# On the distribution of powered numbers

J. Brüdern and O. Robert

###### Abstract.

Asymptotic formulae are established for the number of natural numbers $m$ with
largest square-free divisor not exceeding $m^{\vartheta}$, for any fixed
positive parameter $\vartheta$. Related counting functions are also
considered.

00footnotetext: Keywords: powered numbers, nuclear numbers00footnotetext:
MSC(2020): 11N25, 11N37

##  1\. Introduction

Motivated by questions about diophantine equations and the $abc$-conjecture,
Mazur [1] proposed to smooth out the set of positive $l$-th powers in a
multiplicative way, by what he named powered numbers. To introduce the latter,
let $k(m)$ denote the largest square-free divisor of the natural number $m$,
and let $i(m)={\log m}/{\log k(m)}$. For all $l\in\mathbb{N}$ one has
$i(m^{l})\geq l$. Mazur’s powered numbers (relative to $l$) are the numbers
$m\in\mathbb{N}$ with $i(m)\geq l$. Note here that $l$ need not be integral in
this definition, but for $l\in\mathbb{N}$ the powered numbers (relative to
$l$) contain the $l$-th powers. It is proposed in [1] to replace, within a
given diophantine equation, an $l$-th power by the corresponding powered
numbers, and to consider the resulting equation between powered numbers as the
associated “rounded” diophantine equation.

In this note we analyse the distribution of powered numbers. We find it is
more appropriate to work with the real number $\vartheta=1/l$. The condition
$i(m)\geq l$ is expressed equivalently as $k(m)\leq m^{\vartheta}$, and for
any $\vartheta>0$ we define the set

$\mathscr{A}(\vartheta)=\\{m\in{\mathbb{N}}\colon k(m)\leq m^{\vartheta}\\}.$

Thus, Mazur’s powered numbers (relative to $l$) are exactly the elements of
$\mathscr{A}(1/l)$. For analytic approaches to rounded diophantine problems it
is indispensable to determine the density of the set $\mathscr{A}(\vartheta)$.
Our principal goal are asymptotic formulae for the number $S_{\vartheta}(x)$
of elements in $\mathscr{A}(\vartheta)$ that do not exceed $x$, and for
related counting functions.

It is not difficult to see that for any $0<\vartheta<1$ the number
$S_{\vartheta}(x)$ obeys the inequalities

(1.1) $x^{\vartheta}\ll S_{\vartheta}(x)\ll x^{\vartheta+\varepsilon}$

whenever $\varepsilon$ is a given positive real number and $x$ is large in
terms of $\varepsilon$. It now transpires that for $\vartheta=1/l$ the powered
numbers are not much denser than the $l$-th powers. A weaker version of (1.1)
occurs in Mazur [1] who refers to Granville, showing him an “easy” argument
supposedly confirming the inequalities $x^{\vartheta-\varepsilon}\ll
S_{\vartheta}(x)\ll x^{\vartheta+\varepsilon}$. In Mazur’s article there is no
indication how this would go but the simplest argument we know allows one to
take $\varepsilon=0$ in the lower bound. To substantiate this claim, fix a
number $\vartheta\in(0,1)$. Define $l\in\mathbb{N}$ by $1/l<\vartheta\leq
1/(l-1)$, and $t\in\mathbb{R}$ by $l-t=1/\vartheta$. Then $t\in(0,1]$ and

(1.2) ${1}/{\vartheta}=t(l-1)+(1-t)l.$

Let $W(x)$ be the number of natural numbers $w$ that have a representation

(1.3) $w=8^{l}n^{l-1}m^{l}$

with $n,m$ square-free and constrained to the intervals

(1.4) $\frac{1}{2}\,\Big{(}\frac{x}{8^{l}}\Big{)}^{\vartheta
t}<n\leq\Big{(}\frac{x}{8^{l}}\Big{)}^{\vartheta
t},\quad\frac{1}{2}\,\Big{(}\frac{x}{8^{l}}\Big{)}^{\vartheta(1-t)}<m\leq\Big{(}\frac{x}{8^{l}}\Big{)}^{\vartheta(1-t)}.$

The numbers $w$ counted by $W(x)$ uniquely determine the square-free numbers
$n,m$ with (1.3) and (1.4). This follows from unique factorisation. Hence
$W(x)$ equals the number of pairs $n,m$ of square-free numbers satisfying
(1.4). In particular, we see that $W(x)\gg x^{\vartheta}$.

Next, we observe that (1.3) gives $k(w)\leq 2nm$, and therefore $k(w)\leq
2(8^{-l}x)^{\vartheta}$. By (1.2) and (1.4), we also have $w\geq
2^{-(2l-1)}x$. It follows that $k(w)\leq w^{\vartheta}$. Hence, numbers
counted by $W(x)$ are also counted by $S_{\vartheta}(x)$. This yields
$S_{\vartheta}(x)\geq W(x)$, and the left inequality in (1.1) follows at once.

For the upper bound in (1.1) one may refer to authorities like Tenenbaum [3,
Theorem II.1.15]. However, the shortest argument available to us uses Rankin’s
trick within the following chain of obvious inequalities:

$S_{\vartheta}(x)\leq\sum_{{\scriptstyle m\leq x}\atop{\scriptstyle k(m)\leq
x^{\vartheta}}}1\leq\sum_{m=1}^{\infty}\Big{(}\frac{x^{\vartheta}}{k(m)}\Big{)}\Big{(}\frac{x}{m}\Big{)}^{\varepsilon}\ll
x^{\vartheta+\varepsilon}.$

A finer analysis of the counting function $S_{\vartheta}(x)$ is currently an
undesired lacuna in the literature. As a partial remedy we establish an
asymptotic formula that should be sufficient for many applications. Let $1\leq
y\leq x$. Following Robert and Tenenbaum [2] we write $N(x,y)$ for the number
of $m\in\mathbb{N}$ with $m\leq x$ and $k(m)\leq y$. The numbers counted here
are their nuclear numbers. It seems natural to expect that the trivial upper
bound $S_{\vartheta}(x)\leq N(x,x^{\vartheta})$ should not be very wasteful.
However, it was conjectured by Erdös (1962) and proved by De Bruijn and van
Lint (1963) that

(1.5) $\sum_{m\leq x}\frac{m}{k(m)}=o\Big{(}\sum_{m\leq
x}\frac{x}{k(m)}\Big{)}\qquad(x\to+\infty).$

For a quantitative version of this estimate, see [2, Théorème 4.4]. These
results suggest that the order of magnitude of $S_{\vartheta}(x)$ is somewhat
smaller than that of $N(x,x^{\vartheta})$, and this is indeed the case.

Before we make this precise, we recall an estimate for $N(x,y)$. We are
primarily interested in the case $y=x^{\vartheta}$ with $\vartheta>0$, but we
work in the wider range $y>\exp\big{(}(\log x)^{2/3}\big{)}$. The
multiplicative function

(1.6) $\psi(m)=\prod_{p\mid m}(p+1)\qquad(m\geq 1)$

and the function $F:[0,\infty)\to[0,\infty)$ defined by

$F(v)=\frac{6}{\pi^{2}}\sum_{m\geq
1}\frac{1}{\psi(m)}\min\Big{(}1,\frac{e^{v}}{m}\Big{)}$

are featured in the uniform asymptotic formula

(1.7)
$N(x,y)=\big{(}1+o(1)\big{)}yF\big{(}\log(x/y)\big{)}\qquad\big{(}y>\exp\big{(}(\log
x)^{2/3}\big{)},\thinspace x\to+\infty\big{)}$

that is contained in [2, Proposition 10.1]. As we shall see momentarily, for
each pair $\vartheta,x$ with $0<\vartheta<1$ and $x\geq 2$, there is exactly
one real number $\alpha=\alpha_{\vartheta}(x)>0$ with

(1.8) $\sum_{p}\frac{p^{\alpha}\log
p}{(p^{\alpha}-1)\big{(}1+(p+1)(p^{\alpha}-1)\big{)}}=(1-\vartheta)\log x.$

We are now in a position to state our first result.

######  Theorem 1.

Let $0<\vartheta<1$ fixed. Then, for $x\geq 27$, one has

(1.9) $S_{\vartheta}(x)=x^{\vartheta}F\big{(}(1-\vartheta)\log
x\big{)}\frac{\alpha_{\vartheta}(x)}{\vartheta}\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}\Big{)}\Bigg{)}.$

As $x\to\infty$, one also has

(1.10) $S_{\vartheta}(x)=(1+o(1))x^{\vartheta}F\big{(}(1-\vartheta)\log
x\big{)}\frac{1}{\vartheta}\Big{(}\frac{2}{1-\vartheta}\Big{)}^{1/2}\big{(}(\log
x)\log\log x\big{)}^{-1/2}.$

This result calls for several comments. First we take $y=x^{\vartheta}$ in
(1.7) and substitute the resulting equation

(1.11) $N(x,x^{\vartheta})=(1+o(1))x^{\vartheta}F\big{(}(1-\vartheta)\log
x\big{)}\qquad(x\to+\infty)$

within (1.10) to infer that

$S_{\vartheta}(x)=(1+o(1))N(x,x^{\vartheta})\frac{1}{\vartheta}\Big{(}\frac{2}{1-\vartheta}\Big{)}^{1/2}\big{(}(\log
x)\log\log x\big{)}^{-1/2}\quad(x\to+\infty).$

This is an analogue of (1.5), in quantitative form that is of strength
comparable to [2, Théorème 4.4].

Our second comment concerns the implicitly defined function
$\alpha_{\vartheta}(x)$. It originates in the Dirichlet series

(1.12) $\mathscr{G}(s)=\frac{6}{\pi^{2}}\sum_{m\geq
1}\frac{1}{\psi(m)m^{s}}=\frac{6}{\pi^{2}}\prod_{p}\left(1+\frac{1}{(p+1)(p^{s}-1)}\right)$

that converges absolutely in $\text{Re}\,s>0$, and therefore has no zeros in
this half plane. For real numbers $\sigma>0$ we have $\mathscr{G}(\sigma)>0$.
We may then define

$g(\sigma)=\log\mathscr{G}(\sigma).$

Note that $g$ extends to a holomorphic function on the right half plane, and
one computes the logarithmic derivative of $\mathscr{G}(s)$ from the Euler
product representation (1.12) to

$g^{\prime}(s)=-\sum_{p}\frac{p^{s}\log
p}{(p^{s}-1)\big{(}1+(p+1)(p^{s}-1)\big{)}}.$

Note that the sum on the right hand side here coincides with the sum in (1.8).
On differentiating again, it transpires that the real function
$g^{\prime}:(0,\infty)\to\mathbb{R}$ is increasing. Considering $\sigma\to 0$
and $\sigma\to\infty$ one finds that its range is the open interval
$(-\infty,0)$. We conclude that for a given $v>0$ there is a unique positive
number $\sigma_{v}$ with

$g^{\prime}(\sigma_{v})+v=0.$

In [2, Lemme 6.6] it is shown that

(1.13) $\sigma_{v}=(1+o(1))\sqrt{\frac{2}{v\log v}}\qquad(v\to+\infty).$

Here we choose $v=(1-\vartheta)\log x$ and then have
$\alpha_{\vartheta}(x)=\sigma_{v}$. In particular, we see that (1.9) and
(1.13) imply (1.10). Thus, it only remains to prove (1.9).

Our last comment concerns the actual size of $S_{\vartheta}(x)$. This requires
some more information on the function $F$. From [2, (2.12)] we have the
asymptotic relation

(1.14) $\log F(v)=\big{(}1+o(1)\big{)}\sqrt{\frac{8v}{\log
v}}\qquad(v\to+\infty).$

By inserting (1.14) into (1.10), we deduce that there exists some positive
number $\beta(x;\vartheta)>0$ with

(1.15)
$S_{\vartheta}(x)=x^{\vartheta+\beta(x;\vartheta)}\qquad(0<\vartheta<1,\thinspace
x\geq x_{0}(\vartheta))$

and the property that for any fixed $\vartheta\in(0,1)$ one has

$\beta(x;\vartheta)=(1+o(1))\sqrt{\frac{8(1-\vartheta)}{(\log x)(\log\log
x)}}\qquad(x\to+\infty).$

Note that (1.15) yields another proof of (1.1).

