11institutetext:  Department of Physics, University of Regina, Regina, SK S4S
0A2, Canada 22institutetext: TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T
2A3, Canada 33institutetext: Department of Physics, Queen’s University,
Kingston, ON K7L 3N6, Canada 44institutetext: Department of Physics,
University of Guelph, Guelph, ON N1G 2W1, Canada 55institutetext: Department
of Chemistry, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
66institutetext: Department of Physics and Astronomy, University of British
Columbia, Vancouver, BC V6T 1Z4, Canada 77institutetext: Department of
Physics, Colorado School of Mines, Golden, CO 80401, USA 88institutetext:
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA

# High-precision half-life determination of 14O via direct $\beta$ counting

S. Sharma 11 G.F. Grinyer _Corresponding Author<EMAIL_ADDRESS>11
G.C. Ball 22 J.R. Leslie 33 C.E. Svensson 44 F.A. Ali _Present address:
Department of Physics, College of Education, University of Sulaimani, P.O. Box
334, Sulaimani, Kurdistan Region, Iraq_ 44 C. Andreoiu 55 N. Bernier _Present
address: Department of Physics and Astronomy, University of the Western Cape,
P/B X17, Belleville 7535, South Africa and Department of Physics and
Engineering, University of Zululand, P/B X1001, KwaDlangezwa 3886, South
Africa_ 2266 S.S. Bhattacharjee 22 V. Bildstein 44 C. Burbadge 44 R.
Caballero-Folch 22 R. Coleman 44 A. Diaz Varela 44 M.R. Dunlop 44 R. Dunlop 44
A.B. Garnsworthy 22 E. Gyabeng Fuakye 11 G.M. Huber 11 B. Jigmeddorj 44 K.
Kapoor 11 A.T. Laffoley 44 K.G. Leach 77 J. Long 88 A.D. MacLean 44 C.R.
Natzke 2277 B. Olaizola _Present address: ISOLDE, CERN, CH-1211 Geneva 23,
Switzerland_ 22 A.J. Radich 44 N. Saei 11 J.T. Smallcombe _Present address:
Oliver Lodge Laboratory, University of Liverpool, Liverpool, L69 7ZE, UK_ 22
A. Talebitaher 11 K. Whitmore 55 T. Zidar 44

(Received: date / Revised version: )

###### Abstract

The half-life of the superallowed Fermi $\beta^{+}$ emitter 14O was determined
to high precision via a direct $\beta$ counting experiment performed at the
Isotope Separator and Accelerator (ISAC) facility at TRIUMF. The result,
$T_{1/2}$(14O) = 70619.2(76) ms, is consistent with, but is more precise than,
the world average obtained from 11 previous measurements. Combining the 14O
half-life deduced in the present work with the previous most precise
measurements of this quantity leads to a reduction in the overall uncertainty,
by nearly a factor of 2. The new world average is $T_{1/2}$(14O) = 70619.6(63)
ms with a reduced $\chi^{2}$ value of 0.87 obtained from 8 degrees of freedom.

###### pacs:

21.10.TgLifetimes, widths and 23.40.BwWeak interaction and lepton

##  1 Introduction

High precision measurements of the $ft$ values for superallowed Fermi $\beta$
decays are crucial for providing fundamental tests of the Standard Model
description of electroweak interactions Har20 . All nuclear $\beta$ decays are
characterized by their $ft$ value, which depends on three quantities that can
each be determined experimentally: the energy difference between the parent
and daughter states involved in the $\beta$ transition ($Q$ value), the
branching ratio ($BR$), or fraction of parent decays that populate the
particular daughter state of interest, and the half-life ($T_{1/2}$) of the
parent nucleus.

For the specific case of superallowed Fermi $\beta$ decay between isospin $T$
= 1 isobaric analogue states, the $ft$ value, once corrected for radiative and
isospin-symmetry breaking effects, can be written as a corrected
$\mathcal{F}t$ value according to the expression Har20 :

$\displaystyle\mathcal{F}t=\textit{ft}(1+\delta_{R}^{\prime})(1+\delta_{NS}-\delta_{C})=\frac{K}{2G_{V}^{2}(1+\Delta^{V}_{R})},$
(1)

where $\delta_{R}^{\prime}$, $\delta_{NS}$, $\delta_{C}$ and $\Delta_{R}^{V}$
are all theoretically calculable correction factors that account for radiative
and transition dependent electromagnetic effects in addition to isospin-
symmetry breaking, $\delta_{C}$, that depends sensitively on the structure of
the parent and daughter states. These corrections are small in magnitude
($\leq$ 2.5$\%$) and are described in detail in Ref. Har20 . The term $G_{V}$
in Eqn. 1 is the vector coupling constant for semi-leptonic weak interactions
and $K$ is a group of well-known and fundamental physical constants with a
numerical value that evaluates to $K/{(\hbar c)^{6}}$ =
8120.27648(26)$\times$10-10 GeV-4s Har20 . From Eqn. 1, every experimentally
determined $ft$ value can be used to independently determine a value for
$G_{V}$. According to the conserved vector current (CVC) hypothesis Fey58 ,
this value should be the same for all transitions and implies that the
$\mathcal{F}t$ values themselves should also be constant, within their quoted
uncertainties.

The constancy of the $\mathcal{F}t$ values, and thus the validity of the CVC
hypothesis itself, has been firmly established at the level of 9$\times$10-5
using a set of 15 precisely determined $T$ = 1 superallowed Fermi
$\beta$-decay transitions Har20 . These decays, particularly the lightest
superallowed transitions, are also used to set strict limits on the possible
existence of a scalar term in the weak interaction Har20 . A scalar
interaction, whether fundamental or induced by the vector part, would result
in a deviation to the constancy of the $\mathcal{F}t$ values that would be
inversely proportional to the average decay energy $\langle$1/$W\rangle$.
Since this quantity is largest for decays of 10C and 14O, the $ft$ values for
these decays are essential for setting limits on scalar currents Dun16 .

In this work, we focus on the superallowed decay of 14O, which is already one
of the most precisely determined $ft$ values. Its half-life, $T_{1/2}$ =
70619(11) ms Har20 , is currently obtained from a weighted average of 11
previous measurements. The $Q$-value for this decay was recently measured to
high-precision in Ref. Val15  and the resulting world average, $Q$ =
2831.543(76) keV Har20 , is nearly an order of magnitude more precise than the
value adopted in the previous survey of the superallowed data Har15 . The
recommended value for the superallowed branching ratio to the 0+ isobaric
analogue state at 2.31 MeV excitation in the daughter nucleus, 14N, is $BR$ =
99.446(13)$\%$ Har20 . Combining the above quantities with $P_{EC}$ =
0.088$\%$, that accounts for electron capture, the experimental $ft$ value for
the superallowed decay of 14O is $ft$ = 3042.23(84) s Har20 , a result that
has been determined at the level of $\pm$ 0.027$\%$ precision.

A new, and even more precise, value for the half-life of 14O is presented that
was deduced from an experiment performed at the Isotope Separator and
Accelerator (ISAC) facility at TRIUMF. This new result represents the most
precise half-life ever reported for this decay and is, by itself, more precise
than the world average obtained from all 11 previous measurements.

##  2 Experiment

The experiment was performed at TRIUMF, Canada’s particle accelerator centre.
Radioactive ion beams are produced at the ISAC facility using the Isotope
Separation On-Line (ISOL) technique. The 480 MeV primary proton beam from the
TRIUMF main cyclotron was impinged at 50 $\mu$A intensity onto a thick
silicon-carbide (11.97 g/cm2 Si) production target. Elements produced from
spallation reactions, diffused and effused from the target and entered a
coupled Forced Electron Beam Ion Arc Discharge (FEBIAD) ion source Ame14
where they were ionized. Ions were extracted from the source at 40 keV and
were sent through a low-resolution magnetic pre separator and a high-
resolution ${\Delta}m/m$ = 1/2500 mass separator to select the isotopes of
interest. The production yield of singly ionized 14O+ was measured to be an
order of magnitude less than carbon-monoxide 12C14O+, presumably due to the
presence of carbon in the SiC production target that facilitated the formation
of CO molecules. Since 12C is stable, its presence did not have any impact on
the 14O decay counting measurement. The mass separator was thus tuned to mass
$A$ = 26 to select the 12C14O+ ions and take advantage of the higher 14O yield
at this setting. An average intensity of 105 molecules/s was delivered to the
experiment over a period of approximately 4 days.

