Fusion Basics Cheatsheet

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Matter-Energy Transformation Matter and energy are closely related. Mathematical expression for relationship in a system: $E_{in} + M_{in} \rightarrow E_{out} + M_{out}$ Matter-Energy Accounting Consider a reaction: $a + b \rightarrow d + e$ More complete statement: $_{Z_a}^{A_a}X_a + _{Z_b}^{A_b}X_b \rightarrow _{Z_d}^{A_d}X_d + _{Z_e}^{A_e}X_e$ Conservation of protons and neutrons: $A_a + A_b = A_d + A_e$ $Z_a + Z_b = Z_d + Z_e$ Total energy: $E^* = \sum_j (E_{k,j} + E_{r,j}) = \sum_j (E_{k,j} + m_j c^2)$ Conservation of energy theorem: $E^*_{before} = E^*_{after}$ Energy balance: $(E_{k,a} + m_a c^2) + (E_{k,b} + m_b c^2) = (E_{k,d} + m_d c^2) + (E_{k,e} + m_e c^2)$ Rearranging terms for energy change: $(E_{k,d} + E_{k,e}) - (E_{k,a} + E_{k,b}) = [(m_a + m_b) - (m_d + m_e)] c^2$ Q-value and Mass Defect Definition of Q-value: $Q_{ab} = [(m_a + m_b) - (m_d + m_e)] c^2$ Simpler version: $Q_{ab} = (-\Delta m)_{ab} c^2$ $\Delta m$ is the mass decrement/mass defect: $\Delta m = m_{after} - m_{before}$ Component Energies If reacting particles possess negligible energies compared to Q-value: $E_{k,a} + E_{k,b} \ll Q_{ab}$ Then: $\frac{1}{2}m_d v_d^2 + \frac{1}{2}m_e v_e^2 \approx Q_{ab}$ Momentum conservation: $m_d V_d = m_e V_e$ Solving for kinetic energies: $E_{k,d} \approx \left(\frac{m_e}{m_d + m_e}\right) Q_{ab}$ $E_{k,e} \approx \left(\frac{m_d}{m_d + m_e}\right) Q_{ab}$ Fusion Fuels (Examples) D-T Cycle (most investigated): $d + t \rightarrow n + \alpha + 17.6 \text{ MeV}$ D-D Reactions: $d + d \rightarrow p + t + 4.1 \text{ MeV}$ $d + d \rightarrow n + h + 3.2 \text{ MeV}$ D-3He Reaction: $d + h \rightarrow p + \alpha + 18.3 \text{ MeV}$ D-6Li Reactions: (multiple directions) $d + {^6}\text{Li} \rightarrow {^7}\text{Be} + n + 3.4 \text{ MeV}$ $d + {^6}\text{Li} \rightarrow {^7}\text{Li} + p + 5.0 \text{ MeV}$ $d + {^6}\text{Li} \rightarrow p + \alpha + t + 2.6 \text{ MeV}$ $d + {^6}\text{Li} \rightarrow 2\alpha + 22.3 \text{ MeV}$ $d + {^6}\text{Li} \rightarrow h + \alpha + n + 1.8 \text{ MeV}$ Other reactions (tritium and He-3): $t + t \rightarrow 2n + \alpha + 11.3 \text{ MeV}$ $h + h \rightarrow 2p + \alpha + 12.9 \text{ MeV}$ $t + h \rightarrow n + p + \alpha + 12.1 \text{ MeV}$ Fusion in Nature (Stellar Media) Open Fusion Cycle: $p + p \rightarrow d + \beta^+ + \nu + 1.2 \text{ MeV}$ $p + d \rightarrow h + 5.5 \text{ MeV}$ $h + h \rightarrow \alpha + 2p + 12.9 \text{ MeV}$ $h + \alpha \rightarrow {^7}\text{Be} + 1.6 \text{ MeV}$ ${^7}\text{Be} + \beta^- \rightarrow {^7}\text{Li} + 0.06 \text{ MeV}$ This progression towards heavier elements is called Nucleosynthesis . Closed Cycle (Carbon Cycle): $^{12}\text{C} + p \rightarrow ^{13}\text{N} + 1.9 \text{ MeV}$ $^{13}\text{N} \rightarrow ^{13}\text{C} + \beta^+ + \nu + 1.5 \text{ MeV}$ $^{13}\text{C} + p \rightarrow ^{14}\text{N} + 7.6 \text{ MeV}$ $^{14}\text{N} + p \rightarrow ^{15}\text{O} + 7.3 \text{ MeV}$ $^{15}\text{O} \rightarrow ^{15}\text{N} + \beta^+ + \nu + 1.8 \text{ MeV}$ $^{15}\text{N} + p \rightarrow ^{12}\text{C} + \alpha + 5.0 \text{ MeV}$ C, N, O act as catalysts in the Carbon Cycle. Overall reaction for Carbon Cycle: $4p \rightarrow \alpha + 2\beta^+ + 2\nu + 25.1 \text{ MeV}$ Particles and Forces Gravitational forces are negligible due to small particle masses. Electrostatic force between two isolated