We now turn to local estimates for $S_{\vartheta}(x)$. In our case, this
amounts to comparing the respective behaviour of $S_{\vartheta}(zx)$ and
$S_{\vartheta}(x)$ uniformly for large $x$, when $z$ is in some sense
sufficiently close to $1$. Such estimates are often obtained with the saddle-
point method, and we follow this route here, too. In a suitable range for $z$,
the fraction $S_{\vartheta}(zx)/S_{\vartheta}(x)$ may be approximated by a
simple function of $z$.

######  Theorem 2.

Let $0<\vartheta<1$. Then for $x$ large, we have

$S_{\vartheta}(zx)=z^{\vartheta}S_{\vartheta}(x)\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}\Big{)}\Bigg{)}$

uniformly for $z>0$ with $|\log z|\ll\log\log x$.

Finally, we consider the counting function for a variation of the powered
numbers. For given $\vartheta\in(0,1)$ and $\Theta\in{\mathbb{R}}$, we
consider

$S_{\vartheta,\Theta}(x)=\\#\\{n\leq x\colon k(n)\leq n^{\vartheta}(\log
n)^{\Theta}\\}.$

Note that $S_{\vartheta}(x)=S_{\vartheta,0}(x)$. The set of integers such that
$k(n)\leq n^{\vartheta}(\log n)^{\Theta}$ plays a prominent role in a
forthcoming paper, and therefore, we provide an estimate for
$S_{\vartheta,\Theta}(x)$. It turns out that the conditions $k(n)\leq
n^{\vartheta}$ and $k(n)\leq n^{\vartheta}(\log n)^{\Theta}$ are relatively
close, and that the ratio $S_{\vartheta,\Theta}(x)/S_{\vartheta}(x)$ is
roughly of size $(\log x)^{\Theta}$.

######  Theorem 3.

Let $0<\vartheta<1$ and $\Theta\in{\mathbb{R}}$ be fixed. Then for $x$ large,
one has

$S_{\vartheta,\Theta}(x)=(\log
x)^{\Theta}S_{\vartheta}(x)\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log x}{\log
x}}\Big{)}\Bigg{)}.$

##  2\. Proof of Theorem 1

In this section we derive Theorem 1. Before we embark on the main argument, we
fix some notation and recall a pivotal result concerned with the distribution
of square-free numbers. This involves the function $\psi(m)$ as defined in
(1.6), the Möbius function $\mu(m)$, and for a parameter
$0\leq\gamma\leq\frac{1}{2}$ at our disposal, the product

$r_{\gamma}(m)=\prod_{p\mid m}\big{(}1+4\gamma p^{-1/2}\big{)}.$

One then has the estimate ([2, (10.1)])

(2.1) $\sum_{l\leq
z}\mu^{2}(lk)=\frac{6kz}{\pi^{2}\psi(k)}+O\big{(}r_{\gamma}(k)z^{1-\gamma}\big{)}$

that holds uniformly relative to the square-free number $k$ and the real
parameters $z,\gamma$ in the ranges $z\geq 1$, $0\leq\gamma\leq\frac{1}{2}$.

The first steps of our argument follow the pattern laid out in [2, Sect. 10].
Unique factorisation shows that for all natural numbers $n$ there exists
exactly one pair of coprime natural numbers $l,m$ with $\mu(l)^{2}=1$ and
$n=lmk(m)$. Note that the two conditions $(l,m)=1$ and $\mu(l)^{2}=1$ are
equivalent to the single condition $\mu(lk(m))^{2}=1$. Further, one has
$k(n)=lk(m)$. With $\vartheta\in(0,1)$ now fixed, it follows that
$S_{\vartheta}(x)$ equals the number of $(l,m)\in\mathbb{N}^{2}$ satisfying
the conditions

$\mu(lk(m))^{2}=1,\quad lmk(m)\leq x,\quad lk(m)\leq(lmk(m))^{\vartheta}.$

These last three conditions we recast more compactly as

(2.2) $\mu(lk(m))^{2}=1,\quad lk(m)\leq\min(x/m,m^{\vartheta/(1-\vartheta)}).$

From now on, the number $\kappa=\vartheta/(1-\vartheta)$ features prominently,
and we also put $y=x^{\vartheta}$. Note that

(2.3) $\min(x/m,m^{\kappa})=m^{\kappa}\;\;\text{ if and only if }\,\,m\leq
x/y.$

Hence, we consider the ranges $m\leq x/y$ and $x/y<m\leq x$ separately. By
(2.2) and (2.3), this leads to the decomposition

(2.4) $S_{\vartheta}(x)=S_{\vartheta}^{(1)}(x)+S_{\vartheta}^{(2)}(x)$

in which

$S_{\vartheta}^{(1)}(x)=\\!\sum_{{\scriptstyle m\leq x/y}\atop{\scriptstyle
k(m)\leq m^{\kappa}}}\sum_{l\leq m^{\kappa}/{k(m)}}\\!\mu(lk(m))^{2},\quad
S_{\vartheta}^{(2)}(x)=\\!\sum_{{\scriptstyle x/y<m\leq x}\atop{\scriptstyle
mk(m)\leq x}}\sum_{l\leq{x}/mk(m)}\\!\mu(lk(m))^{2}.$

We apply (2.1) with $k=k(m)$ to both inner sums and obtain

(2.5) $\displaystyle S_{\vartheta}^{(1)}(x)$
$\displaystyle=\frac{6}{\pi^{2}}\sum_{{\scriptstyle m\leq
x/y}\atop{\scriptstyle k(m)\leq
m^{\kappa}}}\frac{m^{\kappa}}{\psi(m)}+O(R_{1}),$ (2.6) $\displaystyle
S_{\vartheta}^{(2)}(x)$ $\displaystyle=\frac{6}{\pi^{2}}\sum_{{\scriptstyle
x/y<m\leq x}\atop{\scriptstyle mk(m)\leq x}}\frac{x}{m\psi(m)}+O(R_{2}).$

where

$R_{1}=\sum_{{\scriptstyle m\leq x/y}\atop{\scriptstyle k(m)\leq
m^{\kappa}}}r_{\gamma}(m)\Big{(}\frac{m^{\kappa}}{k(m)}\Big{)}^{1-\gamma},\qquad
R_{2}=\sum_{{\scriptstyle x/y<m\leq x}\atop{\scriptstyle mk(m)\leq
x}}r_{\gamma}(m)\Big{(}\frac{x}{mk(m)}\Big{)}^{1-\gamma}.$

It turns out that $R_{1}$ and $R_{2}$ are small. In order to couch their
estimation, as well as the analysis of other error terms that arise later,
under the umbrella of a single treatment, we choose parameters $\gamma$ and
$\sigma$ with $0<\gamma<\sigma\leq\frac{1}{2}$ and introduce the series

(2.7)
$E=\sum_{m=1}^{\infty}r_{\gamma}(m)\Big{(}\frac{y}{k(m)}\Big{)}^{1-\gamma}\Big{(}\frac{x/y}{m}\Big{)}^{\sigma}.$

The conditions $\sigma>\gamma>0$ ensure convergence. It is routine to show
that

$R_{1}+R_{2}\ll E.$

In fact, by Rankin’s trick,

$R_{1}\leq\sum_{{\scriptstyle m\leq x/y}\atop{\scriptstyle k(m)\leq
m^{\kappa}}}r_{\gamma}(m)\Big{(}\frac{m^{\kappa}}{k(m)}\Big{)}^{1-\gamma}\Big{(}\frac{x/y}{m}\Big{)}^{\sigma}.$

For $m\leq x/y$ one has $m^{\kappa}\leq y$, and it follows that $R_{1}\leq E$.

Likewise, one confirms $R_{2}\leq E$ by observing that
$1-\gamma>\frac{1}{2}\geq\sigma$ so that

(2.8)
$\Big{(}\frac{x}{mk(m)}\Big{)}^{1-\gamma}=\Big{(}\frac{y}{k(m)}\Big{)}^{1-\gamma}\Big{(}\frac{x/y}{m}\Big{)}^{1-\gamma}\leq\Big{(}\frac{y}{k(m)}\Big{)}^{1-\gamma}\Big{(}\frac{x/y}{m}\Big{)}^{\sigma}.$

The appearance of $k(m)$ in the summation conditions on the right hand sides
of (2.5) and (2.6) is a nuisance, and we proceed by removing these. If the
condition $k(m)\leq m^{\kappa}$ is removed from the sum in (2.5) one imports
an error no larger than

$\sum_{{\scriptstyle m\leq x/y}\atop{\scriptstyle
k(m)>m^{\kappa}}}\frac{m^{\kappa}}{\psi(m)}\leq\sum_{{\scriptstyle m\leq
x/y}\atop{\scriptstyle k(m)>m^{\kappa}}}\frac{m^{\kappa}}{k(m)}\leq\sum_{m\leq
x/y}\Big{(}\frac{m^{\kappa}}{k(m)}\Big{)}^{1-\gamma}\Big{(}\frac{x/y}{m}\Big{)}^{\sigma}\leq
E.$

Here, the last inequality is obtained by the argument that completed the
estimation of $R_{1}$. Similarly, if the condition $mk(m)\leq x$ is removed
from the summation condition in (2.6), then the resulting error does not
exceed

$\sum_{{\scriptstyle x/y<m\leq x}\atop{\scriptstyle
mk(m)>x}}\frac{x}{m\psi(m)}\leq\sum_{{\scriptstyle x/y<m\leq
x}\atop{\scriptstyle mk(m)>x}}\frac{x}{mk(m)}\leq\sum_{{\scriptstyle
m>x/y}\atop{\scriptstyle mk(m)>x}}\frac{x}{mk(m)}=R,$

say. By Rankin’s trick and (2.8),

$R\leq\sum_{m>x/y}\Big{(}\frac{x}{mk(m)}\Big{)}^{1-\gamma}\leq\sum_{m>x/y}\Big{(}\frac{y}{k(m)}\Big{)}^{1-\gamma}\Big{(}\frac{x/y}{m}\Big{)}^{\sigma}\leq
E.$

Collecting together, we deduce from (2.5) and (2.6) the asymptotic relations

$S_{\vartheta}^{(1)}(x)=\frac{6}{\pi^{2}}\sum_{m\leq
x/y}\frac{m^{\kappa}}{\psi(m)}+O(E),\qquad
S_{\vartheta}^{(2)}(x)=\frac{6}{\pi^{2}}\sum_{x/y<m\leq
x}\frac{x}{m\psi(m)}+O(E),$

and by (2.4), we infer that

$S_{\vartheta}(x)=\frac{6}{\pi^{2}}\sum_{m\leq
x}\frac{m^{\kappa}}{\psi(m)}\min(1,xm^{-\kappa-1})+O(E).$

Note that the sum on the right is a partial sum of a convergent series. If one
completes the sum, then it is immediate that the error thus imported is
bounded by $R$, and hence by $E$. We have now reached the provisional
expansion

(2.9)
$S_{\vartheta}(x)=\frac{6}{\pi^{2}}\sum_{m=1}^{\infty}\frac{m^{\kappa}}{\psi(m)}\min(1,xm^{-\kappa-1})+O(E).$

It remains to estimate $E$. In its definition (2.7), we encounter a sum over a
multiplicative function, and so

$\displaystyle E=$ $\displaystyle
y^{1-\gamma}(x/y)^{\sigma}\prod_{p}\Big{(}1+\frac{p^{\gamma}r(p;\gamma)}{p(p^{\sigma}-1)}\Big{)}$
$\displaystyle\leq$ $\displaystyle
y^{1-\gamma}(x/y)^{\sigma}\prod_{p}\Big{(}1+\frac{p^{\gamma}}{p(p^{\sigma}-1)}\Big{)}\prod_{p}\Big{(}1+\frac{4\gamma
p^{\gamma-\frac{1}{2}}}{p(p^{\sigma}-1)}\Big{)}.$

Now, since $p^{\gamma-\frac{1}{2}}\leq 1$, $\gamma<\sigma$,
$p^{\sigma}-1\geq\sigma\log p$, and since the sum $\sum_{p}\frac{1}{p\log p}$
converges,

