# Effective Range Approximation in Variable Phase Approach for Triplet
${}^{3}S_{1}^{\\{np\\}}$ and Singlet ${}^{1}S_{0}^{\\{nn,np,pp\\}}$ State

Anil Khachi1
1 Department of Physics
St. Bedes College, 171002,
Himachal Pradesh, India

###### Abstract

This work is a short communication where phase function method has been
applied to obtain the phase shifts using Effective Range Approximation
potential for ${}^{3}S_{1}-np$, ${}^{1}S_{0}-nn$, ${}^{1}S_{0}-np$, and
${}^{1}S_{0}-pp$ states. No free fitting parameters are used in calculations
and reasonably good match with the experimental phase shifts is observed for E
$\leq$ 20 MeV. Potentials are obtained for n-n, n-p and p-p scattering that
are exponential well shaped.

keywords: Variable phase approach, Effective range approximation, NN
interactions

##  1 Introduction

NN interaction is the most fundamental problem in nuclear physics which is yet
not fully understood. Nuclear physicsts demands precision potentials that has
a capability to obtain the properties of the interacting systems as well has
predictive power. After Yukawa [1] oldest theory of NN interactions which was
based on mesons exchange, certain high class potentials were introduced like:
Av-18, Nijm-I,II and III, CD-Bonn, Paris-Group, Hamada-Johnston, Yale-Group,
Urbana-Group and Sao Paulo-Group CHPT potentials are having the capability to
fit experimental data for other lighter systems [2]. Recently Granada group
[3] has performed partial wave analysis on 6713 published np and pp scattering
data below 350 MeV (from 1950-2013), thus helped to construct the inverse
potentials more accurately. Aoki et al. have extracted the NN, hyperon
potentials and the meson-baryon potentials potentials using lattice QCD.

The experimental setup in neutron-neutron scattering [4] consists of a neutron
beam generated by ${}^{3}H(d,n)^{4}He$ reaction. The target consists of heavy
water enriched in $D_{2}O$. In total the nd-breakup reaction i.e.,
$n+d(deuterium)\rightarrow p+n+n$ is the reaction leading to the appearance of
interacting two neutrons in the final state. In addition to n-d, another
reaction i.e., $\pi^{+}$ $\rightarrow$ $\gamma+n+n$ helps in determinining the
scattering length for nn reaction. For various NN intercations table 1
presents different reactions that produces the desired projectile and
different targets used for the same.

Table 1: Table represents various NN interactions taken up in this paper and different reactions responsible for producing desired projectiles. Last column presents the targets used in the scattering processes. Reaction | Projectile Production by | Projectile | Target
---|---|---|---
n-n [4] |  3H(d,n)4He | n | D2O
n-p [5] |  2H(p,n)2p | n | H in CH2
p-p [6] | Ionizing hydrogen gas | p | H in CH2, W , -(C2H4)n\- or graphite

Neutron-neutron interaction is important because the reaction helps in the
investigation of charge idependence which has been fundamentally interesting
to the nuclear community. It is well known that to investigate the charge
independence in nuclear forces it is quite sufficient to compare the np and pp
interactions. It is also known that to get more accurate information about
these forces one needs to get precise low energy scattering data. Under such
circumstances only S-waves are particularly important. The investigation of
low energy scattering parameters is directly related to test the hypothesis of
charge independence and charge symmetry. Both theoreticians and
experimentalists are keen to precisly determine the low energy parameters for
np, pp and nn interactions.

The nucleon-nucleon interactions at low energies are usually expressend in
terms of effective range ($r_{e}$) and scattering length (a). These parameters
provides information on the charge idependence of the nuclear forces. There
have been various arguments regarding the possibility of violation of charge
independence and charge symmentry and some of them are discussed here:

  * •

Darewych et al. [7] found that there was substantial difference between the np
and nn potential curves with ${}^{1}S_{0}-nn$ ($V_{0}$=40.38 MeV) and for
${}^{1}S_{0}-np$ ($V_{0}$=61.99 MeV) hence observed “violation of charge
independence”. The violation was credited to the unavailability of high energy
nn data. They used high energy np and pp data for fitting the nn SPS curve.

  * •

Henley and others [8] have discussed about the small departure from charge
symmetry and charge indpendence. The departure can be originated by em forces
and is related to the basic understanding nuclear interaction and is
ultimately connected with the fundamental particle principles of elementary
physics. Since departures has been observed to be very small it was imperative
to acquire precise data on the nucleon nucleon interaction.

  * •

Babenko and Petrov [9] observed from a comparison of the low-energy parameters
for the np system and their counterparts pp and nn systems that the charge
dependence of nuclear forces is violated, which is associated with the mass
difference between the charged and neutral pions.

  * •

The values of scattering length and effective range are sensitive to any small
change in n-n potentials. As stated by E.S. Konobeevski et al. [10]: “A change
of 7% in V(r) may lead to change of 20-30% in the scattering length
calculation”.

The above mentioned points are either related to the precise knowledge of data
or related to the fundamental knowledge of elementary physics. Experimentally
it has been observed that the p-p scattering [6] is even qualitatively very
different from the n-p scattering. Because the Pauli principle excludes
triplet states of even orbital angular momentum and singlet states of odd
angular momentum. From the p-p scattering it is not possible to conclude
directly that the n-p and p-p interactions are different. However there are
arguments which make it seem fairly plausible that these interactions are
indeed different. Our main aim in this paper is to use VPA and to look for
energy range where ERA potential is able to fit the experimental SPS data. In
our earlier work we have applied VPA or PFM approach for various scattering
systems like n-n, n-p, $n-\alpha$, n-d, $\alpha-\alpha$ and astrophysically
important reaction $\alpha-^{3}He$ scattering systems
[11][12][13][14][15][16][17] [18][19] where Variational Monte-Carlo methods
have been employed for the optimization process [20][21][22]. We have also
obtained n-p quantum mechanical functions like amplitude and wavefunctions for
various n-p channels and are in good aggreement to that of Av-18 potential
[23].

