# Band-center metal-insulator transition in bond-disordered graphene

Naba P. Nayak  Department of Physics, Indian Institute of Technology Bombay,
Mumbai 400076, India  Surajit Sarkar  Department of Physics, Indian Institute
of Technology Bombay, Mumbai 400076, India  Department of Physics, Concordia
University, Montreal, Quebec, Canada  Kedar Damle  Department of Theoretical
Physics, Tata Institute of Fundamental Research, Mumbai 400005, India  Soumya
Bera  Department of Physics, Indian Institute of Technology Bombay, Mumbai
400076, India

###### Abstract

We study the transport properties of a tight-binding model of non-interacting
fermions with random hopping on the honeycomb lattice. At the particle-hole
symmetric chemical potential, the absence of diagonal disorder (random onsite
potentials) places the system in the well-studied chiral orthogonal
universality class of disordered fermion problems, which are known to exhibit
both a critical metallic phase and a dimerization-induced localized phase.
Here, our focus is the behavior of the two-terminal conductance and the
Lyapunov spectrum in quasi-1D geometry near the dimerization-driven transition
from the metallic to the localized phase. For a staggered dimerization pattern
on the square and honeycomb lattices, we find that the renormalized
localization length $\xi/M$ ($M$ denotes the width of the sample) and the
typical conductance display scaling behavior controlled by a crossover length-
scale that diverges with exponent $\nu\approx 1.05(5)$ as the critical point
is approached. However, for the plaquette dimerization pattern, we observe a
relatively large exponent $\nu\approx 1.55(5)$ revealing an apparent non-
universality of the delocalization-localization transition in the BDI symmetry
class.

##  I Introduction

Quenched disorder plays a significant role in determining electronic transport
properties particularly when quantum interference enhances its effects and
leads to Anderson localization phenomena [1, 2, 3]. Such disorder effects are
controlled crucially by the symmetries of the disordered Hamiltonian. For
instance, in a two-dimensional (2D) electron gas with potential scattering
from random impurities, the sign of the quantum interference correction to the
conductivity depends on whether the electronic system has spin-rotation
symmetry. As a result of the important role played by such symmetry
considerations, our modern understanding of such phenomena relies heavily on a
symmetry-based classification of disordered systems [4, 5].

Tight-binding models of free fermions with real-valued random hopping
amplitudes on the nearest-neighbor links of a bipartite lattice fall in a
particularly interesting symmetry class, labeled BDI by Altland and Zirnbauer
in their ten-fold classification of disordered systems [4]. Due to the absence
of on-site potential energy terms and the bipartite structure of hopping
amplitudes, the free-fermion spectrum in this class is distinguished by the
presence of a particle-hole symmetry which guarantees that each eigenstate at
energy $+\epsilon$ has a partner at energy $-\epsilon$. The band-center energy
$\epsilon=0$ is thus special.

Within the field-theoretical approach pioneered by Gade and Wegner [6, 7], the
bare conductivity at the band center receives no quantum corrections in two
dimensions, while the density of states develops a characteristic ‘Gade-
Wegner’ singularity
$\rho(\epsilon)\sim|\epsilon|^{-1}\exp(-\ln^{1/x}(1/|\epsilon|))$ (with
$x{=}2$) for energies $|\epsilon|$ smaller than a characteristic crossover
energy scale controlled by this conductivity. The conclusion is that such
particle-hole symmetric systems can have a critical metallic phase whose low
energy properties are characterized by a fixed line within the field-
theoretical renormalization group framework.

As is well-known from the work of Dyson and others [8, 9, 10, 11], the
corresponding one-dimensional system has a stronger ‘Dyson singularity’
$\rho(\epsilon)\sim|\epsilon|^{-1}\ln^{-y}(1/|\epsilon|)$ (with $y{=}3$) in
the density of states at the band-center. Generalizations to multichannel
cases and the nature of the zero energy wavefunctions in both one and 2D cases
have also been studied in more recent literature [12, 13, 14, 15].

In the one-dimensional case, this singular behavior can be derived from the
properties of an infinite-disorder fixed point of a real-space strong-disorder
renormalization group approach [16]. In the 2D scenario, it is difficult to
obtain conclusive results from a direct numerical implementation of this
strong-disorder renormalization group approach. However, it motivates a
closely related strong-disorder analysis [17] of the low-energy properties of
the critical metallic phase via a connection to optimal defects in a related
dimer model. This real-space approach predicts a singularity of the Gade-
Wegner form but corrects the associated exponent to $x=3/2$.

This prediction of a modified Gade-Wegner form for the band center singularity
in two dimensions has also been confirmed by subsequent work [18, 19] that
refined the original field theoretical analysis of Gade and Wegner. More
recent work has also studied the effects of vacancy disorder in such 2D
systems, finding that vacancies lead to a stronger Dyson form of the
singularity (albeit with nonuniversal $y$) at intermediate energies before the
system crosses over at the lowest energies to the modified Gade-Wegner form
[20, 21, 22].

The real space approach of Motrunich  _et al._ [17] also predicts that such
two-dimensional systems can realize, in addition to the critical metallic
phase, a localized Griffiths phase with a weaker non-universal power-law
divergence $\rho(\epsilon)\sim|\epsilon|^{-1+2/z}$ (with nonuniversal $z$) of
the density of states near the band center. As noted in Motrunich  _et al._
[17], such a localized phase can be established by strong enough dimerization
in the values of the hopping amplitudes (see Fig. 1). While both the critical
metallic phase and the localized Griffiths phase have been discussed
extensively in the literature, the transition between the two has received
much less attention [17, 23, 24].

Here, we focus on the transport properties of the 2D sample in the vicinity of
this transition via numerical studies of both the two-terminal Landauer
conductance and the full Lyapunov spectrum $\lambda$. The transfer matrix and
the Lyapunov spectrum studies provide complementary evidence in favour of or
against particular scaling results that are obtained by the renormalization
group approach. For small dimerization, the density $\rho(\lambda)$ of
Lyapunov exponents $\lambda$ in quasi-1D geometry is nonzero at $\lambda=0$.
This corresponds [25] to a nonzero conductivity. In this phase, we find that
the conductivity extracted from the typical two-terminal conductance is
roughly independent of the sample geometry and consistent with the
conductivity obtained from the Lyapunov density $\rho(0)$, indicating a
metallic phase. With increasing dimerization, the Lyapunov density of states
develops a gap around $\lambda=0$, signaling the transition to a gapped
insulating phase at a critical dimerization strength $\delta_{c}$. A
qualitative phase diagram is shown in Fig. 1.

