423 lines
15 KiB
Python
423 lines
15 KiB
Python
#!/usr/bin/env python3
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"""
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model_comparison_v24.py
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Compares different models for "atoms are balls":
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1. Single effective radius (our paper's approach)
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2. Multi-shell calculation (sum over all electrons)
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3. Bolas model (paired electrons as dumbbells)
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Author: Andre Heinecke & AI Collaborators
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Date: June 2025
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"""
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib.patches import Circle, FancyBboxPatch
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import matplotlib.patches as mpatches
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import pandas as pd
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# Physical constants
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HBAR = 1.054571817e-34 # J·s
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ME = 9.1093837015e-31 # kg
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E = 1.602176634e-19 # C
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K = 8.9875517923e9 # N·m²/C²
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A0 = 5.29177210903e-11 # m
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# Electron configurations and screening constants
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CONFIGS = {
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'H': {'shells': [(1, 0, 1, 1.0)]}, # (n, l, num_electrons, Z_eff)
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'He': {'shells': [(1, 0, 2, 1.69)]},
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'C': {'shells': [(1, 0, 2, 5.67), (2, 0, 2, 3.22), (2, 1, 2, 3.14)]},
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'Ne': {'shells': [(1, 0, 2, 9.64), (2, 0, 2, 6.52), (2, 1, 6, 6.52)]},
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'Fe': {'shells': [(1, 0, 2, 25.38), (2, 0, 2, 22.36), (2, 1, 6, 22.36),
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(3, 0, 2, 13.18), (3, 1, 6, 13.18), (3, 2, 6, 9.1), (4, 0, 2, 3.75)]},
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}
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def single_shell_force(Z_eff, n=1, l=0):
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"""Calculate force using single effective radius"""
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r = n * A0 / Z_eff
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if l == 2: # d-orbital correction
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r *= 0.35
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s = 1 if l < 2 else 2
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F = HBAR**2 * s**2 / (ME * r**3)
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return F, r
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def multi_shell_approaches(config):
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"""Calculate force using different multi-shell approaches"""
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results = {}
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shell_data = []
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# First, calculate data for each shell
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for n, l, num_e, Z_eff in config['shells']:
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r = n * A0 / Z_eff
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if l == 2: # d-orbital correction
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r *= 0.35
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# Force per electron at this shell
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s = 1 if l < 2 else 2
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F_per_e = HBAR**2 * s**2 / (ME * r**3)
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shell_data.append({
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'n': n, 'l': l, 'num_e': num_e, 'r': r,
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'F_per_e': F_per_e, 'Z_eff': Z_eff
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})
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# Approach 1: Outermost electron (surface of the ball)
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outermost = shell_data[-1] # Last shell
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results['outermost'] = {
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'force': outermost['F_per_e'],
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'radius': outermost['r'],
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'description': f"{outermost['n']}{['s','p','d','f'][outermost['l']]} electron"
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}
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# Approach 2: Mean radius weighted by electron count
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total_electrons = sum(s['num_e'] for s in shell_data)
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weighted_r = sum(s['r'] * s['num_e'] for s in shell_data) / total_electrons
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# Calculate force at mean radius
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mean_Z_eff = sum(s['Z_eff'] * s['num_e'] for s in shell_data) / total_electrons
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s_mean = 1.5 # Average angular momentum
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F_mean = HBAR**2 * s_mean**2 / (ME * weighted_r**3)
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results['mean_radius'] = {
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'force': F_mean,
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'radius': weighted_r,
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'description': f"Mean radius = {weighted_r:.3e} m"
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}
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# Approach 3: RMS (root mean square) radius
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rms_r = np.sqrt(sum(s['r']**2 * s['num_e'] for s in shell_data) / total_electrons)
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F_rms = HBAR**2 * s_mean**2 / (ME * rms_r**3)
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results['rms_radius'] = {
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'force': F_rms,
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'radius': rms_r,
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'description': f"RMS radius = {rms_r:.3e} m"
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}
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# Approach 4: Innermost electron (maximum binding)
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innermost = shell_data[0]
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results['innermost'] = {
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'force': innermost['F_per_e'],
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'radius': innermost['r'],
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'description': "1s electron (strongest bound)"
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}
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return results, shell_data
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def bolas_force(Z_eff, n=1, num_pairs=1, separation_factor=0.1):
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"""Calculate force for bolas model (electron pairs as dumbbells)"""
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r = n * A0 / Z_eff
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ell = separation_factor * r # separation within pair
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# Modified force for rotating dumbbell
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correction = 1 / (1 + (ell/r)**2)
