figure table

# FSDNet-An efficient fire detection network for complex scenarios based on
YOLOv3 and DenseNet

Li Zhu  Nachang University, Cvip Group  Jiahui Xiong  Nachang University, Cvip
Group  Wenxian Wu  Hongyu Yu  Nachang University, Cvip Group

###### Abstract

Fire is one of the common disasters in daily life. To achieve fast and
accurate detection of fires, this paper proposes a detection network called
FSDNet (Fire Smoke Detection Network), which consists of a feature extraction
module, a fire classification module, and a fire detection module. Firstly, a
dense connection structure is introduced in the basic feature extraction
module to enhance the feature extraction ability of the backbone network and
alleviate the gradient disappearance problem. Secondly, a spatial pyramid
pooling structure is introduced in the fire detection module, and the Mosaic
data augmentation method and CIoU loss function are used in the training
process to comprehensively improve the flame feature extraction ability.
Finally, in view of the shortcomings of public fire datasets, a fire dataset
called MS-FS (Multi-scene Fire And Smoke) containing 11938 fire images was
created through data collection, screening, and target annotation. To prove
the effectiveness of the proposed method, the accuracy of the method was
evaluated on two benchmark fire datasets and MS-FS. The experimental results
show that the accuracy of FSDNet on the two benchmark datasets is 99.82$\%$
and 91.15$\%$, respectively, and the average precision on MS-FS is 86.80$\%$,
which is better than the mainstream fire detection methods.

###### keywords:

fire classification, fire detection, spp, feature extraction, dense connection

##  1 Introduction

Fires are characterized by suddenness and spread, and can cause huge economic
losses and seriously threaten human life safety in a short period of time.
According to data released by the Fire and Rescue Bureau of the National
Emergency Management Department in 2022, there were a total of 748,000
reported fires in China in 2021, causing 1,987 deaths, 2,225 injuries, and
direct property losses of up to 6.75 billion yuan. Therefore, researching fire
automatic detection methods that combine real-time and accuracy has important
theoretical and practical significance.

Generally speaking, fire detection methods can be roughly divided into two
types: sensor-based detection methods[1] and vision-based detection methods[2]
. Sensor-based detection methods played a crucial role in early fire
detection, which mainly sense the characteristic changes accompanying a fire,
such as heat, light, and smoke concentration, through sensors[3] . However,
the fire features from the source require a certain amount of time to be
sensed by deployed sensors, which leads to the response of corresponding
sensors being lagging. This characteristic also limits the applicability of
this method to large spaces or public places. In addition, this method also
requires human and other equipment involvement to further determine
information such as the size, location, and degree of burning of the fire[4] .

With the popularization of digital cameras and the rapid development of
various technologies such as image processing, machine learning, and deep
learning, video-based fire detection methods have been rapidly developed. This
type of method captures real-time video image information indoors and outdoors
through cameras, transmits the images to a data processing center, performs
real-time processing and analysis, and finally judges the fire situation on-
site and takes corresponding response measures. In the early days, vision-
based methods were accomplished by manually extracting single or multiple
features from fire images, including static features such as brightness,
color, and texture, and dynamic features such as flicker and motion[5] . For
example, Celik et al.[6] effectively detected fires from video sequences by
combining the YCbCr color space model with foreground object information. Chen
et al.[7] proposed a method that combines the RGB color space model with
dynamic analysis to detect flames and uses the growth characteristics of the
flames to perform secondary judgment on suspected areas. However, detection
methods based on a single static feature are difficult to adapt to various
complex situations and are easily affected by environmental factors.

To overcome this limitation, researchers have considered combining dynamic
features with the static characteristics of flames. Dedeoglu et al.[8]
proposed a flame detection method that combines color, motion features, and
wavelet domain analysis. Time-domain wavelet transforms are used to determine
the boundaries of flames, while spatial-domain wavelet transforms are used to
detect changes in flame color. Toreyin et al.[9] utilized an implicit Markov
model to detect the flickering process of flames, effectively reducing false
alarms caused by dynamic flame color similar objects, but the method is
sensitive to changes in lighting conditions. Habiboglu et al.[10] developed a
video-based wildland fire detection system that obtains the moving area of
flames through background subtraction and color thresholding, and then
extracts relevant features from the motion area. Zhao et al.[11] constructed
static SVM classifiers and dynamic SVM classifiers based on 11 static features
and 27 dynamic features of fire samples to achieve real-time detection of
forest fires, but the false detection rate was high.

Most of the methods proposed by the aforementioned researchers are based on
hand-crafted features and have limitations in their application scenarios.
Therefore, they have gradually been replaced by methods based on deep
learning[12] . Deep learning-based methods can more effectively extract deep
features and semantic information from fire images, further improving the
performance of fire detectors.

In recent years, there have been many fire detection methods based on
Convolutional Neural Networks (CNNs)[13] . Frizzi et al.[14] used a CNN for
fire and smoke detection and tested it with real fire video sequences, showing
superior classification performance compared to traditional fire detection
methods. Gonzalez et al.[15] proposed a fire detection model combining CNNs
and unmanned aerial vehicles based on AlexNet[16] and convolutional sequences,
but the model had a high false alarm rate. In[17], a video fire detection
method was proposed by combining Faster R-CNN with LSTM to extract spatial and
temporal features of flames. In[18], a video fire detection method that
simultaneously considers dynamic features based on motion flicker and depth
static features was proposed. Some researchers have applied CNNs,
Transformers[19], and image segmentation[20] to the field of visual fire
detection to develop trained models for collecting fire image features. Sharma
et al.[21] used VGG16 and ResNet50 to perform fire detection in images and
provided a self-collected fire dataset. Zhang et al.[22] compared the
detection performance of forest fires using object detection algorithms such
as Faster R-CNN[23], YOLO[24, 25, 26] , and SSD[27], where SSD showed better
real-time performance and detection accuracy. They further optimized the
network structure and proposed a new Tiny-YOLO network, demonstrating its
effectiveness.

Although some achievements have been made in the aforementioned fire detection
methods, the overall accuracy is still low, and they have limited
applicability, making it difficult to achieve high robustness in complex real-
life scenarios. To address these issues, this paper proposes a fire detection
network named FSDNet, which combines classification and detection functions
based on YOLOv3 and is used for fire monitoring and prevention. Our main
contributions can be summarized as follows:

(1) A fire detection network named FSDNet based on YOLOv3 and dense
connections is proposed. The network consists of a feature extraction module,
a fire classification module, and a fire detection module, which combines
classification and detection functions and can perform real-time and accurate
fire recognition and detection on image and video data.

(2) In the FSDNet network, a dense connection structure is introduced in the
feature extraction module, which reduces the number of parameters while
strengthening feature propagation through module stacking, small-scale
convolution kernels, and convolutional pooling. The fire detection branch
incorporates spatial pyramid pooling to fuse global and local features. The
CIOU loss function is used as the bounding box regression loss to improve the
precision of predicted bounding box localization.

(3) To address the insufficiency of publicly available fire datasets, we
independently established a fire dataset called MS-FS, which contains 11,938
fire images. Through a large amount of material collection, data screening,
and object annotation work, the dataset covers hundreds of real fire
scenarios.

The experimental results show that the proposed FSDNet network has higher
accuracy than existing mainstream fire detection methods. The overall
accuracy, false alarm rate, and miss rate on two benchmark fire datasets DS1
and DS2 are 99.82$\%$, 0.07$\%$, 0.17$\%$ and 91.15$\%$, 10.28$\%$, 7.56$\%$,
respectively. The mean Average Precision (mAP) on MS-FS reaches 86.80$\%$,
which is 8.5$\%$, 6.4$\%$, and 1.2$\%$ higher than YOLOv3, Gaussian-YOLOv3,
and YOLOv4[28], respectively. It can accurately and real-time accomplish fire
detection tasks. The proposed FSDNet will have a promising application
prospect in community fire monitoring and prevention.

##  2 Method

As shown in Fig. 1, the overall framework of the proposed fire detection
network FSDNet consists of three modules: a dense-connected basic feature
extraction module, a fire classification module, and a fire detection module
with three detection branches. This network takes the flame dataset created in
this paper as input and establishes two models with the same feature
extraction module, respectively used to complete the fire classification task
and real-time fire detection task.

---
Figure 1:  FSDNet algorithm framework diagram.

###  2.1 Feature Extraction Module

DenseNet[29] has shown excellent performance in classification experiments
based on CIFAR and ImageNet datasets, achieving lower error rates with fewer
parameters compared to ResNet[30]. The specific connectivity structure of the
dense connections is shown in Fig. 2. By establishing dense connections such
that the input of each layer is derived from the output of all preceding
layers, feature reuse is achieved in the channel dimension, maximizing
information exchange between preceding and succeeding layers. This
connectivity method has the following advantages: 1. Dense connections are
equivalent to connecting each layer directly to the input and loss,
effectively mitigating the problem of gradient vanishing. 2. Connecting all
preceding layers to succeeding layers enhances feature propagation. 3.
Compared to the common residual connection method of ResNet, the dense
connection method significantly reduces the number of parameters.

Based on the sensitivity of error rates to fire detection and the advantages
of DenseNet, FSDNet chose the dense connection method used in DenseNet for the
convolutional layer connections in the basic feature module. The introduced
dense connection feature extraction module consists of four convolutional
layers, three pooling layers, three Dense Block modules, and one fully
connected layer, which increases the feature reuse, making it highly
meaningful for detecting objects like fire and smoke that do not have fixed
shapes. Additionally, the feature extraction module uses small convolution
kernels for feature extraction, consisting only of small convolution kernels
(1*1, 3*3), to obtain a deeper and wider network with fewer parameters.

The features extracted during the initialization stage are fed into the first
Dense Block, while adjacent Dense Blocks are connected by 1*1 convolutional
layers and max pooling layers for transition. The 1*1 convolutional layer is
used for dimension reduction, and the max pooling layer suppresses local image
noise and reduces interference from background items. The 1x1 convolutional
layer acts as the bottleneck layer, reducing the computational load by
decreasing feature dimensions and merging features from different channels.
The 3*3 convolutional layer captures information from the eight-neighborhood
of pixels, with the smallest kernel size, and cascading multiple 3*3
convolutional layers can capture the same receptive field with fewer
parameters. For example, replacing one 7*7 convolutional layer with three
cascading 3*3 convolutional layers can reduce 45$\%$ of computational load and
parameters. Additionally, the stacking of nonlinear activation functions
increases the network’s ability to perform nonlinear fitting.

