# Low-energy scattering parameters: A theoretical derivation of the effective
range and scattering length for arbitrary angular momentum

Jordi Pera  Jordi Boronat  Departament de Física, Campus Nord B4-B5,
Universitat Politècnica de Catalunya, E-08034 Barcelona, Spain

###### Abstract

The most important parameters in the study of low-energy scattering are the
$s$-wave and $p$-wave scattering lengths, and the $s$-wave effective range. We
solve the scattering problem and find two useful formulas for the scattering
length and the effective range for any angular momentum, as long as the Wigner
threshold law holds. Using that formalism, we obtain a set of useful formulas
for the angular-momentum scattering parameters of four different model
potentials: hard-sphere, soft-sphere, spherical well, and well-barrier
potentials. The behavior of the scattering parameters close to Feshbach
resonances is also analyzed. Our derivations can be useful as hands-on
activities for learning scattering theory.

##  I Introduction

The natural way to learn about the forces acting between particles is to
observe their mutual interaction. Most of what we know about the micro-world
has been established by means of collision processes. In physics, a well-
studied collision process is the scattering of an incident particle from a
stationary target. A free particle (or rather a beam of such particles) with
known characteristics collides with a target particle, interacts with it, and
scatters into a modified free state. One then measures the energy, angular
distribution, and other characteristics of the scattered beam and infers from
them the nature and strength of the forces which, during the collision, acted
between the projectile and the target [1]. In the theoretical analysis of an
scattering process, the asymptotic free states of the scattered particles are
the most relevant. Therefore, it is not necessary to give an interpretation,
or even to have a detailed knowledge, of the state vector of the entire system
when the particles are close and interact strongly. This remark is important
in connection with quantum field theory, where the interpretation of the
states of strongly interacting fields is extremely difficult, if possible at
all. However, it is relatively easy to discuss the asymptotic free states [1].

Low-energy scattering parameters are fundamental in the theoretical
description of interacting many-body systems. By low-energy we mean that the
energy of the particles is much smaller than the typical excitation energy of
the target. The equations of state of dilute Bose and Fermi gases depend on
these parameters, which account for the full inter-atomic potential if the
density is low enough [2, 3], that is, the range of the interaction is much
smaller than the interparticle distance $\sim\rho^{-1/3}$. In particular, the
so called universal regime is the one in which the interaction is fully
described by a single parameter, the $s$-wave scattering length. In the
universal regime, any potential with the same scattering length gives the same
energy, independently of its particular shape. However, when the density of
the system is increased, effects of the shape of the potential arise and we
need to consider more scattering parameters, such as the $s$-wave effective
range. In some recent studies, these effects have been reported [4, 5]. In
this work, we find both scattering lengths and effective ranges. This leads to
a kind of inverse problem; that is, we know a few scattering parameters of the
physical system and the goal is to find a model potential with the same
scattering parameters. Then, we can use this model potential to study the
many-body properties of the system. Obviously, the solution is not unique and
model potentials as simple as possible are sought for this purpose.

This paper presents a well-detailed method that leads to integral expressions
that allow the calculation of the scattering parameters for any angular
momentum of the system, as long as the Wigner threshold law holds, that is,
potentials decaying at large r faster than $1/r^{n}$, with $n>2l+3$ for each
partial wave. This procedure is not only useful for researchers who want to
know the value of the scattering parameters, but also for students learning
scattering theory. The scattering problem has been widely studied and there
are many procedures leading to similar expressions. H. Bethe found the
expressions for $l=0$ scattering [6], and the generalization for any angular
momentum $l$ was done by L. B. Madsen in 2002 [7].

We show that the low-energy scattering parameters can be accurately calculated
by an integral equation method, allowing their determination for any angular
momentum. Using this result, we discuss some properties of the scattering
parameters around a Feshbach resonance [8]. The Feshbach resonance occurs when
the energy of a scattering state is very close to the energy of a bound state
and the scattering length diverges. When the divergence is crossed, a bound
state appears. We apply the formulas for getting analytical results of the
angular momentum scattering lengths and effective ranges for the following
potentials: hard-sphere, soft-sphere, spherical well, and well-barrier
potential.

The rest of the paper is organized as follows. Sec. II introduces basic
relations for low-energy scattering. Sec. III contains the derivation of the
formalism for the calculation of the scattering length and effective range for
any angular momentum $l$. In Sec. IV, the behavior of the parameters and the
Feshbach resonance are addressed. Sec. V is organized around the four
considered potentials: Subsec. A corresponds to the hard-sphere potential,
Subsec. B to the soft-sphere potential, Subsec. C to the spherical well
potential and Subsec. D to the well-barrier potential. Finally, the main
conclusions are discussed in Sec. VI.

##  II Low-energy scattering

The study of low-energy scattering events requires the characterization of
scattering parameters such as the scattering lengths and ranges. One
convenient approach to find these parameters is to study the phase shift due
to the effect of a potential. We need to have in mind that the phase shift is
related to the scattering parameters as they contain information of the
potential. Consider two waves that are initially in phase. One wave is sent
through an interacting system (or interacts with a target), while the other is
sent through free space. When the first wave emerges from the interacting
system (or target), it will no longer be in phase with the second wave. Thus,
we can describe a relation between the potential associated with an
interacting system and the phase shift such a system imparts on particles. Due
to that, the relations we will find will depend on the potential and the wave
function, both known quantities.

And this is precisely what we will find in this section. Afterwards, we will
see that if we expand the expression of the phase shift for low energies, the
coefficients of the expansion are the scattering parameters, which are
obviously related to the potential. We begin with the radial Schrödinger
equation for a central potential $V(r)$:

$u_{l}^{\prime\prime}(r)+\left(k^{2}-U(r)-\frac{l(l+1)}{r^{2}}\right)u_{l}(r)=0\
,$ (1)

with $k=\sqrt{2\mu E/\hbar^{2}}$ and $U(r)=2\mu V(r)/\hbar^{2}$, $\mu$ being
the reduced mass and $u_{l}(r)$ the reduced radial wave function. When there
is no potential and the energy is zero, the solution of Eq. (1) is just a
combination of power functions:

$u_{l}(r)=N\left[r^{l+1}+Mr^{-l}\right]\ ,$ (2)

with $N$ and $M$ two integration constants to determine. If the potential is
zero but the energy is not, the wave function is a combination of spherical
Bessel functions:

$u_{l}(r)=Arj_{l}(kr)+Brn_{l}(kr)\ ,$ (3)

with $A$ and $B$ two integration constants to determine. The boundary
condition that we will always apply is that the wave function must vanish at
$r=0$. This must happen because we are using the reduced wave function,
defined as $rR(r)$, where $R(r)$ is the radial part of the true wave function.
If we want $R(r)$ to be finite at the origin, $u_{l}(0)$ must be zero. If
$u_{l}(0)=0$, then $B=0$. We recall that the spherical Bessel function
$n_{l}(kr)$ is divergent at the origin, while $j_{l}(kr)$ is finite. Then
$u_{l}(r)$ reduces to

$u_{l}(r)=Arj_{l}(kr)\ .$ (4)

At large distances, the potential can be neglected, letting us more easily
compare the solutions in order to find the phase shift. Eq. (3) is the wave
coming out from an interacting system at large distances as it is the general
solution, while Eq. (4) is the wave coming out from the non-interacting system
with the boundary condition explained above as in this case the wave function
exists in all the space. As we want to compare what happens at large distance,
we need to know the asymptotic behavior of the Bessel functions for large
values of $kr$:

$\displaystyle j_{l}(kr)$ $\displaystyle\approx$
$\displaystyle\frac{1}{kr}\sin{\left(kr-\frac{l\pi}{2}\right)}$ (5)
$\displaystyle n_{l}(kr)$ $\displaystyle\approx$
$\displaystyle\frac{-1}{kr}\cos{\left(kr-\frac{l\pi}{2}\right)}\ .$ (6)

If we substitute Eqs. (5) and (6) into Eq. (3), we obtain the asymptotic
behavior of $u_{l}(r)$:

$u_{l}(r)=\overline{N}\frac{1}{k}\left[B_{1}\sin\left(kr-\frac{l\pi}{2}\right)-B_{2}\cos\left(kr-\frac{l\pi}{2}\right)\right]\
.$ (7)

where $A\equiv\overline{N}B_{1}$ and $B\equiv\overline{N}B_{2}$, with
$\overline{N}$ a constant that we will determine later on.

The phase shift $\delta$ is determined by considering that the large-distance
solution of the interacting system is the same as the solution with no
potential at the origin, but with an additional phase. Hence, we include it in
equation (4)):

$u_{l}(r)=\overline{A}rj_{l}(kr+\delta)\approx\overline{A}\frac{1}{k}\left[\sin\left(kr-\frac{l\pi}{2}\right)\cos\delta+\cos\left(kr-\frac{l\pi}{2}\right)\sin\delta\right]\
.$ (8)

In Eq. (8), we use $\overline{A}$ instead of $A$ to keep the constant $A$
coming from Eq. (4) different from the one coming from Eq. (3). If we compare
equations (7) and (8), we can see that $B_{1}=\cos{\delta}$,
$B_{2}=-\sin{\delta}$ and $\overline{N}=\overline{A}$. Therefore, the large-
distance solution of the interacting system in terms of the phase shift is:

$u_{l}(r)=\overline{N}r\left[j_{l}(kr)\cos\delta-n_{l}(kr)\sin\delta\right]\
.$ (9)

We need to be sure that the constant $\overline{N}$ makes this solution (9)
compatible with the zero-energy solution (2). We want this because we want our
large-distance solution to be also valid at zero energy. To this end, we
introduce the $x\to 0$ expansions of the Bessel functions, with $x$ being a
generic coordinate:

$\displaystyle j_{l}(x)$ $\displaystyle\approx$
$\displaystyle\frac{2^{l}l!}{(2l+1)!}x^{l}\bigg{[}1-\frac{x^{2}}{2(2l+3)}\bigg{]}=A_{l}x^{l}\bigg{[}1-\frac{x^{2}}{2(2l+3)}\bigg{]}\
,$ (10) $\displaystyle n_{l}(x)$ $\displaystyle\approx$
$\displaystyle-\frac{(2l)!}{2^{l}l!}x^{-l-1}\bigg{[}1+\frac{x^{2}}{2(2l-1)}\bigg{]}=-B_{l}x^{-l-1}\bigg{[}1+\frac{x^{2}}{2(2l-1)}\bigg{]}\
.$ (11)

Inserting (10) and (11) into Eq. (9) yields the low-energy limit of the
solutions at large distance, we have set $x=kr$:

$u_{l}(r)\approx\overline{N}A_{l}k^{l}\cos{\delta}\left[r^{l+1}+\frac{B_{l}}{A_{l}}r^{-l}k^{-2l-1}\tan{\delta}\right]\
.$ (12)

The previous limit can be done because, although we are in the range of large
$r$, $r$ is supposed to be finite while the energy is tending to zero. If we
compare equations (2) and (12), we can relate the integration constants:

$\overline{N}=\frac{Nk^{-l}}{A_{l}\cos{\delta}}\quad;\quad M=\lim_{k\to
0}\frac{B_{l}}{A_{l}}k^{-2l-1}\tan{\delta}\ .$ (13)

In the limit $k\to 0$, the constant $M$ is not trivial and we need to
introduce the definition of the cotangent expansion:

$k^{2l+1}\cot{\delta}\approx\frac{B_{l}}{A_{l}}\frac{1}{a_{l}^{2l}}\bigg{[}-\frac{1}{a_{l}}+\frac{1}{2}r_{l}^{\text{eff}}k^{2}\bigg{]}\
.$ (14)

This expansion (Eq. (14)) is already known and it can be found elsewhere [1,
3, 9, 10, 11]. As we are in the limit $k\to 0$, if we want Eq. (14) to be
finite, the cotangent must diverge. In Eq. (14), $a_{l}$ is the scattering
length and $r_{l}^{\text{eff}}$ is the effective range. Right now, the
scattering parameters are only the coefficients of the expansion, later on, at
the end of Sec. III and in Sec. IV we will give them a physical meaning.
Although we have introduced here the expansion of the cotangent without
proving it, we show that the tangent, and hence the cotangent, can be expanded
in the manner of Eq. (14) at the beginning of Sec. III in Eq. (33). However,
we point out that Eq. (14) is not usually defined as we have done. In some of
the references given above, it is written as