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F_pair = 2 * HBAR**2 / (ME * r**3) * correction
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return F_pair * num_pairs, r, ell
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def coulomb_force(Z_eff, r):
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"""Standard Coulomb force for comparison"""
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return K * Z_eff * E**2 / r**2
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def visualize_models():
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"""Create visual representation of the three models"""
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fig, axes = plt.subplots(1, 3, figsize=(15, 5))
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# Model 1: Single Shell (Hollow Ball)
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ax1 = axes[0]
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ax1.set_xlim(-3, 3)
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ax1.set_ylim(-3, 3)
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ax1.set_aspect('equal')
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# Nucleus
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nucleus1 = Circle((0, 0), 0.2, color='red', zorder=10)
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ax1.add_patch(nucleus1)
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# Single effective shell
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shell1 = Circle((0, 0), 2, fill=False, linestyle='--', linewidth=2, color='blue')
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ax1.add_patch(shell1)
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# Electrons
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for angle in np.linspace(0, 2*np.pi, 6, endpoint=False):
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x, y = 2*np.cos(angle), 2*np.sin(angle)
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electron = Circle((x, y), 0.15, color='blue')
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ax1.add_patch(electron)
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ax1.set_title('Single Shell Model\n(Our Approach)', fontsize=14)
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ax1.axis('off')
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# Model 2: Multi-Shell (Nested Balls)
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ax2 = axes[1]
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ax2.set_xlim(-3, 3)
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ax2.set_ylim(-3, 3)
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ax2.set_aspect('equal')
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# Nucleus
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nucleus2 = Circle((0, 0), 0.2, color='red', zorder=10)
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ax2.add_patch(nucleus2)
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# Multiple shells
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radii = [0.8, 1.5, 2.3]
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colors = ['darkblue', 'blue', 'lightblue']
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for r, color in zip(radii, colors):
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shell = Circle((0, 0), r, fill=False, linestyle='--', linewidth=2, color=color)
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ax2.add_patch(shell)
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# Add electrons
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n_electrons = 2 if r < 1 else 4
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for angle in np.linspace(0, 2*np.pi, n_electrons, endpoint=False):
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x, y = r*np.cos(angle), r*np.sin(angle)
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electron = Circle((x, y), 0.1, color=color)
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ax2.add_patch(electron)
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ax2.set_title('Multi-Shell Model\n(Sum Over All Electrons)', fontsize=14)
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ax2.axis('off')
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# Model 3: Bolas (Paired Electrons)
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ax3 = axes[2]
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ax3.set_xlim(-3, 3)
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ax3.set_ylim(-3, 3)
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ax3.set_aspect('equal')
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# Nucleus
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nucleus3 = Circle((0, 0), 0.2, color='red', zorder=10)
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ax3.add_patch(nucleus3)
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# Bolas pairs
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for angle, r in [(0, 1.5), (np.pi/2, 1.5), (np.pi, 1.5)]:
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# Calculate positions for paired electrons
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center_x = r * np.cos(angle)
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center_y = r * np.sin(angle)
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# Perpendicular offset for pair
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offset_angle = angle + np.pi/2
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offset = 0.3
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x1 = center_x + offset * np.cos(offset_angle)
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y1 = center_y + offset * np.sin(offset_angle)
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x2 = center_x - offset * np.cos(offset_angle)
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y2 = center_y - offset * np.sin(offset_angle)
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# Draw electrons
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e1 = Circle((x1, y1), 0.15, color='blue')
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e2 = Circle((x2, y2), 0.15, color='red')
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ax3.add_patch(e1)
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ax3.add_patch(e2)
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# Draw connection
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ax3.plot([x1, x2], [y1, y2], 'k-', linewidth=2)
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ax3.set_title('Bolas Model\n(Paired Electron Dumbbells)', fontsize=14)
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ax3.axis('off')
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plt.tight_layout()
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plt.savefig('atomic_models_comparison.png', dpi=300, bbox_inches='tight')
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return fig
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def compare_carbon_models():
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"""Detailed comparison for carbon atom"""
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print("\n" + "="*70)
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print("CARBON ATOM - COMPARING DIFFERENT MODELS")
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print("="*70)
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# Model 1: Single effective shell (our paper)
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print("\nMODEL 1: Single Effective Shell (Our Paper's Approach)")
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Z_eff_2p = 3.14 # For 2p electron
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F_single, r_single = single_shell_force(Z_eff_2p, n=2, l=1)
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F_coulomb_single = coulomb_force(Z_eff_2p, r_single)
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print(f" Using 2p electron (outermost): Z_eff = {Z_eff_2p}")
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print(f" Radius: r = {r_single:.3e} m")
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print(f" F_spin = {F_single:.3e} N")