The 12C14O+ beam was implanted into a 17.2 $\mu$m thick aluminum layer of an
aluminized mylar tape. After implanting the beam for a set period of time to
collect a sufficient amount of activity, the beam was turned off. This was
followed by a pause of 22.2 s in order to give sufficient time for any
possible short-lived beam contaminants to decay before transporting the
remaining 14O sample to the $\beta$ counter. The detector used in this
experiment was a gas-filled proportional counter that has been described in
detail and used in several previous high-precision half-life measurements
Dun16 ; Bal01 ; Gri05 ; Gri08 ; Fin11 .

Signals from the gas counter were amplified using an Ortec 579 amplifier whose
output signals were then sent to an Ortec 436 constant fraction discriminator
(CFD). The CFD was used to set and adjust an energy threshold to trigger on
the $\beta$ particles emitted from the radioactive decays of nuclei in the
sample. Logic pulses from the output of the CFD were then duplicated in a fan-
in-fan-out module and were sent to two inputs of a LeCroy 222 dual gate and
delay generator. This module was used to generate two distinct fixed and non-
extendible dead times that were applied to every trigger event. These were
determined to be 2.9958(42) $\mu$s and 4.0105(43) $\mu$s using the source-
plus-pulser technique Bae65 . Two independent multi-channel scalers (MCS),
referred to as the old and new MCS, were then used to bin the decay data from
the counter into two separate data streams.

###  2.1 Detector Setup

The gas counter was operated in the voltage plateau of the proportional region
to ensure that the counting rate is nearly independent of the gain. The
plateau region was determined by recording the count rate of a long lived 90Sr
source as a function of the applied bias voltage. As the plateau region should
also not depend on the discriminator setting, several measurements were
performed at CFD threshold values of 70 mV, 100 mV and 130 mV. From the data
presented in Fig. 1, the proportional region of the detector was determined to
be between 2300 V and 2480 V (inclusive) and, as shown, is largely independent
of the discriminator thresholds.

Figure 1: (Color online) Measured count rate of a 90Sr calibration source as a
function of applied voltage for different discriminator threshold settings.
The plateau region was determined to be between 2300 V to 2480 V, as indicated
by the vertical bands. The maximum variation in counting rate observed in this
region was less than 0.64$\%$ (horizontal bands).

###  2.2 Data Selection Criteria

Decay data for the 14O experiment were collected in cycles that consisted of a
beam on (collection) period, a tape move to position the 14O activity at the
center of the gas counter, a “cool down” period to allow for any short-lived
contaminants in the beam to decay away, and then the counting measurement to
record the exponential decay activity. The precise values of the collection
time, tape move and decay measurement were varied on a cycle-by-cycle basis.
In general, one complete cycle typically lasted around 27 minutes, with the
total of the collection, “cool down” and tape movement times taking at most 3
minutes to complete. The long decay times of approximately 24 minutes for each
cycle were necessary to observe the entire decay of the 14O samples and to
determine the resulting background level afterwards, that is treated as a free
parameter and potentially contains important information about long-lived
contaminants. Over the course of the 4-day experiment, a total of 142 decay
cycles were recorded.

The first selection criterion that was applied to the experimental data was to
reject those cycles that had very few, or even zero, total counts recorded
during the decay measurement. These corresponded to cycles whose activity
collection period overlapped with an interruption in the ISAC beam delivery. A
total of 17 cycles were removed from the analysis based on this selection
criterion.

For the remaining cycles, a second selection was performed by analyzing the
reduced $\chi^{2}$ values determined from fitting the experimental dead-time
corrected decay curves to a single exponential decay plus a free constant
background. With the exception of one cycle, the reduced $\chi^{2}$ values
ranged between 0.81 and 1.27. The one cycle that was identified in this
analysis, due to its anomalously large $\chi^{2}$ value, had a significant
excess of counts in the middle of the decay counting period. This excess
lasted only seconds in duration and was likely due to a burst of electronics
noise that was generated close to the detector or picked up in the electronics
chain. This cycle was therefore removed from the analysis.

A total of 124 cycles, of the 142 cycles collected, passed these two selection
criteria and were used in the final analysis to determine the half-life of
14O.

###  2.3 Analysis

The 124 cycles that passed the initial selection criteria were corrected for
dead-time losses using the measured fixed and non-extendible dead times
according to the procedure described in Ref. Kos97  and fit using a maximum-
likelihood technique Gri05 . The fit function considered only the exponential
decay of 14O and a constant background according to the expression:

$\displaystyle y_{fit}(t)=a_{1}e^{-a_{2}t}+a_{3},$ (2)

where $a_{1}$ is the initial activity of the 14O sample, $a_{2}$ is the 14O
decay constant, and $a_{3}$ the constant background. All were treated as free
parameters in the fit. A sample decay curve and resulting fit for a single
decay cycle is presented in Fig. 2. Additional parameters were added to the
function to search for, and set limits on, the possibility of trace amounts of
beam contaminants. This analysis is described below in Sec. 3.2.

Figure 2: (Color online) a) Dead-time corrected 14O decay data (points) and
resulting fit (line) for one cycle: 140 s collection, 24.3 min decay, 70 mV
threshold and 2450 V detector bias. b) Residuals (points,
($y_{i}$-$y_{fit}$)/$\sigma_{i}$) obtained from the fit to the experimental
data in a) and zero (line) to guide the eye. For clarity, statistical
uncertainties on the data are not shown.

##  3 Results

The half-life of 14O was determined from the fits to the individual cycles
according to Eqn. 2. For every cycle, the half-life was deduced from the decay
data recorded in each of the MCS modules. The average values obtained from all
124 cycles were $T_{1/2}$ = 70619.0(70) ms from the new MCS and $T_{1/2}$ =
70619.5(70) ms from the old MCS. These results are not statistically
independent because they were obtained from the same input data recorded by
two separate modules. Because they are in excellent agreement with each other,
a simple (non-weighted) average was adopted as the final result.

The MCS averaged half-lives obtained for all 124 decay cycles are plotted in
Fig. 3. A weighted average of these values yields $T_{1/2}$ = 70619.2(70) ms
with a $\chi^{2}$/$\nu$ of 1.13 for 123 degrees of freedom. This result is in
excellent agreement with the previous world average of this quantity $T_{1/2}$
= 70619(11) ms Har20 .

Figure 3: (Color online) Deduced half-life versus cycle number for the 124
decay cycles collected and analyzed in the present experiment. A weighted
average of them yields $T_{1/2}$(14O) = 70619.2(70) ms (solid and dashed
lines) with a $\chi^{2}$/$\nu$ value of 1.13 for 123 degrees of freedom.

###  3.1 Electronic and Experimental Settings

In order to search for possible sources of systematic uncertainty in the final
result, the electronic settings used during the 14O data collection were
modified on a cycle-by-cycle basis. The 124 cycles, described above, were
obtained with different detector voltage settings, different discriminator
settings and with different bin times applied to the MCS data. The two fixed
and non-extendible dead times were swapped repeatedly throughout the course of
the experiment and the beam-on time (initial activity collected) was also
modified to search for potential rate-dependencies in the experimental data.

Before starting a new cycle, the discriminator threshold applied to the gas
counter signals was varied, in 15 mV increments from 70 mV to 115 mV. These
thresholds were within the same range of values that were used to determine
the plateau region (see Fig. 1). Of the 124 cycles collected, 33 were obtained
at the lowest threshold value of 70 mV, 29 were at 85 mV, 31 were at 100 mV
and 31 were at 115 mV. The half-life of 14O deduced from the weighted average
of the cycles collected at each discriminator setting is presented in Table 1.
Because all 124 decay cycles were regrouped into 4 distinct threshold
settings, the weighted average of these values is identical to the average of
the 124 individual cycles. Treating each of the 4 discriminator threshold
settings as an independent measurement of the 14O half-life, yields a
$\chi^{2}$/$\nu$ value of 1.02 (3 degrees of freedom).

A similar procedure was performed for the other electronics settings in the
experiment and the results are presented in Table 1 and in Fig. 4. Four
separate voltages were applied to the gas counter (3 degrees of freedom,
$\chi^{2}$/$\nu$ = 0.14), four discriminator thresholds ($\chi^{2}$/$\nu$ =
1.02), three beam-on times ($\chi^{2}$/$\nu$ = 0.56), two non-extendible dead
times ($\chi^{2}$/$\nu$ = 0.83) and three bin times ($\chi^{2}$/$\nu$ = 0.04)
applied to the MCS.

