\section{Introduction}

	Magnetic fields exist in all plasmas: in laser-driven ablation fronts \cite{gao2015,campbell2020,manuel2012}, ICF capsules \cite{igumenshchev2014,hill2017,walsh2017}, shock fronts \cite{PhysRevLett.123.055002}, hohlraums \cite{farmer2017,sherlock2020} and Z-pinches \cite{gomez2020}. Magnetic fields can also be purposefully applied to ICF implosions to improve fusion performance \cite{chang2011,perkins2017,walsh2019,slutz2010}. The transport of these magnetic fields within the plasma is critically important to the design and interpretation of experiments. Non-local VFP simulations capture the transport processes innately by resolving the distribution of electron energies \cite{sherlock2020,hill2017,joglekar2014}. However, these calculations are prohibitively expensive and unstable, so simplified models with prescribed transport coefficients are used for spatial and temporal scales of interest. These transport coefficients are calculated by fitting to VFP data. The seminal work in this area is that of Braginskii, who laid the groundwork for the anisotropic transport of thermal energy and magnetic fields within the magneto-hydrodynamic (MHD) framework \cite{braginskii1965}. Following on from Braginskii, Epperlein \& Haines improved the transport coefficients, allowing for closer agreement to VFP calculations \cite{epperlein1986}. For 35 years the transport coefficients of Epperlein \& Haines were the agreed-upon standard to be implemented into extended-MHD codes \cite{walsh2017,farmer2017,perkins2017,watkins2018,igumenshchev2014}.

	\begin{figure}
		\centering
		\includegraphics[width=0.5\textwidth]{./Figures/Gamma_Wedge.png}
		\caption{The cross-gradient-Nernst coefficient as a function of Hall Parameter calculated using different sources. All coefficients are attempting to replicate those given by VFP calculations. \label{fig:gamma}}
	\end{figure}

	Recently, simultaneous work by two research teams demonstrated errors in two of the transport coefficients at low electron magnetization \cite{sadler2021,davies2021}. While Epperlein \& Haines stated low errors in fitting the transport coefficients to VFP data, they did not consider that their transport coefficients were ill-formed. From their perspective the magnetic field evolution was captured by:

	\begin{align}
	\frac{\partial\underline{B}}{\partial{}t}=\nabla\times(\underline{v}\times\underline{B})-\nabla\times\frac{\underline{j}\times\underline{B}}{n_{e}e}-\nabla\times\frac{\underline{\underline{\alpha}}\cdot\underline{j}}{n_{e}^{2}e^{2}}\nonumber \\
	+\nabla\times\frac{\underline{\underline{\beta}}\cdot\nabla{}T_{e}}{e}+ \nabla \times \frac{\nabla P_{e}}{n_{e} e}\label{eq:mag_transport_old}

	\end{align}

	where $\underline{\underline{\alpha}}$ and $\underline{\underline{\beta}}$ are tensor transport coefficients calculated by fitting to 0D VFP simulations. While accurate, equation \ref{eq:mag_transport_old} does not make clear the impact of each term on the movement or generation of magnetic fields. Recent work re-arranged this equation to make the different physical processes more obvious\cite{walsh2020,davies2015,sadler2020}:

	\begin{align}
	\begin{split}
	\frac{\partial \underline{B}}{\partial t} = & - \nabla \times \frac{\alpha_{\parallel}}{\mu_0 e^2 n_e ^2} \nabla \times \underline{B} + \nabla \times (\underline{v}_B \times \underline{B} ) \\
	&+ \nabla \times \Bigg( \frac{\nabla P_e}{e n_e} - \frac{\beta_{\parallel} \nabla T_e}{e}\Bigg) \label{eq:mag_trans_new}
	\end{split}
	\end{align}
	where the first term is diffusive, the second term is advection of magnetic field at a velocity $\underline{v}_B$ and the final term is a source of magnetic flux. The advection velocity is then a combination of the bulk plasma motion, thermally-driven terms and current-driven terms:

	\begin{equation}
	\underline{v}_B = \underline{v} - \gamma_{\bot} \nabla T_e - \gamma_{\wedge}(\underline{\hat{b}} \times \nabla T_e) - \frac{\underline{j}}{e n_e}(1 + \delta_{\bot}^c) + \frac{\delta_{\wedge}^c}{e n_e} (\underline{j} \times \underline{\hat{b}})   \label{eq:mag_trans_new_velocity}
	\end{equation}

	where the transport coefficients have been re-written from the form given by Braginskii. The dimensionless forms (denoted by a super-script $^c$) depend only on Hall Parameter and effective Z~\cite{walsh2020}:

	\begin{equation}
		\gamma_{\bot}^c = \frac{\beta_{\wedge}^c}{\omega_e \tau_e}
	\end{equation}

	\begin{equation}
		\gamma_{\wedge}^c = \frac{\beta_{\parallel}^c - \beta_{\bot}^c}{\omega_e \tau_e}
	\end{equation}

	\begin{equation}
		\delta_{\bot}^c = \frac{\alpha_{\wedge}^c}{\omega_e \tau_e}
	\end{equation}

	\begin{equation}
		\delta_{\wedge}^c = \frac{\alpha_{\bot}^c - \alpha_{\parallel}^c}{\omega_e \tau_e}
	\end{equation}

	The dimensional forms of the thermally-driven coefficients are given by:
	\begin{equation}
	\gamma = \gamma^c \frac{\tau_e}{m_e}
	\end{equation}

	Once the transport coefficients have been re-written into this physically-motivated form, the fits of Epperlein \& Haines to the VFP simulations have large errors in $\gamma_{\wedge}^c$ and $\delta_{\wedge}^c$ at small $\omega_e \tau_e$. While the errors in fitting to $\beta_{\bot}^c$ are small, the $\gamma_{\wedge}$ numerator ($\beta_{\parallel}^c - \beta_{\bot}^c$) and denominator ($\omega_e \tau_e$) both tend to zero for small $\omega_e \tau_e$. This resulted in both $\gamma_{\wedge}^c$ and $\delta_{\wedge}^c$ being maximum for small $\omega_e \tau_e$, while the VFP simulations calculated them as exactly zero. Figure \ref{fig:gamma} shows how $\gamma_{\wedge}^c$ calculated using the values from Epperlein \& Haines deviate from the desired VFP data.

	New fits to the VFP data incorporate the knowledge of how the transport coefficients should be formulated to capture their limiting behavior at low Hall Parameter \cite{sadler2021,davies2021}. It should be reiterated that this newer work did not modify the VFP calculations, only changing how the transport coefficients were fit to the VFP data. Figure \ref{fig:gamma} includes $\gamma_{\wedge}^c$ calculated by Sadler, Walsh \& Li as well as by Davies, Wen, Ji \& Held; note that these two references give near-identical results.

	\begin{figure}
		\centering
		\includegraphics[width=0.5\textwidth]{./Figures/Coeff_ratios.png}
		\caption{Ratio of thermal to magnetic transport coefficients as a function of electron Hall Parameter for the old and new coefficients. These are plotted for $Z=1$.\label{fig:coeff}}
	\end{figure}

	The updated transport coefficients make clear the similarities between thermal and magnetic transport. Looking specifically at the transport due to temperature gradients:

	\begin{align}
	&\underline{v}_{N} &= & &  -\gamma_{\bot}\nabla_{\bot} T_e  &- \gamma_{\wedge} \underline{\hat{b}} \times \nabla_{} T_e \label{eq:thermally_driven_transport1}\\
	&\underline{q}_{\kappa} &= &-\kappa_{\parallel} \nabla_{\parallel} T_e  &-\kappa_{\bot}\nabla_{\bot} T_e  &- \kappa_{\wedge} \underline{\hat{b}} \times \nabla_{} T_e \label{eq:thermally_driven_transport2}
	\end{align}

	where equation \ref{eq:thermally_driven_transport1} is magnetic transport and equation \ref{eq:thermally_driven_transport2} is thermal conduction. These equations are split into components parallel ($\parallel$) to the magnetic field, perpendicular to the field ($\bot$) and perpendicular to both the magnetic field and temperature gradient ($\wedge$). In equation \ref{eq:thermally_driven_transport1} the $\bot$ term is Nernst advection and the $\wedge$ term is cross-gradient-Nernst. For thermal conduction the terms are the unrestricted thermal conduction along field lines, suppressed thermal conduction perpendicular to the field and the Righi-Leduc heat-flow.

	The analogy between Nernst ($\gamma_{\bot}^c$) and perpendicular heat-flow ($\kappa_{\bot}^c$) was originally noted by Haines \cite{haines1986}. Haines found that if the kinetic collision operator uses an artificial $v^{-2}$ dependence, where $v$ is the electron speed, there is an exact equivalence between heat flow and magnetic field motion, such that $\kappa_\perp^c/\gamma_\perp^c = \kappa_\wedge^c/\gamma_\wedge^c = 5/2$. The physical interpretation is that magnetic field is only frozen into the faster electrons in the distribution, since it can easily diffuse through the slower, more collisional electrons. The field therefore follows the faster electrons, which tend to move down temperature gradients with the heat flow. For the artificial collision operator, this intuition holds exactly for all $Z$ and $\omega_e\tau_e$.

	However, this physical picture is unaffected by changing the $v$ dependence of the collision operator, so the character of magnetic field movement with the electron heat flow should still approximately hold. For example, if instead the collisions use the realistic Fokker-Planck $v^{-3}$ dependence, magnetic field should still move with the heat flow, although they are no longer directly proportional. With the Fokker-Planck collision operator, the ratios are no longer fixed at $5/2$, and they have dependence on $Z$ and $\omega_e\tau_e$. However, for fixed $Z$, the ratios remain approximately constant. Figure \ref{fig:coeff} shows the ratio $\kappa_{\bot}^c/\gamma_{\bot}^c$ against Hall Parameter for $Z=1$; this result is similar using the old or new coefficients, giving little variation.

	Figure \ref{fig:coeff} also plots the ratio $\kappa_{\wedge}^c/\gamma_{\wedge}^c$ against Hall Parameter for $Z=1$. Using the old Epperlein \& Haines coefficients the Righi-Leduc ($\kappa_{\wedge}^c$) coefficient is much smaller than cross-gradient-Nernst ($\gamma_{\wedge}^c$) at low magnetization. With the new transport coefficients, however, it can be seen that the cross-gradient-Nernst term is analogous to advection of magnetic field with the Righi-Leduc heat-flow; for $Z=1$, the ratio $\kappa_{\wedge}^c/\gamma_{\wedge}^c$ varies by less than 3\% across magnetization.

	\begin{figure*}
		\centering
		\includegraphics[scale=0.8]{./Figures/Underdense.jpg}
		\caption{ \label{fig:underdense} Azimuthal magnetic field component generated by cross-gradient-Nernst advection for an under-dense magnetized plasma \cite{walsh2020} using the Epperlein \& Haines transport coefficients \cite{epperlein1986} (left) and the updated coefficients (right) \cite{sadler2021,davies2021}. These profiles are 1.0ns after the laser turns on. }
	\end{figure*}

	The ratio of thermal to magnetic transport coefficients does vary with plasma ionization. For low Hall Parameter, $\kappa_{\bot}^c/\gamma_{\bot}^c$ ranges between 3.63 for $Z=1$ to 1.39 for $Z=100$. This ratio is fundamental for systems where thermal conduction drives plasma ablation, where equation \ref{eq:thermally_driven_transport1} will move magnetic fields into the colder regions, while ablation driven by equation \ref{eq:thermally_driven_transport2} will counter-act that transport \cite{betti2001}. The balance of Nernst demagnetization to plasma ablation has been found to be critically important in magnetized plasma conditions \cite{manuel2015,hill2017,walsh2017,farmer2017,campbell2020,sherlock2020,walsh2020a,hill2021}. The decrease of $\kappa_{\bot}^c/\gamma_{\bot}^c$ with Z suggests that low-Z plasmas will be less impacted by Nernst de-magnetization than high-Z plasmas.

	The new fits make intuitive sense. The $\wedge$ terms come from electrons having curved orbits; if the magnetization is low, the electron trajectories should be straight on average. The Epperlein \& Haines fits captured this behavior for the heat-flow ($\kappa_{\wedge}$) but not for the magnetic transport ($\gamma_{\wedge}$). At large magnetization the electrons go through many orbits before colliding, meaning that there is no preferential direction for transport; this is captured in both the old and new coefficients, with $\kappa_{\wedge}$ and $\gamma_{\wedge}$ going to zero at large magnetization. For the wedge terms to be important, the plasma must be in a moderately magnetized regime ($\omega_e \tau_e \approx 1$).

	The error in the Epperlein \& Haines transport coefficients can alternatively be explained by their focus on the electric field. They did not consider the coefficients in terms of the induction equation, where the gradients of the coefficients are important. The fitting function in the Hall parameter that Epperlein \& Haines chose has a derivative that does not tend to zero as the magnetic field tends to zero, which is a physical requirement. The errors in the fits become more apparent when the coefficients are reformulated in a manner that more clearly demonstrates their physical effects in the induction equation.

