Chapter 1: Fundamentals of Electron Microscopy

Electron microscopy is a powerful tool that can observe atomic and molecular-level structures, surpassing the resolution limits of optical microscopy. In this chapter, we will learn about electron-matter interactions, the relationship between electron wavelength and resolution, and the fundamental principles of electron microscopy, practicing calculations of electron wavelength, resolution simulation, and scattering cross-section computations using Python.

1.1 Fundamentals of Electron Waves

1.1.1 Wave-Particle Duality

Electrons are quantum mechanical particles possessing both wave properties and particle properties. Louis de Broglie (1924) proposed that a particle with momentum $p$ has a wavelength $\lambda$:

where $h = 6.626 \times 10^{-34}$ J·s (Planck's constant), $p = mv$ (momentum: mass $m$ × velocity $v$).

Relativistic correction: When electrons are accelerated to high speeds, relativistic effects must be considered. The energy of an electron accelerated by voltage $V$ is:

where $e = 1.602 \times 10^{-19}$ C (elementary charge). The relativistic momentum is calculated by:

$m_0 = 9.109 \times 10^{-31}$ kg (electron rest mass), $c = 2.998 \times 10^8$ m/s (speed of light).

Code Example 1-1: Calculation of Electron Wavelength (with Relativistic Correction)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def calculate_electron_wavelength(voltage_kV):
    """
    Calculate electron wavelength from acceleration voltage (with relativistic correction)

    Parameters
    ----------
    voltage_kV : float or array-like
        Acceleration voltage [kV]

    Returns
    -------
    wavelength_pm : float or array-like
        Electron wavelength [pm]
    """
    # Physical constants
    h = 6.62607e-34    # Planck constant [J·s]
    m0 = 9.10938e-31   # Electron mass [kg]
    e = 1.60218e-19    # Elementary charge [C]
    c = 2.99792e8      # Speed of light [m/s]

    # Convert acceleration voltage to J
    V = voltage_kV * 1000  # [V]
    E = e * V              # Energy [J]

    # Relativistic momentum
    p = np.sqrt(2 * m0 * E * (1 + E / (2 * m0 * c**2)))

    # de Broglie wavelength
    wavelength_m = h / p
    wavelength_pm = wavelength_m * 1e12  # [pm]

    return wavelength_pm

# Acceleration voltage range
voltages = np.array([10, 50, 100, 200, 300, 500, 1000])  # [kV]
wavelengths = calculate_electron_wavelength(voltages)

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left plot: Wavelength vs acceleration voltage
ax1.plot(voltages, wavelengths, 'o-', linewidth=2, markersize=8, color='#f093fb')
ax1.set_xlabel('Acceleration Voltage [kV]', fontsize=12)
ax1.set_ylabel('Electron Wavelength [pm]', fontsize=12)
ax1.set_title('Electron Wavelength vs Acceleration Voltage', fontsize=14, fontweight='bold')
ax1.grid(alpha=0.3)
ax1.set_xscale('log')
ax1.set_yscale('log')

# Add visible light wavelength as reference line
ax1.axhline(y=500000, color='orange', linestyle='--', linewidth=2, label='Visible Light (~500 nm)')
ax1.legend(fontsize=10)

# Right plot: Wavelength comparison at representative acceleration voltages
selected_voltages = [100, 200, 300]
selected_wavelengths = calculate_electron_wavelength(selected_voltages)

ax2.bar([f'{v} kV' for v in selected_voltages], selected_wavelengths,
        color=['#f093fb', '#f5576c', '#d07be8'], alpha=0.8, edgecolor='black')
ax2.set_ylabel('Electron Wavelength [pm]', fontsize=12)
ax2.set_title('Typical Operating Voltages', fontsize=14, fontweight='bold')
ax2.grid(axis='y', alpha=0.3)

for i, (v, wl) in enumerate(zip(selected_voltages, selected_wavelengths)):
    ax2.text(i, wl + 0.1, f'{wl:.3f} pm', ha='center', fontsize=10, fontweight='bold')

plt.tight_layout()
plt.show()

# Output specific numerical values
print("Correspondence between acceleration voltage and electron wavelength:")
for v, wl in zip(voltages, wavelengths):
    print(f"  {v:4.0f} kV → λ = {wl:.4f} pm = {wl/100:.4f} Å")
print(f"\nWavelength ratio of 200 kV electron wave to visible light (550 nm): {550000 / wavelengths[3]:.0f}:1")