$\prod_{p}\Big{(}1+\frac{4\gamma
p^{\gamma-\frac{1}{2}}}{p(p^{\sigma}-1)}\Big{)}\leq\prod_{p}\Big{(}1+\frac{4}{p\log
p}\Big{)}\ll 1.$

As on an earlier occasion, we write $v=(1-\vartheta)\log x=\log\frac{x}{y}$,
and then choose $\sigma=\sigma_{v}$ and $\gamma=\sigma_{v}-\frac{1}{\log y}$.
By (1.12) one has

$\prod_{p}\Big{(}1+\frac{p^{\gamma}}{p(p^{\sigma}-1)}\Big{)}=\frac{\pi^{2}}{6}\mathscr{G}(\sigma)\prod_{p}\Big{(}1+\frac{(p^{\gamma}-1)(p+1)+1}{p+p(p+1)(p^{\sigma}-1)}\Big{)}.$

Now, on recalling (1.13),

$\prod_{p\leq
e^{1/\sigma}}\Big{(}1+\frac{(p^{\gamma}-1)(p+1)+1}{p+p(p+1)(p^{\sigma}-1)}\Big{)}\leq\prod_{p\leq
e^{1/\sigma}}\Big{(}1+\frac{1}{p}\Big{)}\ll\frac{1}{\sigma}\ll(v\log v)^{1/2}$

while one also has

$\prod_{p>e^{1/\sigma}}\Big{(}1+\frac{(p^{\gamma}-1)(p+1)+1}{p+p(p+1)(p^{\sigma}-1)}\Big{)}\leq\prod_{p>e^{1/\sigma}}\left(1+\frac{1}{p^{1+\sigma-\gamma}}\right)\ll\zeta\Big{(}1+\frac{1}{\log
y}\Big{)}\ll\log y.$

On collecting together, this shows that

$E\ll y^{1-\sigma_{v}}e^{v\sigma_{v}}\mathscr{G}(\sigma_{v})(v\log
v)^{1/2}\log y.$

From [2, (2.11)] we deduce that

$F(v)\asymp\Big{(}\frac{\log
v}{v}\Big{)}^{1/4}e^{v\sigma_{v}}\mathscr{G}(\sigma_{v})\qquad(v\geq 2),$

and hence

$E\ll y^{1-\sigma_{v}}F(v)v^{3/4}(\log v)^{1/4}\log y.$

With the choice of $y$ and $v$, one has $\sigma_{v}=\alpha_{\vartheta}(x)$.
Moreover, $\log y$ and $v$ have the order of magnitude $\log x$ so that the
last inequality now reads

(2.10) $E\ll x^{\vartheta}F((1-\vartheta)\log
x)x^{-\vartheta\alpha_{\vartheta}(x)}(\log x)^{7/4}(\log\log x)^{1/4}.$

Our final task is to compare our estimate for $E$ with the size of the sum on
the right of (2.9).

Recall that in view of (1.1) and (1.11), $S_{\vartheta}(x)$ and
$N(x,x^{\vartheta})$ are of comparable size. In order to mimick the estimate
(1.11), we introduce the function

$H_{\vartheta}(x)=\frac{6}{\pi^{2}x^{\vartheta}}\sum_{m=1}^{\infty}\frac{m^{\kappa}}{\psi(m)}\min(1,xm^{-\kappa-1})$

so that (2.9) now reads

$S_{\vartheta}(x)=x^{\vartheta}H_{\vartheta}(x)+O(E).$

Our aim is to give an estimate of $H_{\vartheta}(x)$ by using the saddle-point
method, and to describe more precisely $H_{\vartheta}(x)/F((1-\vartheta)\log
x)$ as $x\to+\infty$.

######  Lemma 1.

Fix $\vartheta$ with $0<\vartheta<1$. Then for $x\geq 27$ one has

$H_{\vartheta}(x)=F((1-\vartheta)\log
x)\frac{\alpha_{\vartheta}(x)}{\vartheta}\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}\Big{)}\Bigg{)}.$

Proof. The argument is modelled on [2, Section 8]. Recall the identity

$\frac{1}{2\pi\mathrm{i}}\int_{\sigma+\mathrm{i}{\mathbb{R}}}\frac{y^{s}}{s(1-s)}\,{\mathrm{d}}s=\min(1,y),\qquad(0<\sigma<1,\thinspace
y>0).$

We then have the integral representation

$\displaystyle H_{\vartheta}(x)$
$\displaystyle=\frac{1}{x^{\vartheta}}\frac{6}{\pi^{2}}\sum_{m\geq
1}\frac{m^{\kappa}}{\psi(m)}\frac{1}{2\pi\mathrm{i}}\int_{\sigma+\mathrm{i}{\mathbb{R}}}\frac{1}{s(1-s)}\left(\frac{x}{m^{1/(1-\vartheta)}}\right)^{s}{\mathrm{d}}s$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{\sigma+\mathrm{i}{\mathbb{R}}}\mathscr{G}\Big{(}\frac{s-\vartheta}{1-\vartheta}\Big{)}x^{s-\vartheta}\frac{{\mathrm{d}}s}{s(1-s)}$
$\displaystyle=\frac{1}{2\pi}\int_{{\mathbb{R}}}\mathscr{G}\Big{(}\frac{\sigma+\mathrm{i}t-\vartheta}{1-\vartheta}\Big{)}x^{\sigma+\mathrm{i}t-\vartheta}\frac{{\mathrm{d}}t}{(\sigma+\mathrm{i}t)(1-\sigma-\mathrm{i}t)}.$

After a linear change of variable in $t$, we arrive at

$H_{\vartheta}(x)=\frac{1}{2\pi}\int_{{\mathbb{R}}}\mathscr{G}\Big{(}\frac{\sigma-\vartheta}{1-\vartheta}+\mathrm{i}t\Big{)}x^{\sigma+\mathrm{i}(1-\vartheta)t-\vartheta}\frac{(1-\vartheta)\,{\mathrm{d}}t}{(\sigma+\mathrm{i}(1-\vartheta)t)(1-\sigma-\mathrm{i}(1-\vartheta)t)}.$

Recall again that $v=(1-\vartheta)\log x$, and that
$\alpha_{\vartheta}(x)=\sigma_{v}$. We take
$\sigma=\vartheta+(1-\vartheta)\sigma_{v}$. For large $x$ one then has
$0<\sigma<1$, and the previous formula for $H_{\vartheta}(x)$ becomes

$H_{\vartheta}(x)=\frac{1}{2\pi}\int_{{\mathbb{R}}}\frac{\mathscr{G}(\sigma_{v}+\mathrm{i}t)e^{(\sigma_{v}+\mathrm{i}t)v}}{\big{(}\vartheta+(1-\vartheta)(\sigma_{v}+\mathrm{i}t)\big{)}\big{(}1-\sigma_{v}-\mathrm{i}t\big{)}}\,{{\mathrm{d}}t}.$

After truncation, we have

$H_{\vartheta}(x)=\frac{1}{2\pi}\int_{-v^{2}}^{v^{2}}\frac{\mathscr{G}(\sigma_{v}+\mathrm{i}t)e^{(\sigma_{v}+\mathrm{i}t)v}}{\big{(}\vartheta+(1-\vartheta)(\sigma_{v}+\mathrm{i}t)\big{)}\big{(}1-\sigma_{v}-\mathrm{i}t\big{)}}\,{\mathrm{d}}t+O\left(\frac{\mathscr{G}(\sigma_{v})e^{v\sigma_{v}}}{v^{2}}\right).$

Moreover, following the proof of [2, Théorème 8.6], we set

$\eta_{v}=(\log v)/\sqrt{g^{\prime\prime}(\sigma_{v})}\asymp(\log
v)^{3/4}/v^{3/4}$

and recall [2, Lemme 8.5], asserting that for some $c>0$ we have

$|\mathscr{G}(\sigma_{v}+\mathrm{i}t)|\ll\mathscr{G}(\sigma_{v})e^{-c(\log
v)^{2}}\qquad(\eta_{v}\leq|t|\leq\exp((\log v)^{38/37}).$

It now follows that

$H_{\vartheta}(x)=\frac{1}{2\pi}\int_{-\eta_{v}}^{\eta_{v}}\frac{\mathscr{G}(\sigma_{v}+\mathrm{i}t)e^{(\sigma_{v}+\mathrm{i}t)v}}{\big{(}\vartheta+(1-\vartheta)(\sigma_{v}+\mathrm{i}t)\big{)}\big{(}1-\sigma_{v}-\mathrm{i}t\big{)}}\,{{\mathrm{d}}t}+O\left(\frac{\mathscr{G}(\sigma_{v})e^{v\sigma_{v}}}{v^{2}}\right).$

Setting

$D_{m}=v^{(m+1)/2}(\log v)^{(m-1)/2}\qquad(m\geq 0),$

we have

$H_{\vartheta}(x)=\frac{\mathscr{G}(\sigma_{v})e^{v\sigma_{v}}}{2\pi}\int_{-\eta_{v}}^{\eta_{v}}\Upsilon(t)e^{-g^{\prime\prime}(\sigma_{v})t^{2}/2}{\mathrm{d}}t+O\left(\frac{\mathscr{G}(\sigma_{v})e^{v\sigma_{v}}}{v^{2}}\right)$

where

$\Upsilon(t)=\frac{e^{-\mathrm{i}g^{\prime\prime\prime}(\sigma_{v})t^{3}/6+O(t^{4}D_{4})}}{\big{(}\vartheta+(1-\vartheta)(\sigma_{v}+\mathrm{i}t)\big{)}\big{(}1-\sigma_{v}-\mathrm{i}t\big{)}}.$

By Taylor expansion, we infer

$\Upsilon(t)=\frac{\mathrm{Z}(t)}{(\vartheta+(1-\vartheta)\sigma_{v})(1-\sigma_{v})}$

where

$\mathrm{Z}(t)=1+\mathrm{i}t\Big{(}\frac{1}{1-\sigma_{v}}-\frac{1-\vartheta}{\vartheta+(1-\vartheta)\sigma_{v}}\Big{)}-{\mathrm{i}}g^{\prime\prime\prime}(\sigma_{v})\frac{t^{3}}{6}+O(t^{2}+D_{4}t^{4}+D_{3}^{2}t^{6}).$

Now, still following the pattern of the proof of [2, Théorème 8.6], one is
lead to

$H_{\vartheta}(x)=\frac{x^{(1-\vartheta)\alpha_{\vartheta}(x)}\mathscr{G}\big{(}\alpha_{\vartheta}(x)\big{)}}{\vartheta\sqrt{2\pi
g^{\prime\prime}\big{(}\alpha_{\vartheta}(x)\big{)}}}\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}\Big{)}\Bigg{)}.$

We omit the details. From [2, Théorème 8.6] we import the relation

$F(v)=\frac{e^{v\sigma_{v}}\mathscr{G}(\sigma_{v})}{\sigma_{v}\sqrt{2\pi
g^{\prime\prime}(\sigma_{v})}}\left(1+O\left(\sqrt{\frac{\log
v}{v}}\right)\right)\qquad(v\geq 2),$

and the lemma follows. ∎

We may now complete the proof of Theorem 1. Using the lemma and (1.13), we
obtain

$H_{\vartheta}(x)\asymp F((1-\vartheta)\log x)\big{(}(\log x)\log\log
x\big{)}^{-1/2},$

so that the estimate (2.10) implies

$E\ll x^{\vartheta}H_{\vartheta}(x)x^{-\vartheta\alpha_{\vartheta}(x)}(\log
x)^{9/4}(\log\log x)^{3/4}.$

We then have

$S_{\vartheta}(x)=x^{\vartheta}H_{\vartheta}(x)\big{(}1+O\big{(}x^{-\vartheta\alpha_{\vartheta}(x)}(\log
x)^{9/4}(\log\log x)^{3/4}\big{)}\big{)}.$

We may now replace $H_{\vartheta}(x)$ by the estimate from Lemma 1. Since

$x^{-\vartheta\alpha_{\vartheta}(x)}(\log x)^{9/4}(\log\log
x)^{3/4}\ll\sqrt{\frac{\log\log x}{\log x}},$

this yields (1.9). As remarked earlier, (1.10) follows from (1.13) and (1.9).
The proof of Theorem 1 is complete.