##  2 Methodology:

We know from effective range approximation:

$k\cot(\delta)=-\frac{1}{a}+\frac{1}{2}kr_{e}^{2}$ (1)

where a and $r_{e}$ are scattering length and effective range. If use is made
of the approximation specified by Eq. 1, the S matrix can be written in the
form [9]:

$S(k)=\left(\frac{k+i\alpha}{k-i\alpha}\right)\left(\frac{k+i\beta}{k-i\beta}\right)$
(2)

Where S(k) is related to scattering length a and effective range $r_{e}$ by
following relations

$\alpha=\frac{1}{r_{e}}\bigg{[}1-\big{(}1-\frac{2r_{e}}{a}\big{)}^{1/2}\bigg{]}$
(3)
$\beta=\frac{1}{r_{e}}\bigg{[}1+\big{(}1-\frac{2r_{e}}{a}\big{)}^{1/2}\bigg{]}$
(4)

The ‘velocity independent’ or ‘static’ potential corresponding to effective
range approximation given by [9]

$\boxed{V_{NN}(r)=-V_{NN0}\frac{e^{-r/R_{NN}}}{(1+\gamma e^{-r/R_{NN}})^{2}}}$
(5)

where

$R_{NN}=\frac{1}{2\beta}$ (6)
$V_{NN0}=\frac{\hbar^{2}}{m_{NN}}\frac{\gamma}{R^{2}_{NN}}$ (7)
$\gamma=\frac{1+2\alpha R_{NN}}{1-2\alpha R_{NN}}$ (8)

In above equations $m_{NN}$ is the reduced mass of nucleonic system nn, np and
pp. The effective potential in equation 5 will be introduced inside Variable
phase approach equation as discussed below.

###  2.1 Variable Phase Approach:

The Schr$\ddot{o}$dinger wave equation for a spinless particle with energy E
and orbital angular momentum $\ell$ undergoing scattering is given by

$\frac{\hbar^{2}}{2\mu}\bigg{[}\frac{d^{2}}{dr^{2}}+\big{(}k^{2}-\ell(\ell+1)/r^{2}\big{)}\bigg{]}u_{\ell}(k,r)=V(r)u_{\ell}(k,r)$
(9)

Where $k=\sqrt{E/(\hbar^{2}/2\mu)}$. Second order differential equation Eq.9
has been transformed to the first order non-homogeneous differential equation
of Riccati type [24][25] given by following equation:

$\delta_{\ell}^{\prime}(k,r)=-\frac{V(r)}{k}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$
(10)

Here in Eq. 10 prime denotes differentiation of phase shift with respect to
distance and the Riccati Hankel function of first kind is related to
$\hat{j_{\ell}}(kr)$ and $\hat{\eta_{\ell}}(kr)$ by
$\hat{h}_{\ell}(r)=-\hat{\eta}_{\ell}(r)+\textit{i}~{}\hat{j}_{\ell}(r)$ . In
integral form the above equation can be simply written as:

$\delta(k,r)=\frac{-1}{k}\int_{0}^{r}{V(r)}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j_{\ell}}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta_{\ell}}(kr)\bigg{]}^{2}dr$
(11)

Eq.10 is numerically solved using Runge-Kutta 5th order (RK-5) method with
initial condition $\delta_{\ell}(0)=0$. For $\ell=0$, the Riccati-Bessel and
Riccati-Neumann functions $\hat{j}_{0}$ and $\hat{\eta}_{0}$ get simplified as
$\sin(kr)$ and $-\cos(kr)$, so Eq.10, for $\ell=0$ becomes

$\delta^{\prime}_{0}(k,r)=-\frac{V(r)}{k}\sin^{2}[kr+\delta_{0}(k,r)]$ (12)

In above equation the function $\delta_{0}(k,r)$ was termed “Phase function”
by Morse and Allis [26]. The significant advantage of PFM method is that, the
phase-shifts are directly expressed in terms of the potential and have no
relation to the wavefunction. This has been utilised in this paper to obtain
inverse potentials in an innovative way by implementing a modified VMC in
tandem with PFM. The technique optimizes the model parameters of the potential
to obtain the best match with respect to the experimental SPS values. Also,
rather than solving the second order Schr$\ddot{o}$dinger equation, we only
need to solve the first order non-homogeneous differential equation whose
asymptotic value gives directly the value of SPS.

##  3 Results and Discussion

  * •

For ${}^{3}S_{1}$ state: We can see from figure 1 that the SPS are in
excellent match with analysis data of both Wiringa et al. [27] and Granada
group [3] upto 20 MeV of projectile energy in laboratory frame. The
corresponding potential for ${}^{3}S_{1}$ state is shown in figure 5, where
depth of around 70 MeV is observed.

  * •

For ${}^{1}S_{0}$-nn and np again the SPS are in good match with Wiringa et
al. and Granada as shown in figure 2 and 3. After 20 MeV the deviations can be
clearly seen in figure 2 and 3 indicating the absence of repulsive core which
becomes important at higher energies. The potentials are shown in figure 5.

  * •

Finally for ${}^{1}S_{0}$-pp state the SPS are in reasonably accuracy with
Wiringa et al. and Granada group. The combined potentials for all
${}^{1}S_{0}$-nn, np and pp state can be seen in 5 to be very tightly packed
with each other indicating the charge independence hypothesis of nuclear
force.

It is observed that all the SPS are positive and their corresponding
potentials are negative indicating that positive phase shifts will have
attractive interactions which is in accordance to the main VPA equation. This
point is of great pedagogic value as we can just by observing the nature
($+/-$) of SPS can predict the nature of the interaction potential ($-/+$).
But above statement is not stringent as the results may change for potential
having both attractive part and repulsive core as well as for higher values of
wavenumber k.

Figure 1: Bound triplet state ${}^{3}S_{1}-np$ SPS variation in comparison to
experimental data from WIRINGA-1995 [27] and GRANADA-2016 [3]. Figure 2:
Singlet state ${}^{1}S_{0}-nn$ SPS variation in comparison to experimental
data from WIRINGA-1995 [27]. Figure 3: Singlet state ${}^{1}S_{0}-np$ SPS
variation in comparison to experimental data from WIRINGA-1995 [27] and
GRANADA-2016 [3]. Figure 4: Singlet state ${}^{1}S_{0}-pp$ SPS variation in
comparison to experimental data from WIRINGA-1995 [27] and GRANADA-2016 [3].
Figure 5: Potentials for ${}^{3}S_{1}$-np and ${}^{1}S0$-np, nn and pp
scattering systems.

##  4 Conclusion

Using ERA in VPA we obtained the scattering phase shifts for n-n, n-p and p-p
scattering. Good match in SPS with experiment is found for energy upto 20 MeV
after which deviations are seen for all ${}^{1}S_{0}$-(np, nn & pp) states.
The reason for deviation is that at very small inter-nucleon distances, a
strong repulsive core is expected due to strong anti-corelation between
nucleons hence the absence resulting into deviations observed. As stated by
Breit et al.: “Any of potential shapes like Square well, Gaussian well,
exponential well and yukawa well would give satisfactory agreement with the
experimental data, provided the range and depth of the well are suitably
chosen”. The presented potential is also exponential shaped and good enough to
fit low energy SPS just by giving scattering parameters as inputs within ERA
potential.

