Chapter 3: Magnetic Measurements

Magnetic measurements are techniques for quantitatively evaluating the magnetic properties of materials (magnetization, magnetic moment, magnetic anisotropy). In this chapter, we will learn the principles of VSM (Vibrating Sample Magnetometer) and SQUID (Superconducting Quantum Interference Device), M-H curve analysis (saturation magnetization, coercivity, remanence), fitting with Curie-Weiss law and Langevin function, temperature-dependent measurements (FC/ZFC), and integrated measurement techniques using PPMS (Physical Property Measurement System), and perform practical magnetic data analysis with Python.

3.1 Fundamentals of Magnetism

3.1.1 Relationship Between Magnetization and Magnetic Field

Magnetization $M$ is the magnetic moment per unit volume:

where $\mu_i$ is the magnetic moment of individual atoms/molecules, and $V$ is the volume.

Magnetic susceptibility $\chi$ is the response of magnetization to magnetic field $H$:

Types of magnetic fields:

• Magnetic field $H$ (magnetic field strength): unit A/m or Oe (Oersted)
• Magnetic flux density $B$: unit T (Tesla)
• Relationship: $B = \mu_0(H + M)$ (SI units), $\mu_0 = 4\pi \times 10^{-7}$ H/m

3.1.2 Classification of Magnetism

3.2 VSM (Vibrating Sample Magnetometer)

3.2.1 VSM Measurement Principle

VSM (developed by Simon Foner in 1956) is a method that vibrates the sample and measures the voltage induced in detection coils by its magnetic moment.

Measurement principle:

1. Place the sample in a uniform magnetic field and vibrate it vertically at frequency $f$ (typically 10-100 Hz)
2. When the sample's magnetic moment $\mu$ vibrates, a time-varying magnetic flux $\Phi(t)$ is generated around the detection coils
3. According to Faraday's law of electromagnetic induction, induced voltage $V(t)$ is generated in the coils: $$ V(t) = -\frac{d\Phi}{dt} \propto \mu \cdot f $$
4. Lock-in detection extracts the component at vibration frequency $f$ to quantify the magnetic moment

VSM characteristics:

• Sensitivity: $10^{-6}$ - $10^{-8}$ emu (electromagnetic unit)
• Magnetic field range: 0 - 3 T (typical electromagnet)
• Temperature range: 4 K - 1000 K (with liquid He cryostat)
• Measurement time: 0.1 - 1 second per point (relatively fast)

Code Example 3-1: Curie-Weiss Law Fitting

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

import numpy as np
import matplotlib.pyplot as plt
from lmfit import Model

def curie_weiss(T, C, theta):
    """
    Curie-Weiss law: χ = C / (T - θ)
    
    Parameters
    ----------
    T : array-like
        Temperature [K]
    C : float
        Curie constant
    theta : float
        Curie-Weiss temperature (Weiss constant) [K]
    
    Returns
    -------
    chi : array-like
        Magnetic susceptibility
    """
    return C / (T - theta)

# Generate simulation data (paramagnetic material)
T_range = np.linspace(100, 400, 30)  # [K]
C_true = 1.5  # Curie constant
theta_true = -10  # Curie-Weiss temperature [K] (negative → antiferromagnetic interaction)

chi_data = curie_weiss(T_range, C_true, theta_true)
chi_data_noise = chi_data * (1 + 0.05 * np.random.randn(len(T_range)))  # 5% noise

# Fitting
model = Model(curie_weiss)
params = model.make_params(C=1.0, theta=0)
result = model.fit(chi_data_noise, params, T=T_range)

print("Curie-Weiss Law Fitting Results:")
print(result.fit_report())

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# Left: χ vs T
ax1.scatter(T_range, chi_data_noise, s=80, alpha=0.7, edgecolors='black', linewidths=1.5, label='Measured data', color='#f093fb')
ax1.plot(T_range, result.best_fit, linewidth=2.5, label=f'Fit: C={result.params["C"].value:.2f}, θ={result.params["theta"].value:.1f} K', color='#f5576c')
ax1.set_xlabel('Temperature T [K]', fontsize=12)
ax1.set_ylabel('Magnetic Susceptibility χ', fontsize=12)
ax1.set_title('Curie-Weiss Law: χ vs T', fontsize=14, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)

# Right: 1/χ vs T (Curie-Weiss plot)
ax2.scatter(T_range, 1/chi_data_noise, s=80, alpha=0.7, edgecolors='black', linewidths=1.5, label='Data (1/χ)', color='#ffa500')
T_fit_extended = np.linspace(0, 450, 100)
chi_fit_extended = curie_weiss(T_fit_extended, result.params['C'].value, result.params['theta'].value)
ax2.plot(T_fit_extended, 1/chi_fit_extended, linewidth=2.5, label='Linear fit (extended)', color='#f5576c')

# Highlight θ
theta_fit = result.params['theta'].value
ax2.axvline(theta_fit, color='red', linestyle='--', linewidth=2, label=f'θ = {theta_fit:.1f} K')
ax2.scatter([theta_fit], [0], s=200, c='red', edgecolors='black', linewidth=2, marker='X', zorder=5)

ax2.set_xlabel('Temperature T [K]', fontsize=12)
ax2.set_ylabel('1/χ', fontsize=12)
ax2.set_title('Curie-Weiss Plot: 1/χ vs T', fontsize=14, fontweight='bold')
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)
ax2.set_xlim(0, 450)
ax2.set_ylim(bottom=0)

plt.tight_layout()
plt.show()

print(f"\nPhysical Interpretation:")
print(f"  Curie constant C = {result.params['C'].value:.2f}")
print(f"  Curie-Weiss temperature θ = {result.params['theta'].value:.1f} K")
if result.params['theta'].value < 0:
    print(f"  θ < 0 → Dominant antiferromagnetic interactions")
elif result.params['theta'].value > 0:
    print(f"  θ > 0 → Dominant ferromagnetic interactions")
else:
    print(f"  θ ≈ 0 → Ideal paramagnetism (no interactions)")

3.3 SQUID (Superconducting Quantum Interference Device)

3.3.1 SQUID Measurement Principle

SQUID (Superconducting Quantum Interference Device) is an ultra-high sensitivity magnetic sensor using superconducting rings and Josephson junctions.