Figure 4: (Color online) Systematic plot that groups the half-lives deduced
from the individual cycles data according to the particular electronic
settings used in the experiment (see Table 1 for a detailed description of
each point). The solid and dashed lines correspond to the weighted average
half-life obtained from the entire data set and its statistical uncertainty
$T_{1/2}$ = 70619.2(70) ms. The $\chi^{2}$/$\nu$ value for each of the
electronic settings is indicated.

All of the reduced $\chi^{2}$ values obtained from grouped sets of electronics
settings are close to, or less than, unity. All of them are also smaller than
the $\chi^{2}$/$\nu$ value of 1.13 obtained from the half-life deduced as a
function of cycle number (Fig. 3). While all of these values are consistent
with fluctuations associated with statistical uncertainties, we nevertheless
adopt the method of the Particle Data Group Zyl20  to assign a conservative
systematic uncertainty to the final result. With this method, the overall
statistical uncertainty of 7.0 ms is multiplied by the square root of 1.13
(the largest reduced $\chi^{2}$ value), which increases the total uncertainty
to 7.5 ms. If this total uncertainty is obtained from the quadrature sum of
the statistical uncertainty with the systematic, then the half-life of 14O
deduced in the present experiment is:

$\displaystyle
T_{1/2}\mathrm{(}^{14}\mathrm{O)}=70619.2(70)(26)~{}\mathrm{ms},$ (3)

where the first uncertainty is statistical and the second is the systematic
uncertainty obtained from the method of the Particle Data Group applied to the
deduced half-life as a function of cycle number (Fig. 3).

Table 1: Average values of the 14O half-life deduced from grouping the experimental data according to the particular electronic setting used in the experiment. The second column provides the number of cycles $N_{c}$ collected at each setting. Because each group contains the same experimental data, the sum of all cycles for each group is 124 and the weighted average of each group is $T_{1/2}$(14O) = 70619.2(70) ms.  Electronic Setting |  Nc |  Half-life [ms]
---|---|---
New 3 $\mu$s / Old 4 $\mu$s | 61 | 70625.5(100)
New 4 $\mu$s / Old 3 $\mu$s | 63 | 70612.8(100)
Bias: 2300 V | 22 | 70619.5(157)
Bias: 2350 V | 34 | 70612.7(135)
Bias: 2400 V | 33 | 70625.1(137)
Bias: 2450 V | 35 | 70619.9(135)
Threshold: 70 mV | 33 | 70630.8(131)
Threshold: 85 mV | 29 | 70631.4(152)
Threshold: 100 mV | 31 | 70612.0(140)
Threshold: 115 mV | 31 | 70602.8(140)
Beam on: 0 to 60 s | 44 | 70628.1(116)
Beam on: 60 to 120 s | 56 | 70611.6(105)
Beam on: 120 to 180 s | 24 | 70620.0(162)
Bin time: 2.8 s | 48 | 70621.6(110)
Bin time: 3.0 s | 43 | 70617.8(116)
Bin time: 3.2 s | 33 | 70617.4(145)

###  3.2 Search for Isobaric Contamination

As described above, the 14O activity was delivered to the experiment as a
12C14O+ beam at the mass setting $A/q$ = 26. While the resolution of the mass
separator at ISAC is sufficient to reject all elements and isotopes produced
with $A/q$ $\neq$ 26, it cannot completely eliminate all of the neighbouring
$A/q$ = 26 isobars. From the fit presented in Fig. 2, it is already clear that
significant amounts of contamination can be ruled out because the experimental
data are well described by a single exponential decay. However, because even
trace amounts of isobaric contaminants have the potential to affect a high-
precision half-life determination, it is of crucial importance to evaluate
what effect their possible presence may have on the final result.

The complete list of isobars at $A$ = 26 consists of 26P, 26Si, 26Al, 26Mg,
26Na, 26Ne and 26F. As 26Mg is stable, this isotope cannot produce any counts
in the detector and so its presence cannot affect the measurement. Ignoring
the production yields, and even the mass differences for the remaining
unstable $A$ = 26 isobars, several of these can also be considered completely
negligible based solely on their half-lives. As described above, every sample
collection (beam on) period was followed by a 22.2 s waiting period before
moving the sample into the gas detector. Any radioactive species with a half-
life of 1 s or less, would thus experience a waiting time that is more than 20
times its own half-life. For example, if there was a beam contaminant with a
half-life of 1 s, the fraction of activity remaining after this 22.2 s period
compared to the initial activity implanted is only 2$\times$10-7. This
eliminates 26P, 26Na, 26Ne and 26F as potential contaminants as they all have
half-lives that are $<$1.1 s. Furthermore, based on previous measurements at
ISAC, yields of silicon isotopes (like 26Si) are heavily suppressed when using
a silicon-carbide target since the diffusion and effusion probabilities of
silicon through a silicon target are extremely low. The yield of 10C16O+ from
silicon carbide is also expected to be negligible since this would require
significant amounts of oxygen to be present in the target. This beam has been
produced at ISAC in the past, but only from an oxide target Dun16 .

This leaves 26Al as the only possible isobaric contaminant in the beam that
may still be present in the data after the 22.2 s “cool-down” time. The 5+
ground-state of 26Al is radioactive, but it is very long lived with a half-
life of $T_{1/2}$(26Al) = 7.16$\times$105 years Sam72 . Because this is so
much longer than the 25 min measurement time of each cycle, any 26Al activity
would appear constant in the decay spectra and would therefore already be
included in the free background parameter of Eqn. 2. However, there is a 0+
isomeric state in 26Al located at 228 keV above the 5$+$ ground state that
also decays via a superallowed Fermi $\beta$ transition with a well-known
half-life of $T_{1/2}$ = 6.34602(54) s Har20 .

To determine by how much the 14O half-life deduced in the present experiment
may be affected by trace amounts of 26mAl isobaric contamination in the beam,
the experimental data were re-fit using a fit function that included 14O
decay, the overall constant background as well as an additional exponential
decay component:

$\displaystyle y_{fit}(t)=a_{1}e^{-a_{2}t}+a_{4}e^{-a_{5}t}+a_{3}.$ (4)

In this expression, $a_{1}$ to $a_{3}$ are identical to Eqn.2, $a_{4}$ is the
initial activity of the potential 26mAl contaminant and $a_{5}$ is its decay
constant. The half-life of 26mAl has previously been determined to high-
precision and thus $a_{5}$ was fixed in the analysis. All 124 decay cycles
were re-fit to the function defined in Eqn. 4 with the 26mAl initial activity
$a_{4}$ treated as a free parameter.

The weighted average of the 124 initial 26mAl activities that were deduced in
this analysis was $-$2.9$\pm$4.5 c/s, which is consistent with zero. Based on
the known mass difference between 26mAl and 12C14O and the resolution of the
mass separator at ISAC together with the 22.2 s “cool-down” time, this was
expected to be very small and this analysis provides further confirmation. The
conclusion is that 26mAl isobaric contamination present in the 14O decay data
can be considered negligible.

###  3.3 Leading Channel Removal Analysis

As a final test to search for any other rate-dependent effects in the data,
the half-life of 14O was determined as a function of the starting time of the
fit to the exponential decay activity curves. In this analysis, data from the
leading channels were systematically removed from the fit in steps of 5
channels (average of 15 s/channel) and the 14O half-life was determined, as
described above, from the weighted average of the 124 cycles. The result of
this analysis is presented in Fig. 5. The first point, which corresponds to no
leading channels removed, is the weighted average $T_{1/2}$ = 70619.2(70) ms
that was determined above. Removing up to 60 channels from the start of the
decay curves (approximately 180 s or 2.5 times the half-life of 14O) shows
that the deduced half-life is independent of the initial counting rate. This
confirms that any and all additional sources of rate-dependent effects that
were not accounted for in the analysis, including the possibility that trace
amounts of contaminants may be present, are negligible.

Figure 5: (Color online) Half-life of 14O versus the leading number of
channels removed from the analysis. The weighted average $T_{1/2}$ =
70619.2(70) ms of the 124 cycles with no channels removed (solid and dashed
lines) is overlayed for comparison. As these data are not statistically
independent, a weighted average of them cannot be determined.