$k^{2l+1}\cot{\delta}\approx-\frac{1}{a_{l}^{*}}+\frac{1}{2}r_{l}^{\text{eff},*}k^{2}\
.$ (15)

The reason why we do not use Eq. (15) is the fact that $a_{l}^{*}$ and
$r_{l}^{\text{eff},*}$ only represent lengths for $l=0$. For $l>0$, these
scattering parameters do not have dimensions of length. However, using Eq.
(14), $a_{l}$ and $r_{l}^{\text{eff}}$ always have dimensions of length.
Additionally, we factor out $B_{l}/A_{l}$ because we want $a_{l}=R$ for any
$l$ in the case of a hard-sphere potential with diameter $R$ (see Sec. II of
Supplementary material II [12]). As we mentioned right below Eq. (14), the
scattering parameters are the coefficients of the expansion: $a_{l}$ is the
coefficient of the zero order term, and $r_{l}^{\text{eff}}$ is the one
proportional to $k^{2}$. If we invert equation (14) and we apply the limit
$k\to 0$, we can find $M$:

$M=\frac{B_{l}}{A_{l}}\bigg{(}-\frac{A_{l}}{B_{l}}a_{l}^{2l+1}\bigg{)}=-a_{l}^{2l+1}\
.$ (16)

With that, the solution at zero energy ($k=0$) becomes

$u_{l}(r,k=0)=N\Big{[}r^{l+1}-a_{l}^{2l+1}r^{-l}\Big{]}\ ,$ (17)

and at finite energy,

$u_{l}(r)=\frac{Nk^{-l}}{A_{l}}r\Big{[}j_{l}(kr)-n_{l}(kr)\tan{\delta}\Big{]}\
.$ (18)

The next step is to relate the phase shift $\delta$ to the potential $U(r)$.
In order to do so, we shall combine the true differential equation with the
one that the regular Bessel function satisfies:

$\displaystyle\bigg{[}(rj_{l}(kr))^{\prime\prime}+\bigg{(}k^{2}-\frac{l(l+1)}{r^{2}}\bigg{)}rj_{l}(kr)=0\bigg{]}$
$\displaystyle\times$ $\displaystyle u_{l}(r)\ ,$ (19)
$\displaystyle\bigg{[}u_{l}^{\prime\prime}(r)+\bigg{(}k^{2}-U(r)-\frac{l(l+1)}{r^{2}}\bigg{)}u_{l}(r)=0\bigg{]}$
$\displaystyle\times$ $\displaystyle(rj_{l}(kr))\ .$ (20)

By subtracting the above equations one gets

$\frac{d}{dr}\bigg{(}(rj_{l}(kr))^{\prime}u_{l}(r)-rj_{l}(kr)u_{l}^{\prime}(r)\bigg{)}=-U(r)rj_{l}(kr)u_{l}(r)\
.$ (21)

We can integrate now the entire equation between zero and infinity for regular
(physical) potentials:

$\bigg{[}(rj_{l}(kr))^{\prime}u_{l}(r)-rj_{l}(kr)u_{l}^{\prime}(r)\bigg{]}_{0}^{\infty}=-\int_{0}^{\infty}U(r)rj_{l}(kr)u_{l}(r)dr\
.$ (22)

When evaluating the left-hand side, the value at $r=0$ is zero, so we need
only the value at $r\rightarrow\infty$:

$\bigg{[}(rj_{l}(kr))^{\prime}u_{l}(r)-rj_{l}(kr)u_{l}^{\prime}(r)\bigg{]}_{\infty}=-\int_{0}^{\infty}U(r)rj_{l}(kr)u_{l}(r)dr\
.$ (23)

Using the asymptotic approximation for the Bessel functions (5,6), the wave
function and its derivative at infinity are:

$\displaystyle\lim_{r\rightarrow\infty}u_{l}(r)$ $\displaystyle=$
$\displaystyle\frac{Nk^{-l}}{A_{l}}\frac{1}{k}\bigg{[}\sin{\bigg{(}kr-\frac{l\pi}{2}\bigg{)}}+\cos{\bigg{(}kr-\frac{l\pi}{2}\bigg{)}}\tan{\delta}\bigg{]}_{\infty}\
,$ (24) $\displaystyle\lim_{r\rightarrow\infty}u_{l}^{\prime}(r)$
$\displaystyle=$
$\displaystyle\frac{Nk^{-l}}{A_{l}}\bigg{[}\cos{\bigg{(}kr-\frac{l\pi}{2}\bigg{)}}-\sin{\bigg{(}kr-\frac{l\pi}{2}\bigg{)}}\tan{\delta}\bigg{]}_{\infty}\
.$ (25)

After some algebraic manipulations, we find:

$\bigg{[}(rj_{l}(kr))^{\prime}u_{l}(r)-rj_{l}(kr)u_{l}^{\prime}(r)\bigg{]}_{\infty}=\frac{Nk^{-l}}{A_{l}}\frac{1}{k}\tan{\delta}\
,$ (26)

and the tangent of the phase shift is:

$\tan{\delta}=-A_{l}k^{(l+1)}\int_{0}^{\infty}U(r)rj_{l}(kr)\frac{u_{l}(r)}{N}dr\
.$ (27)

With the expression (27) we can know the phase shift, but we need to relate it
with the cotangent expansion (Eq. (14)) because our goal is to find the
scattering parameters. Therefore, we calculate the cotangent from the
expression (27) by considering an expansion. First, we consider that the
tangent can be written as

$\tan{\delta}\approx-\frac{A_{l}}{B_{l}}k^{2l+1}\big{[}c_{1}+\frac{1}{2}c_{2}k^{2}\big{]}\
,$ (28)

Eq. (28) is valid for low energies (low $k$), and thus,

$\cot{\delta}=-\frac{B_{l}}{A_{l}}\frac{1}{k^{2l+1}}\frac{1}{c_{1}}\frac{1}{1+\cfrac{c_{2}}{2c_{1}}k^{2}}\approx-\frac{B_{l}}{A_{l}}\frac{1}{k^{2l+1}}\frac{1}{c_{1}}\bigg{(}1-\frac{1}{2}\frac{c_{2}}{c_{1}}k^{2}\bigg{)}\
.$ (29)

Finally, by comparing with Eq. (14), we can relate the coefficients $c_{1}$
and $c_{2}$ in expansion (28) to the scattering parameters,

$c_{1}=a_{l}^{2l+1}\quad\quad c_{2}=a_{l}^{2l+2}r_{l}^{\text{eff}}\ .$ (30)

Up to this point, we have related the tangent of the phase shift to the
potential and the wave function through an integral (27). The scattering
parameters appear as coefficients of the low-energy expansion of the tangent
of the phase shift.

##  III Scattering parameters

In this section, we will find the expressions for the scattering parameters in
terms of the potential and the wave function. In Eq. (28), we split the
expression of the tangent into two terms, one proportional to $k^{2l+1}$ and
another proportional to $k^{2l+3}$, whose coefficients have been found at the
end of the previous section. Now, we break Eq. (27) in two parts in order to
compare both expressions and find the coefficients, hence finding the
scattering parameters.

Everything done in this section is under the assumption of a regular
potential, that is, a potential with no divergences (except at the origin)
that decays faster than $1/r^{n}$ with $n>2l+3$ for each partial wave at
infinity (Wigner threshold). We will break the integral:

$\tan{\delta}=-A_{l}k^{l+1}\int_{0}^{\infty}U(r)rj_{l}(kr)\frac{u_{l}(r)}{N}dr\
,$ (31)

in two parts in order to obtain terms proportional to $k^{2l+1}$ and
$k^{2l+3}$. To this end, we expand the Bessel function (Eq. (10)) and the wave
function:

$u_{l}(r)=N\Big{(}u_{l}^{(0)}(r)+k^{2}u_{l}^{(2)}(r)+\ldots\Big{)}\ ,$ (32)

around $k=0$. The super-indices in Eq.(32) indicate the order of the
expansion. The series (32) has only even terms. The odd terms are zero because
the radial Schrödinger equation depends on $k$ as $k^{2}$, causing
$u_{l}(r,k)=u_{l}(r,-k)$. With this, one obtains

$\tan{\delta}\approx-\frac{A_{l}}{B_{l}}k^{2l+1}\bigg{[}A_{l}B_{l}\int_{0}^{\infty}U(r)r^{l+1}u_{l}^{(0)}(r)dr+\frac{1}{2}(2A_{l}B_{l})k^{2}\int_{0}^{\infty}U(r)r^{l+1}\bigg{(}u_{l}^{(2)}(r)-\frac{r^{2}u_{l}^{(0)}(r)}{2(2l+3)}\bigg{)}dr\bigg{]}\
.$ (33)

Comparing the latter equation with Eq. (28), one can identify the coefficients
$c_{1}$ and $c_{2}$ from Eq. (30):

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2l+1}\int_{0}^{\infty}U(r)r^{l+1}u_{l}^{(0)}(r)dr\ ,$
(34) $\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{2l+1}\int_{0}^{\infty}U(r)r^{l+1}\bigg{(}u_{l}^{(2)}(r)-\frac{r^{2}u_{l}^{(0)}(r)}{2(2l+3)}\bigg{)}dr\
,$ (35)

where we have used $A_{l}B_{l}=1/(2l+1)$, which can be derived from Eqs. (10)
and (11). In this way, we already get $a_{l}$ because $c_{1}=a_{l}^{2l+1}$. To
continue with $c_{2}$ and thus to find $r_{l}^{\text{eff}}$, we need to know
what equations $u_{l}^{(0)}(r)$ and $u_{l}^{(2)}(r)$ satisfy. To do so we plug
the expansion of the wave function (32) into the Schrödinger equation:

$u_{l}^{(0)^{\prime\prime}}(r)+k^{2}u_{l}^{(2)^{\prime\prime}}(r)+\bigg{(}k^{2}-U(r)-\frac{l(l+1)}{r^{2}}\bigg{)}\Big{(}u_{l}^{(0)}(r)+k^{2}u_{l}^{(2)}(r)\Big{)}=0$
(36)

At order $k^{0}$ the differential equation is:

$u_{l}^{(0)^{\prime\prime}}(r)-\bigg{[}\frac{l(l+1)}{r^{2}}+U(r)\bigg{]}u_{l}^{(0)}(r)=0$
(37)

and thus $u_{l}^{(0)}(r)$ is nothing else than the wave function at zero
energy. Here, we can understand the Wigner threshold: if we want $c_{1}$ to
exist, the potential must decay faster than $1/r^{n}$ with $n>2l+3$ for each
partial wave. We recall that $u_{l}^{(0)}(r)$ at infinity goes as
$r^{l+1}-a_{l}^{2l+1}r^{-l}$. At order $k^{2}$, one gets:

$u_{l}^{(2)^{\prime\prime}}(r)-\bigg{[}\frac{l(l+1)}{r^{2}}+U(r)\bigg{]}u_{l}^{(2)}(r)=-u_{l}^{(0)}(r)\
.$ (38)

It looks similar to the one for $u_{l}^{(0)}(r)$ (37) but now there is a
source term, which turns out to be $u_{l}^{(0)}(r)$ itself. Combining both
equations:

$\displaystyle\bigg{[}u_{l}^{(0)^{\prime\prime}}(r)-\bigg{(}\frac{l(l+1)}{r^{2}}+U(r)\bigg{)}u_{l}^{(0)}(r)=0\bigg{]}$
$\displaystyle\times$ $\displaystyle u_{l}^{(2)}(r)$ (39)
$\displaystyle\bigg{[}u_{l}^{(2)^{\prime\prime}}(r)-\bigg{(}\frac{l(l+1)}{r^{2}}+U(r)\bigg{)}u_{l}^{(2)}(r)=-u_{l}^{(0)}(r)\bigg{]}$
$\displaystyle\times$ $\displaystyle u_{l}^{(0)}(r)$ (40)

and integrating, one obtains:

$\left[u_{l}^{(0)^{\prime}}(r)u_{l}^{(2)}(r)-u_{l}^{(2)^{\prime}}(r)u_{l}^{(0)}(r)\right]_{0}^{r}=\int_{0}^{r}{u_{l}^{(0)}}^{2}(r)dr\
.$ (41)

