Chapter 18

Radiation Detection

Gas-Filled Detectors

Gas-filled detectors operate by collecting ionization produced when radiation passes through a gas (typically Ar, Xe, or $^3$He). The number of ion pairs created is:

$$N_{\text{pairs}} = \frac{E_{\text{dep}}}{W}$$

where $W$ is the average energy per ion pair (~26 eV for Ar, ~22 eV for Xe). The detector behavior depends on the applied voltage:

Ionization Chambers

At moderate voltages (100-500 V), all ion pairs are collected without multiplication. The current is proportional to the radiation intensity. Used for precise dose measurements and radiation survey instruments. The current for a parallel-plate geometry is:

$$I = \frac{q \cdot \dot{N}_{\text{pairs}} \cdot e}{1 + (d/\mu E)v_r}$$

Proportional Counters

At higher voltages (500-900 V), electrons gain enough energy between collisions to cause secondary ionization (gas multiplication). The multiplication factor $M$can reach $10^3\text{--}10^4$. The output pulse is proportional to the original ionization, enabling energy spectroscopy. The Diethorn formula gives the gas gain:

$$\ln M = \frac{V}{K \ln(b/a)}\left(\ln\frac{V}{pa\ln(b/a)} - \ln K\right)$$

Geiger-Müller Counters

At very high voltages (>1000 V), a single ionization event triggers a complete discharge of the entire gas volume. The output pulse is independent of the initial ionization energy, so GM counters cannot perform spectroscopy. They have a characteristic dead time $\tau \sim 100\text{--}200\;\mu$s after each pulse. The true count rate is corrected by: $n = \frac{m}{1-m\tau}$.

Scintillation Detectors

Scintillation detectors convert radiation energy into visible light, which is then detected by photomultiplier tubes (PMTs) or silicon photomultipliers (SiPMs). The number of scintillation photons is:

$$N_{\text{ph}} = \frac{E_{\text{dep}} \cdot S}{E_{\text{photon}}}$$

where $S$ is the scintillation efficiency. Key scintillator materials include:

NaI(Tl): The standard gamma-ray scintillator. High light yield (~38,000 photons/MeV), $Z_{\text{eff}} = 50$, 7% resolution at 662 keV. Hygroscopic (must be sealed).
BGO (Bi$_4$Ge$_3$O$_{12}$): Very high density (7.13 g/cm$^3$), excellent stopping power for PET. Lower light yield but not hygroscopic.
LaBr$_3$:Ce: Excellent energy resolution (~2.8% at 662 keV) and fast timing (~16 ns decay). Superior to NaI for spectroscopy but more expensive.

Semiconductor Detectors

Semiconductor detectors offer the best energy resolution because the energy per electron-hole pair is much smaller than for gas detectors or scintillators:

$W_{\text{Si}} = 3.62$ eV/pair, $W_{\text{Ge}} = 2.96$ eV/pair (compare $W_{\text{gas}} \sim 26$ eV)
Fano factor: $F_{\text{Si}} = 0.115$, $F_{\text{Ge}} = 0.13$ (sub-Poisson statistics)

The intrinsic energy resolution (FWHM) is limited by charge-carrier statistics:

$$\text{FWHM} = 2.355\sqrt{F \cdot W \cdot E_\gamma}$$

HPGe (High-Purity Germanium)

The gold standard for gamma-ray spectroscopy. Achieves ~0.18% resolution at 1332 keV ($^{60}$Co). Must be cooled to 77 K (liquid nitrogen) to reduce thermal noise. Impurity concentration $< 10^{10}$ atoms/cm$^3$ allows depletion depths of several cm.

Silicon Detectors

Si(Li) detectors are used for X-ray spectroscopy (1-30 keV). Silicon strip and pixel detectors are essential for charged-particle tracking in nuclear and particle physics experiments. Position resolution down to ~10 $\mu$m is achievable.