###  3.4 Uncertainty in the Measured Dead Times

As described above, the fixed and non-extendible dead times that were applied
to the new and old MCS modules and were used to correct the experimental data
were determined to be 2.9958(42) $\mu$s and 4.0105(43) $\mu$s using the
source-plus-pulser method. Their central values were used to correct the 14O
decay data but the impact of their uncertainties on the 14O half-life still
needs to be evaluated. To accomplish this, the 14O decay data were corrected
and re-fit using the dead times of 3.0000 $\mu$s and 4.0148 $\mu$s (nominal
values + 1$\sigma$) and again using 2.9916 $\mu$s and 4.0062 $\mu$s (nominal
$-$ 1$\sigma$). The half-life of 14O obtained from adjusting the fixed dead
times within one standard deviation of their central values, resulted in a
shift to its central value by a maximum of 1.4 ms. This is negligible compared
to the $\pm$ 7.0 ms statistical uncertainty in the 14O half-life measurement,
but it was nevertheless included as an additional source of uncertainty in the
final result.

###  3.5 Final half-life result

A summary of all of the uncertainties (statistical and systematic) that were
used to evaluate the total uncertainty in the 14O half-life are summarized in
Table 2. The two largest contributions are the $\pm$ 7.0 ms statistical
uncertainty and the systematic uncertainty of $\pm$ 2.6 ms, that was assigned
to account for the $\chi^{2}$/$\nu$ value of 1.13 obtained from the half-life
determined as a function of cycle number. Summing all of these uncertainties
in quadrature leads to an overall uncertainty of $\pm$ 7.6 ms. The half-life
of 14O deduced in the present work is therefore:

$\displaystyle T_{1/2}\mathrm{(}^{14}\mathrm{O)}=70619.2\pm
7.6~{}\mathrm{ms}.$ (5)

This result has been determined at the level of $\pm$ 0.011$\%$ and is the
most precise half-life ever determined for 14O. It is also in excellent
agreement with the previous world-average, $T_{1/2}$ = 70619(11) ms, obtained
from 11 previous measurements of this quantity Har20 .

Table 2: Uncertainty budget in the analysis of the half-life of 14O deduced in the present experiment.  Source | Uncertainty (ms)
---|---
Statistical uncertainty | 7.0
Reduced $\chi^{2}$ (Fig. 3) | 2.6
Measured dead times | 1.4
Total (quadrature sum) | 7.6

The half-life of 14O deduced in the present work is compared, in Fig. 6, with
the previous most precise measurements of this quantity Cla73 ; Bec78 ; Wil78
; Gae01 ; Bar04 ; Bur06 ; Tak12 ; Laf13 . The procedure described in Ref.
Har20  for calculating a new world average 14O half-life was adopted. This
method retains only those measurements whose quoted uncertainties are no more
than 10 times larger than the most precise measurement. The weighted average
of the current result with the eight previous measurements that satisfy this
criterion is:

$\displaystyle T_{1/2}\mathrm{(}^{14}\mathrm{O)}=70619.6\pm
6.3~{}\mathrm{ms}.$ (6)

The $\chi^{2}$/$\nu$ value for the 9 measurements plotted in Fig. 6 is 0.87,
which indicates a consistent set. The 14O half-life has now been established
at the level of $\pm$ 0.0089$\%$, which is now one of the most precise half-
lives ever determined for any radioactive decay.

In Ref. Laf13 , a possible systematic discrepancy in the 14O world half-life
data associated with the type of activity measurement (direct $\beta$ counting
or $\gamma$-ray photopeak counting) was described that was based on the
$\chi^{2}$/$\nu$ value of 2.83 obtained when the previous results were grouped
according to the type of decay radiation detected. Adding the direct $\beta$
counting result determined in the present work into this analysis yields the
average values of $T_{1/2}$($\beta$) = 70621(7) ms (from 4 total measurements)
and $T_{1/2}$($\gamma$) = 70603(16) ms (5 measurements) for direct $\beta$ and
$\gamma$-ray photopeak counting, respectively. The $\chi^{2}$/$\nu$ value for
these two values is 1.26, which is consistent with there being no significant
source of systematic uncertainty in the world data that can be attributed to
the type of radiation detected.

Figure 6: (Color online) Comparison of the present result with the previous
most precise measurements of the half-life of 14O from Refs. Cla73 ; Bec78 ;
Wil78 ; Gae01 ; Bar04 ; Bur06 ; Tak12 ; Laf13 . The red solid line is the new
world average that includes the present result and the dashed lines represent
the $\pm$1$\sigma$ uncertainties. The $\chi^{2}$/$\nu$ value for the set is
0.87 for 8 degrees of freedom.

###  3.6 The $ft$ and $\mathcal{F}t$ values for 14O decay

In terms of the $ft$ value for the superallowed decay of 14O, the new world
average 14O half-life leads to an updated value of $ft$ = 3042.25(73) s. This
is a slight improvement compared to the value of $ft$ = 3042.23(84) s from the
most recent survey of superallowed data Har20 . Similarly, the corrected
$\mathcal{F}t$ value for 14O is updated to the new value of $\mathcal{F}t$ =
3070.3(19) s, which is almost unchanged from its previous value $\mathcal{F}t$
= 3070.2(19) s Har20 . The reason for these small differences is because the
new world average half-life is numerically almost identical to the previous
average and because uncertainties in the $Q$ value and the branching ratio are
much larger contributors to the overall uncertainty in the $ft$ value. As a
result, the high-precision half-life that was deduced in the present
measurement is not expected to impact the limit obtained in Ref. Har20  on the
possible presence of scalar currents.

##  4 Conclusion

The half-life of 14O that was deduced in the present work, $T_{1/2}$ =
70619.2(76) ms, is the most precise value ever obtained for this nucleus and
is more precise than the previous world average that was deduced from a
weighted average of 11 independent measurements. The new world average is
entirely consistent with the previous one but the overall uncertainty in the
half-life has been reduced from $\pm$ 11.2 ms to $\pm$ 6.3 ms, or nearly a
factor of 2. Future improvements in the $ft$ value and the corrected
$\mathcal{F}t$ value for the superallowed decay of 14O to improve upon
existing limits on scalar interactions in nuclear $\beta$ decay can only be
achieved through improved measurements of the $Q$ value and branching ratio
for this decay and would require a reduction in the theoretical uncertainty
associated with isospin-symmetry breaking.

## Acknowledgments

We would like to thank the ISAC ion-source and beam development group for
their hard work and dedication to the production and delivery of the high-
quality 12C14O beam that was necessary for this experiment. We acknowledge the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC). Nous remercions le Conseil de recherches en sciences naturelles et en
génie du Canada (CRSNG) de son soutien. This work was supported by the U.S.
Department of Energy, Office of Science under grant nos. DE-FG02-93ER40789 and
DE-SC-0017649. CES acknowledges support from the Canada Research Chairs
Program. TRIUMF receives federal funding via a contribution agreement through
the National Research Council of Canada. The University of Regina is situated
on the territories of the nêhiyawak, Anihšināpēk, Dakota, Lakota, and Nakoda,
and the homeland of the Métis/Michif Nation and TRIUMF is situated in the
traditional, ancestral and unceded territory of the Musqueam people and we
acknowledge that our research was conducted on their lands.

## References

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#  Community, Collaboration, and Climate

Johanna Hardin and Shahriar Shahriari

(April 2022)

###### Abstract

The Department of Mathematics & Statistics at Pomona College has long worked
to create an inclusive and welcoming space for all individuals to study
mathematics. Many years ago, our approach to the lack of diversity we saw in
our majors was remediation through programming which sought to ameliorate
student deficits. More recently, however, we have taken an anti-deficit
approach with focus on changes to the department itself. The programs we have
implemented are described below as enhancing community, collaboration, and
climate within our department.