---
Figure 2:  Dense connection structure in the basic feature extraction module.

###  2.2 Fire Classification Module

As shown in Fig. 1(a), after passing through the basic feature extraction
module, the different layer feature maps generated by the input image are
separately transmitted to the classification module and detection module.

To fully utilize the global information of the input, the classification
module uses a global average pooling layer[31] to receive the feature maps
generated by the final output of the basic feature extraction module. The
global average pooling layer can collect the spatial information of the input
feature maps to obtain the global context information of the object; at the
same time, its introduction can complete regularization throughout the entire
network structure, reduce the number of parameters, and effectively prevent
overfitting.After the global average pooling layer, a fully connected layer is
connected to it, which can complete the learning of global information for the
corresponding feature map, and can selectively emphasize useful features and
suppress useless features. Considering that fire detection is a binary
classification task, the activation function of the output layer is selected
as Sigmoid. The final output of this module is Fire/Normal.

###  2.3 Fire Detection Module

The YOLOv3 detection framework was applied to the fire detection module, which
shows great advantages in accuracy and real-time, suitable for fire detection
tasks. To alleviate the problem of vanishing gradient that is easily occurred
in the feature extraction network Darknet53 in YOLOv3, the feature extraction
network proposed in section 2.1 of this paper is selected as the Backbone. Its
dense connection structure makes full use of the feature information of
different layers in the process of feature forward propagation and
significantly reduces the number of network parameters. In addition, this
paper further improves YOLOv3 in terms of model training and structure
optimization, aiming to complete the fire detection task accurately and in
real-time.

####  2.3.1 Architechture

Similar to the idea of FPN[32], the detection part adopts a multi-scale
detection method, where the feature maps output by the 24th, 44th, and 204th
layers of the feature extraction module are respectively passed to the three
detection branches of the detection part. The small feature map, with a larger
receptive field and richer semantic information, generates small-scale
detectors responsible for detecting large-sized objects. In addition, the
small feature map, after upsampling, is concatenated with the feature map from
the 44th layer of the feature extraction module to obtain medium-sized
detectors responsible for detecting medium-sized object s. Similarly, the
medium-sized feature map, after upsampling, is concatenated with the feature
map from the 24th layer of the feature extraction module to obtain large-sized
feature maps with higher resolution and accurate position information, and the
large-scale detectors are responsible for detecting small-sized objects.

####  2.3.2 Spatial Pyramid Pooling Module and Mosaic

---
Figure 3:  The realization process of the Mosaic data enhancement method.

Due to the morphological characteristics and large differences in size of fire
detection objects (flames, smoke), a spatial pyramid pooling structure[33] is
added to reduce the accuracy loss caused by object morphological differences
in detection. The main purpose of the spatial pyramid pooling structure, first
proposed, is to solve the following two problems: 1. image distortion caused
by operations such as clipping and scaling of the input image; 2. repetitive
extraction of image features by convolutional neural networks.

In this paper, the SPP module is introduced between the basic feature
extraction module and the bottom detection branch of the detection module. The
SPP module consists of four parallel branches, with kernel sizes of 5*5, 9*9,
13*13 max pooling layers and a skip connection. The largest pooling kernel
size in the SPP module is designed to be as consistent as possible with the
size of the input feature map, in order to strengthen the fusion between
features of different sizes. The outputs of the four branches are concatenated
at the feature map level, which enables the fusion of global and local
features in the feature extraction module, greatly enriches the expression
ability of the feature map, improves the recognition accuracy, and is
beneficial for detecting objects with large differences in size, such as
flames and smoke.

Considering the unbalanced proportion of objects with different sizes in the
MS-FS established in section 3 of this paper, we use Mosaic during training.
Mosaic is a data augmentation technique first proposed in YOLOv4, which
randomly scales, crops, and arranges four images and feeds the resulting image
into the training network, equivalent to training with four images at once.
This approach not only speeds up the training process but also greatly
enriches the background of detected objects.

The specific implementation process of Mosaic is as follows: firstly, four
images are randomly selected from the dataset; secondly, each original image
is subjected to operations such as flipping, scaling, color space
transformation (brightness, saturation, hue), and the images are placed in
order from top-left to top-right; finally, adjustments and combinations of
images and object candidate boxes are made. As shown in Fig. 3.

####  2.3.3 Loss Function

The complete form of the loss function for the detection algorithm FSDNet
proposed in this article is shown as follows:

$Loss={L_{box}}+{L_{obj}}+{L_{cls}}\\\ $ (1)

$={\lambda_{coord}}\sum\limits_{i=0}^{{K^{\rm{2}}}}{\sum\limits_{j=0}^{M}{I_{ij}^{obj}(2-{w_{i}}\times{h_{i}})}}[{({x_{i}}-\mathop{{x_{i}}}\limits^{\wedge})^{2}}+{({y_{i}}-\mathop{{y_{i}}}\limits^{\wedge})^{2}}]\\\
$ (2)

$+{\lambda_{coord}}\sum\limits_{i=0}^{{K^{\rm{2}}}}{\sum\limits_{j=0}^{M}{I_{ij}^{obj}(2-{w_{i}}\times{h_{i}})}}[{({w_{i}}-\mathop{{w_{i}}}\limits^{\wedge})^{2}}+{({h_{i}}-\mathop{{h_{i}}}\limits^{\wedge})^{2}}]\\\
$ (3)

$-\sum\limits_{i=0}^{{K^{\rm{2}}}}{\sum\limits_{j=0}^{M}{I_{ij}^{obj}}}[\mathop{{C_{i}}}\limits^{\wedge}\log({C_{i}})+(1-\mathop{{C_{i}}}\limits^{\wedge})\log(1-{C_{i}})]\\\
$ (4)

$-{\lambda_{noobj}}\sum\limits_{i=0}^{{K^{\rm{2}}}}{\sum\limits_{j=0}^{M}{I_{ij}^{noobj}}}[\mathop{{C_{i}}}\limits^{\wedge}\log({C_{i}})+(1-\mathop{{C_{i}}}\limits^{\wedge})\log(1-{C_{i}})]\\\
$ (5)

$-\sum\limits_{i=0}^{{K^{\rm{2}}}}{I_{ij}^{obj}}\sum\limits_{c\in
class}{[\mathop{{p_{i}}}\limits^{\wedge}(c)\log({p_{i}}(c))+(1-\mathop{{p_{i}}}\limits^{\wedge}(c))\log(1-{p_{i}}(c))]}\\\
$ (6)

In this paragraph, $K^{2}$ represents the total number of grids into which the
input image is divided, and $M$ represents the number of candidate boxes
generated by each grid. Each candidate box will produce a corresponding
bounding box after passing through the network, and the total number of
bounding boxes obtained will be $K^{2}\times M$ . $\lambda_{coord}$ and
$\lambda_{noobj}$ are hyperparameters that represent the weights of the
coordinate error and class error, respectively. $I_{ij}^{noobj}$ represents
whether the $j$ th anchor box of the $i$ th grid is responsible for the object
object. If it is responsible, then $I_{ij}^{noobj}$ is 1; otherwise, it is 0.
The responsibility is specifically defined as follows: among all anchor boxes
of the $j$ th anchor box in the $i$ th grid, if the IoU (Intersection over
Union) with the ground truth of the object is the highest, then the anchor box
is responsible for predicting the object. Similarly, $I_{ij}^{noobj}$
represents that the $j$ th anchor box of the $i$ th grid is not responsible
for the object object.

In YOLOv3, the Smooth L1 Loss function is responsible for the position loss,
which overcomes the drawbacks of both MAE and MSE. However, treating (x, y, w,
h) as independent variables separates their relative relationship and ignores
their holistic nature. Therefore, this article introduces CIoU as the
algorithm’s bounding box regression loss. CIoU is an improvement on DIoU loss,
which not only considers the overlap between two bounding boxes and the
distance between their centers but also adds an adjustment parameter. Based on
this, the aspect ratio of the bounding box is also considered, which is more
favorable for detecting objects with large differences in shape and size, such
as flames and smoke.

The formula for the CIoU loss function introduced in this article is shown
below:

${L_{box}}={L_{CIoU}}=1-IoU+\frac{{{\rho^{2}}(b,{b^{gt}})}}{{{c^{2}}}}+\alpha
v$ (7)

Among them, $b$ represents the center point of the predicted bounding box,
$\rho$ represents the center point of the ground truth bounding box, $\rho$
denotes the Euclidean distance between the center points of the two bounding
boxes, $c$ is the diagonal distance of the smallest rectangle that can
simultaneously cover the two bounding boxes. $v$ is responsible for measuring
the similarity of aspect ratios, $\alpha$ is the weight coefficient, and the
specific functional forms of both are shown in the formula.

$v=\frac{4}{{{\pi^{2}}}}{(\arctan\frac{{{\omega^{gt}}}}{{{h^{gt}}}}-\arctan\frac{\omega}{h})^{2}}$
(8) $\alpha=\frac{v}{{(1-IoU)+v}}$ (9)

In the above equation, $h^{gt}$ and $\omega^{gt}$ represent the height and
width of the ground truth bounding box, while $h$ and $\omega$ represent the
height and width of the predicted bounding box, and $IoU$ denotes the
intersection over union between the two boxes.

##  3 Establishment of Fire and Smoke Dataset

Deep learning models have a certain degree of dependency on datasets, and a
high-quality and properly partitioned dataset can accelerate model convergence
speed and improve model performance. Therefore, this article first
investigated and summarized the existing publicly available fire and smoke
datasets. Table 1 shows a comparison between the dataset created in this
article and some of the existing publicly available fire datasets.