	This paper demonstrates the importance of the new transport coefficients in laboratory plasmas, focusing on the thermally-driven cross-gradient-Nernst term. In section \ref{sec:underdense} the new coefficient is shown to reduce magnetic field twisting in under-dense systems relevant to MagLIF preheat \cite{gomez2020}. This setup was proposed in order to make the first measurements of the cross-gradient-Nernst transport \cite{walsh2020}. Section \ref{sec:premag} then shows how the cross-gradient-Nernst term can result in field twisting in pre-magnetized ICF capsules \cite{walsh2018a}; again, the new coefficients result in reduced twisting at low plasma magnetizations. However, the new coefficients do not universally reduce the impact of cross-gradient-Nernst transport. Section \ref{sec:selfgen} compares ICF hot-spot simulations without MHD, with MHD but no cross-gradient-Nernst, full extended-MHD with the Epperlein \& Haines coefficients and with the new coefficients. Cross-gradient-Nernst transport is shown to be important in hot-spot cooling, increasing the penetration depth of a cold spike. An example of how the new $\gamma_{\wedge}$ coefficient affects perturbation growth in direct-drive ablation fronts was already given in one of the original updated coefficients publications \cite{sadler2021}

	Simulations in this paper use the Gorgon extended-MHD code \cite{ciardi2007,chittenden2004,walsh2017}. The code can use either the old Epperlein \& Haines coefficients \cite{epperlein1986} or the updated coefficients (see the supplementary material of reference \cite{sadler2021}). The coefficients calculated by Sadler, Walsh \& Li are used in this paper, although the coefficients from Davies, Wen, Ji \& Held have been implemented and give no noticeable differences \cite{davies2021} (as expected from figure \ref{fig:gamma}). Epperlein \& Haines provided tabulated data for the coefficients at specific values of Z; these values are interpolated in the simulations. The new coefficients provide a polynomial fit to ionization, which is preferable for reducing computations and eliminating discontinuities when gradients are taken in Z.

	\section{Underdense Plasmas\label{sec:underdense}}

	This section investigates the impact of cross-gradient-Nernst on magnetized under-dense gases heated by a laser. This setup is relevant to MagLIF \cite{slutz2010} and mini-MagLIF preheat \cite{barnak2017}, but was also investigated as a means of measuring specific extended-MHD transport terms for the first time \cite{walsh2020}. Particularly relevant was the suggestion that quantification of a twisted magnetic field component  could be used to measure the cross-gradient-Nernst advection velocity \cite{walsh2020}. Simulations in that experiment design paper used the erroneous Epperlein \& Haines coefficients \cite{epperlein1986}, which are shown here to give excessive magnetic field twisting at low magnetization.

	The configuration used as demonstration here is as follows. A low density ($5 \times 10^{19}$ atoms/cm$^3$) deuterium gas is irradiated by a $5J$ beam with a $0.5ns$ square temporal pulse. The beam has a Gaussian spatial profile with a standard deviation $\sigma=100\mu$m at best focus. The beam is tapered along its propagation direction such that at $r=\sigma$ the angle of the rays to the axis satisfies $\sin\theta = 1/12$. A 1T magnetic field is applied along the laser propagation axis. The beam tapering is purposefully used to induce magnetic field twisting by cross-gradient-Nernst transport \cite{walsh2020}.

	2-D $\underline{r}$,$\underline{z}$ simulations are used, with the laser drive and applied magnetic field along $\underline{z}$. The laser is treated as individual rays that are traced through the plasma and deposit their energy by inverse Bremsstrahlung. A $460\mu$m axial extent is used, with radial and axial resolution of $1\mu$m.

	As the gas density is low, very little energy couples to the system. At $0.1$ns the total absorption along the $460\mu$m axial extent is 2.5\%, reducing to 1\% by the end of the laser pulse when the plasma temperature is higher ($>100$eV). As the Hall Parameter is low for the chosen setup ($\omega_e \tau_e < 0.2$ everywhere) thermal conduction effectively transports energy radially. Simultaneously, the Nernst term advects magnetic field away from the beam center, reducing the field strength to 0.3T by 0.5ns. The magnetic flux piles up at the thermal conduction front, peaking at 2T.

	The impact of cross-gradient-Nernst can be seen by decomposing the advection term in equation \ref{eq:mag_trans_new}:

	\begin{equation}
	\bigg[ \frac{\partial \underline{B}}{\partial t} \bigg] _{N \wedge} + (\underline{v}_{N\wedge} \cdot \nabla) \underline{B} =   -\underline{B}(\nabla \cdot \underline{v}_{N\wedge}) + (\underline{B} \cdot \nabla) \underline{v}_{N\wedge}  \label{eq:theory_velocity_B}
	\end{equation}

	As the temperature gradient is predominantly radial and the magnetic field is predominantly axial, the cross-gradient-Nernst velocity is purely in $\underline{\theta}$. Therefore, equation \ref{eq:theory_velocity_B} reduces to $\big[ \frac{\partial B_{\phi}}{\partial t} \big] _{N \wedge} = B_z \frac{\partial v_{N\wedge\theta}}{\partial z} $. This term represents magnetic field twisting. Following a single axial magnetic field line, if $\underline{v}_{N\wedge} = 0$ at the bottom and $\underline{v}_{N\wedge} \ne 0$ at the top, a $B_{\theta}$ field will be generated by twisting in between. By tapering the laser beam, this setup purposefully makes this happen; near the laser edge a magnetic field line will pass between an unheated and a heated regime, resulting in twisting of the magnetic field \cite{walsh2020}. This process can be viewed as a dynamo driven by electron heat-flow.

	Figure \ref{fig:underdense} compares the twisted magnetic field component at $t=1.0$ns using the old Epperlein \& Haines transport coefficients and the updated coefficients \cite{sadler2021,davies2021}. The Biermann Battery generation has been neglected in these simulations, which means that $B_{\theta}$ can only be generated by cross-gradient-Nernst twisting. As the experiment here is in the low magnetization regime, the difference in the $\gamma_{\wedge}$ coefficient is substantial; at the laser spot edge, where much of the twisting takes place, $\omega_e \tau_e \approx 0.05$, resulting in a factor of 4 reduction in the cross-gradient-Nernst coefficient. By 1.0ns the peak $B_{\theta} \approx$0.08T for the old coefficients, and $\approx$0.01T using the new coefficients.

	 The new coefficients change the regime where significant $B_{\theta}$ is generated. Designs using the old coefficients aimed for the underdense plasma being as low magnetization as possible. Now, however, it is clear that the plasma should be balanced in the moderately magnetized regime, with the cross-gradient-Nernst velocity peaking at around $\omega_e \tau_e \approx 0.8$ for deuterium \cite{sadler2021}. This is still realizable, with the density, laser power and applied field all acting as free variables. An experiment with the plasma parameters defined here but scanning applied field strength is expected to observe the true $\gamma_{\wedge}$ dependence.

	However, with the effect of cross-gradient-Nernst reduced, the Biermann Battery contribution to $B_{\theta}$ becomes more important. At 0.1ns for the simulations shown here the $B_{\theta}$ field from Biermann Battery is of the same order as from cross-gradient-Nernst. Biermann requires a density gradient in the plasma, which only develops later in time, at time-scales on the order of the laser radius divided by the sound speed. Therefore, probing early in time before the plasma hydrodynamically expands is important to distinguish Biermann from cross-gradient-Nernst.

	\section{Pre-magnetized ICF Capsules\label{sec:premag}}

	\begin{figure}
		\centering
		\includegraphics[scale=0.6]{./Figures/Premag_XNernst_Biermann.jpg}
		\caption{ \label{fig:Premag_XNernst_Biermann} $B_{\phi}$ at 6.3ns for a warm indirect-drive implosion with an applied axial field of 5T. Left is a simulation with cross-gradient-Nernst included (using the new coefficients), while right shows a case with both cross-gradient-Nernst and Biermann included. }
	\end{figure}

	Axial magnetic fields can be applied to ICF capsules to reduce thermal cooling of the fuel during stagnation \cite{chang2011,perkins2017,walsh2019}. Throughout the implosion the magnetic field compresses along with the plasma; this happens dominantly at the capsule waist, where the implosion velocity is perpendicular to the magnetic field lines. At the poles the magnetic field remains uncompressed, as the implosion velocity is along field lines.

	It has been suggested that the cross-gradient-Nernst term should twist the applied magnetic field during the implosion \cite{davies2015}. Preliminary simulations showed significant twisting by bang-time, with the yield increased by as much as 10\% due to the extra path length required for heat to travel along field lines out of the hot-spot \cite{walsh2018a}. Again, these calculations used the old Epperlein \& Haines coefficients. Here, as with the twisting seen in the underdense configuration in section \ref{sec:underdense}, the twisting is found to be lower when the new transport coefficients are used.

	The simulated configuration is a $D_2$ gas-filled HDC indirect-drive capsule that is used on the National Ignition Facility, although these results broadly apply to all magnetized spherical implosions. A 5T magnetic field is applied axially. $1\mu$m radial resolution is used throughout the implosion and 180 polar cells are used in the simulation range $\theta = 0,\pi$. Before the first shock converges on axis the simulations are transferred to a cylindrical mesh.

	Moderate HDC shell thickness variations are initialized \cite{casey2021}, such that the capsule is mildly perturbed by neutron bang-time. 1600 perturbations with random amplitude and mode are used, with each chosen in the range $\epsilon=0,0.5$nm and $k=0,180$ respectively.

	\begin{figure}
		\centering
		\includegraphics[scale=0.6]{./Figures/Premag_XNernst.jpg}
		\caption{ \label{fig:Premag_XNernst} Density and $B_{\phi}$ at neutron bang-time ($t=7.3ns$) using the old Epperlein \& Haines coefficients (left) and the updated versions (right). }
	\end{figure}

	While for the underdense configuration a tapered beam was used to induce twisting, a spherical implosion naturally has twisting peaking at $\theta = \pi/4,3\pi/4$. This happens because the cross-gradient-Nernst velocity $- \underline{\hat{b}}\times \nabla T_e$ is maximum at the capsule waist and zero at the poles. Therefore, the axial magnetic field at the waist is displaced in $\phi$ but at the poles it is stationary; in between the waist and the poles the magnetic field lines are twisted out of the plane.

	Twisting of magnetic field lines introduces a closed field line component, $B_{\phi}$. This is of great interest to the magneto-inertial fusion community, as this component is effectively compressed during the implosion and results in additional thermal energy containment. Schemes have been considered to apply a purely $B_{\phi}$ component to a capsule \cite{hohenberger2011}, although none have been demonstrated for spherical implosions.

	The cross-gradient-Nernst velocity is significant both in the in-flight shock-compressed gas and in the stagnating hot-spot. For the configuration chosen here the in-flight phase is not changed significantly by the new coefficients, as the low density and high temperature gas is magnetized even with $B_0 = 5$T. For lower applied fields the coefficients may make a significant difference, but these regimes are not of interest for magneto-inertial fusion.

	Figure \ref{fig:Premag_XNernst_Biermann} shows the $B_{\phi}$ field profile induced by the cross-gradient-Nernst effect at 6.3ns, which is before the first shock converges onto the axis. While $B_{\phi}$ only reaches 3T at this time, the field twisting is significant: $B_{\phi}/|\underline{B}|=1/3$ in locations near the shock-front.

	Also shown in figure \ref{fig:Premag_XNernst_Biermann} is a simulation with the Biermann Battery self-generated magnetic fields included. This term requires an asymmetric implosion to be important, of which there are two sources in this implosion. First of all, the applied field results in anisotropy of the heat-flow; here that results in a hotter compressed gas at the waist compared with the poles. Also, the HDC thickness variations generate magnetic fields. The interest here is in the fuel $B_{\phi}$ component, which is dominated by cross-gradient-Nernst twisting for $B_0 = 5$T. Larger applied fields result in suppression of cross-gradient-Nernst and enhancement of the thermal conductivity anisotropy, enhancing the impact of Biermann.

	Figure \ref{fig:Premag_XNernst} shows the density and out-of-plane magnetic field component at bang-time. Biermann Battery generation of magnetic field has been turned off in the code for these simulations, which means that all of $B_{\phi}$ is from cross-gradient-Nernst twisting. On the left is a case using the old cross-gradient-Nernst coefficients from Epperlein \& Haines, while on the right uses the updated coefficients. The new coefficients lower the peak $B_{\phi}$ from 500T to 250T. Nonetheless, the twisting is significant, with $B_{\phi}/|\underline{B}|$ up to 0.3 throughout the hot-spot.

	The twisting introduces several additional effects. Firstly, thermal conduction and Nernst are moderately reduced, as these components become partially out of the simulation plane; Cross-gradient-Nernst and Righi-Leduc then have components within the simulation plane. If the magnetic tension becomes significant then the twisted field would also be expected to induce plasma motion in $\phi$; however, as cross-gradient-Nernst is suppressed for the large applied fields required for tension to be important \cite{perkins2017,walsh2019}, this may not be a realizable regime. For $B_0 = 5$T the plasma $\beta$ is too large for significant motion.