Output interpretation:

• Wavelength of approximately 3.7 pm at 100 kV, 2.5 pm at 200 kV, 2.0 pm at 300 kV
• Approximately 200,000 times shorter wavelength compared to visible light (~500 nm)
• Higher acceleration voltage leads to shorter wavelength and improved resolution

1.1.2 Resolution and Rayleigh Criterion

The resolution of a microscope is defined as the minimum distance at which two points can be distinguished. According to the Rayleigh criterion:

where:

• $d$: resolution (minimum distinguishable distance)
• $\lambda$: wavelength
• $n \sin\alpha$: numerical aperture (NA)
• $\alpha$: half-angle of objective lens

Optical Microscopy vs Electron Microscopy:

Code Example 1-2: Resolution Simulation

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def resolution_rayleigh(wavelength_nm, alpha_deg, n=1.0):
    """
    Calculate resolution by Rayleigh criterion

    Parameters
    ----------
    wavelength_nm : float
        Wavelength [nm]
    alpha_deg : float
        Objective lens half-angle [degrees]
    n : float
        Refractive index (1.0 in vacuum for electron microscopy)

    Returns
    -------
    d_nm : float
        Resolution [nm]
    """
    alpha_rad = np.deg2rad(alpha_deg)
    d_nm = 0.61 * wavelength_nm / (n * np.sin(alpha_rad))
    return d_nm

# Comparison of optical and electron microscopy
alpha_range = np.linspace(0.5, 30, 100)  # [degrees]

# Optical microscopy (visible light)
optical_wavelength = 550  # [nm]
optical_resolution = resolution_rayleigh(optical_wavelength, alpha_range)

# Electron microscopy (100 kV → 3.7 pm = 0.0037 nm)
em_100kV_wavelength = 0.0037  # [nm]
em_100kV_resolution = resolution_rayleigh(em_100kV_wavelength, alpha_range)

# Electron microscopy (200 kV → 2.5 pm = 0.0025 nm)
em_200kV_wavelength = 0.0025  # [nm]
em_200kV_resolution = resolution_rayleigh(em_200kV_wavelength, alpha_range)

# Plot
fig, ax = plt.subplots(figsize=(12, 7))

ax.plot(alpha_range, optical_resolution, linewidth=2.5, label='Optical (550 nm)', color='orange')
ax.plot(alpha_range, em_100kV_resolution, linewidth=2.5, label='EM 100 kV (3.7 pm)', color='#f093fb')
ax.plot(alpha_range, em_200kV_resolution, linewidth=2.5, label='EM 200 kV (2.5 pm)', color='#f5576c')

# Practical resolution limit of optical microscopy
ax.axhline(y=200, color='orange', linestyle='--', linewidth=1.5, alpha=0.7, label='Optical Limit (~200 nm)')

# Practical resolution of electron microscopy (effects of spherical aberration, etc.)
ax.axhline(y=0.1, color='blue', linestyle='--', linewidth=1.5, alpha=0.7, label='Practical EM Limit (~0.1 nm)')

ax.set_xlabel('Objective Lens Half-Angle α [degrees]', fontsize=12)
ax.set_ylabel('Resolution d [nm]', fontsize=12)
ax.set_title('Resolution vs Lens Aperture (Rayleigh Criterion)', fontsize=14, fontweight='bold')
ax.set_yscale('log')
ax.set_ylim(1e-3, 1e3)
ax.legend(fontsize=11, loc='upper right')
ax.grid(alpha=0.3, which='both')

plt.tight_layout()
plt.show()

# Calculate practical values
alpha_typical = 10  # [degrees] (typical objective lens half-angle)
print(f"Resolution at objective lens half-angle α = {alpha_typical}°:")
print(f"  Optical microscopy: {resolution_rayleigh(optical_wavelength, alpha_typical):.1f} nm")
print(f"  Electron microscopy (100 kV): {resolution_rayleigh(em_100kV_wavelength, alpha_typical):.4f} nm")
print(f"  Electron microscopy (200 kV): {resolution_rayleigh(em_200kV_wavelength, alpha_typical):.4f} nm")

1.2 Electron-Matter Interactions

1.2.1 Types of Scattering

When an electron beam enters a specimen, various scattering phenomena occur due to interactions with atomic nuclei and electrons.