Table 1: Comparison with other fire dataset. | Dataset
---
source
Content Description |  | Object
---
Classes
| Pixel level
---
Annotation
Bowfire dataset |  | An image dataset including
---
119 fire images
and 106 normal images.
Fire | False
Bilkent University |  | A video dataset composed of 14
---
fire videos and 17 normal videos.
The average duration of the videos
is142s.
Fire | False
The University of Modena |  | A video dataset consisting of 14
---
outdoor smoke videos.The
average duration of the videos is
12.6s.
Smoke | False
Keimyung University |  | A video dataset including 16
---
indoor and outdoor fire videos.
The average duration of the videos
is 39s.
Fire&Smoke | False
The proposed dataset |  | A dataset composed of 11938 fire
---
images with preprocessing,
covering hundreds of real-time
fire scenarios.
Fire&Smoke | True

The research results show that existing publicly available datasets have
shortcomings such as small data volume, limited coverage of scenarios,
imbalanced data, and lack of annotation information. Therefore, this article
has created a large-scale flame and smoke dataset, MS-FS, with annotation
information. The detailed process is as follows.

(1) Data collection. A web crawler was built to search and download fire-
related videos and images. Ultimately, 70 video clips of real fire scenes and
thousands of fire images were collected, covering complex fire scenarios in
daily life as much as possible, such as city roads, large buildings,
supermarkets, and parking lot fires.

(2) Video frame splitting. The collected video clips were processed by frame
splitting, dividing each second of video into 25 frames. To avoid excessive
similarity between consecutive images, at least one image was extracted as
valid data every 3 frames.

(3) Data filtering. In order to ensure the validity of the data, the data was
screened according to the following criteria. First, according to the
principle of object proportion, filter out the data whose area of flame and
smoke in each scene accounts for less than 30$\%$; secondly, according to the
principle of image signal-to-noise ratio, filter out the data whose signal-to-
noise ratio is less than or equal to 35 dB; finally, according to theRGB color
model, filter out the data whose background RGB value conforms to the
following formula:

$R>{R_{avg}}\\\ $ (10) $G>{G_{avg}}\\\ $ (11) $R>G>{\rm B}$ (12)

(4) Object annotation. Correct category labeling of images is extremely
important for the dataset. The selected data was annotated one by one using
the image annotation tool, LabelImg. The annotated objects were divided into
two categories: flames and smoke. The data format for annotation was the XML
file format of the standard dataset, Pascal VOC. This file contained
annotation information for the original image, including the object position,
object category, and bounding box position information.

MS-FS contains 11,938 images covering hundreds of real fire scenes, including
various backgrounds from simple to complex, environments from indoors to
outdoors, lighting from day to night, and heights from ground level to high-
rise buildings. MS-FS effectively solves the problems of small data volume,
limited coverage of scenarios, imbalanced data, and lack of annotation
information in existing publicly available datasets. MS-FS is divided into
three categories, which are scenes containing only flames, scenes containing
only smoke, and scenes containing both flames and smoke. The annotation
examples for these three types of data are shown in Figure 4, where the purple
and yellow boxes represent the annotations for smoke and flames, respectively.

---
Figure 4:  Labeled examples of the dataset proposed in this paper.

##  4 Experiment and Result

###  4.1 Dataset Description

In order to validate that the proposed FSDNet algorithm can still achieve
excellent performance in different scenarios and to compare it more
intuitively with other fire detection algorithms, this study selected DS1[34]
and DS2[35] as the test set and some of the training set for the FSDNet
classifier, and chose the dataset MS-FS proposed in this paper as the training
and test set for the FSDNet detector.

DS1 is a fire video dataset consisting of 31 video clips captured by cameras
in different scenarios, with a total duration of 74 minutes (4450 seconds).
Among the 31 video clips, the first 14 contain frames with fires, while the
remaining 17 do not and were captured in normal environments. This dataset
imposes a high demand on the model’s performance, as it contains scenes with
heavy smoke, fire-colored objects, and fire objects at different distances.

DS2 consists of 119 fire images and 107 non-fire images, making it a highly
challenging dataset that is commonly used to test the detection accuracy of
fire detection methods.

MS-FS is a self-prepared fire dataset in this paper.

###  4.2 Experimental Results

####  4.2.1 Classification results of FSDNet

This paper uses three commonly used evaluation metrics to assess the
performance of the FSDNet classifier and compare it with other methods: false
positive rate (FP Rate), false negative rate (FN Rate), and accuracy. A lower
FP Rate and FN Rate and a higher accuracy indicate better classifier
performance. In fire detection tasks, FP Rate represents the fire false alarm
rate, while FN Rate represents the fire miss rate.

In the experiments based on DS1, this study compared several relevant methods,
including those based on handcrafted feature extraction and deep learning, and
the comparison results are shown in Table 2.

Table 2: Comparison with other fire detection methods on DS1. Method | FP Rate | FN Rate | Accuracy
---|---|---|---
Rafiee et al. (RGB)[36] | 41.18 | 7.14 | 74.20
Celik et al. | 29.41 | 0 | 83.87
Chen et al. | 11.76 | 14.29 | 87.10
Habiboglu er al. | 5.88 | 14.29 | 90.32
Lascio et al.[37] | 6.67 | 0 | 92.59
Foggia et al. | 11.76 | 0 | 93.55
Muhammad et al.[38] | 8.87 | 2.12 | 94.50
FSDNet | 0.07 | 0.17 | 99.82

(1) The false positive rate (FPR) of fire alarms is discussed in this article.
The FSDNet algorithm proposed in this study has the lowest FPR of only
0.07$\%$, which is 5.81$\%$ to 41.11$\%$ lower than other methods. This
suggests that the FSDNet algorithm proposed in this study is the most reliable
and robust. The second-lowest FPR is Habiboglu’s method, with a rate of
5.88$\%$. However, this method has a higher false negative rate (FNR) of
14.29$\%$, which is the highest value among all the methods in Table 2.

(2) The false negative rate (FNR) was studied and the methods proposed by
Lascio, Foggia, and Celik showed the best performance, achieving zero FNR on
the DS1 test set. However, these three algorithms had high false positive
rates of 6.67$\%$, 11.76$\%$, and 29.41$\%$ respectively. On the other hand,
FSDNet exhibited some false positives but still maintained a relatively low
FNR of 0.17$\%$, making it a superior performing algorithm.

(3) The accuracy of fire detection was studied, and FSDNet achieved the
highest accuracy among all methods with a score of 99.82$\%$, approaching
100$\%$, and obtaining outstanding results. Its accuracy was 5.32$\%$ higher
than the second-ranked algorithm proposed by Muhammad.

In summary, among the three performance metrics compared, the FSDNet algorithm
proposed in this paper showed the highest accuracy (99.82$\%$), the lowest
false positive rate (0.07$\%$), and excellent false negative rate (0.17$\%$).
Its overall performance in fire detection was the best among all methods in
Table 2.

The fire classification results of FSDNet based on DS1 are shown in Fig. 5.

---
Figure 5:  Fire classification results On DS1 of FSDNet. (a)-(d)Conventional
fire scenarios with obvious flames. (e)-(h)Single fire source scenario with a
relatively small object. (i)-(l)Scenarios containing interferences items.

From the fire results in Fig. 5, it can be seen that:

(1) Even in the face of conventional fire scenes with clearly visible flame
objects such as (a)-(d), FSDNet can achieve a detection accuracy of up to
99$\%$ even if the shape of the flame object varies significantly. This means
that FSDNet can easily handle routine fire detection tasks.

(2) When faced with a single fire source scene with a small object area such
as the similar scenes (e)-(h), although the proportion of the fire source area
gradually decreases from the scene (e)-(h), the The detection results of
FSDNet are always kept at a reliable level ($>$90$\%$). Even if the flame
object is relatively small, such as scene (h), the algorithm also guarantees
an accuracy of 74$\%$, which shows that FSDNet can be effectively applied to
the task of early fire detection. By detecting a small area of flame in the
early stage of fire, the fire spread can be curbed, reduce the loss caused by
fire.

(3) In the face of non-fire scenarios with certain interference items such as
similar scenarios (i)-(l), the detection accuracy rate of FSDNet also remains
above 90$\%$, which is also the embodiment of the low false alarm rate in
Table 2. Among them, scene (i) is moving smoke, scene (j) is an indoor light
source, and scenes (k) and (l) are both outdoor light sources at night. This
shows that FSDNet has good anti-jamming performance.

From the above three sets of fire detection results, it can be concluded that
FSDNet maintains a high detection accuracy when faced with conventional fire
scenes, fire scenes with a relatively small flame area, and non-fire scenes
with some interference. The algorithm demonstrates strong resistance to
interference and shows potential for application in early fire detection
tasks.

The performance of the FSDNet algorithm is also compared with five existing
high-performance, lightweight object detection algorithms MobileNet,
ShuffleNet, ShuffleNetV2, MobileNetV2 and SqueezeNet on the DS2 dataset. The
results are shown in Table 3.

Table 3: Comparison with other lightweight networks on DS2.  Method | FP Rate | FN Rate | Accuracy
---|---|---|---
MobileNet | 23.36 | 12.61 | 82.30
ShuffleNet | 21.50 | 10.92 | 84.07
ShuffleNetV2 | 16.82 | 9.24 | 87.17
MobileNetV2 | 15.09 | 9.32 | 87.95
SqueezeNet | 14.15 | 6.72 | 89.78
FSDNet | 10.28 | 7.56 | 91.15

From Table 3, it can be concluded that FSDNet has the lowest false positive
rate (FPR) among all methods, which is 10.28$\%$; SqueezeNet shows the best
false negative rate (FNR) at 6.72$\%$, but its false positive rate is as high
as 14.15$\%$, which is 3.87$\%$ higher than that of FSDNet. In terms of
accuracy, FSDNet still performs the best with an accuracy of 91.15$\%$.
Overall, FSDNet achieves the best combination of the three performance
indicators, showing the lowest false positive rate and the highest accuracy.
Considering that the DS2 dataset covers a wide range of scenes and has high
challenges in terms of interference, the excellent performance of FSDNet on
this test set can to some extent reflect its generalization ability in complex
real-world situations.

An example of the detection results of FSDNet on the DS2 test set is shown in
Fig. 6. From the fire detection results in Figure 6, it can be seen that:

(1) Common fire scenarios. Such as scenarios (a)-(d), the algorithm in this
paper shows excellent performance of 100$\%$ accuracy and 99.65$\%$ accuracy.

(2) Complex fire scene. For example, in scene (e), the object area of the
flame is small; in scene (f), the fire occurs in a building and has obvious
flame color and shape characteristics; scene (g), the object area of the flame
is small and covered by smoke ; Scenario (h), street lights appear as
interference items and small fire sources at the same time, and the accuracy
of FSDNet remains above 80$\%$, which shows that FSDNet has the ability to
handle fire detection tasks in complex environments.