##  3\. Proof of Theorem 2

Subject to the hypotheses of Theorem 2, when $x$ is large, one has $\log
zx\asymp\log x$. Hence, Theorem 1 implies that

(3.1) $S_{\vartheta}(zx)=(zx)^{\vartheta}F\big{(}(1-\vartheta)\log
zx\big{)}\frac{\alpha_{\vartheta}(zx)}{\vartheta}\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}\Big{)}\Bigg{)}$

holds uniformly for $|\log z|\ll\log\log x$. We recall [2, Proposition 8.7].
This asserts that uniformly for $|t|\leq v^{3/4}(\log v)^{1/4}$, one has

$F(v+t)=e^{t\sigma_{v}}F(v)\Big{(}1+O\Big{(}\frac{\log v+t^{2}/v}{\sqrt{v\log
v}}\Big{)}\Big{)}.$

Using this estimate with $v=(1-\vartheta)\log x$ and $t=(1-\vartheta)\log z$,
one finds

$F\big{(}(1-\vartheta)\log zx\big{)}=F\big{(}(1-\vartheta)\log
x\big{)}z^{(1-\vartheta)\alpha_{\vartheta}(x)}\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}\Big{)}\Bigg{)}.$

Moreover, in the ranges for $x$ and $z$ considered here, one has

$z^{(1-\vartheta)\alpha_{\vartheta}(x)}=1+O\big{(}\alpha_{\vartheta}(x)\log
z\big{)},$

which yields an admissible error term.

Finally, [2, (8.9)] implies that uniformly for $|t|\leq v/2$ one has

$\sigma_{v+t}=\sigma_{v}\big{(}1+O(|t|/v)\big{)}.$

Hence

$\alpha_{\vartheta}(zx)=\alpha_{\vartheta}(x)\Big{(}1+O\Big{(}\frac{\log
z}{\log x}\Big{)}\Big{)}$

which again leads to an admissible error term. Inserting these estimates in
(3.1) completes the proof of Theorem 2.

It may be worth pointing out that the above argument actually proves a little
more. A close inspection of the proof of Theorem 2 shows that the estimate

$S_{\vartheta}(zx)=z^{\vartheta+(1-\vartheta)\alpha_{0}(x)}S_{\vartheta}(x)\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log
x}{\log x}}+\frac{(\log z)^{2}}{(\log x)^{3/2}(\log\log
x)^{1/2}}\Big{)}\Bigg{)}$

holds uniformly in the range $z>0$, $x>27$, $|\log z|\leq(\log
x)^{3/4}(\log\log x)^{1/4}$.

##  4\. Proof of Theorem 3

Before proving Theorem 3, we briefly sketch the main steps. We first choose a
suitable real number $U=U(x)$ such that $\log U=(\log x)(1+o(1))$, and count
the integers $n$ not exceeding $x$ sucht that $k(n)\leq n^{\vartheta}(\log
U)^{\Theta}$. The first step is to show that the number of these integers is
essentially $S_{\vartheta}(x)$ multiplied by $(\log x)^{\Theta}$. The second
step is to prove that the number of these integers is close to
$S_{\vartheta,\Theta}(x)$.

In light of this description, for any $x\geq 1$ and any $z>0$, we set

$B(x,z)=\\#\\{n\leq x\colon k(n)\leq zn^{\vartheta}\\}.$

######  Theorem 4.

Let $0<\vartheta<1$ be fixed. Then for $x$ large, one has

$B(x,z)=zS_{\vartheta}(x)\Bigg{(}1+O\Big{(}\sqrt{\frac{\log\log x}{\log
x}}\Big{)}\Bigg{)}$

uniformly for $z>0$ with $|\log z|\ll\log\log x$.

Proof. We follow very closely the proof of Theorem 1 which corresponds to the
case $z=1$. We redefine the meaning of $y$, now set to $y=x^{\vartheta}z$, and
keep the notation $\kappa$ and $E$. Note that in the sequel of this proof, the
error term $E$ is to be interpreted with the current specific choices for $x$
and $y$.

For any $n\geq 1$, recall that there is a unique pair $l,m$ with $n=lmk(m)$
and $\mu^{2}\big{(}lk(m)\big{)}=1$. The conditions $n\leq x$ and $k(n)\leq
z^{\vartheta}$ become

$lmk(n)\leq x\quad\mbox{ and }\quad lk(m)\leq m^{\kappa}z^{1+\kappa}.$

By keeping the same dichotomy $m\leq x/y$ and $m>x/y$ it is straightforward
that

$B(x,z)=\frac{6}{\pi^{2}}\sum_{m=1}^{\infty}\frac{1}{\psi(m)}\min\big{(}m^{\kappa}z^{1+\kappa},x/m\big{)}+O(E).$

Moreover, (2.9) with the parameter $xz^{-\kappa-1}$ reads

$S_{\vartheta}(xz^{-\kappa-1})=\frac{6}{\pi^{2}}\sum_{m=1}^{\infty}\frac{1}{\psi(m)}\min\big{(}m^{\kappa},xz^{-\kappa-1}/m\big{)}+O(z^{-(1+\kappa)(1-\gamma)}E).$

Hence, one has

$B(x,z)=z^{1+\kappa}S_{\vartheta}(xz^{-\kappa-1})+O(z^{(1+\kappa)\gamma}E).$

Still choosing $\gamma=\sigma_{v}-\frac{1}{\log y}$ and $\sigma=\sigma_{v}$,
we have $z^{(1+\kappa)\gamma}\ll 1$. Moreover, (2.10) implies that for some
$c>0$ one has

$E\ll x^{\vartheta}F\big{(}(1-\vartheta)\log
x\big{)}\exp\Big{(}-c\sqrt{\frac{\log x}{\log\log x}}\Big{)}.$

This is sufficient to ensure that

$E\ll zS_{\vartheta}(x)\sqrt{\frac{\log\log x}{\log x}}.$

Finally we estimate the term $S_{\vartheta}(xz^{-\kappa-1})$ by Theorem 2.
Inserting this in the estimate for $B(x,z)$ and noticing that in the main term
the exponent $1+\kappa-(1+\kappa)\vartheta$ of $z$ is equal to $1$ gives the
expected result. ∎

We may now complete the proof of Theorem 3. It is sufficient to prove the
result for $\Theta\neq 0$, since in the case $\Theta=0$ one has
$S_{\vartheta,\Theta}(x)=S_{\vartheta}(x)$. We put

$C=1+(1+|\Theta|)/\vartheta\quad\text{and}\quad U=U(x)=x(\log x)^{-C}.$

Note that we have $C>0$, $C\vartheta\geq 1$ and $\Theta+C\vartheta\geq 1$.

First consider the case $\Theta>0$. Any integer $n$ counted by
$S_{\vartheta,\Theta}(x)$ satisfies $k(n)\leq n^{\vartheta}(\log x)^{\Theta}$,
whence $S_{\vartheta,\Theta}(x)\leq B\big{(}x,(\log x)^{\Theta}\big{)}$. Now,
a lower bound is obtained by noticing that the set of integers $U<n\leq x$
counted in $S_{\vartheta,\Theta}(x)$ contains the integers such that $k(n)\leq
n^{\vartheta}(\log U)^{\Theta}$. These deliberations yield the inequalities

$B\big{(}x,(\log U)^{\Theta}\big{)}-B\big{(}U,(\log U)^{\Theta}\big{)}\leq
S_{\vartheta,\Theta}(x)\leq B\big{(}x,(\log x)^{\Theta}\big{)}.$

Now, using Theorem 4 to estimate $B\big{(}x,(\log U)^{\Theta}\big{)}$ and
$B\big{(}x,(\log x)^{\Theta}\big{)}$, and then replacing $(\log U)^{\Theta}$
by $(\log x)^{\Theta}$ at the price of an admissible error term, one obtains
the main term to estimate $S_{\vartheta,\Theta}(x)$. For the remaining term,
Theorem 4 and the definition of $U$ imply that

$B\big{(}U,(\log U)^{\Theta}\big{)}\ll(\log
U)^{\Theta}S_{\vartheta}(U)\ll(\log x)^{\Theta}S_{\vartheta}(U).$

Finally, Theorem 2 with the choice $z=(\log x)^{-C}$ implies that

$S_{\vartheta}(U)\ll z^{\vartheta}S_{\vartheta}(x)\ll(\log x)^{-\vartheta
C}S_{\vartheta}(x)\ll(\log x)^{-1}S_{\vartheta}(x).$

Gathering these estimates, we obtain an upper bound for $B\big{(}U,(\log
U)^{\Theta}\big{)}$ that provides an admissible error term, and therefore
proves the result for $\Theta>0$.

The case $\Theta<0$ is very similar. Any integer counted by
$S_{\vartheta,\Theta}(x)$ either satisfies $n\leq U$ with $k(n)\leq
n^{\vartheta}$, or $U<n\leq x$ with $k(n)\leq n^{\vartheta}(\log U)^{\Theta}$.
Moreover, any integer $n\leq x$ such that $k(n)\leq n^{\vartheta}(\log
x)^{\Theta}$ is counted in $S_{\vartheta,\Theta}(x)$. We conclude that

$B\big{(}x,(\log x)^{\Theta}\big{)}\leq S_{\vartheta,\Theta}(x)\leq
B\big{(}x,(\log U)^{\Theta}\big{)}+S_{\vartheta}(U).$

As before, one uses Theorem 4 to estimate $B\big{(}x,(\log U)^{\Theta}\big{)}$
and $B\big{(}x,(\log x)^{\Theta}\big{)}$. This provides the main term.
Finally, Theorem 2 with $z=(\log x)^{-C}$ provides

$S_{\vartheta}(U)\ll z^{\vartheta}S_{\vartheta}(x)\ll(\log x)^{-\vartheta
C}S_{\vartheta}(x)\ll(\log x)^{\Theta-1}S_{\vartheta}(x),$

which yields an admissible error term.

## References

  * [1] B. Mazur. Questions about powers of numbers. Notices Amer. Math. Soc. 47 (2000), no. 2, 195–202.
  * [2] O. Robert and G. Tenenbaum. Sur la répartition du noyau d’un entier. Indag. Math. (N.S.) 24 (2013), no. 4, 802–914.
  * [3] G. Tenenbaum. Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, vol 163. American Mathematical Society, Providence, RI, $3^{\mathrm{rd}}$ edn., 2015.

Jörg Brüdern
Universität Göttingen
Mathematisches Institut
Bunsenstrasse 3–5
D 37073 Göttingen
Germany
<EMAIL_ADDRESS>
Olivier Robert
Université de Lyon, Université de Saint-Étienne
Institut Camille Jordan CNRS UMR 5208
25, rue Dr Rémy Annino
F-42000, Saint-Étienne
France
<EMAIL_ADDRESS>


# Analyzing the neutron and $\gamma$-ray emission properties of an americium-
beryllium tagged neutron source

Hiroshi Ito<EMAIL_ADDRESS>Kohei Wada  Takatomi Yano  Yota Hino
Yuga Ommura  Masayuki Harada  Akihiro Minamino  and Masaki Ishitsuka
Department of Physics, Faculty of Science and Technology, Tokyo University of
Science, Noda, Chiba 278-8510, Japan  Department of Physics, Yokohama National
University, Yokohama, Kanagawa, 240-8501, Japan  Kamioka Observatory,
Institute for Cosmic Ray Research, University of Tokyo, Kamioka, Gifu
506-1205, Japan  Department of Physics, Okayama University, Okayama, 700-8530,
Japan

###### Abstract

Americium–beryllium (AmBe), a well-known tagged neutron source, is commonly
used for evaluating the neutron detection efficiency of detectors used in
ultralow background particle physics experiments, such as reactor neutrino and
diffuse supernova neutrino background experiments. In particular, AmBe sources
are used to calibrate neutron tagging by selecting the 4438-keV $\gamma$-ray
signal, which is simultaneously emitted with a neutron signal. Therefore,
analyzing the neutron and $\gamma$-ray emission properties of AmBe sources is
crucial. In this study, we used the theoretical shape of a neutron energy
spectrum, which was divided into three parts, to develop models of the energy
spectrum and verify the results using experimental data. We used an AmBe
source to measure the energy spectra of simultaneously emitted neutrons and
$\gamma$-rays and determine the emission ratio of the neutrons with and
without $\gamma$-ray emission. The measured spectrum was consistent with that
obtained from the simulated result, whereas the measured emission ratio was
significantly different from the corresponding simulated result. Here, we also
discuss the feasibility of determining the neutron emission rates from the
spectra divided into three parts.