The functions $u_{l}^{(0)}(r)$ and $u_{l}^{(2)}(r)$ are zero at the origin for
any energy because $u_{l}(r)$ is zero at the origin due to boundary
conditions. As $u_{l}^{(0)}(r)$ and $u_{l}^{(2)}(r)$ are the terms in the
expansion of $u_{l}(r)$ (Eq. (32)), if $u_{l}(r)$ is zero at the origin for
any energy, then each term of the expansion must be also zero at the origin.
Therefore:

$u_{l}^{(2)^{\prime}}(r)=\frac{u_{l}^{(0)^{\prime}}(r)}{u_{l}^{(0)}(r)}u_{l}^{(2)}(r)-\frac{1}{u_{l}^{(0)}(r)}\int_{0}^{r}{u_{l}^{(0)}}^{2}(r)dr\
.$ (42)

Using the differential equations for $u_{l}^{(0)}(r)$ and $u_{l}^{(2)}(r)$ we
can rewrite the constant $c_{2}$ from Eq. (35) in the following way (see Sec.
I of Supplementary material I [13] for the explicit derivation):

$\frac{(2l+1)c_{2}}{2}=r^{l+1}u_{l}^{(2)^{\prime}}(r)\Big{|}_{0}^{\infty}-(l+1)r^{l}u_{l}^{(2)}(r)\Big{|}_{0}^{\infty}+\frac{1}{2(2l+3)}\bigg{\\{}(l+3)r^{l+2}u_{l}^{(0)}(r)\Big{|}_{0}^{\infty}-r^{l+3}u_{l}^{(0)^{\prime}}(r)\Big{|}_{0}^{\infty}\bigg{\\}}\
.$ (43)

In Eq. (43), everything is zero in the inferior limit $r=0$. Substituting
$u_{l}^{(2)^{\prime}}(r)$ by Eq. (42), we get:

$\frac{(2l+1)c_{2}}{2}=\lim_{r\rightarrow\infty}\bigg{\\{}\frac{(l+3)r^{l+2}u_{l}^{(0)}}{2(2l+3)}-\frac{r^{l+3}u_{l}^{(0)^{\prime}}}{2(2l+3)}-\frac{u_{l}^{(2)}}{u_{l}^{(0)}}[(l+1)r^{l}u_{l}^{(0)}-r^{l+1}u_{l}^{(0)^{\prime}}]-\frac{r^{l+1}}{u_{l}^{(0)}}\int_{0}^{r}{u_{l}^{(0)}}^{2}(r)dr\bigg{\\}}\
.$ (44)

In order to continue, we need to know $u_{l}^{(0)}(r)$ and $u_{l}^{(2)}(r)$ in
the limit $r\to\infty$. To this end, we combine Eqs. (18), (28), (10) and (11)
to write:

$u_{l}(r)=N\bigg{\\{}r^{l+1}-a_{l}^{2l+1}r^{-l}+k^{2}\bigg{[}-\frac{r^{l+3}}{2(2l+3)}-a_{l}^{2l+1}\frac{r^{2-l}}{2(2l-1)}-\frac{1}{2}a_{l}^{2l+2}r_{l}^{\text{eff}}r^{-l}\bigg{]}\bigg{\\}}\
.$ (45)

In Eq. (45), the term proportional to $k^{0}$ is just $u_{l}^{(0)}(r)$ and the
one proportional to $k^{2}$ is $u_{l}^{(2)}(r)$:

$\displaystyle u_{l}^{(0)}(r)$ $\displaystyle=$ $\displaystyle
r^{l+1}-a_{l}^{2l+1}r^{-l}$ (46) $\displaystyle u_{l}^{(2)}(r)$
$\displaystyle=$
$\displaystyle-\frac{r^{l+3}}{2(2l+3)}-a_{l}^{2l+1}\frac{r^{2-l}}{2(2l-1)}-\frac{1}{2}a_{l}^{2l+2}r_{l}^{\text{eff}}r^{-l}\
.$ (47)

The next step is the substitution of $u_{l}^{(0)}$, $u_{l}^{(0)^{\prime}}$ and
$u_{l}^{(2)}$ into the expression of $c_{2}$ (44). After some manipulations
(see Sec. II of Supplementary material I [13] for further details), we obtain:

$c_{2}=\frac{2}{2l+1}\lim_{r\rightarrow\infty}\bigg{\\{}\frac{r^{2l+3}}{2l+3}-a_{l}^{2l+1}r^{2}+\frac{a_{l}^{4l+2}r^{1-2l}}{1-2l}-\int_{0}^{r}{u_{l}^{(0)}}^{2}(r)dr\bigg{\\}}\
.$ (48)

Finally, considering the relation between $c_{2}$ and $r_{l}^{\text{eff}}$, we
obtain the effective range:

$r_{l}^{\text{eff}}=\frac{c_{2}}{a_{l}^{2l+2}}=\frac{2}{(2l+1)a_{l}^{2l+2}}\lim_{r\rightarrow\infty}\bigg{\\{}\frac{r^{2l+3}}{2l+3}-a_{l}^{2l+1}r^{2}+\frac{a_{l}^{4l+2}r^{1-2l}}{1-2l}-\int_{0}^{r}{u_{l}^{(0)}}^{2}(r)dr\bigg{\\}}\
.$ (49)

Applying the limit $r\to\infty$ in the above expression yields two different
expressions depending on whether $l=0$ or $l>0$. The reason for this
difference is the term containing a distance to the power $(1-2l)$. This term
goes to zero at infinity if $l>0$; however if $l=0$, this term diverges. The
expressions after this split are:

$r_{0}^{\text{eff}}=\frac{2}{a_{0}^{2}}\lim_{r\rightarrow\infty}\bigg{[}\frac{r^{3}}{3}-a_{0}r^{2}+a_{0}^{2}r-\int_{0}^{r}{u_{0}^{(0)}}^{2}(r)dr\bigg{]}\
,$ (50)
$r_{l>0}^{\text{eff}}=\frac{2}{(2l+1)a_{l}^{2l+2}}\lim_{r\rightarrow\infty}\bigg{[}\frac{r^{2l+3}}{2l+3}-a_{l}^{2l+1}r^{2}-\int_{0}^{r}{u_{l}^{(0)}}^{2}(r)dr\bigg{]}\
.$ (51)

All the power terms give zero at $r=0$. That means we can write the full
expressions in integral form, containing the zero as the inferior limit, and
take the limit of $r\to\infty$. Finally, the resulting expressions for the
effective range for any $l$ are:

$\boxed{r_{0}^{\text{eff}}=\frac{2}{a_{0}^{2}}\int_{0}^{\infty}\Big{[}(r-a_{0})^{2}-{u_{0}^{(0)}}^{2}(r)\Big{]}dr}$
(52)
$\boxed{r_{l>0}^{\text{eff}}=\frac{2}{(2l+1)a_{l}^{2l+2}}\int_{0}^{\infty}\Big{[}r^{2l+2}-2a_{l}^{2l+1}r-{u_{l}^{(0)}}^{2}(r)\Big{]}dr}$
(53)

For any $l$, the integrand vanishes in the limit $r\rightarrow\infty$ since in
this limit the wave function squared matches the power functions. If the wave
function were to have this behavior everywhere, not only at infinity, the
effective range would be zero. As in general this is not the case, the
effective range takes into account the separation between the true solution
and the dominant terms of the large distance scattering solution.

The above formulas are equivalent to the ones found by L. B. Madsen in Ref.
[7]:

$r_{l}^{\text{eff},*}=\lim_{k\rightarrow
0}2\bigg{(}\int_{0}^{R}dr[v_{l,0}(r)v_{l,k}(r)-u_{l,0}(r)u_{l,k}(r)]-\frac{1}{a_{l}^{*}}\frac{R^{2}}{(2l+1)}-\frac{[(2l-1)!!]^{2}}{(2l-1)R^{2l-1}}\bigg{)}\
.$ (54)

In Ref. [7], $u_{l,k}(r)$ is the solution to the radial Schrödinger equation
for any energy (that is our $u_{l}(r)$); $u_{l,0}(r)$ is the same but at zero
energy (that is our $u_{l}^{(0)}(r)$); $v_{l,k}(r)$ is the solution to the
radial Schrödinger equation for any energy when the potential is zero and that
is finite at the origin (it is the function that we have at Eq. (4)); and
finally, $v_{l,0}(r)$ is the limit of zero energy of the previous function
which is a constant times $r^{l+1}$. All the functions are taken to be zero at
the origin. Differences in our approach and that of Ref. [7] to normalizing
$u_{l,k}(r)$ and $v_{l,k}(r)$ are described in Sec. III of the Supplemental
material I [13].

Our expressions for the effective range are simpler to use than the ones from
Ref. [7] (our Eq. (54)), since they only include integrals that converge,
whereas Eq. (54) additionally requires setting the limits $k\to 0$ and
$R\to\infty$. Apart from that, the effective ranges are defined in two
slightly different ways (see Sec. III of Sup. material I [13]).

For completeness, the formula for the $l$-wave scattering length $a_{l}$ is:

$\boxed{a_{l}^{2l+1}=\frac{1}{2l+1}\int_{0}^{\infty}U(r)r^{l+1}u_{l}^{(0)}(r)dr\
.}$ (55)

Notice that that Eqs. (52,53,55) need the wave function $u_{l}^{(0)}$, which
is the zero-energy solution of the radial Schrödinger equation with the
boundary condition $u_{l}^{(0)}(r=0)=0$ (the wave function must vanish at the
origin) and properly normalized to behave as $r^{l+1}-a_{l}^{2l+1}r^{-l}$ when
$r\rightarrow\infty$.

##  IV Resonance scattering

In this section, we discuss the meaning of positive and negatives values of
the scattering length and when divergences appear.

For any $l$, the scattering length is the point where the large-distance
behavior of the wave function extrapolates to a value of zero, as seen in Fig.
1. This fact comes from the large distance solution
$u_{l}(r)=r^{l+1}-a_{l}^{2l+1}r^{-l}$. If $r=a_{l}$, $u_{l}(r)$ is zero. This
picture allows us to have an idea about the scattering length without looking
at the explicit formulas. For example, if the potential is repulsive, then at
distances close to the origin the wave function will be small, and as we move
away from it, the wave function will increase. We will have a concave up
function and thus the intersection of the large-distance solution with the
horizontal axis will be positive. This can also be verified looking at Eq.
(55): if all the terms inside the integral are positive, the integral, and
hence $a_{l}$, will be positive.

The case of attractive potentials is richer than that of repulsive potentials
since the strength of the attractive potential must be considered. Weakly
attractive potentials exhibit the opposite behavior of repulsive potentials.
In the weakly attractive case, the tendency to be near the origin is larger,
making the wave function to be concave down with a negative scattering length.
If we increase the strength of the potential, the concavity will decrease and
the scattering length will become more negative. Once we reach a critical
value of the potential, the intersection distance will be at negative
infinity. This point is particularly interesting, and we will comment on it
later on (Feshbach resonance). After crossing this point, the scattering
length becomes divergent, but at positive infinity. From this situation, the
evolution will be the same as the one we have explained: with increasing
potential, the scattering length will start to diminish, will become zero, and
then will start approaching negative infinity. After crossing the divergence,
the cycle will be repeated. This process is illustrated in Fig. (1), where we
plot four cases using a soft-sphere potential for (1) and a spherical-well
potential for (1), (1) and (1). The first case (1) corresponds to a repulsive
potential and the intersection with the $r$ axis is positive. In (1), (1) and
(1), we plot the wave function for three attractive potentials of increasing
strength in order to show the regimes explained above. In (1), the potential
is weak and we have not reached the critical value. In (1), we have crossed
the Feshbach resonance. In (1), we show a case which is before the second
Feshbach resonance.

Figure 1: The wave functions (solid blue lines) are shown for four different
potentials, along with linear approximations to their behavior at large $r$
(orange dotted lines). The point at which the linear approximation crosses
zero yields the value of the scattering length, $a_{0}$. The potentials are
all zero for $r>1$ and take the values $V_{0}$=20, -2, -5, and -21.5 for
$r<1$.

