{"id":829,"date":"2025-07-20T05:25:28","date_gmt":"2025-07-20T05:25:28","guid":{"rendered":"https:\/\/synchrotron-light.net\/wp\/?page_id=829"},"modified":"2025-07-21T05:28:00","modified_gmt":"2025-07-21T05:28:00","slug":"appendices","status":"publish","type":"page","link":"https:\/\/synchrotron-light.net\/wp\/?page_id=829","title":{"rendered":"Appendices"},"content":{"rendered":"

Below is a set of python codes associated with the Appendices 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.<\/p>\n

from scipy import special\r\nimport numpy as np\r\nimport matplotlib.pyplot as plt\r\nimport math\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
Appendix A: transformation linking the angular variables \\varphi<\/span> and<\/h2>\n
\\psi<\/span><\/h2>\n
See Fig. A.1<\/p>\n
phi = np.linspace(-np.pi, np.pi, 361)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig = plt.figure(figsize=(8.5,6.5))\r\n    for eps in [0, 0.5, 0.999]:\r\n        plt.plot(phi, phi-eps*np.sin(phi), lw=3)   \r\n    \r\n    plt.xticks([-np.pi, -np.pi\/2, 0, np.pi\/2, np.pi], ['-$\\\\pi$', '-$\\\\pi\/2$', '0', '$\\\\pi\/2$', '$\\\\pi$'], fontsize = 26)\r\n    plt.yticks([-np.pi, -np.pi\/2, 0, np.pi\/2, np.pi], ['-$\\\\pi$', '-$\\\\pi\/2$', '0', '$\\\\pi\/2$', '$\\\\pi$'], fontsize = 26)\r\n    plt.grid(True)\r\n\r\n    plt.xlabel('$\\\\varphi$')\r\n    plt.ylabel('$\\\\psi$')   \r\n    plt.show()<\/pre>\n
\"\"<\/p>\n
Appendix B: Boundary conditions for perfectly reflecting cavity walls<\/h2>\n
See Fig. B.1.<\/p>\n
ell = 1\r\nm = 1\r\nx1 = np.linspace(-0.2, 0, 20)\r\nx0 = np.linspace(0, ell, 100)\r\nx2 = np.linspace(ell, 1.2, 20)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, ax = plt.subplots(1, 1,figsize=(10,8))\r\n    plt.plot(x0, np.sin(np.pi*m*x0\/1), 'b', lw=4, label = \"$m=10$\")\r\n    plt.plot(x0, np.sin(2*np.pi*m*x0\/1), 'r', lw=4, label = \"$m=20$\")\r\n    plt.plot(x2, -np.sin(np.pi*m*x2\/1), 'b--', lw=4)\r\n    plt.plot(x1, -np.sin(np.pi*m*x1\/1), 'b--', lw=4)\r\n    plt.plot(x2, np.sin(2*np.pi*m*x2\/1), 'r--', lw=4)\r\n    plt.plot(x1, np.sin(2*np.pi*m*x1\/1), 'r--', lw=4)\r\n    \r\n    ax.set_xticks([0,0.5,1])\r\n    ax.set_xticklabels([\"0\", \"0.5\", \"1\"], fontsize=30)\r\n    ax.set_yticks([-1,0, 1])\r\n    ax.set_yticklabels([\"-1\", \"0\",\"1\"], fontsize=30)        \r\n    plt.xlabel('$x$', fontsize=20)\r\n    plt.ylabel('$\\\\sin(\\\\pi mx\/\\\\ell)$', fontsize=26)\r\n    plt.legend(fontsize=26, handlelength=1, frameon=True, framealpha=1, \\\r\n               loc = 'center left', bbox_to_anchor=(0.17, 0.2))\r\n    plt.show()<\/pre>\n
\"\"<\/p>\n
Appendix D: Relation between Taylor and Pad\u00e9 coefficients of g(\\tau)<\/span><\/h2>\n
Calculation leading to the plot in Fig. D.1. See also Fig. 4.23.<\/p>\n
