SPECTROSCOPYVerified Reference Parity

Lorentzian (Cauchy-Lorentz) Profile

The characteristic heavy-tailed peak profile describing natural radiative lifetime broadening and collisional damping.

Primary Disciplines:SpectroscopyQuantum OpticsNuclear Magnetic Resonance (NMR)Condensed Matter

Mathematical Formulation

$$y = y_0 + \frac{2A}{\pi} \frac{\gamma}{4(x - \mu)^2 + \gamma^2}$$
Plaintext:y = y0 + (2 * A / pi) * (gamma / (4 * (x - mu)^2 + gamma^2))

Parameters & Physical Meanings

Detailed parameter breakdown, standard units, valid mathematical constraints, and initial guess heuristic algorithms.

SymbolNameUnitConstraintsPhysical InterpretationInitial Guess Heuristic
$y_0$Baseline OffsetIntensity a.u.unconstrainedConstant background intensity levelEstimated from min(y) or median of boundary points
$A$Integrated Peak AreaIntensity · Spectral unitA > 0Total integrated spectral power under the Lorentzian line profileCalculated as (max(y) - y0) * gamma_init * pi / 2
$\mu$Peak Center PositionWavenumber / Frequencymin(x) <= mu <= max(x)Resonance center frequency of the optical or magnetic transitionLocation of maximum peak intensity: x[argmax(y)]
$\gamma$Full Width at Half Maximum (FWHM)Spectral unitsgamma > 0Direct measure of full line width at half-peak amplitude (gamma = FWHM = 2 * HWHM)Direct measurement of width between half-maximum points

When to Choose This Model

  • Fitting homogeneous lifetime-broadened optical transitions in solid-state and molecular physics
  • Deconvolving Nuclear Magnetic Resonance (NMR) chemical shift resonances
  • Analyzing Raman active phonon modes in crystalline lattices with finite damping
  • Modeling Fabry-Perot optical cavity transmission resonance peaks

Typical Scientific Applications

  • NMR Spectroscopy peak integration
  • Crystalline Raman vibrational mode analysis
  • Optical microcavity Q-factor estimation
  • Surface Plasmon Resonance (SPR) absorption fitting

Expected Fit Profile & Curve Morphology

Shape: peak
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 25-point Raman active phonon mode with heavy Lorentzian tails

R²:0.9991RMSE:0.9120

Expected Converged Parameters

  • y05.120
  • A612.400
  • mu520.010
  • gamma4.020

Sample Experimental Vectors (19 points)

#x (Independent)y (Observed)
1480.0005.600
2490.0006.400
3500.0008.200
4505.00010.300
5510.00015.600
6514.00028.400
7516.00045.200
8518.00078.600
+ 11 more points available in AltaiPlot preset

Python / SciPy Reference Implementation

Reference library: scipy.optimize (Levenberg-Marquardt).Exact parameterization parity verified.

import numpy as np
from scipy.optimize import curve_fit

def lorentzian_model(x, y0, A, mu, gamma):
    return y0 + (2.0 * A / np.pi) * (gamma / (4.0 * (x - mu)**2 + gamma**2))

y0_init = np.min(y_data)
mu_init = x_data[np.argmax(y_data)]
peak_height = np.max(y_data) - y0_init
# Approximate FWHM (gamma)
half_height = y0_init + peak_height / 2.0
indices = np.where(y_data >= half_height)[0]
gamma_init = np.max(x_data[indices]) - np.min(x_data[indices]) if len(indices) > 1 else (np.max(x_data) - np.min(x_data)) / 10.0
A_init = peak_height * gamma_init * np.pi / 2.0

p0 = [y0_init, max(A_init, 1e-6), mu_init, max(gamma_init, 1e-6)]
bounds = ([-np.inf, 0, np.min(x_data), 1e-12], [np.inf, np.inf, np.max(x_data), np.inf])

popt, pcov = curve_fit(lorentzian_model, x_data, y_data, p0=p0, bounds=bounds)
Numerical Verification Tolerance: Max Relative Error < 4.1e-6Verified: 2026-08-16

Comparative & Alternative Models

Key decision trade-offs between this model and related functional alternatives:

Alternative Model

Gaussian Peak Profile

The fundamental symmetric bell-shaped peak model describing thermal Doppler broadening and instrumental slit resolution.

$y = y_0 + A \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$
Alternative Model

Pseudo-Voigt Profile

A linear combination of Gaussian and Lorentzian functions providing high-speed approximation of Voigt profile line shapes.

$y = y_0 + A \left[ \eta \frac{2}{\pi \gamma} \frac{1}{1 + 4\left(\frac{x - \mu}{\gamma}\right)^2} + (1 - \eta) \frac{2\sqrt{\ln 2}}{\sqrt{\pi} \gamma} \exp\left( -4\ln 2 \left(\frac{x - \mu}{\gamma}\right)^2 \right) \right]$

Related Mathematical Models

Models sharing mathematical formulations or physical domains in AltaiPlot:

spectroscopy

Gaussian Peak Profile

The fundamental symmetric bell-shaped peak model describing thermal Doppler broadening and instrumental slit resolution.

$y = y_0 + A \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$
spectroscopy

Pseudo-Voigt Profile

A linear combination of Gaussian and Lorentzian functions providing high-speed approximation of Voigt profile line shapes.

$y = y_0 + A \left[ \eta \frac{2}{\pi \gamma} \frac{1}{1 + 4\left(\frac{x - \mu}{\gamma}\right)^2} + (1 - \eta) \frac{2\sqrt{\ln 2}}{\sqrt{\pi} \gamma} \exp\left( -4\ln 2 \left(\frac{x - \mu}{\gamma}\right)^2 \right) \right]$
spectroscopy

True Voigt Profile (Faddeeva Convolution)

The exact physical convolution of Gaussian Doppler broadening and Lorentzian lifetime broadening.

$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx' = \frac{\text{Re}[w(z)]}{\sigma \sqrt{2\pi}}$
spectroscopy

Pearson VII Profile

A versatile continuous peak model bridging pure Gaussian and Lorentzian shapes via a variable exponential exponent (m).

$y = y_0 + A \left[ 1 + 4\left( 2^{1/m} - 1 \right) \left( \frac{x - \mu}{\gamma} \right)^2 \right]^{-m}$

Frequently Asked Questions (Lorentzian (Cauchy-Lorentz) Profile)

Why are Lorentzian fits more sensitive to baseline noise in the tails than Gaussian fits?

Because Lorentzian decay is algebraic (1/x²) compared to Gaussian exponential decay (exp(-x²)). The wide wings extend far from the peak center, making accurate baseline correction essential before fitting.

Scientific References & Citations

  • Marshall, A. G., & Verdun, F. R. (1990). Fourier Transforms in NMR, Optical, and Mass Spectrometry. Elsevier Science.[Source / DOI]