##  5 Acknowledgements

This work is devoted to Prof. Francesco Calogero and my Ph.D supervisor Prof.
O. S. K. S. Sastri.

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# Preprocessing Methods of Lane Detection and Tracking for Autonomous Driving

Akram Heidarizadeh  College of Eng., University of Tehran
Tehran, Iran
Email<EMAIL_ADDRESS>
###### Abstract

In the past few years, researches on advanced driver assistance systems
(ADASs) have been carried out and deployed in intelligent vehicles. Systems
that have been developed can perform different tasks, such as lane keeping
assistance (LKA), lane departure warning (LDW), lane change warning (LCW) and
adaptive cruise control (ACC). Real time lane detection and tracking (LDT) is
one of the most consequential parts to performing the above tasks. Images
which are extracted from the video, contain noise and other unwanted factors
such as variation in lightening, shadow from nearby objects and etc. that
requires robust preprocessing methods for lane marking detection and tracking.
Preprocessing is critical for the subsequent steps and real time performance
because its main function is to remove the irrelevant image parts and enhance
the feature of interest. In this paper, we survey preprocessing methods for
detecting lane marking as well as tracking lane boundaries in real time
focusing on vision-based system.

###### Keywords:

Advanced Driver Assistance System; Lane Detection and Tracking; Preprocessing
Block; Real Time Processing.

##  I Introduction

The increasing number of road accidents is a critical matter that modern
society is confronted with. Driver inattention, fatigue and somnolence are the
most common reasons that cause the accidents with high number of fatalities.
Different measures are being applied by automotive industries in order to
minimize the accidents by incorporating advanced driver assistant system in
vehicles. Nowadays vehicles tend to be more intelligent, intelligent vehicles
are the ones which apply various advanced technologies to capture real time
information which is further analyzed to provide more comfort and safety for
the drivers. Vision-based and Radar-based techniques are the most commonly
used ones in intelligent vehicles. Vision-based techniques are recognized as
robust and impressive methods used for advanced driver assistance system.
Vision-based lane detection, traffic sign detection, pedestrian detection,
vehicle detection, etc. are the most popular features that are extensively
deployed in intelligent vehicles. Efforts have been done by several
researchers all over the world for developing lane detection and tracking
system. Lane detection system faces many challenges and the robust techniques
are needed to be developed to overcome these problems and to increase the
efficiency of detection[1].

Most vision-based lane detection systems are commonly designed based on image
processing techniques within similar frameworks. Vision-based lane detection
systems described in studies usually consist of three main process, which are
preprocessing, lane detection and lane tracking, as shown in Fig.1. Among
these, the preprocessing process, which enhance the input image, is one of the
most significant aspects of the real time lane detection system[2].

A lane detection and tracking system meet too many challenges, as shown in
Fig.2. Strange street materials and different slopes create problems in lane
detection. Not only faded lanes on the road but also sharp and irregular
shaped curves of lanes can reduce the quality of lane detection systems
processing. It has been see that even the merged lanes have leaded to some
detection problems. Shadows made by trees on urban road are one of the most
common conditions which can produce strength edges to mislead the lane
detection. Environmental condition issues also affect on the image clarity
like roads covered with the snow, fog, heavy rains, and reflection on wet
roads with low visibility[3]. On the other hand, in real time lane detection
and tracking systems, it is very important to minimize the processed items to
decrease the processing time. Proper implementation of the preprocessing part
of a lane detection system is great importance in covering these challenges.
This study tries to comprehensively review the lane detection system with
focusing on preprocessing block.

The rest of this paper is organized as below. In section II all preprocessing
methods used in various papers have been exactly investigated. In section III
and IV lane detection and lane tracking methods review respectively. Finally,
the section V highlights the concluding remarks.

Figure 1: General Framework of Lane Detection and Tracking Systems Figure 2: A
Number of Challenges to The Lane Detection and Tracking Systems

##  II Preprocessing

As it mentioned, images which are extracted from the video have various
noises. The task of the preprocessing block is the removal of noise and other
unwanted components of the image and the successful transfer of areas with
useful information to the next stage.

The most common procedures in the preprocessing step include image smoothing,
region of interest (ROI) selection, transferring color image into greyscale
image or a different color format, inverse perspective mapping (IPM) also
known as birds-eye view, segmentation and etc. In the following, we carefully
investigate all of the preprocessing steps.

###  II-A Image Smoothing

In order to recognize lane markings, image smoothing can be performed using
various filters including Median filter, Gaussian filter, Histogram filter,
Dilation and Erosion filters, mean filter and etc. to blur the noisy details.

####  II-A1 Median filter

In [4], [5], [6], the median filter is used for smoothing. By using the median
filter on the image, various noise can be deleted. In [4], the median filter
is used to eliminate the heterogeneity of the road surface. In [4], for the
pixel I (x.y), the median filter is applied as in equation (1).

$I(x,y)=median\\{I(x-k,y-1)|(k,1)\in w\\}$ (1)

Where w is the chosen window. In [7], a medium filter of $3\ast 3$ was used
for smoothing.

####  II-A2 Gaussian filter

In [8], to remove noise in the image, including noises caused by rain and snow
and dirty roads, Equation (2) is used as a Gaussian pattern.

$\frac{1}{16}\times\Bigg{[}\,\begin{tabular}[]{ccc}1&2&1\\\ 2&4&2\\\
1&2&1\end{tabular}\Bigg{]}\,$ (2)

With respect to the part of the image in the form of equation (3), the output
of each pixel is obtained by equation (4).

$A=\Bigg{(}\,\begin{tabular}[]{ccc}a(m-1, n-1)&a(m, n-1)&a(m+1, n-1)\\\ a(m-1,
n)&a(m, n)&a(m+1, n)\\\ a(m-1, n+1)&a(m, n+1)&a(m+1,
n+1)\end{tabular}\Bigg{)}\,$ (3)

$a^{\prime}(m,n)=\frac{1}{4}(1\enspace 2\enspace 1)*A*\frac{1}{4}(1\enspace
2\enspace 1)^{T}$ (4)

In [9] applied a Gaussian filter to smooth the input image based on gradient
information.

####  II-A3 Histogram filter

In [10], the median filter and Gaussian filter have been used to reduce the
random noise and Salt and Pepper noise. They then use the histogram filter to
improve the brightness and contrast of the image. This action is performed
using equation (5).

