# Shell-model study of $\log ft$ values for 139,140,141Ba $\rightarrow$
139,140,141La transitions

Shweta Sharma  Praveen C. Srivastava111Corresponding author:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667, India

( August 30, 2024)

###### Abstract

In the present work, beta-decay properties such as $\log ft$ values and half-
lives have been systematically studied corresponding to Ba isotopes using
large-scale shell-model calculations. An extensive comparison of beta decay
results corresponding to 141Ba$\rightarrow$ 141La using shell-model
calculations is made with the recently available experimental data. In
addition, we have also calculated the nuclear and beta decay properties
corresponding to 139Ba$\rightarrow$ 139La and 140Ba$\rightarrow$ 140La
transitions. The model-space considered here is $Z=50-82$ and $N=82-126$ with
132Sn core, and the interaction employed here is jj56pnb interaction. The beta
decay results using shell-model calculations for all the mentioned isotopes
are compared with the available experimental data. This is the first
theoretical interpretation corresponding to recent experimental data.

###### keywords:

shell-model, beta decay

##  1 Introduction

The beta decay study and its theoretical description are important for better
estimation of nuclear matrix elements [1, 2]. In addition, the beta decay
study provides a deeper insight into the nuclear structure and nuclear
interactions involved in the weak interaction processes. Thus, one needs a
theoretical model that gives an accurate and thorough description of the beta
decay theory. There are various theoretical models that have been utilized for
this purpose, and in the present work, we are using the interacting shell-
model [3, 4]. The nuclear shell-model is very beneficial in the calculation of
several important nuclear properties and matrix elements involved in beta
decay processes, which gives information about the nuclear structure and
corresponding wave functions. The study of beta decay processes is very useful
in understanding astrophysical processes. In Ref. [5], shell-model results for
neutron capture rates for $fp$ shell nuclei are reported. The nuclear
properties for nuclear astrophysical studies have recently been reported in
Ref. [6].

The shell-model calculations usually overestimate the beta-decay strength
functions [7, 8]. Thus, these strength functions are calculated using
quenching factors in the weak coupling constants [7, 8, 9, 10, 11, 12, 13, 14,
15]. Also, the mesonic enhancement factor [16] is needed in the $0^{-}$
transitions of the beta decay using shell-model calculations because of
impulse approximation. This mesonic enhancement factor is evaluated for
different regions in the nuclear chart [17, 18, 19, 20, 21]. Further, the
competition between the first forbidden and allowed transitions in the heavier
mass region from recent experimental data from CERN is reported earlier in
Refs. [22, 23]. Keeping all these factors in mind, earlier we have studied
beta-decay properties of nuclei in the 208Pb region using large-scale shell-
model calculations [24, 25, 26, 27]. Since the nuclei in the 132Sn region show
similarity in nuclear properties with the nuclei in the 208Pb region [28, 29].
Thus, we have also extended our study towards the 132Sn region and calculated
beta decay properties of 135,137Te nuclei using shell-model calculations [30].
Our group has also studied beta decay properties for sd shell nuclei [31, 32,
33] using shell-model calculations.

Recently, several experiments have been carried out to study beta decay
properties for allowed and forbidden transitions in the different mass regions
[10]. In the Sn region, experiments have been carried out to investigate the
excited levels of La isotopes. The results corresponding to beta decay of
139Ba to 139La are reported in Refs. [34, 35, 36, 37, 38, 39]. The decay
scheme of 139Ba has been studied by using NaI(Tl)-Ge(Li) detectors by Berzins
et al in Ref. [34]. Later, this decay scheme along with the decay scheme of
141Ba was studied by Faller et al [35]. They observed that there is no energy
level between 166 and 1209 keV in 139La. Later, Zamboni et al [36] proposed
two new levels at 1524.6 and 1900.3 keV excitation energies corresponding to
beta decay 139Ba. Recently, Danu et al [39] investigated the decay scheme of
139Ba using singles and $\gamma-\gamma$ coincidence spectroscopy techniques
with four Compton suppressed Clover HPGe detectors at BARC, Mumbai. In this
work, various measurements have been taken like energy spectra, $\log ft$
values and half-life. Further, beta decay of 140Ba to 140La has been
investigated by Meyer et al [40] by populating the levels of 140La using
$\gamma$-ray spectroscopy.

Greenwood et al [41] investigated the beta decay corresponding to 141Ba by
measuring beta feeding intensities. Due to the lack of errors in their beta
feeding intensities results, Rufino et al [42] performed a high-sensitivity,
high-resolution $\gamma$-ray measurements to investigate the 141Ba to 141La
beta decay. They carried out these beta decay measurements at Argonne National
Laboratory (ANL) using the Gammasphere array and measured the beta decay
strength functions which were compared with the $\log ft$ results earlier
measured by Faller et al [35]. However, no theoretical interpretation has been
performed.

Thus, motivated by this new experimental data we have performed a systematic
shell-model study for the beta decay properties of 141Ba to 141La transitions.
Along with this we have also studied 139,140Ba to 139,140La transitions.

This work is organized as follows: Section 2 is concerned with the formalism
behind allowed and forbidden beta decay and Section 3 shows the results and
discussion of the present work which includes the calculation of nuclear
structure and beta decay properties corresponding to the aforementioned
nuclei. This discussion is followed by the conclusion in section 4 at the end.

##  2 Formalism

###  2.1 Shell-model Hamiltonian

The shell-model Hamiltonian [43] in terms of single-particle energies and two-
body interaction between the nucleons can be defined as

$H=T+V=\sum_{\alpha}{\epsilon}_{\alpha}c^{\dagger}_{\alpha}c_{\alpha}+\frac{1}{4}\sum_{\alpha\beta\gamma\delta}v_{\alpha\beta\gamma\delta}c^{\dagger}_{\alpha}c^{\dagger}_{\beta}c_{\delta}c_{\gamma},$
(1)

where the term $\alpha=\\{n,l,j,t\\}$ denotes the single-particle orbitals.
The quantity $\epsilon_{\alpha}$ is the single particle energy of the state
$\alpha$. The terms $c^{\dagger}_{\alpha}$ and $c_{\alpha}$ are creation and
annihilation operators, respectively. While,
$v_{\alpha\beta\gamma\delta}=\langle\alpha\beta|V|\gamma\delta\rangle$ are the
antisymmetrized two-body matrix elements (TBMEs).

###  2.2 Beta decay formalism

The streamlined formalism for allowed and forbidden beta decay is given here,
which is based on the concept of impulse approximation. According to impulse
approximation, the decaying nucleon only feels the weak interaction at the
instant of decay and does not interact with the remaining nucleons. It feels
the strong interaction with the remaining nucleons only before and after the
decay. The detailed formalism can be found in Refs. [43, 44, 45].

Figure 1: Comparison between experimental [47] and shell-model energy spectra
of 139La with corresponding energies are in keV.

The partial half-life for a transition in beta decay is given by

$\displaystyle t_{1/2}=\frac{\kappa}{\tilde{C}},$ (2)

where $\kappa=6289$ s is the constant value [46] and $\tilde{C}$ is the
unitless integrated shape factor, given by

$\displaystyle\tilde{C}=\int_{1}^{w_{0}}C(w_{e})pw_{e}(w_{0}-w_{e})^{2}F_{0}(Z,w_{e})dw_{e},$
(3)

where $p$ and $w_{e}$ are the electron momentum and energy, respectively. The
quantity $w_{0}$ is the end-point energy, i.e., the highest amount of energy
acquired by the emitting electron. The term $C(w_{e})$ is the shape factor and
$F_{0}(Z,w_{e})$ is the Fermi function for the daughter nucleus with proton
number $Z$. The shape factor $C(w_{e})$ can be written as

$\displaystyle\begin{split}C(w_{e})=\sum_{k_{e},k_{\nu},K}\lambda_{k_{e}}\Big{[}M_{K}(k_{e},k_{\nu})^{2}+m_{K}(k_{e},k_{\nu})^{2}\\\
-\frac{2\gamma_{k_{e}}}{k_{e}w_{e}}M_{K}(k_{e},k_{\nu})m_{K}(k_{e},k_{\nu})\Big{]}.\end{split}$
(4)

Figure 2: Comparison between experimental [48] and shell-model energy spectra
of 140La with corresponding energies are in keV.

Here, the lowest feasible value of $K$, or forbiddenness order, is chosen
since the terms with the least angular momentum transfer account for the
majority of contributions. As a result, $K=K_{min},K_{min}+1$ where
$K_{min}=|J_{i}-J_{f}|$ with $J_{i}$ and $J_{f}$ represents the initial and
final angular momentum, respectively. The terms $k_{e}$ and $k_{\nu}$ stand
for the positive integers that result from the partial wave expansion of the
electron and neutrino wave functions, respectively. Given that contributions
from higher order terms in the leptonic wave function expansion are minimal,
there are two ways to get the total $k_{e}+k_{\nu}$ to be as small as
possible: $k_{e}+k_{\nu}=K+1$ and $k_{e}+k_{\nu}=K+2$. The quantities
$M_{K}(k_{e},k_{\nu})$ and $m_{K}(k_{e},k_{\nu})$ are the complicated
combinations of the different form factors incorporating nuclear structure
information and the leptonic phase space factors. Additional details on these
quantities can be found in [49, 50]. The impulse approximation can be used to
relate the nuclear form factors to the corresponding nuclear matrix elements
(NMEs), given by

Figure 3: Comparison between experimental [42] and shell-model energy spectra of 141La with corresponding energies are in keV. Table 1: The rms deviations between experimental and shell-model energy levels corresponding to 139,140,141La isotopes.  Isotope |  rms deviation (MeV)
---|---
139La |  0.164
140La |  0.0733
141La |  0.558

$\displaystyle\begin{split}{}^{V/A}\mathcal{M}_{KLS}^{(N)}(pn)(k_{e},m,n,\rho)&\\\
=\frac{\sqrt{4\pi}}{\widehat{J}_{i}}\sum_{pn}\,^{V/A}m_{KLS}^{(N)}(pn)(&k_{e},m,n,\rho)(\Psi_{f}\parallel[c_{p}^{\dagger}\tilde{c}_{n}]_{K}\parallel\Psi_{i}),\end{split}$
(5)

where $m$ is the total power of $(m_{e}R/\hbar)$, $(W_{e}R/\hbar)$, and
$\alpha Z$, $n$ is the total power of $(W_{e}R/\hbar)$ and $\alpha Z$, and
$\rho$ is the power of $\alpha Z$ and ${\widehat{J}_{i}}=\sqrt{2J_{i}+1}$ and
the summation is given over protons and neutrons single-particle states. The
model-independent term ${{}^{V/A}m_{KLS}^{(N)}}(pn)(k_{e},m,n,\rho)$ is
defined as the single particle matrix elements (SPMEs), whereas the model
dependent term $(\Psi_{f}||[c_{p}^{\dagger}\tilde{c}_{n}]_{K}||\Psi_{i})$ is
the one body transition density (OBTD) with $(\Psi_{i})$ and $(\Psi_{f})$
being the initial and final nuclear states, respectively. The OBTDs along with
other nuclear structure properties are obtained from the shell-model
diagonalization codes NuShellX [51] and KSHELL [52]. Further, the auxiliary
quantities $\gamma_{k_{e}}$ and $\lambda_{k_{e}}$ (Coulomb function) in Eqn.
(4) are given as