Energy Resolution

Energy resolution is quantified by the Full Width at Half Maximum (FWHM) of the photopeak, usually expressed as a percentage:

$$R = \frac{\text{FWHM}}{E_0} \times 100\%$$

The resolution has contributions from several independent sources added in quadrature:

$$(\text{FWHM})^2 = (\text{FWHM}_{\text{stat}})^2 + (\text{FWHM}_{\text{noise}})^2 + (\text{FWHM}_{\text{drift}})^2 + (\text{FWHM}_{\text{incomplete}})^2$$

The statistical contribution, dominant in scintillators, follows:$\text{FWHM}_{\text{stat}} = 2.355\sqrt{FWE}$, where $F$ is the Fano factor. Resolution improves with energy as $R \propto 1/\sqrt{E}$.

Pulse Height Analysis & Coincidence Techniques

A multichannel analyzer (MCA) sorts detector pulses by amplitude into energy bins (channels), building an energy spectrum. The channel number is linearly related to pulse height (and thus deposited energy):

$$E = a \cdot \text{channel} + b$$

Coincidence techniques require two or more detectors to register events within a narrow time window $2\tau$. The true coincidence rate is:

$$R_{\text{true}} = R_{\text{measured}} - R_{\text{random}} = R_{\text{measured}} - 2\tau R_1 R_2$$

Gamma-gamma coincidence spectroscopy is crucial for constructing nuclear level schemes. Angular correlation measurements determine spin-parity assignments using the correlation function: $W(\theta) = \sum_k A_k P_k(\cos\theta)$.

Neutron Detection

Neutrons, being electrically neutral, are detected indirectly through nuclear reactions that produce charged particles. The most important reactions for thermal neutron detection are:

$$^3\text{He} + n \to {^3\text{H}} + p + 0.764\;\text{MeV} \qquad (\sigma = 5333\;\text{b})$$

$$^{10}\text{B} + n \to {^7\text{Li}} + \alpha + 2.79\;\text{MeV} \qquad (\sigma = 3840\;\text{b})$$

$$^6\text{Li} + n \to {^3\text{H}} + \alpha + 4.78\;\text{MeV} \qquad (\sigma = 940\;\text{b})$$

For fast neutron detection, proton-recoil methods are used. In elastic n-p scattering, the recoil proton energy distribution is uniform from 0 to $E_n$:

$$E_p = E_n \cos^2\theta_{\text{lab}}, \qquad \frac{d\sigma}{dE_p} = \frac{\sigma_{\text{np}}}{E_n}$$

Time-of-flight (TOF) techniques measure neutron energy by timing the flight over a known distance: $E_n = \frac{1}{2}m_n\left(\frac{L}{t}\right)^2$. This method provides excellent energy resolution for fast neutrons.

Python Simulation: Detector Physics

Gamma-ray spectrum comparison for different detector types, gas detector voltage characteristics, detection efficiency curves, and energy resolution comparison.

Radiation Detection Analysis

Python

Detector response, energy spectra, and resolution comparison

script.py183 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Radiation Detection: Energy Spectra & Resolution
# ──────────────────────────────────────────────

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0a0a1a')

for ax_row in axes:
    for ax in ax_row:
        ax.set_facecolor('#0a0a1a')
        for spine in ax.spines.values():
            spine.set_edgecolor('#334155')

# ─── Panel 1: Gamma-ray spectrum (Cs-137) with different detectors ───
E = np.linspace(0, 800, 2000)  # keV
E_gamma = 661.7  # keV (Cs-137)
E_compton_edge = E_gamma * 2*E_gamma/(511+2*E_gamma)  # ~477 keV
E_backscatter = E_gamma - E_compton_edge  # ~184 keV

def gaussian(x, mu, sigma, amp):
    return amp * np.exp(-(x-mu)**2 / (2*sigma**2))

def spectrum(E, E0, resolution_pct, label_det):
    """Generate approximate gamma spectrum"""
    sigma = E0 * resolution_pct / (2*np.sqrt(2*np.log(2))) / 100
    spec = np.zeros_like(E)