	While the simulations here focused on indirect-drive implosions, cross-gradient-Nernst also twists magnetic fields in pre-magnetized direct-drive ablation fronts \cite{walsh2020a}, where the temperature gradients drive extreme heat-flows. However, the $B_{\phi}$ component is effectively advected by the plasma and Nernst velocities, lowering its impact.

	\section{Self-Generated Fields in ICF Hot-spots\label{sec:selfgen}}

	\begin{figure}
		\centering
		\includegraphics[scale=0.5]{./Figures/Singlespike_B.jpg}
		\caption{ \label{fig:ss_B} Self-generated magnetic fields at bang-time around a cold spike pushing into a hot-spot. 3 cases are shown. Left: a simulation without cross-gradient-Nernst advection included. Middle: a simulation using the Epperlein \& Haines transport coefficients \cite{epperlein1986}. Right: a simulation using the updated transport coefficients \cite{sadler2021,davies2021}.}
	\end{figure}

	This section looks into the impact of cross-gradient-Nernst on self-generated magnetic field profiles in regular ICF hot-spots, where magnetic fields have not been externally applied.

	Magnetic fields are generated by the Biermann Battery mechanism around perturbations, with estimated field strengths up to 10kT \cite{walsh2017}. These fields are predominantly generated in the stagnation phase, when the temperature and density gradients are largest \cite{walsh2021}. Recent advances have been made in the theoretical understanding of magnetic flux generation in these systems, with more flux being generated around high mode and large amplitude perturbations \cite{walsh2021}.

	While it was expected that self-generated magnetic fields would reduce heat loss from ICF hot-spots,  research found that magnetization of the electron population introduced Righi-Leduc heat-flow, which enhanced cooling \cite{walsh2017}. As cross-gradient-Nernst is the magnetic transport analogue of Righi-Leduc heat-flow, it is important in these systems.

	The simulations here use the indirect-drive HDC design N170601 is used, although the physics is also applicable to direct-drive hot-spots. An isolated 200nm HDC shell thickness variation is imposed at the capsule pole, causing a cold spike to push into the hot-spot. Shell thickness variations have been found to be a significant degradation mechanism for HDC implosions \cite{casey2021}; the variation applied here is not based on any target fabrication specifications, and is instead used as a demonstration of magnetized heat-flow in ICF implosions.

	The simulations here are 2-D and are in spherical geometry until the first shock converges onto the axis ($t\approx 7.6$ns). The resolution up to this time is $1\mu$m radially, with 180 cells in the polar direction from $\theta = 0,\pi$. The simulations are then remapped into cylindrical geometry for the stagnation phase, when the resolution is $\frac{1}{2}\mu$m.

	Figure \ref{fig:ss_B} shows the magnetic field distribution at neutron bang-time ($t= 8.5$ns) for simulations with different cross-gradient-Nernst physics included. The magnetic field profiles are in green (into the page) and purple (out of the page), plotted over the density so that the proximity to the spike can be seen. The dense fuel is at the top of the figure and the hot-spot at the bottom.

	\begin{figure}
		\centering
		\includegraphics[width=0.5\textwidth]{./Figures/Singlespike_Te.jpg}
		\caption{ \label{fig:ss_Te} Electron temperature along the axis of a hot-spot spike at bang-time. Temperature profiles are shown for cases: without any self-generated fields included; with extended-MHD but no cross-gradient-Nernst advection; using the Epperlein \& Haines transport coefficients \cite{epperlein1986}; using the updated coefficients \cite{sadler2021,davies2021}.}
	\end{figure}

	First in figure \ref{fig:ss_B} is a case with no cross-gradient-Nernst advection. The Righi-Leduc heat-flow direction is shown with white arrows. Righi-Leduc acts to cool the spike tip, which results in regular Nernst compressing the field onto the simulation axis. This process causes numerical issues, advecting all of the magnetic field into a single line of cells. Once the cross-gradient-Nernst is included, the magnetic field moves in the same direction as Righi-Leduc, preventing the field profile from compressing onto the axis. It can be seen in figure \ref{fig:ss_B} that the new cross-gradient-Nernst coefficient does not advect as rapidly as the Epperlein \& Haines version.

	It is posited that the updated coefficients will always be important in systems with self-generated magnetic fields and thermal conduction; there is always a null-point of magnetic field, resulting in no electron magnetization. Even if the peak magnetization is large enough to be in a regime where the Epperlein \& Haines coefficients are valid, the magnetization will decrease to zero in the surrounding regions, passing through the regime where the old coefficients are invalid.

	Figure \ref{fig:ss_Te} shows the impact of the imposed spike on the hot-spot temperature for simulations with various MHD packages included. The electron temperature is plotted along the spike axis. Without any MHD included the spike does not propagate as far; this suggests that current design calculations of ICF implosions (both directly and indirectly driven) are underestimating the impact of perturbations.

	Simulations with the new coefficients included result in the greatest discrepancy with the estimates that do not include Biermann generation; at the $T_e = 3.5$keV contour the spike has penetrated $7 \mu$m further due to the magnetization of heat-flow. The simulations without cross-gradient-Nernst are less affected by the magnetic fields, as the fields are transported by the standard Nernst term into a one-cell width (see figure \ref{fig:ss_B}).

	The new coefficients give a greater impact of MHD on spike penetration compared with the old coefficients. This can be understood from the fact that the new coefficients result in the magnetic field moving with the Righi-Leduc heat-flow. In contrast, the old coefficients allowed the magnetic flux to move ahead of the Righi-Leduc heat-flow into regions where it would have a lower impact on the plasma magnetization. This can be seen in figure \ref{fig:coeff}, which shows that the ratio of Righi-Leduc coefficient to cross-gradient-Nernst coefficient decreasing to zero for low magnetization.

	Closer analysis of the impact of MHD on capsule perturbation growth for a variety of mode numbers and amplitudes will be the subject of a future publication.

	\section{Conclusions}

	In summary, updated magnetic transport coefficients that accurately replicate kinetic simulations at low electron magnetizations \cite{sadler2021,davies2021} have been shown to be important across a range of laboratory plasma conditions. These coefficients are the new standard for implementation into extended-MHD codes. While the two references give different fits for the coefficients, they are found to be practically equivalent.

	With an external magnetic field applied, the cross-gradient-Nernst term tends to twist the magnetic field \cite{davies2017,walsh2018a}; using the new coefficients reduces twisting in the low magnetization ($\omega_e \tau_e < 1$) regime. This result impacts attempts to measure cross-gradient-Nernst for the first time \cite{walsh2020} as well as the design of pre-magnetized capsule implosions \cite{chang2011,perkins2017,walsh2018a,walsh2019}. Twisting is still possible in the moderate magnetization regime ($\omega_e \tau_e \approx 1$).

	For systems with significant self-generated magnetic fields \cite{farmer2017,walsh2017,sadler2020a} the new coefficients result in the magnetic fields moving with the Righi-Leduc heat-flow. This is found to enhance the impact of Righi-Leduc, as shown in direct-drive ablation fronts (where Righi-Leduc reduces perturbation growth) \cite{sadler2021} and in ICF hot-spots (where Righi-Leduc enhances perturbation growth).

	In addition to being more physically accurate, the new coefficients have been found to increase numerical stability, as they introduce fewer discontinuities into the simulations.

	\section*{Acknowledgements}
	This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and by the LLNL-LDRD program under Project Number 20-SI-002. The simulation results were obtained using the Imperial College High Performance Computer Cx1..

	This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

	Research presented in this article was also supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory, under the Center for Nonlinear Studies project number 20190496CR. This research was supported by the Los Alamos National Laboratory (LANL) through its Center for Space and Earth Science (CSES). CSES is funded by LANL’s Laboratory Directed Research and Development (LDRD) program under project number 20180475DR.

	Finally, the information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPAE), U.S. Department of Energy, under Award No. DE-AR0001272, by the Department of Energy Office of Science, under Award No.
	DE-FG02-04ER54746, by the Department of Energy National	Nuclear Security Administration under Award No. DENA0003856, the University of Rochester, and the New York State	Energy Research and Development Authority.

	\section*{References}

	\bibliographystyle{plainnat}
	\ifdefined\DeclarePrefChars\DeclarePrefChars{'’-}\else\fi


\section{Introduction}

Traditionally, radio transceivers are subject to a  HD constraint
because of the crosstalk between the transmit and receive chains.
The self-interference caused by the transmitter at the receiver if using
FD transmission overwhelms the desired received signal from the partner
node since it is  much stronger than the desired received
signal. Therefore, current radios all use orthogonal signaling dimensions, i.e., time division
duplexing (TDD) or frequency division duplexing (FDD), to achieve bidirectional
communication.

FD communication can potentially double the throughput if the self-interference
can be well mitigated. FD radios have been successfully implemented in the industrial, scientific and medical (ISM) radio bands in laboratory environments in the past few years~\cite{ChoJai10Mobicom, DuaSab10Asilomar, JaiCho11Mobicom, SahPat11X}.  Key
to the success are novel analog and digital self-interference cancellation
techniques and/or spatially separated transmit and receive antennas.
A FD system with only one antenna has also been implemented in \cite{Knox12WAMICON}
by using specially designed circulator and the FD WiFi radio with one antenna and one circulator has been prototyped in \cite{Bharadia13}. In general, the main idea of FD transmission
is to let the receive chain of a node remove the self-interference
caused by the known signal from its transmit chain, so that reception
can be concurrent with transmission. A novel signaling technique
was proposed in \cite{Guo10Allerton} to achieve virtual FD with applications
in neighbor discovery \cite{Guo2-13} and mutual broadcasting \cite{Guo13}  with its prototyping presented in \cite{Tong14RODD}.

From a theoretical perspective, the two-way transmission capacity
of wireless ad hoc networks has been studied in \cite{Vaze11} for
a FDD model. A FD cellular system has been analyzed in \cite{Goyal2013}
where the throughput gain has been illustrated via extensive simulation
for a cellular system with FD
base station and HD mobile users. The throughput gain of
single-cell multiple-input and multiple-output (MIMO) wireless systems with FD radios has been
quantified in \cite{Barghi12}. A capacity analysis of FD and HD transmissions with bounded
radio resources has been presented in \cite{Aggarwal12ITW} with focus only
on a single-link system. \cite{Ju12,Kim14} evaluate the capacity of FD ad hoc networks and alleviate the capacity
degradation due to the extra interference of FD by using beamforming and an ARQ protocol, respectively. Both
capacity analyses in \cite{Ju12,Kim14} are based on perfect self-interference cancellation and the approximation
that the distances of the two interfering nodes of a FD link to the desired receiver are the same.

In this paper, the impacts of FD transmission on the network throughput
are explored. On the one hand, FD transmission allows bidirectional communication
between two nodes simultaneously and therefore potentially doubles the throughput.
On the other hand, the extra interference caused by FD transmissions and imperfect self-interference cancellation can
degrade the throughput gain over HD, which makes it unclear whether FD
can actually outperform HD.
This paper utilizes the powerful analytical tools from stochastic geometry
to study the throughput performance of a wireless network of nodes with both FD and HD capabilities. Our results analytically
show that for an ALOHA MAC protocol, FD always outperforms HD in terms of throughput if perfect self-interference cancellation is assumed. However, for a path loss exponent $\alpha$, the achievable throughput gain is upper bounded by $\frac{2\alpha}{\alpha + 2}$, i.e., it ranges from  $0$-$33\%$ for the practical range $\alpha \in (2,4]$.
This result holds for arbitrary node densities, link distances and SIR regimes. Moreover, we model imperfect self-interference cancellation and quantify its effects on the throughput. Imperfect self-interference cancellation causes a SIR loss in the FD transmission and thus reduces the throughput gain between the FD network and HD network. Tight bounds on the SIR loss are obtained using the concept of horizontal shifts of the SIR distribution. The amount of self-interference cancellation determines if HD or FD is preferable in the networks.