###### keywords:

Neutron tagging, Americium-Beryllium, Gamma-ray detector, Neutron detector,

††journal: Journal of LaTeX Templates

##  1 Introduction

Americium–beryllium (AmBe) is a widely used radioactive neutron source, which
can be tagged with the accompanying 4438 keV $\gamma$-ray emission. It is
commonly used to evaluate the neutron detection efficiency of particle
detectors employed in particle physics experiments. Several experiments
conducted [1, 2, 3, 4, 5, 6, 7] to focus on reactor and astrophysical
neutrinos, such as the diffuse supernova neutrino background search, involved
the use of the neutron tagging technique for distinguishing electron
antineutrino ($\bar{\nu}_{e}$) interactions from other neutrino interactions
or their background events. Coincidental detection of positrons and neutrons
(neutron tagging) is a suitable route for identifying the following charged
current reaction: $\bar{\nu}_{e}+p\to e^{+}+n$. Contrarily, an AmBe source
exhibits several neutron emission channels, which lead to the emission of
neutrons with different kinetic energies accompanied by $\gamma$-ray emission.
Evaluating the fraction of these channels and neutron energies corresponding
to an AmBe source is crucial for precisely measuring the neutron tagging
efficiency.

AmBe is composed of ${}^{241}\rm{Am}$ and ${}^{9}\rm{Be}$ and emits neutrons
according to the following chain reaction:
${}^{241}\rm{Am}\rightarrow{}^{237}\rm{Np}+\alpha$,
${}^{9}\rm{Be}+\alpha\rightarrow{}^{12}\rm{C}^{*}~{}(\rm{or}~{}{}^{12}\rm{C})+n$.
In this process, the carbon nuclei can adopt several excited states, although
only the first excited state of 12C emits $\gamma$-rays. The characteristics
of the neutrons emitted by AmBe, i.e., the energy spectrum, flux, and emission
ratio of neutrons and $\gamma$-rays, are correlated to the ratio of these
excited states. The source size and Be density affect the neutron energy
spectra below 2.5 MeV [8], which corresponds to the second or higher excited
states of 12C, probably owing to the different energy losses of the primary
$\alpha$-particles. This difference can also be attributed to the method of
production of the AmBe source.

To date, several theoretical and experimental studies have been reported on
the energy spectra of neutrons emitted by an AmBe source as well as the
correlation between the excited states of carbon nuclei [9, 8, 10, 11]. Figure
1 shows the recently reported energy spectral measurements [12, 13]. Most
neutrons are paired with the first excited state ($n_{\rm{1st}}$) and ground
state ($n_{\rm GND}$) of the 12C nucleus. The first excited state, with an
energy of 4438 keV, relaxes to its ground state by immediately emitting a
$\gamma$-ray photon with an energy of 4438 keV. Neutrons paired with the
second excited state of carbon ($n_{\rm{2nd}}$) also significantly contribute
to the energy spectra. The second state, with an energy of 7654 keV, is the
Hoyle state [14], which is critical in nuclear astrophysics and decays by
emitting three $\alpha$-particles with a probability of $\sim$99.6%, and only
$\sim$0.4% of the state decays by emitting two $\gamma$-ray photons (energy:
3215 and 4438 keV) [15]. These decayed $\alpha$-particles subsequently
interact with 9Be with a definite probability and simultaneously emit a
$\gamma$-ray photon (energy: 4438 keV) and neutron. However, considering the
ratio of the activity of 241Am to the neutron yield in our source, this
interaction probability is negligible in our source. In addition to the
neutrons paired with higher excited states, break-up neutrons, induced by
following the reaction: ${{}^{9}\rm{Be}}+\alpha\to{{}^{9}\rm{Be}^{*}}+\alpha$,
${{}^{9}\rm{Be}^{*}}\to{{}^{8}\rm{Be}}+n$, contribute to the energy spectra
appreciably below 1 MeV [8]. The kinetic energy of these neutrons is similar
to or lower than that of the $n_{\rm{2nd}}$ neutrons and is well separated
from those of $n_{\rm{1st}}$ and ($n_{\rm{GND}}$). Therefore, these neutrons
have been combined and treated as $n_{\rm{2nd}}$ in the following discussion.

In recent studies on precise measurements of whole neutron energy spectra, 3He
proportional and recoil-proton proportional counters, e.g., stilbene organic
scintillators, have been employed [10, 12, 13, 16]. The partial neutron
spectrum of $n_{\rm 1st}$ is also measured by coincidental detection of 4438
keV $\gamma$-rays using the neutron time-of-flight method [17, 18], although
the energy resolution is worse than those of the proportional counters.
Because of the differences in the experimental setups and energy resolutions,
distinguishing the $n_{\rm GND}$, $n_{\rm 1st}$, and $n_{\rm 2nd}$ spectra via
a simple comparison of the experimental data is challenging. To circumvent
this drawback, we introduced an experimental method to measure the ratio of
$n_{\rm GND}$ and $n_{\rm 1st}$ directly using detector simulations and
neutron spectral models.

In this paper, we propose a model of neutron emission from an AmBe source and
assess its validity by performing experimental measurements of the emission
rates paired with each ${}^{12}\rm{C}^{*}$ level and the neutron energy
spectrum accompanying $\gamma$-ray emission. Neutron energy spectral models,
which are useful for simulating neutrons, combined with experimental
evaluation of the characteristics of an individual AmBe source in a laboratory
are crucial for realizing an efficient neutron tagging. The remainder of this
paper is organized as follows. The experimental details, including the
characteristics of the AmBe source used in our study, components, simulation
details, and experimental setups, are described in section 2. The analysis
results, measured neutron and $\gamma$-ray energy spectra, and emission ratio
of neutrons with/without $\gamma$-ray emission, are described in section 3.
Finally, the major conclusions drawn from this study are elucidated in section
4.

##  2 Material and Methods

We performed two measurements for neutrons and $\gamma$-rays emitted from our
AmBe source using NaI(Tl) crystal and liquid scintillation detectors. The
properties of the AmBe source were determined by comparing the measured data
and simulation results.

###  2.1 Americium-Beryllium source

The AmBe source, developed by the French Alternative Energies and Atomic
Energy Commission in the early 1990s, was supplied by the Kamioka Observatory,
Institute for Cosmic Ray Research, University of Tokyo, Japan. In this source,
241Am, with a strength of $97~{}\mu\rm{Ci}$ (3.6 MBq), is encapsulated with
${}^{9}\rm{Be}$ inside a stainless cylinder with the same outer diameter and
height, i.e., 12.5 mm. The neutron yield measured in March 2022 by the
National Institute of Advanced Industrial Science and Technology (AIST),
Japan, was $236.8\pm 5.0~{}n\cdot\rm s^{-1}$ ($S_{n}$). For the yield
measurements, we compared the National Standard Field of Thermal Neutrons at
AIST under the same conditions, i.e., the AmBe (standard) source located in a
graphite pile. The particles were counted using a ${}^{3}\rm{He}$ proportional
counter [19].

Next, we performed a Monte Carlo simulation to analyze the experimental
results, and the model of the neutron energy spectra paired with the excited
states of 12C is shown in Figure 1. We adopted the International Standard
ISO-8529-1:2001(E) to model the total neutron spectra [16]. The spectra were
divided into three parts based on the theoretical predictions of De Guarrini
and Malaroda [9]. These partial neutron spectra were paired with the states of
12C: ground state ($n_{\rm{GND}}$), first excited state ($n_{\rm{1st}}$), and
the second excited state or higher states ($n_{\rm{2nd}}$).

Figure 1:  Measured neutron energy spectra of an AmBe source based on ISO
8529-1:2001(E) [12, 16], E. Lorch [10], and M. Kostal et al. [13]. These
measurements are conducted without considering the excited states of 12C.
$n_{\rm{GND}}$, $n_{\rm{1st}}$, and $n_{\rm{2nd}}$ represent the partial
neutron spectra with the following states of 12C: ground state, first excited
state, and second or higher excited states, respectively.

####  2.1.1 Measurement of $\gamma$ intensity from the AmBe source

The intensity of the 4438 keV $\gamma$-rays was measured using a high-purity
Ge (HPGe) detector (ORTEC, GEM110P4-ST), whose detection efficiency was
evaluated via a Geant4-based Monte–Carlo simulation (also see section 2.3).
The characteristics of the materials and inner structural dimensions of the
HPGe detector and AmBe source were included in the simulation. The systematic
uncertainty in the simulated $\gamma$-rays peak detection efficiency of the
detector was $\sim$10%, which was determined by comparing the simulated and
calibration data of the detector for the 1173, 1333 (single $\gamma$-ray), and
2505 keV (sum of two $\gamma$-rays) peaks emitted by a 60Co source (19.5 kBq).

Figure 2 (a) and (b) show the energy spectra obtained from our AmBe source;
the spectra measurements were performed at distances of 10 and 50 mm from the
surface of the HPGe detector, respectively. The full widths at half maximum of
the peak at 4438 keV are 119 ($L=10~{}\rm{mm}$) and 114 keV
($L=50~{}\rm{mm}$), which are consistent with previously reported values,
obtained by considering Doppler broadening at the AmBe source [20];
$\gamma$-ray peak broadening was considered in the simulation of the 4438 keV
$\gamma$-ray spectrum for determining the detection efficiency of the
detector. The energy resolution of the HPGe detector was negligible compared
to the broadening caused by the Doppler effect. In our case, the measured
intensity of the 4438-keV $\gamma$-ray photons emitted from our AmBe source
was $110.1\pm 15.5~{}\gamma/\rm s$ ($S_{\gamma}$).

For our source, the emission ratio of the 4438-keV $\gamma$-ray photons to the
neutron yield (${R_{\gamma/n}=S_{\gamma}/S_{n}}$, where $S_{\gamma}$ and
$S_{n}$ indicate the emission rates of the 4438-keV $\gamma$-rays and neutrons
from the AmBe source, respectively) was determined as $0.46\pm 0.07$. This
estimated ratio is consistent with those reported previously
($R_{\gamma/n}=0.57$–-$0.75$) [21] for AmBe sources within experimental
errors. However, the observed differences between the ratio calculated in this
study and those reported previously, albeit small, may reveal the
characteristics of the corresponding AmBe sources used for these measurements.

Figure 2:  Energy spectrum of an AmBe source (black) and background (blue)
measured using an HPGe detector, and the simulated spectrum of 4438-keV
$\gamma$-rays emitted from the AmBe source (red). (a) 10 mm and (b) 50 mm
distance between the center of the AmBe source and the HPGe surface

###  2.2 Equipments

A NaI(Tl) crystal, with dimensions of $\rm 51~{}mm\times 51~{}mm\times
152~{}mm$, was optically connected to a two-inch photomultiplier tube. We used
the assembled detector (OKEN, 8x8S24/1,5.VD(p), Oyo Koken Co. Ltd.) in this
study. The crystal was then surrounded by 1-mm-thick aluminum plates for
optical shielding. The liquid scintillation detector consisted of a liquid
scintillator (diameter: 3”; height: 3”; Saint-Gobain, BC501A) connected to a
three-inch photomultiplier tube. The waveforms obtained from the NaI(Tl) and
liquid scintillation detectors were stored using an oscilloscope (Lecroy,
100MXi) with a sampling rate of $\rm 1~{}GS~{}s^{-1}$.

The NaI(Tl) detector was used as a $\gamma$-ray energy spectroscope, whereas
with the liquid scintillator detector, the pulse-shape discrimination (PSD)
method was applied to separate the neutron and $\gamma$-ray signals. The PSD
and calibration methods are described in detail in Appendix A. The calculated
energy resolution was approximately 2% in the main target energy region above
5 MeV, and the timing resolution was 1.72 ns. Based on the PSD of the pulses
obtained from the liquid scintillation detector, the performance was confirmed
to be $3\sigma$ for the neutron like event selection, and the $\gamma$-like
contamination was less than 0.15%.