Elastic Scattering:

• Electron energy loss is nearly zero ($\Delta E \approx 0$)
• Scattering by Coulomb potential of atomic nuclei
• Scattering angle depends on atomic number $Z$ (heavier elements scatter at larger angles)
• Applications: TEM diffraction, HRTEM, HAADF-STEM (Z-contrast imaging)

Inelastic Scattering:

• Electrons lose energy ($\Delta E > 0$)
• Inner-shell electron excitation, plasmon excitation, phonon excitation, etc.
• Applications: EDS (elemental analysis), EELS (electronic state analysis), SE (surface morphology observation)

1.2.2 Scattering Cross Section and Atomic Number Dependence

Scattering intensity is represented by the scattering cross section $\sigma$. In the Rutherford scattering approximation:

where $Z$ is the atomic number, $E$ is the electron energy, and $\theta$ is the scattering angle.

Code Example 1-3: Atomic Number Dependence of Scattering Cross Section

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def rutherford_scattering_cross_section(Z, E_keV, theta_deg):
    """
    Calculate Rutherford scattering cross section

    Parameters
    ----------
    Z : int or array-like
        Atomic number
    E_keV : float
        Electron energy [keV]
    theta_deg : float
        Scattering angle [degrees]

    Returns
    -------
    cross_section : float or array-like
        Differential scattering cross section [nm²/sr]
    """
    # Constants
    e = 1.60218e-19      # [C]
    epsilon_0 = 8.85419e-12  # [F/m]
    a0 = 0.0529          # Bohr radius [nm]

    # Convert energy to J
    E_J = E_keV * 1000 * e

    # Convert scattering angle to radians
    theta_rad = np.deg2rad(theta_deg)

    # Rutherford scattering cross section (simplified formula)
    # dσ/dΩ ∝ Z² / (E² sin⁴(θ/2))
    Z = np.asarray(Z)
    cross_section = (Z / E_keV)**2 / (np.sin(theta_rad / 2)**4)

    # Normalization factor (arbitrary units)
    cross_section = cross_section * 1e-3

    return cross_section

# Representative elements
elements = {
    'C': 6, 'Al': 13, 'Si': 14, 'Ti': 22,
    'Fe': 26, 'Cu': 29, 'Mo': 42, 'W': 74, 'Pt': 78, 'Au': 79
}

E_keV = 200  # 200 kV
theta_range = np.linspace(1, 30, 100)  # [degrees]

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

# Left plot: Angular dependence
selected_elements = ['C', 'Si', 'Fe', 'Au']
colors = ['#d3d3d3', '#f093fb', '#ffa500', '#ffd700']

for elem, color in zip(selected_elements, colors):
    Z = elements[elem]
    cs = rutherford_scattering_cross_section(Z, E_keV, theta_range)
    ax1.plot(theta_range, cs, linewidth=2.5, label=f'{elem} (Z={Z})', color=color)

ax1.set_xlabel('Scattering Angle θ [degrees]', fontsize=12)
ax1.set_ylabel('d$\sigma$/d$\Omega$ [arb. units]', fontsize=12)
ax1.set_title(f'Scattering Cross Section vs Angle\n(E = {E_keV} keV)', fontsize=14, fontweight='bold')
ax1.set_yscale('log')
ax1.set_ylim(1e-2, 1e4)
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3, which='both')

# Right plot: Atomic number dependence (fixed scattering angle)
Z_values = np.array(list(elements.values()))
element_names = list(elements.keys())
theta_fixed = 10  # [degrees]

cs_Z = rutherford_scattering_cross_section(Z_values, E_keV, theta_fixed)

ax2.scatter(Z_values, cs_Z, s=150, c=Z_values, cmap='plasma',
            edgecolors='black', linewidths=1.5, zorder=3)
ax2.plot(Z_values, cs_Z, 'k--', linewidth=1, alpha=0.5, zorder=1)

for i, (Z, cs, name) in enumerate(zip(Z_values, cs_Z, element_names)):
    ax2.text(Z, cs * 1.2, name, fontsize=9, ha='center', fontweight='bold')

ax2.set_xlabel('Atomic Number Z', fontsize=12)
ax2.set_ylabel('d$\sigma$/d$\Omega$ [arb. units]', fontsize=12)
ax2.set_title(f'Scattering Cross Section vs Z\n(θ = {theta_fixed}°, E = {E_keV} keV)',
              fontsize=14, fontweight='bold')
ax2.set_yscale('log')
ax2.grid(alpha=0.3, which='both')

plt.tight_layout()
plt.show()

print(f"Atomic number dependence of scattering cross section (θ = {theta_fixed}°):")
print(f"  C (Z=6):   {rutherford_scattering_cross_section(6, E_keV, theta_fixed):.3f}")
print(f"  Si (Z=14): {rutherford_scattering_cross_section(14, E_keV, theta_fixed):.3f}")
print(f"  Fe (Z=26): {rutherford_scattering_cross_section(26, E_keV, theta_fixed):.3f}")
print(f"  Au (Z=79): {rutherford_scattering_cross_section(79, E_keV, theta_fixed):.3f}")
print(f"\nAu scattering cross section is approximately {(79/6)**2:.0f} times that of C (Z² dependence)")

