in the MARS focal-plane
detector rather than leaving it free to vary in the fits. The next largest
source of uncertainty was the counting statistics. Our final result for the
half-life of ^{37}K is:

(4) |

This measurement is nearly an order of magnitude improvement over
the previously accepted value SevPRC08 () and now completely dominates
the world average of the ^{37}K half-life.

## Iv -value

We follow the convention used in Refs. NavPRL09 () and SevPRC08 () by defining the corrected value for mirror nuclei which includes the variation due to the ratio of the Fermi to Gamow-Teller matrix elements. Specifically, the value is defined as

(5) |

where experimentally has already been defined in Eq. (1) in terms of the measured value and the small radiative and isospin-mixing corrections as calculated in Ref. SevPRC08 (). The mixing ratio defined by Eq. (3) is experimentally determined from a measurement of one of the angular correlations of the decay.

When calculating the value, we take the branching ratio and
total decay energy of ^{37}K from the survey of Ref. SevPRC08 ():
and .

Using our new value for the half-life of ^{37}K, we find

(6) |

This result is four times more precise than the previous value quoted in Ref. NavPRL09 (), and is now comparable to the precision of the three most precisely measured mirror nuclei. The branching ratio, no longer the half-life, is what now limits the precision of the value. We have already made a new measurement of this branching ratio and are currently in the process of analyzing the data. This should reduce the uncertainty in the value even further.

The only ^{37}K correlation measurement published to date,
MelconianPLB2007 (), can be used to determine a value
for . Using the recoil-order corrections of Holstein holstein (),
we find , where the asymmetric error bars arise from
the fact that the uncertainty in is so large that the uncertainty on
the extracted value of is not Gaussian. Using this value of and
the value above leads to

(7) |

With our new lifetime measurement, the central value of has changed and is now better aligned with other mirror transitions.

Finally, noting that for a mixed transition, is related to by NavPRL09 ()

(8) |

we calculate that from ^{37}K alone

(9) |

This can be compared to the older value of 0.985(12). The uncertainty is barely affected by our improved half-life measurement because it is dominated by the large uncertainty in the correlation parameter measurement.

In Fig. 5 we plot values for the four most
precisely measured nuclear mirror transitions. The point for ^{19}Ne has
been updated from the survey of Ref. SevPRC08 () to include recent lifetime
results triambakPRL2012 (); ujicPRL2013 (); broussardPRL2014 (). The figure shows
that the main effect of the present ^{37}K lifetime measurement is to
shift its value of into better agreement with the other mirror nuclei
and the transitions.

## V Conclusion

We have measured the half-life of ^{37}K, the decay of which proceeds
primarily via a mirror transition to its
analog, the ground state in ^{37}Ar. Our new result for the half-life
is . The precision of this result is nearly
an order of magnitude improvement over the previously accepted world average.
The value for the ^{37}K mirror transition is now
determined to precision, and the corresponding values of
and have shifted into better alignment with values
determined for the other well-measured mirror transitions. Nevertheless, the
uncertainties of and remain completely dominated by
the correlation measurement for the transition. The results from improved
correlation measurements, currently in progress by the Trinat collaboration
at Triumf melconianLASNPA (), are required
before the uncertainty on the mirror-transition value for can be
improved.

## Vi Acknowledgements

We are grateful to the support staff of the Cyclotron Institute, especially Don May and George Kim for providing the primary beam. This work was supported by the U.S. Department of Energy under Grant No. DE-FG02-93ER40773 and Early Career Award ER41747, as well as the Robert A. Welch Foundation under Grant No. A-1397.

## References

- (1) J. C. Hardy and I. S. Towner, Phys. Rev. Lett. 94, 092502 (Mar 2005)
- (2) J. C. Hardy and I. S. Towner, Phys. Rev. C 71, 055501 (May 2005)
- (3) J. C. Hardy and I. S. Towner, Phys. Rev. C 79, 055502 (2009)
- (4) M. Mendenhal et al. (UCNA Collaboration), Phys. Rev. C 87, 032501 (Mar 2013), http://link.aps.org/doi/10.1103/PhysRevC.87.032501
- (5) B. Plaster et al. (UCNA Collaboration), Phys. Rev. C 86, 055501 (Nov 2012), http://link.aps.org/doi/10.1103/PhysRevC.86.055501
- (6) D. Mund et al., Phys. Rev. Lett. 110, 172502 (Apr 2013), http://link.aps.org/doi/10.1103/PhysRevLett.110.172502
- (7) O. Naviliat-Cuncic and N. Severijns, Phys. Rev. Lett. 102, 142302 (Apr 2009)
- (8) N. Severijns, M. Tandecki, T. Phalet, and I. S. Towner, Phys. Rev. C 78, 055501 (Nov 2008)
- (9) D. Melconian et al., Phys. Lett. B 649, 370 (2007), ISSN 0370-2693
- (10) F. Schweizer, Phys. Rev. 110, 1414 (Jun 1958)
- (11) R. W. Kavanagh and D. R. Goosman, Phys. Lett.. 12, 229 (1964), ISSN 0031-9163, erratum Phys. Lett. 13, 358 (1964)
- (12) G. Azuelos, J. E. Kitching, and K. Ramavataram, Phys. Rev. C 15, 1847 (May 1977)
- (13) R. Tribble, C. Gagliardi, and W. Liu, Nucl. Instrum. Methods Phys. Res. B 56-57, 956 (1991)
- (14) R. Tribble et al., Nucl. Phys. A 701, 278 (2002), ISSN 0375-9474, http://www.sciencedirect.com/science/article/pii/S0375947401015974
- (15) V. Koslowsky et al., Nucl. Instrum. Methods Phys. Res. A 401, 289 (1997), ISSN 0168-9002, http://www.sciencedirect.com/science/article/pii/S0168900297010176
- (16) V. Iacob et al., Phys. Rev. C 74, 055502 (Nov 2006), http://link.aps.org/doi/10.1103/PhysRevC.74.055502
- (17) V. Iacob et al., Phys. Rev. C 82, 035502 (Sep 2010), http://link.aps.org/doi/10.1103/PhysRevC.82.035502
- (18) B. Holstein, Rev. Mod. Phys. 46, 789 (1974)
- (19) S. Triambak et al., Phys. Rev. Lett. 109, 042301 (Jul 2012), http://link.aps.org/doi/10.1103/PhysRevLett.109.042301
- (20) P. Ujić et al., Phys. Rev. Lett. 110, 032501 (Jan 2013), http://link.aps.org/doi/10.1103/PhysRevLett.110.032501
- (21) L. J. Broussard et al., Phys. Rev. Lett. 112, 212301 (May 2014), http://link.aps.org/doi/10.1103/PhysRevLett.112.212301
- (22) D. Melconian et al.(2014), submitted to Proceedings of Science for the X Latin American Symposium on Nuclear Physics and Applications (X LASNPA).