# Photopeak
    spec += gaussian(E, E0, sigma, 1.0)

# Compton continuum (simplified)
    mask = (E > 50) & (E < E_compton_edge)
    compton = np.zeros_like(E)
    compton[mask] = 0.3 * (1 + (E[mask]/E0)**2) / (1 + E[mask]/E0)
    spec += compton

# Compton edge
    spec += gaussian(E, E_compton_edge, sigma*3, 0.15)

# Backscatter peak
    spec += gaussian(E, E_backscatter, sigma*2, 0.08)

return spec

# NaI(Tl) - 7% resolution
spec_nai = spectrum(E, E_gamma, 7.0, 'NaI(Tl)')
axes[0,0].plot(E, spec_nai, color='#ef4444', linewidth=1.5, label='NaI(Tl) (7%)')

# HPGe - 0.2% resolution
spec_hpge = spectrum(E, E_gamma, 0.2, 'HPGe')
axes[0,0].plot(E, spec_hpge * 0.8, color='#22c55e', linewidth=1.5, label='HPGe (0.2%)')

# LaBr3 - 3% resolution
spec_labr = spectrum(E, E_gamma, 3.0, 'LaBr3')
axes[0,0].plot(E, spec_labr * 0.9, color='#3b82f6', linewidth=1.5, label='LaBr3 (3%)')

axes[0,0].axvline(E_gamma, color='#64748b', linewidth=0.5, linestyle=':', alpha=0.5)
axes[0,0].text(E_gamma+10, 0.9, '661.7 keV', color='#94a3b8', fontsize=8)
axes[0,0].set_xlabel('Energy [keV]', color='#94a3b8', fontsize=11)
axes[0,0].set_ylabel('Counts (arb.)', color='#94a3b8', fontsize=11)
axes[0,0].set_title('Cs-137 Spectrum: Detector Comparison', color='white', fontsize=13)
axes[0,0].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0,0].tick_params(colors='#94a3b8')
axes[0,0].set_xlim(0, 800)

# ─── Panel 2: Gas detector regions (voltage curve) ───
V = np.linspace(0, 1500, 500)

# Simplified characteristic curve for gas-filled detector
def gas_response(V):
    resp = np.zeros_like(V)
    # Recombination region
    mask1 = V < 200
    resp[mask1] = V[mask1] / 200 * 0.3

# Ionization plateau
    mask2 = (V >= 200) & (V < 500)
    resp[mask2] = 0.3 + 0.02 * (V[mask2] - 200) / 300

# Proportional region
    mask3 = (V >= 500) & (V < 900)
    resp[mask3] = 0.32 * np.exp(0.003 * (V[mask3] - 500))

# Limited proportional
    mask4 = (V >= 900) & (V < 1100)
    resp[mask4] = resp[mask3][-1] * (1 + 0.5*(V[mask4]-900)/200)

# GM region
    mask5 = (V >= 1100) & (V < 1350)
    resp[mask5] = 100

# Continuous discharge
    mask6 = V >= 1350
    resp[mask6] = 100 * np.exp(0.01*(V[mask6]-1350))
    return resp

# Alpha particles (more ionization)
resp_alpha = gas_response(V) * 3
# Beta particles (less ionization)
resp_beta = gas_response(V)

axes[0,1].semilogy(V, resp_alpha, color='#ef4444', linewidth=2, label='Alpha')
axes[0,1].semilogy(V, resp_beta, color='#3b82f6', linewidth=2, label='Beta')