$\displaystyle\gamma_{k_{e}}$ $\displaystyle=\sqrt{k_{e}^{2}-(\alpha{Z})^{2}}$
$\displaystyle\mbox{and}\,\,\,\,\lambda_{k_{e}}$
$\displaystyle={F_{k_{e}-1}(Z,w_{e})}/{F_{0}(Z,w_{e})},$

where the quantity $\alpha=1/137$ is the fine structure constant.
$F_{k_{e}-1}(Z,w_{e})$ is the generalized Fermi function which is given as

$\displaystyle F_{k_{e}-1}(Z,w_{e})$
$\displaystyle=4^{k_{e}-1}(2k_{e})(k_{e}+\gamma_{k_{e}})[(2k_{e}-1)!!]^{2}e^{\pi{y}}$
(6)
$\displaystyle\times\left(\frac{2p_{e}R}{\hbar}\right)^{2(\gamma_{k_{e}}-k_{e})}\left(\frac{|\Gamma(\gamma_{k_{e}}+iy)|}{\Gamma(1+2\gamma_{k_{e}})}\right)^{2},$

where $y=(\alpha{Zw_{e}}/p_{e}c)$. In the case of allowed beta decay, the
average shape factor becomes independent of electron energy and is given by
$C(w_{e})=B(\text{F})+B(\text{GT})$ where $B(F)$ and $B(GT)$ are Fermi and
Gamow-Teller reduced transition probabilities [9]. Further, in $0^{-}$
transitions the axial-vector matrix element gets improved over the impulse
approximation with the aid of mesonic enhancement factor, i.e.,
$\epsilon_{\textrm{MEC}}$ [16]. The value of the mesonic enhancement factor is
taken to be 2.0 in our work from Ref. [21].

Table 2:  Comparison between experimental [53, 54] and shell-model calculated electromagnetic observables corresponding to 139,140,141La. In the shell-model calculations, we have taken $e_{p}=1.5e,e_{n}=0.5e$ and $g_{s}^{eff}=g_{s}^{free}$. Nucleus |  $\mu(\mu_{N})$ |  $Q(eb)$
---|---|---
| Expt. | SM | Expt. | SM
139La($7/2^{+}_{1}$) | 2.7830455(9) | 1.760 | +0.200(6) | +0.127
140La($3^{-}_{1}$) | 0.730(15) | 0.130 | +0.094(10) | +0.152
141La($7/2^{+}_{1}$) | NA | 1.726 | NA | +0.030
|  $B(M1)(\mu_{N}^{2})$ |  $B(E2)(e^{2}fm^{4})$
| Expt. | SM | Expt. | SM
139La($5/2^{+}_{1}\rightarrow 7/2^{+}_{1}$) | 0.00460(7) | 0.0003 |  $<$85.54 | 1.62
140La($2^{-}_{1}\rightarrow 3^{-}_{1}$) | 0.913${}^{+197}_{-143}$ | 0.813 |  $<$1434 | 349
141La($5/2^{+}_{1}\rightarrow 7/2^{+}_{1}$) | 0.00381${}^{+34}_{-18}$ | 0.005 | NA | 137

Figure 4: Comparison of experimental [42, 53] and calculated average shape
factor calculated via (a) bare values of weak coupling constants (b) effective
values of weak coupling constants.

The half-life for a specific transition can vary in a wide range. Thus, it is
generally given in terms of $\log ft$ values.

$\log ft\equiv\log(f_{0}t_{1/2}).$ (7)

In this expression, $t_{1/2}$ can be obtained from Eq. (2) and the phase space
factor $f_{0}$ can be expressed as

$f_{0}=\int_{1}^{w_{0}}pw_{e}(w_{0}-w_{e})^{2}F_{0}(Z,w_{e})dw_{e}.$ (8)

Further, in the case of forbidden unique transitions, the phase space factor
is given by $f_{Ku}$ which is expressed as

$\displaystyle f_{Ku}=\Big{(}\frac{3}{4}\Big{)}^{K}\frac{(2K)!!}{(2K+1)!!}$
$\displaystyle\times\int_{1}^{w_{0}}C_{Ku}(w_{e})pw_{e}(w_{0}-w_{e})^{2}F_{0}(Z_{f},w_{e})dw_{e},$
(9)

where $C_{Ku}(w_{e})$ is the shape function for $K^{\text{th}}$ forbidden
unique transition which is expressed as

$C_{Ku}(w_{e})=\sum_{k_{e}+k_{\nu}=K+2}\frac{\lambda_{k_{e}}p^{2(k_{e}-1)}(w_{0}-w_{e})^{2(k_{\nu}-1)}}{(2k_{e}-1)!(2k_{\nu}-1)!}.$
(10)

and $f_{1u}=12f_{K=1,u}$ for the first forbidden unique transitions.

Further, the experimental Gamow-Teller strength and forbidden transition
probabilities are usually smaller in value as compared to the values
calculated with the help of the shell-model because of various factors such as
truncated model-space, nuclear medium effects, and impulse approximation used
in the beta decay formalism [10]. Thus, there is a need to quench these
values. This quenching is performed via the renormalization of the vector and
axial vector coupling constant and is calculated by comparing theoretical and
experimental average shape factors. These average shape factor values are
given as

$\overline{C(w_{e})}(\text{fm}^{2K})=\frac{6289{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{\text{Ce}}^{2K}}{ft},$ (11)

where ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{\text{Ce}}=386.159$ fm stands for the reduced
Compton wavelength of the electrons and $ft$ is obtained from the Eq. (7).