###  2.3 Monte Carlo simulation

In this study, we used the GEANT4 Monte Carlo simulation tool [22]
(Geant4.9.6p04 + G4NDL4.2) in order to evaluate the neutron and $\gamma$
emission of the AmBe source and take into account the detector response to the
measurements. In particular, neutron interactions were carefully treated
because of the uncertainty in several models.

We prepared a geometrical setup for the simulation in which the source capsule
was treated as a uniform stainless-steel cylinder to reproduce the size and
total weight of our AmBe source. Further, the material characteristics and
geometries of the NaI(Tl) and liquid scintillators were reproduced. In
addition, to incorporate the contribution of scattered neutron background, the
distance between the detector and the wall and the dimensions of the test
bench (see section 2.4.1) used in the experiment were simulated. Although the
details of the detector, such as the inner parts of the photomultiplier tubes,
were approximated, the uncertainty of the material composition was considered
in the results. To determine the energy resolution of our NaI(Tl) detector, a
$\gamma$-ray source was used to irradiate the detector (for calibration), and
the resulting output spectra of the detector were fitted with Gaussian
functions. Further, the position of the $\gamma$-ray source was varied during
the calibration process to determine the non-uniformity in the light yield of
the detector, which was ultimately found to be negligible (see Appendix A.1).
In addition, the timing resolution of the detector/experimental setup was
evaluated for the subsequent neutron energy measurements, which were performed
using the time-of-flight method (see Appendix A.2).

Fast neutrons with kinetic energies on the MeV scale mostly interact with the
sodium or iodine nuclei in the NaI(Tl) scintillator. In this study, we refer
to the recent quenching factor reported by Joo et al. [23] as a function of
the recoil energy for sodium and iodine.

In this study, we used models of the neutron energy spectra to simulate $②$
$n_{\rm 1st}$ and $③$ $n_{\rm GND}$ as shown in Fig. 1. In particular, we
validated the $n_{\rm 1st}$ spectral shape by considering the detector
response. We considered only $n_{\rm GND}$ and n1st because the setups
(explained in the next subsection) were designed to minimize the contribution
of $n_{\rm 2nd}$; the simultaneous emission of $\gamma$-rays and $n_{\rm 2nd}$
is rare, and the amount of energy deposited in the NaI(Tl) scintillator is
small.

###  2.4 Setups

The experimental setups used for measuring the neutron and $\gamma$-ray energy
spectra are presented in this section, and the corresponding measurement
results and their analyses are discussed in section 3.

####  2.4.1 Energy spectrum of the neutron with $\gamma$ emission

To determine the kinetic energy of the neutrons, we used the time-of-flight
method, in which the neutron kinetic energy was estimated from the difference
between the arrival times of the neutrons and the $\gamma$-ray photons due to
their different velocities. The NaI(Tl) and liquid scintillation detectors
were set at a distance from each other; the AmBe source was placed near the
NaI(Tl) detector and 63.21 cm away from the center of the liquid scintillation
detector as shown in Figure 3. Both the detectors and the source were placed
on a stainless-mesh shelf (height: 120 cm) to reduce measurement
uncertainties, which may have different origins; for example, the uncertainty
caused by the event rate of the neutrons scattered by the surrounding
materials (such errors were not observed in the simulation results).
Furthermore, the detectors were set at distances of greater than 1 m from the
walls and ceiling, and the coincidence signals were stored using an
oscilloscope. The neutron emission events were determined by tagging the
4.4-MeV peak in the energy spectrum obtained from the NaI(Tl) detector. The
neutron and $\gamma$-ray background signals were distinguished via PSD using
the liquid scintillation detector.

Figure 3:  Schematics of the (a) experimental setup and (b) front-end data
acquisition system used to measure the neutron energy spectrum based on the
time-of-flight method.

####  2.4.2 Determination of event ratio of higher energy tail to 4.4 MeV peak

We determined the neutron emission ratio of $n_{\rm GND}$ to $n_{\rm 1st}$
with $\gamma$-ray emission ($R_{n0/n1}$) from the AmBe source using the setup
depicted in Figure 4. To measure the energy spectra using the NaI(Tl)
detector, the distance ($L$) between the centers of the AmBe source and the
NaI(Tl) detector was varied to 3.5, 4.5, 5.5, 8.5, 10.0, 13.5, 16.0, 18.5,
28.5, and 38.5 cm. The ratio of the number of events in the high-energy tail
components (5–11 MeV) of the 4.4 MeV peak was measured. The experiments were
performed in the energy region of 3–10 MeV to ensure that the neutrons of the
second or higher excited states remain almost unaffected. The emission ratio
of the neutrons with and without $\gamma$-ray emission was determined by
comparing the observed and simulated event ratios of the peak and tail
components.

Figure 4:  Schematic of the experimental setup and front-end data acquisition
system used to determine the neutron emission ratio of $n_{\rm GND}$ to
$n_{\rm 1st}$ with $\gamma$-ray emission by varying the distance $L$.

##  3 Result and Discussion

###  3.1 Neutron with $\gamma$-ray energy spectra measured using the time-of-
flight method

The neutron-like events were discriminated by analyzing the pulse shapes of
the signals obtained from the liquid scintillator. The arrival-time
distribution is shown in Figure 5 (a). The black solid and red dashed
histograms represent the experimental data and the simulation based on the ISO
model with smearing by the measured timing resolution, respectively. Then,
based on the time of flight, the neutron kinetic energy is converted from the
arrival time difference between the NaI(Tl) and liquid scintillation
detectors, given as

$\displaystyle E_{\rm TOF}=m_{n}\left(\frac{1}{\sqrt{1-(L/ct)^{2}}}-1\right),$
(1)

where $m_{\rm n}$ is the neutron mass, $L$ is the path length of the neutron
(63.21 cm), and $ct$ is the path length of light during flight.

The measured neutron energy spectra and the expectations from the simulation
are shown in Figure 5 (b). The simulation band indicates systematic
uncertainty, which is dominated by the model difference of the cross-section
of neutron scattering with iodine (Appendix B). In comparison with the
simulation, $\chi^{2}/dof=28.7/40$ is calculated, which corresponds to the
p-value of 0.908; this value indicates that the measured energy spectrum is in
good agreement with that simulated using the prediction model (shown in Figure
1).

Figure 5:  Arrival time distribution (a) and neutron energy spectra of data
and simulation based on time-of-flight (b). Both events are selected by
requiring neutron identified by PSD of the liquid scintillation detector and
$\gamma$ detection in the NaI(Tl) detector. The black dots and red lines are
data and the simulation, respectively. Figure 6:  Distribution of visible
energy in the NaI(Tl) detector for a distance between the center of the AmBe
source and the NaI(Tl) detector ($L$). Data and BG represent measured spectra
with the AmBe source and without the source. MC $n_{\rm GND}$ and MC $n_{\rm
1st}$ indicate simulated spectra for neutrons with the ground state and 1st
exited state of carbon. In a case of 1st exited state, 4438 keV $\gamma$ ray
also emitted simultaneously.

###  3.2 Emission ratio of neutron with/without $\gamma$-ray emission

The distribution of the visible energy for each distance $L$ is shown in
Figure 6. The high-energy tail beyond the 4438 keV $\gamma$-ray peak can be
attributed to the neutrons emitted from the source. A fraction of the emitted
neutrons interact with the nuclei in the NaI(Tl) detector and excite them via
inelastic scattering. The $\gamma$-rays emitted by these nuclei upon
deexcitation are detected by the detector via the detection of energy-
deposition-induced scintillations in the scintillator medium: e.g.,
$n+{{}^{22}\rm Na}\to n+{{}^{22}\rm Na}^{*}$, ${}^{22}{\rm
Na}^{*}\to{{}^{22}\rm Na}+\gamma(\rm s)$. The nuclei may also recoil without
undergoing excitation upon interacting with the incident neutrons. However,
the energy deposited during this elastic scattering process (nuclear recoil)
is smaller by an order of magnitude. This neutron inelastic scattering process
is utilized to evaluate the emission ratio of $n_{\rm GND}$ and $n_{\rm 1st}$
by considering the different acceptances of the detector for a single neutron,
pair of neutrons, and the 4438 keV $\gamma$-rays. These acceptances are
approximately described as follows:proportional to $L^{-2}$, $L^{-2}$, and
$L^{-4}$ for a single $\gamma$-ray (4438 keV peak), single neutron (${E_{\rm
vis}>5~{}{\rm MeV}}$), and coincidentally detected $\gamma$-ray and neutron
(${E_{\rm vis}>5~{}{\rm MeV}}$), respectively.

The ratio of the high-energy tail (${E_{\rm vis}>5~{}{\rm MeV}}$) to the 4438
keV peak, $R_{\rm tail/peak}$, is calculated for each value of distance ($L$)
between the source and the NaI(Tl) detector (see Figure 6) using the following
equation:

$\displaystyle R_{\rm tail/peak}(L)=\frac{\int^{10,000~{}\rm keV}_{4,750~{}\rm
keV}\Bigl{\\{}N(E_{\rm vis},L)-N_{\rm BG}(E_{\rm vis})\Bigr{\\}}~{}dE_{\rm
vis}}{\int^{4,536~{}\rm keV}_{4,321~{}\rm keV}\Bigl{\\{}N(E_{\rm
vis},L)-N_{\rm BG}(E_{\rm vis})\Bigr{\\}}~{}dE_{\rm vis}},$ (2)

where $N(E_{\rm vis},L)$ is the event rate for distance $L$ and $E_{\rm vis}$
bin, and $N_{\rm BG}(E_{\rm vis})$ are the background rates for the $E_{\rm
vis}$ bin.

The ratio ($R_{\rm tail/peak}$) as a function of $L$ is shown in Figure 7,
which indicates the simulated $R_{\rm tail/peak}$ also varies with $L$ because
of the different $L$ dependences of the peak and tail components. As a result,
we determined $R_{\rm n0/n1}$ to be $0.68\pm 0.01_{\rm stat.}\pm 0.05_{\rm
syst.}$ by fitting the $R_{\rm tail/peak}$ distribution from the simulation to
the data. The systematic uncertainty was dominated by the model difference in
the cross-section of neutron scattering with iodine (Appendix B). The best fit
value of Rn0/n1 was determined with a minimum $\chi^{2}$ of $\chi^{2}$ of
$\chi^{2}/dof=9.46/9$.

Figure 7:  Correlation between distance $L$ and $R_{\rm tail/peak}$ used to
determine the neutron emission ratio of $n_{\rm GND}$ to $n_{\rm 1st}$ with
$\gamma$-ray emission. Black dots are the measured data, and the red line is
the best fit simulated data. The red band shows a total systematic
uncertainty, where a difference of cross-section models for neutron
interaction is a dominant contribution in this study.

###  3.3 Comprehensive profile of the AmBe source

The measured emission parameters of neutrons and $\gamma$ are summarized in
Table 1. Based on this information, we deduced the emission rates of neutrons
from the AmBe source, that is, $n_{\rm 1st}$, $n_{\rm GND}$, and $n_{\rm
2nd}$.