particles: $F_{c,a} = \frac{1}{4\pi\varepsilon_0} \frac{q_a q_b}{r^3} \vec{r}$ This force is crucial in fusion (repulsive for like, attractive for unlike charges). Potential energy associated with charged particles: $U(r) = \int_r^\infty F_{c,a}(r') dr'$ $U(r) = \frac{1}{4\pi\varepsilon_0} \frac{q_a q_b}{r}$ Restriction: For $r \geq (R_A + R_B)$, nuclear force dominates. Coulomb Barrier: If $R_0 = R_a + R_b$, then $U(R_0) = \frac{1}{4\pi\varepsilon_0} \frac{q_a q_b}{(R_a + R_b)}$ Approximation for $R_0$: $R_0 = R_p(A_a^{1/3} + A_b^{1/3})$ $R_p = (1.3 - 1.7) \times 10^{-15} \text{ m}$ Quantum Mechanical Tunneling: Probability of tunneling through Coulomb Barrier: $Pr(\text{tunneling}) \propto \frac{1}{v_r} \exp\left(-\gamma \frac{q_a q_b}{v_r}\right)$ $v_r$ is relative speed, $\gamma$ is a constant. This explains small fusion probability at room temperature. Thermal Kinetics Coulomb Barrier for p, d, t is $\sim 0.4 \text{ MeV}$. Heating to overcome this barrier is the logical approach. Due to tunneling, fusion occurs even at tens-of-keV kinetic energy. This still results in a temperature of $\sim 10^8 \text{ K}$. Hydrogen's ionizing potential is $13.6 \text{ eV}$, so the gas is completely ionized (plasma). Energy release from a fusion reactor must exceed: Energy to heat and ionize the gas. Energy to confine the plasma. Sustaining high plasma temperature and confinement is key. Attempts to produce more energy than consumed have been unsuccessful so far. Kinetic Theory of Gases Used to study plasma where fusion occurs. For N atoms with proton number $Z_i$, complete ionization yields $N_i$ ions and $N_e = Z_i N_i$ electrons. $N \rightarrow N_i + N_e = N_i + Z_i N_i$ For Hydrogen (H-1) atom, $N_i = N_e$. In thermodynamic equilibrium, local pressures for ions and electrons: $P_i = \frac{1}{3} N_i m_i \overline{v_i^2}$ $P_e = \frac{1}{3} N_e m_e \overline{v_e^2}$ Introducing average kinetic energies: $P_i = \frac{2}{3} N_i \left[\frac{1}{2}m_i \overline{v_i^2}\right] = \frac{2}{3} N_i \overline{E_i}$ $P_e = \frac{2}{3} N_e \left[\frac{1}{2}m_e \overline{v_e^2}\right] = \frac{2}{3} N_e \overline{E_e}$ If $N^*$ particles are uniformly distributed: $N(\xi) = N^* M(\xi)$ $M(\xi)$ is a normalized distribution function: $\int_{-\infty}^\infty M(\xi)d\xi = 1$ Distribution functions for a plasma in thermodynamic equilibrium (Maxwell-Boltzmann): Velocity: $M(v) = \left(\frac{m}{2\pi kT}\right)^{3/2} \exp\left(-\frac{1}{2}\frac{mv^2}{kT}\right)$ Speed: $M(v) = \left(\frac{2}{\pi}\right)^{1/2} \left(\frac{m}{kT}\right)^{3/2} v^2 \exp\left(-\frac{mv^2}{2kT}\right)$, for $0 Energy: $M(E) = \frac{2}{\sqrt{\pi}} \frac{1}{(kT)^{3/2}} E^{1/2} \exp\left(-\frac{E}{kT}\right)$, for $0 Distribution Parameters Distribution function depends on a single parameter: temperature. In a mixture of particles, each may have a different distribution function. Magnetic fields can cause temperature differences in different regions. Most probable value of a distribution parameter (obtained by differentiation): $\left.\frac{\partial M(\xi)}{\partial \xi}\right|_{\xi=\hat{\xi}} = 0$ $\left.\frac{\partial M(v_x)}{\partial v_x}\right|_{v_x=\hat{v}_x} = 0 \implies \hat{v}_x = 0$ $\left.\frac{dM(E)}{dE}\right|_{E=\hat{E}} = 0 \implies \hat{E} = \frac{1}{2}kT$ $\left.