Table 3:  Comparison between theoretical and experimental [47] $\log ft$ values for 139Ba$(7/2^{-})$ to the different excited states in 139La($J_{f}$) transitions using effective value of axial-vector coupling constant i.e. $(g_{A}^{eff}=0.66\pm 0.03)$ and $\epsilon_{MEC}=2.0$. Final state | Decay mode | Energy (keV) |  $\log ft$ |  ${[\overline{C(w_{e})}]}^{1/2}$
---|---|---|---|---
$J_{f}^{\pi}$ (Expt.) |  $J_{f}^{\pi}$ (SM) |  |  | Expt. | SM | Expt. | SM
$7/2^{+}$ | $7/2^{+}_{1}$ | 1st FNU | 0.0 | 6.845(3) | 6.774 | 11.576(40) | 12.562
$5/2^{+}$ | $5/2^{+}_{1}$ | 1st FNU | 165.859(7) | 7.087(5) | 6.254 | 8.761(50) | 22.859
$9/2^{+}$ | $9/2^{+}_{1}$ | 1st FNU | 1219.047(10) | 9.68(1) | 9.332 | 0.443(5) | 0.661
$(5/2)^{+}$ | $5/2^{+}_{2}$ | 1st FNU | 1256.797(11) | 10.24(1) | 8.557 | 0.232(3) | 1.613
$(9/2^{+})$ | $9/2^{+}_{2}$ | 1st FNU | 1381.409(14) | 9.65(1) | 8.689 | 0.458(5) | 1.385
$(5/2^{+},7/2^{+})$ | $5/2^{+}_{3}$ | 1st FNU | 1420.491(8) | 7.64(1) | 8.708 | 4.635(53) | 1.355
$(5/2^{+},7/2^{+})$ | $7/2^{+}_{2}$ | 1st FNU | 1420.491(8) | 7.64(1) | 7.805 | 4.635(53) | 3.833
$(9/2^{+})$ | $9/2^{+}_{3}$ | 1st FNU | 1476.489(8) | 8.78(2) | 10.763 | 1.248(29) | 0.127
$7/2^{+}$ | $7/2^{+}_{3}$ | 1st FNU | 1536.388(8) | 9.19(2) | 9.208 | 0.778(18) | 0.762
$(3/2^{+},5/2^{+})$ | $3/2^{+}_{1}$ | 1st FU | 1558.72(3) | 10.21(2) | 9.371 | 0.240(6) | 0.632
$(3/2^{+},5/2^{+})$ | $5/2^{+}_{4}$ | 1st FNU | 1558.72(3) | 10.21(2) | 9.292 | 0.240(6) | 0.692
$(5/2^{+},7/2^{+})$ | $5/2^{+}_{5}$ | 1st FNU | 1578.156(14) | 10.00(1) | 9.317 | 0.306(4) | 0.672
$(5/2^{+},7/2^{+})$ | $7/2^{+}_{4}$ | 1st FNU | 1578.156(14) | 10.00(1) | 8.322 | 0.306(4) | 2.114
$7/2^{+}$ | $7/2^{+}_{5}$ | 1st FNU | 1683.145(10) | 9.08(1) | 8.431 | 0.883(10) | 1.864
$(5/2)^{+}$ | $5/2^{+}_{7}$ | 1st FNU | 1766.432(25) | 9.59(2) | 9.660 | 0.491(11) | 0.453
$(3/2^{+},5/2^{+})$ | $3/2^{+}_{2}$ | 1st FU | 1856.62(4) | 10.1(3) | 10.485 | 0.273(94) | 0.175
$(3/2^{+},5/2^{+})$ | $5/2^{+}_{8}$ | 1st FNU | 1856.62(4) | 10.1(3) | 10.801 | 0.273(94) | 0.122
$(7/2^{+})$ | $7/2^{+}_{6}$ | 1st FNU | 1920.43(4) | 9.43(2) | 8.894 | 0.590(14) | 1.094
$(5/2^{+})$ | $5/2^{+}_{9}$ | 1st FNU | 1962.84(11) | 9.88(6) | 10.187 | 0.352(24) | 0.247
Table 4:  Comparison between theoretical and experimental [48] $\log ft$ values for 140Ba$(0^{+})$ to the different excited states in 140La($J_{f}$) transitions using effective value of axial-vector coupling constant i.e. $(g_{A}^{eff}=0.66\pm 0.03)$ and $\epsilon_{MEC}=2.0$. Here, ’$u$’ denotes forbidden unique decay. Final state | Decay mode | Energy (keV) |  $\log ft$ |  ${[\overline{C(w_{e})}]}^{1/2}$
---|---|---|---|---
$J_{f}^{\pi}$ (Expt.) |  $J_{f}^{\pi}$ (SM) |  |  | Expt. | SM | Expt. | SM
$2^{-}$ | $2^{-}_{1}$ | 1st FU | 29.9677(8) | 8.911u(24) | 9.2721u | 1.074(297) | 0.708
$1^{-}$ | $1^{-}_{1}$ | 1st FNU | 43.8132(18) | 8.03(11) | 6.634 | 2.958(375) | 14.759
$2^{-}$ | $2^{-}_{2}$ | 1st FU | 162.6585(8) | 9.301u(3) | 9.1661u | 0.686(24) | 0.800
$1^{-}$ | $1^{-}_{2}$ | 1st FNU | 467.5350(17) | 7.805(23) | 6.769 | 3.833(102) | 12.634
$0^{-}$ | $0^{-}_{1}$ | 1st FNU | 581.073(9) | 7.07(3) | 7.775 | 8.934(309) | 3.968
Table 5:  Comparison between theoretical and experimental [42] $\log ft$ values for 141Ba$(3/2^{-})$ to the different excited states in 141La($J_{f}$) transitions using effective value of axial-vector coupling constant i.e. $(g_{A}^{eff}=0.66\pm 0.03)$ and $\epsilon_{MEC}=2.0$. Here, ’$u$’ denotes forbidden unique decay. Final state | Decay mode | Energy (keV) |  $\log ft$ |  ${[\overline{C(w_{e})}]}^{1/2}$
---|---|---|---|---
$J_{f}^{\pi}$ (Expt.) |  $J_{f}^{\pi}$ (SM) |  |  | Expt. | SM | Expt. | SM
$7/2^{(+)}$ | $7/2^{+}_{1}$ | 1st FU | 0.0 | 9.41u(5) | 12.9741u | 0.611(352) | 0.010
$5/2^{(+)}$ | $5/2^{+}_{1}$ | 1st FNU | 190.40(20) | 7.63(8) | 6.608 | 4.689(432) | 15.207
$3/2^{(+)}$ | $3/2^{+}_{1}$ | 1st FNU | 467.37(8) | 7.05(3) | 6.035 | 9.142(316) | 29.414
$1/2^{(+)}$ | $1/2^{+}_{1}$ | 1st FNU | 580.17(10) | 8.45(5) | 9.033 | 1.824(105) | 0.932
$3/2^{(+)}$ | $3/2^{+}_{2}$ | 1st FNU | 647.94(8) | 6.805(24) | 6.589 | 12.122(335) | 15.544
$3/2^{(+)}$ | $3/2^{+}_{3}$ | 1st FNU | 685.40(10) | 8.66(3) | 7.507 | 1.432(49) | 5.402
$5/2^{(+)}$ | $5/2^{+}_{3}$ | 1st FNU | 826.42(10) | 8.17(3) | 8.121 | 2.518(87) | 2.664
$3/2^{(+)}$ | $3/2^{+}_{4}$ | 1st FNU | 831.66(8) | 7.720(24) | 7.969 | 4.227(117) | 3.174
$5/2^{(+)}$ | $5/2^{+}_{4}$ | 1st FNU | 929.44(9) | 6.896(25) | 6.699 | 10.916(314) | 13.695
$3/2^{(-)}$ | $3/2^{-}_{1}$ | Allowed | 991.97(10) | 9.7(3) | 6.277 | 0.0011(4) | 0.058
$5/2^{(+)}$ | $5/2^{+}_{5}$ | 1st FNU | 1039.48(9) | 8.94(6) | 8.091 | 1.038(72) | 2.758
$3/2^{(-)}$ | $3/2^{-}_{2}$ | Allowed | 1066.57(10) | 7.317(25) | 6.373 | 0.017(1) | 0.052
$1/2^{(+)}$ | $1/2^{+}_{2}$ | 1st FNU | 1171.99(9) | 7.678(24) | 8.540 | 4.437(123) | 1.645
$3/2^{(-)}$ | $3/2^{-}_{3}$ | Allowed | 1426.36(10) | 7.910(25) | 8.073 | 0.0088(2) | 0.007
$5/2^{(+)}$ | $5/2^{+}_{6}$ | 1st FNU | 1501.56(9) | 6.701(23) | 8.091 | 13.663(362) | 2.758
$1/2^{(+)}$ | $1/2^{+}_{3}$ | 1st FNU | 1547.69(17) | 8.22(3) | 8.726 | 2.377(82) | 1.328
$3/2^{(-)}$ | $3/2^{-}_{4}$ | Allowed | 1628.16(10) | 6.899(24) | 9.745 | 0.028(1) | 0.001
$5/2^{(+)}$ | $5/2^{+}_{7}$ | 1st FNU | 1740.69(10) | 6.949(24) | 9.602 | 10.270(284) | 0.484
$3/2^{(-)}$ | $3/2^{-}_{5}$ | Allowed | 1844.30(11) | 6.749(24) | 8.606 | 0.035(1) | 0.004
$1/2^{(+)}$ | $1/2^{+}_{4}$ | 1st FNU | 1872.60(9) | 6.518(24) | 9.845 | 16.868(466) | 0.366
Table 5:  Continued. Final state | Decay mode | Energy (keV) |  $\log ft$ |  ${[\overline{C(w_{e})}]}^{1/2}$
---|---|---|---|---
$J_{f}^{\pi}$ (Expt.) |  $J_{f}^{\pi}$ (SM) |  |  | Expt. | SM | Expt. | SM
$3/2^{(-)}$ | $3/2^{-}_{6}$ | Allowed | 1926.01(10) | 6.724(25) | 7.000 | 0.034(1) | 0.025
$3/2^{(-)}$ | $3/2^{-}_{7}$ | Allowed | 2180.38(12) | 6.964(25) | 10.241 | 0.026(1) | 0.001
$1/2^{(+)}$ | $1/2^{+}_{5}$ | 1st FNU | 2216.56(11) | 6.77(3) | 7.610 | 12.620(436) | 4.798
$3/2^{(-)}$ | $3/2^{-}_{8}$ | Allowed | 2327.22(14) | 7.06(3) | 8.927 | 0.023(1) | 0.003
$3/2^{(-)}$ | $3/2^{-}_{9}$ | Allowed | 2375.85(12) | 6.46(3) | 8.570 | 0.047(2) | 0.004
$3/2^{(-)}$ | $3/2^{-}_{10}$ | Allowed | 2385.68(11) | 6.87(3) | 7.379 | 0.029(1) | 1.621
$5/2^{(+)}$ | $5/2^{+}_{8}$ | 1st FNU | 2468.74(9) | 6.24(3) | 8.677 | 23.230(802) | 1.405

##  3 Results and discussion

In this section, the nuclear structure and beta decay properties corresponding
to 139,140,141Ba $\rightarrow$ 139,140,141La transitions are reported using
large-scale shell-model calculations. For these shell-model calculations, a
particular model-space is chosen above 132Sn core. The model-space considered
here includes five orbitals for protons, i.e., $0g_{7/2}$, $1d_{5/2}$,
$1d_{3/2}$, $2s_{1/2}$, $0h_{11/2}$ ($50\leq Z\leq 82$) and six orbitals for
neutrons, i.e., $0h_{9/2}$, $1f_{7/2}$, $1f_{5/2}$, $2p_{3/2}$, $2p_{1/2}$,
$0i_{13/2}$ ($82\leq N\leq 126$). The effective interaction used for the above
model space is the jj56pnb interaction [21, 30, 55], which includes the
Coulomb interaction part in the proton-proton interaction part. Now using this
effective interaction, first we have calculated the nuclear structure
properties such as energy spectra and electromagnetic observables. Further,
using the obtained wave functions we have performed calculations for the beta
decay properties such as $\log ft$, average shape factor values, and half-
lives. The average shape factors have been calculated using both bare and
effective values of weak coupling constants. The comparison of experimental
and shell-model average shape factor values is shown in this work for both
bare and effective values of weak coupling constants. The effective value of
weak axial-vector coupling constant ($g_{A}^{eff}=0.66\pm 0.03$) is taken from
Ref. [21] to calculate $\log ft$ values and half-lives in the present work.
These beta decay properties, such as the $\log ft$, average shape factor
values, and half-lives are reported in this work and compared with the
corresponding experimental data. The deviation in the shell-model results with
respect to the experimental data is also reported in this section.

Figs. 1, 2 and 3 shows the energy spectra for ground and excited states of
139,140,141La isotopes, respectively using jj56pnb interaction and comparison
with the experimental spectra for the same is also shown in these figures. In
Fig. 1, shell-model and experimental energy spectra for ground and excited
states of 139La are shown. It can be seen from the figure that the jj56pnb
interaction correctly reproduces the ground state (g.s.), i.e., $7/2_{1}^{+}$,
and the first excited state as $5/2_{1}^{+}$. Further, there is a large gap
between 72 and 1010 keV energy levels similar to the gap between 166 and 1219
keV experimentally. In Fig. 2, the experimental and shell-model energy spectra
corresponding to the ground and excited states of 140La is shown. In this
spectra, the ground state, i.e., $3_{1}^{-}$ and other excited states are
correctly reproduced by jj56pnb interaction. For instance, the first excited
state is $2_{1}^{-}$ experimentally, and the same is reproduced by our shell-
model calculation. Similarly, Fig. 3 shows the experimental and shell-model
energy spectra corresponding to the ground and excited states of 141La
isotope. It also correctly reproduced the ground state and reasonably other
excited states. Furthermore, the root mean square (rms) deviations between the
experimental and the shell-model energy levels corresponding to 139,140,141La
isotopes are computed in order to quantify the predictive power of jj56pnb
interaction in the shell-model calculations. The root mean square deviation
can be defined as

${\rm rmsd}=\Big{[}\frac{1}{n}\sum_{i=1}^{n}(E_{i}({\rm expt})-E_{i}({\rm
SM}))^{2}\Big{]}^{1/2},$ (12)

where $E_{i}({\rm expt})$ is the experimental energy and $E_{i}({\rm SM})$ is
the shell-model energy of the corresponding state. Here, $n$ is the total
number of states considered for the calculation of rms deviation. These rms
deviations are computed for non-degenerate states only, and they are reported
in Table 1. While calculating the rms deviation for 139La, the states up to
1476 keV experimental energy are taken into account and the rms deviation
comes out to be 0.164 MeV. Similarly, in the case of 140La, the $2^{-}$ and
$3^{-}$ states at experimental energies 592 and 658 keV are not considered for
rms deviation calculation because these states are above unconfirmed states
$(2^{-},3^{-})$ at energy 576 keV. The rms deviation for 140La is 0.0733 MeV,
and it is maximum for 141La, i.e, 0.558 MeV.