Keywords: anti-deficit; diversity, equity, and inclusion; access;

##  1 Introduction

The murders of George Floyd, Breonna Taylor, Ahmaud Arbery, Tony McDade,
Philando Castile, Tamir Rice, Michael Brown, Eric Garner, Trayvon Martin, and
countless others, as well as the drastic disparities in death rates due to
COVID-19 between whites and communities of color has brought to sharp focus
the systematic and structural racism in the US. As a related byproduct, the
importance of providing access to a quality education for all is ever more
self-evident. Our math and stats department—like many others—has struggled
with its role in providing an inclusive and inviting climate for students from
groups that are traditionally underrepresented in mathematics. With the hope
that small and concrete changes can make a difference, and that sharing our
experiences may be helpful to others, we are eager to join in the discussion.
While we still have a long way to go, we have, over the years, implemented a
number of concrete programs, that have been positive overall. We are not
speaking as the voice of our department, but we hope that our personal
perspective on the modest and concrete programs in our department over the
past decade can be of some use to others.

Consider the following two different perspectives on the job of a mathematics
or statistics faculty at a selective undergraduate college: (1) Our main job
is to identify talent, to encourage the talented to pursue the mathematical
sciences, and to lead the rest to other, more “appropriate”, disciplines, and
(2) we should have the philosophy that _everyone_ can do math, and that our
job is to facilitate _every_ student’s success in mathematics. In the math and
stats department at Pomona College, we never explicitly discussed these points
of view, but some variation of the tension between them has always been in the
background, and, if you take a bird’s eye view of the last thirty years,
despite the range of personalities and continued disagreements, we have slowly
but decisively moved from the first point of view toward the second.

The opening dichotomy is, of course, somewhat rhetorical. Ideally, students
find success in mathematics at different levels; some continue in the
mathematical sciences and some don’t. What is not rhetorical is the data that,
over a long period of time, we had very little success in attracting students
of color (particularly Black, Hispanic, and indigenous students), low-income
students, first generation students, and even women.

Pomona College is a small liberal arts college with enviable resources, our
faculty do their best to be active researchers while being dedicated teachers
and mentors, and our student to faculty ratio is low. Even so, consistently
and for many years, the department had attracted fewer students of color into
the major than their proportional representation at the College. Even more
alarming was the fact that often students of color, students from low income
backgrounds, and first-generation students did poorly in our beginning
courses. It is not surprising that such students, by and large, did not
continue taking mathematics, and steered clear of STEM fields. Whether we
liked it or not, we acted as an effective filter across STEM.

Our first response, starting about three decades ago, was curricular. We
worried about the students’ prior preparation, the quality of their high
school work, and the mismatch with our courses. Various faculty developed new
courses that relied less on students’ prior courses, were interesting,
strengthened problem solving skills, and prepared students for future
work.111Our colleague Erica Flapan’s _Problem Solving for the Sciences_ —a
course organized around doing non-calculus “word problems”—and the second
author’s _Approximately Calculus_ —later turned into a book—were attempts in
this direction. Many of us introduced or more fully embraced active student
learning—with or without technology—in our classes. We reorganized the
curriculum to create multiple pathways through the introductory
material.222Thirty years ago, at Pomona, if you wanted to do math, you took
three semesters of calculus followed by two semesters of a combined linear
algebra/differential equations class before you had any choice of topics or
classes. All of these changes were, on balance, positive. The students
responded, and our major and courses grew. The problem of too few students of
color, low-income students, and first generation students who continued on to
the next math class, however, persisted.

About ten years ago, two students of color, who had done superbly in the
second author’s own honors calculus class, were failing his linear algebra
class. This was one among many data points suggesting that a focus on student
preparation cannot possibly be enough. It seemed to us that background
preparation was not a determining factor as students of color with all
different backgrounds were underperforming what their background preparation
alone would have predicted. The college’s work with the Posse Foundation and
the work of Uri Treisman333Our colleague Ami Radunskaya had worked directly
with Uri Treisman, the second author was Pomona’s inaugural Posse faculty
mentor, and the first author has advised a Posse cohort also. (Asera, 2001)
helped us shift our focus to the “climate” in the department. We moved the
spotlight from the students and their preparation to us, the faculty, and the
environment in the department. We slowly started to understand that some of
our students did not feel that they belonged, and, as a result, they were not
using the available resources (mentor sessions, office hours, tutors, etc.) as
effectively as possible. The stress of tackling a difficult discipline in an
unfriendly environment—which extended far beyond the math and stats
department—adversely affected the students’ successes in the classroom and
their desire to continue in mathematics. The conclusion was that we had to
create extracurricular programs to challenge the dominant climate in the
department.

At this point it is worth clarifying that while our goal is to reach students
who are minoritized or marginalized for any of a variety of reasons,
generally, depending on the program, we include students across the board who
self-identify as an individual who wants to benefit from the programming. That
is, our programs are populated with students who do not see themselves in
mathematics (are minoritized); feel dismissed in the mathematical space (are
marginalized); or want to learn and grow in a supportive environment designed
to create equity in mathematics. With our approach, the vast majority (but not
all) of students in our programs are students of color, low-income students,
and/or first generation students. Is our approach a strength or a weakness? We
don’t have the answer. But we do know that we firmly believe that mathematics
is for everyone, and our inclusive attitude is integrated in to all that we
do.

##  2 What have others done?

As posed above, there are generally two approaches to broadening participation
in the mathematical sciences. Substantial work has been done to implement
creative ideas in the college mathematics classroom surrounding each of these.
Our paper is not meant to be a comprehensive review of different programming,
but we provide a few examples below to contextualize our work with that of the
larger mathematics community.

###  2.1 A deficit approach

A deficit approach is typically connected to the amount of previous
mathematical training a student has had prior to their college mathematics
classes. Peck (2021) provides a comprehensive overview of the problems
surrounding a deficit model, including a succinct definition of a deficit
perspective:

> Definition: The deficit perspective is a propensity to locate the source of
> academic problems in deficiencies within students, their families, their
> communities, or their membership in social categories (such as race and
> gender).

One possible reaction to the definition is that it indicates a personal
failing of the student which teachers of mathematics surely do not mean to
imply. However, from the perspective of a student of mathematics, an
insufficient background is felt as a personal failing.

Online programs have potential to fill in gaps of understanding either
throughout the semester (remediation through online just-in-time teaching in
order to prevent pre-calculus and calculus from being barriers toward all the
STEM fields (Bertrand et al., 2021)) or prior to college in order to avoid
taking remedial mathematics classes (using the online software ALEKS during
high school to develop mathematical skills (Fine et al., 2009)).

In-person programs can also be effective. A one-credit supplemental co-
requisite Calculus I course (Hancock et al., 2021) allows for active learning
in the classroom as well as reinforcement of the calculus concepts and
community within the course. As has been well-documented, working with faculty
on research projects is associated with retention (Hernandez et al., 2018) and
achievement (Parker, 2018).

###  2.2 An anti-deficit approach

An anti-deficit approach tackles issues which are outside of the individual
student’s purview. Again, as defined by Peck (2021):

> Definition: Anti-deficit perspectives locate the source of academic problems
> within institutional structures that work to limit access to educational
> opportunities. The focus is on the assets that students bring to the
> classroom, rather than what they lack.

Peck (2021) provides an extensive review of issues and practical suggestions
toward creating anti-deficit mathematics programs. Additionally, Hagman (2021)
identifies diversity, equity, and inclusion as one of the core components
which make up a successful calculus program. Working through a classroom
activity, an anti-deficit approach to teaching is developed by Adiredja et al.
(2020).

In retrospect, many of the steps we have taken at Pomona College to create a
math and stats department that is more diverse, equitable, and inclusive
follow an anti-deficit model. We describe these next.

##  3 What did we implement?

The Department of Mathematics & Statistics has always valued community. Since
the early 1990s, we have organized, for all of our lower division classes,
evening peer mentoring sessions where students work together on their
assignments under the supervision of a more experienced undergraduate. These
mentor sessions have encouraged collaboration and have resulted in a buzz of
mathematical activity in our building in the evenings. As a department, we
encourage collaboration and want the students to compete with the material
instead of with each other. Many of our faculty assign challenging projects
that are meant for collaborative work, and mentor sessions play a crucial role
in bringing the students together to collaborate. But student experience at
the mentor sessions has been very uneven. While many of our students loved
collaborating in the mentor sessions, and credited these evening sessions with
a strong sense of community in the department, others felt left out. Somewhat
unsurprisingly, students of color, low-income students, and first generation
students were more often than not in the latter group. They felt out of place,
did not participate in the discussions, were not comfortable asking questions,
and often found themselves isolated.