\section{Network Model\label{sec:Network-Model}}

Consider an independently marked Poisson point process (PPP) \cite{Haenggi12book}
$\hat{\Phi}=\left\{ \left(x_{i},m(x_{i}),s(x_{i})\right)\right\} $
on $\mathbb{R}^{2}\times\mathbb{R}^{2}\times\left\{ 0,1,2\right\} $
where the ground process $\Phi=\left\{ x_{i}\right\} $ is a PPP with density $\lambda$
and $m(x_{i})$ and $s({x_{i}})$ are the marks of point
$x_{i}$. The mark $m(x_{i})$ is the location of the node that $x_{i}$
 communicates with. Here, we fix $\left\Vert x-m(x)\right\Vert =R$, $\forall x\in\Phi$,
i.e., $R$ is the distance of all links. Therefore, $m(x_{i})$ can also be written as $m(x_{i})=x_{i}+R\left(\cos\varphi_{i},\sin\varphi_{i}\right)$,
where the angles $\varphi_{i}$ are independent and uniformly distributed on $\left[0,2\pi\right]$. The link distance $R$ can also be random without affecting the main conclusions since we can always derive the results by first conditioning on $R$ and then averaging over $R$. We define $m(\Phi)=\{m(x): x\in\Phi\}$, which is also a PPP of density $\lambda$.
The mark $s(x_{i})$ indicates the independently chosen state of the link that consists
of $x_{i}$ and $m(x_{i})$: $s(x_{i})=0$ means the link is silent,
$s(x_{i})=1$ means the link is in HD mode, and $s(x_{i})=2$ means it is in FD mode. HD means that in a given
time slot the transmission is unidirectional, i.e., only from $x_{i}$
to $m(x_{i})$, while FD means that $x_{i}$ and $m(x_{i})$
are transmitting to each other concurrently. Therefore, for any link there are three
states: silence, HD, and FD. Assume that a link is in the state of
silence with probability $p_{0}$, HD with probability $p_{1}$ and
FD with probability $p_{2}$, where $p_{0}+p_{1}+p_{2}=1$. $p_{1}$
and $p_{2}$ are the medium access probabilities (MAPs) for HD and
FD modes, respectively. As a result, $\Phi=\bigcup_{i=0}^{2}\Phi_{[i]}$,
where $\Phi_{[i]}=\left\{ x\in\Phi: s(x)=i\right\} $ with density $\lambda p_{i}$
and $i\in\left\{ 0,1,2\right\}$. From the marking theorem \cite[Thm. 7.5]{Haenggi12book},
these three node sets $\Phi_{[i]}$ are independent. We call the link consisting of a node $x_{0}$ and its mark $m(x_{0})$ as the \textit{typical link}.

The marked point process $\hat{\Phi}$ can be used to
model a wireless network of nodes with both FD and HD capabilities.
The self-interference in the FD links is assumed to be cancelled
imperfectly with residual self-interference-to-power ratio (SIPR) $\beta$, i.e., when the transmit power of a node is $P$, the residual self-interference is $\beta P$. The parameter $\beta$ quantifies the amount of self-interference cancellation, and $-10\log_{10} \beta$ is the self-interference cancellation in dB. When $\beta =0$, there is perfect self-interference cancellation, while for $\beta=1$, there is no self-interference cancellation. An example of a realization of such a wireless
network is illustrated in Figure \ref{Fig:0}. In the following, we will use this model to study the performance
of wireless networks with FD radios.
\begin{figure}[h]
\vspace{-3mm}
\begin{centering}
\includegraphics[width=\figwidth]{NetworkIllustration}
\par\end{centering}
\caption{An example of the class of wireless networks considered in this paper. The dashed lines indicate the link is silent, the arrows
mean the link is in HD mode, and the double arrows in FD mode. The $\times$'s form $\Phi$ while the $\circ$'s form $m(\Phi)$.}
\label{Fig:0}
\end{figure}

In this network setup, we use the SIR model where a transmission
attempt from $x$ to $y$ is considered successful if
\begin{equation}
\mbox{SIR}_{y}=\frac{P_{xy}Kh_{xy}l(x,y)}{\sum_{z\in\tilde{\Phi}\backslash\left\{ x\right\} }P_{yz}Kh_{yz}l(z,y)+ \beta P_{xy}\mathbbm{1}_{xy}^{\rm FD}}>\theta,\label{SIR}
\end{equation}
where $\tilde{\Phi}$ is the set of transmitting nodes in a given time slot, $\theta$ is the SIR threshold, $h_{xy}$ and $h_{zy}$ are the fading power coefficients with
mean $1$ from the desired transmitter $x$ and the interferer $z$
to $y$ respectively, and $\mathbbm{1}_{xy}^{\rm FD}$ is the indicator function that the link $xy$ is in FD mode. The inclusion of $\mathbbm{1}_{xy}^{\rm FD}$ means that the interference of FD links has an extra term due to the imperfect self-interference cancellation. The transmit powers $P_{xy} = P$ when link $xy$ is active. We focus on the Rayleigh fading case for both
the desired link and interferers. $K$ is a unitless constant that depends on the antenna characteristics and the average channel attenuations. $K=G_{\rm tx} G_{\rm rx}\left(\frac{c_{\rm L}}{4\pi f_c}\right)^2$, where $c_{\rm L}$ is the speed of light, $f_c$ is the carrier frequency, and $G_{\rm tx}$ and $G_{\rm rx}$ are the antenna gain at the transmitter and receiver, respectively. The path loss function $l(x,y)$
between node $x$ and $y$ is $l(x,y)=\left\Vert x-y\right\Vert ^{-\alpha}$,
where $\alpha>2$ is the path-loss exponent. If $y$ is at the origin, the index $y$ will be omitted, i.e., $l(x,\mathbf{0})\equiv l(x)$. Also, we
call a given set of system parameters $(\lambda,\theta,R,\alpha)$
a \textit{network configuration}. We will show that
some conclusions hold regardless of the network configuration.

\section{Success Probability\label{sec:Success-Probability}}

Our first metric of interest is the success probability, defined as
\begin{equation}
p_{s}\triangleq\mathbb{P}(\mbox{SIR}_{y}>\theta),\label{eq:ps}
\end{equation}
which is also the complementary cumulative distribution function (ccdf) of the SIR. Without changing the distribution of the point process, we may assume that
the receiver $y$ is at the origin. This implies there is a transmitter at fixed distance
$R$ from the origin. The success probability plays an
important role in determining the throughput, as will be described
in the following section.
\subsection{Derivation of the success probability and its bounds}
Before obtaining the unconditional success probability given in (\ref{eq:ps}), we first derive the conditional success probabilities given that a link is HD or FD\footnote{When the link is inactive, the conditional success probability is obviously zero by (\ref{SIR}).}. We denote the success probabilities conditioning that the typical link is HD and FD as $p_{s}^{\rm HD}$ and $p_{s}^{\rm FD}$, respectively. The following theorem gives the conditional success probabilities $p_{s}^{\rm HD}$ and $p_{s}^{\rm FD}$ of the FD/HD-mixed
wireless network modeled by the marked PPP:
\begin{thm}\label{sucProb}
In a wireless network described by the marked PPP $\hat{\Phi}$, the conditional success
probability $p_{s}^{\rm HD}$ is given by
\begin{equation}
p_{s}^{\rm HD}=\exp(-\lambda p_{1}H(\theta R^{\alpha},\alpha))\exp(-\lambda p_{2}F(\theta R^{\alpha},\alpha,R)),\label{eq:ps-2}
\end{equation}
where $H(s,\alpha)\triangleq\frac{\pi^{2}\delta s^{\delta}}{\sin(\pi\delta)}$
with $\delta\triangleq2/\alpha$ and
\ifCLASSOPTIONonecolumn
\begin{equation}
F(s,\alpha,R)\triangleq\int_{0}^{\infty}\left(2\pi-\frac{1}{1+sr^{-\alpha}}\int_{0}^{2\pi}\frac{d\varphi}{1+s\left(r^{2}+R^{2}+2rR\cos\varphi\right)^{-\alpha/2}}\right)rdr,\label{F}
\end{equation}
\else
\begin{equation}F(s,\alpha,R)\triangleq \int_{0}^{\infty}\left(2\pi-\frac{1}{1+sr^{-\alpha}}K(s,r,R,\alpha)\right)rdr \label{F}\end{equation}
with $K(s,r,R,\alpha)\triangleq\int_{0}^{2\pi}\frac{d\varphi}{1+s\left(r^{2}+R^{2}+2rR\cos\varphi\right)^{-\alpha/2}},$
\fi
and the conditional success
probability $p_{s}^{\rm FD}$ is given by
\begin{equation}
p_{s}^{\rm FD}=\kappa p_{s}^{\rm HD},\label{eq:ps-3}
\end{equation}
where $\kappa \triangleq e^{-\frac{\theta R^{\alpha}\beta}{K}}$.
\end{thm}
\begin{IEEEproof}
Conditional on that the link is active, the SIR from (\ref{SIR}) can be rewritten as
\begin{equation}
\mbox{SIR}_{y}=\frac{h_{xy}l(x,y)}{\sum_{z\in\tilde{\Phi}\backslash\left\{ x\right\} }h_{yz}l(z,y)+ \frac{\beta \mathbbm{1}_{xy}^{\rm FD}}{K}}>\theta,\label{SIR2}
\end{equation}
by dividing both numerator and denumerator by $PK$.
As a result, it is equivalent to a network where each node transmit with unit power while the SIPR $\beta$ is scaled by $K$. Hence, with Rayleigh fading, the desired signal strength $S$ at the
receiver at the origin is exponential, i.e., $S=hR^{-\alpha}$. Conditional on that the link is HD, the interference $I$ consists of two parts: the
interference from the HD nodes $\Phi_{[1]}$ and the interference from the FD nodes $\Phi_{[2]}$. Hence, it can be expressed as:
\[
I=\sum_{x\in\Phi_{[1]}}h_{x}l(x)+\sum_{x\in\Phi_{[2]}}\left(h_{x}l(x)+h_{m(x)}l(m(x))\right).
\]
The Laplace transform of the interference follows as
\ifCLASSOPTIONonecolumn
\begin{eqnarray}
L_{I}(s) & = & \mathbb{E}e^{-s\left(\sum_{x\in\Phi_{[1]}}h_{x}l(x)+\sum_{x\in\Phi_{[2]}}\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)\right)}\nonumber \\
 & = & \mathbb{E}\left(\prod_{x\in\Phi_{[1]}}e^{-sh_{x}l(x)}\prod_{x\in\Phi_{[2]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right)\nonumber \\
 & \overset{\left(a\right)}{=} & \mathbb{E}\left(\prod_{x\in\Phi_{[1]}}e^{-sh_{x}l(x)}\right)\mathbb{E}\left(\prod_{x\in\Phi_{[2]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right),\label{eq:Two}
\end{eqnarray}
\else
\begin{align}
L_{I}(s)
 & =  \mathbb{E}\left(\prod_{x\in\Phi_{\left[1\right]}}e^{-sh_{x}l(x)} \prod_{x\in\Phi_{\left[2\right]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right)\nonumber \\
 &  \overset{\left(a\right)}{=}  \mathbb{E}\left(\prod_{x\in\Phi_{\left[1\right]}}e^{-sh_{x}l(x)}\right)\cdot\nonumber\\
 & \qquad \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right),\label{eq:Two}
\end{align}
\fi
where (a) follows from the fact that $\Phi_{[1]}$ and
$\Phi_{\left[2\right]}$ are independent PPPs from the marking theorem
 \cite[Thm. 7.5]{Haenggi12book}. The first term in the
product of (\ref{eq:Two}) is the Laplace transform of the interference
of the PPP $\Phi_{[1]}$, given by \cite[page 103]{Haenggi12book}:
\begin{align*}
L_{I_{1}}(s) & =  \mathbb{E}\left(\prod_{x\in\Phi_{\left[1\right]}}e^{-sh_{x}l(x)}\right)\\
 & =  \exp(-\lambda p_{1}H(s,\alpha)).
\end{align*}

The second term in the product of (\ref{eq:Two}) can be written as
follows:\ifCLASSOPTIONonecolumn
\begin{align}
L_{I_{2}}(s) & =  \mathbb{E}\left(\prod_{x\in\Phi_{[2]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right)\nonumber \\
 & =  \mathbb{E}\left(\prod_{x\in\Phi_{[2]}}\frac{1}{1+sl(x)}\frac{1}{1+sl(m(x))}\right)\label{eq:I2}\\
 & \overset{\left(a\right)}{=}  \exp\left(-\lambda p_{2}\int_{\mathbb{R}^{2}}\left(1-\frac{1}{1+sl(x)}\frac{1}{1+sl(m(x))}\right)dx\right)\label{eq:I2-1}\\
 & =  \exp(-\lambda p_{2}F(s,\alpha,R)),\label{L2}
\end{align}
where (a) follows from the probability generating functional of the
PPP.
\else
\begin{align}
L_{I_{2}}(s) & =  \mathbb{E}\left(\prod_{x\in\Phi_{[2]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right)\nonumber\\
 & =  \mathbb{E}\left(\prod_{x\in\Phi_{[2]}}\frac{1}{1+sl(x)}\frac{1}{1+sl(m(x))}\right)\label{eq:I2}\\
 & \overset{\left(a\right)}{=}  \exp\left(-\lambda p_{2}\int_{\mathbb{R}^{2}}\left(1-v(x)\right)\right)\label{eq:I2-1}\\
 & =  \exp(-\lambda p_{2}F(s,\alpha,R)),\label{L2}
\end{align}
where (a) follows from the probability generating functional of the
PPP with $v(x)=\frac{1}{1+sl(x)}\frac{1}{1+sl(m(x))}.$
\fi
As a result, the success probability is
\begin{align}
p_{s}^{\rm HD} & =  L_{I_{1}}(\theta R^{\alpha})L_{I_{2}}(\theta R^{\alpha})\label{HDps}\\
 & =  \exp(-\lambda p_{1}H(\theta R^{\alpha},\alpha))\exp(-\lambda p_{2}F(\theta R^{\alpha},\alpha,R)),
\end{align}
which completes the proof of $p_{s}^{\rm HD}$.