In addition to the energy spectra, the electromagnetic observables such as
magnetic moment, quadrupole moment, B(E2) and B(M1) are also calculated in
this work. The calculated electromagnetic observable values are also compared
with the experimental data wherever it is available. In the case of 139La, the
magnetic moment value calculated from shell-model for $7/2_{1}^{+}$ state is
1.760 $\mu_{N}$ whereas it is 2.7830455(9) $\mu_{N}$ experimentally. Likewise,
in the case of 140La for $3^{-}_{1}$ state, this value comes out to be 0.130
$\mu_{N}$ from shell-model and 0.730(15) $\mu_{N}$ experimentally. In the case
of 141La for $7/2_{1}^{+}$, the experimental value of the magnetic moment is
not yet observed although the calculated value from the shell-model is 1.726
$\mu_{N}$. The quadrupole moment is also calculated in this work and compared
with the experimental data. We are getting good shell-model results for
quadrupole moment for all of the isotopes, and both shell-model and
experimental values are positive which means these nuclei show prolate shape.
For instance, the quadrupole moment value for 139La($7/2_{1}^{+}$) is +0.127
$eb$ theoretically, and it is +0.200(6) $eb$ experimentally. Similarly, in the
case of 140La isotope for $3^{-}_{1}$, it is +0.152 $eb$ theoretically whereas
it is +0.094(10) $eb$ experimentally. Moving forward, the experimental value
of quadrupole moment for 141La($7/2_{1}^{+}$) isotope is not available, but
theoretically, this value comes out to be +0.030 $eb$. In addition to these
quantities, B(E2) and B(M1) are also evaluated. The shell-model results for
B(M1) agree pretty well with the experimental data. For instance, B(M1) value
for 139La($5/2^{+}_{1}\rightarrow 7/2^{+}_{1}$) is 0.0003 $\mu_{N}^{2}$
theoretically whereas it is 0.00460(7) $\mu_{N}^{2}$ experimentally.
Similarly, B(M1) value for 140La($2^{-}_{1}\rightarrow 3^{-}_{1}$) is 0.813
$\mu_{N}^{2}$ theoretically which is close to the experimental value, i.e.,
0.913${}^{+197}_{-143}$ $\mu_{N}^{2}$. In the case of
141La($5/2^{+}_{1}\rightarrow 7/2^{+}_{1}$), this value comes out to be 0.005
$\mu_{N}^{2}$ theoretically, and experimentally it is 0.00381${}^{+34}_{-18}$
$\mu_{N}^{2}$. Further, B(E2) values are also evaluated using shell-model for
all the mentioned isotopes. Corresponding results are shown in the table, the
calculated result for 140La($2^{-}_{1}\rightarrow 3^{-}_{1}$) transition is
larger in comparison to the available experimental data.

After testing the wave functions through nuclear structure properties
calculations, then we have calculated beta decay properties corresponding to
139,140,141Ba transitions. At first, the shell-model $\log ft$ and average
shape factor values are calculated using the bare values of weak coupling
constants. Similarly, the experimental average shape factor values are also
extracted from the experimental $\log ft$ values, and then both shell-model
and experimental results for average shape factor values are compared. This
comparison is shown in Fig. 4(a). In this figure, the data points are
scattered. Following this, the beta decay properties such as $\log ft$ and
average shape factor values are computed using the effective value of axial-
vector coupling constant $g_{A}^{eff}=0.66\pm 0.03$. The comparison of these
average shape factor values with the experimental average shape factor values
is shown in Fig. 4(b) where the data points are less scattered.

Tables 3, 4, 5 show the shell-model results for beta decay properties such as
the $\log ft$ values, average shape factor values reported using the effective
value of axial-vector coupling constant corresponding to 139,140,141Ba
isotopes, respectively. The experimental data are also given in the tables.
The decay mode and spin parity of the final state along with the experimental
excitation energy are also listed in these tables. In the tables, FU denotes
the forbidden unique beta decay, and FNU denotes the forbidden non-unique beta
decay.

In Table 3, the shell-model $\log ft$ and the average shape factor values
corresponding to beta decay of 139Ba using effective values of axial-vector
coupling constant are reported and compared with the experimental ones. Since
the accurate computation of $\log ft$ values requires a precise Q-value, the
experimental Q-value is taken, i.e.,
2314.6(23) keV. We are getting a good agreement between the shell-model
results and experimental data for these transitions. For instance, at ground
state energy, the shell-model $\log ft$ value is 6.774, which is close to the
experimental value, i.e., 6.845(3). Also, at energy level 1219.047(10) keV,
the shell-model $\log ft$ value is 9.332, which is close to the experimental
$\log ft$ value, i.e. 9.68(1). There are some energy levels where tentative
spin parities are given experimentally. The shell-model $\log ft$ values for
these states are also calculated and compared with the experimental $\log ft$
values. Based on the calculated $\log ft$ values from shell-model it might be
possible to assign one particular spin parity state. For instance, at energy
level 1420.491(8) keV, there are two tentative spin parities
($5/2^{+},7/2^{+}$) given experimentally and the experimental $\log ft$ value
is 7.64(1). The shell-model $\log ft$ values for both of the spin parity
states are computed which comes out to be 8.708 for $5/2_{3}^{+}$ and 7.805
for $7/2_{2}^{+}$. The shell-model $\log ft$ value for $7/2_{2}^{+}$ state is
closer to the experimental $\log ft$ value as compared to the other one. Thus,
the $7/2_{2}^{+}$ state can be proposed at 1420.491(8) keV energy level.
Further, at energy level 1578.156(14) keV, the $5/2_{5}^{+}$ state can be
proposed since its shell-model $\log ft$ value is closer to the experimental
value as compared to the other. Further, in other cases, the shell-model $\log
ft$ values for tentative spin parity states as shown in the table, differ
slightly from each other corresponding to two tentative experimental states.
Thus, it is difficult to proposed one particular spin parity state at these
levels from shell model. For instance, at energy level 1558.72(3) keV, there
are two tentative spin parities ($3/2^{+},5/2^{+}$) given experimentally and
the experimental $\log ft$ value is 10.21(2). The shell-model $\log ft$ values
for $3/2_{1}^{+}$ state come out to be 9.371, and it is 9.292 for
$5/2_{4}^{+}$. Here, both values are very close to each other. Thus, we are
unable to propose one particular spin parity state at this energy level from
the shell-model. A similar situation occurs for other transitions also.

In Table 4, the experimental and shell-model $\log ft$ and average shape
factor values using effective values of axial-vector coupling constant
corresponding to the beta decay of 140Ba using experimental Q-value, i.e.,
1047(8) keV are given. The shell-model $\log ft$ values match quite well with
the experimental data. For instance, at energy level 29.9677(8) keV, the
shell-model $\log ft$ value is 9.2721u, which is close to the experimental
$\log ft$ value, i.e., 8.911u(24). Similarly, at energy level 162.6585(8) keV,
the shell-model $\log ft$ value is 9.1661u, and the experimental $\log ft$
value is 9.301u(3). In conclusion the shell model prediction for $\log ft$
values corresponding to 140Ba $\rightarrow$ 140La transitions are fairly well.

Table 6: Comparison between theoretical and experimental half-lives [42, 47, 48] for 139,140,141Ba isotopes using effective value of axial-vector coupling constant i.e. $(g_{A}^{eff}=0.66\pm 0.03)$ and $\epsilon_{MEC}=2.0$. Isotope |  Half-life
---|---
| Expt. | SM
139Ba | 82.93(9) min | 28.1 min
140Ba | 12.751(4) d | 1.10 d
141Ba | 18.27(7) min | 4.16 min
Figure 5: Comparison of experimental [42, 47, 48] and shell-model half-life
calculated using effective values of weak coupling constants.

Figure 6: Shell-model results for strength functions calculated via effective
values of weak coupling constants. Table 7: Error in shell-model half-life
values in comparison to experimental half-lives [42, 47, 48].

$\bar{r}$ -0.726 $\sigma$ 0.250 $10^{\bar{r}}$ 0.188 $10^{\sigma}$ 1.776

In Table 5, the shell-model results for $\log ft$ and average shape factor
values have been listed corresponding to the beta decay of 141Ba along with
the experimental data from ANL [42]. The experimental Q-value is used here,
i.e., 3197(7) keV. The shell-model results for the $\log ft$ values are in
reasonable agreement with the experimental data. For instance, the shell-model
$\log ft$ value at energy level 826.42(10) is 8.121, whereas the corresponding
experimental value is 8.17(3). Similarly, at energy level 831.66(8) keV, the
shell-model $\log ft$ value is 7.969, which is close to the experimental $\log
ft$ value, i.e., 7.720(24). There are some additional energy levels above
1171.99(9) keV experimental excitation energy for which the spin parity is not
mentioned experimentally, but these states are involved in the beta decay
process and thus contribute to the half-life of the nucleus. Since we can not
calculate $\log ft$ values of states without knowing their spin parity with
shell-model, thus, these states are excluded from the table. Because of this
fact the shell-model results above 1628.16(10) keV energy levels do not agree
well with the experimental data. Hence, additional experimental data is
needed.

Table 6 shows the calculated half-life values for the beta decay of
139,140,141Ba using shell-model and corresponding experimental data. The same
has been plotted in Fig. 5 in seconds against neutron number on the x-axis,
and one can interpret from this graph that our shell-model results for half-
life match fairly well with the corresponding experimental half-life values.
For instance, the shell-model half-life value for 141Ba decay is 4.16 min
which is not very far from the corresponding experimental half-life value,
i.e., 18.27(7) min.

Fig. 6 shows strength function plots [56] of 139,140,141Ba with respect to the
excitation energy of daughter nuclei. These plots also display the
corresponding Q-value. Moreover, the contribution of allowed and forbidden
transitions can be studied through these plots. In 141Ba decay, there are some
GT transitions involved, i.e., 141Ba($3/2^{-}$) $\rightarrow$
141La($3/2^{-}$). These GT transitions include the transition of the neutron
from $0h_{9/2}$ orbital to the $0h_{11/2}$ proton orbital and these
transitions are typically seen towards higher excitation energy.