Figure 1: Top: Schematic phase diagram as a function of dimerization. Without
disorder, there is a dimerization-driven transition from semi-metal to band
insulator. With bond disorder, there is a corresponding transition from a
critical metal to a Griffiths insulator [17]. Bottom left: shows the staggered
dimer pattern on a honeycomb flake. Right: $E{=}0$ wavefunction amplitudes
$\log|\psi|^{2}$, for a typical disorder realization of disorder with
staggered dimer pattern (1). The system size is $64\times 64$ with periodic
boundary conditions in both directions. The wavefunction shows more spreading
in the critical metal phase compared to the Griffiths phase as expected.

In the gapped insulating phase close to $\delta_{c}$, we perform a finite size
scaling analysis of the renormalized localization length $\xi/M$ (where $M\gg
1$ is the width of the quasi-1D sample whose length $L\gg M$). For staggered
dimerization on both the square and honeycomb lattice, we argue that the rigid
shift [17] of the Lyapunov spectrum, characteristic of staggered dimerization,
implies that $\xi/M$ diverges as $\sim(\delta-\delta_{c})^{-\nu}$ with $\nu=1$
as $\delta$ approaches $\delta_{c}$ from above. The value of $\nu$ we obtain
from both the Lyapunov spectrum study and direct measurement of the typical
two-terminal conductance in the wide-sample geometry is consistent with this
prediction within the numerical errors associated with these calculations. A
calculation of the transport properties in the direction perpendicular to the
strong bonds of the staggered dimerization pattern also yields the same value
of $\nu$ within numerical uncertainties. However, on the honeycomb lattice,
for a more symmetric plaquette dimerization that does not have this rigidity
property, we observe a larger exponent $\nu\approx 1.55(5)$. This apparent
non-universality in the value of the exponent is one of our main findings.

Intriguingly, the larger value of $\nu$ that we find for plaquette
dimerization on the honeycomb lattice is consistent, within our numerical
errors, with a recent prediction [26] of the same exponent in the closely
related problem of random bipartite hopping with complex hopping amplitudes on
the square lattice with the same kind of plaquette dimerization. This falls in
the Altland-Zirnbauer class AIII, while the systems we study are in class BDI.
Since this study also used a plaquette dimerization pattern for their
numerical work, it would be interesting in follow-up work to ask if a similar
non-universality is also present in the problem with complex hopping
amplitudes and explore this apparent nonuniversality within the framework of
the theory developed in Ref. 26.

##  II Model and observables

We consider a model of free fermions hopping on the honeycomb lattice, and an
analogous model on the square lattice. In this model, the real-valued nearest-
neighbor hopping amplitudes are independent random variables sampled from a
uniform distribution. For conductance calculation, we take a clean one-
dimensional lead attached to the disordered sample. The Hamiltonian of the
system thus reads

$\displaystyle\hat{\mathscr{H}}=-\sum_{\langle
ij\rangle}t_{ij}(c_{i}^{\dagger}c_{j}+\mathrm{h.c.}),$ (1)

where $c_{i}$($c_{i}^{\dagger}$) is the fermion annihilation (creation)
operator at site $i$, and the sum is over all nearest-neighbor links $\langle
ij\rangle$. We define a dimerization pattern by choosing the hopping strength
of selected “dimerized” bonds to be independent random variables uniformly
distributed in the range $[0,e^{\delta}]$, while the hopping amplitudes on
other “non-dimerized” bonds are random variables drawn uniformly in the range
$[0,1]$. The parameter $\delta$ thus represents a dimerization in the mean
value of the hopping amplitudes. Since the ratio of the width of the
distribution to its mean remains the same, tuning $\delta$ changes the
dimerization without changing the strength of the bare disorder. In all our
numerical simulations, typically, we average over ${\sim}10^{3}$ disorder
configurations for the correlation length calculations, and for conductance
typically we average over ${\sim}10^{4}$ samples unless specified otherwise.

We study both the staggered and plaquette dimerization patterns (described
below) on the honeycomb lattice, and the staggered pattern on the square
lattice to access the dimerization-driven transitions from metallic to
localized behavior in this model on both lattices at $E=0$. The staggered
anisotropic pattern was chosen as a representative of patterns that break the
rotational symmetry between the transverse direction and the direction of the
current and leads to a rigid shift of the Lyapunov spectrum. The other
plaquette pattern was chosen because it maintains the rotational symmetry and
does not lead to a rigid shift of the Lyapunov spectrum. Further, note that
the staggered dimerization pattern can choose any one out of the three
equivalent bond orientations and enhance the corresponding bond strengths.
Given the symmetry of the honeycomb lattice, all these choices are equivalent,
and the only physical distinction then is in the direction of the current flow
relative to the chosen orientation for the staggered pattern.

In the staggered dimerization pattern on the honeycomb lattice (shown in Fig.
3(b) inset) the horizontal bonds along the direction of the arrow are
dimerized, while along the other direction the bonds are non-dimerized in the
way defined above. The plaquettes dimerization pattern on the honeycomb
lattice (shown in fig 5(b) inset) is a more symmetric pattern that does not
single out a particular direction as in the staggered pattern. In this
pattern, the strong “dimerized” bonds are ordered at the three-sublattice
wavevector of the underlying triangular Bravais lattice. On the square
lattice, we mainly study the staggered pattern, in which the strong
“dimerized” bonds are all of one orientation and arranged in a pattern
corresponding to wavevector $(\pi,\pi)$. The plaquette dimerization pattern on
the square lattice on the other hand has strong “dimerized” bonds that form
perimeters of elementary plaquettes that are ordered at wavevectors $(\pi,0)$
and $(0,\pi)$, and has been studied in Ref. [26] with complex hopping
amplitudes.

Figure 2: (a) The Lyapunov spectral density $\rho(\lambda)$ for A-sublattice
transfer along the direction of the arrow in the staggered graphene lattice as
shown in Fig. 3(b) inset. With increasing $\delta$, the spectral density
shifts rigidly, and the horizontal dotted lines indicate the value where the
density curves cut the $\lambda=0$ axis, i.e. $\rho(\lambda=0)$ and this gives
an estimate of the conductivity (upto a trivial constant) of the sample in the
metal phase [25]. The right panel shows the collapse of all curves with the
rigid shift by $\delta$. (c) Shows the conductivity $\sigma=G(L,M)L/M$ as a
function of $\delta$ estimated from the Lyapunov density of states in
quasi-$1$D geometry and the scattering matrix formalism for a wide sample
$L/M\sim 1/4$ (see further details in the text). One-dimensional lattice leads
are used for the conductance calculation.  Figure 3:  Results for staggered
dimerization pattern on arm-chair graphene shown in Fig 3(b) inset. The
normalized localization length $\xi/M$ for a transfer along the direction of
the arrow of the lattice as a function of dimerization $\delta$ for different
values of width $M=\\{64-768\\}$ and length $L{=}10^{5}$ in the localized
phase. (b) Shows the corresponding finite size data collapse with critical
exponent $\nu\approx 1.05$, and a critical dimerization strength
$\delta_{\mathrm{c}}\approx 1.431$. (c) The $\langle\ln G\rangle$ is
calculated within the two terminal setup for the same lattice with 1D clean
lattice leads connected along the direction of the arrow in the localized
phase for various system lengths $L=\\{32-112\\}$ for $L/M\sim 1/4$. (d) The
corresponding approximate scaling collapse is shown for $\langle\ln G\rangle$
which yields $\nu\approx 1.1$.