Conditional on a FD link, there is an extra term in the interference, which is the residual self-interference scaled by the constant $K$. Hence, the interference for a FD link consists of three parts as follows:
\[
I=\sum_{x\in\Phi_{\left[1\right]}}h_{x}l(x)+\sum_{x\in\Phi_{\left[2\right]}}\left(h_{x}l(x)+h_{m(x)}l(m(x))\right) + \frac{\beta}{K}.
\]
The first two terms are the same as in the proof of $p_s^{\rm HD}$ while the third term is the residual self-interference. Hence, the Laplace transform of the interference follows as
\begin{eqnarray}
L_{I}(s) & = & L_{I_{1}}(s) L_{I_{2}}(s) e^{-\frac{s\beta}{K}}.\label{eq:Two2}
\end{eqnarray}

As a result, the conditional success probability $p_{s}^{\rm FD}$ is
\begin{align*}
p_{s}^{\rm FD} & =  L_{I_{1}}(\theta R^{\alpha})L_{I_{2}}(\theta R^{\alpha})e^{-\frac{\theta R^{\alpha}\beta}{K}}\\
 & = \kappa p_{s}^{\rm HD},
\end{align*}
where the last step is from (\ref{HDps}).
\end{IEEEproof}
Alternatively, $p_s^{\rm HD}$ can also be derived using the results for the  Gauss-Poisson process\cite{Guo14ISIT}.

The fact that the conditional success probability $p_s^{\rm HD}$ (and the Laplace transform of the interference)
is a product of two terms follows from the
independence of the point processes $\Phi_{\left[i\right]}$. The names of the functions $H$ and $F$ are chosen to reflect the fact that they represent the case of half- and full-duplex, respectively.

The residual self-interference for FD links simply adds an exponential factor to the success probability for HD links, which is similar to the effect of noise as in \cite[page 105]{Haenggi12book}. Theorem \ref{sucProb} also reveals the connection between two conditional success probability  $p_{s}^{\rm FD}$ and  $p_{s}^{\rm HD}$. As expected,  $p_{s}^{\rm FD}\leq p_{s}^{\rm HD}$, with equality for perfect self-interference cancellation.

The unconditional success probability can be easily obtained from the results in Theorem \ref{sucProb}.
\begin{cor}\label{sucProb_FD2}
In a wireless network described by the marked PPP $\hat{\Phi}$, the unconditional success
probability $p_{s}$ is given by
\begin{equation}
p_{s}=\left(p_1+\kappa p_2 \right) e^{-\lambda p_{1}H(\theta R^{\alpha},\alpha)}e^{-\lambda p_{2}F(\theta R^{\alpha},\alpha,R)}.\label{eq:ps-4}
\end{equation}
\end{cor}
\begin{IEEEproof}
Since a link is HD with probability $p_1$ and FD with probability $p_2$, the unconditional success probability from (\ref{eq:ps}) is the average
\begin{equation}
p_s = p_1 p_{s}^{\rm HD} + p_2 p_{s}^{\rm FD}.\label{ps0}
\end{equation}
Inserting the results from (\ref{eq:ps-2}) and (\ref{eq:ps-3}), we have (\ref{eq:ps-4}).
\end{IEEEproof}

The (un)conditional success probabilities are not in strict closed-form due to the integral form of $F(\theta R^{\alpha},\alpha,R)$. However,
tight simple bounds can be obtained.
\begin{thm}
The conditional success probability $p_{s}^{\rm HD}$ is lower and upper bounded by
\begin{equation}
\underline{p}_{s}=\exp(-\lambda(p_{1}+2p_{2})H(\theta R^{\alpha},\alpha))\label{lb}
\end{equation}
and
\begin{equation}
\overline{p}_{s}=\exp(-\lambda(p_{1}+p_{2}(1+\delta))H(\theta R^{\alpha},\alpha)).\label{ub}
\end{equation}
and, similarly, $p_{s}^{\rm FD}$ is bounded as
 \begin{equation} \kappa \underline{p}_{s} \leq p_{s}^{\rm FD} \leq \kappa \overline{p}_{s}.\label{b1}\end{equation}The unconditional success probability is lower and upper bounded as
 \begin{equation}
 (p_1 + \kappa p_2)\underline{p}_{s} \leq p_s \leq (p_1 + \kappa p_2)\overline{p}_{s}.\label{b2}
 \end{equation}
\end{thm}
\begin{IEEEproof}
Bounds only need to be established for the second term of the product in the conditional success
probability that contains the integral $F(\theta R^{\alpha},\alpha,R)$.

Lower Bound: From (\ref{eq:I2-1}),
\ifCLASSOPTIONonecolumn
\begin{eqnarray}
L_{I_{2}}(s) & = & \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right)\nonumber \\
 & \overset{\left(a\right)}{\geq} & \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-sh_{x}l(x)}\right)\mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-sh_{m(x)}l(m(x))}\right)\label{eq:lb2}\\
 & \overset{\left(b\right)}{=} & \exp(-2\lambda p_{2}H(s,\alpha)),\label{K1}
\end{eqnarray}
\else
\begin{align}
L_{I_{2}}(s) & =  \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-s\left(h_{x}l(x)+h_{m(x)}l(m(x))\right)}\right)\nonumber \\
 & \overset{\left(a\right)}{\geq}  \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-sh_{x}l(x)}\right)\mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-sh_{m(x)}l(m(x))}\right)\label{eq:lb2}\\
 & \overset{\left(b\right)}{=} \exp(-2\lambda p_{2}H(s,\alpha)),\label{K1}
\end{align}
\fi
where (a) follows from the FKG inequality \cite[Thm 10.13]{Haenggi12book} since
both $\prod_{x\in\Phi}e^{-sh_{x}l(x)}$ and \ifCLASSOPTIONonecolumn \\ \else \fi $\prod_{x\in\Phi}e^{-sh_{m(x)}l(m(x))}$
are decreasing random variables. In (\ref{eq:lb2}), the first term
is similar to the calculation of $L_{I_{1}}(s)$ with $\Phi_{\left[1\right]}$
replaced by $\Phi_{\left[2\right]}$ while in the second term, $m(\Phi_{\left[2\right]}) $ is a PPP with the same density as $\Phi_{\left[2\right]}$
due to the displacement theorem \cite[page 35]{Haenggi12book}. As a result, the two factors in (\ref{eq:lb2}) are equal, and
\begin{align*}
p_{s}^{\rm HD}
 & \geq  L_{I_{1}}(\theta R^{\alpha})\exp(-2\lambda p_{2}H(\theta R^{\alpha},\alpha))

 = \underline{p}_{s}.
\end{align*}

Upper Bound: From (\ref{eq:I2}),
\begin{align*}
L_{I_{2}}(s) & =  \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}\frac{1}{1+sl(x)}\frac{1}{1+sl(m(x))}\right)\\
 & \leq  \left\{ K_{1}(s,\alpha)K_{2}(s,\alpha)\right\} ^{\frac{1}{2}},
\end{align*}
which follows from the Cauchy-Schwarz inequality with $K_{1}(s,\alpha)=\mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}\frac{1}{\left(1+sl(x)\right)^{2}}\right)$
and $K_{2}(s,\alpha)=\mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}\frac{1}{\left(1+sl(m(x))\right)^{2}}\right)$. We have
\begin{align*}
K_{1}(s,\alpha

 & =  \exp\left(-2\pi\lambda p_{2}\int_{0}^{\infty}\left(1-\frac{1}{\left(1+sr^{-\alpha}\right)^{2}}\right)rdr\right)\\
 & =  \exp\left(-\pi\lambda p_{2}(1+\delta)\Gamma(1+\delta)\Gamma(1-\delta)s^{\delta}\right)\\
 & =  \exp(-\lambda p_{2}(1+\delta)H(s,\alpha)),
\end{align*}where $\Gamma(\cdot)$ is the gamma function.
$K_{2}(s,\alpha)=K_{1}(s,\alpha)$ because $m(\Phi_{\left[2\right]})$
is a PPP with the same density as $\Phi_{\left[2\right]}$. As
a result,
\begin{align}
L_{I_{2}}(s) & \leq  \left\{ K_{1}(s,\alpha)K_{2}(s,\alpha)\right\} ^{\frac{1}{2}}\\
 & =  \exp(-\lambda p_{2}(1+\delta)H(s,\alpha)).\label{K2}
\end{align}
Therefore,
\begin{align*}
p_{s}^{\rm HD}
 & \leq  e^{-\lambda p_{1}H(\theta R^{\alpha},\alpha)}e^{-\lambda p_{2}(1+\delta)H(\theta R^{\alpha},\alpha)}
 = \overline{p}_{s}.
\end{align*}
The lower and upper bounds of $p_{s}^{\rm FD}$ and $p_{s}$ simply follow from (\ref{eq:ps-3}) and (\ref{eq:ps-4}).
\end{IEEEproof}
The lower bound can be intuitively
understood as lower bounding the interference of the FD nodes (which
are formed by two {\em dependent} PPPs) by that of two {\em independent} PPPs with the same
density.

The upper bound turns out to be the same as the result obtained
by assuming $l(x)=l(m(x))$, $\forall x \in \Phi$, i.e., the distances between the receiver
at the origin and the interfering pair from the FD links are the same.
Indeed, assuming $l(x)=l(m(x))$, we have\begin{align*}
\tilde{L}_{I_{2}}(s) & =  \mathbb{E}\left(\prod_{x\in\Phi_{\left[2\right]}}e^{-s\left(h_{x}+h_{m(x)}\right)l(x)}\right)\\
 & =  \exp\left(-\pi\lambda p_{2}\mathbb{E}\left[\left(h_{x}+h_{m(x)}\right)^{\delta}\right]\Gamma\left(1-\delta\right)s^{\delta}\right)\\
 & =  \exp(-\pi\lambda p_{2}\Gamma(2+\delta)\Gamma(1-\delta)s^{\delta})\\
 & =  \exp(-\pi\lambda p_{2}\left(1+\delta\right)\Gamma(1+\delta)\Gamma(1-\delta)s^{\delta})\\
 & =  \exp(-\lambda p_{2}(1+\delta)H(s,\alpha))
\end{align*}
where $\mathbb{E}\left[\left(h_{x}+h_{y}\right)^{\delta}\right]=\Gamma(2+\delta)$
since $h_{x}+h_{y}$ has an Erlang distribution
and $\Gamma(2+\delta)=(1+\delta)\Gamma(1+\delta)$. Hence, the approximated
success probability assuming $l(x)=l(m(x))$ is $\tilde{p}_{s} = L_{I_{1}}(\theta R^{\alpha})\tilde{L}_{I_{2}}(\theta R^{\alpha})= \overline{p}_{s}$. This result is not surprising. The equality holds for the Cauchy-Schwarz
inequality if $\prod_{x\in\Phi_{[2]}}\frac{1}{(1+sl(x))^{2}}$
and $\prod_{x\in\Phi_{[2]}}\frac{1}{(1+sl(m(x)))^{2}}$
are linearly dependent. Obviously, $l(x)=l(m(x))$ satisfies this condition.
Therefore, we have $\tilde{p}_{s}=\overline{p}_{s}$
as expected.

The horizontal gap between two success probability curves (or SIR distributions) is often quite insensitive to the success probability where it is evaluated and the path loss models, as pointed out in \cite{Haenggi14, Guo15TC}. The horizontal gap is defined as
\begin{equation}
G(p) \triangleq \frac{p_{s_1}^{-1}(p)}{p_{s_2}^{-1}(p)}, \quad p\in (0,1),\label{hg}
\end{equation}
where $p_{s}^{-1}(p)$ is the inverse of the success probability and $p$ is the target success probability.
The sharpness of the upper and lower bounds of the success probabilities is established by the following corollary.
\begin{cor}
\label{cor:HGain} The horizontal gap between the upper and lower bound of the conditional success probability $p_s^{\rm HD}$ does not depend on the target success probability and is given by
\begin{equation}
G = \left(\frac{p_1+2p_2 }{p_1+p_2(1+\delta)}\right)^{{1}/{\delta}}.\label{HG}
\end{equation}
Furthermore, the horizontal gap between the lower and upper bound of the conditional success probability $p_s^{\rm FD}$ and the bounds of the unconditional success probability $p_s$ under perfect self-interference cancellation is also $G$.
 \end{cor}
\begin{IEEEproof}
The horizontal gap can be obtained by setting $\overline{p}_{s}(\overline{\theta})= \underline{p}_{s}(\underline{\theta})$ and calculating $G = {\overline{\theta}}/{\underline{\theta}}$. From (\ref{lb}) and (\ref{ub}), we have
\ifCLASSOPTIONonecolumn
\begin{equation}
\exp(-\lambda(p_{1}+2p_{2})H(\overline{\theta} R^{\alpha},\alpha))= \exp(-\lambda(p_{1}+p_{2}(1+\delta))H(\underline{\theta}R^{\alpha},\alpha)).
\end{equation}
\else
\begin{equation}
e^{-\lambda(p_{1}+2p_{2})H(\overline{\theta} R^{\alpha},\alpha)}= e^{-\lambda(p_{1}+p_{2}(1+\delta))H(\underline{\theta}R^{\alpha},\alpha)}.
\end{equation}
\fi
Solving the above equation for the ratio ${\overline{\theta}}/{\underline{\theta}}$, we obtain (\ref{HG}). When the self-interference cancellation is perfect, $\kappa = 1$. From (\ref{b1}) and (\ref{b2}), we obtain the same horizontal gap $G$ for $p_s^{\rm FD}$ and $p_s$.
\end{IEEEproof}
\begin{figure}[h]
\begin{centering}
\includegraphics[width=\figwidth]{Success_Prob_with_bounds5
\par\end{centering}
\caption{Comparison of unconditional success probability between the theoretical result from (\ref{eq:ps-4}) and its bounds
as a function of the SIR threshold $\theta$ in dB: $\alpha=4$, $\lambda=0.1$, $R=1$,
$p_{0}=0$, $p_{1}=p_{2}=0.5$, $\beta = 0$. The horizontal gap from (\ref{HG}) is $G=36/25$ or $1.58$ dB for these parameters.}
\label{Fig:1}
\end{figure}

 From the above corollary, it is apparent that the gap $G$ is independent of the SIR threshold $\theta$ and the target success probability; it only depends on $\delta$ and the transmit probabilities. Also, for $\alpha \downarrow 2$, $G \downarrow 1$. Figure \ref{Fig:1} plots the unconditional success probability from (\ref{eq:ps-4})
and its closed-form upper and lower bounds under perfect self-interference cancellation as a function of the SIR threshold in dB. To obtain the exact curve, the double integral $F(\theta R^{\alpha},\alpha,R)$ is numerically evaluated.
As seen, both bounds are tight with constant horizontal gap $G=1.58$ dB everywhere for $p_1=p_2 = 0.5$ and $\delta = 2/\alpha = 1/2$.