We want to analyze the critical point, that is, the situation in which the
scattering length diverges, going from negative to positive infinity. The
Schrödinger equation always has two kind of solutions: scattering states and
bound states. Scattering states can have any positive energy and at infinity
behave like a non-normalizable wave. Bound states form a discrete set of
solutions that have negative energy, vanish at infinity, and are normalizable.
Upon increasing the strength of an attractive potential, as bound states are
discrete, there will be certain values in which a scattering state with very
low energy will become a bound state with negative energy close to zero. As
bound states vanish at infinity, they cannot be described with the large-
distance solution ($u_{l}(r)=r^{l+1}-a_{l}^{2l+1}r^{-l}$) that we have used to
this point. The large-distance solution of a bound state is $Nr^{-l}$, with
$N=0$ if $l=0$. In order to be able to normalize the mode, the term
$a_{l}^{2l+1}r^{-l}$ should dominate $r^{l+1}$. This anomaly causes the
scattering length to diverge. In fact, in this very limit (called the unitary
limit) the scattering length is undefined because it can be $\pm\infty$. This
process in which a scattering state transitions to a bound state as the
scattering length diverges from negative to positive infinity due to an
increasingly attractive potential is called a Feshbach resonance [8].

This divergent behavior around certain values of the potential can be modelled
if we Taylor-expand the inverse of the scattering length around these values:

$\frac{1}{a_{l}}(V_{0})=\frac{1}{a_{l}}\bigg{|}_{V_{c}}+\bigg{(}\frac{1}{a_{l}}\bigg{)}^{\prime}\bigg{|}_{V_{c}}(V_{0}-V_{c})+\frac{1}{2}\bigg{(}\frac{1}{a_{l}}\bigg{)}^{\prime\prime}\bigg{|}_{V_{c}}(V_{0}-V_{c})^{2}+\ldots$
(56)

Requiring that the inverse of the scattering length at these values is zero,
and ignoring orders higher than 1, we obtain:

$a_{l}=\frac{C}{V_{0}-V_{c}}\quad\text{where}\quad
C=\bigg{[}\bigg{(}\frac{1}{a_{l}}\bigg{)}^{\prime}\bigg{|}_{V_{c}}\bigg{]}^{-1}$
(57)

Feshbach resonances appear when the energy of a scattering state matches the
energy of a bound state. Although in the procedure explained above we have
crossed the resonance by means of increasing the strength of the potential, we
can leave the potential unchanged and make use of external sources such as
magnetic fields to vary the effective interaction and cross the resonance
either-way. Magnetically tunable Feshbach resonances are very important in
ultracold quantum gases. Their use provides means of varying the scattering
length almost at will. [14] For instance, one can go from a Fermi gas with
pairing (BCS) with $a<0$ to a Bose-Einstein condensate (BEC) with $a>0$ by
crossing a Feshbach resonance. [15]

As we have found a compact formula for the effective range, it is interesting
to explore the behavior of the effective range whenever $a_{l}$ is zero or
$\pm\infty$. The case $a_{l}=0$ is only interesting from a mathematical point
of view. Physically, it is not relevant because what matters is the product of
$r_{l}^{\text{eff}}$ times $a_{l}^{2l+2}$. This product is the constant
$c_{2}$ introduced in Sec. II, which is one of the coefficients of the
expansion of the tangent of the phase shift. On the other hand, the case
$a_{l}=\pm\infty$ is physically important, since this corresponds to the
behavior around the Feshbach resonance. To this end, we write a compact
expression of $r_{l}^{\text{eff}}$:

$r_{l}^{\text{eff}}=\frac{2}{(2l+1)a_{l}^{2l+2}}\int_{0}^{\infty}\Big{[}r^{2l+2}-2a_{l}^{2l+1}r+a_{l}^{4l+2}r^{-2l}\delta_{l,0}-{u_{l}^{(0)}}^{2}(r)\Big{]}dr\
.$ (58)

First, we analyze its behavior when $a_{l}\to 0$. By looking at the dominant
terms in this limit, one gets:

$r_{l}^{\text{eff}}\rightarrow\frac{2}{(2l+1)a_{l}^{2l+2}}\int_{0}^{\infty}\Big{[}r^{2l+2}-{u_{l}^{(0)}}^{2}(r)\Big{]}dr\rightarrow\pm\infty\
.$ (59)

The integral in Eq. (59) is always finite unless $a_{l}$ diverges. For the
case where $a_{l}$ does not diverge, the effective range can only diverge if
the fraction preceding the integral diverges, which would necessarily be due
to $a_{l}^{2l+2}$. However, since the product of $a_{l}^{2l+2}$ and
$r_{l}^{\text{eff}}$ is finite,

$r_{l}^{\text{eff}}a_{l}^{2l+2}\rightarrow\frac{2}{(2l+1)}\int_{0}^{\infty}\Big{[}r^{2l+2}-{u_{l}^{(0)}}^{2}(r)\Big{]}dr\
,$ (60)

this limit (Eq. (59)) is not physically relevant. We recall that the product
$r_{l}^{\text{eff}}a_{l}^{2l+2}$ is what appears in the expansion of the
tangent of delta, see Eqs. (28) and (30).

When considering the case in which $a_{l}\to\pm\infty$, it is useful to split
the wave function into two terms (the derivation of which is found in Sec. IV
of the Supplementary material I [13]):

$u_{l}^{(0)}(r)=\frac{lu_{l}^{(0)}(r)+r{u^{\prime}}_{l}^{(0)}(r)}{1+2l}-\frac{r{u^{\prime}}_{l}^{(0)}(r)-(l+1)u_{l}^{(0)}(r)}{1+2l}\
.$ (61)

The first term goes to positive infinity as $r^{l+1}$ while the second term
goes to positive infinity as $a_{l}^{2l+1}r^{-l}$. We recall that at infinity
$u_{l}^{(0)}(r)$ is $r^{l+1}-a_{l}^{2l+1}r^{-l}$. As long as $a_{l}$ is non-
zero, we can redefine the two parts as two functions:

$\frac{lu_{l}^{(0)}(r)+r{u^{\prime}}_{l}^{(0)}(r)}{1+2l}\equiv
f_{l}(r)\quad;\quad\frac{r{u^{\prime}}_{l}^{(0)}(r)-(l+1)u_{l}^{(0)}(r)}{1+2l}\equiv
a_{l}^{2l+1}h_{l}(r)\ .$ (62)

Both $f_{l}(r)$ and $h_{l}(r)$ at $r=0$ are zero. And at $r\to\infty$, they
are $r^{l+1}$ and $r^{-l}$ respectively. The differential equations that
$f_{l}(r)$ and $h_{l}(r)$ satisfy can be found in Sec. IV of the Supplementary
material I [13]. With this, the wave function can be rewritten as
$u_{l}^{(0)}(r)=f_{l}(r)-a_{l}^{2l+1}h_{l}(r)$. If we apply this
transformation to the effective range, we obtain:

$r_{l}^{\text{eff}}=\frac{2}{(2l+1)a_{l}^{2l+2}}\bigg{\\{}\int_{0}^{\infty}\Big{[}r^{2l+2}-f_{l}^{2}(r)\Big{]}dr-2a_{l}^{2l+1}\int_{0}^{\infty}\Big{[}r-f_{l}(r)h_{l}(r)\Big{]}+a_{l}^{4l+2}\int_{0}^{\infty}\Big{[}r^{-2l}\delta_{l,0}-h_{l}^{2}(r)\Big{]}dr\bigg{\\}}\
.$ (63)

The above integrals are finite. This can be seen by noting that
$f_{l}^{2}(0)$, $f_{l}(0)h_{l}(0)$, and $h_{l}^{2}(0)$ are zero, and that
$\lim_{r\rightarrow\infty}\Big{[}r^{2l+2}-f_{l}^{2}(r)\Big{]}=0$,
$\lim_{r\rightarrow\infty}\Big{[}r-f_{l}(r)h_{l}(r)\Big{]}=0$, and
$\lim_{r\rightarrow\infty}\Big{[}r^{-2l}\delta_{l,0}-h_{l}^{2}(r)\Big{]}=0$.

We distinguish two cases, $l=0$ and $l>0$. In the first case, when
$a_{0}\to\pm\infty$:

$r_{0}^{\text{eff}}\rightarrow
2\int_{0}^{\infty}\Big{[}1-h_{l}^{2}(r)\Big{]}dr\rightarrow\text{finite
value}\ .$ (64)

Instead, for $l>0$ and $a_{l}\to\pm\infty$:

$r_{l}^{\text{eff}}\rightarrow-\frac{2}{(2l+1)}a_{l}^{2l}\int_{0}^{\infty}h_{l}^{2}(r)dr\rightarrow-\infty\
.$ (65)

As a summary, whenever $a_{l}$ is zero, $r_{l}^{\text{eff}}$ diverges. For a
diverging scattering length, we have two situations: for $l=0$, the effective
range is finite, while for $l>0$, it diverges to $-\infty$.

##  V Examples

In this section, we apply the scattering formalism to obtain the scattering
length and effective ranges of model potentials that can be analytically
integrated. We just present the final formulas, the development can be found
in the Supplementary material II. [12] Moreover, in Sup. Mat. II [12], we have
computed the parameters for the Pöschl-Teller potential. However, this case
has been done numerically, not analytically, and this is why it is not
presented here.

###  V.1 Hard-sphere potential

The hard-sphere potential is one of the simplest potential models. It only has
one parameter, which is the size of the repulsive core. We point out that the
formulas derived in Sec. III do not work for this particular potential due to
its divergent behavior at the core. The well-known hard-sphere potential is
given by

$V(r)=\begin{cases}\infty&\mbox{if}\quad r\leq R\\\ 0&\mbox{if}\quad r>R\
.\end{cases}$ (66)

The wave function is zero for $r\leq R$ and

$u_{l}(r>R)=\frac{Nk^{-l}}{A_{l}}r\bigg{[}j_{l}(kr)-n_{l}(kr)\tan{\delta}\bigg{]}$
(67)

for $r>R$. We recall that $k=\sqrt{2\mu E/\hbar^{2}}$. The scattering length
and effective range for any angular momentum $l$ are:

$\displaystyle a_{l}$ $\displaystyle=$ $\displaystyle R$ (68) $\displaystyle
r_{l}$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{1}{(2l+3)}+\frac{1}{(2l-1)}\bigg{)}R\ ,$ (69)

with $a_{l}$ being the scattering length and $r_{l}$ the effective range
(hereafter, we have eliminated the superscript “eff” in the effective range to
simplify the notation). As we can see, for hard spheres the scattering length
is $R$ at any $l$, whereas the effective range is also proportional to $R$
with a value that is positive only for $l=0$, $r_{0}=2/3R$.