$O(\,x,y)\,=\frac{I(\,x,y)\,-min[\,I(\,x,y)\,]\,}{max[\,I(\,x,y)\,]\,-min[\,I(\,x,y)\,]\,}*255$
(5)

####  II-A4 Dilation and Erosion filters

In [11], in order to remove noise and simplify the computation in subsequent
steps, basic mathematical morphology operators, Dilation and Erosion filters,
are used.

####  II-A5 Average filter

In [13], a temporal blurring creates an average image $\overline{I}_{K}(n)$ to
make the line markings appear long and continuous. This smoothing helps
connect dashed lane markers to form a near continuous line. Consider a video
clip that contains $N$ images where each image is indexed by $n$ and
$n\in[1,N]$. The average image is created by computing the average between the
current image $I(n)$ and $K$ images from the past. It is formed as follows:

$\overline{I}_{K}(n)=\sum^{K}_{i=0}\frac{I(n-i.\bigtriangleup)}{K}$ (6)

where $i$ and $\triangle$ are used to create the offset from the current image
index $n$.

###  II-B Region of Interest (ROI)

The extraction of region of interest (ROI) is an important task of
preprocessing stage, aiming at reducing the computational cost due to the
processing time. It is unnecessary to process the entire pixels of images.
Computation should be focused on regions which contain important information,
as shown in Fig.3. To get the ROI, three main approaches can be found in
literature, which are vanishing point detection, perspective analysis and
projective model, and sub-sampling.

Figure 3: Region of Interest

####  II-B1 Vanishing Point Extraction

A host of researchers utilized vanishing point for road following or as a
directional constraint for defining the path boundaries. Vanishing point is
defined as the point of intersection of the perspective projections of a set
of parallel lines in 3D scene onto the image plane. For a straight road
segment,vanishing point is obtained as the intersection point of the lines
that characterize the lane, as shown in Fig.4(a). In the case of a curved
road, vanishing point is approximated by the lane borders, markings etc. in
the vicinity of the vehicle, as shown in Fig.4(b). Its distance components to
the image center yield the relative direction (yaw and pitch angles) of the
vehicle with respect to the path. Therefore, estimated vanishing point can be
utilized to as a directional constraint for region of interest segmentation
[12] or steer the vehicle [13], lane marking detection [14] etc.

Over the past decades, vanishing point detection has been intensively studied.
Some methods have been proposed to detect road vanishing point based on road
edge [15]. In this way, road boundaries or lane markings are first extracted
by edge detector, and then straight lines are detected by Hough transform,
from the intersections of line pairs, vanishing point can be detected. This
kind of approaches may work well in structured roads with lane markings,
however, these methods do not appropriately behave in unstructured roads with
complex geometric characteristics.

In order to overcome the shortcomings of the above methods, texture-based
methods have been proposed recently. The methods do not rely on road
boundaries or lane markings, and their performances in unstructured roads are
much better than that of edge-based methods. Some researchers employed Gabor
filters to estimate local textures orientations for the voting of road
vanishing point [13], [12], [16]. However, the local dominant texture
orientation in every pixel of the road image is estimated, and the vanishing
point candidates are obtained in a greedy way. Thus, the computational cost is
very high, so that these algorithms cannot be realized in real time.

Figure 4: Vanishing Point Extraction

####  II-B2 Perspective analysis and projective model

The lane marking width and shape change as a result of perspective effect.
Parallel lane markings in real world plane intersects at vanishing point in
the image plane. Usually, by analysing the perspective effect, detection range
can be focused on a certain area, which can be region of interest. With a
reasonable projection applied between image plane, real world plane and camera
plane, not only the region of interest can be extracted, it also benefits
overcoming the perspective distortion, getting the position of vanishing point
and bird’s eye view.

A projective model can be determined by computing the homography between real
world plane and image plane. Road plane needs to be transformed to image
coordinate system through projection matrix given by camera calibration. It is
noticeable that, as an essential component of a homography, the calibration
matrix of camera is usually computed off line. As described in [17], a pinhole
camera projection model is used to compute the position of vanishing point, by
computing the unknown parameters of the rotation matrix.

Rapidly Adapting Lateral Position Handler (RALPH) constructed a very basic
projection model to get region of interest, which is a trapezoid, to focus on
lane marking areas, so that irrelevant area (road surface) can be eliminated.
Due to the perspective effect, the bottom is shorter than the top of the
trapezoid, with their length vary based on vehicle velocity
[pomerleau1995ralph].

####  II-B3 Sub-Sampling

It is also reasonable to split ROIs by sub-sampling. A predefined or adaptive
percentage of the image can be used to determine the size of region of
interest [8]. Besides, some researchers divide the image horizontally or
vertically into small parts [18]. By conducting different strategies for rural
and urban areas, in [19] applied predefined percentage to split ROIs for rural
ways and adaptive percentage for highways.

###  II-C Inverse Perspective Map (IPM)

Inverse perspective mapping (IPM) is one of the most important parts of
preprocessing that is commonly used. This may be realized by remapping each
pixel towards a different position, usually birds’ eye view. Experimentally,
it has already been proved that Lane detection within the image given by IPM
performs much better than methods without IPM. Many researchers have used IPM
to transform an image from a real world plane to a bird’s eye view [20], [21],
[22]. As shown in Fig. 5, by eliminating the perspective effect, the desired
lane marking candidates present in the form of straight and parallel lines in
the remapped image.

Figure 5: Inverse perspective mapping

###  II-D Segmentation

To prepare images for detection stage, segmentation can be accomplished by
extracting certain features from the input image. Colour and edge are two main
features which are considered for lane detection segmentation. The major
problem with edge segmentation is that a number of thresholds need to be
adjusted, and the adjustments depend only on the image undergoing testing,
while colour segmentation techniques are sensitive to lightening conditions,
especially those occurring on sunny days.

####  II-D1 Edge-based Segmentation

Edge-based segmentation usually consists of edge detection and binarization.
Different edge detection techniques are proposed based on edge features, which
usually include shape (distribution of edge points) and gradient information.
Canny algorithm is used in [23] and [9]. Steerable filter is another effective
approach for edge segmentation. As demonstrated in [24], edges are extracted
based on lane orientation characteristics with applying steerable filter.
Prweitt filter is another edge detector used in [22].

A straightforward step after edge detection is to apply binarization, by doing
this properly, effective details can be reserved while noisy points being
neglected. Different methods have been proposed to find suitable threshold for
binarization. A predefined threshold based on massive experimental results is
used for this task in [4]. Because of the dependence on massive experimental
results and previous experience, predefining the threshold can be inaccurate.
Hence, adaptive threshold is proposed as a more effective approach for
segmentation to binarize images [25].

