{"id":202,"date":"2025-05-24T10:24:27","date_gmt":"2025-05-24T10:24:27","guid":{"rendered":"https:\/\/synchrotron-light.net\/wp\/?page_id=202"},"modified":"2025-07-20T05:13:10","modified_gmt":"2025-07-20T05:13:10","slug":"chapter-6-quantum-effects-in-synchrotron-radiation","status":"publish","type":"page","link":"https:\/\/synchrotron-light.net\/wp\/?page_id=202","title":{"rendered":"Chapter 6: Quantum effects in synchrotron radiation"},"content":{"rendered":"

Below is a set of python codes associated with Chapter 6 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\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
Compton scattering<\/h2>\n
Increase in wavelength of a photon undergoing Compton scattering, as in Fig. 6.4<\/p>\n
h = 6.62607015e-34 # J\/Hz\r\nee = 1.602176634e-19  C\r\nhbar = h\/(2*np.pi)\r\nme = 9.11e-31 # kg\r\nc = 2.99792458e8 # m\/s, speed of light \r\nlambda_c = h\/(me*c)\r\n\r\nB_star = me**2 * c**2 \/ (ee*hbar)\r\n\r\nphi = np.linspace(0, np.pi, 100)\r\ndelta_l = 2*lambda_c*np.sin(phi\/2)**2\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1, 1,figsize=(8,8))\r\n    axs.plot(phi, delta_l, 'b', lw = 4)\r\n    \r\n    axs.tick_params(axis='x', labelsize=20)\r\n    axs.tick_params(axis='y', labelsize=20)\r\n    \r\n    axs.set_xlabel('$\\\\varphi$ (rad)', fontsize=32, labelpad = 5)  \r\n    axs.set_xticks([0, np.pi\/4, np.pi\/2, 3*np.pi\/4, np.pi])\r\n    axs.set_xticklabels([\"0\", \"$\\\\pi\/4$\",\"$\\\\pi\/2$\",\"$3\\\\pi\/4$\",\"$\\\\pi$\"], fontsize=28)\r\n    \r\n    axs.set_yticks([0, lambda_c, 2*lambda_c])\r\n    axs.set_yticklabels([\"0\", \"$\\\\lambda_{C}$\",\"$2\\\\lambda_{C}$\"], fontsize=28)\r\n    axs.set_ylim(0, 2.5*lambda_c)   \r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Fractional energy change for a photon undergoing Compton scattering, as in Fig. 6.5.<\/p>\n
phi = np.linspace(0, np.pi, 100)\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1, 1,figsize=(12,8))\r\n\r\n    for E_ph in [5,10,20,40,60,80,100]:\r\n        axs.plot(phi, 1.0\/(1+2*E_ph\/511*np.sin(phi\/2)**2), lw = 4, label=\"$\\\\mathcal{E}_{ph}^{(i)}=$\"+str(E_ph)+\" keV\")\r\n\r\n    axs.tick_params(axis='x', labelsize=24)\r\n    axs.tick_params(axis='y', labelsize=24)        \r\n    \r\n    axs.set_xlabel('$\\\\varphi$ (rad)', fontsize=26, labelpad = 5)  \r\n    axs.set_xticks([0, np.pi\/4, np.pi\/2, 3*np.pi\/4, np.pi])\r\n    axs.set_xticklabels([\"0\", \"$\\\\pi\/4$\",\"$\\\\pi\/2$\",\"$3\\\\pi\/4$\",\"$\\\\pi$\"], fontsize=20)\r\nplt.legend(fontsize=19, framealpha=1, handlelength=1) \r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Inverse Compton scattering<\/h2>\n
Contour plots of the relative energy boost for a photon undergoing inverse Compton scattering, as in Fig. 6.7.<\/p>\n
gamma_e = 2 # Change this value to generate the other panels in Fig. 6.7\r\nif gamma_e == 2:\r\n    phi = np.linspace(0, np.pi, 2000)\r\nif gamma_e == 1000:\r\n    phi = np.linspace(0, 0.005, 2000)\r\nif gamma_e == 10000:\r\n    phi = np.linspace(0, 0.0007, 2000)\r\n\r\nlambda_ratio = np.logspace(2,9,1000)\r\ne_ratio = np.zeros((lambda_ratio.shape[0],phi.shape[0]))\r\nfor i in range(lambda_ratio.shape[0]):\r\n    for j in range(phi.shape[0]):\r\n        e_ratio[i,j] = (np.sqrt(1-1\/gamma_e**2) + 1) \/ \\\r\n                  (1- np.cos(phi[j])*np.sqrt(1-1\/gamma_e**2) + (1+np.cos(phi[j]))\/(gamma_e*lambda_ratio[i])  )\r\n\r\nif gamma_e == 2:\r\n    xscale = phi\r\nelse:\r\n    xscale = 1000*phi\r\n\r\nwith plt.style.context(('default')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1, 1,figsize=(18,6))\r\n    cs = axs.contourf(xscale,lambda_ratio,  e_ratio, cmap='cividis',levels=20)\r\n    axs.contour(xscale,lambda_ratio,  e_ratio, levels=20, colors='k')\r\n\r\n    axs.set_yscale('log')\r\n    axs.tick_params(axis='x', labelsize=30)\r\n    axs.tick_params(axis='y', labelsize=30) \r\n    #axs.yaxis.get_offset_text().set_fontsize(20)\r\n    plt.title(\"$\\\\times 10^{6}$\")\r\n    cbar = fig.colorbar(cs)\r\n    cbar.ax.tick_params(labelsize=30) \r\n    plt.show()<\/pre>\n
\"\"<\/h1>\n
Simulating the energy distribution of an electron bunch undergoing inverse Compton scattering<\/h2>\n
Computational exercise on page 199 and Fig. 6.8.<\/p>\n
lambdaC = 2.42631023867e-3 #nm\r\nlambda_i = 632.8\r\nsig_laser = 0.002\r\ngamma_e = 1000\r\nsigma_gamma = 1\r\nphi = 0.1\r\nsigma_phi = 0.01\r\n\r\nn_events = 100000\r\n\r\nlambda_laser = lambda_i+sig_laser*np.random.normal(size=n_events)\r\ngamma_electr = gamma_e+sigma_gamma*np.random.normal(size=n_events)\r\nangle = phi + sigma_phi*np.random.normal(size=n_events)\r\n\r\nen_boost = []\r\nfor i in range(n_events):\r\n    sq = np.sqrt(1-1\/gamma_electr[i]**2)\r\n    sf = (1 + np.cos(angle[i])) \/ (gamma_electr[i]*lambda_laser[i]\/lambdaC)\r\n    en_boost.append( ((sq + 1) \/ (1-sq*np.cos(angle[i]) + sf)))\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=(10,10))\r\n    plt.hist(en_boost, bins=100, log=True)\r\n\r\n    plt.xticks(fontsize = 36)\r\n    plt.yticks(fontsize = 36)\r\n    plt.xlabel(\"Relative energy boost\")\r\n    plt.ylabel(\"Number of events\")\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Parameter-space plots for domain of validity of classical synchrotron-light theory<\/h2>\n
Bohr synchrotron (see Fig. 6.10).<\/p>\n
alpha = 0.007297\r\ngamma = np.logspace(0,6, 2000)\r\nB_ratio = alpha**2 *gamma\/10\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1, 1,figsize=(15,8))\r\n    #axs.loglog(gamma, B_ratio,lw = 4)\r\n    \r\n    axs.fill_between(gamma, B_ratio, y2=0, lw=4, edgecolor = 'k',\r\n                             alpha=0.5, facecolor='red') #zorder=2,\r\n    \r\n    axs.set_xscale('log')\r\n    axs.set_yscale('log')\r\n    axs.tick_params(axis='x', labelsize=34, pad=10)\r\n    axs.tick_params(axis='y', labelsize=34) \r\n    \r\n    axs.set_xlim(1,900000)\r\n    \r\n    axs.set_xlabel('$\\\\gamma$', fontsize=26, labelpad = 5)\r\n    axs.set_ylabel('$B\/B_{*}$', fontsize=26, labelpad = 5)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Electron recoil (see Fig. 6.11).<\/p>\n
alpha = 0.007297\r\ngamma = np.logspace(0,6, 2000)\r\nB_ratio = 0.1*np.sqrt(1. - 1.0\/gamma**2)\/gamma \r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1, 1,figsize=(15,8))\r\n    \r\n    axs.plot(gamma, 0.1*(1\/(gamma)), lw=4, color='k' )\r\n    axs.fill_between(gamma, B_ratio, y2=0, lw=4, edgecolor = 'k',\r\n                             alpha=0.5, facecolor='red') #zorder=2,\r\n    \r\n    axs.set_xscale('log')\r\n    axs.set_yscale('log')\r\n    axs.tick_params(axis='x', labelsize=34, pad=10)\r\n    axs.tick_params(axis='y', labelsize=34) \r\n    \r\n    axs.set_xlim(1,900000)\r\n    axs.set_ylim(0.0000001,1)\r\n    \r\n    axs.set_xlabel('$\\\\gamma$', fontsize=26, labelpad = 5)\r\n    axs.set_ylabel('$B\/B_{*}$', fontsize=26, labelpad = 5)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Compton wavelength (see Fig. 6.12).<\/p>\n