###  II.1 Lyapunov analysis

The standard Lyapunov spectrum $\lambda_{i}$ of the transfer matrix
$T=\prod_{n}T_{n}$ ($T_{n}$ is the transfer matrix for the $n$-th slice) for a
quasi-$1$D geometry of width $M$ is used to study the metal-insulator
transition (MIT) [27]. In the quasi-1D limit, the inverse of the smallest
Lyapunov exponent in the limit of large $L$ in the localized phase defines the
localization length $\xi=1/\lambda_{\text{min}}$ [25]. The normalized
localization length $\xi/M$ [25] is expected to exhibit finite size scaling
across an MIT, allowing one to extract the exponent $\nu$ from a finite-size
scaling analysis of $\xi$ approaching the transition from the localized side.
Moreover, the limiting spectral density of Lyapunov exponents {$\lambda_{i}$}
in such a quasi-1D geometry,

$\rho{(\lambda)}=\lim_{M\rightarrow\infty}1/M\sum_{i}\delta(\lambda-\lambda_{i}),$
(2)

contains information about the transport properties of the sample (here the
width $M$ of the sample is assumed to increase while maintaining the quasi-1D
geometry with $L\gg M$). In particular, the $\lambda=0$ density of states
$\rho(\lambda{=}0)$ is finite in a metal and proportional to the conductivity,
while it is strictly zero in an insulating phase [25]. Therefore, at a
transition to a localized insulating phase, we expect the Lyapunov spectral
density to develop a gap in the spectrum at $\lambda=0$. In our calculation,
we use the fact that the Hamiltonian has chiral symmetry. Due to this, the
wavefunctions at $E=0$ can be chosen to have support only on one sublattice,
and the transfer of such a wavefunction from one side to the other can be
performed just on this sublattice; this decoupling allows a trivial factor of
two increases in the system width $M$ [17, 28], since one can assemble the
full Lyapunov spectrum from such a calculation for just one sublattice. The
full spectrum $\rho(\lambda)$ is symmetric about $\lambda=0$, and this
symmetry implies that a Lyapunov mode at $+\lambda$ on one sublattice has a
partner at $-\lambda$ on the other sublattice, allowing a reconstruction of
the full spectrum from this sublattice calculation.

As has been emphasized earlier [17, 28], the staggered dimerization pattern
leads to an interesting simplification: The two sublattice Lyapunov spectra at
$E=0$ shift rigidly in opposite directions with increasing $\delta$, all the
while maintaining the $\lambda\rightarrow-\lambda$ symmetry of the full
spectrum (see Appendix A for more details). It is therefore possible to
determine the critical value $\delta_{c}$ at which the metal gives way to the
insulator simply by knowing the spectrum for one sublattice at $\delta=0$. For
$\delta>\delta_{c}$ in the insulator, this ridigity also implies that
$\lambda_{\rm min}\sim(\delta-\delta_{c})$, implying that $\xi/M\sim
1/(M(\delta-\delta_{c})$. This argument suggests that the correlation length
exponent $\nu$ takes on the value $\nu=1$ for an MIT driven by such a rigid
shift in the sublattice Lyapunov spectrum. It is interesting to note that such
an argument is also valid for complex hoppings with staggered dimerization
pattern, but we have not studied it further here.

Below we explore the validity of this argument and check if this also controls
the scaling of the two-terminal conductance calculated from the scattering
wavefunctions (described below). We also ask if a generic transition, not
driven by dimerization patterns that involve a rigid shift, has a different
value of $\nu$ characterizing its critical behavior.

###  II.2 Two terminal conductance

The two terminal linear conductance at zero temperature is calculated using
the Landauer formula

$G=\frac{e^{2}}{h}\sum_{nm}|S_{nm}|^{2},$

where $G$ is the dimensionless conductance (in units of $e^{2}/h$) and
$S_{nm}$ is the scattering matrix element between the scattering channel $n$
and $m$ obtained in terms of the zero energy wavefunctions. It relates
electrical conductance to the total transmission probability of electron waves
through a region of random scatters. As is well known, a closely related
analysis [29] in terms of Green functions also leads to the Landauer formula,
which relates electrical conductance to the total transmission probability of
electron waves through a region of random scatters; in this sense, our results
are expected to be equivalent to those obtained from explicit use of the two-
terminal Landauer formula.

We used open source Kwant package [30], a wavefunction-based method for the
conductivity calculation. The ‘effective’ disorder in the system is large, and
thus the conductivity is small $\mathcal{O}(1)$. To avoid small numbers we use
a wide geometry $M>L$, which has more conducting channels and ensures a larger
value of the conductance. This is useful because the divergent density of
states at zero energy gives rise to numerical instability in the opposite
limit of a quasi-1D sample. In all our conductance calculations we use one-
dimensional lattice leads which allow having a finite density of state in the
lead at $E=0$. The sample width $M$ is chosen in such a way that the clean
band structure with armchair edges has linear energy dispersion.

Figure 4: (a) Renormalized localization length as a function of the
dimerization strength $\delta$ for staggered dimerization pattern on the
honeycomb lattice in the direction of the arrow as shown in the inset. (b)
Shows the scaling collapse, which yields $\nu\approx 1.05(5)$. (c) The
renormalized localization length $\xi/M$ for stagger dimerization pattern on a
square lattice for a transfer along the direction of the arrow. Details are
similar to (a). (d) The corresponding data collapse yields a critical exponent
$\nu\approx 1.01(3)$.

In a generic Anderson delocalization-localization transition, the typical
conductance $\exp(\langle\ln G\rangle)$ acts as an order parameter and
displays scaling behavior [31]. With this in mind, we study the $L$ and
$\delta$ dependence of $\langle\ln G\rangle$, and also we monitor the
probability distribution of $\ln G$ for a range of $L$ and $\delta$ in the
metallic phase. We obtain data for the staggered dimerization patterns on both
the square and honeycomb lattice, and honeycomb lattice data for the plaquette
dimerization pattern (Fig. 3, Fig. 4, and insets).