(3) Non-fire scenarios with strong interference items. For example, in
scenarios (i)-(l), the interference items are sunset, dim indoor light,
morning glow and night light. FSDNet also guarantees an accuracy rate higher
than 85$\%$ and has strong antiinterference ability.

In general, the proposed fire detection algorithm FSDNet shows that the FSDNet
has high accuracy, low false alarm rate and low false alarm rate in the face
of common fire scenes, complex fire scenes and non-fire scenes with strong
interference items. The false alarm rate can be better able to perform the
task of fire detection.

---
Figure 6:  Fire classification results On DS2 of FSDNet. (a)-(d)Common fire
scenarios occurred in daily life (e)-(h)Fire scenarios that are more complex
and challenging. (i)-(l)Normal scenarios with extremely disturbing objects.

####  4.2.2 Detection results of FSDNet

To verify the effectiveness of the FSDNet detector, FSDNet was compared with
other deep learning algorithms, including the original YOLOv3, YOLOv3-dense,
and the high-performance YOLOv4. All experiments in this section were based on
the MS-FS dataset proposed in this paper, and the Pascal VOC’s mean average
precision (mAP) calculation method was used as the evaluation metric to assess
the performance of each method. The experimental results are shown in Table 4.

Table 4: Comparison with other fire detection methods on DS3.  Method | Backbone | Fire | Smoke | mAP
---|---|---|---|---
YOLOv3-dense | Densenet201 | 0.685 | 0.711 | 0.698
YOLOv3 | Darknet53 | 0.775 | 0.791 | 0.783
YOLOv4 | CSPResNext50 | 0.853 | 0.858 | 0.856
FSDNet | Feature Extraction Module | 0.934 | 0.801 | 0.868

It can be seen from Table 4 that the mAP of the FSDNet algorithm on the DS3
dataset is higher than that of other algorithms, which is 86.8$\%$, which is
17$\%$, 8.5$\%$ and 1.2$\%$ higher than that of YOLOv3-dense, YOLOv3 and
YOLOv4, respectively. In particular, the detection accuracy of FSDNet for
flame category objects is significantly improved, which is 24.9$\%$ and
15.9$\%$ higher than YOLOv3-dense and YOLOv3. Even compared to the high-
performance YOLOv4, the improvement is as high as 8.1$\%$, which means that
the FSDNet algorithm has better fire detection performance and can detect
flame objects more accurately.

---
Figure 7:  Fire and smoke detection results On DS3 of FSDNet.(a)(b)Fire that
often occur in cities. (c)(d)Fire that often occur in nature. (e)(f)Fire that
often occur in indoor scenarios.

However, objects in fire detection tasks have uniqueness in that they undergo
dynamic changes in morphology and color features over time, leading to
corresponding changes in the overlap between the predicted object box and the
true label box. For example, the true label box and the detected bounding box
only partially overlap, but this does not have a significant impact on the
detection performance; there may be multiple detected bounding boxes, but the
true label box is only one.

The above two points can greatly affect the IOU between the predicted bounding
box and the true label box, which in turn affects the algorithm’s mAP.
However, in the initial stage of fire detection, it is usually expected to
determine whether there is a fire as quickly as possible, and in this stage,
the relationship between the object box and the true box can be ignored, and
the AP value is not calculated. Therefore, in this paper, the detection rate
is used as an indicator to further verify the algorithm’s performance by
focusing on detecting the presence of fire. The specific detection results of
FSDNet are presented to further demonstrate the reliability of the FSDNet
detector.

The fire and smoke detection results of the FSDNet algorithm based on the MS-
FS dataset are shown in Fig. 7, which further demonstrates the good
performance of the FSDNet detector. (a)-(f) are some common fire scenarios in
daily life, including both outdoor and indoor scenes, and all of them have
certain detection difficulties. In scenes (b) and (f), the object shapes are
varied and irregular; in scenes (c) and (e), there are significant differences
in the proportion of the objects; in scene (d), there are many objects with
significant differences in proportions. Nevertheless, FSDNet exhibits
excellent performance in the specific categories, bounding box positions, and
accuracy of detection.

In order to more intuitively observe the fire detection performance of the
FSDNet algorithm in complex scenes, this section compares the detection
results of the FSDNet algorithm and the YOLOv3 algorithm on selected images.
The specific results are shown in Fig. 8.

---
Figure 8:  Fire and smoke detection results comparison between YOLOv3 and
FSDNet under same scenarios. The first and the second columns are the results
of YOLOv3 and FSDNet, respectively.

Fig. 8 shows the comparison of the detection results of the FSDNet algorithm
and the YOLOv3 algorithm on the same data. The left side is the detection
result of the YOLOv3 algorithm, and the right side is the detection result of
the FSDNet algorithm. There are 4 sets of scene comparisons.

For scene (a), the FSDNet algorithm detects more flame objects, and for the
same flame objects, the predicted probability value is higher; for scene (b),
when the same number of flame objects are detected, the FSDNet algorithm shows
that More precise positioning accuracy and classification effect; for scenes
(c) and (d), the FSDNet algorithm not only showed higher predicted probability
values on the same object, but also detected smaller flame objects.

It can be clearly seen that compared with the YOLOv3 algorithm, the FSDNet
algorithm can detect more and smaller flame objects in various scenarios,
which is of great significance for fire detection and early fire monitoring in
complex scenarios.

##  5 Conclusions

In this article, a fire detection network called FSDNet is proposed to improve
detection accuracy while maintaining real-time performance. In the FSDNet
network, a basic feature extraction module consisting solely of small
convolutional kernels is further proposed, and a dense connection structure of
convolutional layers is applied to strengthen feature propagation while
reducing the number of parameters, aimed at enhancing the feature extraction
ability of the backbone network and alleviating the problem of gradient
disappearance. In addition, a spatial pyramid pooling structure and a Mosaic
data augmentation method are introduced to improve the accuracy loss caused by
the shape differences of flame objects and the imbalance of object proportions
in the dataset, respectively. A new loss function is adopted to accelerate
model training. Furthermore, a large-scale fire dataset covering diverse
scenes, MS-FS, is independently constructed to provide strong data support.

Experimental results on the DS1, DS2, and MS-FS datasets demonstrate that
FSDNet achieves higher classification accuracy than previous methods, and also
outperforms YOLOv3 and high-performance YOLOv4 in detection accuracy, enabling
precise and real-time fire detection. Overall, FSDNet proposed in this article
is expected to have a promising application prospect in community fire
monitoring and prevention.

### Acknowledgments

My mentor, Associate Professor Julie

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First Author is a postgraduate student majoring in communication engineering
at Nanchang University. His research topics during his postgraduate period are
deep learning and computer vision.

###### List of Figures

  1. 1  FSDNet algorithm framework diagram.
  2. 2  Dense connection structure in the basic feature extraction module.
  3. 3  The realization process of the Mosaic data enhancement method.
  4. 4  Labeled examples of the dataset proposed in this paper.
  5. 5  Fire classification results On DS1 of FSDNet. (a)-(d)Conventional fire scenarios with obvious flames. (e)-(h)Single fire source scenario with a relatively small object. (i)-(l)Scenarios containing interferences items.
  6. 6  Fire classification results On DS2 of FSDNet. (a)-(d)Common fire scenarios occurred in daily life (e)-(h)Fire scenarios that are more complex and challenging. (i)-(l)Normal scenarios with extremely disturbing objects.
  7. 7  Fire and smoke detection results On DS3 of FSDNet.(a)(b)Fire that often occur in cities. (c)(d)Fire that often occur in nature. (e)(f)Fire that often occur in indoor scenarios.
  8. 8  Fire and smoke detection results comparison between YOLOv3 and FSDNet under same scenarios. The first and the second columns are the results of YOLOv3 and FSDNet, respectively.

###### List of Tables

  1. 1 Comparison with other fire dataset.
  2. 2 Comparison with other fire detection methods on DS1.
  3. 3 Comparison with other lightweight networks on DS2.
  4. 4 Comparison with other fire detection methods on DS3.


# Chirality-controlled spin scattering through quantum interference

Jan M. van Ruitenbeek<EMAIL_ADDRESS>Huygens-Kamerlingh
Onnes Laboratory, Leiden University, NL-2333CA Leiden, Netherlands  Richard
Korytár  Department of Condensed Matter Physics, Charles University, 121 16
Praha 2, Czech Republic  Ferdinand Evers  Institute of Theoretical Physics,
University of Regensburg, D-93050 Regensburg, Germany

###### Abstract

Chirality-induced spin selectivity has been reported in many experiments, but
a generally accepted theoretical explanation has not yet been proposed. Here,
we introduce a simple model system of a straight cylindrical free-electron
wire, containing a helical string of atomic scattering centers, with spin-
orbit interaction. The advantage of this simple model is that it allows
deriving analytical expressions for the spin scattering rates, such that the
origin of the effect can be easily followed. We find that spin-selective
scattering can be viewed as resulting from constructive interference of
partial waves scattered by the spin-orbit terms. We demonstrate that forward
scattering rates are independent of spin, while back scattering is spin
dependent over wide windows of energy. Although the model does not represent
the full details of electron transmission through chiral molecules, it clearly
reveals a mechanism that could operate in chiral systems.