# Label regions
regions = [(100, 'Recomb.'), (350, 'Ion.\nChamber'),
           (700, 'Propor-\ntional'), (1000, 'Limited'),
           (1225, 'G-M'), (1425, 'Disch.')]
for x, label in regions:
    axes[0,1].text(x, 0.15, label, color='#94a3b8', fontsize=7, ha='center', va='bottom')

axes[0,1].set_xlabel('Applied Voltage [V]', color='#94a3b8', fontsize=11)
axes[0,1].set_ylabel('Pulse Height (arb.)', color='#94a3b8', fontsize=11)
axes[0,1].set_title('Gas Detector Characteristic Curve', color='white', fontsize=13)
axes[0,1].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0,1].tick_params(colors='#94a3b8')
axes[0,1].set_ylim(0.1, 1e4)

# ─── Panel 3: Detection efficiency vs energy ───
E_eff = np.linspace(10, 3000, 500)  # keV

# Photoelectric efficiency (dominant at low E)
eff_pe = 100 * np.exp(-E_eff/150)

# Compton efficiency (peaks ~200-500 keV)
eff_cs = 50 * (E_eff/300) * np.exp(-E_eff/800)

# Pair production (above 1022 keV)
eff_pp = np.where(E_eff > 1022, 10 * (1 - np.exp(-(E_eff-1022)/500)), 0)

total_eff = eff_pe + eff_cs + eff_pp

axes[1,0].plot(E_eff, total_eff, color='white', linewidth=2.5, label='Total')
axes[1,0].plot(E_eff, eff_pe, color='#ef4444', linewidth=1.5, linestyle='--', label='Photoelectric')
axes[1,0].plot(E_eff, eff_cs, color='#22c55e', linewidth=1.5, linestyle='--', label='Compton')
axes[1,0].plot(E_eff, eff_pp, color='#3b82f6', linewidth=1.5, linestyle='--', label='Pair prod.')

axes[1,0].set_xlabel('Gamma Energy [keV]', color='#94a3b8', fontsize=11)
axes[1,0].set_ylabel('Relative Efficiency [%]', color='#94a3b8', fontsize=11)
axes[1,0].set_title('HPGe Detection Efficiency', color='white', fontsize=13)
axes[1,0].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[1,0].tick_params(colors='#94a3b8')

# ─── Panel 4: Energy resolution comparison ───
detectors = ['NaI(Tl)', 'BGO', 'LaBr3', 'HPGe', 'Si(Li)', 'CdZnTe']
res_662 = [7.0, 12.0, 2.8, 0.18, 0.5, 1.5]  # % FWHM at 662 keV
colors_det = ['#ef4444', '#f59e0b', '#3b82f6', '#22c55e', '#a855f7', '#ec4899']

bars = axes[1,1].barh(range(len(detectors)), res_662, color=colors_det, height=0.6)
axes[1,1].set_yticks(range(len(detectors)))
axes[1,1].set_yticklabels(detectors, color='#94a3b8', fontsize=10)
axes[1,1].set_xlabel('Energy Resolution at 662 keV [% FWHM]', color='#94a3b8', fontsize=11)
axes[1,1].set_title('Detector Energy Resolution', color='white', fontsize=13)
axes[1,1].tick_params(colors='#94a3b8')

for i, (bar, val) in enumerate(zip(bars, res_662)):
    axes[1,1].text(bar.get_width() + 0.2, bar.get_y() + bar.get_height()/2,
                   f'{val}%', color='#94a3b8', fontsize=9, va='center')

plt.tight_layout()
plt.savefig('radiation_detection.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())

print("=== Radiation Detection Summary ===")
print()
print("Cs-137 gamma spectrum features:")
print(f"  Photopeak: {E_gamma:.1f} keV")
print(f"  Compton edge: {E_compton_edge:.1f} keV")
print(f"  Backscatter peak: {E_backscatter:.1f} keV")
print()
print("Energy resolution at 662 keV (FWHM):")
for name, res in zip(detectors, res_662):
    fwhm_keV = res * E_gamma / 100
    print(f"  {name:10s}: {res:5.2f}%  ({fwhm_keV:.1f} keV)")
print()
print("Compton edge formula: E_ce = E * 2E/(m_e*c^2 + 2E)")
print("Backscatter peak: E_bs = E - E_ce = E * m_e*c^2/(m_e*c^2 + 2E)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation: Detector Calculations

Compton scattering kinematics, energy resolution analysis for semiconductor and scintillation detectors, and neutron detection cross sections.

