{"id":220,"date":"2025-05-25T05:32:54","date_gmt":"2025-05-25T05:32:54","guid":{"rendered":"https:\/\/synchrotron-light.net\/wp\/?page_id=220"},"modified":"2025-07-20T05:13:20","modified_gmt":"2025-07-20T05:13:20","slug":"chapter-6-quantum-effects-in-relativistic-charged-particle-orbits","status":"publish","type":"page","link":"https:\/\/synchrotron-light.net\/wp\/?page_id=220","title":{"rendered":"Chapter 7: Quantum effects in relativistic charged-particle orbits"},"content":{"rendered":"

Below is a set of python codes associated with Chapter 7 of Daniele Pelliccia and David M. Paganin, “Synchrotron Light: A Physics Journey from Laboratory to Cosmos” (Oxford University Press, 2025).<\/p>\n

In order to run any of these python codes, you will need to include the following header file.<\/p>\n

import numpy as np\r\nimport matplotlib.pyplot as plt\r\nfrom scipy.special import genlaguerre, factorial\r\n\r\nsynchrotron_style = {\r\n    'font.family': 'FreeSerif',\r\n    'font.size': 30,\r\n    'axes.linewidth': 2.0,\r\n    'axes.edgecolor': 'black',\r\n    'grid.color': '#5b5b5b',\r\n    'grid.linewidth': 1.2,\r\n}<\/pre>\n
Allowed energy levels<\/h2>\n
See Fig. 7.2.<\/p>\n
bb = np.logspace(-2,-10, 9)\r\nn = np.logspace(0,12,1000)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(25,15))\r\n\r\n    for ratio in bb:\r\n\r\n        ax.scatter(0.1\/ratio, np.sqrt(1+2*ratio*(0.1\/ratio+0.5)), s = 150, c='g',zorder=10)\r\n        ax.scatter(10\/ratio, np.sqrt(1+2*ratio*(10\/ratio+0.5)), s = 150, c='r',zorder=10)\r\n        ax.loglog(n, np.sqrt(1+2*ratio*(n+0.5)), lw = 4, label = str(ratio), zorder=0)\r\n\r\n    ax.tick_params(axis='x', labelsize=40, pad=10)\r\n    plt.yticks(fontsize = 40)\r\n\r\n    plt.xlabel('n', fontsize=40, labelpad=10)\r\n    plt.ylabel(\"$\\\\mathcal{E}\/ (m_{e}c^{2})$\", rotation=90, fontsize=40, labelpad=-1)\r\n    plt.legend(fontsize = 40, frameon=True, title=\"$B\/B_{*}$\")\r\n    plt.show()<\/pre>\n
\"\"<\/p>\n
“Fuzzy ring” thickness<\/h2>\n
See Fig. 7.4.<\/p>\n
bb = np.logspace(-8,8, 1000)\r\nhbar = 1.054571817e-34\r\nee = 1.602176634e-19\r\n\r\nW = np.sqrt(2*hbar\/(ee*bb))\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(12,10))\r\n    ax.loglog(bb, W, lw = 8)\r\n\r\n    plt.xticks(fontsize = 34)\r\n    plt.yticks(fontsize = 34)\r\n    ax.tick_params(axis='x', pad=10)\r\n\r\n    plt.xlabel('$B$ (T)', fontsize=30, labelpad=10)\r\n    plt.ylabel('$W$ (m)', fontsize=30, labelpad=10)\r\n    # plt.savefig('ring-thickness-versus-magnetic-field.png', dpi=600, transparent=False)\r\n    plt.show()<\/pre>\n
\"\"<\/p>\n
“Fuzzy ring” probability density<\/h2>\n
See Fig. 7.5.<\/p>\n
r = np.linspace(0,20,5000)\r\ntheta = np.linspace(0, 2*np.pi, 1000)\r\nbb=0.4\r\nn=50\r\nR, Th = np.meshgrid(r, theta)\r\npkg = np.exp(-bb*R**2)*R**(2*n)\r\nX = R*np.cos(Th)\r\nY = R*np.sin(Th)\r\n\r\nfig = plt.figure(figsize=(15,15))\r\nplt.contourf(X,Y,pkg, levels=50, cmap='Purples')\r\nplt.axis('off')\r\nplt.show()<\/pre>\n
\"\"<\/h1>\n
Radial line profiles of Klein-Gordon wavefunctions<\/h2>\n
See Fig. 7.6.<\/p>\n