####  II-D2 Colour-based Segmentation

Most of the case, different colour features can be presented based on
different colour spaces (RGB, HSV, gray scale, etc.). In the literature, data
structure of the images can be changed in order for better performance based
on different colour spaces. For structured roads (mostly highways), it is
better to use gray images for colour representation of images to be detected,
because gray images make lane markings with light colours (white or yellow)
distinguished from road surface (with dark gray for most of the time). Hence,
gray-level conversion is considered in many papers, such as [26] and [17]. As
in [17], a parametric multiple-class likelihood model of the road is proposed
based on the gray level of pavement, lane markings and objects. Parameters are
estimated by the Expectation-Maximization (EM) algorithm.

In [27], different from directly converting coloured images into gray level,
only uses R and G channels of coloured images to form gray images. Furtherly,
to make full use of the colour information of RGB images, some researchers
deployed colour format conversion in their algorithm. For example, in [28],
RGB images are converted to HSV. However, format conversion of coloured images
may inevitably lose a certain amount of detail information in real scenarios.
To solve this problem, in [29] converted RGB to HSI. In in [29] revealed the
advantage of combining HSI model with loosen thresholds, in terms of keeping
useful information and highlighting the difference between lane markings and
road surface. The method proposed in [30] is based on the colour consistency
of the road surface. The road colour is assumed to be Gaussian distributed, by
which the road surface colour and relevant variance can be computed.

####  II-D3 MSER-based Segmentation

In [31] proposed a method of describing MSER blobs using SIFT-based
descriptors, followed by a graphical model in order to localize lane markings.
This method combines unsupervised learning algorithm with off-line training.
It consists of 3 main steps: 1) obtain descriptors by extracting the feature
of lane markings; 2) quantize descriptors into which relies much on visual
words retrieving and 3) the image is described as a combination of visual
words.
There are two drawbacks for this method: 1) the processing time varies in
different scenarios, as the number of features detected varies; 2) the visual
words for the feature of lane markings mostly consist of line segments, which
takes both salient lane markings and noisy lines into account. This might
increase the false positive results.

##  III Lane Detection

The second stage, detection, extracts lane markings from the output of the
preprocessing step. In this section, lane detection methods are reviewed from
the scope of conventional image processing and novel machine learning
techniques. The first part will focus on the review of generally used
conventional image processing methods. In the second part, lane detection
algorithms based on machine learning and deep learning techniques will be
discussed.

###  III-A Image Processing Based Lane Detection Algorithms

This approach extracts lane markings by using feature extraction methods.
Three main feature extraction approaches can be categorized in the literature:
edge-based methods, colour-based methods and hybrid methods.

####  III-A1 Edge-Based Methods

One of the most commonly edge extractors used in this task is Hough transform
[32],[20]. However, Hough transform has some disadvantages such as its
computational complexity and the unavoidably high false positive rate. Facing
with this issue, variants of Hough Transform are used. Adaptive random Hough
transform (ARHT)[27] and probabilistic Hough transform (PHT) [9] are both
impressive methods in terms of reducing false positive rate. Another edge-
based method based on Steerable filter is used in many research papers such as
in [26],[21]. A Steerable filter is convolved with an input image to select
the maximum response in the direction of lane marking. This method has good
effects when road markings are clearly painted and totally smooth. However,
Steerable filter is not adapted to heavy traffic where the orientation of lane
markings is not always dominant of all directions.

####  III-A2 Colours-Based Methods

These methods are quite efficient for unstructured roads, or rural roads
without clear lane boundaries. However, unlike edge-based methods, colour-
based methods are not widely used by researchers. Colour information has its
own drawbacks as it is influenced by lightening. Therefore, Detection based on
colour does not work well for night scenes or heavy and complex traffic. In
[33] used a colour-based method in the HIS color space by computing the
cylindrical distribution of colour features. Lane detection based on colour
methods on gray level is used in [34].

####  III-A3 Hybrid Information Based Methods

Hybrid information is usually composed of edge information and colour
information. This type of feature extraction scheme usually combines width,
length, location (coordinates of pixels) of lanes with gray level and
brightness intensity of groups of points, which makes the extraction results
better. The example of using hybrid information is in [35], lane boundaries
are deemed as a combination of four lines, which are left and right boundaries
of left and right lane marking, respectively. lane marking is treated as a
region with a certain thickness (space between left and right boundaries). The
grading criterion for the hypothesis is first calculated separately for the
left and right line pairs, and then the two are combined together. The grading
scheme are based on both edge and gray-level information, which are mean
support-size of a pair, mean perpendicular gradient of a pair, similarity of a
pair, mean continuity of a pair and mean lateral accuracy.

###  III-B Machine Learning Based Lane Detection Algorithms

In spite of using conventional image processing based methods to detect lane
markings, some investigators focus on detecting lane marking using novel
machine learning and deep learning methods. In the last decade Deep Learning
became one of the hottest topics and trended research areas due to the
development of deep network theories, parallel computing techniques, and large
scale data. Many deep learning algorithms show great advantages in computer
vision tasks and the detection and recognition performance increases
dramatically compared to conventional approaches. The convolution neural
network (CNN) is one of the most popular approaches used for the object
recognition research. CNN provides some impressive properties such as high
detection accuracy, automatic feature learning, and end-to-end recognition.
Recently, some researchers have successfully applied CNN and other deep
learning techniques to lane detection. It was reported that by using CNN
model, the lane detection accuracy increased dramatically from 80% to 90%
compared with traditional image processing methods [36],[2].

##  IV Lane Tracking

To enable the following of lane marking over time, a tracking stage is usually
incorporated. The two most common tracking techniques used in lane position
detection systems are Kalman filtering [20], [37] and Particle filtering [38],
[39]. In these systems, feature extraction and position tracking are often
combined into a closed-loop feedback system in which the tracked lane position
is defined as an initial estimation of the location and orientation of the
extracted features. This stage has the ability to reduce the detection of the
misleading parts by comparing the current detection with previous frames and
then improve the detection accuracy.

###  IV-A Kalman Filter

Previous work has been done on tracking the parameters of Hough transform by
using Kalman Filter as in [20] and [37]. Usually, as the output of standard
Hough transform, ($\rho$, $\theta$) is used to initialize the Kalman filter.
With proper setting of state transition matrix, the state vector can be
updated, which consists of ($\rho$, $\theta$) and its derivative. In spite of
the advantages of Kalman filter in terms of real time performance, it is
unable to reject outliers that cause failure of tracking, which in turn
decreases the accuracy of detection. To alleviate these problems, an extended
Kalman filter (EKF) for road tracking was introduced in [40]. EKF is designed
for the tracking stage of a non-linear dynamic systems, such as lane detection
in complex road environment where the noise might pose as a non-Gaussian
distribution.