###  V.2 Soft-sphere potential

In contrast to the hard-sphere interaction, the soft-sphere potential takes
into account that the core, although being repulsive, may be penetrated. This
correction results in the model potential having two parameters: the strength
of the potential and the core size. The soft-sphere potential is given by

$V(r)=\begin{cases}V_{0}&\mbox{if}\quad r\leq R\\\ 0&\mbox{if}\quad r>R\
.\end{cases}$ (70)

For $r\leq R$ , the Schrödinger equation is:

$u_{l}^{\prime\prime}(r)+\bigg{(}\overline{k}^{2}-\frac{l(l+1)}{r^{2}}\bigg{)}u_{l}(r)=0\quad\text{where}\quad\overline{k}^{2}=\frac{2\mu}{\hbar^{2}}(E-V_{0})\
,$ (71)

with $\overline{k}=\sqrt{k^{2}-k_{0}^{2}}$, a complex number for energies
$E<V_{0}$. We have defined $k_{0}$ as $\sqrt{2\mu V_{0}/\hbar^{2}}$. In order
to have a real function, we need to multiply the Bessel function with an
imaginary argument by $i^{-l}$:

$u_{l}(r)=Nrj_{l}(\overline{k}r)i^{-l}\ .$ (72)

For $r>R$ and $k\to 0$, we know the asymptotic solution (see Sec II) is:

$u_{l}(r,k=0)=r^{l+1}-a_{l}^{2l+1}r^{-l}\ .$ (73)

By imposing that the wave function is continuous at $r=R$, we can find the
normalization constant $N$:

$N=\frac{R^{l+1}-a_{l}^{2l+1}R^{-l}}{Ri^{-l}j_{l}(ik_{0}R)}\ .$ (74)

After applying the formulas we have derived in Sec. III, we can obtain the
scattering length $a_{l}$:

$a_{l}^{2l+1}=R^{2l+1}\frac{k_{0}Ri^{-l-1}j_{l+1}(ik_{0}R)}{(2l+1)i^{-l}j_{l}(ik_{0}R)+k_{0}Ri^{-l-1}j_{l+1}(ik_{0}R)}\
.$ (75)

The effective range for any $l$ is:

$r_{l}=\frac{2}{(2l+1)a_{l}^{2l+2}}\bigg{\\{}\frac{R^{2l+3}}{2l+3}-a_{l}^{2l+1}R^{2}+\frac{a_{l}^{4l+2}R^{1-2l}}{1-2l}-N^{2}(i^{-l})^{2}\frac{1}{2}R^{3}\Big{(}j_{l}^{2}(ik_{0}R)-j_{l-1}(ik_{0}R)j_{l+1}(ik_{0}R))\Big{)}\bigg{\\}}\
.$ (76)

From the general results (75) and (76), we can find the expressions for $l=0$
and $l=1$ by introducing the Bessel functions explicitly. The scattering
lengths are:

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle
R\bigg{[}1-\frac{\tanh{(k_{0}R)}}{k_{0}R}\bigg{]}\ ,$ (77) $\displaystyle
a_{1}^{3}$ $\displaystyle=$ $\displaystyle
R^{3}\bigg{[}\frac{3+(k_{0}R)^{2}-3k_{0}R\coth{(k_{0}R)}}{(k_{0}R)^{2}}\bigg{]}\
,$ (78)

and the effective ranges are:

$\displaystyle r_{0}$ $\displaystyle=$ $\displaystyle
R-R\frac{(k_{0}R)^{2}}{3(k_{0}R)^{2}\bigg{(}1-\cfrac{\tanh{(k_{0}R)}}{k_{0}R}\bigg{)}^{2}}+R\frac{1}{(k_{0}R)^{2}\bigg{(}1-\cfrac{\tanh{(k_{0}R)}}{k_{0}R}\bigg{)}}\
,$ (79) $\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle-3k_{0}a_{1}^{2}\frac{-2k_{0}R(-5+(k_{0}R)^{2})+2k_{0}R(10+(k_{0}R)^{2})\cosh{(2k_{0}R)}-5(3+2(k_{0}R)^{2})\sinh{(2k_{0}R)}}{10\bigg{(}-3k_{0}R\cosh{(k_{0}R)+(3+(k_{0}R)^{2})\sinh{(k_{0}R)}}\bigg{)}^{2}}\
.$ (80)

In Fig. 2, we plot the functions (77,78,79,80). The scattering lengths $a_{0}$
and $a_{1}$ are zero when $k_{0}R$ is zero, and are equal to $R$ (hard-spheres
solution) when $k_{0}R$ becomes large. In contrast, $r_{0}$ and $r_{1}$
diverge for $k_{0}R$ equal to zero, and when $k_{0}R$ increases we recover the
value of hard-spheres, which is $2R/3$ and $-6R/5$, respectively.

Figure 2:  $a_{0}$, $r_{0}$, $a_{1}$ and $r_{1}$ for a soft-sphere potential
as a function of $k_{0}R$. The red dashed lines correspond to the hard-sphere
limit.

An alternate way of finding the scattering parameters of the soft-sphere
potential is shown in the Appendix of the Supplemental Material II [12].

###  V.3 Spherical well potential

In the previous Section, we solved the soft-sphere model potential, where we
have a potential barrier. The spherical well potential is the same idea of the
soft-sphere potential but taking it as an attractive potential,

$V(r)=\begin{cases}-V_{0}&\mbox{if}\quad r\leq R\\\ 0&\mbox{if}\quad r>R\
.\end{cases}$ (81)

For the spherical well, where the potential is negative, we can use the
expressions that we have obtained for soft-spheres by simply changing $k_{0}$
to $-ik_{0}$. Note that since $k_{0}$ depends on the square root of $V_{0}$, a
sign change in $V_{0}$ results in $k_{0}$ becoming imaginary. If we apply this
change in sign to $V_{0}$, then we arrive easily to the general formulas for
any $l$. For the lowest partial waves $l=0$ and $l=1$, the scattering lengths
are:

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle
R\bigg{[}1-\frac{\tan{(k_{0}R)}}{k_{0}R}\bigg{]}\ ,$ (82) $\displaystyle
a_{1}^{3}$ $\displaystyle=$ $\displaystyle
R^{3}\bigg{[}\frac{(k_{0}R)^{2}-3+3k_{0}R\cot{(k_{0}R)}}{(k_{0}R)^{2}}\bigg{]}\
,$ (83)

and the effective ranges are:

$\displaystyle r_{0}$ $\displaystyle=$ $\displaystyle
R-R\frac{(k_{0}R)^{2}}{3(k_{0}R)^{2}\bigg{(}1-\cfrac{\tan{(k_{0}R)}}{k_{0}R}\bigg{)}^{2}}-R\frac{1}{(k_{0}R)^{2}\bigg{(}1-\cfrac{\tan{(k_{0}R)}}{k_{0}R}\bigg{)}}\
,$ (84) $\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle-3k_{0}a_{1}^{2}\frac{2k_{0}R(5+(k_{0}R)^{2})+2k_{0}R(10-(k_{0}R)^{2})\cos{(2k_{0}R)}-5(3-2(k_{0}R)^{2})\sin{(2k_{0}R)}}{10\bigg{(}3k_{0}R\cos{(k_{0}R)-(3-(k_{0}R)^{2})\sin{(k_{0}R)}}\bigg{)}^{2}}\
.$ (85)

As we discussed in Sec. IV, in an attractive potential and at certain values
of the potential strength, the scattering length diverges (Feshbach
resonance). That means that a bound state appears. For the spherical well
potential, the resonance for $l=0$ occurs at $k_{0}R=(2n-1)\pi/2$ and for l=1
at $k_{0}R=n\pi$, with $n\geq 1$. In Fig. 3, we plot the scattering lengths
and effective ranges for $l=0$ and $l=1$. We observe the predicted
divergences: $a_{0}$ diverges at $k_{0}R=$ $\pi$/2, 3$\pi$/2 and 5$\pi$/2.
$a_{1}$ diverges at $k_{0}R=$ $\pi$, 2$\pi$ and 3$\pi$.

Figure 3: Representation of $a_{0}$, $r_{0}$, $a_{1}$ and $r_{1}$ for a
spherical well.

If one looks to the shape of the effective ranges, one can see some
divergences appearing. As we proved in the end of Sec. IV, the divergences of
the effective ranges are related to the divergences of the scattering length
and to its zero value. In particular, for $r_{0}$ the divergences occur when
$a_{0}=0$, while for $r_{1}$, we see divergences coming from both limits, that
is when $a_{1}=0$ and $a_{1}=\pm\infty$. From that, we can also understand why
we were not seeing a divergence in the effective range, $r_{0}$, near the
region of the first divergence of the scattering length. This is because,
before the first divergence of $a_{0}$, $a_{0}$ is already negative and we do
not see a change in the sign (hence, passing through zero) until we go to the
second divergence of $a_{0}$.

###  V.4 Well-barrier potential

This model potential is more involved than the previous ones. It is a
combination of a spherical-well potential and a soft-sphere (barrier)
potential. Due to this combination, with this model potential we have four
parameters at our disposal: the strengths and the sizes of the spherical-well
and the barrier. This model potential is very well suited when working with
perturbative expansions of many-body systems that contain the three most
important scattering parameters ($a_{0}$, $a_{1}$ and $r_{0}$) as it more
easily fits them than in the previous model potentials. It is given by

$V(r)=\begin{cases}-V_{1}&\mbox{if}\quad r\leq R_{1}\\\ V_{2}&\mbox{if}\quad
R_{1}<r\leq R_{2}\\\ 0&\mbox{if}\quad r>R_{2}\ .\end{cases}$ (86)

In the internal region we have a well potential, in which the wave function
will be the regular Bessel function with a real argument:

$u_{l}^{(0)}(r)=N_{1}rj_{l}(k_{1}r)\quad\text{if}\quad r\leq
R_{1},\quad\text{where}\quad k_{1}=\sqrt{2\mu V_{1}/\hbar^{2}}\ .$ (87)

In the second region, the solution is a combination of Bessel functions with
imaginary arguments, just like in the barrier potential:

$u_{l}^{(0)}(r)=N_{2}r\Big{[}j_{l}(ik_{2}r)+M_{2}n_{l}(ik_{2}r)\Big{]}\quad\text{if}\quad
R_{1}<r\leq R_{2},\quad\text{where}\quad k_{2}=\sqrt{2\mu V_{2}/\hbar^{2}}\ .$
(88)

And finally, beyond $R_{2}$, we have the long-distance solution:

$u_{l}^{(0)}(r)=r^{l+1}-a_{l}^{2l+1}r^{-l}\quad\text{if}\quad r>R_{2}\ .$ (89)

We need the constants of integration:

$N_{1}=N_{2}\frac{j_{l}(ik_{2}R_{1})+M_{2}n_{l}(ik_{2}R_{1})}{j_{l}(k_{1}R_{1})}\quad;\quad
N_{2}=\frac{R_{2}^{l+1}-a_{l}^{2l+1}{R_{2}}^{-l}}{R_{2}\big{[}j_{l}(ik_{2}R_{2})+M_{2}n_{l}(ik_{2}R_{2})\big{]}}$
(90) $\text{and}\quad
M_{2}=-\frac{ik_{2}R_{1}j_{l}(k_{1}R_{1})j_{l+1}(ik_{2}R_{1})-k_{1}R_{1}j_{l}(ik_{2}R_{1})j_{l+1}(k_{1}R_{1})}{ik_{2}R_{1}j_{l}(k_{1}R_{1})n_{l+1}(ik_{2}R_{1})-k_{1}R_{1}n_{l}(ik_{2}R_{1})j_{l+1}(k_{1}R_{1})}\
.$ (91)

The $l$-wave scattering length is:

$a_{l}^{2l+1}=\frac{R_{2}^{2l+1}\displaystyle\int_{0}^{R_{2}}U(r)r^{l+1}u_{l}^{(0)}(r)dr}{(2l+1)R_{2}^{l}u_{l}^{(0)}(R_{2})+\displaystyle\int_{0}^{R_{2}}U(r)r^{l+1}u_{l}^{(0)}(r)dr}\
,$ (92)

where $u_{l}^{(0)}(R_{2})$ is
$N_{2}R_{2}\Big{[}j_{l}(ik_{2}R_{2})+M_{2}n_{l}(ik_{2}R_{2})\Big{]}$ and the
integrals are:

$\int_{0}^{R_{2}}U(r)r^{l+1}u_{l}^{(0)}(r)dr=-N_{1}k_{1}R_{1}^{l+2}j_{l+1}(k_{1}r)-iN_{2}k_{2}\bigg{[}r^{l+2}j_{l+1}(ik_{2}r)+M_{2}(-1)^{l}r^{l+2}j_{-l-2}(ik_{2}r)\bigg{]}_{R_{1}}^{R_{2}}\
.$ (93)

Finally, the effective range can be written as

$r_{l}=\frac{2}{(2l+1)a_{l}^{2l+2}}\bigg{\\{}\frac{R_{2}^{2l+3}}{2l+3}-a_{l}^{2l+1}R_{2}^{2}+\frac{a_{l}^{4l+2}R_{2}^{1-2l}}{1-2l}-N_{1}^{2}I_{jj}(k_{1}R_{1})-N_{2}^{2}\bigg{[}I_{jj}(ik_{2}r)+M_{2}^{2}I_{nn}(ik_{2}r)+2M_{2}I_{jn}(ik_{2}r)\bigg{]}_{R_{1}}^{R_{2}}\bigg{\\}}\
.$ (94)

The functions $I_{jj}(kr)$, $I_{nn}(kr)$ and $I_{jn}(kr)$ in Eq. (94) are:

$I_{jj}(kr)=\int
r^{2}j_{l}^{2}(kr)dr=\frac{r^{3}}{2}\big{(}j_{l}^{2}(kr)-j_{l-1}(kr)j_{l+1}(kr)\big{)}\
,$ (95) $I_{nn}(kr)=\int
r^{2}n_{l}^{2}(kr)dr=\frac{r^{3}}{2}\big{(}n_{l}^{2}(kr)-n_{l-1}(kr)n_{l+1}(kr)\big{)}\
,$ (96) $\displaystyle I_{jn}(kr)=\int
r^{2}j_{l}(kr)n_{l}(kr)dr=\frac{(-1)^{l}r^{3}}{2}j_{-l}(kr)j_{l-1}(kr)-\frac{(-1)^{l}r^{3}}{2}j_{-l-1}(kr)j_{l}(kr)$
(97)
$\displaystyle-\frac{(1+2l)(-1)^{l}r^{2}}{4k}j_{-l}(kr)j_{l}(kr)+\frac{(1+2l)(-1)^{l}r^{2}}{4k}j_{-l-1}(kr)j_{l-1}(kr)\
.$

The scattering length for $l=0$ is:

$a_{0}=\frac{\cosh{(k_{2}R_{12})}[-k_{1}k_{2}R_{2}\cos{(k_{1}R_{1})+k_{2}\sin{(k_{1}R_{1})}}]+\sinh{(k_{2}R_{12})}[k_{2}^{2}R_{2}\sin{(k_{1}R_{1})}-k_{1}\cos{(k_{1}R_{1})}]}{-k_{1}k_{2}\cos{(k_{1}R_{1})}\cosh{(k_{2}R_{12})}+k_{2}^{2}\sin{(k_{1}R_{1})}\sinh{(k_{2}R_{12})}}\
,$ (98)

with $R_{12}=R_{1}-R_{2}$. For $a_{1}$ and $r_{0}$, the expressions are
complicated but still manageable, and hence, we can split them into parts. For
$a_{1}$ we get:

$a_{1}^{3}=\frac{p_{1}+p_{2}}{d_{1}}\ ,$ (99)

with

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\cosh{(k_{2}R_{12})}\\{3k_{1}k_{2}^{3}R_{1}R_{2}^{2}\cos{(k_{1}R_{1})+k_{2}R_{2}[-3k_{2}^{2}R_{2}+k_{1}^{2}(3R_{1}-3R_{2}+k_{2}^{2}R_{1}R_{2}^{2})]\sin{(k_{1}R_{1})}}\\}\
,\hskip 52.63777pt$ (100) $\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle\sinh{(k_{2}R_{12})}\\{k_{1}k_{2}^{2}R_{1}R_{2}[3+k_{2}^{2}R_{2}^{2}]\cos{(k_{1}R_{1})}$
(101) $\displaystyle-
R_{2}[3k_{2}^{2}+3k_{1}^{2}+k_{2}^{4}R_{2}^{2}+k_{1}^{2}k_{2}^{2}R_{2}(-3R_{1}+R_{2})]\sin{(k_{1}R_{1})}\\}\
,$ $\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle
k_{1}^{2}k_{2}^{3}R_{1}\sin{(k_{1}R_{1})}\cosh{(k_{2}R_{12})}+[k_{1}k_{2}^{4}R_{1}\cos{(k_{1}R_{1})}-k_{2}^{2}(k_{1}^{2}+k_{2}^{2})\sin{(k_{1}R_{1})}]\sinh{(k_{2}R_{12})}\
.$ (102)

Similarly, for $r_{0}$:

$r_{0}=\frac{p_{3}+p_{4}+p_{5}}{d_{2}}\ ,$ (103)

with

$\displaystyle p_{3}$ $\displaystyle=$ $\displaystyle
2k_{1}k_{2}\cosh{(2k_{2}R_{12})}\\{R_{2}(3+k_{2}^{2}R_{2}^{2})[k_{1}^{2}+k_{2}^{2}+(k_{1}^{2}-k_{2}^{2})\cos{(2k_{1}R_{1})}]-3k_{1}(1+2k_{2}^{2}R_{2}^{2})\sin{(2k_{1}R_{1})}\\}\
,\hskip 28.45274pt$ (104) $\displaystyle p_{4}$ $\displaystyle=$
$\displaystyle
2k_{2}\\{-3k_{1}(k_{1}^{2}+k_{2}^{2})R_{1}+k_{1}(k_{1}^{2}-k_{2}^{2})k_{2}^{2}R_{2}^{3}+(k_{1}^{2}+k_{2}^{2})[k_{1}(-3R_{1}+k_{2}^{2}R_{2}^{3})\cos{(2k_{1}R_{1})}+3\sin{(2k_{1}R_{1})}]\\}\
,$ (105) $\displaystyle p_{5}$ $\displaystyle=$ $\displaystyle
k_{1}\\{3(1+2k_{2}^{2}R_{2}^{2})[k_{1}^{2}+k_{2}^{2}+(k_{1}^{2}-k_{2}^{2})\cos{(2k_{1}R_{1})}]-4k_{1}k_{2}^{2}R_{2}(3+K_{2}^{2}R_{2}^{2})\sin{(2k_{1}R_{1})}\\}\sinh{(2k_{2}R_{12})}\
,$ (106) $\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle
12k_{1}k_{2}\\{k_{2}\cosh{(k_{2}R_{12})}[k_{1}R_{2}\cos{(k_{1}R_{1})}-\sin{(k_{1}R_{1})}]$
(107)
$\displaystyle+[k_{1}\cos{(k_{1}R_{1})}-k_{2}^{2}R_{2}\sin{(k_{1}R_{1})}]\sinh{(k_{2}R_{12})}\\}^{2}\
.$

The formulas found for $a_{0}$ and $r_{0}$ coincide with the ones in Ref.
[16]. The analytic expressions for the rest of parameters are too large and we
do not show them in this work.

##  VI Conclusions

In this work, we have re-analyzed the scattering problem to find expressions
for obtaining the scattering length and effective range for any angular
momentum value $l$ as long as the potential decays faster than $1/r^{n}$ with
$n>2l+3$ for each partial wave at infinity. We have first worked out the
procedure that leads to integral formulas allowing the calculation of the
scattering parameters for any angular momentum. The procedure consists of
relating two apparently different formulas that calculate the phase shift.
After lengthy manipulations, one obtains the expressions for the scattering
parameters. Also, we discuss the Feshbach resonance and how the scattering
parameters behave around it, with special emphasis on the importance of the
resonance in the study of ultracold quantum gases.

As model examples, we have obtained the exact expressions of the scattering
length and the effective range for any angular momentum $l$ for the cases of
hard-sphere, soft-sphere, spherical well and well-barrier potentials. As the
more useful parameters are $a_{0}$, $a_{1}$ and $r_{0}$, we also particularize
the general formulas for these cases.

The results obtained for the analyzed model potentials can be useful for
researchers working on perturbative expansions used in many-body theory when
the interaction strength is small [2, 11, 3]. They can also be useful when
working the inverse problem, where one tries to build potentials which fulfill
some a-priori known low-energy scattering parameters. Finally, Sections II,
III and IV can be used as a a guide for students studying scattering theory,
and the model problems in Sec. V can be used as hands-on applied exercises.

###### Acknowledgements.

This work has been supported by AEI (Spain) under grant No.
PID2020-113565GB-C21. We also acknowledge financial support from Secretaria
d’Universitats i Recerca del Departament d’Empresa i Coneixement de la
Generalitat de Catalunya, co-funded by the European Union Regional Development
Fund within the ERDF Operational Program of Catalunya (project QuantumCat,
ref. 001-P-001644).

## References

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# Spatial Transformer Network YOLO Model for Agricultural Object Detection

Yash Vivek Zambre1, Ekdev Rajkitkul2, Akshatha Mohan1, Joshua Peeples1
2Department of Computer Science and Engineering, Texas A&M University, College
Station, TX, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Electrical and Computer Engineering, Texas
A&M University, College Station, TX, USA

###### Abstract

Object detection plays a crucial role in the field of computer vision by
autonomously identifying and locating objects of interest. The You Only Look
Once (YOLO) model is an effective single-shot detector. However, YOLO faces
challenges in cluttered or partially occluded scenes and can struggle with
small, low-contrast objects. We propose a new method that integrates spatial
transformer networks (STNs) into YOLO to improve performance. The proposed
STN-YOLO aims to enhance the model’s effectiveness by focusing on important
areas of the image and improving the spatial invariance of the model before
the detection process. Our proposed method improved object detection
performance both qualitatively and quantitatively. We explore the impact of
different localization networks within the STN module as well as the
robustness of the model across different spatial transformations. We apply the
STN-YOLO on benchmark datasets for Agricultural object detection as well as a
new dataset from a state-of-the-art plant phenotyping greenhouse facility. Our
code and dataset are publicly available 111https://github.com/Advanced-Vision-
and-Learning-Lab/STN-YOLO.

###### Index Terms:

Spatial transformer network, Object detection, YOLO, plant phenotyping

##  I Introduction

Plant phenotyping is critical for crop improvement [1], yield optimization
[2], and sustainable practices [3]. Artificial intelligence (AI), particularly
object detection algorithms, has transformed plant phenotyping, enhancing
efficiency and performance [4]. The You Only Look Once (YOLO) has been used
effectively in various Agricultural applications such as pest detection [5],
crop disease detection [6], and crop harvesting [7]. Despite the vast number
of use cases, YOLO has some limitations due to various spatial transformations
[8]. Spatial transformer networks [9] are an approach to improve an artificial
neural network’s robustness to spatial transformations. We propose to
integrate STNs with the YOLO model to incorporate spatial invariance. The STN
applies a learnable affine transformations to the images that will help with
object detection. The STN-YOLO model demonstrates spatial invariance and
outperforms the baseline YOLO model on several Agricultural benchmark
datasets.

Figure 1: Our object detection framework leverages STNs to improve spatial
invariance in the network for object detection. STNs are comprised of three
components: localization network, grid generator, and sampler. The
localization network outputs affine transform parameters ($\theta$) that are
passed to the grid generator. The grid transformation, $T_{\theta}(G)$, is
applied to the input image resulting in the new output image.

Additionally, we present a new, high quality, annotated dataset of various
plants to advance the study of agriculture plant detection and phenotyping.
The new Plant Growth and Phenotyping (PGP) dataset differs from other
available image datasets in the following aspects: (1) multi-spectral images
captured across varying heights; (2) challenging images of multiple crops
across varying illumination conditions; (3) precise annotations assisted with
the use of Segment Anything model [10], (4) large size and shape variations of
plants. The features of the PGP dataset present new challenges for
agricultural object detection. We perform extensive experiments on the
proposed database and new model (STN-YOLO) for object detection. The key
contributions of our work are the following:

  * •

Integration of STN within YOLO for improved object detection performance

  * •

New benchmark dataset for plant object detection.

##  II Related Work

###  II-A Object Detection Models

Object detection is a prominent challenge in computer vision, where
researchers use deep learning to boost performance [11, 12, 13]. Two-stage
detectors, such as Faster R-CNN [14], involve distinct stages with a region
proposal network (RPN) and region of interest (ROI) pooling for candidate
bounding boxes [11]. In contrast, single-stage detectors such as single-shot
detectors (SSD) predict bounding boxes directly by utilizing grid boxes and
anchors, as described in [15]. Notably, YOLO has emerged as a prominent
example of a single-stage detector. The advancements of CNNs architectures
introduces innovations such as anchor boxes to enhance object detection [13].
Key advancements include R-CNN, integrating region proposals with CNN using
support vector machine (SVM) classification and bounding-box regression [16].
However, R-CNN incurs substantial computational costs and information loss.
Fast R-CNN extends this with ROI pooling and proposal refinement, improving
test speed and algorithm precision [17]. Faster R-CNN further reduces
computational requirements by introducing the RPN for regional proposals,
selecting anchors based on specific criteria for outstanding recognition
precision [14]. Ongoing advancements focus on speed optimization, including
anchor-free detection [18].