r = np.linspace(0,20,2000)\r\nell = 100\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(3,2, figsize=(16,12))\r\n    fig.subplots_adjust(wspace=5)\r\n    \r\n    for ax, s in zip(axs.flat, [0,1,2,5,10,20]):\r\n   \r\n        func = factorial(s)\/(factorial(s+ell))*np.exp(-r**2)*r**(2*ell)*(genlaguerre(s, ell)(r**2))**2\r\n    \r\n        ax.plot(r, func, lw=3)\r\n        ax.set_title(\"$s= $\"+str(s)+\", $\\\\ell= $100\", fontsize=24)\r\n        ax.set_xlabel('$\\\\breve{r}$', fontsize=28, labelpad=1)\r\n        \r\n        ax.tick_params(axis='x', labelsize=30)\r\n        ax.tick_params(axis='y', labelsize=30)\r\n        \r\nplt.tight_layout()    \r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Density plots of Klein-Gordon wavefunctions<\/h2>\n
See Fig. 7.7.<\/p>\n
x = np.linspace(-17,17,1000)\r\ny = np.linspace(-17,17,1000)\r\nX,Y = np.meshgrid(x,y)\r\nell = 100\r\n\r\nwith plt.style.context(('default')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(3,2, figsize=(11.5,16))\r\n\r\n    for ax, s in zip(axs.flat, [0,1,2,5,10,20]):\r\n\r\n        func = factorial(s)\/(factorial(s+ell))*np.exp(-X**2 - Y**2)*(X**2 + Y**2)**(ell)*(genlaguerre(s, ell)(X**2 + Y**2))**2\r\n\r\n        ax.contourf(X,Y,func, levels=50)\r\n        ax.set_title(\"$s=$\"+str(s), fontsize=24, loc='center', y=0.9, color='w')\r\n\r\n        ax.set_xticks(ticks = [-10,0,10])\r\n        ax.tick_params(axis='x',direction='in', color='w', length=5, width=2)\r\n        ax.set_yticks(ticks = [-10,0,10])\r\n        ax.tick_params(axis='y',direction='in', color='w', length=5, width=2)\r\n        \r\nplt.tight_layout()    \r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Klein-Gordon wavefunctions and classical orbits<\/h2>\n
See Fig. 7.10.<\/p>\n
def plot_arrows(offsetx, offsety, color):\r\n    plt.arrow(4.3+np.sqrt(ell+s)+offsetx,0+offsety, 0, 1, head_length=1,head_width=1, length_includes_head=True, \\\r\n          linewidth=0, fc=color, overhang=0.25)\r\n    plt.arrow(4.3-np.sqrt(ell+s)+offsetx,0+offsety, 0, -1, head_length=1,head_width=1, length_includes_head=True, \\\r\n              linewidth=0, fc=color, overhang=0.25)\r\n    plt.arrow(np.sqrt(ell+s)\/2+offsetx,np.sqrt(ell+s)+offsety, -1, 0, head_length=1,head_width=1, length_includes_head=True, \\\r\n              linewidth=0, fc=color, overhang=0.25)\r\n    plt.arrow(np.sqrt(ell+s)\/2+offsetx,-np.sqrt(ell+s)+offsety, 0.1, 0, head_length=1,head_width=1, length_includes_head=True, \\\r\n              linewidth=0, fc=color, overhang=0.25)\r\n\r\nx = np.linspace(-17,17,1000)\r\ny = np.linspace(-17,17,1000)\r\nX,Y = np.meshgrid(x,y)\r\nell = 100\r\ns=20\r\n\r\ncircle1 = plt.Circle((4.3, 0), np.sqrt(s+ell), color=None, edgecolor='r', fill=False, lw=3)\r\ncircle2 = plt.Circle((0, 4.3), np.sqrt(s+ell), color=None, edgecolor='lawngreen', fill=False, lw=3)\r\ncircle3 = plt.Circle((-4.3, 0), np.sqrt(s+ell), color=None, edgecolor='white', fill=False, lw=3)\r\ncircle4 = plt.Circle((0, -4.3), np.sqrt(s+ell), color=None, edgecolor='magenta', fill=False, lw=3)\r\n\r\nfunc = factorial(s)\/(factorial(s+ell))*np.exp(-X**2 - Y**2)*(X**2 + Y**2)**(ell)\\\r\n       *(genlaguerre(s, ell)(X**2 + Y**2))**2\r\n\r\nfig, ax = plt.subplots(figsize=(15,15))\r\nax.contourf(X,Y,func, levels=50)\r\nax.add_patch(circle1)\r\nax.add_patch(circle2)\r\nax.add_patch(circle3)\r\nax.add_patch(circle4)\r\n\r\nplot_arrows(0,0,'r')\r\nplot_arrows(-4.3,4.3,'lawngreen')\r\nplot_arrows(-2*4.3,0,'white')\r\nplot_arrows(-4.3,-4.3,'magenta')\r\n\r\nplt.axis('off')\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Linear combination of Klein-Gordon solutions<\/h2>\n