###  IV-B Particle Filter

Particle filter is another reliable option for lane tracking. It has been used
in [38] and [39]. In [38] proposed a method, which combines lane border
hypotheses obtained by RANSAC with hypotheses from a particle filter. In [39]
takes more information into account, which are horizontal and vertical
curvature, lateral offset and lane width. In terms of visual cues for robust
lane tracking, [39] uses curbs and road edges as the cues, while [41] uses
more than those, which consists of lane marking, road edge, road and non-road
colour and road width. Usually, more reliable cues make the lane tracking with
particle filter performs well in more complex situations.

In [39] with a comparison between kalman fiter and particle filter pointed out
that, particle filter and Kalman filter have their own advantages and
disadvantages. Unlike a Kalman filter, a particle filter does not need
initialization or measurement validation before updating the state matrix.
Particles can evolve by themselves, and it might have the chance to cluster
around the best lane estimation. However, the premise is the tracking system
must be well designed and the measurement cues are relevant, which requires
the system must know how to validate a lane candidate.

##  V Conclusion

Lane detection is a fundamental aspect of most current Advanced Driver
Assistance Systems. Several intelligent transportation system applications,
such as automatic steering systems, lane departure warning systems, lane
change warning, adaptive cruise control and lane keeping assistance systems,
depend on obtaining accurate estimates of the vehicle’s trajectory with
respect to the lane the vehicle is driving in. The critical challenge that
lane detection systems are faced with is the demand for high reliability and
the diverse working conditions. In this paper we have studied the different
approaches of lane detection and tracking system. We focused on preprocessing
block and investigated the details of different methods. The purpose of
preprocessing stage is to enhance the input image in order to increase the
likelihood of the successful delivery of areas with useful information to
subsequent stages. With proper implementation and appropriate selection of
preprocessing methods, real time lane detector can be implemented with proper
accuracy.

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# Dose rate dependence of TID damage to 65 nm CMOS transistors in X-ray
irradiations of the ATLAS ITk Pixel ASIC (ITkPix)

D. Bortoletto  A. Dimitrievska  M. Garcia-Sciveres  T. Heim  M. Mironova  R.
Plackett  I. Shipsey  J. Xiong  University of Oxford (UK)  University of
Birmingham (UK)  LBNL (US)

###### Abstract

The ATLAS Inner Tracker (ITk) upgrade for the High-Luminosity LHC (HL-LHC)
requires a radiation-tolerant pixel readout chip, which must withstand a total
ionising dose (TID) of up to 1 Grad. The readout ASIC for the ITk upgrade has
been designed by the RD53 collaboration using 65 nm CMOS technology. In order
to characterise the radiation tolerance of the chip digital logic, the RD53
ASICs include ring oscillators, which can be used to measure gate delay
degradation. Extensive X-ray irradiation studies of the ring oscillators have
been performed on the ITk Pixel pre-production readout ASIC, ITkPixV1. A
dependence of radiation damage on dose rate has been observed in 65 nm CMOS
technology. This paper aims to quantify the dose rate dependence of TID damage
to the ITkPix ring oscillators and, therefore, the ITkPix ASIC digital logic.
X-ray irradiations at different dose rates between 20 krad/h and 30 Mrad/h are
compared. A dose rate dependence is observed, with 2-3 times more damage at
the lowest dose rate of 20 krad/h, compared to 4 Mrad/h. The dose rate
dependence was also observed to be dependent on transistor size and type.

##  1 Introduction

The ATLAS experiment will upgrade its inner detector for operation during
High-Luminosity LHC (HL-LHC) to cope with the increased instantaneous and
total luminosity and increased collisions per beam-crossing. New pixel readout
chips were designed by the RD53 collaboration for the ATLAS and CMS
experiments to satisfy the harsher requirements on the radiation tolerance and
the efficient and fast detection of signals. Following the RD53 design effort,
the ITk pre-production ASIC ITkPixV1 was submitted in 2020, followed by the
final production ASIC ITkPixV2 in 2023.

The readout chip is required to withstand a total ionising dose (TID) of up to
1 Grad in the innermost ITk layers, and the radiation tolerance of the chip
has to be validated experimentally [1]. For radiation testing, the chip
includes 42 ring oscillators, using different types and sizes of logic gates,
which allow the characterisation of gate delay increase with irradiation.
Generally, TID damage can occur in different parts of transistors, and for the
small transistors in 65 nm technology, channel edge effects make up the most
significant contribution [2]. For instance, charge build-up in the Shallow
Trench Isolation (STI) and spacers can significantly change the leakage
current and threshold voltage of the transistor, phenomena known as Radiation-
Induced Short Channel (RISCE) and Narrow Channel (RINCE) effects [3, 4].
Irradiation studies of single transistors show that the radiation tolerance is
worse for narrow-channel transistors, and more damage is observed during
irradiations using low dose rates and at higher temperatures [5, 6].

This dose rate dependence needs to be considered when performing radiation
studies for ITk. While it is possible to achieve the expected ITk TID of 1
Grad on the timescale of weeks when irradiating at dose rates of several
Mrad/h, the dose during ITk operation will be delivered at a much lower dose
rate. The highest dose rate in the innermost ITk layers is expected to be
around 20 krad/h. Therefore, understanding the scaling of TID damage with dose
rate is crucial to determine whether the ITkPix ASIC is sufficiently radiation
tolerant to withstand the ITk environment. This paper aims to quantify the
dose rate dependence of TID damage in small transistor gates by performing
X-ray irradiation campaigns of the ITkPixV1 ring oscillators at different
doses ranging from 20 krad/h to 30 Mrad/h and comparing the gate delay
degradation at the different dose rates, after total doses of at least 10
Mrad.

All measurements presented in this paper have been performed using the pre-
production readout chip ITkPixV1.

##  2 Setup

This section introduces the ring oscillators in the ITkPixV1 chip, which are
used to perform the radiation studies, along with the X-ray irradiation setup
and calibration measurements.

###  2.1 Ring oscillators in the ITkPixV1 chip

The ITkPixV1 chip includes various ring oscillators, which can be used to
characterise radiation damage to different kinds of gates used in the chip
digital logic. These ring oscillators are made with varying types of logic
cells and using a range of transistor sizes. Each oscillator drives a 12-bit
counter, while a 4-bit counter counts the number of start/stop pulses
received. From this, it is possible to calculate the frequency of the ring
oscillator and the gate delay.

