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PhaseSpaceOfSmallSignalFEL.html
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PhaseSpaceOfSmallSignalFEL.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge,IE=9,chrome=1"><meta name="generator" content="MATLAB 2021a"><title>Phase space of a small-signal free-electron laser (Section 10.3)</title><style type="text/css">.rtcContent { padding: 30px; } .S0 { margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: normal; text-align: left; }
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.S8 { margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: normal; text-align: left; }
.S9 { margin: 3px 10px 5px 4px; padding: 0px; line-height: 18px; min-height: 0px; white-space: pre-wrap; color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 15px; font-weight: bold; text-align: left; }
.S10 { margin: 3px 10px 5px 4px; padding: 0px; line-height: 20px; min-height: 0px; white-space: pre-wrap; color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 20px; font-weight: bold; text-align: left; }</style></head><body><div class = rtcContent><div class = 'S0'><span>Companion software for "Volker Ziemann, </span><span style=' font-style: italic;'>Hands-on Accelerator physics using MATLAB, CRCPress, 2019</span><span>" (https://www.crcpress.com/9781138589940)</span></div><h1 class = 'S1'><span>Phase space of a small-signal free-electron laser (Section 10.3)</span></h1><div class = 'S0'><span>Volker Ziemann, 211124, CC-BY-SA-4.0</span></div><div class = 'S0'><span style=' font-weight: bold;'>Important:</span><span> requires the elliptic package from </span><a href = "https://github.com/moiseevigor/elliptic"><span>https://github.com/moiseevigor/elliptic</span></a><span> located in a subdirectory below the present one (for the fast evaluation of elliptic functions).</span></div><div class = 'S0'><span>The dynamics of an electron moving in an undulator magnet and, at the same time, exposed to the field of an external laser can be described by a mathematical pendulum, given by Equation 10.30, which reads </span><span texencoding="\ddot\psi+\Omega^2\sin\psi=0" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="105" height="20" /></span><span>. Here </span><span texencoding="\eta=\dot\psi" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="39" height="20" /></span><span> describes the energy difference with respect to the energy at which the electron motion and laser are synchronous, which is is encoded in the FEL resonance condition in Equation 10.28. Since the electron bunches are much longer than the laser wavelength, we can assume that the electrons are uniformly distributed in phase and follow these uniformly distributed electrons for some time and calculate where they are at the end of the FEL.