##  III Numerical Results

###  III.1 Staggered dimerization on the honeycomb lattice

Figure 5: (a) Results for plaquette dimerization pattern on arm-chair graphene
as shown in Fig 5(b) inset. The normalized localization length $\xi/M$ as a
function of dimerization $\delta$ for different values of width
$M=\\{30-384\\}$ and length $L{=}10^{5}$ in the localized phase. (b) Shows the
corresponding finite size data collapse with critical exponent $\nu\approx
1.55$, and a critical dimerization strength $\delta_{\mathrm{c}}\approx
0.015$. (c) The $\langle\ln G\rangle$ is calculated within the two terminal
setups for the same lattice with 1D leads connected along the direction of the
arrow in the localized phase for various system lengths $L=\\{32-128\\}$ for
$L/M\sim 1/4$. (d) The corresponding approximate scaling collapse is shown for
$\langle\ln G\rangle$ which yields $\nu\approx 1.47$.

#### Lyapunov spectrum and $\delta_{\mathrm{c}}$:

The Lyapunov spectral density (2) for $A$ sublattice transfer along the
direction of the arrow in the staggered quasi-1D sample (shown in fig 3(b)
inset) of width $M=768$ is shown in Fig. 2(a). The curves show a finite
density at $\lambda=0$ in the metallic phase.

Increasing $\delta$ further creates a finite gap around the center i.e. zero
spectral density at $\lambda=0$. This marks the onset of the localized phase
of the system and that allows a precise determination of
$\delta_{\mathrm{c}}$. The estimated critical disorder strength is found to be
$\delta_{c}\approx 1.431$. In this case, the spectrum does shift rigidly,
consistent with the general argument made earlier.

In the Griffiths insulator phase, we see that $\xi/M$ decreases with
increasing $\delta-\delta_{c}$ and $M$ showing increasing localization as seen
in Fig. 3(a). A one-parameter scaling of $\xi/M$ is performed with following
scaling form $\xi/M{=}\mathcal{F}\left(\chi(w)M^{1/\nu},\phi(w)M^{y}\right)$
(see Appendix B for more details), where $y$ is the irrelevant scaling
exponent. The corresponding data collapse is shown in Fig. 3(b). The estimated
critical disorder strength from the analysis is
$\delta_{\mathrm{c}}{=}1.431(2)$, which is consistent with the critical
dimerization value extracted from the Lyapunov density of state (see in Fig.
2). The localization length exponent is found to be $\nu{=}1.05(5)$,
consistent with the theoretical prediction of $\nu=1$ for cases when the
dimerization leads to a rigid shift in the sublattice Lyapunov spectrum.

#### Conductance calculation in localized phase

We compute $\langle\ln G\rangle$ via scaterring matrix formulation as
described earlier. In the localized phase, this provides an independent window
to these localization properties.

We find that the conductance is exponentially small for $\delta>\delta_{c}$,
i.e. $G\ll 1$ and decreasing with system length $L\gg\xi$ as well as with
increasing dimerization $\delta$ as shown in Fig. 3(c). We investigate the
dependence on $\delta$ and $M$ via a finite size scaling analysis at fixed
$L/M$ for wide samples as shown in Fig. 3(d). The scaling collapse gives a
critical disorder to be $\delta_{c}{=}1.46(2)$ and the exponent $\nu=1.10(4)$.
The marginally higher critical parameters can be attributed to a slightly
different geometry $L/M\approx 1/4$ used for the numerical simulation. Within
the available computational resources, we could not directly probe the true 2D
limit $L/M{=}1$ as it would require simulation of large sample sizes. The
estimate of $\nu$ obtained in this way is also very close to the value
obtained from the Lyapunov exponent analysis; indeed the theoretical
prediction of $\nu=1$ is just outside the error bars of this numerical
estimate.

#### Additional confirmation of scaling behavior for staggered dimerization:

To explore this further, we also determine $\nu$ using a Lyapunov spectrum
analysis for transfer in the perpendicular direction i.e., along the zigzag
edge of the honeycomb lattice (shown in Fig. 4(a) inset). The results are
shown in Fig. 4(a). A one-parameter scaling collapse gives a localization
length exponent of $\nu\approx 1.05$ in agreement with the transfer along the
other direction. The critical disorder strength is estimated to be
$\delta_{c}\approx 1.38$. The smaller critical disorder strength in this
direction could be related to the strong anisotropy of the bond arrangement.
We also observed (data not shown) an extremely slow flow towards the larger
$\delta_{\text{c}}\sim 1.43$ with increasing system sizes $M$.

Additionally, we also verify our results for staggered dimer patterns on a
square lattice. The main difference here is that the density of states at
$E=0$ is finite in the clean limit compared to the Dirac density of states.
The results for the transfer matrix are shown in Fig. 4. A finite size
analysis again predicts the exponent to be $\nu\approx 1.01$ in agreement with
all our previous estimates of staggered dimer patterns on the honeycomb
lattice. Again, we attribute this value of $\nu$ to the fact that the Lyapunov
spectrum on each sublattice shifts rigidly with $\delta$ for this kind of
staggered dimerization.

Figure 6: (a)-(b) Conductance distribution $\mathscr{P}(G)$ at $\delta{=}0$ is
calculated within the two terminal geometry for the finite staggered honeycomb
sample shown in Fig. 3(b) inset, with two different aspect ratios $L/M\sim
1/2,1/4$. The corresponding $L,M$, and the average conductance $G[e^{2}/h]$
along with the mean conductivity $\sigma[e^{2}/(\pi h)]\approx 0.85(2)$ are
indicated in the legend. (c) Shows the $\ln G$ distribution in the localized
phase $\delta{=}1.55$ for different sample length $L=\\{64-112\\}$. Open
armchair boundary is used for these calculations.

#### Critical metal phase:

Following Chalker and Bernhardt [25] we estimate the average conductivity in
the critical metal phase $\sigma=G(L,M)L/M=b\rho(0)$, where $b$ is an order
one number, and $\rho(0)$ is the Lyapunov density, which is shown in Fig. 2.
The two terminal conductance calculated in the wide sample limit $L/M\ll 1$ is
also shown in Fig. 2 as a function of dimerization $\delta$ and compares well
to the estimate obtained above from the Lyapunov spectrum. In the metallic
phase, it increases with $\delta$ until the metal-insulator transition point
is reached, perhaps reflecting the fact that the strengthened dimerization
facilitates transport in the measured direction.

###  III.2 Plaquette dimer configuration

The plaquette dimerization pattern on the honeycomb lattice is shown in Fig.
5(b) inset. This pattern does not single out a particular direction of the
lattice as in the case of a staggered pattern. The plaquette dimerization also
drives a dimerization-driven critical metal-insulator transition, however,
with a larger localization length exponent, which could point towards a
different fixed point.

The normalized localization length $\xi/M$ and log-conductance $\langle\ln
G\rangle$ data is shown in Fig 5(a) and in Fig 5(c) respectively. The finite-
size analysis of both of our observables (Fig 5(b) and Fig 5(d)) points
towards a higher localization length exponent in this dimer configuration. The
localization length predicts the exponent to be $\nu\approx 1.55(2)$ which
agrees within error bars to the conductance data which predicts $\nu\approx
1.4(1)$. As noted earlier, recent work in the other chiral class AIII indeed
predicts a larger exponent $\nu\approx 1.55(1)$ with a similar dimer pattern
on the square lattice with complex hopping [26].

