Part 6, Chapter 7

Fusion Reactors

ITER and reactor concepts

7.1 Introduction

This chapter provides comprehensive coverage of ITER and reactor concepts.

Topics include fundamental theory, mathematical derivations, physical interpretations, and practical applications in fusion energy, space physics, and astrophysics.

7.2 Key Concepts

Essential concepts and equations covered in this chapter with detailed explanations and examples.

MIT: Principles of Plasma Diagnostics

Lecture on nuclear diagnostics from MIT's Principles of Plasma Diagnostics course.

Lecture 19: Nuclear Diagnostics

Interactive Simulations

Lawson Criterion and Fusion Triple Product

Python

script.py70 lines

import numpy as np

# Lawson criterion for D-T fusion ignition:
# n * tau_E > 1.5e20 m^-3 s (at T ~ 15 keV)
# Triple product: n * T * tau_E > 5e21 m^-3 keV s

e = 1.602e-19
k_B = 1.381e-23

# D-T fusion reaction rate <sigma*v> parameterization
def sigma_v_DT(T_keV):
    """D-T fusion reactivity in m^3/s"""
    # Bosch-Hale parameterization (approximate)
    T = T_keV
    if T < 0.5:
        return 1e-30
    xi = 34.3827 / T**(1/3)
    sv = 3.7e-18 * T**(-2/3) * np.exp(-xi)  # simplified
    return sv

print("Fusion Triple Product: Path to Ignition")
print("=" * 60)
print()

# Scan temperature
T_values = np.array([1, 2, 5, 8, 10, 15, 20, 30, 50, 100])

print("D-T Reactivity and Lawson Criterion:")
print(f"{'T (keV)':>8} {'<sv> (m3/s)':>14} {'n*tau_E':>14} {'n*T*tau_E':>16} {'P_fus/V':>12}")
print(f"{'':>8} {'':>14} {'(m-3 s)':>14} {'(m-3 keV s)':>16} {'(MW/m3)':>12}")
print("-" * 66)

for T in T_values:
    sv = sigma_v_DT(T)

# Lawson: radiation losses = fusion heating
    # n*tau_E = 12*k*T / (E_alpha * <sv> - C_B*sqrt(T))
    E_alpha = 3.5e6 * e   # 3.5 MeV alpha energy
    # Simplified: P_fusion = n^2/4 * <sv> * 17.6 MeV
    # Break-even: P_fusion = P_loss = 3*n*k*T/tau_E

# Required n*tau_E for break-even
    if sv > 1e-30:
        n_tau = 12 * T * 1000 * e / (E_alpha * sv)
        n_T_tau = n_tau * T
    else:
        n_tau = 1e30
        n_T_tau = 1e30

# Fusion power density at n = 1e20
    n_ref = 1e20
    P_fus = n_ref**2 / 4 * sv * 17.6e6 * e / 1e6

print(f"{T:8.0f} {sv:14.3e} {n_tau:14.3e} {n_T_tau:16.3e} {P_fus:12.3f}")

print()
print("Key milestones:")
experiments = [
    ("JET (1997)", 0.7e20, 3.5, 0.9, "Q=0.67"),
    ("JT-60U", 0.9e20, 4.5, 1.0, "Q_eq~1.25"),
    ("NIF (2022)", 5e25, 0.001, 0.01, "Q=1.5"),
    ("ITER (target)", 1e20, 15, 3.7, "Q=10"),
    ("DEMO", 1e20, 15, 5.0, "Q>25"),
]
print(f"{'Experiment':>15} {'n(m-3)':>10} {'T(keV)':>8} {'nTtau':>12} {'Note':>10}")
print("-" * 57)
for name, n, T, tau, note in experiments:
    nTtau = n * T * tau
    print(f"{name:>15} {n:10.1e} {T:8.1f} {nTtau:12.2e} {note:>10}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

D-T Fusion Cross Section and Gamow Peak

Fortran

program.f9071 lines

program fusion_cross_section
  implicit none
  ! D-T fusion cross section parameterization
  ! sigma(E) = S(E)/E * exp(-sqrt(E_G/E))
  ! Gamow energy E_G = (pi * alpha * Z1 * Z2)^2 * 2 * m_r * c^2
  real(8) :: e_charge, m_p, c, pi, alpha, k_B
  real(8) :: m_D, m_T, m_r, E_G
  real(8) :: E, sigma, S_factor, sv
  real(8) :: T_keV, kT, v_max, dv, v, f_MB
  integer :: i, j
  real(8) :: E_keV

e_charge = 1.602d-19
  m_p = 1.673d-27
  c = 3.0d8
  pi = 4.0d0 * atan(1.0d0)
  alpha = 1.0d0/137.036d0
  k_B = 1.381d-23

m_D = 2.0d0 * m_p
  m_T = 3.0d0 * m_p
  m_r = m_D * m_T / (m_D + m_T)

! Gamow energy for D-T (Z1=Z2=1)
  E_G = (pi * alpha)**2 * 2.0d0 * m_r * c**2
  write(*,'(A)') "D-T Fusion Cross Section"
  write(*,'(A)') "========================"
  write(*,'(A,F8.2,A)') "Gamow energy: ", E_G/e_charge/1000, " keV"
  write(*,'(A,ES10.3,A)') "Reduced mass: ", m_r, " kg"
  write(*,*)

! S-factor for D-T (approximately constant ~ 25 MeV.barn)
  S_factor = 25.0d0 * 1.0d6 * e_charge * 1.0d-28  ! convert to J.m^2

write(*,'(A)') "Cross section vs center-of-mass energy:"
  write(*,'(A10,A14,A14,A14)') &
    "E (keV)", "sigma (barn)", "Gamow exp", "S(E) (keVb)"
  write(*,'(A)') repeat("-", 54)

do i = 1, 15
    E_keV = dble(i) * 10.0d0
    E = E_keV * 1000.0d0 * e_charge

! Gamow tunneling factor
    sigma = S_factor / E * exp(-sqrt(E_G/E))

write(*,'(F10.1,ES14.3,F14.4,F14.1)') &
      E_keV, sigma/1.0d-28, exp(-sqrt(E_G/E)), &
      S_factor/(1000.0d0*e_charge*1.0d-28)
  end do

! Gamow peak for different temperatures
  write(*,*)
  write(*,'(A)') "Gamow peak energy (most probable reaction energy):"
  write(*,'(A10,A14,A14)') "T (keV)", "E_peak(keV)", "Width(keV)"
  write(*,'(A)') repeat("-", 40)

do i = 1, 8
    T_keV = dble(i) * 5.0d0
    kT = T_keV * 1000.0d0 * e_charge

! Gamow peak: E_0 = (E_G * kT^2 / 4)^(1/3)
    E = (E_G * kT**2 / 4.0d0)**(1.0d0/3.0d0)
    ! Width: Delta = 4 * sqrt(E_0 * kT / 3)
    sigma = 4.0d0 * sqrt(E * kT / 3.0d0)

write(*,'(F10.1,F14.2,F14.2)') &
      T_keV, E/(1000.0d0*e_charge), sigma/(1000.0d0*e_charge)
  end do
end program fusion_cross_section

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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