Here, we report on three on-going programs—two in the department and one
college wide—that we believe have had a positive impact on both our department
and the students we are hoping to target. The first one, _Learning
Communities_ (abbreviated LiCMath) are organized for individual classes. The
second one is a department-wide cohort program called the _Pomona Scholars of
Mathematics_ (PSM), and the third one, _Pomona Academy for Youth Success_
(PAYS) is a college wide outreach program for local high school students. Our
experience with these programs are of varying lengths. The first group of PAYS
students arrived on campus in the summer of 2003. The first LiCMath group was
organized in the fall of 2010, and the first cohort of PSM students began in
the fall of 2014. We are not presenting an empirical study (as the programs
were not set up with a particular experimental design); instead, we present
the programs and the evidence we have that they are working to bring about
change in our department.

###  3.1 LiCMath

The literature describes learning community structures to support mathematics
programs. However, while aligned in goals of improved mathematics content
delivery, the programs often have different goals in terms of who and how they
serve the students. For example, learning communities play a role in
encouraging women in mathematics as described by Evans et al. (2018). Linking
mathematics and English courses, Piercey and Cullen (2017) have found learning
communities to be effective at improving problem solving; linking biology and
statistics courses Ryu and Zhu (2022) have found that learning communities
improve learning across both disciplines. General college sponsored learning
communities can be quite effective and efficient when using peer mentors
(Rieske and Benjamin, 2015). A set of qualitative interviews demonstrated that
(college wide) learning communities help students interact at many different
levels and that the first year was an ideal time for learning community
involvement (Firmin et al., 2013).

At Pomona College in the Department of Mathematics & Statistics, a learning
community is a small group of students who are all enrolled in one particular
class. They commit to work together and to support each other throughout the
semester. A learning community is meant to become a supportive and safe space
to work on homework problems, to discuss the lectures, and to bounce ideas off
each other. The meetings take place outside of class times, and, at least
initially, under the supervision of an upper class undergraduate student
mentor.

The targeted students are those who do not feel at home in the math and stats
department—or in the college in general—and have doubts about whether they
belong. Such students could have a strong mathematics background or a weak one
but may try to go through the class in isolation. They may, consciously or
unconsciously, believe that using the available resources (faculty office
hours, mentor sessions, discussions with peers) is a sign of weakness and that
they should be able to succeed by sitting alone in their room and working
doubly hard on the mathematics. It may not be easy for such students to create
a safe collaborative learning environment early in the semester. Such students
can come from all walks of life, but first-generation college students, women,
students of color, students from less prestigious high schools, and students
from less privileged families tend to fall in this category most often.
Learning communities are open to all students, but we make a special effort to
encourage the students in our targeted audience to participate.

A faculty member who decides to organize a learning community in their class,
announces the aim and structure of LiCMath to the whole class and invites
everyone who is interested to apply. In addition, targeted students are sent
customized individual emails encouraging them to participate. In our pitch, we
consistently emphasize the value of collaboration, and we explain that every
student needs a supportive group of peers in order to get the most out of a
mathematics class. Some students already have such a support group; others
will find it in their dorms or in the mentor sessions. LiCMath is an
alternative for students who want a more structured collaborative environment.
We explicitly make it clear that students can participate in LiCMath
regardless of their perception of their strengths or weaknesses in
mathematics. All applicants who apply and who accept the responsibilities of
membership are accepted and organized into small groups with 3 to 7 members
each.

The learning communities meet for at least two hours a week, and the students
commit to going to all of the meetings regardless of whether they have
specific course related questions. If a student believes that they already
know the material for that week, then they are still expected to attend the
weekly meeting to provide support for others. To build community and to start
getting to know each other, we suggest that the students begin each meeting
with a short check-in about their week. We also encourage the students not to
focus solely on the next assignments. Rather, they should spend some time
going over the main points of the material recently covered in class, and
discuss their returned homework assignments. By communicating to each other
what is interesting, important, challenging, or unclear, they start the
process of learning from each other.

While, on paper, it seems as though LiCMath adds hours of work each week to a
student’s plate, we almost never get feedback that LiCMath is a burden.
Instead, the time spent in a learning community typically replaces time that a
student would have been working alone, possibly frustrated about not making
progress. Additionally, working in the learning community increases both
confidence and communication skills, making the class mentor sessions much
more efficient for the student.

Our concept of a learning community owes much to the Posse Foundation’s idea
of creating a peer support group for student success. It also builds on Uri
Treisman’s pioneering ideas for “Emerging Scholars” workshops for calculus
classes. We are a small liberal arts college where most students know each
other, and where most of our classes are not overburdened with students from
other departments simply fulfilling requirements, and, hence, already
incorporate many challenging projects and experiences. Our specific
circumstances have meant some possibly important modifications to previous
models. For example, our learning communities are not marketed as “honors”
sections, and they do not usually incorporate additional more challenging
course material. Nor are the learning communities remedial tutoring sessions
for students who are falling behind in the class. Our students are told that
they should join learning communities because isolation is not a path to
success in mathematics, and everyone needs a supportive group of peers through
the semester to be able to get the most out of any of our classes.

Learning communities are a relatively low-cost structure for creating
community. The interested faculty member initiates the process, recruits the
students and the mentor, and provides some early guidance. While it takes some
time to think through the organization, and striking the right tone in
communicating with the students can sometimes be challenging, there is very
little on-going work for the faculty during the semester.

We count our students’ participation and even enthusiasm for LiCMath as
circumstantial evidence for its success. Initially, LiCMath sessions were
organized only for lower division classes, but, in a number of upper division
classes, the students themselves asked for and organized LiCMath as well. In
fact, both Pomona Physics and Chemistry departments started experimenting with
learning communities because students, who had been part of LiCMath, asked for
them.

###  3.2 PSM

The first semester at college, especially as a full time student living away
from family, can be very stressful. Learning to navigate the expectations and
the resources of a university while adjusting to a new social landscape is
challenging. This is especially so for first generation college students or
for individuals who find the physical space, the bureaucratic structure, the
social scene, or the academic expectations unfamiliar. To counter the sense of
isolation in facing the new environment, and to level the academic playing
field, Pomona College has been experimenting with creating cohorts of incoming
students centered around academic disciplines. As a part of our efforts, the
authors, beginning in the fall of 2014, started the Pomona Scholars in
Mathematics (PSM). The goal of PSM is two fold. On the one hand, we want to
create a community of students who experience the first year of college and
its challenges and opportunities together, and, on the other hand, we hoped
that, over time, there would be a spill over effect, and the sense of
inclusion and community in the whole department would be enhanced.

Recognizing the special role of mathematics as a gatekeeper or conduit to
other fields, PSM aims to support students who express an interest in fields
where mathematics plays an important role (math, statistics, computer science,
physics, economics, engineering, etc). In the summer before their first year
at Pomona, targeted students—students of color, low-income students, and first
generation students—are invited to join PSM. Approximately 15 students do so,
and they are advised by two faculty members (usually from the math and stats
department but sometimes from aligned disciplines like computer science or
physics). The program is built on three pillars:

  * •

early intensive one-on-one faculty-student advising. Each of the two faculty
members are assigned to be the faculty academic advisor for half of the
students, and the faculty have biweekly one-on-one meetings with their PSM
advisees throughout the first year. Taking a wholistic approach to mentoring,
these meetings go beyond just academics. We check in with the students about
their college experience and home life as well as how their classes are going.

  * •

weekly group lunch meetings. The PSM group of first year students and faculty
mentors meet once a week over lunch for the duration of the students’ first
year in college. While the primary purpose of these meetings is to build
community and camaraderie—we routinely have the students check in and share
their ongoing challenges, successes, and insights—we also introduce the
students to college resources, and we regularly invite faculty members from a
variety of disciplines to come in and share their academic journey. Our hope
is that demystifying the academy and normalizing the students’ experiences
help counter the sense of isolation in a new and challenging academic
environment.

  * •

within-course clusters. Our PSM students have a wide range of mathematical
background, and they do not all start in the same first mathematics course as
first year students. Even so, we encourage the PSM students to take classes
together when they can, and, in keeping with our mathematics-is-a-
collaborative-enterprise mantra, we urge them to form small study clusters in
various sections of math.

After the first year, the PSM students join what we call PSM+. PSM+ is a
loosely organized community of sophomore, junior, and senior students that
aims to continue the goals of PSM. The larger group meets about once a month
over lunch, and organizes other department-wide events.