††preprint: AIP/123-QED

> The first observations of, what is now known as chirality induced spin
> selectivity (CISS), were already reported in 1999 by Ray, Ananthavel,
> Waldeck and Naaman.Ray  _et al._ (1999) Since, many papers have appeared
> that confirm the general picture: transmission of electrons through chiral
> molecules selectively favors one of the two spin directions, depending on
> the handedness of the molecule. The effect has been found in photo-emission
> experiments Ray  _et al._ (1999); Göhler  _et al._ (2011); Mishra  _et al._
> (2013); Kettner  _et al._ (2018); Abendroth  _et al._ (2019) in electron
> transport experiments, either for small numbers of molecules,Mishra  _et
> al._ (2020); Kiran  _et al._ (2016, 2017); Aragonès  _et al._ (2017); Xie
> _et al._ (2011) or in cross-bar configurations across a self-assembled
> monolayer (SAM) of chiral molecules.Mishra  _et al._ (2020); Liu  _et al._
> (2020); Mathew  _et al._ (2014); Al-Bustami  _et al._ (2022) Spin
> polarization efficiencies have been reported to approach even 100%,Lu  _et
> al._ (2019); Al-Bustami  _et al._ (2022) and the effects are observed under
> ambient conditions at room temperature. Even more surprising are the related
> observations that the direction of the magnetization of a thin magnetic film
> can be determined by the handedness of a monolayer of chiral molecules Ben
> Dor  _et al._ (2017) and, conversely, a magnetic surface selectively binds
> one out of the two enantiomers in a racemic mixture.Banerjee-Ghosh  _et al._
> (2018); Reza Safari  _et al._ (2022) Recent reviews are given in Refs.
> Waldeck  _et al._ , 2021; Naaman  _et al._ , 2019, and the connection with
> chiral spin currents in condensed matter systems is made by Yang et al.Yang
> _et al._ (2021)

The various attempts at capturing these observations in a theoretical
description have recently been summarized by Evers et al.Evers  _et al._
(2022) Although several possible mechanisms have been discussed that lead to
spin selectivity, the discrepancy between the observations and the theory is
large. The main difficulty that theories encounter is the fact that the spin-
orbit coupling in molecules composed of only light elements is extremely
small, leaving a gap of many orders of magnitude in the size of the effects
between theory and experiment. Although claims have been put forward that the
effects can be understood from a combination of spin-orbit interaction and
inelastic scattering,Fransson (2020); Das  _et al._ (2022) it remains unclear
how the smallness of the spin-orbit interaction can be addressed by including
a, presumably small, correction to it. Until this question is resolved we take
the point of view that currently  few of the proposed ideas are capable of
explaining the observations.  A favorable exception is the work by Dalum and
Hedegård,Dalum and Hedegård (2019) where an amplification of spin-orbit
interaction was identified associated with level crossing at multiple energies
in chiral molecules.

Three groups of experiments: Following Evers  _et al._ (2022) we  categorize
the experimental evidence in three groups, of increasing level of difficulty
met in explaining all the observations. The first group of experiments
involves the detection of a difference in transmission of electrons of
opposite spin through chiral molecules, such as the photo-emission experiments
in Refs. Göhler  _et al._ , 2011; Mishra  _et al._ , 2013; Kettner  _et al._ ,
2018. These experiments detect the spin of the electrons directly.

The second and largest group of experimental reports considers changes in
electrical resistance of a junction or device as a function of either magnetic
field or magnetization of a component.Mishra  _et al._ (2020); Kiran  _et al._
(2016, 2017); Aragonès  _et al._ (2017); Xie  _et al._ (2011); Mishra  _et
al._ (2020); Liu  _et al._ (2020); Mathew  _et al._ (2014); Al-Bustami  _et
al._ (2022) As pointed out by Yang et al., the resistance of such set-ups is
expected to be insensitive to reversal in the magnetization or the magnetic
field as known from Onsager’s relations, based on fundamental limits imposed
by time reversal symmetry.Yang  _et al._ (2019, 2020); Yang and van Wees
(2021) Therefore, magnetoresistance reveals itself in the nonlinear regime,
which is much more challenging to understand.

The third and final group of experiments comprises observations of
(near-)equilibrium properties that are controlled by the handedness of the
molecules involved, such as the direction of magnetization of a thin
ferromagnet,Ben Dor  _et al._ (2017) or the selective adhesion of enantiomers
to a magnetized surface.Banerjee-Ghosh  _et al._ (2018); Reza Safari  _et al._
(2022) Here, the gap between available theoretical ideas and the experimental
observations is largest, and only a few sketchy proposals have been put
forward.Evers  _et al._ (2022); Wu and Subotnik (2021)

Outline of this paper:  Here, we want to focus on the first group of
experimental observations only, in the hope that clarification of possible
mechanisms for those will lead the way to also resolve the more complicated
problems involved in the second and third categories of experiments. Rather
than constructing detailed models for describing any specific experiment or
chiral molecule, we focus on simple analytically tractable models, which can
guide us in our understanding of the principles involved in CISS.

One such model has already been introduced by Michaeli and Naaman:Michaeli and
Naaman (2019) free electrons inside a helical tube with quadratic confining
potential. By including the spin-orbit interaction resulting from the smooth
confining potential an anti-crossing gap opens in the energy spectrum, which
results in spin-selective transport through the helical tube. This is an
important result, because it shows that spin-selective transmission can be
obtained generically. However, it falls short in explaining the experiments in
that it allows for spin-dependent scattering only in a very narrow energy
window of a few meV around the anti-crossing. It is important to stress that
the model does not predict magnetoresistance (needed for explanation of the
second class of experiments) because for evaluation of the full current one
needs to include a magnetic electrode.Korytár  _et al._ (2022)

The model we consider here is that of free electrons traveling inside a
straight cylindrical tube, see Fig. 1. We place an atomic scattering potential
off-axis inside this tube, and analyse the spin-dependent scattering due to
the spin-orbit interaction at this site by Fermi’s golden rule. The simplicity
of this model permits obtaining analytical expressions, and analyzing the
various mechanisms of scattering systematically. Chirality is introduced into
the problem by arranging a number of atomic scattering potentials as a helical
string inside the tube.  By adding the contributions from the individual atoms
we find that quantum interference produces large differences in back
scattering into the two spin channels, and leads to spin-dependent reflection
over wide energy windows. The coupling between momentum scattering and spin
scattering is seen to result from the non-symmorphic character of the
scattering potential.

##  I Model and analytic results

We consider the spin-dependent electronic conduction of chiral molecules as a
scattering problem.

Figure 1: Illustration of the model, which consists of free electrons inside a
cylindrical tube, containing a helical string of atomic-like scattering
centers.

We use a model based on a helical string of atomic-like scattering potentials,
sitting inside a perfectly straight wire, carrying Landauer-type conduction
channels. The axis of the tube coincides with the $z$-axis in a system of
cylindrical coordinates. Before introducing the localized scattering
potentials, the conductance channels are those for a perfectly smooth and
straight cylindrical free-electron-like conductor with hard-wall boundary
conditions, and radius $R_{b}$ (Fig. 1). The eigenchannels for this perfect
wire are given by,

$\Psi_{\mu nk}(r,\varphi,z)=J_{\mu}(\gamma_{\mu
n}r)e^{\mathfrak{i}kz}e^{\mathfrak{i}\mu\varphi}$ (1)

Here, $\gamma_{\mu n}=x_{\mu n}/R_{b}$, with $x_{\mu n}$ the $n^{\rm th}$ zero
of $J_{\mu}$, the $\mu^{\rm th}$ Bessel function ($n\geq 1$). The energy of
this state is

$\epsilon_{\mu nk}=\frac{\hbar^{2}}{2m}\gamma_{\mu
n}^{2}+\frac{\hbar^{2}k^{2}}{2m}.$ (2)

The interaction with the atomic-like spin-orbit terms is introduced as a
perturbation of these states, by means of Fermi’s golden rule. Although the
Coulomb potential of the atoms will be the dominant source of scattering, we
propose to focus on the role of the spin-orbit interaction term only, and
ignore potential scattering. Such a model could be a reasonable approximation
for a metallic wire, where the electronic states are described by Bloch waves,
containing a helical arrangement of heavy ions. At this point it is important
to stress that we do not aim at a realistic description of a molecular wire.
Yet, the model will help us to trace similar mechanisms in more realistic
models. The advantage of having an explicit expression for the unperturbed
states is that we may evaluate the matrix elements explicitly, which allows us
to trace the origin of spin dependent scattering in our model.

For the spin-orbit interaction term we have,

$\displaystyle\hat{h}_{0}=\frac{\mathfrak{i}\hbar^{2}}{4(mc)^{2}}\bm{\sigma}\left(\frac{\partial
v(\mathbf{r})}{\partial\mathbf{r}}\times\bm{\nabla}\right).$ (3)

Here, the differential operator $\partial/\partial_{\mathbf{r}}$ only acts on
the electrostatic potential of the atomic nucleus, $v(\mathbf{r})$, while the
operator $\bm{\nabla}$ on the right acts on the electron wave function.
Although the cylindrical confining potential will also produce a contribution
to the spin-orbit interaction, we will ignore this in the following, because
the symmetry of this potential prohibits any back scattering. Forward
scattering cannot lead to spin polarization, as we show in Section I.A.6,
below.

The spin-flip rates induced by spin-orbit scattering are given by Fermi’s
golden rule as,

$\displaystyle\left.\frac{1}{\tau_{0}}\right|_{\mu
nk\sigma}=2\pi\sum_{{\mu^{\prime}}\,{n^{\prime}}\,{k^{\prime}},{\sigma^{\prime}}}$
$\displaystyle|\langle{\mu^{\prime}}{n^{\prime}}{k^{\prime}}{\sigma^{\prime}}|\hat{h}_{0}|\mu
nk\sigma\rangle|^{2}\,\delta(E_{f}-E_{i}).$ (4)

Here, $E_{i}=\epsilon_{\mu nk}$ and
$E_{f}=\epsilon_{{\mu^{\prime}}{n^{\prime}}{k^{\prime}}}$ are the total energy
of the initial and final states.