###  II-B Spatial Transformer Network (STN)

STN is a module with differentiable properties that performs learnable spatial
transformation on input feature maps [9]. The transformation is dependent on
the specific input, generating a singular output feature map. In the case of
multi-channel inputs, an identical transformation is applied on each channel.
The STN module is comprised of three components, as illustrated in the Figure
1. The first component is a localization network, which passes input feature
maps through several hidden layers to generate the parameters for a learnable
affine transformation that is applied to each feature map. Subsequently, the
predicted transformation parameters $\theta$ are used to construct a sampling
grid—a set of points indicating where the input feature maps should be sampled
to generate the transformed output. Finally, the feature maps and the sampling
grid serve as inputs to the sampler that generates the output map sampled from
the input at the specified grid points.

Localization Networks The localization network ($f_{loc}$) takes the input
feature maps $\mathbf{X}\in\mathbb{R}^{H\times W\times C}$ with width $W$,
height $H$, and $C$ channels. The features extracted by the localization
network are then passed into a fully connected layer that outputs $\theta$,
the parameters of the transformation $T_{\theta}$ to be applied to each
feature map: $\theta=f_{loc}(X)$. The size of $\theta$ can vary depending on
the transformation type that is parameterized [9]. The localization network
function $f_{\text{loc}}()$ can take any form, such as a fully-connected
network or a CNN such as ResNet18 [19], but should include a final regression
layer (i.e., fully connected layer) to produce the transformation parameters
$\theta$ (e.g., affine transformation have six parameters).

Parameterized Sampling Grid To perform a transformation on the input feature
map, each resulting “pixel” is determined by applying a sampling kernel
centered at a specific location in the input feature map. The term “pixel” in
this context refers to an element of a general feature map, not necessarily an
image. In a general context, the resulting “pixels” are positioned on a
regular grid $G=\\{G_{i}\\}$ of pixels $G_{i}=(x^{t}_{i},y^{t}_{i})$, creating
an output feature map $V\in\mathbb{R}^{H^{\prime}\times W^{\prime}\times C}$.
Here, $H^{\prime}$ and $W^{\prime}$ represent the height and width of the
grid, respectively, and $C$ is the number of channels, which is same in the
input and output feature maps.

For clarity of exposition, assume that $T_{\theta}$ is a 2D affine
transformation $A_{\theta}$. In the affine case, the point-wise transformation
is shown in Equation 1:

$\begin{bmatrix}x_{i}^{s}\\\
y_{i}^{s}\end{bmatrix}=T_{\theta}(G_{i})=A_{\theta}\begin{bmatrix}x_{i}^{t}\\\
y_{i}^{t}\\\
1\end{bmatrix}=\begin{bmatrix}\theta_{11}&\theta_{12}&\theta_{13}\\\
\theta_{21}&\theta_{22}&\theta_{23}\end{bmatrix}\begin{bmatrix}x_{i}^{t}\\\
y_{i}^{t}\\\ 1\end{bmatrix}$ (1)

In this scenario, $(x_{i}^{t},y_{i}^{t})$ denotes the target coordinates of
the regular grid in the output feature map, and $(x_{i}^{s},y_{i}^{s})$
represents the source coordinates in the input feature map that define the
sample points. The relationship between these coordinates is determined by the
affine transformation matrix $A_{\theta}$ [9]. The STN is self-contained
module that is versatile and can be seamlessly integrated at any point in an
artificial neural network architecture. The inclusion of an STN at the
beginning of a CNN enables the network to learn how to dynamically transform
input images to account for spatial variations.

##  III Method

###  III-A Plant Growth and Phenotyping Dataset

(a) Corn

(b) Cotton

(c) Rice

(d) Corn Label

(e) Cotton Label

(f) Rice Label

Figure 2: Example images from PGP dataset. The top rows (2(a)-2(c)) are input
images and the bottom rows (2(e)-2(f)) are the associated labels.

The Plant Growth and Phenotyping (PGP) dataset is a publicly available, high-
resolution ($512\times 512$ pixels), multi-spectral collection of plant images
captured from a state-of-the-art greenhouse. PGP consists of 1137 images from
different crops: 442 corn, 246 cotton, and 449 rice plants. The dataset
includes images taken from various heights of the same plants.

Figure 3: Architecture of STN-YOLO model is shown. The STN module (shown in
light orange and green) is added at the beginning of the network to account
for spatial invariances. After the STN, the image is passed into the YOLO
model where the layers P1 - P5 are the layers of the YOLO backbone, HEAD part
is responsible for generating the final output and the CLS + BBOX are losses
to perform object detection.

The plant phenotyping facility consisted of a camera mounted on a robotic arm,
capturing side and top views of the plants (only side view images are used in
this work). The Multispectral Imaging System (MSIS-AGRI-1-A), integrating an
MSC2-AGRI-1-A snapshot multispectral camera and a 4-channel LED illuminator,
was used to collect images across four spectral bands: 580 nm (Green), 660 nm
(Red), 730 nm (Red Edge), and 820 nm (Near Infrared).

For this work, the Red, Red Edge, and Green bands were combined to create
pseudo “RGB” images (as shown in Figure 2) compatible with pre-trained machine
learning models. The individual bands were superimposed, and min-max
normalization was applied to the final image. These plant images were captured
between 2022 and 2024 in controlled greenhouse environments. Data augmentation
techniques (e.g., rotation, cropping) can be applied using frameworks like
Roboflow [20] to expand the dataset. The same framework is used for labeling,
a two-step process involving auto-generation of initial labels using the
Segment Anything Model (SAM) [10], followed by manual refining of the auto-
generated labels.

###  III-B STN-YOLO MODEL

For the proposed STN-YOLO model, the STN module is first integrated in the
beginning of the YOLO model to account for spatial transformations in the
input images. After the images passes through STN module, the YOLO model is
then used to perform the task of object detection. As stated in Section II-B,
the STN has can use any backbone for the localization networks. The
localization network within the STN module comprises various convolutional
layers, linear layers, and activation functions.

Localization in STN model The STN models have the localization network which
takes the input feature maps with width W and height H and C channels and
outputs $\theta$ which are the transformation parameters. The localization
network is selected by the user and can vary from a shallow or deeper network.
The performances of different localization network are compared to evaluate if
a shallow or deep, pre-trained model will lead to better object detection
performance quantitatively (e.g., accuracy, precision, and recall) and
computationally. The shallow network is comprised of one convolution layer
with max pooling and ReLU activation function was used in contrast to the deep
pretrained models. Following the convolution layer in the shallow network, an
adaptive average pooling layer was used to aggregate the spatial resolution of
the feature maps to a desired size (e.g., $28\times 28$).

YOLO Backbone The output of the STN module is then given to the YOLO model
where, P1, P2, P3, P4, and P5 typically refer to different levels or stages in
the feature pyramid and these are the different convolutional layers as shown
in the Figure 3. Output features are obtained by extracting features at
different scales from the input image to detect objects of various sizes and
capture both fine-grained details and global context. P1 corresponds to the
finest level with the highest resolution, while P5 is the coarsest level with
lower resolution but capturing more context. The head in the YOLO model
usually refers to the part of the network responsible for making predictions.
The head takes features from multiple pyramid levels and produces predictions
for object classes, bounding box coordinates, and other relevant information.

Loss in YOLOv8 A two term loss function is used during training. The objective
function consists of two components: classification loss (CLS), which
penalizes errors in predicting object classes, and bounding box (BBOX) loss,
which penalizes errors in predicting the coordinates of the bounding boxes
around objects. Combining these losses helps train the model to simultaneously
improve both classification and localization aspects of object detection.
Class imbalance in single-class detection refers to the discrepancy between
detection target objects and the background, typically considered the negative
class. Addressing this imbalance is crucial for improving detection
performance. Bounding box loss in YOLOv8 uses the complete intersection over
union (CIOU) loss [21], which considers box overlap and aspect ratio
differences, enhancing regression performance. Distributed focal loss (DFL)
[22] effectively tackles class discrepancy by dynamically adjusting the loss
for each class during training to counteract biases caused by class
imbalances.

TABLE I: Performance metrics for different localization network with the average value and +/- 1 standard deviation are shown across three experimental runs of 100 epochs each with random initialization. The best average metric is bolded. Localization | Accuracy | Precision | Recall | mAP
---|---|---|---|---
Shallow | 80.34 $\pm$ 0.92 | 94.33 $\pm$ 0.74 | 87.62 $\pm$ 0.71 | 71.82 $\pm$ 0.58
VGG16 | 81.05 $\pm$ 0.08 | 92.94 $\pm$ 0.39 | 88.50 $\pm$ 0.92 | 72.87 $\pm$ 0.66
Resnet18 | 80.97 $\pm$ 0.74 | 91.51 $\pm$ 0.09 | 88.03 $\pm$ 0.69 | 72.04 $\pm$ 0.71
TABLE II: Metrics for different spatial resolutions of the STN shallow localization network with the average value and +/- 1 standard deviation are shown across three experimental runs of 100 epochs each with random initialization. The best average metric is bolded. Feature Map Size | Accuracy | Precision | Recall | mAP
---|---|---|---|---
$1\times 1$ | 80.34 $\pm$ 0.92 | 94.33 $\pm$ 0.74 | 87.62 $\pm$ 0.71 | 71.82 $\pm$ 0.58
$7\times 7$ | 80.73 $\pm$ 0.57 | 95.17 $\pm$ 0.80 | 88.71 $\pm$ 0.46 | 71.39 $\pm$ 0.82
$28\times 28$ | 81.63 $\pm$ 1.53 |  95.34 $\pm$ 0.76 |  89.52 $\pm$ 0.57 |  72.56 $\pm$ 0.90

##  IV Experimental Results and Discussion

###  IV-A Experimental Setup

Three experimental runs of random initialization were used for each model and
the YOLO backbones were pretrained on the COCO dataset. We used the AdamW
optimizer with an initial learning rate of 0.002. The optimizer was configured
with distinct parameter groups, each with specific weight decay values
assigned to all parameters. The batch size for all experiments was set to 16.
We used 100 epochs with early stopping (patience of 50) for each model. All
the experiments were performed using the Ultralytics [23] framework and
performed on an NVIDIA A100 GPU.

In our study, we used various benchmark datasets to assess the effectiveness
of our proposed plant detection models. The GlobalWheat2020 dataset contains
4,000 images, with 3,000 for training and 1,000 for testing, focusing on wheat
detection. The PlantDoc dataset comprises 2,569 images, with 2,330 for
training and 239 for testing, addressing plant identification across varying
environments. The MelonFlower dataset includes 288 images, with 193 for
training and 95 for testing, specifically targeting melon flower
identification. These datasets collectively provide a robust evaluation
framework for assessing model generalization across diverse plant species and
environments.

###  IV-B Ablation Studies using PGP Dataset

Impact of Localization Networks In Table I, we outline the performance metrics
of various localization networks, emphasizing the comparable quantitative
results achieved by both the shallow localization network and deep pre-trained
models. Both the shallow and deep networks used a global average pooling layer
after the convolutional layers. Based on these findings, we opted for the
shallow network as the localization network in the STN-YOLO model. This
decision stemmed from the observed effectiveness of the shallow network in
extracting features suitable for learning affine transformations comparable to
the capabilities of the pre-trained models (based on similar quantitative
measures). The shallow network and the pretrained networks were finetuned with
the baseline YOLO model. The number of learnable parameters for the deeper
networks, VGG16 and ResNet18 (approximately 82 million and 49 million
respectively), were greater than the shallow network (approximately 5 million)
for marginal improvements in performance hence the shallow network was chosen
for the remaining experiments.

Impact of Spatial Information To assess the impact of retaining more spatial
information in the STN-YOLO model, modifications were applied to the last
layer of the shallow localization network. Specifically, the adaptive average
pooling layer, was evaluated for various spatial sizes of $1\times 1$,
$7\times 7$, and $28\times 28$ as shown in Table II. This investigation aims
to evaluate the model’s sensitivity to variations in spatial information
preservation and identify the optimal configuration for improved performance
in object detection tasks. We wanted to look at no spatial information
($1\times 1$) and retaining some spatial information ($7\times 7$, $28\times
28$). The results demonstrated that retaining the most amount of spatial
information ($28\times 28$) resulted in the best object detection performance.
If the feature maps have more spatial information, then there is more context
that the model will be able to use for maximizing performance. However, as the
spatial resolution is increased, so does the computational cost as the number
of parameters will increase.