While it is difficult to tease out confounding variables, our impression is
that PSM has been successful in changing the feel of the department. Not only
are more of the targeted students continuing in mathematics courses, but more
students of color, low-income students, and first generation students have
been taking on leadership roles in the department, and, in general, have been
more visible as part of Pomona’s mathematics community.

Unlike learning communities, the downside of PSM is the need for heavy faculty
commitment. Weekly group meetings and biweekly advising meetings are extremely
time consuming. The faculty members receive a small stipend for their work but
no other compensation.

Of all our programs, PSM is the one which embraces an anti-deficit approach
most fully. In our weekly meetings we center the experience of the PSM
students and who they are as a whole person. In community, they are able to
see that their experiences are sometimes not unique, and they are able to
process how to be successful in college. We also bring in older students,
alumni, and other role models (e.g., staff and faculty) who have had non-
traditional paths. Although the majority of guests (and students) are POC,
their more important connection is that to a person, they’ve had to overcome
systemic barriers. Our goal in centering their experiences is to have them see
themselves being successful, in their own way. The PSM community empowers our
students to understand and challenge the oppressive structures they encounter
(see Section 5.1).

###  3.3 PAYS

Pomona Academy for Youth Success
(PAYS444https://www.pomona.edu/administration/draper-center/pays) is a Pomona
College summer academic program for high school students. The program, which
began in 2003, has an enviable record of its participants continuing onto
selective colleges and universities. The program brings 30 rising tenth
graders from local under-served high schools to spend 4 weeks in residence at
the College. These same students come back for another 4 weeks as rising
eleventh graders and again as rising twelfth graders.

Each four week session is intensely academic, and its core consists of each
cohort taking two classes that are taught by regular college faculty: a math
class and a humanities/social sciences class. The classes meet four days a
week in the mornings. Afternoons are focused on working collaboratively on
homework, workshops on college application processes, and enrichment
activities. The faculty are aided by a group of dedicated undergraduate
students who serve as teaching assistants to the faculty and as mentors to the
students.

The Department of Mathematics & Statistics has been instrumental in the design
and the implementation of the PAYS program.555The second author was the lead
author of the institutional grant from the Irvine Foundation that started the
program. Every summer five faculty members (mostly from within Pomona’s
Department of Mathematics & Statistics) teach math in the program. While the
PAYS program is a college-wide program, it has directly and positively
affected the climate in the math and stats department in many ways, including
the following.

  * •

The myriad faculty who, over the years, have taught in the program have had a
chance to think deeply about questions of inclusion and access as well as the
value of community and collaboration. They have also gained a better
appreciation of the lived experiences of our students of color, low-income
students, and first generation students and the grit and tenacity that they
bring to the classroom.

  * •

Many of our math students, including our minoritized math students,
participate in the PAYS program as TAs and mentors for the high school
participants. The college students work closely with faculty and are trained
as facilitators of collaborative learning. For the college students, the
experience is worthwhile and socially meaningful. Moreover, the work is
related to, and, in fact, requires, their mathematical training. These
students feel connected to the department and its mission, and they are often
departmental student leaders.

  * •

Regularly, PAYS students matriculate at Pomona College, and some have become
math majors. Not only has the presence of these students helped diversify the
department, they have also brought the collaborative and inclusive culture of
PAYS to the department.

Through PSM, LiCMath, and PAYS, we endeavor to create a safe space for
students of color, low-income students, and first generation students where
their experience, their voice, and their story takes center stage. At the same
time, we want to involve as broad a swath of our students as possible for a
number of reasons. First, without some work on the part of the (typically
white) students who already feel safe in mathematics, it is hard to imagine a
positive change in the climate. Second, we are concerned about stigmatizing a
single group of students further, and third, rightly or wrongly, our
assessment was that an approach that used a broader language would give us
more room to operate. As a result, we have not been entirely consistent about
“who” the programs are for. For PSM, our targeted students are students of
color, first-generation students, and low income students. In the PSM space,
we often explicitly talk about race, gender, and ethnicity, and we try to
center the experiences of the students involved. In contrast, for learning
communities, while we specifically encourage students of color, first-
generation students, and low income students to participate, we invite all
students to join.

##  4 Departmental Involvement

Like any other department, over the years, we have had disagreements over
priorities. Even so, one of our shared principles has been to empower
individual faculty to pursue their passions. As an example, we don’t choose a
common textbook or agree on a common syllabus for our calculus courses. The
hope is that what we gain in faculty enthusiasm outweighs the advantages of
common standards. At no time has any department member been directly obligated
to participate in diversity initiatives, and, as you may expect, everyone
participates at different levels. Even so, at this juncture, the whole
department is supportive of the inclusion efforts, and, as a result, in
addition to the programs outlined above, our faculty contribute to the sense
of community and to our diversity work in myriad of other ways. For example,
many routinely incorporate meals or social time with their students or with
their research groups, and in choosing students for research and for
leadership positions issues of openness and access are always considered.

The consensus over the importance of issues of access is the result of
discussions—sometimes very heated—over a long period of time. Our first point
of agreement—many years ago and with some pressure from our administration—was
to agree that issues of access mattered in our job searches for new faculty.
We have been including some version of the following in our job ads and in our
discussions of individual candidates:

> The department has directed much effort in creating a supporting community
> for all students, and is particularly interested in candidates who have
> experience working with students from diverse backgrounds and a demonstrated
> commitment to improving access to and success in higher education for
> students of color, low-income students, and first generation students.

Our deliberate discussion of diversity and access in the hiring process has
meant that most of our current colleagues were hired with a clear
understanding that this work is important to the department. As a _PRIMUS_
Associate Editor pointed out to us in the refereeing process, thinking of
Diversity, Equity, and Inclusion work as a voluntary add-on to our jobs is
actually one of the sources of the problems we face. While we don’t have a
mechanism for distributing the work, and we don’t force anyone to participate
in any of the specific programs outlined here, most everyone in the department
does see the DEI work as an important part of what our department does, and
many contribute to it in one way or another.

Among many other department initiatives, we also want to highlight three other
efforts by our colleagues. The first is a community seminar on identity,
culture, and mathematics organized in the Spring of 2018 by our colleague
Gizem Karaali. The weekly seminar—with no credits or grades and open to
all—brought together a group of students and faculty from our department. Each
week the seminar discussed a new issue of interest to the community based on
selected readings. Topics included _Math Anxiety_ , _Microagressions_ ,
_Mathematics and Class_ , _Mathematical Inqueery_ , _Critical Mathematics
Education_ , _Intentional Learning Communities and cohorts_ , _Role Models in
Math_ , _Algorithmic Bias_ , and _Chinese Math, why do we think proofs are
Greek?_ In addition to providing a venue for in-depth discussion between
faculty and students, the seminar came up with a number of action items. For
example, the students, with faculty input, came up with a list of guidelines
for our mentor sessions (see Figure 1). The guidelines are posted in every
classroom as large posters.

Figure 1: Mentor Session Guidelines – posted in every classroom as a large
poster. The original set of guidelines was created by one of our student
mentors, Chris Donnay class of 2018.

The second is the summer undergraduate research program
PRiME666https://ehgoins.wixsite.com/prime run by our colleague Edray Goins.
Combining funding from the NSF and the College, Edray’s program targets groups
traditionally underrepresented in algebraic geometry–such as women and
minoritized students, and, every summer, brings to campus a number of talented
undergraduates who, along with a number of Pomona College students, do
mathematical research. While it is too early to herald the results and impact
on our department, it has been widely documented that undergraduate research
is one of the best practices for retaining students in mathematics, and having
that opportunity locally is invaluable.

The third is the EDGE program for women in
mathematics777https://www.edgeforwomen.org/, co-directed by our colleague Ami
Radunskaya and hosted in Claremont three times in recent years. While this
program is for students who are already heading to graduate school, its
availability, the programming around it, and the energy and enthusiasm of the
participants is not lost to our undergraduate students.

While the connection between the myriad initiatives may not be immediately
clear, we believe that the synergy across the programs is vital to the success
of all of them. Our students see the faculty involved and committed to
diversity, equity, and inclusion. Indeed, some of our students become involved
in the satellite initiatives (even after graduating). Our efforts do not (and
cannot) have a directly linear trajectory or a simple cause and effect path,
but we recognize and point out that our individual efforts have effectively
combined to create many different spaces and venues where our students feel
supported.