See Fig. 7.11.<\/p>\n
x = np.linspace(-11,11,1000)\r\ny = np.linspace(-11,11,1000)\r\nX,Y = np.meshgrid(x,y)\r\nr = np.sqrt(X**2 + Y**2)\r\nphi = np.arctan2(Y,X)\r\nell = 50\r\ns=0\r\n\r\ncircle1 = plt.Circle((0, 0), np.sqrt(s+ell), color=None, edgecolor='lawngreen', fill=False, lw=4, ls='--')\r\n\r\nfunc1 = np.sqrt(factorial(s)\/(factorial(s+ell)))*\\\r\n        np.exp(1j*ell*phi)*\\\r\n        np.exp(-0.5*r**2)*\\\r\n        r**(ell)\\\r\n       *(genlaguerre(s, ell)(r**2))\r\n\r\nfunc2 = np.sqrt(factorial(s+1)\/(factorial(s+ell)))*\\\r\n        np.exp(1j*(ell-1)*phi)*\\\r\n        np.exp(-0.5*r**2)*\\\r\n        r**(ell-1)\\\r\n       *(genlaguerre(s+1, ell-1)(r**2))\r\n\r\nwith plt.style.context(('default')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(10,10))\r\n    ax.contourf(X,Y,np.abs(func1+func2)**2, levels=50)\r\n    ax.add_patch(circle1)\r\n\r\n    ax.set_xticks(ticks = [-10,0,10])\r\n    ax.tick_params(axis='x',direction='in', color='w', length=5, width=2, labelsize=42)\r\n    ax.set_yticks(ticks = [-10,0,10])\r\n    ax.tick_params(axis='y',direction='in', color='w', length=5, width=2, labelsize=42)\r\n\r\n# plt.savefig(\"KleinGordonWavefunctions4.png\", dpi=600)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Spin-relaxation timescale<\/h2>\n
See Fig. 7.19.<\/p>\n
hbar = 1.054571817e-34 # J*s\r\nm_e = 9.1093837015e-31 # kg\r\nc = 299792458 # m\/s\r\nalpha = 1.0\/137.035999084\r\nBstar = 4.41e9 # T (tesla)\r\n\r\nB = np.logspace(-1, 2, 1000)\r\ngs = ['$10^{2}$', '$10^{3}$', '$10^{4}$', '$10^{5}$']\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(9,10))\r\n\r\n    for i, gamma in enumerate([100,1000,10000,100000]):\r\n        Tspinflip = (hbar\/(m_e *c**2))*(Bstar\/B)**3 * (gamma**(-2)\/alpha)        \r\n        ax.loglog(B, Tspinflip, lw = 4, label = '$\\\\gamma=$'+gs[i])\r\n\r\nplt.xticks(fontsize = 30)\r\nplt.yticks(fontsize = 30)\r\nax.tick_params(axis='x', pad=5)\r\n\r\nplt.xlabel('$B$ (T)', fontsize=30, labelpad=10)\r\nplt.legend(fontsize=28, frameon=True, framealpha=1, handlelength=1)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Degree of radiative polarisation<\/h2>\n
Gradual build-up of the degree of radiative polarisation, as in Fig. 7.20(a).<\/p>\n
\"\"<\/p>\n
Maximum attainable spin polarisation, as in Fig. 7.20(b).<\/p>\n
\"\"<\/p>\n
Ratio of spin-light versus synchrotron-radiation power<\/h2>\n
See Fig. 7.22.<\/p>\n
hbar = 1.054571817e-34 # J*s\r\nee = 1.602176634e-19 # C\r\nm_e = 9.1093837015e-31 # kg\r\nc = 299792458 #m\/s\r\nB = np.logspace(0,4,500) # T (tesla)\r\n\r\ngg = [\"$\\\\gamma = 10^{2}$\", \"$\\\\gamma = 10^{3}$\", \"$\\\\gamma = 10^{4}$\", \"$\\\\gamma = 10^{5}$\"]\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(9,11))\r\n    for i, gamma in enumerate(np.logspace(2,5,4)):\r\n        R =  (gamma*B*ee*hbar\/(m_e**2 * c**2))**2\r\n        ax.loglog(B,R, lw = 4, label=gg[i])\r\n\r\n        plt.legend(fontsize=27, frameon=True, framealpha=1, handlelength=1)\r\nplt.xticks(fontsize = 34)\r\nplt.yticks(fontsize = 34)\r\nax.tick_params(axis='x', pad=5) \r\nax.set_xlabel(\"B(T)\", fontsize=50)\r\nax.set_ylabel(\"R\", fontsize=50)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Quantum corrections to emitted synchrotron power<\/h2>\n
See Fig. 7.23.<\/p>\n