We have also calculated the total deviation [57, 58, 59] in the shell-model
half-life values from the experimental half-lives. Suppose $r$ is the measure
of deviation, which can be calculated from the theoretical and experimental
total half-lives by the expression

$r=\log_{10}(T_{1/2}^{\rm{SM}}/T_{1/2}^{\rm{expt}}),$ (13)

where $T_{1/2}^{\rm{SM}}$ is the total half-life value calculated from shell-
model and the experimental half-life value is denoted by
$T_{1/2}^{\rm{expt}}$. The total half-life can be obtained from partial half-
lives by using the expression,

$\frac{1}{T_{1/2}}=\sum_{k}\frac{1}{t_{1/2}^{(k)}},$ (14)

where summation is given over the total number of transitions ($k$) involved
in the beta-decay. The deviation is calculated in the logarithmic scale, and
$\bar{r}$ and $\sigma$ are termed as the mean value in the deviation and its
standard deviation, respectively which are given in the form

$\bar{r}=\frac{1}{n}\sum_{i=1}^{n}r_{i},$ (15)

$\sigma=\Big{[}\frac{1}{n}\sum_{i=1}^{n}(r_{i}-\bar{r})^{2}\Big{]}^{1/2},$
(16)

where $n$ is the total number of beta-decaying isotopes considered in this
work for which the deviation is being calculated. The mean deviation and
standard deviation should be close to zero as these are calculated in the
logarithmic scale. These quantities are also given in the power of 10 in the
table. From Table 7, it can be concluded that our shell-model results are
pretty well according to the experimental data as $\sigma=0.250$ which means
these values approach zero.

##  4 Conclusion

Motivated by recent experimental data for beta decay properties of 141La from
Argonne National Laboratory (ANL), in this work, the nuclear structure and the
beta decay properties are calculated using shell-model and compared with the
corresponding experimental data. The nuclear properties such as energy spectra
and electromagnetic observables are computed using jj56pnb interaction. The
energy spectra of ground and excited states of 139,140,141La are also shown.
The shell-model calculations correctly reproduce the spin parity of the ground
state of all the mentioned isotopes and also reasonably describe the overall
energy spectra. In addition, the electromagnetic observables such as magnetic
moment, quadrupole moment, B(M1) and B(E2) are also calculated After
calculating nuclear structure properties, the beta decay properties are
calculated using these wave functions. At first, the $\log ft$ and average
shape factor values are computed using bare values of coupling constants.
Further, the effective values of axial-vector coupling constant is used to
calculate the $\log ft$ values and average shape factor values which are
listed and compared with the experimental data. Now, the total half-life
values are calculated using these $\log ft$ values. In order to establish a
better comparison, the error in the calculated half-life values is computed
with respect to the experimental half-life values. Overall our results are
fairly close to the experimental data.

## Acknowledgement

This work is supported by a research grant from SERB (India), CRG/2022/005167.
S. S. would like to thank CSIR-HRDG (India) for the financial support for her
Ph.D. thesis work. We acknowledge the National Supercomputing Mission (NSM)
for providing computing resources of ‘PARAM Ganga’ at the Indian Institute of
Technology Roorkee.

## References

  * [1] J. Suhonen, and O. Civitarese, “Weak-interaction and nuclear-structure aspects of nuclear double beta decay”, Phys. Rep. 300, 123 (1998).
  * [2] J. Suhonen, “Nuclear matrix elements of $\beta\beta$ decay from $\beta$-decay data”, Phys. Lett. B 607, 87 (2005).
  * [3] D. J. Dean, T. Engeland, M. Hjorth-Jensen, M. P. Kartamyshev, and E. Osnes, “Effective interactions and the nuclear shell-model”, Prog. Part. Nucl. Phys. 53, 419 (2004).
  * [4] E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, “The shell model as a unified view of nuclear structure”, Rev. Mod. Phys. 77, 427 (2005).
  * [5] K. Sieja, S. Goriely, “Shell-model based study of the direct capture in neutron-rich nuclei”, Eur. Phys. J. A 57, 110 (2021).
  * [6] S. Goriely, “Nuclear properties for nuclear astrophysics studies”, Eur. Phys. J. A 59, 16 (2023).
  * [7] Q. Zhi et al., “Shell-model half-lives including first-forbidden contributions for $r$-process waiting-point nuclei”, Phys. Rev. C 87, 025803 (2013).
  * [8] G. Martínez-Pinedo, A. Poves, E. Caurier, and A. P. Zuker, “Effective $g_{A}$ in the $pf$ shell”, Phys. Rev. C 53, R2602(R) (1996).
  * [9] B. A. Brown, and B. H. Wildenthal, “Experimental and theoretical Gamow-Teller beta-decay observables for the sd-shell nuclei”, At. Data Nucl. Data Tables 33, 347 (1985).
  * [10] J. Suhonen, “Value of the axial-vector coupling strength in $\beta$ and $\beta\beta$ decays: A review”, Frontiers in Physics 5, 55 (2017).
  * [11] J. Kostensalo, M. Haaranen, and J. Suhonen, “Electron spectra in forbidden $\beta$ decays and the quenching of the weak axial-vector coupling constant $\text{g}_{A}$”, Phys. Rev. C 95, 044313 (2017).
  * [12] J. Kostensalo, and J. Suhonen, “$\text{g}_{A}$-driven shapes of electron spectra of forbidden $\beta$ decays in the nuclear shell-model”, Phys. Rev. C 96, 024317 (2017).
  * [13] V. Kumar, P. C. Srivastava, and H. Li, “ Nuclear $\beta^{-}$-decay half-lives for ${fp}$ and ${fpg}$ shell nuclei”, J. Phys. G: Nucl. Part. Phys. 43 105104 (2016).
  * [14] V. Kumar, and P. C. Srivastava, “ Shell model description of Gamow-Teller strengths in $pf$-shell nuclei”, Eur. Phys. J. A 52 181 (2016).
  * [15] V. Kumar, P. C. Srivastava, and H. Li, “ Shell-model study of $\beta^{+}/$EC-decay half-lives for $Z=21-30$ nuclei”, Eur. Phys. J. A 59 237 (2023).
  * [16] I. S. Towner, “Enhancement in axial-charge matrix elements from meson-exchange currents”, Nucl. Phys. A 542, 631 (1992).
  * [17] E. K. Warburton, “First-forbidden decay in the lead region and mesonic enhancement weak axial current”, Phys. Rev. C 44, 233 (1991).
  * [18] E. K. Warburton, “Mesonic enhancement of the weak axial-vector current evaluated from $\beta$ decay in the lead region”, Phys. Rev. Lett. 66, 1823 (1991).
  * [19] E. K. Warburton, J. A. Becker, B. A. Brown, and D. J. Millener, “First-forbidden beta decay near A = 40”, Annals of Phys. 187, 471 (1988).
  * [20] E. K. Warburton, I. S. Towner, and B. A. Brown, “First-forbidden $\beta$ decay: Meson-exchange enhancement of the axial charge at $A\sim 16$”, Phys. Rev. C 49, 824 (1994).
  * [21] J. Kostensalo, and J. Suhonen, ”Mesonic enhancement of the weak axial charge and its effect on the half-lives and spectral shapes of first- forbidden J+ $\leftrightarrow$ J- decays”, Phys. Lett. B 781, 480 (2018).
  * [22] R. J. Carroll et al., “Competition between allowed and first-Forbidden $\beta$ decay: The case of ${}^{208}\mathrm{Hg}\rightarrow^{208}\mathrm{Tl}$ ”, Phys. Rev. Lett. 125, 192501 (2020).
  * [23] M. Brunet et al., “Competition between allowed and first-forbidden $\beta$ decays of 208At and expansion of the 208Po level scheme”, Phys. Rev. C 103, 054327 (2021).
  * [24] S. Sharma, P. C. Srivastava, A. Kumar, and T. Suzuki, ”Shell-model description for the properties of the forbidden $\beta^{-}$ decay in the region ’northeast’ of 208Pb”, Phys. Rev. C 106, 024333 (2022).
  * [25] P. C. Srivastava, S. Sharma, A. Kumar, and P. Choudhary, “Shell model study of first-forbidden beta decay around 208Pb”, J. Phys.: Conf. Ser. 2586, 012026 (2023).
  * [26] A. Kumar, and P. C. Srivastava, “Shell-model description for the first-forbidden $\beta^{-}$ decay of 207Hg into the one-proton-hole nucleus 207Tl”, Nucl. Phys. A 1014, 122255 (2021).
  * [27] S. Sharma, P. C. Srivastava, A. Kumar, T. Suzuki, C. Yuan, and N. Shimizu, “Shell-model study for allowed and forbidden $\beta^{-}$ decay properties in the ”south” region of 208Pb”, arXiv:2309.07903 [nucl-th] (2023).
  * [28] L. Coraggio, A. Covello, A. Gargano, and N. Itaco, “Similarity of nuclear structure in the 132Sn and 208Pb regions: Proton-neutron multiplets”, Phys. Rev. C 80, 021305(R) (2009).
  * [29] B. Bhoy, and P. C. Srivastava, “Systematic shell model study for $N=82$ and $N=126$ isotones and nuclear isomers”, J. Phys. G: Nucl. Part. Phys. 49, 085101 (2020).
  * [30] S. Sharma, P. C. Srivastava, and A. Kumar, “Forbidden beta decay properties of 135,137Te using shell-model”, Nucl. Phys. A 1031, 122596 (2023)
  * [31] A. Kumar, P. C. Srivastava, J. Kostensalo, and J. Suhonen, “Second-forbidden nonunique $\beta^{-}$ decays of 24Na and 36Cl assessed by the nuclear shell-model”, Phys. Rev. C 101, 064304 (2020).
  * [32] A. Kumar, P. C. Srivastava, and T. Suzuki, “Shell model results for nuclear $\beta^{-}$-decay properties of sd-shell nuclei”, Prog. Theor. Exp. Phys. 3, 033D01 (2020).
  * [33] P. Choudhary, A. Kumar, P. C. Srivastava, and T. Suzuki, “Structure of 46,47Ca from the $\beta^{-}$ decay of 46,47K in the framework of the nuclear shell-model”, Phys. Rev. C. 103, 064325 (2021).
  * [34] G. Berzins, M. E. Bunker, and J. W. Starner, “The decay scheme of 139Ba”, Nucl. Phys. A 128, 294 (1969)
  * [35] S. H. Falleret al., “Decay of 83-min 139Ba to levels of 139La and the decay of 18-min 141Ba to levels of 141La”, Phys. Rev. C. 34, 2 (1986).
  * [36] C. B. Zamboni, J. A. G. Medeiros et al., “Energy levels in 139La from the decay of 139Ba”, Appl. Radiat. Isot. 55, 4 (2001).
  * [37] J. C. Hill, and M. L. Wiedenbeck, “Levels in the $N=82$ nucleus 139La populated in 139Ba decay”, Nucl. Phys. A 119, 53 (1968).
  * [38] R. J. Gehrke, “Gamma-ray emission probabilities and half-life of 139Ba”, Appl. Radiat. Isot. 31, 37 (1980).
  * [39] L. S. Danu et al., “Revisiting the decay scheme and half-life of 139Ba”, Eur. Phys. J. A 48, 186 (2012).
  * [40] R. A. Meyer, K. V. Marsh, H. Seyfarth, S. Brant, M. Bogdanović, and V. Paar, “Interacting-boson-fermion-fermion model description of 140La83 and comparison with levels populated by beta decay and neutron capture”, Phys. Rev. C. 41, 1172 (1990).
  * [41] R. C. Greenwood, R. G. Helmer, M. H. Putnam, and K. D. Watts, “Measurement of $\beta^{-}$-decay intensity distributions of several fission-product isotopes using a total absorption $\gamma$-ray spectrometer”, Nucl. Instrum. Methods Phys. Res. Sect. A 390, 95 (1997).
  * [42] J. Rufino et al., “$\beta$ decay of 141Ba”, Phys. Rev. C. 106, 034318 (2022).
  * [43] J. Suhonen, From Nucleons to Nucleus: Concept of Microscopic Nuclear Theory, (Springer, Berlin 2007).
  * [44] H. Behrens, and W. B$\ddot{\text{u}}$hring, Electron Radial Wave Functions and Nuclear Beta-Decay (Clarendon, Oxford, 1982).
  * [45] H. F. Schopper, Weak interaction and Nuclear Beta Deacy (North-Holland, Amsterdam, 1996).
  * [46] C. Patrignami (Particle Data Group), “Review of particle physics”, Chin. Phys. C 40, 100001 (2016).
  * [47] P. K. Joshi, B. Singh, S. Singh, A. K. Jain , “Nuclear Data Sheets for A = 139”, Nucl. Data Sheets 138, 1 (2016).
  * [48] N. Nica , “Nuclear Data Sheets for A=140”, Nucl. Data Sheets 154, 1 (2018).
  * [49] M. Haaranen, J. Kotila, and J. Suhonen, “Spectrum-shape method and the next -to-leading-order terms of the $\beta$-decay shape factor”, Phys. Rev. C 95 024327 (2017).
  * [50] M. T. Mustonen, M. Aunola, and J. Suhonen, “Theoretical description of the fourth-forbidden non-unique $\beta$ decays of 113Cd and 115In”, Phys. Rev. C 73, 054301 (2006).
  * [51] B. A. Brown, and W. D. M. Rae, “The shell-model code NuShellX@MSU”, Nucl. Data Sheets 120, 115 (2014).
  * [52] N. Shimizu, T. Mizusaki, Y. Utsuno, and Y. Tsunoda, “Thick-restart block Lanczos method for large-scale shell-model calculations”, Computer Physics Communications 244, 372 (2019).
  * [53] ENSDF database, http://www.nndc.bnl.gov/ensdf/.
  * [54] IAEA, https://www-nds.iaea.org/nuclearmoments/.
  * [55] B. A. Brown, (unpublished).
  * [56] I. N. Borzov, “Beta-decay of nuclei near the neutron shell $N=126$”, Phys. At. Nucl. 74, 1435 (2011).
  * [57] S. Yoshida, Y. Utsuno, N. Shimizu, and T. Otsuka, “Systematic shell-model study of $\beta$-decay properties and Gamow-Teller strength distributions in A $\approx$ 40 neutron-rich nuclei”, Phys. Rev. C 97, 054321 (2018).
  * [58] P. M$\ddot{\text{o}}$ller, B. Pfeiffer, and K. L. Kratz, “New calculations of gross $\beta$-decay properties for astrophysical applications: Speeding-up the classical r process”, Phys. Rev. C 67, 055802 (2003).
  * [59] T. Marketin, L. Huther, and G. Martínez-Pinedo, “Large-scale evaluation of $\beta$-decay rates of $r$-process nuclei with the inclusion of first-forbidden transitions”, Phys. Rev. C 93, 025805 (2016).