###  III.3 Conductance distribution

In Fig. 6 we show the distribution of the conductance $G(L,M)$ for the
honeycomb lattice, in the metallic phase at $\delta=0$. The data is shown for
two different aspect ratios $L/M=1/2,1/4$ Fig. 6(a,b). For both the aspect
ratios the width of the $\mathscr{P}(G)$ is independent of the sample $L$,
indicating its scale-invariant properties. While the average conductance of
course depends on the sample geometry, the conductivity $\sigma{\approx}0.85$
(in the units of $e^{2}/(\pi h)$) for $\delta{=}0$ is independent of the
sample geometry indicating that the value is close to its true 2D limit. The
distribution $\mathscr{P}(G)$ has a long tail with a singularity at $G\approx
1$ (in the units of $e^{2}/h$), which is a reminiscence of the critical
conductance distribution at a 3D metal-insulator transition [32].

In the localized insulating phase, the mean conductance is small $G(L,M)\ll
1$; therefore we show the log-conductance $\ln G$ distribution in Fig. 6(c) at
$\delta=1.55$, which is quite close to $\delta_{c}$. Deep in the localized
phase, the conductance becomes extremely small and suffers from numerical
instability. The $\mathscr{P}(\ln G)$ is far from a normal Gaussian form,
which one would expect in the localized phase [27]. However, we observe the
peak at $G\approx 1$ decreases rapidly with $L$ in this regime and the mean
conductance becomes smaller. The scaling analysis of the mean $G$ has already
been presented in Fig. 3.

##  IV Discussion and Outlook

We present a numerical study of transport properties in a bond-disordered
tight-binding model of honeycomb and square lattice. The disordered version of
the phase transition is allied with the opening of a band gap in the clean
model due to dimerization.

From our results for the critical behavior of the mean log-conductance and
normalized localization length, we find an apparent non-universality of
critical exponents in the BDI symmetry class. In our analysis we estimated it
to be $\nu\approx 1.05(5)$ for staggered dimer pattern driven localization
transition, and the critical exponent to be $\nu\approx 1.55(5)$ for plaquette
dimer configuration. Important to note that with particle-hole symmetry, the
localization transition is actually preceded by the critical metal phase that
we have discussed here, and not by the weak localized phase that is usually
observed in 2D standard Anderson transition. The critical phase itself is a
strongly disordered phase as signalled by small conductivity $\sigma[e^{2}/\pi
h]\lesssim 1$.

At the critical point, we find that the distribution of the conductance is
scale invariant. In the range of sizes accessible to our study, this scale-
invariance remains approximately valid for a range of $\delta$ on the metallic
side of the transition.

In the immediate vicinity of the transition in the insulating side, the
distribution shows a significant deviation from the critical distribution.
However, the expected log-gaussian distribution of log-conductance is yet to
develop fully in our calculation.

Recently, similar transport statistics have been studied in quasi-1$d$
armchair graphene with bond disorder [33]. In this work, the focus was to
understand the crossover phenomenon of the quasi-localized (chiral) critical
point to an exponentially localized regime by two parameters scaling with
respect to the energy and the system sizes. On the contrary, we are in the 2D
limit, where the shape of the conductance fluctuations remains log-Gaussian
across the phase transition unlike in the 1D model. In particular, we observe
system size-independent conductance fluctuations close to the critical point,
which is absent in the quasi-1D limit.

In the future, it would be relevant to investigate the effects of temperature
and interaction on the chiral symmetric critical point. Exploring the
stability of the phase transition and the interaction effects on the
universality of the exponent would be important to investigate. Similarly, it
would be important to understand the origin of such non-universality in the
exponent for different dimerization patterns, possibly, using the field
theoretical framework that has been developed in Ref. [26] and in particular,
when the symmetry of lattice rotations is broken by the dimerization pattern.

##  V ACKNOWLEDGMENTS

We are grateful to Sasha Mirlin for pointing out a crucial numerical error in
an earlier version of the manuscript, and for several discussions. SB
acknowledges support from SERB-DST, India, through Matrics (No.
MTR/2019/000566), and MPG for funding through the Max Planck Partner Group at
IITB. NN would like to thank DST-INSPIRE fellowship No. IF- 190078 for
funding. KD was supported at the TIFR by DAE, India, and in part by a J.C.
Bose Fellowship (JCB/2020/000047) of SERB, DST India, and by the Infosys-
Chandrasekharan Random Geometry Center (TIFR).

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##  Appendix A Transfer Matrix and Rigid Spectrum

Figure 7: Shows the 2D square lattice geometry for transfer matrix
calculation. Blue and gray circles represent two different sub-lattices in
this bipartite lattice.

Here we discuss the rigid shift in Lyapunov spectrum observed in Fig. 2 in
detail. Figure 7 shows a part of a 2D square lattice with vertical slices
denoted by $(n-1,n,n+1)$ and horizontal slices are denoted by $(a,b,c)$. The
hopping $t^{b}_{n,n+1}$ defines a connection between the lattice points
$(n,b)$ and $(n+1,b)$.

The wave-amplitudes at the slice $n+1$ at $E=0$ are given by the following
transfer matrix [27]

$\displaystyle\left(\begin{array}[]{c}\psi_{n+1}\\\
\psi_{n}\end{array}\right)=\left(\begin{array}[]{cc}\frac{-h_{n,n}}{t_{n,n+1}}&-\frac{t_{n,n-1}}{t_{n,n+1}}\\\
\mathds{1}&0\end{array}\right)\left(\begin{array}[]{c}\psi_{n}\\\
\psi_{n-1}\end{array}\right),$ (9)

where $h_{n,n}$ is the Hamiltonian of the $n$-th slice. In a bipartite
lattice, the sites can be separated into A (blue circle) and B sites (grey
circle) (see Fig. 7). For the nearest neighbour Hamiltonian defined in Eq. 1,
the transfer matrix equation (9) decouples in individual sublattices at $E=0$
and the wavefunction transfer is restricted to one sublattice. The transfer
matrix in Eq. (9) gives the wave amplitude at the lattice site $(n+1,b)$ as

$\displaystyle\psi_{n+1,b}{=}-\frac{t^{a,b}_{n}}{t^{b}_{n,n+1}}\psi_{n,a}-\frac{t^{b,c}_{n}}{t^{b}_{n,n+1}}\psi_{n,c}-\frac{t^{b}_{n-1,n}}{t^{b}_{n,n+1}}\psi_{n-1,b}$
(10)

In particular, for the brickwall (equivalent to honeycomb) lattice the bond
$t^{b}_{n-1,n}=0$ (see e.g. Fig. 1) and the Eq. (10) reduces to

$\displaystyle\psi_{n+1,b}=-\frac{t^{a,b}_{n}}{t^{b}_{n,n+1}}\psi_{n,a}-\frac{t^{b,c}_{n}}{t^{b}_{n,n+1}}\psi_{n,c}$
(11)

i.e., the transfer only involves two slices $(n,n+1)$ in one iteration. Here,
the bonds are distributed equally between $[0,1]$ with an extra multiplicative
factor $e^{\delta}$ for the horizontal bonds. Therefore, the multiplicative
factor comes out of the transfer matrix multiplication, which is the rigid
shift of the spectrum as seen in Fig. 2. The magnitude of the shift is
proportional to $\delta$ and in this sense the exponent is $\nu=1.0$ by
construction as explained in the main text.