\frac{dM(v)}{dv}\right|_{v=\hat{v}} = 0 \implies \hat{v} = \sqrt{\frac{2kT}{m}}$ Average value of distribution parameters: $\bar{\xi} = \frac{\int \xi M(\xi)d\xi}{\int M(\xi)d\xi}$ $\overline{v_x} = \int_{-\infty}^\infty v_x M(\mathbf{v}) dv_x = 0$ (i.e. $\overline{\mathbf{v}} = \mathbf{0}$) $\overline{v} = \int_0^\infty v M(v)dv = \sqrt{\frac{8kT}{m\pi}}$ $\overline{E} = \int_0^\infty E M(E)dE = \frac{3}{2}kT$ Kinetic Temperature ($kT$): $kT = \text{(kinetic) temperature of a plasma}$ $\frac{3}{2}kT = \text{average energy of Maxwellian-distributed particles}$ $kT = \text{most frequently occurring particle energy of Maxwellian-distributed particles}$ $\sqrt{\frac{8}{m\pi}} \sqrt{kT} = \text{average particle speed of Maxwellian-distributed particles}$ Power and Reaction Rates Fusion reactor core power: $P_{fu} = R_{fu} Q_{fu}$ $R_{fu}$ is fusion reaction rate density, $Q_{fu}$ is energy released per reaction. If two reactants 'a' and 'b' have constant speeds, $R_{fu} \propto N_a N_b v_r$. More precisely: $R_{fu} = \sigma_{ab}(v_r) N_a N_b v_r$ $\sigma_{ab}$ is the cross-section of the fusion reaction, unit is barns . $1 \text{ b} = 10^{-24} \text{ cm}^2 = 10^{-28} \text{ m}^2$ Sigma-V Parameter In reality, reactant speeds are not constant; they have a range of velocities. This is handled by extending the equation to include summation/integration over particle energies and directions. Distribution functions are normalized: $\int F_a(\mathbf{v}_a)d^3v_a = 1$ and $\int F_b(\mathbf{v}_b)d^3v_b = 1$ Relative speed: $v_r = |\mathbf{v}_a - \mathbf{v}_b|$ Extended reaction rate density: $R_{fu} = \int \int \sigma_{fu}(|\mathbf{v}_a - \mathbf{v}_b|)|\mathbf{v}_a - \mathbf{v}_b| N_a F_a(\mathbf{v}_a) N_b F_b(\mathbf{v}_b) d^3v_a d^3v_b$ Simplified expression using Sigma-V parameter (reaction rate parameter): $R_{fu} = N_a N_b \langle \sigma v \rangle_{ab}$ $\langle \sigma v \rangle_{ab} = \int \int \sigma_{ab}(|\mathbf{v}_a - \mathbf{v}_b|)|\mathbf{v}_a - \mathbf{v}_b| F_a(\mathbf{v}_a) F_b(\mathbf{v}_b) d^3v_a d^3v_b$ Collision Processes Collisions can be atomic, nuclear, or sub-nuclear. Fusion: Discrete inelastic process of nucleon rearrangement. Coulomb Scattering: Continuous changes in direction and kinetic energy of ions and electrons. Coulomb scattering probability for ions is much larger than fusion. Deflection due to scattering can lead to bremsstrahlung radiation power losses, lowering plasma temperature. Differential Scattering Cross-section Force equation: $F_c = \frac{1}{4\pi\varepsilon_0} \frac{q_a q_b}{r^2}$ To specify cross-section, use impact parameter $r_0$ and scattering angle $\theta_s$. In center of mass system, scattering is simple antiparallel motion. In laboratory system, $\theta_s = \theta_c$. Area of axisymmetric ring: $dA = 2\pi r_0 dr_0$ Solid angle of scattering: $d\Omega^* = 2\pi \sin(\theta_c) d\theta_c$ Differential scattering cross-section: $\sigma_s(\theta_c) = \frac{d\sigma}{d\Omega^*}$ Total scattering cross-section: $\sigma_s = \int_0^{4\pi} \sigma_s(\theta_c) d^2\Omega = \int_0^\pi \sigma_s(\theta_c) d\Omega^*$ Relationship between $r_0$ and $\theta_c$: $\tan\left(\frac{\theta_c}{2}\right) = \frac{q_a q_b}{4\pi\varepsilon_0 m_r v_r^2 r_0}$ Differential cross-section for Coulomb scattering: $\sigma_s(\theta_c) = \frac{K^2}{4\sin^4(\theta_c/2)} = \left(\frac{q_a q_b}{4\pi\varepsilon_0 m_r v_r^2}\right)^2 \frac{1}{\sin^4(\theta_c/2)}$ As $r_0 \rightarrow 0$ (head-on collision), $\sigma_s(\theta_c) \rightarrow K^2/4$. Debye Length Plasma is globally neutral but has