###  I.1 Evaluation of the matrix elements

####  I.1.1 Spin-orbit interaction in cylinder coordinates

In order to evaluate the matrix elements in (4) we need to work out the
structure of the perturbing Hamiltonian (3). We express the spin operator
$\bm{\sigma}$ in cylindrical coordinates as,

$\displaystyle\bm{\sigma}$
$\displaystyle=\sigma_{r}\bm{e}_{r}+\sigma_{\varphi}\bm{e}_{\varphi}+\sigma_{z}\bm{e}_{z}$
(5)
$\displaystyle=(e^{-\mathfrak{i}\varphi}\sigma_{+}+e^{+\mathfrak{i}\varphi}\sigma_{-})\bm{e}_{r}-\mathfrak{i}(e^{-\mathfrak{i}\varphi}\sigma_{+}-e^{+\mathfrak{i}\varphi}\sigma_{-})\bm{e}_{\varphi}+\sigma_{z}\bm{e}_{z},$

where  $\bm{e}_{r}$, $\bm{e}_{\varphi}$, and $\bm{e}_{z}$ are the unit vectors
in cylindrical coordinates and,

$\sigma_{+}=\hbar\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\qquad\sigma_{-}=\hbar\begin{pmatrix}0&0\\\
1&0\end{pmatrix}=\sigma_{+}^{\dagger}.$ (6)

Using coordinates $(r,\varphi,z)$ also for the electrons we can work out the
vector products in the expression for $\hat{h}_{0}$ as,

$\displaystyle\bm{\sigma}\left(\frac{\partial
v(\mathbf{r})}{\partial\mathbf{r}}\times\bm{\nabla}\right)=\hat{h}_{z}\sigma_{z}+e^{-\mathfrak{i}\varphi}\hat{h}_{+}\sigma_{+}+e^{\mathfrak{i}\varphi}\hat{h}_{-}\sigma_{-}$
(7)

where we have introduced the real-space operators

$\displaystyle\hat{h}_{z}(r,\varphi,z)$
$\displaystyle\coloneqq\frac{1}{r}\frac{\partial v}{\partial
r}\frac{\partial}{\partial\varphi}-\frac{1}{r}\frac{\partial
v}{\partial\varphi}\frac{\partial}{\partial r}$
$\displaystyle\hat{h}_{+}(r,\varphi,z)$
$\displaystyle\coloneqq\frac{1}{r}\frac{\partial
v}{\partial\varphi}\frac{\partial}{\partial z}-\frac{1}{r}\frac{\partial
v}{\partial
z}\frac{\partial}{\partial\varphi}-\frac{1}{\mathfrak{i}}\left(\frac{\partial
v}{\partial r}\frac{\partial}{\partial z}-\frac{\partial v}{\partial
z}\frac{\partial}{\partial r}\right)$

and $\hat{h}_{-}\coloneqq\hat{h}_{+}^{*}$.

####  I.1.2 General structure of matrix elements

Due to (7), two kinds of matrix element appear in Eq. (4), spin flipping and
non-flipping. The spin-conserving one is given by

$\displaystyle\langle{\mu^{\prime}}{n^{\prime}}{k^{\prime}}{\sigma^{\prime}}|\hat{h}_{z}\sigma_{z}|\mu
nk\sigma\rangle=\langle{\mu^{\prime}}{n^{\prime}}{k^{\prime}}|\hat{h}_{z}|\mu
nk\rangle\langle{\sigma^{\prime}}|\sigma_{z}|\sigma\rangle$ (8)

The second matrix element on the rhs is trivially evaluated as

$\displaystyle\langle{\sigma^{\prime}}|\sigma_{z}|\sigma\rangle=\text{sign}({\sigma})\delta_{{\sigma^{\prime}}\sigma},$
(9)

while the first matrix element on the rhs of (8) takes the explicit form

$\displaystyle I_{z}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}=\int_{0}^{R_{b}}$ $\displaystyle
dr\int_{-\infty}^{+\infty}dz\int_{0}^{2\pi}rd\varphi\
e^{-\mathfrak{i}{\mu^{\prime}}\varphi}\ e^{-\mathfrak{i}{k^{\prime}}z}$
$\displaystyle\times J_{{\mu^{\prime}}}(\gamma_{{\mu^{\prime}}{n^{\prime}}}r)\
$ $\displaystyle\hat{h}_{z}(r,\varphi,z)\ e^{\mathfrak{i}\mu\varphi}\
e^{\mathfrak{i}kz}J_{\mu}(\gamma_{\mu n}r).$ (10)

The spin-non-conserving matrix elements are given by

$\displaystyle\langle{\mu^{\prime}}{n^{\prime}}{k^{\prime}}{\sigma^{\prime}}|$
$\displaystyle e^{\mp\mathfrak{i}\varphi}\hat{h}_{\pm}\sigma_{\pm}|\mu
nk\sigma\rangle=$
$\displaystyle\langle{\mu^{\prime}}{n^{\prime}}{k^{\prime}}|e^{\mp\mathfrak{i}\varphi}\hat{h}_{\pm}|\mu
nk\rangle\langle{\sigma^{\prime}}|\sigma_{\pm}|\sigma\rangle$ (11)

The second matrix element on the rhs for $\sigma_{+}$ and $\sigma_{-}$ is non-
vanishing only for $\sigma=\downarrow$ and $\sigma=\uparrow$, respectively:

$\displaystyle\langle{\sigma^{\prime}}|\sigma_{+}|\sigma\rangle$
$\displaystyle=\delta_{{\sigma^{\prime}}\uparrow}\delta_{\sigma\downarrow},$
$\displaystyle\langle{\sigma^{\prime}}|\sigma_{-}|\sigma\rangle$
$\displaystyle=\delta_{{\sigma^{\prime}}\downarrow}\delta_{\sigma\uparrow},$
(12)

while the first matrix element takes the explicit form

$\displaystyle I_{\pm}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}=\int_{0}^{R_{b}}$ $\displaystyle
dr\int_{-\infty}^{+\infty}dz\int_{0}^{2\pi}rd\varphi\
e^{-\mathfrak{i}({\mu^{\prime}}\pm 1)\varphi}\ e^{-\mathfrak{i}{k^{\prime}}z}$
$\displaystyle\times J_{{\mu^{\prime}}}(\gamma_{{\mu^{\prime}}{n^{\prime}}}r)\
$ $\displaystyle\hat{h}_{\pm}(r,\varphi,z)\ e^{\mathfrak{i}\mu\varphi}\
e^{\mathfrak{i}kz}\ J_{\mu}(\gamma_{\mu n}r).$ (13)

Based on these expressions we introduce rates for spin-conserving and spin-
flipping processes

$\displaystyle\left.\frac{1}{\tau_{z}}\right|_{\mu
nk\sigma}{=}2\pi\xi^{2}\sum_{{\mu^{\prime}}\,{n^{\prime}}\,{k^{\prime}}}$
$\displaystyle|I^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}_{z}|^{2}\delta(E_{f}-E_{i})$
$\displaystyle\left.\frac{1}{\tau_{+}}\right|_{\mu
nk\downarrow}{=}2\pi\xi^{2}\sum_{{\mu^{\prime}}\,{n^{\prime}}\,{k^{\prime}}}$
$\displaystyle|I^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}_{+}|^{2}\delta(E_{f}{-}E_{i})$
$\displaystyle\left.\frac{1}{\tau_{-}}\right|_{\mu
nk\uparrow}{=}2\pi\xi^{2}\sum_{{\mu^{\prime}}\,{n^{\prime}}\,{k^{\prime}}}$
$\displaystyle|I^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}_{-}|^{2}\delta(E_{f}{-}E_{i}),$ (14)

with $\xi=\hbar^{2}/(4m^{2}c^{2})$. These three scattering rates follow
directly from the three terms in Eq. (7), and can be considered separately
because the processes do not mutually interfere. The most interesting
situations arise when the rates for up- or down-conversion of the spins, i.e.
the lower two lines of (14), differ.

####  I.1.3 Evaluation of the integrals

The integral for the spatial coordinates in (10) has two terms inherited from
the structure of $\hat{h}_{z}$. The partial derivative
$\frac{\partial}{\partial\varphi}$ at the right in the operator $\hat{h}_{z}$,
acting on a wave function $|\mu nk\sigma\rangle$, produces $\mathfrak{i}\mu$.
The two terms can be combined through partial integration. For the first term
we use for the integral over $r$,

$\displaystyle\mathfrak{i}\mu\int_{0}^{R_{b}}$ $\displaystyle\,\frac{\partial
v}{\partial r}\,J_{{\mu^{\prime}}}J_{\mu}\,dr=$ (15)
$\displaystyle-\mathfrak{i}\mu\int_{0}^{R_{b}}\,v\,\left(J_{\mu}\frac{dJ_{{\mu^{\prime}}}}{dr}+J_{{\mu^{\prime}}}\frac{dJ_{\mu}}{dr}\right)\,dr.$

where we write $J_{\mu}$ and $J_{{\mu^{\prime}}}$ as short for
$J_{\mu}(\gamma_{\mu n}r)$ and
$J_{{\mu^{\prime}}}(\gamma_{{\mu^{\prime}}{n^{\prime}}}r)$.

For the second term we use,

$\displaystyle\int_{0}^{2\pi}\,e^{-\mathfrak{i}(\mu-{\mu^{\prime}})\varphi}\frac{\partial
v}{\partial\varphi}\,$ $\displaystyle d\varphi=$ (16)
$\displaystyle-\mathfrak{i}(\mu-{\mu^{\prime}})\int_{0}^{2\pi}v\,e^{-\mathfrak{i}(\mu-{\mu^{\prime}})}\,d\varphi.$

The two terms combine into,

$\displaystyle I_{z}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu nk}$
$\displaystyle=-\mathfrak{i}\int_{0}^{R_{b}}dr\int_{-\infty}^{+\infty}dz\,\int_{0}^{2\pi}d\varphi\
\,v(\mathbf{r})\,e^{\mathfrak{i}(\mu-{\mu^{\prime}})\varphi}$
$\displaystyle\times e^{\mathfrak{i}(k-{k^{\prime}})z}\,\left(\mu
J_{\mu}\frac{dJ_{{\mu^{\prime}}}}{dr}+{\mu^{\prime}}J_{{\mu^{\prime}}}\frac{dJ_{\mu}}{dr}\right)$
(17)

By similar steps we obtain for the corresponding integrals for the operators
$\hat{h}_{\pm}$,

$\displaystyle I_{\pm}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu nk}$
$\displaystyle=\int_{0}^{R_{b}}dr\int_{-\infty}^{+\infty}dz\int_{0}^{2\pi}d\varphi\,v(\mathbf{r})\,e^{\mathfrak{i}(\mu-{\mu^{\prime}}\mp
1)\varphi}\times$ $\displaystyle e^{\mathfrak{i}(k-{k^{\prime}})z}$
$\displaystyle\left[(\mu{k^{\prime}}-{\mu^{\prime}}k)J_{{\mu^{\prime}}}J_{\mu}\pm
r\left(kJ_{\mu}\frac{dJ_{{\mu^{\prime}}}}{dr}+{k^{\prime}}J_{{\mu^{\prime}}}\frac{dJ_{\mu}}{dr}\right)\right].$
(18)

Here, we have assumed that the potential $v(\mathbf{r})$ vanishes for
$z\rightarrow\pm\infty$, but otherwise it has not yet been specified.