# Image Classification on Accelerated Neural Networks

İlkay Sıkdokur1, İnci M. Baytaş2, Arda Yurdakul1,2 1Department of
Computational Science and Engineering, Bogazici University, Turkey
2Department of Computer Engineering, Bogazici University, Turkey
{ilkay.sikdokur, inci.baytas<EMAIL_ADDRESS>
###### Abstract

For image classification problems, various neural network models are commonly
used due to their success in yielding high accuracies. Convolutional Neural
Network (CNN) is one of the most frequently used deep learning methods for
image classification applications. It may produce extraordinarily accurate
results with regard to its complexity. However, the more complex the model is
the longer it takes to train. In this paper, an acceleration design that uses
the power of FPGA is given for a basic CNN model which consists of one
convolutional layer and one fully connected layer for the training phase of
the fully connected layer. Nonetheless, inference phase is also accelerated
automatically due to the fact that training phase includes inference. In this
design, the convolutional layer is calculated by the host computer and the
fully connected layer is calculated by an FPGA board. It should be noted that
the training of convolutional layer is not taken into account in this design
and is left for future research. The results are quite encouraging as this
FPGA design tops the performance of some of the state-of-the-art deep learning
platforms such as Tensorflow on the host computer approximately 2 times in
both training and inference.

##  I Introduction

Figure 1: Sketch of the design

Image classification is one of the core tasks in many artificial intelligence
tasks, such as autonomous driving, object recognition, and robotics
applications. Such applications require real-time image classification where
the latency and the size of the model become some of the important factors
that define the performance of the task. Convolutional Neural Network (CNN), a
deep learning architecture for grid-like inputs (e.g., image, graph),
constitutes the state-of-the-art in image classification. Although CNN is
successfully used for image classification, high model complexity of CNN
training leads to high computational time and memory consumption. For this
reason, CNN architectures may be cumbersome when it comes to using them in
real-time applications. This situation rises the importance of acceleration of
CNN models that are used in image classification problems.

For the acceleration of deep neural networks, different types of hardware
units are used such as Graphic Processing Unit (GPU) and Field-Programmable
Gate Array (FPGA). FPGA makes an outstanding candidate for this acceleration
process in various ways. One of the reasons of this is that FPGA is a fully
configurable system which consists of numerous fully programmable logic gates.
This feature of FPGA makes it easier and cheaper to specialize by the designer
for the acceleration of any desired calculation. However, the programmable
logic gates need to be programmed via a low level language such as Hardware
Description Language (HDL) in general like the other hardware units. But this
problem can be solved by the second advantage of FPGA usage: High Level
Synthesis (HLS). HLS method lets the designer to write the code piece that is
sought to be accelerated in various high level languages such as C++,
transforms it into low level hardware configuration languages such as HDL and
simulates a numerical circuit as if the logic gates are programmed to do the
desired calculations as stated in the high level. Besides, it lets the
designer to be more flexible in the design of the circuit for acceleration by
giving options to the designer on how the acceleration is sought to be done
like the parallelization methods and factors, memory usage on the unit, etc.
It is much more easier to change the acceleration design in numerous aspects
of it compared to other hardware units. By considering all of these reasons,
the acceleration of CNN training is decided to be done on an FPGA board in
this study. The target FPGA board is chosen as FFV1156 package of Zynq-7000
board family in this study. Moreover, the host computer that the FPGA is
connected to runs a 64 bit operating system and contains a CPU as AMD Ryzen 5
3550H and 8 GB DDR4 RAM.

In this study, the goal is to offer a hardware design that is to accelerate
the training phase of a basic CNN model that has one convolutional layer, one
fully connected layer and one output layer as shown in Fig. 1. Since the
inference phase forward propagation is contained in the training phase forward
and backward propagation, this design accelerates the inference phase also.
The results of acceleration of inference phase is also given in the paper with
the acceleration of the training phase. It should be noted that in this study,
training includes learning the parameters of the fully connected layers. The
convolution kernels are not learned from data. However, the convolution
operation is still performed in the forward propagation such that the
complexity of the convolution is included in the inference phase. Training of
the convolution layer is left for future work. In this sense, convolutional
kernel is set to be a constant array that is $3\times 3$ sharpening kernel
being commonly used in image processing applications. The kernel can be seen
in Eq. (1):

$kernel=\begin{bmatrix}0&-1&0\\\ -1&5&-1\\\ 0&-1&0\end{bmatrix}$ (1)

The kernel is illustrated with kernel in the Fig. 1. The convolution operation
is calculated in the host computer whereas the fully connected layer and the
output layer is calculated in the FPGA board.

To describe the model and the dataset, the fully connected layer consists of
128 neurons. The input images used in this problem are 28x28 pixel MNIST
handwritten digits (LeCun et al., 1998). Therefore, the output layer is of 10
neurons for each digit from 0 to 9. It should be noted that mini-batch
normalization is used that is of 32 images. In Fig. 1, these mini-batches are
illustrated with v_raw and the classes for every image in the mini-batches are
illustrated with outActual. Firstly, a 2D convolution is applied on each mini-
batch by the 3x3 kernel with stride of 1 and padding of 0 in both directions
and then their sized are reduced into half by 2x2 max-pooling. The convolved
mini-batches are flattened and inserted into the model. These flattened images
are illustrated with v in Fig. 1. The outputs obtained after these flattened
images are inserted into the model is illustrated as h2 in Fig. 1. MNIST
dataset consists of 60,000 images for training and 10,000 images for test
purpose (LeCun et al., 1998). For training, both forward and backward passes
are done for 60,000 training images and for testing, only forward pass is done
for 10,000 testing images. Adam optimizer (Kingma & Ba, 2014) is used during
training phase. Firstly, step-wise moving momentums ($m_{t}$) are calculated
by using the momentum values of previous step ($m_{t-1}$) and the gradient of
the loss of the current mini-batch ($g_{t}$) as in Eq. (2):

$\displaystyle m_{t}=\beta_{1}m_{t-1}+(1-\beta_{1})g_{t}$ (2) $\displaystyle
v_{t}=\beta_{2}v_{t-1}+(1-\beta_{2})g_{t}^{2}$

And then, in order to make momentums move and converge slower as the steps
increase, a step-wise correction is applied on the momentums as in Eq. (3):

$\displaystyle\widehat{m_{t}}=\frac{m_{t}}{1-\beta_{1}^{t}}$ (3)
$\displaystyle\widehat{v_{t}}=\frac{v_{t}}{1-\beta_{2}^{t}}$

These corrected momentums are illustrated as mW1, vW1, mW2 and vW2 in Fig. 1.
Finally, the weights are updated by using the updated and correction applied
momentum values as in Eq. (4):

$W_{t}=W_{t-1}-\eta\frac{\widehat{m}_{t}}{\sqrt{\widehat{v_{t}}}+\epsilon}$
(4)

In Fig. 1, the weights are illustrated as W1 and W2.