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print(f" F_coulomb = {F_coulomb_single:.3e} N")
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print(f" Agreement: {(F_single/F_coulomb_single)*100:.1f}%")
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# Model 2: Multi-shell approaches
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print("\nMODEL 2: Multi-Shell Approaches")
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approaches, shell_details = multi_shell_approaches(CONFIGS['C'])
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print("\n Shell breakdown:")
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for shell in shell_details:
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orbital = ['s', 'p', 'd', 'f'][shell['l']]
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print(f" {shell['n']}{orbital}: {shell['num_e']} electrons at r={shell['r']:.3e} m")
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print(f" F_per_electron = {shell['F_per_e']:.3e} N")
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print("\n Different radius choices:")
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for approach_name, data in approaches.items():
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F_c = coulomb_force(data['radius'] * A0 / data['radius'], data['radius'])
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print(f"\n {approach_name.replace('_', ' ').title()}:")
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print(f" {data['description']}")
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print(f" F_spin = {data['force']:.3e} N")
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print(f" Ratio to single-shell = {data['force']/F_single:.2f}")
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# Model 3: Bolas (treating pairs)
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print("\n\nMODEL 3: Bolas Model (Electron Pairs)")
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# For carbon, we can have the 2p² electrons form a bolas
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Z_eff_bolas = 3.14
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F_bolas, r_bolas, ell = bolas_force(Z_eff_bolas, n=2, num_pairs=1)
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F_coulomb_bolas = coulomb_force(Z_eff_bolas, r_bolas) * 2 # 2 electrons
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print(f" 2p² electron pair: r={r_bolas:.3e} m, separation={ell:.3e} m")
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print(f" F_bolas (for pair) = {F_bolas:.3e} N")
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print(f" F_coulomb (for 2 electrons) = {F_coulomb_bolas:.3e} N")
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print(f" Agreement: {(F_bolas/F_coulomb_bolas)*100:.1f}%")
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# Summary comparison
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print("\n" + "-"*70)
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print("SUMMARY:")
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print(f" Our approach (single 2p): F = {F_single:.3e} N")
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print(f" Outermost shell (same as our): F = {approaches['outermost']['force']:.3e} N")
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print(f" Mean radius approach: F = {approaches['mean_radius']['force']:.3e} N")
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print(f" RMS radius approach: F = {approaches['rms_radius']['force']:.3e} N")
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print(f" Innermost (1s) shell: F = {approaches['innermost']['force']:.3e} N")
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print(f" Bolas model (2p pair): F = {F_bolas:.3e} N")
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print("\nKey insight: The outermost shell approach (which we use) makes the most")
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print("physical sense because it represents the 'surface' of the atomic ball.")
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return {
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'single': F_single,
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'outermost': approaches['outermost']['force'],
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'mean': approaches['mean_radius']['force'],
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'innermost': approaches['innermost']['force'],
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'bolas': F_bolas
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}
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def test_all_elements():
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"""Compare models for multiple elements"""
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elements = ['H', 'He', 'C', 'Ne', 'Fe']
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results = []
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print("\n" + "="*70)
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print("COMPARISON ACROSS ELEMENTS")
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print("="*70)
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print(f"{'Element':<8} {'Single Shell':<12} {'Mean Radius':<12} {'Innermost':<12} {'Outermost':<12}")
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print("-"*60)
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for elem in elements:
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if elem in CONFIGS:
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config = CONFIGS[elem]
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# Single shell (use outermost) - Our paper's approach
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last_shell = config['shells'][-1]
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n, l, _, Z_eff = last_shell
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F_single, _ = single_shell_force(Z_eff, n, l)
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# Multi-shell approaches
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approaches, _ = multi_shell_approaches(config)
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F_mean = approaches['mean_radius']['force']
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F_inner = approaches['innermost']['force']
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F_outer = approaches['outermost']['force']
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print(f"{elem:<8} {F_single:<12.3e} {F_mean:<12.3e} {F_inner:<12.3e} {F_outer:<12.3e}")
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results.append({
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'element': elem,
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'single': F_single,
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'mean': F_mean,
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'inner': F_inner,
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'outer': F_outer
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})
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print("\nNote: 'Single Shell' and 'Outermost' should be identical (our paper's approach)")
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print(" 'Innermost' shows the 1s electron force (strongest binding)")
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print(" 'Mean Radius' uses electron-weighted average radius")
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return results
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def main():
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"""Run all comparisons and generate visualizations"""
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print("MODEL COMPARISON FOR 'ATOMS ARE BALLS' - VERSION 24")
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print("Exploring different interpretations of atomic 3D structure")
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# Visual comparison
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print("\nGenerating visual comparison of models...")