In contrast, for the square lattice, the wave amplitude $\psi_{n+1,b}$
involves three slices at one iteration of transfer and the spectrum is non-
rigid (not shown here). In this scenario, it is not entirely obvious that
$\nu\approx 1$ and therefore, we resort to numerical simulation.

##  Appendix B Finite size scaling analysis

Table 1: Finite size analysis for Transfer matrix data of staggered dimerized
graphene as shown in Fig 3(a)

$L$ |  $n_{R}$ |  $n_{I}$ |  $m_{R}$ |  $m_{I}$ |  N |  $\delta_{c}$ |  $\nu$ |  GOF |
---|---|---|---|---|---|---|---|---|---
32-384 |  6 |  0 |  1 |  0 |  10 |  $1.430\pm 0.002$ |  $1.07\pm 0.04$ |  0.03 |
32-384 |  6 |  0 |  2 |  0 |  11 |  $1.431\pm 0.001$ |  $1.05\pm 0.05$ |  0.04 |
32-384 |  4 |  0 |  1 |  0 |  8 |  $1.428\pm 0.005$ |  $1.13\pm 0.05$ |  0.16 |
32-384 |  4 |  0 |  2 |  0 |  9 |  $1.431\pm 0.002$ |  $1.10\pm 0.06$ |  0.13 |
32-384 |  4 |  1 |  1 |  0 |  11 |  $1.430\pm 0.005$ |  $1.04\pm 0.04$ |  0.46 |
64-384 |  4 |  1 |  1 |  0 |  11 |  $1.430\pm 0.003$ |  $1.04\pm 0.09$ |  0.51 |
128-384 |  4 |  1 |  1 |  0 |  11 |  $1.431\pm 0.003$ |  $1.03\pm 0.05$ |  0.43 |

Table 2: Finite size analysis for conductance data in staggered dimerized
graphene as shown Fig 3(c)

$L$ |  $n_{R}$ |  $n_{I}$ |  $m_{R}$ |  $m_{I}$ |  N |  $\delta_{c}$ |  $\nu$ |  GOF |
---|---|---|---|---|---|---|---|---|---
32-112 |  6 |  0 |  1 |  0 |  10 |  $1.457\pm 0.002$ |  $1.11\pm 0.04$ |  0.45 |
32-112 |  4 |  0 |  1 |  0 |  8 |  $1.456\pm 0.002$ |  $1.12\pm 0.05$ |  0.46 |
48-112 |  4 |  1 |  1 |  0 |  11 |  $1.455\pm 0.003$ |  $1.15\pm 0.09$ |  0.39 |
64-112 |  4 |  1 |  1 |  0 |  11 |  $1.456\pm 0.003$ |  $1.13\pm 0.07$ |  0.29 |

Table 3: Finite size scaling analysis table for data in Fig 3. $N$ is the
number of parameters to fit. The critical parameters $(\delta_{c},\nu)$ are
found by fitting the data in Fig. (3) to the scaling form in Eq. (14). The
bootstrap error has been obtained by repeated re-sampling (10000 data points)
of the original data and the error is the standard deviation of the resulting
sampling distribution. The goodness of fit parameter as defined in Eq 15 is
also given in the table. Table 4: Finite size analysis table for transfer
matrix data in plaquette dimerized graphene as shown in Fig 5(a)

$L$ |  $n_{R}$ |  $n_{I}$ |  $m_{R}$ |  $m_{I}$ |  N |  $\delta_{c}$ |  $\nu$ |  GOF |
---|---|---|---|---|---|---|---|---|---
30-384 |  6 |  0 |  1 |  0 |  10 |  $0.005\pm 0.002$ |  $1.64\pm 0.04$ |  0.82 |
30-384 |  4 |  0 |  1 |  0 |  8 |  $0.009\pm 0.003$ |  $1.58\pm 0.04$ |  0.26 |
30-384 |  4 |  0 |  2 |  0 |  9 |  $0.005\pm 0.002$ |  $1.55\pm 0.05$ |  0.76 |
66-384 |  4 |  1 |  1 |  0 |  11 |  $0.007\pm 0.003$ |  $1.54\pm 0.06$ |  0.28 |
126-384 |  4 |  1 |  1 |  0 |  11 |  $0.009\pm 0.003$ |  $1.59\pm 0.09$ |  0.43 |

Table 5: Finite size analysis for conductance data in staggered dimerized
graphene as shown Fig 5(c)

$L$ |  $n_{R}$ |  $n_{I}$ |  $m_{R}$ |  $m_{I}$ |  N |  $\delta_{c}$ |  $\nu$ |  GOF |
---|---|---|---|---|---|---|---|---|---
32-128 |  6 |  0 |  1 |  0 |  10 |  $0.005\pm 0.002$ |  $1.47\pm 0.04$ |  0.24 |
32-128 |  4 |  0 |  1 |  0 |  8 |  $0.003\pm 0.002$ |  $1.45\pm 0.05$ |  0.56 |
48-128 |  6 |  0 |  1 |  0 |  11 |  $0.015\pm 0.003$ |  $1.45\pm 0.09$ |  0.51 |
64-128 |  6 |  0 |  1 |  0 |  11 |  $0.010\pm 0.005$ |  $1.44\pm 0.05$ |  0.15 |
80-128 |  6 |  0 |  1 |  0 |  11 |  $0.009\pm 0.003$ |  $1.47\pm 0.08$ |  0.18 |

Table 6: Finite size scaling analysis for data in Fig. 5. The critical
parameters $(\delta_{c},\nu)$ are found by fitting the data in Fig. (5) to the
scaling form in Eq. (14). The bootstrap error has been obtained by repeated
re-sampling (10000 data points) of the original data and the error is the
standard deviation of the resulting sampling distribution. The goodness of fit
parameter as defined in Eq. (15) is also given in the table.