Radiation Detection Calculator

Fortran

Compton kinematics, resolution analysis, and detector comparison

program.f90124 lines

program radiation_detection
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979323846_dp
  real(dp), parameter :: me_c2 = 511.0_dp  ! keV

integer :: i
  real(dp) :: E_gamma, E_ce, E_bs, theta, E_scattered
  real(dp) :: FWHM, sigma, R_pct
  real(dp) :: N_pairs, F_fano, E_pair

write(*,'(A)') '=================================================='
  write(*,'(A)') ' Radiation Detection Calculations'
  write(*,'(A)') '=================================================='
  write(*,'(A)') ''

! ─── Compton Scattering Kinematics ───
  write(*,'(A)') '--- Compton Scattering for Common Sources ---'
  write(*,'(A)') ''
  write(*,'(A15,A12,A12,A15,A15)') 'Source', 'E_gamma', 'E_CE', &
       'E_backscatter', 'E(90 deg)'
  write(*,'(A15,A12,A12,A15,A15)') '', '(keV)', '(keV)', '(keV)', '(keV)'
  write(*,'(A)') '─────────────────────────────────────────────────' // &
       '────────────────'

! Cs-137
  E_gamma = 661.7_dp
  E_ce = E_gamma * 2.0_dp*E_gamma / (me_c2 + 2.0_dp*E_gamma)
  E_bs = E_gamma - E_ce
  E_scattered = E_gamma / (1.0_dp + E_gamma/me_c2)  ! 90 degrees
  write(*,'(A15,F12.1,F12.1,F15.1,F15.1)') 'Cs-137', E_gamma, E_ce, &
       E_bs, E_scattered

! Co-60 (1173 keV)
  E_gamma = 1173.2_dp
  E_ce = E_gamma * 2.0_dp*E_gamma / (me_c2 + 2.0_dp*E_gamma)
  E_bs = E_gamma - E_ce
  E_scattered = E_gamma / (1.0_dp + E_gamma/me_c2)
  write(*,'(A15,F12.1,F12.1,F15.1,F15.1)') 'Co-60 (1)', E_gamma, &
       E_ce, E_bs, E_scattered

! Co-60 (1332 keV)
  E_gamma = 1332.5_dp
  E_ce = E_gamma * 2.0_dp*E_gamma / (me_c2 + 2.0_dp*E_gamma)
  E_bs = E_gamma - E_ce
  E_scattered = E_gamma / (1.0_dp + E_gamma/me_c2)
  write(*,'(A15,F12.1,F12.1,F15.1,F15.1)') 'Co-60 (2)', E_gamma, &
       E_ce, E_bs, E_scattered

! Na-22 (511 keV)
  E_gamma = 511.0_dp
  E_ce = E_gamma * 2.0_dp*E_gamma / (me_c2 + 2.0_dp*E_gamma)
  E_bs = E_gamma - E_ce
  E_scattered = E_gamma / (1.0_dp + E_gamma/me_c2)
  write(*,'(A15,F12.1,F12.1,F15.1,F15.1)') 'Na-22 (ann)', E_gamma, &
       E_ce, E_bs, E_scattered

write(*,'(A)') ''