</span></div><div class = 'S0'><span>The intensity of the laser and the parameters of the undulator are encoded in the oscillation frequency </span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: normal; color: rgb(0, 0, 0);">Ω</span><span> and is given in Equation 10.30. We set the duration of the FEL process to 1/10 of the full oscillation period and then define a range of starting phase </span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: normal; color: rgb(0, 0, 0);">ϕ</span><span>.</span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span >clear</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >addpath </span><span style="color: rgb(170, 4, 249);">./elliptic</span><span > </span></span></div></div><div class="inlineWrapper outputs"><div class = 'S4'><span style="white-space: pre"><span >Omega=</span></span><span>0.25</span><span style="white-space: pre"><span > </span><span style="color: rgb(2, 128, 9);">% Slider to set Omega</span></span></div><div class = 'S5'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>Omega = 0.2500</div></div></div><div class="inlineWrapper"><div class = 'S6'><span style="white-space: pre"><span >Ts=2*pi/Omega;</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >dt=0.1*Ts; </span><span style="color: rgb(2, 128, 9);">% time step</span></span></div></div><div class="inlineWrapper"><div class = 'S7'><span style="white-space: pre"><span >phi=-pi:2*pi/201:pi;</span></span></div></div></div><div class = 'S8'><span>We define the separatrix, which separates the periodic from the non-periodic trajectories, and plot both its upper and the lower branch. Then we scale the axes and annotate the plot. </span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span >separatrix=2*Omega*cos(0.5*phi); </span><span style="color: rgb(2, 128, 9);">% eq. 5.47</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >clf</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >plot(phi,separatrix,</span><span style="color: rgb(170, 4, 249);">'k'</span><span >,phi,-separatrix,</span><span style="color: rgb(170, 4, 249);">'k'</span><span >)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >axis([-pi,pi,-0.85,0.85]); </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >set(gca,</span><span style="color: rgb(170, 4, 249);">'xtick'</span><span >,[-pi,-pi/2,0,pi/2,pi],</span><span style="color: rgb(170, 4, 249);">'fontsize'</span><span >,14, </span><span style="color: rgb(14, 0, 255);">...