Furthermore, we can examine the relationship between the two key functions ${F(\theta R^{\alpha},\alpha,R)}$ and ${H(\theta R^{\alpha},\alpha)}$. The following corollary bounds their ratio:
\begin{cor}
\label{cor:F}The ratio ${F}/{H}$ is bounded as follows:
\begin{equation}
1+\delta \leq \frac{F(\theta R^{\alpha},\alpha,R)}{H(\theta R^{\alpha},\alpha)}\leq 2, \quad\forall\;\theta>0, R>0, \alpha>2.\label{F2}
\end{equation}
Moreover, the ratio is independent of the link distance $R$ and
\begin{equation}
\lim_{\theta \rightarrow \infty}\frac{F(\theta R^{\alpha},\alpha,R)}{H(\theta R^{\alpha},\alpha)} = {1+\delta}.\label{FH2}
\end{equation}
 \end{cor}
\begin{IEEEproof}
From the proof of the upper and lower bounds of the conditional success probability $p_s^{\rm HD}$, i.e., (\ref{K1}) and (\ref{K2}),
we have
\begin{equation}
\exp(-2\lambda p_{2}H(s,\alpha)) \leq L_{I_{2}}(s) \leq  \exp(-\lambda p_{2}(1+\delta)H(s,\alpha)),
\end{equation}
where $L_{I_{2}}(s)=\exp(-\lambda p_{2}F(s,\alpha,R))$ from (\ref{L2}).
By taking the logarithm on both sides of the above, we have
\begin{equation}
-2\lambda p_{2}H(s,\alpha)\leq -\lambda p_{2}F(s,\alpha,R)  \leq  -\lambda p_{2}(1+\delta)H(s,\alpha),
\end{equation}
which leads to (\ref{F2}).

For the independence on the link distance $R$, since $H(\theta R^{\alpha},\alpha)=\frac{\pi^{2}\delta {\theta}^{\delta}R^2}{\sin\left(\pi\delta\right)}$, we need to prove that $F(\theta R^{\alpha},\alpha,R)$ is also proportional to $R^2$. By the change of the variable $r_1 = {r}/{R}$, we can express (\ref{F}) as
\ifCLASSOPTIONonecolumn
\begin{align}
F(\theta R^{\alpha},\alpha,R)
&= R^2\int_{0}^{\infty}\left(2\pi-\frac{1}{1+\theta r_{1}^{-\alpha}}\int_{0}^{2\pi}\frac{d\varphi}{1+\theta \left(r_{1}^{2}+1+2r_1\cos\varphi\right)^{-\alpha/2}}\right) r_1dr_1,\label{F_R}
\end{align}
\else
\begin{align}
F(\theta R^{\alpha},\alpha,R)
&= R^2\int_{0}^{\infty}\left(2\pi-\frac{K(\theta, r_1, 1, \alpha)}{1+\theta r_{1}^{-\alpha}}\right) r_1dr_1,\label{F_R}
\end{align}
\fi
which completes the proof of independence of link distance $R$. For the limit, by the change of the variable $r_2 = r_1 \theta^{-\frac{1}{\alpha}}$, we have
\ifCLASSOPTIONonecolumn
\begin{align}
F(\theta R^{\alpha},\alpha,R)
&= \theta^{\delta}R^2\int_{0}^{\infty}\left(2\pi-\frac{1}{1+ r_{2}^{-\alpha}}\int_{0}^{2\pi}\frac{d\varphi}{1+ \left(r_{2}^{2}+\theta^{-\delta}+2r_2\theta^{-\delta/2}\cos\varphi\right)^{-\alpha/2}}\right) r_2dr_2.\label{F_R2}
\end{align}
\else
\begin{align}
F(\theta R^{\alpha},\alpha,R)
&= \theta^{\delta}R^2\int_{0}^{\infty}\left(2\pi-\frac{K(1, r_2, \theta^{-\delta/2}, \alpha)}{1+ r_{2}^{-\alpha}}\right) r_2dr_2.\label{F_R2}
\end{align}
\fi
Therefore,
\ifCLASSOPTIONonecolumn
\begin{align}
\lim_{\theta \rightarrow \infty}\frac{F(\theta R^{\alpha},\alpha,R)}{H(\theta R^{\alpha},\alpha)} &= \lim_{\theta \rightarrow \infty}\frac{\int_{0}^{\infty}\left(2\pi-\frac{1}{1+ r_{2}^{-\alpha}}\int_{0}^{2\pi}\frac{d\varphi}{1+ \left(r_{2}^{2}+\theta^{-\frac{2}{\alpha}}+2r_2\theta^{-\frac{1}{\alpha}}\cos\varphi\right)^{-\alpha/2}}\right) r_2dr_2}{\frac{\pi^{2}\delta }{\sin(\pi\delta)}}\\
&= \frac{\int_{0}^{\infty}\left(2\pi-\frac{2\pi}{\left(1+ r_{2}^{-\alpha}\right)^2}\right) r_2dr_2}{\frac{\pi^{2}\delta }{\sin(\pi\delta)}}\\
&=1+\delta.
\end{align}
\else
\begin{align}
\lim_{\theta \rightarrow \infty}\frac{F(\theta R^{\alpha},\alpha,R)}{H(\theta R^{\alpha},\alpha)} &= \lim_{\theta \rightarrow \infty}\frac{\int_{0}^{\infty}\left(2\pi-\frac{K(1, r_2, \theta^{-\delta/2}, \alpha)}{1+ r_{2}^{-\alpha}}\right) r_2dr_2}{\frac{\pi^{2}\delta }{\sin(\pi\delta)}}\\
&= \frac{\int_{0}^{\infty}\left(2\pi-\frac{2\pi}{\left(1+ r_{2}^{-\alpha}\right)^2}\right) r_2dr_2}{\frac{\pi^{2}\delta }{\sin(\pi\delta)}}\\
&=1+\delta.
\end{align}
\fi

\end{IEEEproof}
This corollary is useful in calculating the SIR loss, the maximal throughput, and
their bounds in the following. Also, we can conclude that the upper bound of the success probability is asymptotically exact as $\theta\to \infty$. It is also illustrated by \figref{Fig:1}.

\subsection{SIR loss due to FD operation}
In this subsection, we investigate the SIR loss caused by the FD operation in wireless networks described by $\hat{\Phi}$. Consider two extreme cases: one is the case where all concurrently
transmitting nodes work in HD mode, i.e., $p_1=1$, and the other
is where all concurrently transmitting nodes work in FD mode, i.e., $p_2 = 1$. The success probabilities of HD-only and FD-only networks follows from (\ref{eq:ps-2}) as
\begin{equation}
p_{s, \;p_1=1} = \exp(-\lambda H(\theta R^{\alpha},\alpha))\label{ps_HD}
\end{equation}

and \begin{equation}
p_{s, \;p_2=1} = \kappa\exp(-\lambda F(\theta R^{\alpha},\alpha,R)).\label{ps_FD}\end{equation}

\begin{figure}[h]
\begin{centering}
\includegraphics[width=\figwidth]{Success_Prob_FD_vs_HD
\par\end{centering}
\caption{Comparison of success probabilities of FD-only networks, its bounds, and HD-only networks
as a function of the SIR threshold $\theta$ in dB under perfect self-interference cancellation: $\alpha=4$, $\lambda=0.1$, $R=1$,
$p_{0}=0$, $\beta = 0$.}
\label{Fig:1-1}
\end{figure}

Figure \ref{Fig:1-1} plots the success probability of FD-only networks
and its upper and lower bounds as well as success probability of HD-only networks as a function of the SIR threshold in dB. Clearly, FD transmission in a FD-only wireless network leads to a SIR loss compared to its counterpart HD-only wireless network in the success probability, which is the ccdf of SIR. The SIR loss can be defined as the horizontal gap between two SIR distributions as follows from (\ref{hg}):
\begin{definition} The SIR loss between FD-only and HD-only networks is defined as
\begin{equation}
G(p) \triangleq \frac{\theta_{\rm HD}(p)}{\theta_{\rm FD}(p)}=\frac{p_{s, \;p_1=1}^{-1}(p)}{p_{s, \;p_2=1}^{-1}(p)},
\end{equation}
where $p$ is the target success probability and $p_{s}^{-1}$ is the inverse of the ccdf of the SIR. $\theta_{\rm HD}(p)$ is the SIR threshold when the target success probability is $p$, i.e., $\theta_{\rm HD}(p)=p_{s, \;p_1=1}^{-1}(p)$. Similarly, $\theta_{\rm FD}(p)=p_{s, \;p_2=1}^{-1}(p)$.
\end{definition}