##  5 Results

Unquestionably, the Department of Mathematics & Statistics at Pomona College
looks and feels different now than it did when we started teaching (in 2002
and 1989, respectively). The ongoing discussions we have throughout the
building: in department meetings, in our classrooms, in the hallways, and one-
on-one with students tell us that, while we have not found a magic bullet, we
are headed in the right direction. We had zero Black math majors from 2003 to
2011 (when there were roughly 19 math majors per year); in 2020 out of our 45
graduating math majors, 5 (11.1%) were Black and 5 (11.1%) were Hispanic. It
is also worth noting that in the same class of 2020 math majors, 19 of 45 were
women (42.2%). During a time in which the math major has been growing
substantially, we are happy that the number of women majors has increased
correspondingly while the total percentage of students of color has increased
noticeably.

We are fortunate to work at an institution that is committed to higher
educational access. Over the years since initiating our programs, the overall
student body at the College has diversified quite a bit. We have seen parallel
trends in the math and stats department. The sample sizes are too small to do
full comparative analyses, but we are encouraged by the direction of the
trend, particularly with students of color becoming math majors. The
institution has long been approximately half women. Unfortunately, in the last
two decades, we have seen little to no change in the proportion of women
mathematics majors, suggesting that we should possibly think even more
carefully about our female students and likely our female students of color.
We hope that our initiatives are part of the mechanism encouraging women and
marginalized students to pursue math. However, without a better experimental
design, we are not in a position to claim that our programs are causing the
changing demographics in the math and stats department.

###  5.1 Qualitative Student Feedback

As they were graduating, we surveyed the inaugural PSM class (c/o 2018). Here,
we provide their qualitative feedback as a way to give a sense of their
experiences with PSM.

  * •

“Having PSM made it easier to acknowledge when I was struggling the most.”

  * •

“This group is probably the main reason I chose math.”

  * •

“Thank you so, so much. This community really keeps me going. Isolation is
easy to come by at Pomona.”

  * •

“More open/safe conversations… re: imposter syndrome/stereotype threat.”

  * •

“I had a particularly rough time my first semester because only 2 PSM members
were in my calc class and they both dropped, leaving me alone. I know it’s
difficult to ensure there will be PSM members together since you cannot force
people to take a course, but maybe some form of additional communication could
better help facilitate that.”

  * •

“[PSM] is very math-centric which mades sense b/c the word math is in there
but it might push some away from being in it.”

  * •

“the relationships that I formed through this group as both a mentor and as a
mentee have allowed me to be successful as a low-income first-gen woman in
STEM.”

  * •

“PSM could improve by discussing non-academic topics during meetings and one-
on-one advising. Many of the PSM peers, myself included, are the first in
their family to attend college. This whole experience, the different social
rules, and the full new atmosphere are brand new. I think that spending more
time discussing this and potential responses / coping strategies / lessons
learned would be helpful.”

  * •

“PSM was great and I really really appreciated it. It truly supported me and
helped me during my first year.”

The quantitative information we have on diversification of our department is
neither comprehensive nor striking enough to make bold claims about having
figured out the perfect blend of programs to create an ideal space for all of
our students. However, the qualitative information and our sense for how the
department has changed over the last two decades does give us hope that we are
moving in the right direction. Our students seem to be well served by the
programs, and we can tell that their engagement is likely due in a large part
to our engagement. That said, we are well aware that it is imperative for us
to reach each new group of students as they matriculate at Pomona College.
Even after the programs have been set up and put into motion, we are obligated
to continually revisit and revise their effectiveness.

We have stayed in touch with many of the students who have passed through the
programs described above. Some of the students are in graduate school or
finished with PhDs or master’s degrees. Others are working in tech or finance
or education. They are a heterogeneous group who have taken their Pomona
educations in many different directions. In our efforts to bring their voices
into this manuscript, we have reached out to our alumni and reconfirmed the
variety of student experiences. We certainly continue to hear from some of our
alumni who are still navigating their own identities in mathematics, and we
sometimes hear from students that their voices and their experiences are not
sufficiently valued in the math and stats department and that they
continuously feel that they have to carry a much heavier burden than their
peers. Other students, however, are able to directly connect the efforts from
our programming to their post-Pomona trajectory as evidenced by the recent
reflection from a PSM student who is currently in a PhD program:

> I think Pomona math’s diversity programming made a huge difference in the
> way that I conceived of and conceptualized my mathematical self at the time.
> I gained so much confidence, I was comfortable being wrong and asking for
> help, and I trusted myself to contribute to collaborative settings. This
> part of me wilted and at one point had disappeared at [University XXX]888A
> top research institution that has been redacted for privacy reasons.. I
> really struggled to feel competent or even smart during the first two years
> of my PhD, when none of this support was available to me, in a program with
> $<$15% women and $<$5% POC (not including east-Asian backgrounds), being
> constantly undermined by peers and professors, interrupted, ignored in group
> settings. I really leaned on the spirit of Pomona’s math community from my
> past to figure out how to handle this. …[T]he foundation was available for
> me to have just enough disbelief in the new voices [University XXX] had
> cultivated in me (I am stupid, I don’t belong here, I faked my way in, no
> one takes me seriously) to talk back to them and wrangle them out of the
> driver’s seat.

##  6 Challenges

In recent years, we have put quite a bit of effort into issues of diversity
and access. Our department indeed feels and looks different than earlier
times, and we are excited about the changes. Even so, many challenges remain.
Here are a scattering of questions with which we continue to wrestle.

  * •

What are best practices in targeting students for our programs, and how do we
avoid unintentionally siloing students and creating new barriers between
different groups of students?

  * •

How much of a priority should diversity issues be? Are there aspects of the
work that should be considered a core part of our mission? If so, how should
the department support these efforts?

  * •

How do we meaningfully assess what we do? In the past several years, we have
thrown everything and the kitchen sink at the lack of diversity among our
students. We have seen some improvement, but are the changes due to our
efforts or to the changes in demographics brought about by the Pomona College
admissions staff? Which aspects of our programs are most effective? Some of
what we do is incredibly labor intensive. Do the outcomes justify the energy
spent?

  * •

How do we put a structure in place which assures that we learn from past
experience, that we pass along best practices, and that we achieve our goals
efficiently? As an example, what kind of training makes sense for our faculty
and student mentors?

  * •

How can we learn from the vast literature on education research? We do not
have the in-house expertise to effectively evaluate what we do and to separate
out the essential from the superfluous.

##  7 Takeaways

As we said from the onset, neither one of us are experts in research on
education or in the cutting edge debates on diversity in the academy. We are
run of the mill mathematicians and statisticians who believe that we have to
play a role. In this decades long process, we ourselves have been learning and
often have adjusted our thinking, our language, and our approaches as we
gained more understanding. Our main take aways from this experience are:

  1. 1.

There is no sliver bullet,

> try things! try implementing our ideas. try implementing your ideas. ask
> your students what they think would work for them.

  2. 2.

change takes time,

> time is likely the most important factor. not only is change slow, but it
> also requires sustained commitment. that is, we spend many hours on the
> programming described above. it is difficult to outsource or scale the work.
> you just need to put in the time.

  3. 3.

it is possible and necessary to make progress,

> anything is better than nothing. whatever you do will be appreciated, and
> the effects of your programs will infuse into all aspects of your own work
> and community.

  4. 4.

everyone plays a role,

> the work isn’t only done by the faculty members. one of our big keys to
> success was our long time administrative assistant who was able to help
> create community in myriad ways. another group who led to our success were
> older or graduated students. we were able to lean on their experiences as
> examples for our more junior students.

  5. 5.

but not everyone needs to be on board to start, and

> it’s okay (and expected!) if your department has different levels of
> commitment to this work. join with the people in your department who want to
> do the work, and you may find your group expanding over time.

  6. 6.

no progress is permanent, you have to keep reproducing it.

> we have a long way to go, and we are in it for the long haul. we hope that
> you are, too.

In conclusion, there is no question that our efforts to create an inclusive
community in the Department of Mathematics & Statistics will be and should be
an ongoing pursuit. We must continue to adapt our programs and our approach to
the changes in our student body and to the challenges and uncertainties posed
by the outside world. Our students have been a source of inspiration, and we
are impelled to engage them and to learn from them. We are also happy to share
our experiences in greater detail. Feel free to contact us, especially if you
are interested in starting a program and have logistical questions.

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