x = np.logspace(-5,-2, 1000)\r\nRplus = 1 - (1.5 + 55*np.sqrt(3)\/16.0)*x\r\nRminus = 1 - (-1.5 + 55*np.sqrt(3)\/16.0)*x\r\nRavg = 0.5*(Rplus+Rminus)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(12,12))\r\n    \r\n    ax.semilogx(x,Rplus, lw = 4,  label=\"$\\\\mathcal{R}_{+}$\")\r\n    ax.semilogx(x,Rminus, lw = 4, label=\"$\\\\mathcal{R}_{-}$\")\r\n    ax.semilogx(x,Ravg, lw = 4,   label=\"$\\\\frac{1}{2} \\\\left( \\\\mathcal{R}_{+} + \\\\mathcal{R}_{-}  \\\\right)$\")\r\n    \r\n    plt.legend(fontsize=42 , frameon=True, framealpha=1, handlelength=1, labelspacing=1.0, borderpad=0.9)\r\n    ax.set_yticks(ticks = [0.94,0.96,0.98, 1.0])\r\n    plt.xticks(fontsize = 48)\r\n    plt.yticks(fontsize = 48)   \r\n    ax.tick_params(axis='x', pad=10) \r\n\r\n    ax.set_ylabel('$\\\\mathcal{R}$', fontsize=50)\r\n    ax.set_xlabel('$\\\\gamma B\/B_{*}$', fontsize=50, labelpad=-10)\r\n\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Characteristic timescales<\/h2>\n
See Fig. 7.25.<\/p>\n
hbar = 1.054571817e-34 # J*s\r\nm_e = 9.1093837015e-31 # kg\r\nee = 1.602176634e-19 # C\r\neps_0 = 8.8541878128e-12 \r\nc = 299792458 # m\/s\r\nalpha = 1.0\/137.035999084\r\nBstar = 4.41e9 # T (tesla)\r\ngamma = 2000\r\nB = np.linspace(0.1,15000,10000)\r\n\r\nTspinflip = (hbar\/(m_e *c**2))*(Bstar\/B)**3 * (gamma**(-2)\/alpha)\r\nTbuildup = 2*m_e \/ (ee*B)\r\nTdamping = (6*np.pi*eps_0 * m_e**3 * c**3) \/ (ee**4 * B**2 * gamma)\r\nTemission = (2 * np.pi * m_e) \/ (ee * alpha * B)\r\nTrevolution = (2 * np.pi * gamma * m_e) \/ (ee * B)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(figsize=(14,10))\r\n    \r\n    ax.loglog(B,Tspinflip, lw = 4,  label=\"$T_{spin-flip}$\")\r\n    ax.loglog(B,Tbuildup, lw = 4, label=\"$T_{buildup}$\")\r\n    ax.loglog(B,Tdamping, lw = 4, label=\"$T_{damping}$\")\r\n    ax.loglog(B,Temission, lw = 4, label=\"$T_{emission}$\")\r\n    ax.loglog(B,Trevolution, lw = 4, label=\"$T_{revolution}$\")\r\n    \r\n    plt.legend(fontsize=31, frameon=True, loc = 'upper right', framealpha=1, handlelength=1)\r\n    plt.xticks(fontsize = 32)\r\n    plt.yticks(fontsize = 32)   \r\n    ax.tick_params(axis='x', pad=5) \r\n    ax.set_ylabel('Timescale (s)', fontsize=40)\r\n    ax.set_xlabel('B(T)', fontsize=40)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"
Below is a set of python codes associated with Chapter 7 of Daniele Pelliccia and David M. Paganin, “Synchrotron Light: A Physics Journey from Laboratory to Cosmos” (Oxford University Press, 2025). In order to run any of these python codes, you will need to include the following header file. import numpy as np import matplotlib.pyplot […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":79,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"class_list":["post-220","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/220","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=220"}],"version-history":[{"count":10,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/220\/revisions"}],"predecessor-version":[{"id":790,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/220\/revisions\/790"}],"up":[{"embeddable":true,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/79"}],"wp:attachment":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=220"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}