</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > </span><span style="color: rgb(170, 4, 249);">'xticklabels'</span><span >,{</span><span style="color: rgb(170, 4, 249);">'-\pi'</span><span >,</span><span style="color: rgb(170, 4, 249);">'-\pi/2'</span><span >,</span><span style="color: rgb(170, 4, 249);">'0'</span><span >,</span><span style="color: rgb(170, 4, 249);">'\pi/2'</span><span >,</span><span style="color: rgb(170, 4, 249);">'\pi'</span><span >})</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >xlabel(</span><span style="color: rgb(170, 4, 249);">'\psi'</span><span >); </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >ylabel(</span><span style="color: rgb(170, 4, 249);">'$$\dot\psi$$'</span><span >,</span><span style="color: rgb(170, 4, 249);">'interpreter'</span><span >,</span><span style="color: rgb(170, 4, 249);">'latex'</span><span >)</span></span></div></div><div class="inlineWrapper"><div class = 'S7'><span style="white-space: pre"><span >hold </span><span style="color: rgb(170, 4, 249);">on</span><span >; </span></span></div></div></div><div class = 'S8'><span>Now we define a range of initial starting values for </span><span texencoding="\eta=\dot\psi" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="39" height="20" /></span><span> and loop over the two values. Inside that loop we loop over the uniformly distributed starting phases of the electron that we mark with a green dot and use </span><span style=' font-family: monospace;'>pendulumtracker()</span><span>, defined in the Appendix, to find the final position after time </span><span style=' font-family: monospace;'>dt</span><span>, which we mark by a red dot. In this way we follow where each electron starts and ends it journey through the FEL.</span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span >eta2=</span></span><span>0.4</span><span style="white-space: pre"><span >; </span><span style="color: rgb(2, 128, 9);">% Slider for the second value of eta</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >eta0=[0,eta2];</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">for </span><span >m=1:length(eta0)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > </span><span style="color: rgb(14, 0, 255);">for </span><span >k=1:length(phi)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > plot(phi(k),eta0(m),</span><span style="color: rgb(170, 4, 249);">'g.'</span><span >) </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > x=pendulumtracker([phi(k),eta0(m)],Omega,dt);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > plot(x(1),x(2),</span><span style="color: rgb(170, 4, 249);">'r.'