Important observations:

• Scattering cross section is proportional to $Z^2$ → heavier elements scatter more strongly
• Small-angle scattering dominates (proportional to inverse of $\sin^4(\theta/2)$)
• HAADF-STEM (high-angle annular dark field) detects large-angle scattering to obtain Z-contrast images

1.3 Components of Electron Microscopy

1.3.1 Electron Gun

The electron gun is a device that generates a stable electron beam. Main types:

Advantages of Field Emission Gun (FEG):

• High brightness → enables fine probes (below 0.1 nm)
• Narrow energy spread → high energy resolution in EELS
• High coherence → high resolution in HRTEM

1.3.2 Electron Lenses and Aberrations

Electron lenses focus the electron beam using magnetic fields (electromagnetic lenses) or electrostatic fields.

Major aberrations:

• Spherical Aberration ($C_s$): Electrons passing through lens edges deviate from focus
• Chromatic Aberration ($C_c$): Focus shift due to energy spread
• Astigmatism: Image distortion due to lens asymmetry

Modern TEMs use aberration correctors (Cs-correctors) to correct spherical aberration to nearly zero, achieving resolutions below 0.05 nm.

Code Example 1-4: Basics of Contrast Transfer Function (CTF)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def ctf_simple(k, wavelength_A, defocus_nm, Cs_mm):
    """
    Simplified contrast transfer function (CTF)

    Parameters
    ----------
    k : array-like
        Spatial frequency [1/Å]
    wavelength_A : float
        Electron wavelength [Å]
    defocus_nm : float
        Defocus [nm] (negative: underfocus)
    Cs_mm : float
        Spherical aberration coefficient [mm]

    Returns
    -------
    ctf : array-like
        CTF values
    """
    # Unit conversion
    defocus_A = defocus_nm * 10
    Cs_A = Cs_mm * 1e7

    # Phase shift χ(k)
    chi = (np.pi * wavelength_A / 2) * k**2 * (defocus_A + 0.5 * wavelength_A**2 * k**2 * Cs_A)

    # CTF = sin(χ)
    ctf = np.sin(chi)

    return ctf

# Parameters
voltage_kV = 200
wavelength_pm = 2.508  # Wavelength at 200 kV
wavelength_A = wavelength_pm / 100

k = np.linspace(0, 10, 500)  # Spatial frequency [1/Å]

# CTF at different defocus and Cs values
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

# Left plot: Defocus dependence
Cs = 1.0  # [mm]
defocus_values = [-50, -100, -200]  # [nm]
colors = ['#f093fb', '#f5576c', '#d07be8']

for df, color in zip(defocus_values, colors):
    ctf = ctf_simple(k, wavelength_A, df, Cs)
    ax1.plot(k, ctf, linewidth=2, label=f'Δf = {df} nm', color=color)

ax1.axhline(0, color='black', linestyle='--', linewidth=0.8)
ax1.set_xlabel('Spatial Frequency k [1/Å]', fontsize=12)
ax1.set_ylabel('CTF', fontsize=12)
ax1.set_title(f'CTF vs Defocus\n(200 kV, Cs = {Cs} mm)', fontsize=14, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)
ax1.set_xlim(0, 10)
ax1.set_ylim(-1, 1)

# Right plot: Spherical aberration coefficient dependence
defocus = -100  # [nm]
Cs_values = [0.5, 1.0, 2.0]  # [mm]
colors2 = ['#99ccff', '#ffa500', '#ff6347']

for Cs_val, color in zip(Cs_values, colors2):
    ctf = ctf_simple(k, wavelength_A, defocus, Cs_val)
    ax2.plot(k, ctf, linewidth=2, label=f'Cs = {Cs_val} mm', color=color)

ax2.axhline(0, color='black', linestyle='--', linewidth=0.8)
ax2.set_xlabel('Spatial Frequency k [1/Å]', fontsize=12)
ax2.set_ylabel('CTF', fontsize=12)
ax2.set_title(f'CTF vs Cs\n(200 kV, Δf = {defocus} nm)', fontsize=14, fontweight='bold')
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)
ax2.set_xlim(0, 10)
ax2.set_ylim(-1, 1)

plt.tight_layout()
plt.show()

print("CTF interpretation:")
print("  - Positive CTF region: Spatial frequency components contribute to image with positive contrast")
print("  - Negative CTF region: Contrast is inverted")
print("  - Zero CTF: Frequency components do not contribute to image (information loss)")