(a) Cotton

(b) YOLO

(c) STN-YOLO (ours)

(d) YOLO

(e) STN-YOLO (ours)

Figure 4: Example results of YOLO and STN-YOLO on an example image from PGP
dataset (Figure 4(a)). We show the detection differences in the YOLO and the
STN-YOLO models in Figure 4(b) and 4(c). The YOLO model does not capture
portions of the leaf in the pot and falsely detects a portion of the
background. EigenCAM [24] was used to show which areas of the image each model
focused on for object detection. STN-YOLO focused on the area of the image
containing a majority of the plant as opposed to YOLO emphasizing the pot of
the plant.

Model Analysis using Explainable AI The proposed model is better on average at
reducing the detection of false positives, as the precision of the STN-YOLO
model is better than the baseline YOLO model as shown in Table III. The STN-
YOLO model also exhibits improved recall and mean average precision (mAP),
which demonstrates a better object detection model as illustrated in Figure
(4(b)-4(c)), where the baseline YOLO model fails to eliminate the bottom right
vase area in comparison to the proposed STN-YOLO model. In addition to these
metrics from Table III, we also provide insights from Eigen class activation
maps (EignenCAMs) [24] as seen in Figure (4(d)-4(e)). These EigenCAMs offer
valuable visualization for model explainability, as they highlight significant
aspects of the images that contribute to the model’s predictions. The results
support the hypothesis that incorporating the spatial invariance in the input
image can result in better object detection both qualitatively and
quantitatively as the STN-YOLO emphasizes more of the plant area.

TABLE III: Object detection metrics for different augmentations individually and together with the average value and +/- 1 standard deviation are shown across three experimental runs of 100 epochs each with random initialization. The best average metric is bolded. Rotation | Shear | Crop | Model | Accuracy | Precision | Recall | mAP
---|---|---|---|---|---|---|---
|  |  | YOLO | 84.86 $\pm$ 0.47 | 94.30 $\pm$ 0.56 | 89.21 $\pm$ 0.53 | 71.76 $\pm$ 1.03
|  |  | STN-YOLO | 81.63 $\pm$ 1.53 |  95.34 $\pm$ 0.76 |  89.52 $\pm$ 0.57 |  72.56 $\pm$ 0.90
|  | ✓ | YOLO | 83.81 $\pm$ 2.29 | 94.87 $\pm$ 0.87 | 87.90 $\pm$ 0.84 | 71.26 $\pm$ 1.56
|  |  | STN-YOLO | 84.52 $\pm$ 2.37 | 94.58 $\pm$ 1.01 | 88.44 $\pm$ 0.74 | 70.06 $\pm$ 0.64
| ✓ |  | YOLO | 84.86 $\pm$ 0.97 | 93.06 $\pm$ 1.23 | 88.70 $\pm$ 0.30 | 72.35 $\pm$ 1.22
|  |  | STN-YOLO | 83.38 $\pm$ 1.61 |  94.87 $\pm$ 0.85 | 89.64 $\pm$ 0.54 | 68.10 $\pm$ 1.56
| ✓ | ✓ | YOLO | 82.73 $\pm$ 2.60 | 93.94 $\pm$ 0.31 | 88.96 $\pm$ 0.70 | 71.97 $\pm$ 1.08
|  |  | STN-YOLO | 83.36 $\pm$ 0.55 |  94.58 $\pm$ 0.82 | 87.67 $\pm$ 0.20 | 69.09 $\pm$ 0.28
✓ |  |  | YOLO | 83.46 $\pm$ 0.71 | 94.87 $\pm$ 0.82 | 88.71 $\pm$ 0.31 | 70.65 $\pm$ 0.68
|  |  | STN-YOLO | 85.39 $\pm$ 0.69 |  95.47 $\pm$ 1.38 | 89.91 $\pm$ 0.72 | 71.54 $\pm$ 0.43
✓ |  | ✓ | YOLO | 85.43 $\pm$ 0.48 |  95.09 $\pm$ 1.61 | 89.91 $\pm$ 0.74 | 70.67 $\pm$ 0.57
|  |  | STN-YOLO | 82.66 $\pm$ 1.67 | 94.54 $\pm$ 1.12 | 88.09 $\pm$ 0.19 | 68.69 $\pm$ 0.63
✓ | ✓ |  | YOLO | 88.59 $\pm$ 2.04 | 94.87 $\pm$ 0.84 | 89.02 $\pm$ 0.61 | 72.97 $\pm$ 1.12
|  |  | STN-YOLO | 82.78 $\pm$ 1.46 |  96.54 $\pm$ 1.52 | 89.02 $\pm$ 0.69 | 71.31 $\pm$ 0.61
✓ | ✓ | ✓ | YOLO | 84.86 $\pm$ 0.91 | 93.02 $\pm$ 0.43 | 88.49 $\pm$ 0.31 | 72.35 $\pm$ 1.46
|  |  | STN-YOLO | 84.60 $\pm$ 1.28 |  94.62 $\pm$ 1.84 | 89.53 $\pm$ 0.65 | 69.83 $\pm$ 0.73
TABLE IV: Object detection performance metrics for each model on benchmark Agricultural datasets with the average value and +/- 1 standard deviation are shown across three experimental runs of 100 epochs each with random initialization. The best average metric is bolded for each dataset. Datasets | Model | Accuracy | Precision | Recall | mAP
---|---|---|---|---|---
PGP | YOLO | 84.86 $\pm$ 0.47 | 94.30 $\pm$ 0.56 | 89.21 $\pm$ 0.53 | 71.76 $\pm$ 1.03
| STN-YOLO | 81.63 $\pm$ 1.53 |  95.34 $\pm$ 0.76 |  89.52 $\pm$ 0.57 |  72.56 $\pm$ 0.90
Global Wheat 2020 | YOLO | 94.29 $\pm$ 0.10 | 92.17 $\pm$ 0.03 | 96.69 $\pm$ 0.29 | 62.67 $\pm$ 0.25
| STN-YOLO | 93.67 $\pm$ 0.79 | 92.24 $\pm$ 0.24 | 97.53 $\pm$ 0.02 | 63.32 $\pm$ 0.71
PlantDoc | YOLO | 47.57 $\pm$ 0.40 | 47.21 $\pm$ 0.15 | 47.13 $\pm$ 1.19 | 30.29 $\pm$ 0.84
| STN-YOLO | 49.57 $\pm$ 0.74 | 52.01 $\pm$ 2.76 | 48.47 $\pm$ 1.16 | 34.79 $\pm$ 2.02
MelonFlower | YOLO | 86.47 $\pm$ 2.37 | 73.63 $\pm$ 2.83 | 85.27 $\pm$ 0.82 | 44.91 $\pm$ 0.47
| STN-YOLO | 89.78 $\pm$ 1.15 | 88.09 $\pm$ 3.44 | 86.97 $\pm$ 0.72 | 45.09 $\pm$ 3.31

Augmentations Testing One of the most important set of experiments performed
was the testing of models on the unseen augmented data as shown in Table III.
To do this, no data augmentation was added to the training dataset, but data
augmentations were performed on the test dataset to evaluate the robustness of
the STN-YOLO model across various transformations. The augmentations used on
the test dataset were random cropping (15% zoom), shear ($\pm$ $10^{\circ}$
horizontal and $\pm$ $10^{\circ}$ vertical), and rotation (between $\pm$
$10^{\circ}$) to simulate image conditions that may happen in the facility. We
observed that the proposed STN-YOLO model gave better quantitative results in
case of rotation and shear for PGP dataset compared to the baseline YOLO
model. This result is intuitive as rotation and shear are affine
transformations that can be learned by the STN-YOLO model. However, in the
cases where cropping is used, the performance is comparable to the baseline
YOLO method as cropping is not an affine transformation. Furthermore, for the
combinations of the different augmentations (e.g., rotation with shear,
rotation with crop), we observed that for most of the data augmentation cases
(six out of eight), the STN-YOLO model performs better on average than the
YOLO baseline for the precision values indicating that the STN-YOLO was
effective at reducing the number of false positives by incorporating the added
spatial invariance properties of the STNs.

Ground Truth YOLO STN-YOLO

(a) GlobalWheat2020

(b) PlantDoc

(c) MelonFlower

Figure 5: Example images from benchmark dataset. The top row (5) are the
GlobalWheat2020 dataset illustrations with labels followed the YOLO and STN-
YOLO detections. (5) are associated with the plantdoc dataset and (5) are the
melonflower dataset results for the labels, YOLO and the STN-YOLO detections.

###  IV-C Benchmark Datasets Results

GlobalWheat2020 Dataset The GlobalWheat2020 dataset [25] is a comprehensive
Agricultural dataset containing images of wheat heads sourced from various
countries. From the results in Table IV, the STN-YOLO model has a higher
average value for most metrics (except accuracy), but the baseline YOLO model
is comparable to the average value when you take the standard deviation in
comparison. We can also qualitatively look at this observation from Figure 5
where the number of detections by YOLO in the GlobalWheat2020 dataset are less
as compared than the detections of STN-YOLO. For the GlobalWheat2020 dataset,
the images consisted of “zoomed”-in views of the wheatheads. In this case, the
spatial invariance would not impact the object detection results and this is
indicated by the slight improvements in the average object detection metrics
(except for accuracy) shown in Table IV.

PlantDoc Dataset The PlantDoc dataset [26] is derived from the Plant Village
dataset (PVD) [27]. The dataset was created using an automated system
leveraging GoogleNet [28] and AlexNet [29]. Interesting observations can also
be seen from the Table IV such as unlike the GlobalWheat2020 dataset, there is
an increase in all the object detection metrics in the STN-YOLO model as
compared to the baseline model. Particularly, the precision for STN-YOLO model
is higher than the baseline model indicating the model reduced the number of
false positives detected as noted for the PGP dataset. Qualitatively, we can
see that in Figure 5 where the YOLO detects the leaf correctly but wrongly
classifies the leaf as tomato early blight leaf. However, the STN-YOLO detects
and correctly classifies the leaf as Tomato Septoria. In the PlantDoc dataset,
the emphasis remains on various plants and zoomed-in images. However, spatial
invariance remains insignificant due to the absence of pre-translated or
rotated input images. Despite this, spatial invariance could potentially offer
some benefits with this dataset, particularly as certain images provide a more
“zoomed-out” perspective of the plants compared to the GlobalWheat2020
dataset.

MelonFlower Dataset The MelonFlower dataset [30] is dataset available on
Roboflow and the images were captured in a greenhouse (similar to the PGP
dataset). This dataset was used to demonstrate the model’s capability in
detecting small objects (e.g., flowers) and different viewing geometries of
the plants. As seen from Table IV, there is a significant difference in the
precision values with the STN-YOLO model as compared to the baseline model.
The precision metrics reveal a substantial improvement, showcasing the model’s
enhanced ability to accurately detect objects. Visually, we can see this
difference in the Figure 5 where in the YOLO detections of the Melonflower
dataset, the model detects the melon flower with an IOU of $0.50$ and the STN-
YOLO detections of the Melonflower dataset has a better improved detection
with an IOU value of $0.60$. This difference not only emphasizes the
effectiveness of the STN-YOLO model but also underscores its potential for
applications where precision is a critical performance indicator. Since the
MelonFlower dataset had the most spatial variance among all datasets evaluated
in this work, the STN-YOLO model exhibited notably improved performance, as
evidenced by the precision values and other object detection metrics presented
in Table IV.

##  V Conclusion

This work focused on integrating STN with YOLO, creating the STN-YOLO model to
address spatial invariance challenges. The model aimed to enhance plant image
detection quality and can be used for downstream applications such as
phenotypic feature extraction. Results demonstrated that STN improved model
robustness and reduced number of false positives in the datasets as indicated
by the higher precision scores. The STN-YOLO model improved performance on
benchmark datasets, showcasing potential in handling real-world spatial
transformations. Future work includes integrating STNs into other object
detection models (e.g., future versions of YOLO [31]), developing new
objective functions to improve the learning of the model with STNs, expanding
our PGP dataset to include more images across multiple crops with different
image conditions (e.g., illumination), and incorporating the near infrared
channel.

##  VI Acknowledgement

This material is based upon work supported by Texas A&M AgriLife. Portions of
this research were conducted with the advanced computing resources provided by
Texas A&M High Performance Research Computing.

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