This section summarizes the finite size scaling analysis used to estimate
$\nu$ and $\delta_{c}$ in the localized phase of the model. We follow a
approach similar to [34, 31] and expand the scaling function in the leading
relevant ($\chi$) and irrelevant $(\phi)$ scaling variables as follows,

$\displaystyle\frac{\xi}{M}=\mathcal{F}\left(\chi(w)~{}M^{1/\nu},\phi(w)M^{y}\right),w=(\delta_{c}-\delta)/\delta_{c},$
(12)

where $\mathcal{F}$ is a generic scaling function and M is the width of the
quasi-1D sample. The irrelevant exponent $y<0$ characterizes correction to
finite-size scaling. The scaling variables are expanded in terms $w$ as,

$\displaystyle\chi(w)=\sum_{n=0}^{m_{R}}a_{n}w^{n},\phi(w)=\sum_{n=0}^{m_{I}}b_{n}w^{n},a_{0}=0.$
(13)

A further Taylor expansion of $\mathcal{F}$ leads to,

$\displaystyle\frac{\xi}{M}=\sum_{m}^{n_{R}}\sum_{n}^{n_{I}}a_{mn}\chi^{m}L^{m/\nu}\phi^{n}L^{ny}\mathbb{F}_{mn}.$
(14)

An expansion of Eq. (14) to order $(m_{R},m_{I},n_{R},n_{I})$ gives
$N=2+m_{R}+m_{I}+(n_{R}+1)(n_{I}+1)$ number of parameters to fit. An analogous
expansion also used for the observable $\langle\ln G\rangle$ used in the main
text. We use non-linear least squares to fit the function to the available
data. In the main text, we kept the order of expansion
$(m_{R},m_{I},n_{R},n_{I})=(1,0,4,0)$ to avoid over-fitting of the model. But
we checked the stability of the critical exponents to various other orders of
expansion $(m_{R},m_{I},n_{R},n_{I})$ and the results are summarized in table
LABEL:table1 along with the goodness of fit (GOF) defined as follows

$\mathrm{GOF}=\sum_{i=1}^{N_{D}}(F_{i}-\Gamma_{i})^{2}/\Gamma_{i},$ (15)

where $F_{i}$ is the scaling function evaluated at $i^{th}$ data point,
$\Gamma_{i}$ is numerically observed value at the same data point and $N_{D}$
is the number of such data points. The error bars are estimated using the
bootstrap resampling of the data set. The irrelevant exponent $y$ is found to
be rather large.


# Neutron-induced fission reaction studies with minor actinides using a double
Frisch grid fission detector

C. Yadav1 D. Ben Simhon1 M. Friedman1<EMAIL_ADDRESS>1Racah
Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

###### Abstract

Neutron-induced fission cross-section studies are being pursued with a double
Frisch grid fission detector. We are working towards conducting neutron-
induced fission cross-section measurements of high priority 241Am($n$, $f$)
and 244,245Cm($n$, $f$) fission reactions in the energy range of 1 – 2000 keV.
These measurements of high priority ($n$, $f$) reactions cross-sections from
the OECD Nuclear Energy Agency High Priority Request List (HPRL), are
identified as reactions that require improved data in the energy range of
1-2000 keV for the safe and economic design of next-generation fast nuclear
reactors. The 7Li($p$, $n$) reaction at various proton energies from 1.9 MeV
to 3.6 MeV will be used to produce neutron flux to conduct neutron-induced
fission cross-section studies. In the present special issue contribution
paper, we are presenting the characterization study of the double Frisch grid
fission detector. The detector is characterized with a 252Cf (spontaneous
fission) source. The chamber consists of two cylindrical ionization sections
in a back-to-back geometry sharing a common cathode. Each section consists of
a cathode, guard rings, a grid, and an anode. The grids are made as a wire
plane. The separation between the cathode and grid is 5.3 cm, and the grid to
the anode is 1.2 cm. The source is mounted on a common cathode, and the
chamber is operated at approximately 800 Torr pressure of P-10 gas mixture.
The data acquisition with the digital data acquisition module CAEN N6725 is
being set up and the preliminary analysis of the chamber performance suggests
that the detector operation is good and stable. Further work with regard to
employing trigger from the cathode and incorporating energy loss corrections
are being pursued.

###### keywords:

Neutron-induced fission studies, Double Frisch grid fission detector

##  1 Introduction

Neutron-induced fission reactions have been extensively studied since their
discovery for their importance in basic and applied nuclear sciences. In basic
nuclear physics, the fission cross sections provide important information on
the properties of nuclear matter and help in improving the theoretical
description of the fission process, while in nuclear technology they are at
the basis of present and future reactor designs. Furthermore, there is a
renewed interest in fission reactions in nuclear astrophysics, especially due
to the important role of fission in $r$-process nucleosynthesis.

Nuclear technology - The neutron-induced fission cross-section of the
actinides is highly significant in the criticality and safety evaluation in
nuclear reactors [1], transmutation of nuclear waste, and investigation of the
nuclear fuel cycle. The design of new and innovative reactors, for safer and
cleaner energy, also requires new and higher accuracy fission cross section
[2]. The new Generation-IV reactors are expected to have, among other
advantages, the capability to use nuclear waste from currently operating
reactors as fuel [3]. The waste that is planned to be used contains mainly
minor actinides, like plutonium, neptunium, americium, and curium isotopes.
Gen-IV systems are expected to incinerate these isotopes and minimize nuclear
waste. An alternative innovative nuclear system that can incinerate and
transmute long-lived nuclear waste is the sub-critical accelerator-driven
system (ADS), which uses high-energy accelerators to produce high-energy
neutrons, which allows the operation of the reactor in sub-critical
conditions. Apart from new reactor designs, the Th/U fuel cycle is considered
for future nuclear power options, due to the limited availability of natural
uranium resources. All the above solutions, all in their R $\&$ D phase, can
make nuclear power more attractive and efficient. The study of these future
systems requires accurate knowledge of the cross-section of all relevant
reactions, mainly the neutron-induced reactions of the minor actinides, at
energies from thermal to 10s of MeV. However, significant discrepancies exist
in the evaluated and measured cross-section data. The importance of accurate
nuclear data is described in the High Priority Request List (HPRL) of the
Nuclear Energy Agency (NEA) [4], and the required accuracies are summarized in
[1]. The HPRL identified several high-priority ($n$,$f$) reaction cross
sections where improved data is required. Some of those within the energy
range of 1-2000 keV, such as 241,242mAm($n$, $f$) and 244,245Cm($n$, $f$).
241Am (t1/2 = 433 y) for example is one of the most important fissionable
isotopes to be considered as a potential candidate for
incineration/transmutation which represents about 1.8% the actinide mass in
spent fuel [5].