1.3.3 Detectors

In electron microscopy, multiple detectors are used to detect various signals:

• Fluorescent screen: Converts electron beam to visible light (real-time observation)
• CCD camera: High sensitivity, digital image acquisition
• Direct Electron Detector (DED): Direct electron detection, fast and high sensitivity
• Energy Dispersive X-ray Spectroscopy (EDS/EDX): Elemental analysis
• Electron Energy Loss Spectrometer (EELS): Electronic state analysis

1.4 Signal Intensity and Statistics

1.4.1 Signal-to-Noise Ratio (S/N Ratio)

The quality of electron microscopy images is evaluated by the signal-to-noise ratio (S/N):

Since the electron beam follows Poisson statistics, for $N$ detected electrons, the noise standard deviation is $\sqrt{N}$. Therefore:

Strategies for improving S/N ratio:

• Increase electron dose (increase $N$) → S/N $\propto \sqrt{N}$
• Average multiple images → S/N $\propto \sqrt{M}$ ($M$: number of images)
• However, be careful of specimen damage (beam damage)

Code Example 1-5: Poisson Statistics and Signal-to-Noise Ratio

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def simulate_electron_image(size=256, signal_level=100, noise_factor=1.0):
    """
    Simulate electron microscopy image following Poisson statistics

    Parameters
    ----------
    size : int
        Image size [pixels]
    signal_level : float
        Average electron count [counts/pixel]
    noise_factor : float
        Noise scaling coefficient

    Returns
    -------
    image : ndarray
        Simulated image
    snr : float
        Signal-to-noise ratio
    """
    # Generate electron count following Poisson distribution
    image = np.random.poisson(signal_level, (size, size)).astype(float)

    # Additional Gaussian noise (readout noise, etc.)
    image += np.random.normal(0, noise_factor * np.sqrt(signal_level), (size, size))

    # Calculate S/N ratio
    signal = signal_level
    noise = np.sqrt(signal_level + noise_factor**2 * signal_level)
    snr = signal / noise

    return image, snr

# Simulation at different electron doses
dose_levels = [10, 50, 200, 1000]  # [electrons/pixel]

fig, axes = plt.subplots(2, 4, figsize=(16, 8))

for i, dose in enumerate(dose_levels):
    image, snr = simulate_electron_image(size=128, signal_level=dose, noise_factor=0.5)

    # Display image
    axes[0, i].imshow(image, cmap='gray', vmin=0, vmax=1200)
    axes[0, i].set_title(f'Dose: {dose} e⁻/px\nS/N: {snr:.2f}', fontsize=11, fontweight='bold')
    axes[0, i].axis('off')

    # Histogram
    axes[1, i].hist(image.ravel(), bins=50, color='#f093fb', alpha=0.7, edgecolor='black')
    axes[1, i].set_xlabel('Intensity [counts]', fontsize=10)
    axes[1, i].set_ylabel('Frequency', fontsize=10)
    axes[1, i].set_title(f'Histogram (σ={np.std(image):.1f})', fontsize=10)
    axes[1, i].grid(alpha=0.3)

plt.tight_layout()
plt.show()

# Compare theoretical and measured S/N ratios
print("Relationship between electron dose and S/N ratio:")
for dose in dose_levels:
    _, snr = simulate_electron_image(size=128, signal_level=dose, noise_factor=0.5)
    theoretical_snr = np.sqrt(dose)
    print(f"  Dose {dose:4d} e⁻/px: Theoretical S/N={theoretical_snr:.2f}, Measured S/N={snr:.2f}")

1.5 Acceleration Voltage Selection and Specimen Damage

1.5.1 Effects of Acceleration Voltage

Acceleration voltage directly affects both electron microscope performance and specimen effects:

Mechanisms of specimen damage:

• Knock-on damage: High-energy electrons knock out atoms (above threshold energy)
• Radiolysis: Chemical bond breaking due to electron beam irradiation
• Thermal damage: Specimen heating due to inelastic scattering

Code Example 1-6: Threshold Energy for Knock-on Damage

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def knock_on_threshold_voltage(element_mass, displacement_energy_eV=25):
    """
    Calculate threshold acceleration voltage for knock-on damage

    Parameters
    ----------
    element_mass : float
        Atomic mass of element [u]
    displacement_energy_eV : float
        Displacement energy of atom [eV] (typical: 25 eV)

    Returns
    -------
    V_threshold : float
        Threshold acceleration voltage [kV]
    """
    # Constants
    m_e = 9.10938e-31   # Electron mass [kg]
    u = 1.66054e-27     # Atomic mass unit [kg]
    e = 1.60218e-19     # Elementary charge [C]
    c = 2.99792e8       # Speed of light [m/s]

    # Atomic mass [kg]
    M = element_mass * u

    # Knock-on threshold energy (relativistic)
    # E_threshold ≈ (M + 2m_e) / (2M) * E_d * (1 + E_d / (2 M c²))
    E_d = displacement_energy_eV * e  # [J]

    # Simplified formula (non-relativistic approximation)
    E_threshold_J = (M / (2 * m_e)) * E_d
    V_threshold_kV = E_threshold_J / (e * 1000)

    return V_threshold_kV

# Knock-on thresholds for representative elements
elements = {
    'C': 12, 'Si': 28, 'Fe': 56, 'Cu': 64, 'Mo': 96, 'W': 184, 'Au': 197
}

element_names = list(elements.keys())
masses = list(elements.values())
thresholds = [knock_on_threshold_voltage(m) for m in masses]

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

# Left plot: Thresholds by element
colors = plt.cm.plasma(np.linspace(0.2, 0.9, len(elements)))
bars = ax1.bar(element_names, thresholds, color=colors, alpha=0.8, edgecolor='black', linewidth=1.5)

# Add typical acceleration voltages as reference lines
ax1.axhline(y=80, color='green', linestyle='--', linewidth=2, label='Low Voltage TEM (80 kV)')
ax1.axhline(y=200, color='orange', linestyle='--', linewidth=2, label='Standard TEM (200 kV)')
ax1.axhline(y=300, color='red', linestyle='--', linewidth=2, label='High Voltage TEM (300 kV)')

ax1.set_ylabel('Knock-on Threshold Voltage [kV]', fontsize=12)
ax1.set_title('Knock-on Damage Threshold for Elements', fontsize=14, fontweight='bold')
ax1.legend(fontsize=10, loc='upper left')
ax1.grid(axis='y', alpha=0.3)
ax1.set_ylim(0, 350)

# Numerical labels
for bar, threshold in zip(bars, thresholds):
    height = bar.get_height()
    ax1.text(bar.get_x() + bar.get_width()/2, height + 5, f'{threshold:.0f}',
             ha='center', fontsize=9, fontweight='bold')

# Right plot: Atomic mass vs threshold
ax2.scatter(masses, thresholds, s=200, c=masses, cmap='plasma',
            edgecolors='black', linewidths=2, zorder=3)
ax2.plot(masses, thresholds, 'k--', linewidth=1.5, alpha=0.5, zorder=1)

for name, mass, threshold in zip(element_names, masses, thresholds):
    ax2.text(mass, threshold + 10, name, fontsize=10, ha='center', fontweight='bold')

ax2.set_xlabel('Atomic Mass [u]', fontsize=12)
ax2.set_ylabel('Knock-on Threshold Voltage [kV]', fontsize=12)
ax2.set_title('Threshold Voltage vs Atomic Mass', fontsize=14, fontweight='bold')
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print("Knock-on damage threshold voltages:")
for name, mass, threshold in zip(element_names, masses, thresholds):
    print(f"  {name:2s} (M={mass:3.0f} u): {threshold:6.1f} kV")
print("\nCarbon (graphene, etc.) can reduce damage by observing below 80 kV")

1.6 Exercise Problems

Exercise 1-1: Calculating Electron Wavelength (Easy)

Problem: Calculate the electron wavelengths at acceleration voltages of 100 kV, 200 kV, and 300 kV, and find the wavelength ratio to visible light (550 nm).