Table 1: Summary of measured emission rate and ratio for the AmBe source. Neutron yield | $S_{\rm n}=236.8\pm 5.0~{}n\cdot{\rm s^{-1}}$
---|---
Intensity of 4.4 MeV $\gamma$ emission | $S_{\rm\gamma}=110.1\pm 15.5~{}\gamma\cdot\rm s^{-1}$
Emission ratio of $n_{\rm GND}$ to $n_{\rm 1st}$ | $R_{\rm n0/n1}=0.68\pm 0.01_{\rm stat.}\pm 0.05_{\rm syst.}$

The $n_{\rm 1st}$ emission rate is assumed to be equal to $S_{\gamma}$.
Second, the $n_{\rm GND}$ emission rate is calculated by using $R_{\rm
n0/n1}\cdot S_{\rm\gamma}$ and is found to be ${74.9\pm 11.9~{}n\cdot{\rm
s^{-1}}}$. Finally, the $n_{\rm 2nd}$ emission rate is estimated by
subtracting $n_{\rm GND}$ and $n_{\rm 1st}$ from the total neutron yield and
found to be ${51.8\pm 17.2~{}n\cdot{\rm s^{-1}}}$. The results are consistent
with the expectations for the $n_{\rm GND}$ and $n_{\rm 2nd}$ emission rates
of $59.0\pm 1.3~{}n\cdot{\rm s^{-1}}$ ($n_{\rm GND}$) and $54.2\pm
1.1~{}n\cdot{\rm s^{-1}}$ ($n_{\rm 2nd}$), respectively. These expectations
are given by our neutron energy spectrum models shown in Figure 1 and the
measured total neutron and 4438 keV $\gamma$ emission. As described above, our
method can be used to determine the emission rates of $n_{\rm GND}$, $n_{\rm
1st}$, and $n_{\rm 2nd}$.

##  4 Conclusion

AmBe, which emits neutrons accompanied by $\gamma$-rays, is widely used to
calibrate neutron detectors. We examined the energy spectra of the neutrons
and $\gamma$-rays emitted from our AmBe source using NaI(Tl) and liquid
scintillation detectors. We modeled the neutron energy spectra paired with the
respective excited states of carbon, divided into three parts, by referring to
the theoretical prediction. The energy spectrum of $n_{\rm 1st}$ was evaluated
based on the time of flight, which was consistent with our prediction
considering its neutron energy spectrum and our detector response. The
emission ratio of neutrons with/without $\gamma$ emission ($R_{n0/n1}$) was
measured based on the calorimetric method with the NaI(Tl) detector by
changing the distance between the source and the detector. It was determined
to be $R_{n0/n1}=0.68\pm 0.01_{\rm stat.}\pm 0.05_{\rm syst.}$ This method was
established for the first time. The $\gamma$ intensity of $110.1\pm
15.5~{}\gamma\cdot\rm s^{-1}$ from the AmBe source with a neutron fluence of
$236.8\pm 5.0~{}n\cdot\rm s^{-1}$ was obtained by performing independent
measurements. The emission rates of $n_{\rm GND}$ and $n_{\rm 2nd}$ were
determined to be $74.9\pm 11.9$ and $51.8\pm 17.2~{}n\cdot\rm s^{-1}$,
respectively; these are reasonably consistent with the expectations of our
model. Therefore, this method comprehensively determined the neutron
emissivity of the excited states of Carbon in the AmBe source. Our neutron
spectral model showed good consistency in this study. The characteristics of
the AmBe source can be understood by integrating the experimental measurement
of the emission ratio of neutrons and the modeled energy spectra. Thus, these
will contribute to the general use of AmBe sources, such as in measurements
and calibrations.

## Acknowledgments

This work was partially supported by the joint research program of the
Institute for Cosmic Ray Research (ICRR) at The University of Tokyo. This work
was supported by the Neutron Measurement Consortium for Underground Physics,
JSPS KAKENHI Grant Grants, Grant-in-Aid for Scientific Research (C) [grant
number 20K03998], and Grant-in-Aid for Scientific Research on Innovative Areas
[grant number 19H05808].

## Appendix A. Calibration and performance check of the detectors

### A.1. Energy calibration of NaI(Tl) detector

We calibrated the energy of the NaI(Tl) detector using $\gamma$ emission
sources, 22Na, 137Cs, and 60Co. In addition, the environmental $\gamma$
sources 40K and 208Tl were used for the energy calibration. The relationship
between the visible energy $E_{\rm vis}$ (MeV) and the integrated charge $q$
of the signal waveform (pC) was determined as a function of the energy as
follows:

$\displaystyle q(E_{\rm vis})=a\biggl{[}1-\exp\left(-E_{\rm
vis}/b\right)\biggr{]},$ (3)

where $a$ and $b$ are the fit parameters. A saturation effect for the
phototubes was found at approximately a few MeV, and a fit function with an
exponential function was employed to account for this effect.

Figure 8:  Energy resolution of the NaI(Tl) detector as a function of visible
energy with a fit line.

Then, we measured the energy resolution ($\sigma_{E}(E_{\rm vis})/E_{\rm
vis}$) of the detectors given as

$\displaystyle\sigma_{E}(E_{\rm vis})/E_{\rm
vis}=\sigma_{0}+\frac{\sigma_{1}}{\sqrt{E_{\rm vis}}}$ (4)

where $\sigma_{0}$ and $\sigma_{1}$ are fit parameters. The energy dependence
of resolution $\sigma_{E}/E_{\rm vis}$ follows the function of Eq. 4, as shown
in Figure 8, and the function with the best-fit parameters was used to
reproduce the detector response in the simulation, as explained in Section
2.3.

The dependence of the light yield’s uniformity on the position of the
$\gamma$-ray source was investigated by measuring the integrated charge of the
662 keV $\gamma$-rays at different positions of a 137Cs source. The light
yield was found to be uniform within an error of 0.4%, and error propagation
analysis showed that this error is negligible compared to the experimental
energy resolution (Figure 8).

### A.2. Timing resolution of the NaI(Tl) and liquid scintillation detectors

To measure the timing resolution of the setup as shown in Fig. 3 using two
$\gamma$’s from annihilation, a 22Na source was placed in the center between
two detectors, at a distance of 30 cm from each detector. Events in the 511
keV peaks were selected, and the time resolution and offset between the two
detectors were determined based on the time difference.

In this study, the time walk effect (pulse height dependence of arrival time)
was considered using the constant fraction discrimination (CFD) method in the
waveform analysis. We optimized the CFD delay times for the signals of the
NaI(Tl) and liquid scintillation detectors to 10 and 2 ns, respectively.

The timing resolution was measured to be 1.2 ns, and this value was used to
smear the arrival time in the simulation to compare the measurements of
neutrons and $\gamma$’s from the AmBe source. The possible bias was estimated
to be 0.03 ns to determine the time offset and is negligibly small for the
conversion of the time of flight of neutrons to kinetic energy.

Figure 9:  Typical waveforms of (a) $\gamma$-like and (b) neutron-like
signals.  Figure 10:  Relation between $Q_{total}$ and $Q_{\rm slow}/Q_{\rm
total}$. Two components appear and black solid and dashed lines include
3$\sigma$ of neutron-like and $\gamma$-like events, respectively. In this
study, as the neutron-like selection, the PSD selection region is within the
gray solid line.

### A.3. Discrimination of neutron and $\gamma$

To discriminate neutron events from $\gamma$ events, the 252Cf source was
placed at a distance of 63.21 cm from the liquid scintillation detector.
Neutron and $\gamma$ events are discriminated by using the signal waveform,
called PSD, because neutron events tend to have light emission in late timing
(tail in the signal waveform), whereas the waveform of $\gamma$ events tends
to be sharp. Figure 9 shows the typical waveforms of $\gamma$-like and
neutron-like signals. In particular, variable $Q_{\rm slow}/Q_{\rm total}$ is
defined as the ratio of the integrated charges of a signal in a liquid
scintillation detector with different time windows, where $Q_{\rm slow}$ and
$Q_{\rm total}$ are respectively given as

$\displaystyle Q_{\rm slow}=\int_{t_{\rm r}+30~{}\rm ns}^{t_{\rm r}+t_{\rm
w}}\frac{V(t)}{z}~{}dt,$ (5) $\displaystyle Q_{\rm total}=\int_{t_{\rm
r}}^{t_{\rm r}+t_{\rm w}}\frac{V(t)}{z}~{}dt,$ (6)

$V(t)$ is the amplitude of the waveform (mV) of the signal as a function of
time t in a unit of ns), and ${z=50~{}\Omega}$ is the impedance of the cables.
In the integral region, $t_{\rm r}$ and $t_{\rm w}$ are the rising times
satisfying $V(t_{\rm r})=V_{\rm max}/3$ and the gate time of the window, which
was set to 350 ns in this study.

Figure 10 shows the relationship between $Q_{\rm total}$ and $Q_{\rm
slow}/Q_{\rm total}$, where two clear components appear at $Q_{\rm
total}>50~{}\rm pC$. The criteria are determined to select neutron-like events
with $3\sigma$ range of $Q_{\rm slow}/Q_{\rm total}$ distribution sliced into
$Q_{\rm total}$ in a region of $50<Q_{\rm total}<300~{}\rm pC$. The
$\gamma$-like contamination in any $Q_{\rm total}$ is estimated to be less
than 0.15% in the selected region. We calibrated the conversion factor from
energy deposition to $Q_{\rm total}$ of $96~{}\rm pC/MeV_{ee}$.

## Appendix B. Systematic uncertainties

We considered the following systematic uncertainties: The main contribution is
the difference in the physical model of neutron interaction with the material.
Other sources that are estimated to have negligible effects are as follows:
(1) the uncertainty of material components, such as the inner parts of
photomultiplier tubes and the stainless mesh shelf, (2) the effect of bias to
determine the timing offset for the NaI(Tl) and the liquid scintillation
detector, and (3) the quenching factor uncertainty of nuclear recoil energy in
the NaI(Tl) scintillator.

### B.1. Neutron interaction with the material

We adopted the physical models of thermal and fast neutron interactions with
the materials listed in Table 2. These models are included in the default
GEANT4 package and are reasonably consistent with typical fast neutron
experiments, for example, the cross-section of elastic/inelastic scattering
and nuclei capture. The ENDF-VII.0 model is adopted as a data set of the
cross-section for neutrons [24], whereas the cross-sections for elastic and
inelastic neutrons with 23Na and 127 311 I in the BROND [25], JEFF [26], and
JENDL [27] models, which are often used in simulations, show some
discrepancies from our data. Figure 11 shows the difference between cross-
sections $\Delta\sigma_{n}$ and those obtained using the ENDF-VII.0 model as a
function of neutron energy for elastic/inelastic scattering with iodine and
sodium nuclei. We consider the difference between the models in the simulation
to be a systematic uncertainty.

Table 2: Physics model of neutron interaction Thermal neutron | G4NeutronHPThermalScattering
---|---
($<4$ eV) |
Fast neutron | G4NeutronHPorLElastic
($4$ eV $\sim$ 20 MeV) | G4NeutronHPorLEInelastic
| G4NeutronHPorLCapture
Data set | ENDF-VII.0 [24]
Figure 11:  Difference of cross section from ENDF-VII.0 model as a function of
neutron energy for elastic/inelastic scattering with iodine and sodium nuclei.
Red, green, blue-dashed, magenta, and cyan lines are ENDF-VII.1 [24],
BROND-3.1 [25], ENDF-VIII.0 [24], JEFF-3.3 [26], and JENDL-4.0u [27] models,
respectively.

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# Architecture Matters: Investigating the Influence of Differential Privacy on
Neural Network Design

Felix Morsbach
Institute AIFB
Karlsruhe Institute of Technology
Karlsruhe, Germany
<EMAIL_ADDRESS>
Tobias Dehling
Institute AIFB, KASTEL Security Research Labs
Karlsruhe Institute of Technology
Karlsruhe, Germany
<EMAIL_ADDRESS>
Ali Sunyaev
Institute AIFB, KASTEL Security Research Labs
Karlsruhe Institute of Technology
Karlsruhe, Germany
<EMAIL_ADDRESS>

###### Abstract

One barrier to more widespread adoption of differentially private neural
networks is the entailed accuracy loss. To address this issue, the
relationship between neural network architectures and model accuracy under
differential privacy constraints needs to be better understood. As a first
step, we test whether extant knowledge on architecture design also holds in
the differentially private setting. Our findings show that it does not;
architectures that perform well without differential privacy, do not
necessarily do so with differential privacy. Consequently, extant knowledge on
neural network architecture design cannot be seamlessly translated into the
differential privacy context. Future research is required to better understand
the relationship between neural network architectures and model accuracy to
enable better architecture design choices under differential privacy
constraints.