TABLE I: ADAM constant values Constant | Value
---|---
$\beta_{1}$ | $0.9$
$\beta_{2}$ | $0.999$
$\eta$ | $0.01$
$\epsilon$ | $10^{-7}$

Hyperparameters of Adam optimizer are chosen in agreement with the literature
as given in Table 1. Nonetheless, being one of the common practices, the
initialization of the weights are done by Gaussian noise where the mean is 0
and the standard deviation is 0.1.

For the acceleration operations, matrix multiplication and addition loops that
are mainly used during training are unrolled in order to grant parallelization
by a specific factor. In order to grant parallelization, the arrays to be
multiplied and added need to be available for parallel access. In order to
create parallel accesses to the arrays, they are partitioned in cyclic manner
by the same factor. Moreover, the calculations that are already done are
stored in the memory in order to prevent a re-calculation.

During the progress of this paper, a literature review of related works will
take place in Section 2. In Section 3, the design will be described. In this
description, it will be detailed how the design is structured for the model,
how the parallelization is supplied and how the memory storage is handled in
order to maintain this parallelization. Finally, final thoughts and some
future work insights will be given in Section 4.

##  II Literature Review

Current works in hardware acceleration of neural networks offer various
techniques and designs of implementations of trained models on FPGA. Chitty-
Venkata and Somani (2020) offer a method to reduce the size of the weight
matrices into systolic arrays by pruning various nodes of weight arrays so
they achieve a faster inference in less cycles. Arora, Wei and John (2020)
propose adding additional matrix multiplier blocks of systolic arrays on FPGA
in order to accelerate the process of FPGAs on machine learning applications.
They obtain almost 4 times faster results than reference design. In another
paper, Zhang, Zheng and Prasanna (2020) propose a design that aims to increase
throughput in Graph Convolutional Networks that are based on data traffic of
web sites. They part the data by blocks and cut unnecessary parts off so they
achieve faster results. Sestito et al. (2020) offer a new method of
convolution by applying a sliding window FIFO structure where multiple
convolution layers work in parallel. They state that this module that they
offer works approximately 650 times faster with 70 times less memory usage and
13 times less energy consumption. Meng et al. (2020) offer packing data and
using ensemble models on the packages of data. They state that they obtain a
speed-up between 2 and 7 times and there could even be a possible increase in
accuracy due to usage of ensemble methods. Moreover, Gibson et al. (2020)
offer a method in which they divide the kernel and images into parts, apply
convolution separately on each part and consolidate the results afterwards.
They state that they obtain up to 4 times speed-up than reference designs.
Ting et al. (2020) offer a method to reschedule the works of multiple Deep
Neural Network modules in multiple accelerators by deciding how accurate
results they are sought to produce. By rescheduling the modules in an optimal
way, they vanish the waiting periods between each module process so accelerate
the whole process in overall. Dias et al. (2020) offer to process each
dimension of multi-dimensional data in parallel and sum up the results
afterwards in order to increase throughput.

Results of some current papers show that FPGAs are very good candidates for
implementation of the training of neural networks. Courbariaux et al. (2016)
use Binarized Neural Networks to speed-up training and inference in which
weight matrices consist of only -1 and 1 values. In addition to binarized
values, Blott et al. (2018) propose usage of Quantized Neural Networks in
which weights are limited in an interval which gives a better accuracy on
FPGA. Prost-Boucle, Bourge and Pétrot (2018) offer usage of -1, 0 and 1 values
for weights on FGPA in which they obtained remarkable speed-ups.

##  III Design

###  III-A Description

The sketch of the design can be seen in Fig. 1. In the figure, input and
output arrays, different modules of the CNN model and the memory interface
connection between the host and the FPGA is shown. The size mappings for this
problem can be seen in Table 2. The definitions and sizes of the arrays
illustrated in the figures are given in Table 3. The system is available to
use any dataset so the sizes can be changed freely.

TABLE II: Definitions and values of the model constants Constant | Definition | Value
---|---|---
BATCHSIZE |  | Image amount
---
in a mini-batch
32
| IMAGEX
---
IMAGEY
| Width and height
---
of input image
| 28
---
28
| KERNELX
---
KERNELY
| Width and height
---
of the kernel
| 3
---
3
POOLMAPLENGTH |  | Size of the flattened
---
product in the end
169
LAYERSIZE |  | Size of the
---
hidden layer
128
CLASSSIZE |  | Amount of possible
---
classes in the dataset
10
TABLE III: Variables, definitions and sizes Array | Definition | Size
---|---|---
v_raw | Images from dataset |  | BATCHSIZE$\times$
---
IMAGEX$\times$
IMAGEY
kernel | Convolution kernel |  | KERNELX$\times$
---
KERNELY
outActual | Classes of images |  | BATCHSIZE$\times$
---
CLASSSIZE
v | Convolution product |  | BATCHSIZE$\times$
---
POOLMAPLENGTH
W1 |  | Weights of fully
---
connected layer
| POOLMAPLENGTH$\times$
---
LAYERSIZE
W2 | Weights of output layer |  | LAYERSIZE$\times$
---
CLASSSIZE
h2 |  | Output layer values
---
after activation
| BATCHSIZE$\times$
---
CLASSSIZE
| mW1
---
vW1
| ADAM momentums of fully
---
connected layer weights
| POOLMAPLENGTH$\times$
---
LAYERSIZE
| mW2
---
vW2
| ADAM momentums of
---
output layer weights
| LAYERSIZE$\times$
---
CLASSSIZE

It can be seen from Fig. 1 that calculations commence on host side as
convolution is applied on raw images taken from the dataset. Then, for the
sake of application of supervised learning, the actual class vectors are
prepared in order to calculate loss between them and obtained output values of
the model. The output of the model is kept in the host side in order to be
used in testing phase to test how accurate the model predicts the classes
since this design allows an acceleration in inference as well. This is the
flow of the host side in brief. In addition, the both of the weights are
instantiated in host part.

It is seen that host part and FPGA part are connected with an S_AXILITE
interface. This part allows the usage of AXI-4 LITE inter-chip communication
interface. The advantage of using AXI-4 interfaces is that they build a
connection between the different parts of the design by creating registers
between the parts. One of the main advantages of AXI-4 interfaces is that they
also provide data burst features, i.e more than one pack of data could be read
of written through the interface which provides a data stream. However, AXI-4
LITE interface provides only one pack is transferred from one part to the
other. The reason of using this limited interface is that the other interfaces
stream the data from one part to the other to be stored in an extra storage in
order to be used more than once for the sake of keeping the data stream
continuous. In this model, for example the products of convolution and output
layer values are used more than once. By keeping aligned with the target FPGA
choice, it is aimed to reduce storage usage as well in order to fit into the
chip. Nonetheless, in future work the other interfaces could be used with a
bigger and more complex FPGA chip to increase throughput of data even more.

It can be seen in Fig. 1 that the host sends a signal illustrated as
$isTraining$ in the figure to the FPGA. This signal distinguishes whether the
process is desired to be ended after the forward pass that is inference phase
or kept going for whole training phase including backward pass. This design
grants that the inference phase is accelerated as well since the whole
training is aimed to be accelerated.

Also, it can be noticed that almost every part of the modules in the FPGA use
some values in its calculations that are calculated in the part before. This
is a feature of the simplicity of this design in the FPGA part also.
Unnecessary calculations are sought to be avoided so this decreases the amount
of computations a lot. For example, RELU activation operations are calculated
right in the end of fully connected layer calculations. Another example is
that the sum of all exponentials of output layer values is calculated
concurrently in output layer calculations for softmax activation operations.
In the softmax module, cross-entropy loss is calculated in parallel. These are
the main calculations done in forward pass module.

The same less computation approach could be seen in backward pass also. Since
some of the values that are needed for the ADAM calculation on the fully
connected layer weights are calculated before in ADAM calculation on the
output layer weights, they are passed through one side through the other. The
step-wise correction values consist of power operation which is a
computationally expensive operation. In order to reduce the application of
this operation, they are calculated once and used in the next ADAM calculation
because they are based on the index of mini-batches that are passed through
the model and both ADAM calculations are done for the same mini-batch. It can
be seen that momentum values are seen to be updated. This is because ADAM is
an adaptive optimization method and this leads the momentums are to be stored
in FPGA side. And then it is seen that weights of both fully connected layer
and output layer are updated in the FPGA as well which means weight values are
stored in the FPGA too. It should be stressed that update processes of both
momentums and weights are shown separately for the sake of clarity on the
image. Each update is done in its own ADAM module in order to reduce loops
inside the FPGA.

###  III-B Parallelization

One of the advantages of FPGA is that its parallelization features are quite
abundant and grants the designer very useful tools for acceleration in
computing. This is one of the main reasons that an FPGA is used in this
design. The training of a deep learning model consists of a huge number of
matrix multiplication and addition which causes a long execution time. In
order to prevail this problem, several parallelizations could be applied on
these matrix operations in order to lessen the long execution times.

This is the main approach of the acceleration process of this design. Briefly,
the arrays are partitioned into several blocks and they are calculated in
parallel. An example of a parallel matrix multiplication loop in this design
can be seen in Listing 1.

⬇

for (i=0 ; i<BATCHSIZE ; ++i){

#pragma HLS UNROLL factor=4

for (j=0 ; j<LAYERSIZE ; ++j){

#pragma HLS UNROLL factor=4

#pragma HLS PIPELINE

for (k=0 ; k<POOLMAPLENGTH ; ++k){

h1[i][j] += v[i][k] * W1[k][j];

}

}

}

Listing 1: Fully connected layer

It can be seen that the first dimension of the convolution products and the
second dimension of the weights of the fully connected layer are read by parts
in parallel by unrolling the regarding loops by the factor of 4. On top of
this, the multiplication part is calculated in pipelined manner. A more
explicit code piece of this calculation can be visualized in Listing 2.