Table 1 summarises the different kinds of ring oscillators in ITkPixV1, their
transistor drive strength and the length of the oscillators (i.e. the number
of gates in each circuit). The transistor drive strength is related to the
size of some of the transistors in the circuit, scaling as $W/L$ for those
transistors, where $W$ and $L$ are the transistor widths and lengths,
respectively. In particular, strength 0 gates use minimum size transistors,
while strength $>$0 gates do not have minimum size transistors.

Table 1: Summary of the 42 ring oscillators available in ITkPixV1 with their gate type, drive strength and length. Ring oscillator number | Type | Strength | Length
---|---|---|---
Bank A
0 | Inv. CLK driver | 0 | 55
1 | Inv. CLK driver | 4 | 51
2 | Inverter | 0 | 55
3 | Inverter | 4 | 55
4 | 4-input NAND | 0 | 19
5 | 4-input NAND | 4 | 19
6 | 4-input NOR | 0 | 19
7 | 4-input NOR | 4 | 19
Bank B left/right
0 & 1 | Inv. CLK driver | 0 | 38.2
2 & 3 | Inv. CLK driver | 4 | 44.5
4 & 5 | Inverter | 0 | 38.1
6 & 7 | Inverter | 4 | 44.3
8 & 9 | 4-input NAND | 0 | 12.6
10 & 11 | 4-input NAND | 4 | 16
12 & 13 | 4-input NOR | 0 | 14.5
14 & 15 | 4-input NOR | 4 | 14.5
Bank B FF
16 & 17 | Scan D-flip-flop | 0 | 6.1
18 & 19 | D-flip-flop | 1 | 6.2
20 & 21 | Neg. edge D-flip-flop | 1 | 5
Bank B LVT
22 | LVT inverter | 0 | 40.6
23 | LVT inverter | 4 | 56
24 | LVT 4-input NAND | 0 | 16.5
25 | LVT 4-input NAND | 4 | 22.8
Bank B CAPA
26-33 | Inj-cap loaded 4-input NAND | 4 | 16.8

For each ring oscillator, it is possible to count the number of oscillations
in a given time, which then can be used to determine the ring oscillator
frequency $f$, or the delay of the individual gates as $T=1/(N\times f)$,
where $N$ is the length of the ring oscillator. It is also possible to
calculate the relative increase in gate delay as $\Delta T=1/(N\times
f)-1/(N\times f_{\mathrm{start}})$, where $f_{\mathrm{start}}$ is the ring
oscillator frequency before irradiation. The gate delay will increase with
increasing radiation damage, and the ring oscillator frequency will decrease,
which is the primary measure of radiation damage to the digital logic.
Simulations of the chip determined that the gate delay increase has to stay
below 200% to ensure sufficiently good timing in the digital logic of the
chip. Before irradiation, the logic gate delays are between 0.03 to 0.4 ns.

###  2.2 X-ray irradiation setup and calibration

Most irradiation campaigns presented in this paper were performed at the
Oxford Microstructure Detector facility (OPMD) using a Euroteck X-ray system
[7]. The X-rays are produced with a tungsten target and then filtered by a 0.8
mm Beryllium window when exiting the tube. The tube can be operated with an
anode voltage of 5 kV to 60 kV and a current between 1 and 60 mA, with the
maximum achievable power being 3 kW. For the ITkPixV1 irradiations, the X-rays
are additionally filtered by 150 $\mu m$ of Aluminium to remove low-energy
components. The dose rate (in $SiO_{2}$) delivered by the X-ray tube at a
particular setting was measured before each irradiation campaign using a diode
calibrated at CERN [8].

The X-ray tube was further characterised by measuring its spectrum with a
silicon drift detector, with the results shown in Figure 1, with and without
the Aluminium filter. The beam spot profile was measured using a Medipix
Merlin detector [9], by counting the number of photon hits per
$\mathrm{10\;\mu s}$ window at a distance of 10 cm from the tube exit. The
resulting map of hits was then scaled to the measured dose rate at this
distance from the tube and is also shown in Figure 1. At 10 cm from the tube
exit, the maximum achievable dose rate is 4 Mrad/h, and the beam spot has a
size of 8.7 cm, defined as the region where the dose rate is at least 80% of
its maximum value. The beam spot is significantly larger than the size of the
ITkPixV1 chip. If the ITkPixV1 chip is placed at the centre of the beam spot,
the delivered dose is expected to be uniform within 5%.

One additional X-ray irradiation at a dose rate of 0.5 Mrad/h was performed at
LBNL, using a Seifert X-ray system, but otherwise using the same setup for
both X-ray machine settings and chip setup, and the dose rate was calibrated
using the same calibration diode.

In this paper, the TID received by the ring oscillators is quoted. This means
the dose rate measured by the calibration diode is reduced by 20% to account
for the absorption of X-rays by metal layers covering the transistors in the
chip. This correction factor was derived based on the X-ray spectrum and
thickness of the metallisation layers and was previously studied in a
dedicated measurement. Note that the results determined in this paper do not
significantly depend on the assumptions made about the exact dose rate, as the
main reported number are ratios of radiation damage at high and low dose
rates.

Figure 1: Energy spectrum of the X-ray tube at an operational voltage of 40 kV
(left), with and without a $\mathrm{150\;\mu m}$ Aluminium filter. Beam spot
10 cm from the exit of the tube (right).

###  2.3 Irradiation setup

Figure 2: Schematic (left) and picture (right) of the setup used in
irradiations of the ITkPixV1 chips.

The ITkPixV1 chips were irradiated in the X-ray setup described above. The
ITkPixV1 chip was mounted on a dedicated PCB (single-chip card), which is used
to supply the input voltage, connect to the YARR DAQ system [10], and has pins
accessible for monitoring the voltages and currents in the chip. The X-ray
tube was operated at a voltage of 40 kV, with a 150 $\mu$m Aluminium filter
placed at the tube exit. The desired dose rate was achieved by adjusting the
tube current and placing the chip at a certain distance from the tube exit.
The setup is illustrated in Figure 2.

The irradiations were performed while the chip was cooled by placing the chip
on a chuck connected to a Huber CC-508 chiller, which was able to keep the
chip at a constant temperature of -10∘C (+/- 1∘C), measured using an NTC
thermistor on the chip PCB. This corresponds to a temperature of -7.5∘C on the
chip, which is the maximum expected temperature during ITk operation. The
maximum expected temperature was chosen, as radiation damage has been seen to
be more severe at high temperatures. The temperature was monitored by reading
out the resistance of an NTC thermistor on the single-chip card [11]. The chip
was powered by supplying 1.6 V to the chip. This voltage is distributed to the
analog and digital domains of the chip via the chip internal linear
regulators, leading to a nominal 1.2 V supplied on each. The voltage provided
to the digital domain (VDDD) was monitored, as the frequency of the ring
oscillators is sensitive to it.