Measurement principle:

1. When magnetic flux $\Phi$ penetrates a superconducting ring, quantized superconducting current flows
2. The critical current through the Josephson junction changes periodically with magnetic flux: $$ I_c(\Phi) = I_0 \left|\cos\left(\frac{\pi\Phi}{\Phi_0}\right)\right| $$ where $\Phi_0 = h/(2e) \approx 2.07 \times 10^{-15}$ Wb is the magnetic flux quantum
3. Minute changes in magnetic flux are detected with extremely high sensitivity as changes in critical current

SQUID characteristics:

• Sensitivity: $10^{-10}$ - $10^{-12}$ emu (over 1000 times better than VSM)
• Magnetic field range: 0 - 7 T (superconducting magnet)
• Temperature range: 1.8 K - 400 K (with He gas)
• Measurement time: 1 - 10 seconds per point (slower than VSM, but higher precision)
• Noise level: $10^{-14}$ T/√Hz (world's highest sensitivity magnetic sensor)

Code Example 3-2: Langevin Function Fitting (Paramagnetism)

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

import numpy as np
import matplotlib.pyplot as plt
from lmfit import Model

def langevin(H, M_s, T, mu):
    """
    Langevin function (classical paramagnetism)
    
    Parameters
    ----------
    H : array-like
        Magnetic field [A/m or Oe]
    M_s : float
        Saturation magnetization [emu/g or Am^2/kg]
    T : float
        Temperature [K]
    mu : float
        Magnetic moment per particle [Bohr magneton units]
    
    Returns
    -------
    M : array-like
        Magnetization [emu/g]
    """
    mu_B = 9.274e-24  # Bohr magneton [J/T]
    mu_0 = 4 * np.pi * 1e-7  # Vacuum permeability [H/m]
    k_B = 1.38065e-23  # Boltzmann constant [J/K]
    
    # Convert field to SI units (Oe → T)
    H_T = H * 1e-4  # 1 Oe = 10^-4 T
    
    # Langevin parameter
    xi = mu * mu_B * H_T / (k_B * T)
    
    # Langevin function: L(ξ) = coth(ξ) - 1/ξ
    L = np.where(np.abs(xi) < 1e-3, xi/3, 1/np.tanh(xi) - 1/xi)  # Avoid divergence for small ξ
    
    M = M_s * L
    return M

# Generate simulation data (superparamagnetic nanoparticles)
H_range = np.linspace(-10000, 10000, 100)  # [Oe]
M_s_true = 50  # Saturation magnetization [emu/g]
T_true = 300  # Temperature [K]
mu_true = 5000  # Particle magnetic moment [μ_B]

M_data = langevin(H_range, M_s_true, T_true, mu_true)
M_data_noise = M_data + 0.5 * np.random.randn(len(H_range))  # Add noise

# Fitting
model = Model(langevin)
params = model.make_params(M_s=40, T=300, mu=3000)
params['T'].vary = False  # Fix temperature

result = model.fit(M_data_noise, params, H=H_range)

print("Langevin Function Fitting Results:")
print(result.fit_report())

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# Left: M vs H
ax1.scatter(H_range, M_data_noise, s=50, alpha=0.6, edgecolors='black', linewidths=1, label='Data (with noise)', color='#f093fb')
ax1.plot(H_range, result.best_fit, linewidth=2.5, label=f'Fit: M_s={result.params["M_s"].value:.1f} emu/g, μ={result.params["mu"].value:.0f} μ_B', color='#f5576c')
ax1.axhline(M_s_true, color='green', linestyle='--', linewidth=1.5, label=f'True M_s = {M_s_true} emu/g')
ax1.axhline(-M_s_true, color='green', linestyle='--', linewidth=1.5)
ax1.set_xlabel('Magnetic Field H [Oe]', fontsize=12)
ax1.set_ylabel('Magnetization M [emu/g]', fontsize=12)
ax1.set_title('Langevin Function Fit (Paramagnetism)', fontsize=14, fontweight='bold')
ax1.legend(fontsize=10)
ax1.grid(alpha=0.3)

# Right: Temperature dependence (Langevin curves at different temperatures)
T_list = [50, 100, 200, 300, 400]
colors = plt.cm.plasma(np.linspace(0, 0.9, len(T_list)))

for T, color in zip(T_list, colors):
    M_T = langevin(H_range, result.params['M_s'].value, T, result.params['mu'].value)
    ax2.plot(H_range, M_T, linewidth=2.5, label=f'T = {T} K', color=color)

ax2.set_xlabel('Magnetic Field H [Oe]', fontsize=12)
ax2.set_ylabel('Magnetization M [emu/g]', fontsize=12)
ax2.set_title('Temperature Dependence of Langevin Curve', fontsize=14, fontweight='bold')
ax2.legend(fontsize=10, loc='lower right')
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.show()

print(f"\nPhysical Interpretation:")
print(f"  Saturation magnetization M_s = {result.params['M_s'].value:.1f} emu/g")
print(f"  Particle magnetic moment μ = {result.params['mu'].value:.0f} μ_B")
print(f"  Lower temperature approaches saturation at lower field (reduced thermal fluctuation)")