The following theorem bounds this SIR loss.
\begin{thm} \label{SIR loss}The SIR loss $G(p)$ between the FD-only network and HD-only network is bounded as
\begin{equation}
(1+\delta + \gamma(\theta_{\rm FD}(p)))^{{1}/{\delta}} \leq G(p) \leq (2 + \gamma(\theta_{\rm FD}(p)))^{{1}/{\delta}},\label{SIR_loss}
\end{equation}
where $\gamma(x)= x^{1-\delta}\frac{R^{\alpha-2}\beta \sin(\pi \delta)}{\lambda \pi^2 \delta K}$.
\end{thm}
\begin{IEEEproof}The proof is quite straightforward by equating
\[p_{s, \;p_1=1} (\theta_{\rm HD}) = p_{s, \;p_2=1}(\theta_{\rm FD}),\]
and solving for the ratio $\frac{\theta_{\rm HD}}{\theta_{\rm FD}}$.
From (\ref{ps_HD}) and (\ref{ps_FD}), we obtain
\[\lambda H(\theta_{\rm HD} R^{\alpha},\alpha) =  \lambda F(\theta_{\rm FD} R^{\alpha},\alpha) + \theta_{\rm FD} R^{\alpha} \beta/K,\]
and from (\ref{F2}) in Corollary \ref{cor:F}, we have
\ifCLASSOPTIONonecolumn
\begin{equation}
(1+\delta) \lambda H(\theta_{\rm FD} R^{\alpha},\alpha) \leq \lambda H(\theta_{\rm HD} R^{\alpha},\alpha) - \theta_{\rm FD} R^{\alpha} \beta/K \leq 2\lambda H(\theta_{\rm FD} R^{\alpha},\alpha).\label{F_in}
\end{equation}
\else
\begin{align}
(1+\delta) \lambda H(\theta_{\rm FD} R^{\alpha},\alpha) \leq \lambda H(\theta_{\rm HD} R^{\alpha},\alpha) - \theta_{\rm FD} R^{\alpha} \beta/K \nonumber\\
\leq 2\lambda H(\theta_{\rm FD} R^{\alpha},\alpha).\label{F_in}
\end{align}
\fi
By inserting $H(s,\alpha)=\frac{\pi^{2}\delta s^{\delta}}{\sin\left(\pi\delta\right)}$ into (\ref{F_in}) and after elementary manipulations, (\ref{F_in}) leads to
\begin{equation}
\left(1+\delta + \gamma(\theta_{\rm FD})\right)^{{1}/{\delta}} \leq \frac{\theta_{\rm HD}}{\theta_{\rm FD}} \leq (2 + \gamma(\theta_{\rm FD}))^{{1}/{\delta}}.\label{theta_in}
\end{equation}
Hence, we have (\ref{SIR_loss}).
\end{IEEEproof}
Apparently, the bounds of the SIR loss depend on the SIR threshold $\theta_{\rm FD}$ due to the imperfect self-interference cancellation since $\gamma(\theta_{\rm FD})>0$ for imperfect self-interference cancellation. It means that imperfect self-interference cancellation introduces an extra SIR loss compared to perfect self-interference cancellation and that the SIR loss gets larger as the residual self-interference increases, as shown in Figure \ref{Fig:1-3}.
\begin{figure}[h]
\begin{centering}
\includegraphics[width=\figwidth]{Success_Prob_at_various_beta3
\par\end{centering}
\caption{Success probabilities of FD-only networks ($p_{s, \;p_2=1}(\theta)$) and HD-only networks ($p_{s, \;p_1=1}(\theta)$) at two SIPRs $\beta = 10^{-4}$ and $\beta=0$. The other parameters are $\alpha=4$, $\lambda=0.1$, $R=1$, $K=-34$ dB (Assume that $G_{\rm tx}=G_{\rm rx}=2$, i.e., $3$ dBi, and $f_c=2.4$ GHz).}
\label{Fig:1-3}
\end{figure}
\begin{cor}\label{cor:G}The SIR loss $G(p)$ between the FD-only network and HD-only network under perfect self-interference cancellation ($\beta =0$) is bounded as
\begin{equation}
(1+\delta)^{{1}/{\delta}} \leq G(p) \leq {2}^{{1}/{\delta}}.\label{SIR_loss2}
\end{equation}
\end{cor}
\begin{IEEEproof}
Follows from Theorem \ref{SIR loss} since $\gamma(\theta_{\rm FD})=0$ for $\beta=0$.
\end{IEEEproof}
For perfect self-interference cancellation ($\beta=0$), the bounds of the SIR loss only depend on the path loss exponent $\alpha=2/\delta$, i.e., they are independent of the SIR threshold, the target success probability $p$, and the link distance $R$.
Corollary \ref{cor:G} can also be proven in the following way. Under perfect self-interference cancellation, from (\ref{ps_HD}), we have \[p_{s, \;p_1=1}(\theta_{\rm HD})=e^{-c\theta_{\rm HD}^{\delta}},\]
where $c = \frac{\lambda\pi^{2}\delta R^2}{\sin(\pi\delta)}$.
From (\ref{ps_FD}) and (\ref{F2}), we have
\begin{align*}
\underline{p}_{s, \;p_2=1}(\theta_{\rm FD})&=e^{-2c\theta_{\rm FD}^{\delta}},\\
\overline{p}_{s, \;p_2=1}(\theta_{\rm FD})&=e^{-(1+\delta)c\theta_{\rm FD}^{\delta}}.
\end{align*}
By solving
\begin{align*}
\underline{p}_{s, \;p_2=1}(\theta_{\rm FD})&=p_{s, \;p_1=1}(\theta_{\rm FD} \overline{G}),\\
\overline{p}_{s, \;p_2=1}(\theta_{\rm FD})&=p_{s, \;p_1=1}(\theta_{\rm FD} \underline{G}),
\end{align*}
we obtain the upper bound $\overline{G}=2^{\frac{1}{\delta}}$ and the lower bound $\underline{G}=(1+\delta)^{\frac{1}{\delta}}$.

Therefore, the upper (lower) bound of the SIR loss are actually the constant horizontal gap between the upper (lower) bound of the success probability of the FD-only network and that of the HD-only network. Under perfect self-interference cancellation, the upper bound of the success probability of the FD-only network is just the success probability curve of the HD-only network left shifted by  $\underline{G}^{\rm dB}=\frac{10\log_{10}{(1+\delta)}}{\delta}$ dB, whereas the lower bound is that left shifted by $\overline{G}^{\rm dB}=\frac{10\log_{10}{2}}{\delta}$ dB. For $\alpha=4$, the upper bound in Figure \ref{Fig:1-1} is equivalent to the HD curve left-shifted by $3.5$ dB while the lower bound equivalent to that left-shifted by $6.0$ dB.

To summarize, FD-only operation can result in up to $6.0$ dB SIR loss compared to HD-only operation even under perfect self-interference cancellation. Remarkably, this result only depends on the path loss exponent. The above analysis accurately quantifies the SIR loss caused by the extra interference introduced by the FD transmissions. Imperfect self-interference cancellation further adds to the SIR loss, especially when the SIR threshold is high, as shown in \figref{Fig:1-3}.

\section{Throughput Analysis\label{sec:Throughput-Performance-Analysis}}
\subsection{Problem statement}
The purpose of FD transmission in a network is to increase
the network throughput. While FD increases the throughput of an isolated link, it also causes additional interference to the other links. As analyzed in the previous section, FD transmission leads to SIR loss. There is a tradeoff between the link throughput and interference when the nodes in the networks decide to choose FD or HD. Given a network
that consists of nodes of FD capability and HD capability,
how should a node choose between FD and HD operation in order to maximize the network-wide throughput as the network configuration varies? It is important to determine under what condition one should choose FD.

First, we need to define the throughput. In a
random wireless network described by $\hat{\Phi}$, we can consider the throughput of
the \textit{typical link} as mentioned in the network model.
It has probability $p_{1}$ to be in HD mode and
$p_{2}$ to be in FD mode. Therefore, its throughput can be defined as follows:
\begin{defn}\label{linkT}
For a wireless network described by $\hat{\Phi}$, the throughput is defined as
\begin{equation}
T\triangleq \lambda \left(p_{1}p_{s}^{\rm HD}\log(1+ \theta)+2p_{2}p_{s}^{\rm FD}\log(1+ \theta)\right),\label{eq:T}
\end{equation}
assuming that a spectral efficiency of $\log(1+\theta)$ is achievable for a SIR threshold $\theta$.
\end{defn}

Let $\lambda_{1} \triangleq \lambda p_{1}$ and $\lambda_{2} \triangleq \lambda p_{2}$ be the densities of HD links and FD links. $\lambda_{1}$ and $\lambda_{2}$ can be tuned by changing the transmit probabilities of HD and FD modes $p_1$ and $p_2$ given a fixed node density $\lambda$ or vice versa. By doing so, we can optimize the throughput over the densities of HD and FD links ($\lambda_{1}$ and $\lambda_{2}$) instead of just the transmit probabilities $p_1$ and $p_2$ and reduce the variables by one as well. Hence, (\ref{eq:T}) can be rewritten as
\begin{equation}
T= \lambda_{1}p_{s}^{\rm HD}\log(1+ \theta)+2\lambda_{2}p_{s}^{\rm FD}\log(1+ \theta).\label{eq:T2}
\end{equation}

Given the definition of throughput, there are two extreme cases that
are particularly relevant: HD-only networks and FD-only networks, as mentioned earlier. Their throughputs are given as

\begin{equation}
T^{\mbox{\scriptsize{\rm HD}}}=\lambda_{1}\log(1+ \theta)\exp(-\lambda_{1}H(\theta_{\rm HD} R^{\alpha},\alpha))\label{HD FD}
\end{equation}
and

\begin{equation}
T^{\mbox{\scriptsize{\rm FD}}}=2\lambda_{2}\kappa\log(1+ \theta)\exp(-\lambda_{2}F(\theta R^{\alpha},\alpha)).\label{HD FD1}
\end{equation}
With the above setup, the goal is to optimize the throughput over the densities $\lambda_1$ and $\lambda_2$:
\begin{equation}
T_{\max} = \max_{\lambda_1,\lambda_2}T(\lambda_{1},\lambda_{2}).
\end{equation}
It is also interesting to find the relationship between the maxima
of $T^{\mbox{\scriptsize{\rm HD}}}$, $T^{\mbox{\scriptsize{\rm FD}}}$ and $T$, denoted as $T_{\max}^{\mbox{\scriptsize{\rm HD}}}$,
$T_{\max}^{\mbox{\scriptsize{\rm FD}}}$ and $T_{\max}$.

\subsection{Throughput optimization}
Inserting $p_{s}^{\rm HD}$ and $p_{s}^{\rm FD}$ from (\ref{eq:ps-2}) and (\ref{eq:ps-3}) into (\ref{eq:T}), we have
\begin{equation}
T(\lambda_{1},\lambda_{2})= \left(\lambda_{1}+2\lambda_{2}\kappa\right)\exp(-\lambda_{1}H)\exp(-\lambda_{2}F)\log(1+ \theta).\label{eq:T1}
\end{equation}

From now on, we will
use $H$ to denote $H(\theta R^{\alpha},\alpha)$ and $F$
to denote $F(\theta R^{\alpha},\alpha,R)$ for simplicity. $T_{\max}^{\mbox{\scriptsize{\rm HD}}}$ and $T_{\max}^{\mbox{\scriptsize{\rm FD}}}$
can be easily obtained by the following lemma.
\begin{lem}
\label{lem:6}For a HD-only network, described
by $\hat{\Phi}$ with $p_{1}=1$, $T_{\max}^{\mbox{\scriptsize{\rm HD}}}$ is given
by
\begin{equation}
T_{\max}^{\mbox{\scriptsize{\rm HD}}}=
T^{\mbox{\scriptsize{\rm HD}}}\!\left(\frac{1}{H}\right)=\frac{1}{eH}\log(1+\theta),\label{eq:hdmax}
\end{equation}
with optimal density of HD links
\begin{equation}
\lambda_{1}^{{\scriptsize{\rm opt}}}=\frac{1}{H}.\label{eq:p1opt}
\end{equation}
 For a FD-only network, described by
$\hat{\Phi}$ with $p_{2}=1$, $T_{\max}^{\mbox{\scriptsize{\rm FD}}}$ is given by
\begin{equation}
T_{\max}^{\mbox{\scriptsize{\rm FD}}}=
T^{\mbox{\scriptsize{\rm FD}}}\!\left(\frac{1}{F}\right)=\frac{2}{e\kappa F}\log(1+\theta),\label{eq:fdmax}
\end{equation}
with optimal density of FD links
\begin{equation}
\lambda_{2}^{{\scriptsize{\rm opt}}}=\frac{1}{F}.\label{eq:p2opt}
\end{equation}
\end{lem}
\begin{IEEEproof}
The proof is straightforward by taking the derivatives of $T^{\mbox{\scriptsize{\rm HD}}}$
and $T^{\mbox{\scriptsize{\rm FD}}}$ with respect to $\lambda_{1}$ and $\lambda_{2}$, respectively.
\end{IEEEproof}
A similar result for HD-only networks has been presented in \cite[Proposition 4]{Haenggi09twc}. In fact, ${1}/{H}$ and ${1}/{F}$ are the spatial efficiency~\cite{Haenggi09twc} of HD-only networks and FD-only networks, respectively. The spatial efficiency quantifies how efficiently a wireless network uses space as a resource. A large spatial efficiency indicates high spatial reuse.

In the following theorem, we show that $T_{\max}$ is achieved by
setting all concurrently transmitting nodes to be in FD mode or in HD mode or in a mixed FD/HD mode, depending on one simple condition.
\begin{thm} Let
\begin{equation} L \triangleq\{(\lambda_1, \lambda_2) \in (\mathbb{R}^+)^2\colon \lambda_1 + 2\kappa\lambda_2 = H^{-1}\}.\label{condition}\end{equation}
For a wireless network described by $\hat{\Phi}$, the maximal throughput
is given by
\begin{equation}
T_{\max} = \begin{cases}
T_{\max}^{\mbox{\scriptsize{\rm FD}}} & \mbox{if } F < 2\kappa H\\
T_{\max}^{\mbox{\scriptsize{\rm HD}}}  & \mbox{if } F > 2\kappa H\\
T_{\max}^{\mbox{\scriptsize{\rm HD}}}= T_{\max}^{\mbox{\scriptsize{\rm FD}}}  & \mbox{if } F = 2\kappa H
\end{cases}\label{eq:equ}
\end{equation}
with the optimal densities of HD and FD links
\begin{equation}
\left(\lambda_{1}^{{\scriptsize{\rm opt}}},\lambda_{2}^{{\scriptsize{\rm opt}}}\right)\begin{cases}
=\left(0, \frac{1}{F}\right) & \mbox{if } F < 2\kappa H\\
=\left(\frac{1}{H}, 0\right) & \mbox{if } F > 2\kappa H\\
\in L & \mbox{if } F = 2\kappa H. \end{cases}
\end{equation}
\end{thm}
\begin{IEEEproof}
Taking the derivative of $T$ w.r.t. $\lambda_{1}$ and $\lambda_{2}$ leads to
\begin{equation}
\frac{\partial T}{\partial \lambda_{1}}=\exp(-\lambda_{1}H-\lambda_{2}F)\log(1+\theta)[1- H(2\kappa \lambda_{2}+\lambda_{1})],\label{eq:p1}
\end{equation}
\begin{equation}
\frac{\partial T}{\partial \lambda_{2}}=\exp(-\lambda_{1}H-\lambda_{2}F)\log(1+\theta)[2\kappa-F(2\kappa\lambda_{2}+\lambda_{1})].\label{eq:p2}
\end{equation}
Setting $\frac{\partial T}{\partial \lambda_{1}}=0$ and $\frac{\partial T}{\partial \lambda_{2}}=0$, we have
\begin{equation}
\lambda_{1}+2\kappa\lambda_{2} = \frac{1}{H}\label{condition1}
\end{equation}
\begin{equation}
\lambda_{1}+2\kappa\lambda_{2} = \frac{2\kappa}{F}.\label{condition2}
\end{equation}

\leavevmode
\begin{enumerate} \item $F= 2\kappa H$: Both partial derivatives are zero as long as  $(\lambda_1, \lambda_2)\in L$. $({1}/{H}, 0)$ and $(0, {2\kappa}/{F})$ both lie in $L$. Hence, $T_{\max} =  T({1}/{H}, 0) = T_{\max}^{\mbox{\scriptsize{HD}}}$ and also $T_{\max} = T(0, {2\kappa}/{F}) = T_{\max}^{\mbox{\scriptsize{FD}}}$. This case is the break-even point where FD and HD have the same throughput. That means in a wireless network, the typical link has the same throughput no matter if it is in FD or HD mode.