The primary measurement during the irradiations was the frequency of the ring
oscillators. The ring oscillators were read out at least after every 0.1 Mrad
of delivered dose. When the ring oscillators were not measured, digital
injection tests were run to simulate operation in a data-taking environment.
Irradiations were performed at dose rates ranging between 20 krad/h and 30
Mrad/h. For the irradiation at a dose rate of 30 Mrad/h, the chip was placed
as close as possible to the X-ray tube exit, and the beam spot was centred on
the ring oscillators in the chip. For the other irradiation campaigns, the
entire chip was irradiated.

##  3 Results

This sections outlines the calibration measurements performed on the ring
oscillators, followed by the analysis and results of the X-ray irradiation
campaigns.

###  3.1 Ring oscillator calibration measurements

Figure 3: Examples of the VDDD (left) and temperature (right) calibrations of
the ring oscillators in ITkPixV1 before irradiation.

Before performing irradiations, preliminary measurements of the ring
oscillators were taken. There is a strong dependence of ring oscillator
frequency on the supplied voltage and a slight dependence on the temperature.
The dependence on the digital voltage was characterised for each chip before
irradiation.

First, the frequency of the ring oscillators was measured while varying the
VDDD voltage. The resulting dependence is shown in Figure 3 for a subset of
the ring oscillators. A linear behaviour is expected and visible, and the ring
oscillator frequency increases by up to 300 MHz when increasing VDDD from 1.0
to 1.3 V. The slopes and offsets are determined for each ring oscillator. They
are then used to account for changes in VDDD during the irradiation.

Next, the dependence of the ring oscillators on temperature was measured. This
was done by slowly cooling down the setup from room temperature to -12∘C and
monitoring the ring oscillator frequency while accounting for any changes in
VDDD using the parameterisations derived above. The results are also shown in
Figure 3. A clear linear frequency dependence on the temperature can be seen.
The frequency changes by up to 20 MHz when changing the temperature from -12∘C
to room temperature. The temperature dependence does not need to be accounted
for during irradiation, as the temperature is kept constant.

###  3.2 Dose rate dependence of TID damage

Figure 4: Gate delay increases at different dose rates for the ring oscillator
made of Inverter gates, for the strength 0 (left) and strength 4 (right)
gates.

Seven irradiation campaigns were performed, with dose rates of 20 krad/h, 40
krad/h, 60 krad/h, 80 krad/h, 0.4 Mrad/h, 4 Mrad/h and 30 Mrad/h, collecting
at least 10 Mrad of TID during each irradiation campaign. The lowest dose rate
of 20 krad/h corresponds roughly to the dose rate expected in the innermost
layers of ITk. The frequencies of each ring oscillator were analysed and
corrected for the variations in digital voltage (VDDD), using the calibration
curves derived above. The ring oscillator frequencies were then translated
into the delays of the individual gates, and the relative increase in gate
delay was calculated. Figure 4 shows these relative increases in delay for the
different irradiation campaigns for some examples of the strength 0 and 4
inverter gates. After an initial decrease in gate delay, corresponding to an
initial drop of $V_{gs}$ in the transistor, the increase follows a linear
trend, up to total doses of 20 Mrad. This trend is parametrised for each
irradiation campaign by fitting a linear function to the data from a total
dose of 3 Mrad onwards. The resulting fitted functions are also shown in
Figure 4.

Figure 5: Fitted increase in relative gate delay per TID as a function of dose
rate for all types of gates used in the ITkPixV1 ring oscillators.

It is also possible to consider the increase in gate delay per TID as a
function of dose rate. Figure 5 shows the gate delay increase per TID (slopes
of the fitted linear functions) as a function of the dose rate for each type
of ring oscillator. The top panel in each plot shows the fitted values. The
bottom panel shows the ratio of the values compared to the irradiation at a
dose rate of 4 Mrad/h. This estimates the conversion factors needed for
translating results from irradiations at high dose rates, commonly performed
to high TID, and the expected gate delay degradation at the same TID delivered
at a low dose rate. The results are averaged between the Bank A and Bank B
ring oscillators of the same type. A systematic uncertainty on the dose rate
is considered, depending on the precision with which it could be measured
using the calibrated diode and the accuracy of the chip position in the beam
spot. This uncertainty is usually of the order of 5-10%. A systematic
uncertainty on the fitted ring oscillator delays is also considered by varying
the range used for the linear fit.

There is a clear trend of increased gate delay degradation at low dose rates.
This increase depends on the transistor size, with the gate delay increase
being roughly 3 times larger at low dose rates for the strength 0 gates, and
only 2.5 times larger for the strength 4 gates. However, the conversion factor
also depends on the exact type of gate used. It is also worth noting that the
behaviour of increasing gate delay increase seems to flatten out at low dose
rates, which is consistent with what was observed in single transistor
measurements [6]. This suggests that the gate delay degradation per TID will
not be significantly worse at even lower dose rates, for instance, in the
outer layers of ITk.

##  4 Conclusions

This paper characterised the dose rate dependence of radiation damage in 65 nm
CMOS transistors in X-ray irradiation of the ATLAS ITk pixel pre-production
readout chip ITkPixV1. Several X-ray irradiation campaigns were performed at
dose rates varying between 20 krad/h and 30 Mrad/h. The resulting gate delay
degradation was compared for total doses of at least 10 Mrad and more
radiation damage was found for the same delivered TID at low dose rates, as
expected from existing single transistor measurements [6]. The exact increase
in damage was found to depend on the size and type of gate used, and the gate
delay degradation was observed to be 2-3 times worse at a dose rate of 20
krad/h, compared to 4 Mrad/h.

This paper only covers dose rate effects at low total doses. To confirm the
actual radiation hardness of the ITkPix digital logic, significantly longer
irradiation campaigns at low dose rates are necessary, which require a
different kind of irradiation method. Such an irradiation campaign with long
exposure is possible using a Krypton-85 source and will be subject of a
separate study.

##  5 Acknowledgements

The research presented here was funded by a grant from the STFC (Oxford CG
ST/S000933/1 and ST/W000628/1). Maria Mironova gratefully acknowledges the
support of the STFC (grant number ST/S505638/1) and the Scatcherd Scholarship
during her PhD. This work was supported by the Office of High Energy Physics
of the U.S. Department of Energy under contract DE-AC02-05CH11231.

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