def cn_recursive(n):\r\n    # Base case: c_0 = 0\r\n    if n == 0:\r\n        return 0\r\n    # Recursive case: c_n = c_{n-1}\/2 + 1\/2^(n-1)\r\n    else:\r\n        return 0.5*cn_recursive(n-1) + 1\/2**(n-1)\r\n\r\ndef derivative_K(z, N):\r\n    if N == 0:\r\n        return z*special.kv(2\/3,z\/2)\r\n    else:\r\n        return cn_recursive(N)*special.kvp(2\/3,z\/2,n=N-1) + (z\/2**N)*special.kvp(2\/3,z\/2,n=N)<\/pre>\n
 <\/p>\n
x = np.linspace(0.25,10.25,401)\r\n\r\n# Point of maximum\r\npoint_max = np.argmax(x*special.kv(2\/3,x\/2))\r\n\r\n# Calculate the Taylor series up the the order ord\r\nord = 4\r\na = np.zeros(ord)\r\ntayl_exp = np.zeros_like(x)\r\nfor i in range(ord):\r\n    test = derivative_K(x, i)\r\n    a[i] = test[30]\/math.factorial(i)\r\n    tayl_exp += a[i]*(x-x[30])**i\r\n\r\nA = a[0]\r\nD = (a[2]**2 - a[3]*a[1]) \/ (a[1]**2 -a[2]*a[0])\r\nC = -a[0]*D\/a[1] - a[2]\/a[1]\r\nB = a[1]+a[0]*C\r\nxmax = 1.0+(np.sqrt(A**2*D**2 + B*D*(B-A*C)) - A*D)\/(B*D)\r\npade = (A+B*(x-1)) \/ (1+C*(x-1)+D*(x-1)**2)\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig, axs = plt.subplots(1,2, figsize=(21,8))\r\n    axs[0].plot(x, x*special.kv(2\/3,x\/2), 'k', lw=3,label = \"Exact\")\r\n    axs[0].plot(x, tayl_exp, 'r', lw=3,label=\"Taylor\")\r\n    axs[0].plot(x, pade, 'b', lw=3,label=\"Pade'\")\r\n    axs[0].set_xlim([0.18, 1.8])\r\n    axs[0].set_ylim([1, 1.25])\r\n    \r\n    axs[0].tick_params(axis='x', labelsize=36)\r\n    axs[0].tick_params(axis='y', labelsize=36)\r\n    axs[0].set_xlabel('$\\\\tau$', fontsize=36)\r\n    axs[0].set_ylabel(\"$g(\\\\tau)$\", fontsize=36)  \r\n    axs[0].legend(fontsize = 34, frameon=True, framealpha=0.9, loc = 'upper right', labelspacing=0.3, handlelength=1)\r\n    \r\n    axs[1].plot(x, x*special.kv(2\/3,x\/2), 'k', lw=3,label = \"Exact\")\r\n    axs[1].plot(x, tayl_exp, 'r', lw=3,label=\"Taylor\")\r\n    axs[1].plot(x, pade, 'b', lw=3,label=\"Pade'\")\r\n    axs[1].set_xlim([0.18, 10])\r\n    axs[1].set_ylim([0.05, 1.25])\r\n    \r\n    axs[1].tick_params(axis='x', labelsize=32)\r\n    axs[1].tick_params(axis='y', labelsize=32)\r\n    axs[1].set_xlabel('$\\\\tau$', fontsize=36)\r\n    axs[1].set_ylabel(\"$g(\\\\tau)$\", fontsize=36)  \r\n    axs[1].legend(fontsize = 34, frameon=True, framealpha=0.9, labelspacing=0.3, handlelength=1)\r\n\r\nplt.show()<\/pre>\n
\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"
Below is a set of python codes associated with the Appendices 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. from scipy import special import numpy as […]<\/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-829","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/829","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=829"}],"version-history":[{"count":2,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/829\/revisions"}],"predecessor-version":[{"id":834,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/829\/revisions\/834"}],"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=829"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}