####  I.1.4 Limiting case: a single ‘atom’

Note that the potential due to a single atomic-like scatterer does not define
a chiral structure. Yet, we find a difference between the integrals for
$\tau_{+}$ and $\tau_{-}$ in (I.1.3) even for a single scattering center. This
spin-dependent scattering arises from the combination of the finite orbital
angular momentum $\hbar\mu$ and $\hbar{\mu^{\prime}}$ of the incoming and
scattered waves, in combination with the off-axis position of the ‘atom’.
Experiments that permit selection of the angular momentum of incoming waves
may observe this spin-dependent scattering, but typical experiments select the
incoming waves by energy only.

This chiral effect  that is not associated with a chiral potential is removed
by summation of contributions of all incoming electrons at the same energy.
This can be seen from the fact that
$I_{+}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}=-I_{-}^{-{\mu^{\prime}}{n^{\prime}}{k^{\prime}},-\mu nk}$, so that
$|I_{+}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}|^{2}=|I_{-}^{-{\mu^{\prime}}{n^{\prime}}{k^{\prime}},-\mu nk}|^{2}$.
Therefore,  for a single ‘atom’,

$\displaystyle\left.\frac{1}{\tau_{+}}\right|_{\mu
nk\downarrow}=\left.\frac{1}{\tau_{-}}\right|_{-\mu nk\uparrow}$ (19)

Since the states with quantum number $\mu$ are degenerate with those having
$-\mu$ we should consider the sum of the scattering rates for $\mu$ and $-\mu$
and, thus, this sum is equal for $\tau_{+}$ and $\tau_{-}$.

Concluding, as we should expect, a single atomic-like potential does not
represent a chiral structure and  only produces spin-dependent scattering when
we can conceive of an experiment that selects the orbital states of the
electron waves.  Below we will combine the scattering of a string of ‘atoms’
and demonstrate that spin dependence results from a helical arrangement of
single scatterers.

The potential $v(\mathbf{r})$ in (3) describes the electrostatic potential due
to the ‘atoms’ inside the tube.  As a first step, we limit this to a single
site at $(R,\Phi,Z)$. We calculate the matrix elements explicitly by adopting
a minimal model for the scattering potential $v(\mathbf{r})$. The potential is
normally given by the Coulomb interaction between the electrons and the
effective core. For the purpose of the minimal model we have the freedom of
adjusting the actual form of this potential, and for computational convenience
we choose it to be a delta function

$v(r,\varphi,z)=K_{0}\delta(r-R)\frac{1}{r}\delta(\varphi-\Phi)\delta(z-Z).$
(20)

With this, the integrals for the spatial coordinates in matrix elements for
$\hat{h}_{z}$, $\hat{h}_{+}$ and $\hat{h}_{-}$ become,

$\displaystyle\quad I_{z}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}=-\mathfrak{i}K_{0}e^{\mathfrak{i}(\mu-{\mu^{\prime}})\Phi}e^{\mathfrak{i}(k-{k^{\prime}})Z}\times$
(21) $\displaystyle\qquad\qquad\left[\frac{1}{r}\left(\mu
J_{\mu}\frac{dJ_{{\mu^{\prime}}}}{dr}+{\mu^{\prime}}J_{{\mu^{\prime}}}\frac{dJ_{\mu}}{dr}\right)\right]_{r=R},$
$\displaystyle\quad I_{\pm}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}=K_{0}e^{\mathfrak{i}(\mu-{\mu^{\prime}}\mp
1)\Phi}e^{\mathfrak{i}(k-{k^{\prime}})Z}\times$
$\displaystyle\qquad\left[\frac{1}{r}(\mu{k^{\prime}}-{\mu^{\prime}}k)J_{\mu}J_{{\mu^{\prime}}}\pm\left(kJ_{\mu}\frac{dJ_{{\mu^{\prime}}}}{dr}+{k^{\prime}}J_{{\mu^{\prime}}}\frac{dJ_{\mu}}{dr}\right)\right]_{r=R}.$

The terms in square brackets depend on the quantum numbers, but otherwise only
on the radial distance $R$ for the position of the atomic potential. The
dependence on $Z$ and $\Phi$ is completely covered by the phase factors in the
first lines of (21).

####  I.1.5 A helical string of atomic-like scatterers

In order to implement a chiral structure we arrange $2N+1$ identical ‘atoms’
along a helix inside the cylindrical conductor at positions $(R,\nu\theta,\nu
s)$ for $\nu=-N,\ldots,N$, where the constants $\theta$ and $s$ describe the
chirality of the ‘molecule’. Thus, we write the potential $v$ in the form of a
sum over the potentials of the individual nuclei,

$v_{N}(\mathbf{r})=\sum_{\nu=-N}^{N}v_{0}(r-R,\,\varphi-\nu\theta,\,z-\nu s),$
(22)

where $v_{0}$ is the potential due to a single scattering center.

In order to evaluate the matrix elements for this potential, we insert it in
Eqs. (I.1.3) and (I.1.3). The scattering amplitudes from each of the ‘atoms’
in the string receive phase factors according to (21), which allows us to
express the integrals as,

$\displaystyle I_{z}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu nk}$
$\displaystyle=I_{z}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}(v_{0})\sum_{\nu=-N}^{N}e^{\mathfrak{i}\nu(s\Delta k+\theta\Delta\mu)}$
$\displaystyle=$ $\displaystyle
I_{z}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}(v_{0})\quad\frac{\sin{((N+\tfrac{1}{2})(s\Delta
k+\theta\Delta\mu))}}{\sin{(\tfrac{1}{2}(s\Delta k+\theta\Delta\mu))}}$
$\displaystyle I_{\pm}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu nk}$
$\displaystyle=I_{\pm}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu
nk}(v_{0})\sum_{\nu=-N}^{N}e^{\mathfrak{i}\nu(s\Delta k+\theta(\Delta\mu\mp
1))}$ $\displaystyle=I_{\pm}^{{\mu^{\prime}}{n^{\prime}}{k^{\prime}},\mu nk}$
$\displaystyle(v_{0})\quad\frac{\sin{((N+\tfrac{1}{2})(s\Delta
k+\theta(\Delta\mu\mp 1)))}}{\sin{(\tfrac{1}{2}(s\Delta k+\theta(\Delta\mu\mp
1)))}},$ (23)

where $\Delta k={k^{\prime}}-k$ and $\Delta\mu=\mu-{\mu^{\prime}}$. The
integrals on the rhs are evaluated for the potential $v_{0}$ due to a single
scattering center. The sums in (I.1.5) generally lead to near-cancellations,
unless the argument $s({k^{\prime}}-k)+\theta(\mu-{\mu^{\prime}}\mp 1)$ is
close to a multiple of $2\pi$. This quantum interference effect strongly
selects specific channels for spin scattering.

This is the central result of this paper: quantum interference breaks the
symmetry between spin-up and spin-down scattering. For example, consider
initial and final states with ${\mu^{\prime}}=\mu$ and ${n^{\prime}}=n$.
Energy conservation requires ${k^{\prime}}=\pm k$. For back scattering we have
${k^{\prime}}=-k$, so that the arguments are $x=-2ks-\theta$ for $I_{+}$ and
$x=-2ks+\theta$ for $I_{-}$. This breaks the symmetry of (19), in particular
for the energy range for which $k\simeq\theta/2s$ so that $x$ approaches zero
for $I_{-}$, while $x\simeq 2\theta$ for $I_{+}$, which will be typically far
from zero. The other spin channel is selected when $x=-2ks-\theta\simeq-2\pi$.

At higher energy many combinations of quantum numbers lead to constructive
interference, and in order to evaluate this we need to resort to numerical
evaluation of the expressions.

####  I.1.6 Absence of spin-dependent forward scattering

In order to suppress  scattering that is not associated with the chiral shape
of the potential  we consider the average of scattering of all states
available at a given energy,

$\displaystyle\left.\frac{1}{\tau_{\pm}}\right|_{E}{=}\frac{1}{n_{\tau}}\sum_{\mu\,n}\sum_{k>0}\left.\frac{1}{\tau_{\pm}}\right|_{\mu
nk\sigma}\delta(E-\epsilon_{\mu nk}),$ (24)

where the energy for each of the states $\epsilon_{\mu nk}$ is given by (2),
and $n_{\tau}$ counts the numbers of states available at this energy. The
polarization of the spin scattering can then be defined as,

$P(E)=\frac{\left.\frac{1}{\tau_{+}}\right|_{E}-\left.\frac{1}{\tau_{-}}\right|_{E}}{\left.\frac{1}{\tau_{+}}\right|_{E}+\left.\frac{1}{\tau_{-}}\right|_{E}+\left.\frac{1}{\tau_{z}}\right|_{E}}$
(25)

Remarkably, we find that the polarization for forward scattering
(${k^{\prime}}>0$) is identical to zero. In order to demonstrate this let us
consider incoming states with energy $E$ and quantum numbers $\mu,n,k>0$. The
spin-flip scattering rates averaged over states at energy $E$ can be written
as,

$\displaystyle\left.\frac{1}{\tau_{+}}\right|_{E}^{\text{fwd.}}$
$\displaystyle=\frac{2\pi\xi^{2}}{n_{\tau}}\,\sum_{\mu
nk}\sum_{{\mu^{\prime}}{n^{\prime}}{k^{\prime}}}\
\left|\bra{{\mu^{\prime}}{n^{\prime}}{k^{\prime}}\downarrow}\,\hat{h}_{0}\,\ket{\mu
nk\uparrow}\right|^{2}$
$\displaystyle\qquad\qquad\times\delta(E-\epsilon_{\mu^{\prime}n^{\prime}k^{\prime}})\delta(E-\epsilon_{\mu
nk})$ (26a) $\displaystyle\left.\frac{1}{\tau_{-}}\right|_{E}^{\text{fwd.}}$
$\displaystyle=\frac{2\pi\xi^{2}}{n_{\tau}}\,\sum_{\mu
nk}\sum_{{\mu^{\prime}}{n^{\prime}}{k^{\prime}}}\
\left|\bra{{\mu^{\prime}}{n^{\prime}}{k^{\prime}}\uparrow}\,\hat{h}_{0}\,\ket{\mu
nk\downarrow}\right|^{2}$
$\displaystyle\qquad\qquad\times\delta(E-\epsilon_{\mu^{\prime}n^{\prime}k^{\prime}})\delta(E-\epsilon_{\mu
nk}).$ (26b)

In order to account for forward scattering only we restrict the final-state
momenta to positive values, $k^{\prime}>0$. In this case, the indices
$\mu,n,k$ and ${\mu^{\prime}},{n^{\prime}},{k^{\prime}}$ run through identical
sets of quantum numbers in the (26). When one uses the fact that $\hat{h}_{0}$
in (26) is hermitian, and relabels the indices, it it easy to see that

$\left.\frac{\hbar}{\tau_{+}}\right|_{E}^{\text{fwd.}}=\left.\frac{\hbar}{\tau_{-}}\right|_{E}^{\text{fwd.}}.$
(27)

This identity does not apply for back scattering, because in this case the
summation over $k$ and ${k^{\prime}}$ cannot be interchanged. In the following
we will focus on the properties of back scattered electron spins.