alpha = 0.007297\r\ngamma = np.logspace(0,6, 2000)\r\nB_ratio = 0.1*gamma*np.sqrt(1. - 1.0\/gamma**2) \/(2*np.pi)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1, 1,figsize=(15,8))\r\n    \r\n    axs.plot(gamma, 0.1*(gamma\/(2*np.pi)), lw=4, color='k' )\r\n    axs.fill_between(gamma, B_ratio, y2=0, lw=4, edgecolor = 'k',\r\n                             alpha=0.5, facecolor='red') #zorder=2,\r\n    \r\n    axs.set_xscale('log')\r\n    axs.set_yscale('log')\r\n    axs.tick_params(axis='x', labelsize=34, pad=10)\r\n    axs.tick_params(axis='y', labelsize=34) \r\n    \r\n    axs.set_xlim(1,900000)\r\n    axs.set_ylim(0.00001,1000)\r\n    \r\n    axs.set_xlabel('$\\\\gamma$', fontsize=26, labelpad = 5)\r\n    axs.set_ylabel('$B\/B_{*}$', fontsize=26, labelpad = 5)\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Parameter space plots of all curves which indicate the region where classical synchrotron theory is valid, as in Fig. 6.13.<\/p>\n
def ratio2Bstar(x):\r\n    return x * B_star\r\ndef Bstar2ratio(x):\r\n    return x \/ B_star\r\n\r\nalpha = 0.007297\r\ngamma = np.logspace(0,7, 2000)\r\nmask = gamma > 10\/alpha\r\ncurve_I = 0.1*np.sqrt(1. - 1.0\/gamma**2)\/gamma \r\ncurve_III = 0.1*gamma*np.sqrt(1. - 1.0\/gamma**2) \/(2*np.pi)\r\n\r\ncurve_VI = 0.1*1.0\/(alpha*gamma)\r\ncurve_VII = 0.1*1.0\/(alpha*gamma**2)\r\ncurve_VIII = 0.1*alpha**2 *gamma\r\ncurve_IX = 0.1*np.ones_like(gamma)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(figsize=(15,10))\r\n    \r\n    axs.plot(gamma, curve_I, lw=4, color='k' )\r\n    axs.plot(gamma, curve_III, lw=4, color='b', linestyle='dashed' )\r\n    axs.plot([10\/alpha, 10\/alpha], [1e-12, 100], lw=4, color='red', linestyle='dashed'  ) # curve_V\r\n    axs.plot(gamma, curve_VI, lw=4, color='k', linestyle='dashed' )\r\n    axs.fill_between(gamma, curve_VII, lw=4, color='red', alpha=0.5, where=mask)\r\n    axs.plot(gamma, curve_VII, lw=4, color='r')\r\n    axs.plot(gamma, curve_VIII, lw=4, color='b' )\r\n    axs.plot(gamma, curve_IX, lw=4, color='brown' )\r\n    \r\n    axs.set_xscale('log')\r\n    axs.set_yscale('log')\r\n    axs.tick_params(axis='x', labelsize=34, pad=10)\r\n    axs.tick_params(axis='y', labelsize=34) \r\n    \r\n    axs.set_xlim(1,1000000)\r\n    axs.set_ylim(1e-11,100)\r\n    \r\n    axs.set_ylabel(\"$B\/B_{*}$\", fontsize = 26)\r\n    axs.set_xlabel(\"$\\gamma$\", fontsize = 26)\r\n    \r\n    secax = axs.secondary_yaxis('right', functions=(ratio2Bstar, Bstar2ratio))\r\n    secax.set_ylabel(\"$B$ (T)\", fontsize = 26)\r\n    secax.tick_params(axis='y', labelsize=34) \r\nplt.show()<\/pre>\n
\"\"<\/p>\n
 <\/p>\n","protected":false},"excerpt":{"rendered":"
Below is a set of python codes associated with Chapter 6 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-202","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/202","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=202"}],"version-history":[{"count":7,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/202\/revisions"}],"predecessor-version":[{"id":797,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/202\/revisions\/797"}],"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=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}