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Problem: Calculate the electron wavelengths at acceleration 

Purpose: Demonstrate core concepts and implementation patterns
Target: Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

def calc_wavelength(V_kV):
    h = 6.62607e-34
    m0 = 9.10938e-31
    e = 1.60218e-19
    c = 2.99792e8
    E = V_kV * 1000 * e
    p = np.sqrt(2 * m0 * E * (1 + E / (2 * m0 * c**2)))
    return (h / p) * 1e12  # [pm]

voltages = [100, 200, 300]
visible_light_nm = 550

print("Comparison of electron wavelength with visible light:")
for V in voltages:
    wl_pm = calc_wavelength(V)
    wl_nm = wl_pm / 1000
    ratio = visible_light_nm / wl_nm
    print(f"  {V} kV: λ = {wl_pm:.3f} pm = {wl_nm:.6f} nm, ratio = {ratio:.0f}:1")

Exercise 1-2: Comparing Resolution (Medium)

Problem: Compare the resolution of optical microscopy (λ=550 nm, α=30°) and electron microscopy (200 kV, α=10°) using the Rayleigh criterion.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Problem: Compare the resolution of optical microscopy (λ=550

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

def resolution(wavelength_nm, alpha_deg):
    alpha_rad = np.deg2rad(alpha_deg)
    return 0.61 * wavelength_nm / np.sin(alpha_rad)

# Optical microscopy
optical_res = resolution(550, 30)

# Electron microscopy (200 kV → 2.508 pm = 0.002508 nm)
em_res = resolution(0.002508, 10)

print(f"Optical microscopy resolution: {optical_res:.1f} nm")
print(f"Electron microscopy resolution: {em_res:.5f} nm = {em_res * 10:.3f} Å")
print(f"Resolution improvement: {optical_res / em_res:.0f}×")

Exercise 1-3: Calculating Scattering Cross Section (Medium)

Problem: For a 200 kV electron beam, find the scattering cross section ratio at a scattering angle of 10° between Si (Z=14) and Au (Z=79).

Z_Si = 14
Z_Au = 79
E_keV = 200
theta = 10

# Scattering cross section is proportional to Z²
ratio = (Z_Au / Z_Si)**2

print(f"Scattering cross section ratio between Si (Z={Z_Si}) and Au (Z={Z_Au}):")
print(f"  σ_Au / σ_Si = ({Z_Au}/{Z_Si})² = {ratio:.1f}")
print(f"  Au scattering cross section is approximately {ratio:.0f}× that of Si")

Exercise 1-4: Improving Signal-to-Noise Ratio (Hard)

Problem: How much electron dose is needed to improve an electron microscopy image with a current S/N ratio of 10 to an S/N ratio of 100? Also calculate the S/N ratio improvement effect when averaging 10 images.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Problem: How much electron dose is needed to improve an elec

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

# Since S/N = √N, increasing S/N ratio by 10× requires 100× dose
current_snr = 10
target_snr = 100
dose_increase = (target_snr / current_snr)**2

print(f"Dose increase needed to improve S/N ratio from {current_snr} to {target_snr}:")
print(f"  Required dose increase: {dose_increase}×")

# Effect of averaging 10 images
num_images = 10
snr_improvement = np.sqrt(num_images)
new_snr = current_snr * snr_improvement

print(f"\nWhen averaging 10 images:")
print(f"  S/N ratio improvement factor: √{num_images} = {snr_improvement:.2f}")
print(f"  New S/N ratio: {current_snr} × {snr_improvement:.2f} = {new_snr:.1f}")

Exercise 1-5: Evaluating Knock-on Damage (Hard)

Problem: Evaluate whether knock-on damage occurs when observing graphene (carbon, M=12) with 200 kV TEM. Also propose the optimal acceleration voltage to avoid damage (displacement energy 25 eV).

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Problem: Evaluate whether knock-on damage occurs when observ

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

def knock_on_threshold(M, E_d=25):
    m_e = 9.10938e-31
    u = 1.66054e-27
    e = 1.60218e-19
    M_kg = M * u
    E_d_J = E_d * e
    E_threshold = (M_kg / (2 * m_e)) * E_d_J
    return E_threshold / (e * 1000)  # [kV]

M_C = 12
threshold_kV = knock_on_threshold(M_C, E_d=25)
operating_voltage = 200

print(f"Knock-on threshold for graphene (carbon, M={M_C}):")
print(f"  Threshold voltage: {threshold_kV:.1f} kV")
print(f"  Operating voltage: {operating_voltage} kV")

if operating_voltage > threshold_kV:
    print(f"  ⚠️ Observation at {operating_voltage} kV will cause knock-on damage")
    print(f"  Recommended voltage: {threshold_kV * 0.8:.0f} kV or below (20% safety margin)")
else:
    print(f"  ✅ Observation at {operating_voltage} kV is safe")