##  II Numerical evaluation

The many contributions to the sums in (14) require numerical evaluation. For
back scattering we find the result plotted in Fig. 2. The important result we
obtain is that $P(E)$ is typically large, and even approaches complete
polarization in some regions of energy. When we change the sign of the
chirality (by $s\rightarrow-s$, or $\theta\rightarrow-\theta$) the graph is
reflected around the horizontal axis, as expected. We find that the spin-
conserving term $1/\tau_{z}$ only makes a minor contribution to the total
scattering rates.

Figure 2: Constructive interference of the spin scattering in a string of 21
‘atoms’ (N=10). Here $P$ is the ratio of the rates for back scattering spins
from up to down and the reverse process, as given by Eq. (25), as a function
of energy $E$, in units $\hbar^{2}/(2mR_{b}^{2})$. The angle between
successive scattering centers in the helical string is set at
$\theta=2\pi/10$, and the distance between the sites $s=0.85$ in units of
$R_{b}$. The vertical lines mark the points of opening of new conductance
channels for $n=1$ (solid), $n=2$ (dashed), and $n=3$ (dotted). Successive
lines in the same style have increasing values of $\mu$. Changing the sign of
the chirality produces a curve with the opposite sign.

Initially, $P(E)$ is zero, because below $(\gamma_{01}/R_{b})^{2}=5.78$ there
are no conduction channels available. After that point the first channel
opens, with $\mu=0,\,n=1$, ${\mu^{\prime}}=0,\,{n^{\prime}}=1$, and this
remains the only channel until $(\gamma_{11}/R_{b})^{2}=14.68$. In this range
of energies the transmission and reflection of spins is exactly balanced, in
accordance with Kramers’ degeneracy for a single-channel conductor.Bardarson
(2008) In our expressions, this absence of spin scattering for the lowest
conductance channel can be read from (21): the first term in the square
brackets vanishes for $\mu={\mu^{\prime}}=0$, and the second term cancels
because the Bessel functions for $\mu$ and ${\mu^{\prime}}$ are the same, and
because ${k^{\prime}}=-k$ for back scattering into the same channel.

Once we cross $E=14.68$ additional conductance channels become accessible,
having non-zero angular momentum quantum numbers $\mu,\,{\mu^{\prime}}=\pm 1$,
with $n={n^{\prime}}=1$. The contributions of these channels allow for the
integrals (21) to become finite, and the gradual increase of $k$ and
${k^{\prime}}$ above $E=14.68$ leads to rapid oscillations of $P(E)$ due to
the phase factors in (I.1.5). The polarization continues to fluctuate with
energy, but longer-period components take over. In some cases it is possible
to trace the saturation of $P$ near 1 to a contribution for which the argument
of the phase factor in the interference
$x=s(k-{k^{\prime}})+\theta(\mu-{\mu^{\prime}}\pm 1)$ becomes very small, so
that all terms add constructively. When this happens the spin scattering rate
for these terms shows a very long period oscillation as a function of energy
and as a function of the numbers of ‘atoms’ in the string.

Importantly, we find that $P(E)$ maintains predominantly the same sign over
wide ranges of energy. For example, $P$ is almost exclusively positive between
$E=17.5$ and $E=27.5$. Since many experiments do not select a sharply defined
energy, but a finite range of energies contributes to the signal, it is useful
to integrate the scattering rates over an energy window.

Figure 3 shows the rates $\left.1/\tau_{+}\right|_{E}$ (red),
$\left.1/\tau_{-}\right|_{E}$ (blue), and $\left.1/\tau_{z}\right|_{E}$
(green) for $2N+1=41$ atomic scattering centers, integrated from E = 17.5 to
E. Clearly the difference between the two spin scattering directions is large,
in particular at the lower energy end. At higher energies the predominance of
spin-up vs. spin-down scattering alternates.

Figure 3: Rates $\left.1/\tau_{+}\right|_{E}$ (red),
$\left.1/\tau_{-}\right|_{E}$ (blue), and $\left.1/\tau_{z}\right|_{E}$
(green) integrated over energy from $E=17.5$, for $2N+1=41$. The rates are
given in arbitrary units. The inset shows the low-energy range on an expanded
scale.

One of the hallmark observations in the experiments is a roughly linear
dependence of the spin polarization with the length of the molecule. In our
model, at any given energy we find that the scattering rates oscillate with
the number of ‘atoms’ $2N+1$ included in the helical string. This oscillation
is often rapid, but long-period oscillations are found near the points where
$P(E)$ approaches 1 or -1 (Fig. 2). However, as noted above, experiments
typically measure the signals due to a finite range of energies. Figure 4
shows the two spin scattering rates, integrated from $E=17.5$ to $E=27.5$, as
a function of the length of the helical string, where the number of atoms is
$2N+1$. The observed dependence is close to linear, while the polarization $P$
remains approximately constant. The linear dependence of the spin scattering
rates persists (at least up to $N=100$), so that the spin scattering can, in
principle, dominate other sources of scattering for long molecules. The linear
dependence will ultimately saturate due to interactions that are not included
in our model, such as multiple scattering and inelastic scattering.

Figure 4: Energy-integrated back scattering rates
$\left.1/\tau_{+}\right|_{E}$ (red), $\left.1/\tau_{-}\right|_{E}$ (blue), and
$\left.1/\tau_{z}\right|_{E}$ (green) as a function of the length of the
helical string. The integration over energy is taken from $E=17.5$ to
$E=27.5$. The number of atoms in the string is $2N+1$. At $N=0$ the tube
contains one atom, and the blue and red rates are equal, as expected.  The
inset plots the rates devided by $N$, as a function of $1/N$, which
demonstrates the saturation to a constant slope of the rates for large $N$.
The rates are given in arbitrary units.

##  III  Discussion and conclusion

The model presented above was investigated in order to trace a possible origin
of the chirality-induced spin selectivity. We find that quantum interference
of partial waves scattered off atomic spin-orbit interactions leads to
selective back scattering of one spin component over the other. Although this
mechanism appears to be intuitively appealing, we cannot claim that we offer a
quantitative explanation of the CISS effect. The nature of the unperturbed
electronic states is idealized in our model, and the spin-orbit interaction at
the atomic sites is represented by a delta-function potential, which is
clearly not realistic. Therefore, the quantitative outcomes for the scattering
rates cannot be taken literally for a comparison with experiments.

The strong points of the mechanism proposed here is that it is robust,
conceptually simple, and may be applicable to a wider range of unperturbed
molecular wave functions. In our model, spin selectivity is found in wide
ranges of energy, despite the smallness of the spin-orbit interaction. This
contrasts with the model proposed by Michaeli,Michaeli and Naaman (2019) which
produces spin selectivity of order unity only in a narrow window of a few meV
at low energies. In our case it is found for all energies above a certain
threshold value. Furthermore, we find that the spin scattering rates increase
almost linearly with the length of the ’molecule’, in agreement with
observations.Göhler  _et al._ (2011)

An interesting feature of our model is that the sign of the spin scattering is
not uniquely determined by the handedness, as shown by the plot in Fig. 2, but
also by the helicity $\theta/s$ and by the energy range that we consider. Most
experiments have compared the effects of the sign of chirality by comparing
the two enantiomers, or have studied similar molecular structures as a
function of length, all under the same experimental conditions. In a system
that can be described by this quantum interference mechanism one would expect
to observe sign changes when varying the helical pitch, even when maintaining
the same sign of chirality, or when probing the system in a different energy
range.

The model considered here is consistent with Landauer’s picture of a phase-
coherent scattering problem. It transitions to a classical resistance only
when we add inelastic scattering to the description, and consider the limit of
very long helical wires. Even at room temperature, for most chiral molecules
probed in experiments the inelastic scattering length is much longer than the
length of the molecule. On the other hand, the long DNA strands tested by
Gohler et al.Göhler  _et al._ (2011) are possibly long enough for temperature-
induced dephasing to become observable.

In conclusion, we propose a simple model of constructive quantum interference
as a mechanism giving rise to spin-selective electron reflection. The
mechanism proposed here may guide the design of experiments and the analysis
of more realistic computations. We have limited our discussions to possible
explanations of spin-selectivity in the transmission properties of chiral
molecules, and the model presented here offers a simple and intuitive
mechanism that may be transferable to actual molecular systems. We have
refrained from touching upon experiments that involve charge detection, rather
than spin directly, because of the additional complications involved in
describing the spin-to-charge conversion (both in terms of the modeling, and
regarding the proper design of the experimental conditions). The observation
of the third class of experiments, revolving around near-equilibrium
properties of enantiomer absorption on magnetized surfaces, pose even greater
difficulties for explanation, and we have not attempted at addressing those. A
proper understanding of spin-selective transmission may form a solid basis for
proceeding with developing an explanation for the second two classes of
experiments.

###### Acknowledgements.

FE acknowledges support by the German Science Foundation. The work by JMvR is
part of the research program of the Netherlands Organisation for Fundamental
Research, NWO. RK acknowledges the Czech Science Foundation (project no.
22-22419S). We gratefully acknowledge lively discussions with Per Hedegård,
Jȩdrzej Tepper, Sense Jan van der Molen, Peter Neu, Tjerk Oosterkamp, and
Julian Skolaut.

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