##  1 Introduction

Differential privacy has become the de facto standard for achieving data
confidentiality in machine learning settings. The most prominent way to train
differentially private neural networks is by clipping gradients in order to
limit the impact of each data point and by adding noise to the gradient
updates abadi_deep_2016 . While gradient clipping is also used in non-private
training to avoid overfitting zhang_why_2020 , adding noise inherently reduces
the accuracy of the machine learning model jayaraman_evaluating_2019 . This is
considered as a trade-off between utility and differential privacy guarantees
and is usually measured as accuracy loss, describing the difference between
the accuracy with and without differential privacy constraints. Even on
comparably simple tasks, such as the CIFAR-10 classification task, the
accuracy loss can be significant. This hinders the adoption of differentially
private neural networks.

It is well known, that the architecture of a neural network can have a
significant influence on the accuracy of the model. For example, expanding the
size of a network in depth rather than width is usually said to increase
accuracy goodfellow_multidigit_2014 , but also novel architectural features
such as residual or dense connections can vastly improve model accuracy
he_deep_2016 ; huang_densely_2017 . Yet, early works on differentially private
neural networks do not seem to account for this in great detail. They either
do not include different architectures in their benchmarks
jayaraman_evaluating_2019  or find that the effect is not significant
abadi_deep_2016 . There is, however, also contrary evidence. For instance,
expanding the overall size of the network in a differentially private setting
exhibits an inflection point papernot_making_2019 . Increasing the network
size beyond this inflection point reduces model accuracy, which is not the
case in the plain setting. Furthermore, the choice of an activation function
plays an important role. Bounded activation functions (e. g., tanh)
consistently outperform unbounded ones (e. g., ReLU) due to the phenomena of
exploding activations during differentially private training
papernot_tempered_2021 .

Nevertheless, even with these findings taken into account, there still remains
a significant gap between the accuracy of differentially private and non-
private neural networks, even on simple tasks. Extant research has already
shown that the choice of an appropriate activation function differs between
the non-private and differentially private setting and that choosing an
activation function appropriate for differentially private neural networks can
reduce the incurred accuracy loss papernot_tempered_2021 . However, there are
a multitude of other architectural features, such as the number, ordering,
type and configuration of layers, whose impact on model accuracy under
differential privacy constraints has not been really understood. Consequently,
we investigate whether architectural features other than the activation
function of a neural network affect the accuracy loss incurred by differential
privacy constraints.

If we can show that more architectural features than the activation function
impact model accuracy, this would imply that the accuracy of differentially
private models can be improved by carefully tuning a neural network
architecture for differential privacy instead of simply copying architectures
and hyperparameters that work well without differential privacy constraints
vanderveen_three_2018 .

We designed and carried out an experiment to test whether the architecture
affects accuracy loss under differential privacy constraints. Our findings
show, not only, that the network architecture has an influence on the accuracy
loss incurred by differential privacy constraints, but also, that the
suitability of architecture choices is sensitive to variations in the targeted
level of differential privacy. We elaborate the implications of our findings
and argue what future research is necessary to reduce the accuracy loss
incurred by differential privacy constraints in order to improve the
applicability of privacy-preserving machine learning.

##  2 Background

Differentially private stochastic gradient descent (DP-SGD) is an algorithm
for training neural networks with $(\epsilon,\delta)$-differential privacy
guarantees abadi_deep_2016 ; dwork_algorithmic_2014 . DP-SGD extends classic
stochastic gradient descent in two ways. First, gradients are clipped on a
per-example basis to a $l_{2}$ norm, which is set through a clipping threshold
$C$. Second, random noise is added to the gradient update calibrated via the
standard deviation $\sigma$, also called noise multiplier. Thus, DP-SGD adds
two additional hyperparameters to the training algorithm.

The privacy level $(\epsilon,\delta)$ of a model trained with DP-SGD is
dependent on the batch size, the noise multiplier, the number of epochs, and
the number of training examples. Hence, the privacy level for a given set of
these hyperparameters can be calculated upfront without the need to actually
train the model. Since the privacy level is independent of the architecture of
a given model, different architectures can be compared at a fixed privacy
level.

##  3 Experiments

In order to answer our research question, we need to show that the
architecture of a neural network does have an effect on the model’s accuracy
loss under differential privacy constraints. We hypothesize that given a set
of neural architectures for a given machine learning task, the architecture
which performs best without differential privacy constraints does not
necessarily also performs best under differential privacy constraints. Or
formulated differently, let there be two neural network architectures $A_{1}$
and $A_{2}$ and let $U(A)$ denote the accuracy of architecture $A$ without
differential privacy and $U_{d}(A)$ the accuracy with differential privacy. We
assume the case exists that $U(A_{1})>U(A_{2})$ and
$U_{d}(A_{1})<U_{d}(A_{2})$.

To test our hypothesis, we conducted an experiment on the standard CIFAR-10
image classification task which contains $60000$ color images in $10$ classes
krizhevsky_learning_2009 . We used Tensorflow abadi_tensorflow_2016  and the
Tensorflow-Privacy extension andrew_tensorflow_2018  as machine learning
libraries for the implementation. For our experiments we chose $8$
convolutional neural network architectures, including prominent examples from
the literature such as the LeNet-5 lecun_gradientbased_1998 , but also
architectures from other well-cited works arachchige_local_2019 ;
mcmahan_communicationefficient_2017 ; phan_adaptive_2017  and own creations.
The architectures differ mostly in the size and number of convolution,
pooling, and fully-connected layers. We chose convolutional architectures as
an initial setting, as their architecture configuration space is more complex
and interesting compared to simple feedforward networks with only fully
connected layers. See Table 1 for a full definition of the architectures used.
We trained each model architecture for $100$ epochs with DP-SGD, with a batch
size of $250$, $5$ micro batches, a fixed learning rate of $0.1$, a clipping
threshold of $1.0$ and at two noise multipliers of $0.01$ and $0.1$.

Table 1: The architectures used for the experiment $\\#1$ | $\\#2$
---|---
Layer type | Parameters | Layer type | Parameters
Convolution | 32 filters of 3x3, ReLU | Convolution | 32 filters of 3x3, ReLU
Max-Pooling | 2x2 | Max-Pooling | 2x2
Convolution | 64 filters of 3x3, ReLU | Convolution | 64 filters of 3x3, ReLU
Fully connected | 64 units, ReLU | Max-Pooling | 2x2
Softmax | 10 units | Convolution | 64 filters of 3x3, ReLU
|  | Fully connected | 64 units, ReLU
|  | Softmax | 10 units
$\\#3$ |  $\\#4$, lecun_gradientbased_1998  but with ReLU
Layer type | Parameters | Layer type | Parameters
Convolution | 32 filters of 3x3, ReLU | Convolution | 6 filters of 5x5, ReLU
Max-Pooling | 2x2 | Avg-Pooling | 2x2, stride 2
Convolution | 64 filters of 3x3, ReLU | Convolution | 16 filters of 5x5, ReLU
Max-Pooling | 2x2 | Avg-Pooling | 2x2, stride 2
Convolution | 64 filters of 3x3, ReLU | Fully connected | 120 units, ReLU
Fully connected | 128 units, ReLU | Fully connected | 84 units, ReLU
Softmax | 10 units | Softmax | 10 units
$\\#5$ |  $\\#6$, arachchige_local_2019
Layer type | Parameters | Layer type | Parameters
Convolution | 32 filters of 5x5, ReLU | Convolution | 32 filters of 3x3, ReLU
Avg-Pooling | 2x2, stride 2 | Convolution | 32 filters of 3x3, ReLU
Convolution | 64 filters of 5x5, ReLU | Max-Pooling | 2x2
Avg-Pooling | 2x2, stride 2 | Dropout | $0.25$
Fully connected | 200 units, ReLU | Convolution | 64 filters of 3x3, ReLU
Fully connected | 100 units, ReLU | Convolution | 64 filters of 3x3, ReLU
Softmax | 10 units | Max-Pooling | 2x2
|  | Fully connected | 512 units, ReLU
|  | Dropout | $0.5$
|  | Softmax | 10 units
$\\#7$, mcmahan_communicationefficient_2017  |  $\\#8$, phan_adaptive_2017
Layer type | Parameters | Layer type | Parameters
Convolution | 32 filters of 5x5, ReLU | Convolution | 32 filters of 5x5, ReLU
Avg-Pooling | 2x2, stride 2 | Convolution | 64 filters of 5x5, ReLU
Convolution | 64 filters of 5x5, ReLU | Fully connected | 384 units, ReLU
Avg-Pooling | 2x2, stride 2 | Fully connected | 192 units, ReLU
Fully connected | 512 units, ReLU | Softmax | 10 units
Softmax | 10 units |  |

##  4 Results

Figure 1 shows the training results of the $8$ different architectures on the
CIFAR-10 machine learning task, repeated three times in identical settings
with two different noise multipliers. We can clearly see that with a low noise
multiplier model $6$ performs best, but with a high noise multiplier model $6$
comes in second to last while model $2$ performs best. As the noise multiplier
is one of the hyperparameters that determines the privacy level (besides
epochs, batch size, and training set size, which are identical for all
results), this means that at different privacy levels, different architectures
perform best. In turn, this shows that, in a given set of architectures, it is
not always the case that a single architecture will perform best for all
differential privacy settings. Therefore, it is also not guaranteed that an
architecture that performs best without differential privacy necessarily
performs best in the differential privacy setting since its performance will
be dependent on the targeted privacy level. The raw results and the code used
to generate the graph can be found on GitHub
111https://github.com/FMorsbach/ArchitectureMatters.

Figure 1: Results of training 8 neural architectures for 100 epochs on the
CIFAR-10 image classification task with DP-SGD and a batch size of $250$, $5$
micro batches, a fixed learning rate of $0.1$ and a clipping threshold of
$1.0$, at two different noise multipliers ($0.01$ and $0.1$). (Best viewed in
color.)

##  5 Conclusion

In this paper, we investigated whether the architecture of a neural network
can influence the accuracy loss incurred through differential privacy
constraints. Our findings show that this is the case; neural network
architectures that perform well in the non-private setting will not
necessarily perform well in the differentially private setting. Furthermore,
we found that which architecture performs best also depends on the chosen
level of differential privacy.

The implications of our findings are two-fold: First, for research, our
findings show that architectures are only comparable at the same privacy
level, as the relative ranking of architectures might differ across privacy
levels. Second, for practitioners, our findings show that best practices,
experiences, or architectures from the non-private setting cannot be easily
transferred to the design of differentially private neural networks. Rather,
the model architecture has to be designed and tuned for specific differential
privacy settings; differential privacy cannot be treated as an afterthought.

Moreover, the design of neural networks is a challenging task that either
takes a lot of resources to try many different architectures through neural
architecture search elsken_neural_2019  or requires a good understanding of
neural network design from the modeler. Doing a neural architecture search in
the plain setting requires additional computations, but an extensive search
will not hurt the final accuracy of the model. Searching for an architecture
or optimizing hyperparameters in the differential privacy setting does,
however, consume privacy budget vanderveen_three_2018 . Therefore, spending
more time on architecture search or hyperparameter optimization will decrease
the privacy budget available for the actual training, which will decrease the
number of epochs available for training; hence, it will probably reduce the
accuracy of the final model. As a consequence, neural architecture search is
not easily applicable in the differential privacy setting.

Instead, a good understanding of how to design neural network architectures in
the differential privacy setting is needed in order to maximize the available
privacy budget for the actual training. But as shown by our findings, the
experience from the design of neural network architectures without
differential privacy constraints can only be transferred to the design of
differentially private neural network architectures to a limited degree.
Therefore, we need to derive new best practices for the design of
architectures tailored specifically for differentially private neural
networks, in order to improve the applicability of privacy-preserving machine
learning by reducing the accuracy loss incurred by differential privacy
constraints.

In our future work, we will derive best practices for the design of neural
architectures under differential privacy constraints. We will set up
additional experiments across multiple classic machine learning benchmarks.
Subsequently, we will analyse the results, derive a set of candidate best
practices and test those on different benchmarks. We aim to also incorporate
qualitative data from expert interviews to even further advance the best
practices for designing neural network architectures that perform well under
differential privacy constraints.

## Acknowledgments and Disclosure of Funding

This work was supported by funding from the topic Engineering Secure Systems
of the Helmholtz Association (HGF) and by KASTEL Security Research Labs.

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