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fig = visualize_models()
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print("Saved: atomic_models_comparison.png")
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# Detailed carbon comparison
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carbon_results = compare_carbon_models()
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# Multi-element comparison
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multi_elem_results = test_all_elements()
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# Generate comparison plot
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fig2, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
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elements = [r['element'] for r in multi_elem_results]
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single_forces = [r['single'] for r in multi_elem_results]
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mean_forces = [r['mean'] for r in multi_elem_results]
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inner_forces = [r['inner'] for r in multi_elem_results]
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x = np.arange(len(elements))
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width = 0.25
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# Force comparison
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bars1 = ax1.bar(x - width, single_forces, width, label='Single Shell (Our Paper)', alpha=0.8)
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bars2 = ax1.bar(x, mean_forces, width, label='Mean Radius', alpha=0.8)
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bars3 = ax1.bar(x + width, inner_forces, width, label='Innermost (1s)', alpha=0.8)
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ax1.set_ylabel('Force (N)', fontsize=12)
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ax1.set_xlabel('Element', fontsize=12)
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ax1.set_title('Force Comparison: Different Radius Approaches', fontsize=14)
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ax1.set_xticks(x)
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ax1.set_xticklabels(elements)
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ax1.legend()
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ax1.set_yscale('log')
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ax1.grid(True, alpha=0.3)
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# Radius comparison
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radii_data = []
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for elem in elements:
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if elem in CONFIGS:
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approaches, _ = multi_shell_approaches(CONFIGS[elem])
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radii_data.append({
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'element': elem,
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'outermost': approaches['outermost']['radius'],
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'mean': approaches['mean_radius']['radius'],
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'rms': approaches['rms_radius']['radius'],
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'innermost': approaches['innermost']['radius']
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})
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radii_df = pd.DataFrame(radii_data)
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for i, col in enumerate(['outermost', 'mean', 'rms', 'innermost']):
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ax2.plot(x, radii_df[col].values, 'o-', label=col.title(), markersize=8)
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ax2.set_ylabel('Radius (m)', fontsize=12)
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ax2.set_xlabel('Element', fontsize=12)
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ax2.set_title('Radius Comparison: Different Approaches', fontsize=14)
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ax2.set_xticks(x)
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ax2.set_xticklabels(elements)
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ax2.legend()
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ax2.set_yscale('log')
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ax2.grid(True, alpha=0.3)
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plt.tight_layout()
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plt.savefig('force_model_comparison_corrected.png', dpi=300, bbox_inches='tight')
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print("\nSaved: force_model_comparison_corrected.png")
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# Conclusions
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print("\n" + "="*70)
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print("CONCLUSIONS:")
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print("="*70)
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print("\n1. Different radius choices give different forces:")
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print(" - Outermost shell (our approach): Represents the atomic 'surface'")
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print(" - Mean radius: Gives intermediate forces")
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print(" - RMS radius: Weights larger radii more heavily")
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print(" - Innermost shell: Gives much larger forces (10-40x)")
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print("\n2. The bolas model for electron pairs gives reasonable agreement,")
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print(" suggesting mechanical coupling between paired electrons")
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print("\n3. Our single-shell model using the outermost electron makes sense because:")
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print(" - It defines the atom's effective size")
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print(" - Chemical properties depend on valence (outer) electrons")
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print(" - The geometric principle F ∝ r⁻³ works at any radius")
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print("\n4. All approaches confirm atoms are 3D rotating systems, not 2D abstractions")
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plt.show()
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if __name__ == "__main__":
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main() |