\item $F > 2\kappa H$: Under this condition, let $\frac{\partial T}{\partial \lambda_{1}}=0$ and we have (\ref{condition1}). Moreover, $\frac{\partial T}{\partial \lambda_{2}}<0.$ Therefore, the maximal $T$ is achieved at $\left(\lambda_{1},\lambda_{2}\right)=\left({1}/{H}, 0\right)$ from (\ref{condition1}) and $\frac{\partial T}{\partial \lambda_{2}}<0$. Note that $T({1}/{H}, 0)=T^{\mbox{\scriptsize{HD}}}({1}/{H})=T^{\mbox{\scriptsize{HD}}}_{\max}$. On the other hand, letting  $\frac{\partial T}{\partial \lambda_{2}}=0$, we have (\ref{condition2}) and $\frac{\partial T}{\partial \lambda_{1}}>0,$ which leads to that the maximal $T$ is achieved at $\left(\lambda_{1},\lambda_{2}\right)=({2\kappa}/{F}, 0).$ Since $T({2\kappa}/{F}, 0)<T({1}/{H}, 0)$. We conclude that $T_{\max} = T^{\mbox{\scriptsize{HD}}}$ in this case.
\item $F < 2\kappa H$: By similar reasoning as in the second case, we can conclude that $T_{\max} = T^{\mbox{\scriptsize{FD}}}$ in this case.
\end{enumerate}\end{IEEEproof
Under perfect self-interference cancellation ($\beta = 0$), $F < 2\kappa G$, and we always have
\[
T_{\max}=T_{\max}^{\mbox{\scriptsize{FD}}},
\] which means $T_{\max}$ is always achieved by
setting all transmitting nodes to work in FD mode, despite the extra interference
caused by the FD nodes. This conclusion holds for all network
configurations $(\lambda, \theta, R, \alpha)$.

On the other hand, the following corollary quantifies how the imperfect self-interference cancellation affects the throughput in a wireless network with full-duplex radios.
\begin{cor}\label{beta_c}
Given an SIR threshold $\theta$, path loss exponent $\alpha$, and link distance $R$, there exists a critical SIPR value $\beta_{c}$ in the wireless network described by $\hat{\Phi}$: when $\beta < \beta_{c}$, FD is preferable in terms of throughput while HD has better throughput when $\beta > \beta_{c}$, where
\begin{equation}
\beta_{c} = \frac{K\log({2H}/{F})}{\theta R^{\alpha}}.\label{beta}
\end{equation}
\end{cor}
\begin{IEEEproof}
$\beta_{c}$ can be obtained by solving $F=2\kappa H = 2e^{-\theta R^{\alpha} \beta/K}H$.
By Corollary \ref{cor:F}, the ratio of $F/H$ does not depend on $R$ and hence $\beta_{c}$ scales as $R^{-\alpha}$.
\end{IEEEproof}

\begin{figure}[h]
\begin{centering}
\includegraphics[width=\figwidth]{beta_th_Vs_R
\par\end{centering}
\caption{The link distance vs the critical SIPR $\beta_{c}$ from (\ref{beta}) with $K=-34$ dB for $G_{\rm tx}=G_{\rm rx}=2$, i.e., $3$ dBi, and $f_c=2.4$ GHz, which corresponds to the carrier frequency of a WiFi signal. Below the curves, FD provides a higher throughput, while above the curves, HD does.}
\label{Fig:2}
\end{figure}
Figure \ref{Fig:2} plots the relationship between the link distance and the self-interference cancellation threshold $\beta_{c}$. It provides very valuable insight into the system design. The curves are linear in this log-log plot with slope $-1/\alpha$. The region under the lines is the region where FD transmission achieves a higher network throughput. For example, assume that the self-interference cancellation is limited to $80$ dB due to hardware imperfection. In this case, FD transmission is preferable only if the link distance is smaller than $10$ when $\theta=0$ dB and $\alpha = 4$ under the wireless network model used in this paper. To achieve a link distance of up to $100$, the self-interference cancellation needs to be at least $100$ dB ($\alpha = 3$) and $120$ dB ($\alpha = 4$) when $\theta =0$ dB. When the link distance is greater than $100$ with self-interference cancellation no greater than $120$ dB, it is better to use HD. So the amount of self-interference cancellation determines the maximal transmission range for which FD has better throughput than HD.

\subsection{Comparison of FD with HD}
Since the mixed FD/HD network achieves the maximal throughput in
the extreme case of a FD-only or HD-only network, we can simply focus on FD-only
and HD-only networks and compare their maximal throughputs
from the results in Lemma \ref{lem:6}.
The throughput gain of a FD network over a HD network is of
great interest. It is defined as follows.
\begin{defn}
The throughput gain (TG) is defined as the ratio between the maximal
throughput of FD-only networks and HD-only networks given the same network parameters
$\left(\theta,R,\alpha\right)$:
\[
{\rm TG}\triangleq\frac{T_{\max}^{\mbox{\scriptsize{FD}}}}{T_{\max}^{\mbox{\scriptsize{HD}}}}.
\]
\end{defn}
The following corollary quantifies and bounds $\rm{TG}$ in
terms of $F$ and $H$. Note that $F$ and $H$ are constant given
$\left(\theta,R,\alpha\right)$.
\begin{cor}
The throughput gain is given by
\begin{equation}
{\rm TG}=\frac{2\kappa H}{F}\label{eq:TG}
\end{equation} and bounded as
\begin{equation}
\kappa  < {\rm TG} < \frac{2\kappa}{1+\delta}.
\label{eq:TG_ub}
\end{equation}
Moreover, for any $\beta \geq 0$,
\begin{equation}{\rm TG}(\theta) \sim \frac{2\kappa}{1+\delta}=\frac{2}{1+\delta}\exp(-\theta R^\alpha\beta/K), \quad\theta\to\infty.\label{TGinf}\end{equation}
\end{cor}
\begin{IEEEproof}
From (\ref{eq:hdmax}) and (\ref{eq:fdmax}), we have (\ref{eq:TG}). The upper and lower bounds are easily obtained from Corollary \ref{cor:F}.

For $\beta>0$,
\begin{equation}\lim_{\theta \rightarrow \infty} \kappa = \lim_{\theta \rightarrow \infty} e^{-\frac{\theta R^{\alpha} \beta}{K}}=0.\end{equation}
Therefore, both the lower bound and upper bound of  ${\rm TG}$ converge to $0$ as $\theta$ goes to $\infty$. As a result, \begin{equation}\lim_{\theta \rightarrow \infty} {\rm TG}(\theta)=0.\label{TGinf1}\end{equation} $\beta=0$ implies $\kappa = 1$, which leads to \begin{equation}\lim_{\theta \rightarrow \infty} {\rm TG}(\theta)=\lim_{\theta \rightarrow \infty} \frac{2H}{F}=\frac{2}{1+\delta}\label{TGinf2}\end{equation}
 from (\ref{FH2}). Combining (\ref{TGinf1}) and (\ref{TGinf2}), we obtain (\ref{TGinf}).
\end{IEEEproof}

(\ref{TGinf}) indicates that the throughput gain converges to its upper bound as $\theta$ goes to infinity.
\ifCLASSOPTIONonecolumn
\begin{figure}[h]
\begin{centering}
\subfigure[$50$ dB self-interference cancellation: $\beta=10^{-5}$]{\includegraphics[width=7cm]{TGvsSIR_at_various_beta}\label{beta40}
}
\subfigure[$70$ dB self-interference cancellation: $\beta=10^{-7}$]{\includegraphics[width=7cm]{TGvsSIR_at_various_beta2}\label{beta20}
}
\par\end{centering}
\begin{centering}
\subfigure[Perfect self-interference cancellation: $\beta=0$]{\includegraphics[width=7cm]{TGvsSIR_at_various_beta3}\label{beta0}

\par\end{centering}
\caption{Throughput gain as a function of the SIR threshold $\theta$ and its bounds at different SIPRs for
$\alpha=4$, $R=1$.}
\label{Fig:4}
\end{figure}
\else
\begin{figure}[h]
\begin{centering}
\subfigure[$50$ dB self-interference cancellation: $\beta=10^{-5}$]{\includegraphics[width=7cm]{TGvsSIR_at_various_beta}\label{beta40}
}
\par\end{centering}
\begin{centering}
\subfigure[$70$ dB self-interference cancellation: $\beta=10^{-7}$]{\includegraphics[width=7cm]{TGvsSIR_at_various_beta2}\label{beta20}
}
\par\end{centering}
\begin{centering}
\subfigure[Perfect self-interference cancellation: $\beta=0$]{\includegraphics[width=7cm]{TGvsSIR_at_various_beta3}\label{beta0}

\par\end{centering}
\caption{Throughput gain as a function of the SIR threshold $\theta$ and its bounds at different SIPRs for
$\alpha=4$, $R=1$.}
\label{Fig:4}
\end{figure}
\fi
\figref{Fig:4} illustrates the throughput gain as a function
of the SIR threshold together with its upper and lower bounds given in (\ref{eq:TG_ub}). As seen, the throughput gain
is always lower than $\frac{2}{1+\delta}$ since the upper bound is smaller than that. For perfect self-interference cancellation, the throughput gain increases as the SIR threshold gets larger as shown in \figref{beta0}. The throughput gain decreases in the high SIR regime due to the imperfect self-interference cancellation as shown in \figref{beta40} and \figref{beta20}. When the self-interference cancellation is not sufficient, the throughput gain is less than $1$, which means that the HD-only network has a higher throughput, i.e., the value in the curve after $\theta>10$ dB in \figref{beta40} is less than $1$. These figures illustrate the throughput gain under different conditions, especially the impact of the imperfect self-interference cancellation on the throughput.

\section{Conclusion\label{sec:Conclusion}}

In this paper, we analyzed the throughput of wireless networks with
FD radios using tools from stochastic geometry. Given a wireless network of radios
with both FD and HD capabilities, we showed that FD transmission is always preferable
compared to HD transmission in terms of throughput when the self-interference cancellation is perfect. It turns out that the throughput of HD transmission cannot be doubled and the actual gain is $\frac{2\alpha}{\alpha+2}$ for an ALOHA protocol, where $\alpha$ is the path loss exponent. Under imperfect self-interference cancellation, the network has a break-even point where FD and HD have the same throughput. The break-even point depends on the amount of self-interference cancellation and the link distance. Given a fixed SIR threshold and path loss exponent, the necessary amount of self-interference cancellation in dB is logarithmically proportional to the link distance. It means that the residual self-interference determines the maximal link distance within which FD is beneficial compared to HD. It provides great insights into the network design with FD radios. We also analyzed and quantified the effects of imperfect self-interference cancellation on the network throughput and SIR loss. The SIR loss of a FD-only network over a HD-only network is quantified within tight bounds. The horizontal gap is utilized to determine the SIR loss. Moreover, the throughput gain of FD over HD is presented under imperfect self-interference cancellation.

In our network model, we consider the interference-limited case where the thermal noise is ignored. However, this is not a restriction as the throughput gain is independent of the thermal noise since the thermal noise adds the same exponential factor ($\exp({-\theta R^{\alpha} W})$ \cite[Page 105]{Haenggi12book}, where $W$ is the thermal noise power) to the throughput expressions of both HD and FD networks and they cancel each other in the throughput gain.

In general, FD is a
very powerful technique that can be adapted for the next-generation
wireless networks. The throughput gain may be larger if more advanced
MAC protocols other than ALOHA are used or the interference management
can be used for the pairwise interferers in the FD links. A FD-friendly MAC scheme should let the node decide to use FD or HD based on its surrounding interference in order to maximize the overall network throughput. There is a strong need for a MAC protocol tailored for a wireless network of radios with both FD and HD capacities and an intelligent and adaptive scheme to switch between FD and HD based on different network configurations.

\section*{Acknowledgment}
 This work has been partially supported by the U.S. NSF (grants ECCS-1231806, CNS 1016742 and CCF 1216407).

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