</span><span >) </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > pause(0.001)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > </span><span style="color: rgb(14, 0, 255);">end</span></span></div></div><div class="inlineWrapper outputs"><div class = 'S4'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">end</span></span></div><div class = 'S5'><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="41222690" data-scroll-top="null" data-scroll-left="null" data-testid="output_1" style="width: 1405px;"><div class="figureElement"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="uniqName_333_422" widgetid="uniqName_333_422" style="width: 100%; height: auto;"><div class="ImageView" id="uniqName_333_424" widgetid="uniqName_333_424" style="width: 100%; height: auto;">
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BQoBUqBUiBG8B68Z71orRFCYEZVERHMyTPGzL0YIzln+r4npYQxZg6JgAh0HcQIpYAqqEIpMJlA1/GqWoPWYBhgGGAygb6HECAECAGcg9EIRiMYjWA0gtEInAPnwDlwDpwD52A0gtEIRiMYjcA5cA5CgL6HvofJBCYTqBVqhVqhNV6RCHgP3sNkApMJqIIqlAI5Q4zgPXjPetRaI6VECIGcMzFGzMk3xmwI3ntUlRnnHLVWjDEbgAh4DzFCzrC2BmtroAqlwGQCkwl0HXgPIrwhrUFr0Bq0Bq1Ba7wuIiACIuA9eA85Q86gCqqgCqVAKRAjxAgiIMI8aK0RQmBGVfHeY06NMWZDiTGScyaEQEoJY8wGJQLeQ4wQI+QMpYAqrK2BKpQCpUDOkDNMJjCZQNdB10HXQdeB9+A9eA/eg/fgPXgPXQddB10HXQddB5MJTCZQCpQCqqAKqqAKqlAKlAJdB10HIiDCvGqtkVIihEDXdcQYMafWmDny8MMPc9111/Hxj3+c3bt38/zzz3OsDh06xNe//nWuu+46duzYwac//Wn+8z//k43Ie4+qMuOco9aKMeY0IwLeg/fQddB1ECPECDlDzpAz5AylQClQCpQCpUApUArkDDlDzpAz5AwxQozgPXgPIiDCRtVaI4TAjKoSY8ScemPmxPXXX88HP/hBHn30UZ577jluvPFGLr/8cvbv389ref755/mTP/kTPvaxj/H444/zP//zP9xxxx28//3v54477mAjEhFijOScCSGQUsIYY8yxa62RUiKEQNd1xBgx68eYObC6usrtt9/Ohz70Ie655x5uueUWvvrVr/Liiy9y7bXX8lpuuukmHnvsMW6++Wa+9KUv8U//9E+srq7ynve8h5QSTz75JBuV9x5VZcY5R60VY4wxr661RgiBGVUlxohZX8bMgdtuu41NmzZxzTXX8JItW7awvLzMnj17ePLJJzmaQ4cO8eUvf5mLL76Y9773vbzkrLPO4sMf/jAzpRQ2MhEhxkjOmRACKSWMMcb8vNYaKSVCCHRdR4wRsz6NmQMPPfQQl1xyCQsLC7zcBRdcwMyePXs4mrW1Nf7hH/6BK6+8kiMtLCww88ILL3A68N6jqsw456i1Yowx5v9qrRFCYEZViTFi1q8x69wLL7zAT37yE975zndypAsvvJCZxx9/nKM544wzuPTSS3n3u9/NkVZXV5n57d/+bU4XIkKMkZwzIQRSShhjzOmstUZKiRACXdcRY8Ssf2PWuccee4yZt771rRzpzDPPZObHP/4xr9eDDz7IMAz85m/+JhdddBGvZXFxkcXFRRYXF9m5cyfzznuPqjLjnKPWijHGnG5aa4QQmFFVYozMu507d7K4uMji4iKLi4tsVGPWuYMHD/JaDh48yOvx4IMP8hd/8Rf80i/9Ep/+9Kc5FtPplOl0ynQ65eqrr2YjEBFijOSc6fuelBLGGHM6aK2RUiKEQM6ZGCMbxdVXX810OmU6nTKdTtmoxqwT3/rWt1hZWWFlZYWVlRVWVla4+eab2bx5M0dz6NAhZhYWFjhWX/nKV/jzP/9zzjvvPL7whS/wi7/4i5zuvPeoKjPOOWqtGGPMRlVrpe97ZlQV7z1m/oxZJ5599ln27t3L3r172bt3L3v37uWpp55CRJg5cOAAR9q/fz8z55xzDsfixhtv5Nprr2Xbtm188YtfZPPmzZj/J8ZIKYW+7wkhYIwxG0lrjZQSfd8TYyTGiJlfY9aJSy+9lH379rFv3z727dvHvn37uOmmm1hYWODcc8/lmWee4UhPPPEEM1u3buW1XHfddfzLv/wLl19+Obt37+Yd73gH5ueJCKqK9x7nHCkljDFm3qWU6PueGVXFe4+Zb2PmwGWXXcYjjzxCa42Xu+uuu9i0aRMXX3wxr2bXrl3cddddfOADH+Cmm27ijDPOwLy6GCOlFGqthBBorWGMMfOmtUYIgWEYyDkTY8RsDGPmwPbt2zn77LPZvn07DzzwAK01brjhBu6//36uuOIKzjrrLF7y4IMPsm3bNj75yU8y81//9V989rOfZeYHP/gBO3bsYMeOHezYsYMdO3awY8cO/u3f/g3z80SEUgree5xzpJQwxph50FojpUQIAe89qoqIYDaOMXPgvPPO4/Of/zwz27dv533vex933nknV111FVdeeSUvd/DgQQ4cOMAPf/hDZr75zW/yox/9iJm7776bu+++m7vvvpu7776bu+++m7vvvptvf/vbmKOLMaKq1FpxzlFrxRhj1qvWGiEEaq2UUogxYjaeMXNi27Zt3Hfffdx7773ceuutfOc73+GjH/0oR1paWmI6nXLjjTcy8/73v5/pdMp0OmU6nTKdTplOp0ynU6bTKdPplBtuuAHz6kSEUgo5Z0IIpJQwxpj1pLVGSokQAl3XUUpBRDAb05g5s2XLFi666CLOOOMMzMnnvUdVmXHOUWvFGGNOtWEY6PueGVUlxojZ2MYY8zqJCDFGSimklAgh0FrDGGNOttYafd+TUiLGSIwRc3oYY8wbJCKUUvDe45wjpYQxxpwMrTVSSoQQEBFUFe895vQxxpg3KcaIqlJrxTlHrRVjjDlRWmuEEGitUUohxog5/Ywx5jgQEUop5Jzp+54QAsYYczy11kgpEUIgxkjOGRHBnJ7GGHMcee9RVbz3jEYjUkoYY8yb0VojpUQIgRlVpes6zOltjDEnQIwRVaXWinOOWivGGPN6tdYIIdBao5RCjBFjZsYYc4KICKUUcs70fU8IgdYaxhjzWlprpJQIIRBjJOeMiGDMS8YYc4J571FVvPc450gpYYwxr6S1RkqJEAIzqkrXdRhzpDHGnCQxRlSV1hrOOYZhwBhjXjIMAyEEWmuUUogxYszRjDGZYL0vAAAgAElEQVTmJBIRcs6UUti9ezfOOVprGGNOX601QgiklMg5k3NGRDDm1Ywx5hQQEUopdF2Hc46UEsaY00trjZQSIQS896gq3nuMORZjjDmFYoyoKjOj0YiUEsaYja21RkqJEAIzqkqMEWNejzHGnGIiQowRVaW1hnOOlBLGmI1nGAZCCLTWKKUQY8SYN2KMMeuEiJBzppRCrRXnHLVWjDHzr7VGCIGUEjlncs6ICMa8UWOMWWdEhFIKOWf6vieEQGsNY8z8aa3R9z0hBLz3qCree4x5s8YYs05571FVvPc450gpYYyZD601UkqEEBARVJUYI8YcL2OMWedijKgqM6PRiJQSxpj1qbVGSokQAjOqSowRY463McbMAREhxoiqMuOcI6WEMWZ9aK2RUiKEwEwphRgjxpwoY4yZIyJCjJFSCjPOOVJKGGNOjdYaKSVCCLTWKKUQY0REMOZEGmPMHBIRYoyUUmit4ZwjpYQx5uRorZFSIoRAa42cMzlnRARjToYxxswxESHnTCmFWivOOVJKGGNOjNYawzAQQqC1Rs6ZnDPee4w5mcYYswGICKUUSinUWnHOkVLCGHN8tNYYhoEQAiklcs7knPHeY8ypMMaYDUREKKVQSqHWinOOlBLGmDemtcYwDIQQSCmRc0ZV8d5jzKk0Zo48/PDDXHfddXz84x9n9+7dPP/887xROWdyzpiNSUQopVBKobWGc46UEsaYY9NaI6VECIHdu3eTc0ZV8d5jzHowZk5cf/31fPCDH+TRRx/lueee48Ybb+Tyyy9n//79vF6lFD71qU9x//33YzY2ESHnTCmF1hrOOVJKGGNeWWuNlBIhBFpr5JwppeC9x5j1ZMwcWF1d5fbbb+dDH/oQ99xzD7fccgtf/epXefHFF7n22mt5PZ599ln+9m//FnN6ERFyzpRSmHHOkVLCGPN/tdZIKRFCoLVGzpmcM957jFmPxsyB2267jU2bNnHNNdfwki1btrC8vMyePXt48sknOVZ/8zd/w+bNm3nHO96BOf2ICDFGSinMjEYjUkq01jDmdNRaI6VECIHWGqUUcs547zFmPRszBx566CEuueQSFhYWeLkLLriAmT179nAsbrvtNh566CE+85nPcMYZZ2BOXyJCjBFVZcY5RwiBWivGnA5aa/R9TwiBmVIKOWdEBGPmwZh17oUXXuAnP/kJ73znOznShRdeyMzjjz/Oa3n66af5+7//e/7qr/4KEcGYGREhxsja2hree/q+xzlHSgljNprWGsMwEEIghICIoKrEGBERjJknY9a5xx57jJm3vvWtHOnMM89k5sc//jGv5tChQ3zsYx9j69atLC8vY8wriTGiquScqbXinCOlhDHzrrVGSokQAikllpeXUVVijBgzr8ascwcPHuS1HDx4kFfzj//4j3zve9/jpptu4o1aXFxkcXGRxcVFdu7cidm4vPeUUiilMDMajUgp0VrDmHnSWqPve0IItNbIOaOqdF2H2bh27tzJ4uIii4uLLC4uslGNWSe+9a1vsbKywsrKCisrK6ysrHDzzTezefNmjubQoUPMLCwscDR79uzhlltuIcbIeeedxxs1nU6ZTqdMp1OuvvpqzMYnIsQYUVVmQgg450gpYcx61VpjGAZCCIQQEBFUlZwz3nvMxnf11VcznU6ZTqdMp1M2qjHrxLPPPsvevXvZu3cve/fuZe/evTz11FOICDMHDhzgSPv372fmnHPO4WhyzrzlLW/ha1/7GisrK6ysrLCyssKBAwf47ne/y8rKCrfccgvGHI2IEGNEVSml0FrDOUdKidYaxqwHrTVSSjjnWF1dZXl5GVUlxogxG9GYdeLSSy9l37597Nu3j3379rFv3z5uuukmFhYWOPfcc3nmmWc40hNPPMHM1q1bOZpf+7Vf4+KLL8aY40FEyDlTSmHGOYdzjpQSxpxsrTWGYSCEQAiBGVUl50zXdRizkY2ZA5dddhmPPPIIrTVe7q677mLTpk1cfPHFHM1HPvIRdu3axa5du9i1axe