谢谢你！

Question

拟合参数：1.417070339751493E-44，b = 1.075485244260637E-15。

函数的定义与纸张完全相同：方程式 （

Https：//i.sstatic.net/nww2hypn.png

）

curve拟合

import numpy as np
from scipy.integrate import quad
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt

# Constants
kb = 1.380649e-23  # Boltzmann constant (J/K)
hbar = 1.0545718e-34  # Reduced Planck's constant (J*s)
theta_D = 280  # Debye temperature (K)
v = 2300  # Average phonon group velocity (m/s) -


# Define the inverse relaxation times
def tau_PD_inv(omega, A):
    return A * omega**4

def tau_U_inv(omega, B, T):
    return B * omega**2 * np.exp(-theta_D / (3 * T))

def tau_GB_inv(omega):
    D = 0.7e-6  # Grain boundary characteristic length (m)
    return v / D

def tau_total_inv(omega, A, B, T):
    return tau_PD_inv(omega, A) + tau_U_inv(omega, B, T) + tau_GB_inv(omega) 

# Debye Callaway model for thermal conductivity
def kappa_lattice(T, A, B):
    def integrand(x):
        omega = x * kb * T / hbar
        tau_total = 1 / tau_total_inv(omega, A, B, T)
        return x**4 * np.exp(x) / (np.exp(x) - 1)**2 * tau_total

    integral, _ = quad(integrand, 0, theta_D / T)
    prefactor = kb**4 * T**3 / (2 * np.pi**2 * v * hbar**3)
    return prefactor * integral

# Vectorize the model for curve fitting
def kappa_lattice_vectorized(T, A, B ):
    return np.array([kappa_lattice(Ti, A, B ) for Ti in T])

# Example thermal conductivity and temperature data for Ge WQ
T_data = np.array([3.0748730372823156, 3.1962737114707203, 3.4441443065638015, 3.785528026732028,
    4.043999986888003, 4.524033859711473, 4.902329259251676, 5.270E+00,
    5.902534836759174, 6.721645553370004, 7.690683028568884, 9.031E+00,
    1.103E+01, 12.811754075061538, 16.07373842258417, 19.67675447459597,
    23.951202854256202, 29.710517518825778, 3.684E+01, 45.030707655028536,
    56.92469139605285, 6.929E+01, 85.62796640159988, 105.41803279520694,
    129.5096325800293, 159.77670578646953, 200.9314386671018, 240.34130109997898,
    291.63127697234717])  # Temperature in K
kappa_data = np.array([ 0.5021456742832973, 0.5907837911587943, 0.7331309897165834, 0.9570228747931655,
    1.1281866396926783, 1.4904675008427906, 1.8682223847710366, 2.1856825810901013,
    2.8515011626042894, 4.057791636291224, 5.328111104734368, 7.645627510566391,
    11.045409412114584, 14.277055520040301, 19.54065663430087, 24.738891215975602,
    28.776422188514683, 32.562485211039274, 33.37572711709385, 32.455084248250884,
    28.56802589740088, 24.937473590597776, 20.887192910896978, 17.37073097716312,
    14.783541083880793, 12.26010669954394, 9.991973525044475, 8.531955488387288,
    7.347906481962733])  # Thermal conductivity in W/mK

# Initial guess for parameters A, B, and C
initial_guess = [4.75E-42, 1.054E-17]

# Curve fitting
popt, pcov = curve_fit(kappa_lattice_vectorized, T_data, kappa_data, p0=initial_guess, maxfev=8_000)
#popt, pcov = curve_fit(kappa_lattice_vectorized, T_data, kappa_data, p0=initial_guess, bounds=(0, np.inf))

# Extract fitted parameters
A_fit, B_fit = popt

# Print the fitted parameters
print(f"Fitted parameters: A = {A_fit}, B = {B_fit}")

# Generate curve fit data
T_fit = np.linspace(min(T_data), max(T_data), 100)
kappa_fit = kappa_lattice_vectorized(T_fit, *popt)


# Plot the original data, fitted data, and extrapolated data
plt.figure(figsize=(8, 6))
plt.plot(T_data, kappa_data, 'o', label='Experimental data')
plt.plot(T_fit, kappa_fit, '-', label='Fitted model')
#plt.plot(T_extrap, kappa_extrap, '--', label='Extrapolated (400-600K)')
plt.xlabel('Temperature (K)')
plt.ylabel('Thermal Conductivity (W/mK)')
plt.legend()
plt.title('Thermal Conductivity Fitting ')
plt.show()

您的功能忘了忘记考虑tau_U_inv。

校正版本应该是：

T

当我进行此更改时，参数显示

def tau_U_inv(omega, B, T):
    return B * omega**2 * T * np.exp(-theta_D / (3 * T))

Answer 1

似乎更靠近原件。