local charge variations. These variations establish electric potential and field. Thermal motion of ions and electrons influences these effects. The spatial extent of this effect is represented by Debye length . Plasma self-shields from applied electric fields; effects are confined to immediate surroundings. In hot plasma, ions/electrons have kinetic energy, allowing some particles to escape. Local electric field due to charge density: $\nabla \cdot \mathbf{E} = \frac{\rho^c}{\varepsilon_0}$ Scalar potential function: $\mathbf{E} = -\nabla\Phi$ Poisson's equation: $\nabla \cdot (-\nabla\Phi) = \frac{\rho^c}{\varepsilon_0}$ Introducing definition: $\rho^c = \sum_{j=e,i} q_j N_j$ For electrons, $N_e = C \exp\left(-\frac{q_e\Phi}{kT_e}\right)$ (Boltzmann equation) Simplified differential equation: $\nabla^2\Phi - \frac{\Phi}{\lambda_D^2} = 0$ Debye length ($\lambda_D$): $\lambda_D = \sqrt{\frac{\varepsilon_0 k T_e}{q_e^2 N}}$ For shielding ions by electrons: $\lambda_{De} = \sqrt{\frac{\varepsilon_0 k T_e}{q_e^2 N_e(r\rightarrow\infty)}}$ For screening electrons by ions: $\lambda_{Di} = \sqrt{\frac{\varepsilon_0 k T_i}{q_i^2 N_i(r\rightarrow\infty)}}$ In equilibrium, for p, d, or t: $T_e = T_i = T$ and $N_i(r\rightarrow\infty) = N_e(r\rightarrow\infty) = N$. Simplified Debye length: $\lambda_D = \sqrt{\frac{\varepsilon_0 k T}{q_e^2 N}}$ Potential in plasma: $\Phi_{plasma} \propto \frac{\exp(-r/\lambda_D)}{r}$ Free space potential: $\Phi_{free\ space} \propto \frac{1}{r}$ $\Phi$ is attenuated in plasma depending on $\lambda_D$. Scattering Limit To avoid singularity at $\theta_c \rightarrow 0$, consider a finite non-zero bound $\theta_{min}$. The value of $\theta_{min}$ must be chosen such that Coulomb scattering is relatively small. For plasma, $r_{0,max} = \lambda_D$. Minimum angle: $\theta_{min} = 2 \tan^{-1}\left(\frac{K}{\lambda_D}\right)$ The scattering cross-section is always higher than fusion cross-section for d-t for any given particle energy. Thus, the more likely reaction between d and t is scattering, not fusion. Bremsstrahlung Radiation Emission of radiation when a charged particle accelerates or decelerates. Transforms particle kinetic energy into radiation energy. High frequency (X-ray wavelength $\sim 10^{-9} \text{ m}$), readily escapes plasma. Leads to plasma cooling; requires compensating energy supply. Emission power of a charged particle $q_e$ moving with time-varying velocity $v(t)$: $P \propto q_e^2 \left|\frac{d\mathbf{v}}{dt}\right|^2$ Average acceleration: $\bar{a} \approx \frac{F_c(r_0)}{m} = \frac{Z q_e^2}{4\pi\varepsilon_0 r_0^2 m}$ Electrons are main contributors due to their small mass. Energy released per collision: $E_{br,collision} \approx P \Delta t_0 \propto \frac{Z^2 q_e^6}{m_e r_0^3 v_r}$ Number of electrons colliding per unit time with an ion: $dR_0 \approx N_e v_r 2\pi r_0 dr_0$ Electron-ion collision rate density: $dR_0 \approx N_i N_e v_r 2\pi r_0 dr_0$ Bremsstrahlung power density: $P_{br} \propto N_i N_e \frac{Z^2 q_e^6}{m_e^2} \int_{r_{0,min}}^{r_{0,max}} \frac{dr_0}{r_0^2}$ Lower limit of integrand: $r_{0,min} = \frac{h}{m_e v_e}$ (de-Broglie wavelength) Average speed of electron: $v_e = \sqrt{\frac{8kT_e}{m_e\pi}}$ Final expression for electron bremsstrahlung radiation power density: $P_{br} = A_{br} N_i N_e Z^2 \sqrt{kT}$ Where $A_{br} \approx 1.6 \times 10^{-38} \frac{\text{m}^3 \text{J}}{\sqrt{\text{eV s}}}$