7du1i165dnH322fzqr/4qu3bt4sMf/jDGvB4iQoyRtbU1Yoy01nDOkVKi1ooxJ1Jrjb7vCSGwurrK8vIyqkqMERHBmNPBmDmwfft2zj77bLZv384DDzxAa40bbriB+++/nyuuuIKzzjqLlzz44INs27aNT37ykxhzMnRdR86ZUgozfd/jnCOlRGsNY46H1hopJZxzhBAQEUop5Jzpug5jTjdj5sB5553H5z//eWa2b9/O+973Pu68806uuuoqrrzySl7u4MGDHDhwgB/+8IcYczKJCDFGVJVSCjPOOZxzpJQw5vVqrTEMAyEEQgi01sg5o6rEGBERjDldjZkT27Zt47777uPee+/l1ltv5Tvf+Q4f/ehHOdLS0hLT6ZQbb7yRV/PNb36TnDPGnAgiQoyRtbU1cs601hiNRjjnSClhzNG01hiGgRACIQRWV1dZXl5GVck5473HGANj5syWLVu46KKLOOOMMzBmHnjvyTmjqnRdR2uN0WiEc46UEsa01hiGgRACIQRWV1dZXl5GVck503UdxpifNcYYc1KICDFGcs6oKl3X0VpjNBrhnCOlhDl9tNYYhoEQAiEEVldXWV5eRlXJOdN1HcaYoxtjjDnpRIQYIzlnVJWu65gZjUY450gpUWvFbCytNVJKhBAIIbC6usry8jKqSs6ZruswxhybMcaYU0pEiDESY2RtbY1SCjN93+OcI6VESgkzf1prDMNA3/eMRiNCCLTWiDGiquSc6boOY8zrN8YYs66ICDFGVJVSCjO1VkajEc45UkrUWjHrT2uNWispJUIIhBDYvXs3S0tLlFJQVXLOeO8xxrw5Y4wx65aIEGOklMLa2hqlFGZSSoxGI5xzpJSotWJOvtYatVZSSoQQcM7R9z2tNWKMqCqlFLquw3uPMeb4GWOMmRsiQoyRUgqqSs6ZmZQSo9EI5xwpJVJKmOOvtcYwDPR9TwiBEAJ939NaY3l5mbW1NVSVnDPee4wxJ84YY8xcEhG898QYKaWgquScmam1MhqNcM4RQiClRK0Vc+xaa9RaGYaBEAKj0YgQArt370ZEiDGiqqgqOWe6rsMYc/KMMcZsCCKC954YI6UU1tbWKKWwvLzMTEqJ0WiEc44QAiklUkrUWjmdtdZorTEMA33fE0JgNBoRQiClxOrqKsvLy5RSUFVKKcQY8d5jjDl1xhhjNiwRoes6YoyUUlhbW6OUQoyRmdYaKSVGoxHOOUIIpJRIKVFrpdbKRtBao7VGrZVhGOj7nhACzjlCCIQQWF1dRURYXl5GVVFVSinknOm6Du89xpj1Y4wx5rQiInjviTGSc6aUwtraGqUUYoy8JKVE3/eMRiOcczjn6PuelBIpJYZhoNZKrZXWGqdKa43WGrVWhmFgGAb6vqfve0IIOOdwzhFCIKXE6uoqS0tLLC8vk3NGVVFVcs7EGOm6DhHBGLO+jTHGmP9DRPDeE2MkxkgpBVVlbW2NUgo5Z5aWlnjJ6uoqKSX6vsc5x2g0YjQa4ZzDOUcIgRACfd/T9z1935NSIqVESomUEiklUkqklEgpkVIipURKib7v6fuevu/p+54QAiEEQgg45xiNRoxGI0IIhBBIKbG6usrq6ipLS0ssLS0RY6SUwtraGqpKKYWcM13X0XUd3nuMMfNpjDHGvAYRwXtP13XEGIkxknOmlIKqsra2xtraGqpKKYVSCjFGYowsLS2xtLTE0tISr4eIsLS0xNLSEktLS8QYiTESY6SUwtraGmtra6gqqkophZwzOWe6rqPrOrz3iAjGmI1njDHGHCcigoggInjv8d7TdR1d19F1HTFGYozEGIkxEmMkxkiMkRgjMUZijMQYiTHSdR1d19F1Hd57vPd47xERjDGntzHGGGOMMXNmjDHGGGPMnBljjDHGGDNnxhhjjDHGzJkxxhhjjDFzZowxxhhjzJwZY4wxxhgzZ8YYY4wxxsyZMcYYY4wxc2aMMcYYY8ycGWOMMcYYM2fGzJGHH36Y6667jo9//OPs3r2b559/ntfj6aef5tOf/jQ7duzghhtu4Lvf/S7GGGOMmT9j5sT111/PBz/4QR599FGee+45brzxRi6//HL279/Psbjzzjt5//vfz5e+9CWef/55/v3f/50/+IM/4NZbb8UYY4wx82XMHFhdXeX222/nQx/6EPfccw+33HILX/3qV3nxxRe59tpreS2PPfYYk8mE9773vXzjG9/g5ptv5r777uPXf/3X+dSnPsX3v/99jDHGGDM/xsyB2267jU2bNnHNNdfwki1btrC8vMyePXt48skneTWf+9zn+IVf+AU+9alPsbCwwMyZZ57JRz7yEd7znvfw/e9/H2OMMcbMjzFz4KGHHuKSSy5hYWGBl7vggguY2bNnD6/mG9/4Br/7u7/L29/+dl7ukksuYffu3bz73e/GGGOMMfNjzDr3wgsv8JOf/IR3vvOdHOnCCy9k5vHHH+do9u/fz49+9CMuuOACnnzyST7xiU/wZ3/2Z1x11VV87WtfwxhjjDHzZ8w699hjjzHz1re+lSOdeeaZzPz4xz/maJ588klmnn76af74j/+Yp556ire//e1897vf5S//8i/5u7/7O47F4uIii4uLLC4usnPnTowxxpj1aOfOnSwuLrK4uMji4iIb1Zh17uDBg7yWgwcPcjSHDh1i5vbbb+ev//qv+cIXvsBnPvMZ/uM//oPf+q3fYhgGHn30UV7LdDplOp0ynU65+uqrMcYYY