Exercise 1-6: CTF Zero Crossing (Hard)

Problem: Find the spatial frequency of the first CTF zero crossing for 200 kV TEM (Cs=1.0 mm, defocus -100 nm). This corresponds to real-space resolution.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

"""
Example: Problem: Find the spatial frequency of the first CTF zero cr

Purpose: Demonstrate optimization techniques
Target: Intermediate
Execution time: 10-30 seconds
Dependencies: None
"""

import numpy as np
from scipy.optimize import fsolve

voltage_kV = 200
Cs_mm = 1.0
defocus_nm = -100

# Electron wavelength
wavelength_pm = 2.508
wavelength_A = wavelength_pm / 100

# Unit conversion
Cs_A = Cs_mm * 1e7
defocus_A = defocus_nm * 10

# Find first non-trivial solution where CTF phase shift χ(k) = 0
def chi(k):
    return (np.pi * wavelength_A / 2) * k**2 * (defocus_A + 0.5 * wavelength_A**2 * k**2 * Cs_A)

# First zero crossing: sin(χ) = 0 → χ = π
def equation(k):
    return chi(k) - np.pi

k_zero = fsolve(equation, 2.0)[0]  # Initial estimate 2.0
resolution_A = 1 / k_zero

print(f"200 kV TEM (Cs={Cs_mm} mm, Δf={defocus_nm} nm):")
print(f"  First zero crossing spatial frequency: {k_zero:.3f} 1/Å")
print(f"  Corresponding real-space resolution: {resolution_A:.3f} Å = {resolution_A/10:.3f} nm")

Exercise 1-7: Optimizing Acceleration Voltage (Hard)

Problem: When observing a 100 nm thick Si specimen, discuss which acceleration voltage (80 kV or 200 kV) provides better transmission images, considering mean free path.

Exercise 1-8: Experimental Planning (Hard)

Problem: Plan an experimental approach to analyze unknown metal nanoparticles (10 nm diameter) by electron microscopy. Explain your choices of acceleration voltage, observation mode (TEM/SEM/STEM), and analytical methods (EDS/EELS).

1.7 Learning Check

Check your understanding by answering the following questions:

1. Can you derive de Broglie's wavelength equation and explain its relationship with acceleration voltage?
2. Can you understand the definition of resolution by Rayleigh criterion and explain the differences between optical and electron microscopy?
3. Can you explain the physical mechanisms of elastic and inelastic scattering and their detection signals?
4. Do you understand the atomic number dependence of scattering cross section ($Z^2$)?
5. Can you explain the types of electron guns (tungsten, LaB₆, FEG) and their characteristics?
6. Do you understand how spherical and chromatic aberrations affect images?
7. Can you explain the relationship between signal-to-noise ratio and Poisson statistics?
8. Do you understand the threshold energy for knock-on damage and strategies for reducing specimen damage?

1.8 References

1. Williams, D. B., & Carter, C. B. (2009). Transmission Electron Microscopy: A Textbook for Materials Science (2nd ed.). Springer. - Comprehensive electron microscopy textbook
2. Reimer, L., & Kohl, H. (2008). Transmission Electron Microscopy: Physics of Image Formation (5th ed.). Springer. - Detailed image formation theory
3. Goldstein, J. I., et al. (2017). Scanning Electron Microscopy and X-Ray Microanalysis (4th ed.). Springer. - Standard text for SEM and EDS analysis
4. Egerton, R. F. (2011). Electron Energy-Loss Spectroscopy in the Electron Microscope (3rd ed.). Springer. - EELS technique details
5. Spence, J. C. H. (2013). High-Resolution Electron Microscopy (4th ed.). Oxford University Press. - HRTEM theory
6. Kirkland, E. J. (2020). Advanced Computing in Electron Microscopy (3rd ed.). Springer. - Image simulation
7. De Graef, M. (2003). Introduction to Conventional Transmission Electron Microscopy. Cambridge University Press. - TEM introductory text

1.9 Next Chapter

In the next chapter, we will learn about scanning electron microscopy (SEM) principles, secondary electron (SE) and backscattered electron (BSE) imaging, energy dispersive X-ray analysis (EDS), and image analysis basics. SEM is a versatile technique for surface morphology observation and elemental analysis at high spatial resolution.