⬇

for (i=0 ; i<BATCHSIZE ; i+=4){

for (j=0 ; j<LAYERSIZE ; j+=4){

#pragma HLS PIPELINE

for (k=0 ; k<POOLMAPLENGTH ; k++){

h1[i][j] += v[i][k] * W1[k][j];

h1[i][j+1] += v[i][k] * W1[k][j+1];

h1[i][j+2] += v[i][k] * W1[k][j+2];

h1[i][j+3] += v[i][k] * W1[k][j+3];

h1[i+1][j] += v[i+1][k] * W1[k][j];

h1[i+1][j+1] += v[i+1][k] * W1[k][j+1];

h1[i+1][j+2] += v[i+1][k] * W1[k][j+2];

h1[i+1][j+3] += v[i+1][k] * W1[k][j+3];

h1[i+2][j] += v[i+2][k] * W1[k][j];

h1[i+2][j+1] += v[i+2][k] * W1[k][j+1];

h1[i+2][j+2] += v[i+2][k] * W1[k][j+2];

h1[i+2][j+3] += v[i+2][k] * W1[k][j+3];

h1[i+3][j] += v[i+3][k] * W1[k][j];

h1[i+3][j+1] += v[i+3][k] * W1[k][j+1];

h1[i+3][j+2] += v[i+3][k] * W1[k][j+2];

h1[i+3][j+3] += v[i+3][k] * W1[k][j+3];

}

}

}

Listing 2: Explicit code of unrolling fully connected layer

It can be seen that the arrays have 4 accesses in the same loop body that is
parallelized by pipelining.

Arrays on an FPGA are stored in memory blocks that have ports for input and
output providing access to the arrays. If this body loop is desired to be
computed in one computation cycle, all of the accesses are needed to be open
for input and output. This could be done by either increasing ports of the
block memories or partition the arrays so that each access is ready in memory
at the time of need. Because of the port amount to these block memories is
limited and cannot be increased without physical effect, arrays need to be
partitioned by the desired access amount. In our case, the arrays need to be
parted into 4 in order to get ready for parallel computation. The more the
arrays are partitioned, the more parallel they can be accessed and used in
calculations.

Another code piece from the design can be seen in Listing 3.

⬇

for (i=0 ; i<BATCHSIZE ; i++){

#pragma HLS UNROLL factor=4

for (j=0 ; j<CLASSSIZE ; j++){

#pragma HLS UNROLL factor=10

#pragma HLS pipeline

for (k=0 ; k<LAYERSIZE ; k++){

h2[i][j] += h1[i][k] * W2[k][j];

}

}

}

Listing 3: Output layer TABLE IV: Execution times for one epoch System |  | Execution Time
---
(sec)
| Training | Test
| Host | FPGA |  | Total
---
(Sequential)
| Total
---
(System Level Pipelined)
Host | FPGA |  | Total
---
(Sequential)
| Total
---
(System Level Pipelined)
Tensorflow | 5.3 | - | 5.3 | - | 0.8 | - | 0.8 | -
This Work | 1.9 | 2.3 | 4.2 | 2.3 | 0.3 | 0.02 | 0.32 | 0.3
TABLE V: Resource utilizations for FPGA | Training | Test
---|---|---
| BRAM | DSP | FF | LUT | URAM | BRAM | DSP | FF | LUT | URAM
Amount | 1,421 | 510 | 514,899 | 186,926 | 56 | 424 | 188 | 81,170 | 126,238 | 0
| Utilization
---
(%)
94 | 25 | 92 | 67 | 100 | 28 | 9 | 14 | 45 | 0

In this example, output layer in the forward pass is calculated in parallel
similar to fully connected layer. This is the main parallelization method
throughout the whole design. As also it can be seen from Listing 1 and Listing
3, it should be stressed that the dimension of mini-batch size, convolution
product and fully connected layer are partitioned into 4 and the dimension of
class size is partitioned into 10 which is a complete partition since class
size of this problem is 10.

###  III-C Memory Access

In this subsection, it is aimed to clarify in which way the arrays are
partitioned and stored inside FPGA.

As it can be seen from Listing 2, the arrays are accessed in parallel however
the indices of the accesses are sequential. Hence, the partitions need to be
done so that each index can be accessed at the same time, i.e they need to be
at the same level of the partitioned arrays. This is done by the feature of
cyclic partition. A basic demonstration of cyclic partition can be seen in
Fig. 4.

Figure 2: Cyclic partition

It can be seen that all partitions have the values of unroll factor of 4 at
the same level. The example is solely in dimension 1; nonetheless, it can be
applied in dimension 2 as well.

In order to fit into the target FPGA, the partitions are distributed in
different types of RAM blocks. The weights are stored in Unified RAM(URAM)
pieces. They are quite scarce in the FPGA yet they provide a faster access
than regular Block RAMs (BRAM). S_AXILITE interface reduces the BRAM memory
usage for the arrays passed from host to FPGA so BRAM memory usage is used for
the other arrays on FPGA. In order to increase throughput while doing so, dual
port BRAMs are used. In dual port BRAMs, one port is designated for input and
one port is designated for output whereas in single port BRAMs, the only port
is used for both input and output.

Finally, in order to limit the LUT register usage to fit into the board,
multiplier and adder cores are limited to 25, which may have a decreasing
effect on performance, caused almost no material effect on the performance.

##  IV Results

In this section, the execution times for reference with Tensorflow on host
computer and this design are compared. Also, the resource utilities for these
results are given for this design on FPGA side. It is to be highlighted that
the model trained on Tensorflow is identical to the model trained on the
design in every aspect including untrained constant kernel in convolution. In
addition, the model on Tensorflow is built with the Python version. 64-bit
floating point double is used as datatype in the demonstrations. Moreover, the
execution times are calculated for just one epoch on the whole dataset. The
results can be seen in Table 4.

In the table, two kinds of final execution times can be seen: sequential and
system level pipelined. Sequential execution is obtained when each of the
parts of the design waits until the other completes its process. In other
words, host part applies the convolution, prepares the images and the passes
them to FPGA part. It does not prepare the next mini-batch until FPGA is done
with its training or inference. Thus their execution times are added on top of
each other. System level pipelined version is obtained when the parts are
allowed to work in parallel. In this approach, host does not wait until FPGA
finishes its process. It continuously produces its convolved mini-batches and
sends it to FPGA when it is ready to take the next mini-batch to train or
infer. Thus, the execution time of the whole system depends on the slower side
of the system. For the training, it completes its whole task faster than FPGA
so the latency of the system is the latency of the FPGA. For the testing, the
opposite is valid. The latency of the system is the latency of host.

It can be seen that this design passes Tensorflow on host both in sequential
and system level pipelined version. Furthermore, as system level pipelining is
valid, the system is approximately 2 times faster than Tensorflow on host.

In Table 5, the resource utilities of FPGA side is given. It can be seen that
the resources are utilized in the upper limits in training yet is fits into
the chip. The utility usage is remarkably less than training.

This paper leans on acceleration of the training; however, it is not solely
enough to accelerate the system. It is expected to give accurate results after
the acceleration is done. The accuracy results can be seen in Table 6 for both
Tensorflow on host computer and this system.

TABLE VI: Accuracy System | Accuracy (%)
---|---
| 1 Epoch | 5 Epoch | 10 Epoch
Tensorflow | 91 | 93 | 94
This Work | 90 | 92 | 94

It can be seen that both of the systems converge to the same level of accuracy
after 10 epochs. The difference that is seen in early epochs arises from that
the weights are randomly initialized in both of the systems. Moreover, the
training set is not shuffled between the epochs for both of the systems.

##  V Discussions

This work shows the potential of FPGA usage in deep learning problems. The
results show that the execution times can be drastically lowered without so
much accuracy loss in the end. Since acceleration on whole training contains
inference, the same hardware piece may accelerate inference as well.

There are improvements that can be done on this work such as acceleration of
convolution where convolution kernel is trained too, usage of stream interface
and dataflow and different data types such as 16-bit floating point half or
fixed point data types. The usage of less precision data types may end up with
a decrease in accuracy but increase in performance a lot. In addition,
different systems like GPU-FPGA hybrids could be designed in order to improve
performance.

## References

  * [1] LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. (1998). Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86, 2278–2324.
  * [2] Kingma, D., Ba, J.. (2014). Adam: A Method for Stochastic Optimization. International Conference on Learning Representations.
  * [3] Haidar, A., Bayraktar, H., Tomov, S., Dongarra, J., Higham, N. J. (2020), Mixed-Precision Solution of Linear Systems Using Accelerator-Based Computing. Innovative Computing Laboratory Technical Report, no. ICL-UT-20-05, University of Tennessee.
  * [4] Chitty-Venkata, K. T., Somani, A. K., (2020), Array Aware Training/Pruning: Methods for Efficient Forward Propagation on Array-based Neural Network Accelerators [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [5] Arora, A., Wei, Z., John, L. K. (2020), Hamamu: Specializing FPGAs for ML Applications by Adding Hard Matrix Multiplier Blocks [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [6] Zhang, B., Zeng, H., Prasanna, V. (2020), Hardware Acceleration of Large Scale GCN Inference [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [7] Sestito, C., Spagnolo, F., Corsonello, P., Perri, S. (2020), An Efficient Convolution Engine based on the A-trous Spatial Pyramid Pooling [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [8] Meng, J., Guo, C., Gebara, N., Luk, W. (2020), Fast and Accurate Training of Ensemble Models with FPGA-based Switch [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [9] Gibson, P., Cano, J., Turner, J., Crowley, E. J., O’Boyle, M., Storkey, A. (2020), Optimizing Grouped Convolutions on Edge Devices [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [10] Ting, H., Giyahchi, T., Sani, A. A., Bozorgzadeh, E. (2020), Dynamic Sharing in Multi-accelerators of Neural Networks on an FPGA Edge Device [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [11] Dias, L. A., Coutinho, M. G. F., Gaura, E., Fernandes, M. A. C. (2020), A New Hardware Approach to Self-Organizing Maps [Paper presentation]. The 31st IEEE International Conference on Application-specific Systems, Architectures and Processors, Manchester.
  * [12] Courbariaux, M., Hubara, I., Soudry, D., El-Yaniv, R., Bengio, Y. (2016), Binarized Neural Networks: Training Deep Neural Networks with Weights and Activations Constrained to +1 or -1. doi: arXiv:1602.02830.
  * [13] Blott, M., Preusser, T., Fraser, N., Gambardella, G., O’Brien, K., Umuroglu, Y. (2018), FINN-R: An End-to-End Deep-Learning Framework for Fast Exploration of Quantized Neural Networks. doi: arXiv:1809.04570.
  * [14] Prost-Boucle, A., Bourge, A., Pétrot, F., (2018), F High-Efficiency Convolutional Ternary NeuralNetworks with Custom Adder Trees and Weight Compression. ACM Transactions on ReconfigurableTechnology and Systems (TRETS), ACM, 2018, Special Issue on Deep learning on FPGAs, 11 (3),pp.1-24. 0.1145/3294768, al-01686718v2.