Stellar nucleosynthesis - In explosive scenarios such as supernovae or neutron
star mergers (NSM) the $r$-process nucleosynthesis takes place. During the
$r$-process heavy nuclei, all the way up to unstable actinides, are formed.
Spontaneous and neutron-induced fission of these actinides produce fission
fragments that in turn act as seeds to a new cycle of rapid neutron capture
reactions. This process is called ”fission recycling” and it is considered
highly significant in the $r$-process abundance distribution of the heavy
elements. While the spontaneous $\beta$-delayed fission cannot be studied in
the laboratory, neutron-induced fission is experimentally possible.

The neutron spectrum produced via 7Li($p$, $n$)7Be reaction at a proton energy
of 1.912 MeV has been extensively studied and employed reaction for neutron-
induced reaction studies [6]. We are working towards using this reaction at
various proton energies from 1.9 MeV to 3.6 MeV to produce neutron flux in the
energy range of a few keV to around 2 MeV. An important advantage of this
approach is a substantial increase in the neutron intensity up to $\approx$109
neutrons$/$second for 100 $\mu$A proton beam. The intense proton beam from the
superconducting linear RF accelerator SARAF (Soreq Applied Research
Accelerator Facility) which is under upgrade (SARAF Phase II ) provides a beam
current of a few mA and thus allows measurements with even higher neutron
intensity on the order of 1010 neutrons$/$second [7, 8, 9]. The four-order of
magnitude increase in neutron intensity allows measurements of high-priority
($n$, $f$) reactions with more accuracy and precision.

A double Frisch grid fission detector [10, 11], courtesy of K. E. Rehm (ANL)
and M. Paul (HUJI) is being characterized to be used for these fission
reaction studies. Frisch grids ionization chambers have been very successfully
and widely used for neutron-induced ﬁssion-fragment studies during the past
three decades, for their relative simplicity, versatility, and radiation
hardness and for their capability in determining the energy and emission angle
of fission-fragments with relatively good resolution [12, 13, 14, 15, 16]. We
plan to conduct our experiments at the PTB, Germany facility with a 10-µA
proton beam at energies of 1.9-3.6 MeV on a thick Li target producing neutrons
via the 7Li($p$, $n$)7Be reaction. The target for the ($n$, $f$) reactions
will be placed in the detector. The details of the detector and
characterization study are described in the following sections ahead.

##  2 The detector and experimental set-up

The detector consists of two cylindrical ionization sections in a back-to-back
geometry sharing a common cathode. Each section consists of a cathode, guard
rings, a grid, and an anode. The grids are made as a wire plane. The
separation between the cathode and grid is 5.3 cm, and the grid to the anode
is 1.2 cm. The Frisch grid was produced by soldering 20 $\mu$m thick Au-plated
tungsten wires to an annular stainless steel frame. The spacing between the
wires is 1 mm. Fig. 1 shows a schematic illustration of the detector.

Figure 1: Schematic of a double Frisch grid fission detector

Figure 2: (Top) The picture of a double Frisch grid fission detector. (Bottom
left) A 252Cf (sf) source. (Bottom right) 252Cf(sf) source being assembled on
a common cathode Figure 3: The detector set-up.

The double Frisch grid fission detector is characterized with a 4000 Bq 252Cf
(spontaneous fission) source deposited on a 100 $\mu$g/cm2 carbon backing, a
kind contribution by the Actinide Target Preparation Laboratory at Oregon
State University. Fig. 2 shows the pictures of the fission detector along with
a 252Cf source and the source being assembled on a common cathode. The chamber
and gas handling system were then evacuated by pumping out using an nXDS
scroll pump. After reaching the suitable vacuum ($\simeq$ 0.004 Torr), gas
flow to the detector was started. The chamber is operated at approximately 800
Torr pressure of P-10 gas mixture in flow mode with a flow rate of around 50
cc/min. Once the required pressure and flow rate of P-10 gas were achieved,
voltages were applied gradually to the electrodes while observing the signals
from the pre-amplifier through the oscilloscope. In the present special issue
contribution paper, we are reporting our preliminary results for the voltage
configuration for the cathode -2200 V, the anode around +1200 V, and the grid
plane was kept grounded. Furthermore, to minimize electron losses on the
Frisch grid, the electric field in the grid-anode region is chosen to be of
increasing strength with respect to the cathode-grid region [17]. Fig. 3 shows
the picture of the detector testing set-up showing the detector wrapped in
copper sheets and experimental set-up.

##  3 Data acquisition and preliminary results

Figure 4: The schematic of the digital data acquisition scheme.  Figure 5:
Coincidence fission signals from the anodes.

The signals from the detector are fed to the 8-channel charge-sensitive pre-
amplifier CAEN A1422C-D. The output from the pre-amplifier is further fed to
the 8-channel 14-bit 250 MS/s waveform digitizer module CAEN N6725SB. Fig. 4
shows the schematic of the digital data acquisition scheme. Fig. 5 shows a
typical pre-amplifier coincidence fission signal from the anodes. The
Digitizer runs a DPP-PHA (Digital Pulse Processing for Pulse Height Analysis)
firmware. It uses an algorithm based on the Jordanov trapezoidal filter [18].
The trapezoidal filter transforms a signal such as an exponential decay input
signal from a charge-sensitive pre-amplifier into a trapezoid shape with a
flat top height directly proportional to the amplitude of the input pulse. The
role of the trapezoid filter is similar to that of a shaping amplifier in an
analog data acquisition system. The CoMPASS [19] digitizer user interface was
used for communicating with the DPP-PHA firmware. The interface input can be
categorized into three sections: Input Signal Settings, Trigger Settings
(Timing Filter), and Energy Filter Settings. The first settings to be
configured are those related to the input signal. While the trigger settings
determine the criteria for a signal to be recorded and the energy filter
settings control the trapezoidal fit for the incoming signals. Fig. 6 shows an
example of a typical waveform from a fission event for the source backing side
anode along with timing (top panel) and energy (bottom panel) filters
respectively. Fig. 7 shows pulse height distribution from the sample side
anode and the backing side anode respectively at the configuration of the
voltage mentioned in the previous section. Further work is being pursued with
regard to connecting all the channels of the detector and employing a trigger
from the cathode for data acquisition. Further work with regard to
incorporating energy loss corrections for the sample and backing side
spectrums is also being pursued.

Figure 6: An example of digitized far anode signal waveform with timing (top)
and energy (bottom) filters.

Figure 7: (Top) Pulse height distribution from the sample side anode. (Bottom)
Pulse height distribution from the backing side anode.

##  4 Summary and conclusions

Neutron-induced fission cross-section studies for 241Am($n$, $f$) and
244,245Cm($n$, $f$) reactions are being pursued. In the present special issue
contributory paper, our present work with regard to the testing and
characterization of a double Frisch grid fission detector with a 252Cf(SF)
source is discussed. The analysis of the chamber performance suggests that the
detector operation is good and stable and can be employed for neutron-induced
fission reaction studies. Further work with regard to connecting all the
channels of the detector and employing a trigger from the cathode and
incorporating the energy loss corrections is being pursued.

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