! ─── Energy Resolution ───
  write(*,'(A)') '--- Energy Resolution Analysis ---'
  write(*,'(A)') ''
  write(*,'(A)') 'HPGe at 1332 keV:'
  E_gamma = 1332.5_dp
  E_pair = 2.96_dp  ! eV per electron-hole pair in Ge
  F_fano = 0.13_dp  ! Fano factor for Ge
  N_pairs = E_gamma * 1000.0_dp / E_pair  ! convert keV to eV
  sigma = sqrt(F_fano * N_pairs) * E_pair / 1000.0_dp  ! keV
  FWHM = 2.355_dp * sigma
  R_pct = FWHM / E_gamma * 100.0_dp

write(*,'(A,F12.0)') '  Number of e-h pairs: ', N_pairs
  write(*,'(A,F8.2,A)') '  Fano-limited FWHM: ', FWHM, ' keV'
  write(*,'(A,F8.4,A)') '  Resolution: ', R_pct, ' %'
  write(*,'(A)') ''

write(*,'(A)') 'NaI(Tl) at 662 keV:'
  E_gamma = 661.7_dp
  E_pair = 25.0_dp  ! eV per scintillation photon (approximate)
  F_fano = 1.0_dp   ! Poisson statistics (F=1)
  N_pairs = E_gamma * 1000.0_dp / E_pair
  sigma = sqrt(F_fano * N_pairs) * E_pair / 1000.0_dp
  FWHM = 2.355_dp * sigma
  R_pct = FWHM / E_gamma * 100.0_dp

write(*,'(A,F12.0)') '  Number of photons: ', N_pairs
  write(*,'(A,F8.2,A)') '  Statistical FWHM: ', FWHM, ' keV'
  write(*,'(A,F8.2,A)') '  Resolution (stat): ', R_pct, ' %'
  write(*,'(A,A)') '  Actual resolution ~7%', ' (includes PMT, non-uniformity)'
  write(*,'(A)') ''

! ─── Detector Comparison Table ───
  write(*,'(A)') '--- Detector Type Comparison ---'
  write(*,'(A)') ''
  write(*,'(A18,A12,A12,A18,A12)') 'Detector', 'Type', 'Z_eff', &
       'Best For', 'Cost'
  write(*,'(A)') '────────────────────────────────────────────────────' // &
       '────────────────'
  write(*,'(A18,A12,A12,A18,A12)') 'NaI(Tl)', 'Scint.', '50', &
       'Survey/SPECT', 'Low'
  write(*,'(A18,A12,A12,A18,A12)') 'BGO', 'Scint.', '74', &
       'PET', 'Medium'
  write(*,'(A18,A12,A12,A18,A12)') 'LaBr3:Ce', 'Scint.', '47', &
       'Timing/spec.', 'Medium'
  write(*,'(A18,A12,A12,A18,A12)') 'HPGe', 'Semicond.', '32', &
       'Spectroscopy', 'High'
  write(*,'(A18,A12,A12,A18,A12)') 'Si(Li)', 'Semicond.', '14', &
       'Low-E X-ray', 'Medium'
  write(*,'(A18,A12,A12,A18,A12)') 'CdZnTe', 'Semicond.', '49', &
       'Room-T spec.', 'High'
  write(*,'(A)') ''

! ─── Neutron Detection Methods ───
  write(*,'(A)') '--- Neutron Detection Reactions ---'
  write(*,'(A)') ''
  write(*,'(A30,A15,A20)') 'Reaction', 'Q-value(MeV)', 'Cross-sec(b)'
  write(*,'(A)') '─────────────────────────────────────────────────────' // &
       '──────────'
  write(*,'(A30,F15.3,F20.1)') 'He-3(n,p)H-3', 0.764_dp, 5333.0_dp
  write(*,'(A30,F15.3,F20.1)') 'Li-6(n,a)H-3', 4.780_dp, 940.0_dp
  write(*,'(A30,F15.3,F20.1)') 'B-10(n,a)Li-7', 2.790_dp, 3840.0_dp
  write(*,'(A30,A15,F20.1)') 'U-235(n,f)', '~200', 585.0_dp

end program radiation_detection

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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