9ajq6++mul0ynQ6ZTqdslGNWSe+9a1vsbKywsrKCisrK6ysrHDzzTezefNmjubQoUPMLCwscDTj8ZiZrVu38oEPfICXLCws8IlPfIKZr3zlKxhjjDFmfoxZJ5599ln27t3L3r172bt3L3v37uWpp55CRJg5cOAAR9q/fz8z55xzDkdz/vnnM/Mrv/IrHGlxcZGZ5557DmOMMcbMjzHrxKWXXsq+ffvYt28f+/btY9++fdx0000sLCxw7rnn8swzz3CkJ554gpmtW7dyNL/8y7/MW97yFv77v/+bI7344ovMvO1tb8MYY4wx82PMHLjssst45JFHaK3xcnfddRebNm3i4osv5mjG4zG///u/z8MPP0xrjZf74he/yMxll12GMcYYY+bHmDmwfft2zj77bLZv384DDzxAa40bbriB+++/nyuuuIKzzjqLlzz44INs27aNT37yk7zkqquu4uyzz2Z5eZmvf/3rfP/7/197cB9bdWG3cfiTX4cPAmuIUtlqDMZ1uyEZTjp5SW15MRs4R//ZmhAn1hxXrWgMk0SabcT4QrI0wRACsTE6HYolrqJFXjIzAsWyptLTNuXkgDdiwtIE3GCmQRBUODw5fzRpGpFtkcJh3+s6woYNG3juuee49dZbmTNnDiGEEEIoHAkFYOLEibz00kvk1dXVsWDBAjZu3MgjjzzCkiVLGOrcuXOcOnWKzz//nEE33XQTzc3NXHfddTz++OPMmzePZ599lqqqKl588UVCCCGEUFgSCkR5eTk7duxg27ZtvPrqq+zbt4+lS5cy3Jw5c7BNY2MjQ33/+9/n7bff5m9/+xuvvPIK3d3drFu3jvHjxxNCCCGEwpJQYMrKypg5cyZFRUX8NyZMmEBFRQXjxo0jhBBCCIUpIYQQQgihwCSEEEIIIRSYhBBCCCGEApMQQgghhFBgEkIIIYQQCkxCCCGEEEKBSQghhBBCKDAJIYQQQggFJiGEEEIIocAkhBBCCCEUmIQQQgghhAKTEEIIIYRQYBJCCCGEEApMQgghhBBCgUkIIYQQQigwCSGEEEIIBSYhhBBCCKHAJIQQQgghFJiEEEIIIYQCkxBCCCGEUGASQgghhBAKTEIIIYQQQoFJCCGEEEIoMAkF5P3332fFihU88cQTrF+/nhMnTvCf2LFjB7/73e944oknaGpq4tixY4QQQgih8CQUiGeeeYba2lr6+voYGBigsbGR6upqjh49yr+jvr6eRx99lH379nHy5EnWrVvH3XffTW9vLyGEEEIoLAkFYPfu3bz++us88MADbNmyhRdffJF33nmHzz77jOXLl3Mxb775Jm1tbTzwwANs3bqVpqYm/vrXv1JUVERDQwMhhBBCKCwJBWDDhg2MHj2aZcuWMaisrIz777+fvXv3cujQIb5OZ2cneb/5zW8YVFpayrx58/j73//OsWPHCCGEEELhSCgAHR0dVFVVMWrUKIaaOnUqeXv37uXrjB49mrx//vOfDPXll1+SV1xcTAghhBAKR8IV7uTJk5w9e5bx48cz3LRp08jbv38/X+fee+9l9OjRPPnkkxw9epRcLsfmzZvZsmULP//5z/m///s/QgghhFA4Eq5w2WyWvGuuuYbhrr32WvK+/PJLvs6UKVP44x//SF9fH3PnzmXKlCksX76cBQsWsGrVKkIIIYRQWBKucOfOneNizp07x9d5//33efjhh7nhhht46qmnWL16Nb/4xS949913WbFiBf8OSUhCEmvXriWEEEK4Eq1duxZJSEISV6uEK0Q6naa+vp76+nrq6+upr6+nqamJkpISLiSXy5E3atQoLiSXy7F8+XK+/e1v8+c//5l77rmHu+++mz/84Q8sW7aMTZs2sXnzZi7GNraxzWOPPUYIIYRwJXrsscewjW1sc7VKuEJ88skndHV10dXVRVdXF11dXXz00UfcfPPN5J06dYrhjh49St7111/PhWSzWT7++GPuuusuiouLGaquro6ioiJ2795NCCGEEApHwhVi/vz59PT00NPTQ09PDz09PaxatYpRo0Zxww030N/fz3AHDx4k79Zbb+VCzp49S94111zDcEVFReSdPXuWEEIIIRSOhAJw11130d3dzeHDhxmqpaWF0aNHU1lZyYX86Ec/YuzYsezcuZNcLsdQO3fu5Ny5c0yZMoUQQgghFI6EAlBXV8fYsWOpq6ujvb2dw4cPs3LlSt577z0efvhhxowZw6A9e/ZQXl7Ok08+SV6SJCxbtoyDBw/y61//mo6ODv7xj3/wxhtv0NDQQGlpKffddx/h6rF27VpCYVm7di2hcKxdu5YQLreEAjBx4kReeukl8urq6liwYAEbN27kkUceYcmSJQx17tw5Tp06xeeff86gxYsX88wzz2CbVCrF7NmzefLJJ/nhD39Ic3Mz48aNI1w91q1bRygs69atIxSOdevWEcLlllAgysvL2bFjB9u2bePVV19l3759LF26lOHmzJmDbRobGxlq0aJFdHR0sG3bNl555RW6u7t55ZVX+O53v0sIIYQQCktCgSkrK2PmzJkUFRXx3ygrK6OiooJx48YRQgghhMKUEC5qxowZSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSyJOEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCGJPElIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEnmSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCQkIQlJSEISkpCEJCQhCUlIQhKSkIQkJCEJSUhCEpKQhCRmzJjB1SghXNRrr72GbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trGNbWxjG9vYxja2sY1tbGMb29jGNraxjW1sYxvb2MY2trHNa6+9xtUoIYQQQgihwCSEEEIIIRSYhBBCCCGEApMQQgghhFBgEkIIIYQQCkxCCCGEEEKBSQghhBBCKDAJIYQQQggFJiGEEEIIocAkhHCV6+/vJ5VKcezYMfJyuRzbt29nxYoVNDQ0sHr1aj788EPCyHv//fdZsWIFTzzxBOvXr+fEiROEwtDf308qleLYsWPk5XI5tm/fzooVK2hoaGD16tV8+OGHhHCpJIRwlevo6ODgwYOUlJRw4sQJampqePzxx9m/fz+ffvopzc3NLFy4kObmZsLIeeaZZ6itraWvr4+BgQEaGxuprq7m6NGjhCtfR0cHBw8epKSkhBMnTlBTU8Pjjz/O/v37+fTTT2lubmbhwoU0NzcTwqWQEMJVrr29ndmzZ5O3atUqstksTU1NvPXWWzz//PPs3r2b6dOn8/TTT3Po0CHCpbd7925ef/11HnjgAbZs2cKLL77IO++8w2effcby5csJV7729nZmz55N3qpVq8hmszQ1NfHWW2/x/PPPs3v3bqZPn87TTz/NoUOHCOGblhDCVSyXy7Fr1y7mzZtHLpfj7bffprKykjvvvJNBY8aM4cEHHyRv165dhEtvw4YNjB49mmXLljGorKyM+++/n71793Lo0CHClSuXy7Fr1y7mzZtHLpfj7bffprKykjvvvJNBY8aM4cEHHyRv165dhPBNSwjhKtbZ2cn58+epqKjg/PnzPPfccyxZsoThRo0aRd7JkycJl15HRwdVVVWMGjWKoaZOnUre3r17CVeuzs5Ozp8/T0VFBefPn+e5555jyZIlDDdq1CjyTp48SQjftIRwUblcjt7eXtLpNOl0mnQ6TTqdJp1Ok06nOXLkCGHk5HI5ent7SafTpNNp0uk06XSadDpNOp3myJEjDGpvb+e2225j3LhxFBUVMX/+fG6//XaG2717N3kVFRWES+vkyZOcPXuW8ePHM9y0adPI279/P2Fk5XI5ent7SafTpNNp0uk06XSadDpNOp3myJEjDGpvb+e2225j3LhxFBUVMX/+fG6//XaG2717N3kVFRWEkZXL5ejt7SWdTpNOp0mn06TTadLpNOl0miNHjlDoEsLXOnDgAFVVVSxevJja2lpSqRSpVIra2lpSqRSpVIo1a9YQRsaBAweoqqpi8eLF1NbWkkqlSKVS1NbWkkqlSKVSrFmzhkEdHR1UVFTwdfbs2cOf/vQnZsyYwcyZMwmXVjabJe+aa65huGuvvZa8L7/8kjByDhw4QFVVFYsXL6a2tpZUKkUqlaK2tpZUKkUqlWLNmjUM6ujooKKigq+zZ88e/vSnPzFjxgxmzpxJGDkHDhygqqqKxYsXU1tbSyqVIpVKUVtbSyqVIpVKsWbNGgpdQriggYEBmpqaaGlpIZvNUlNTQyaToa+vj5qaGjKZDJlMhsbGRsKlNzAwQFNTEy0tLWSzWWpqashkMvT19VFTU0MmkyGTydDY2Eje8ePH+eCDD7jjjju4kD179vDoo49y4403snr1asKld+7cOS7m3LlzhJExMDBAU1MTLS0tZLNZampqyGQy9PX1UVNTQyaTIZPJ0NjYSN7x48f54IMPuOOOO7iQPXv28Oijj3LjjTeyevVqwsgZGBigqamJlpYWstksNTU1ZDIZ+vr6qKmpIZPJkMlkaGxspNAl/I9Lp9PU19dTX19PfX099fX1NDU1kZfJZFi5ciWlpaXkcjnOnj1LXjabZfLkyYRvXjqdpr6+nvr6eurr66mvr6epqYm8TCbDypUrKS0tJZfLcfbsWfKy2SyTJ09muM7OTsaOHcu0adP4Kps3b+ahhx5i4sSJvPHGG0yYMIFw6ZWUlHAhuVyOvFGjRhFGRiaTYeXKlZSWlpLL5Th79ix52WyWyZMnM1xnZydjx45l2rRpfJXNmzfz0EMPMXHiRN544w0mTJhAGDmZTIaVK1dSWlpKLpfj7Nmz5GWzWSZPnszVJOF/3CeffEJXVxddXV10dXXR1dXFRx99RMJQqg4AAActSURBVF5VVRXFxcXktbW1MXXqVPIOHTrE+PHjCd+8Tz75hK6uLrq6uujq6qKrq4uPPvqIvKqqKoqLi8lra2tj6tSp5B06dIjx48cz3M6dO5k7dy5fpbGxkeXLl1NeXs6bb75JSUkJYWTcfPPN5J06dYrhjh49St71119PGBlVVVUUFxeT19bWxtSpU8k7dOgQ48ePZ7idO3cyd+5cvkpjYyPLly+nvLycN998k5KSEsLIqqqqori4mLy2tjamTp1K3qFDhxg/fjxXk4T/cfPnz6enp4eenh56enro6elh1apVDLdx40amT59OXnd3N7lcjvDNmz9/Pj09PfT09NDT00NPTw+rVq1iuI0bNzJ9+nTyuru7yeVyDNfW1sa8efMYbsWKFbz88stUV1ezfv16iouLCSNn1KhR3HDDDfT39zPcwYMHybv11lsJI2/jxo1Mnz6dvO7ubnK5HMO1tbUxb948hluxYgUvv/wy1dXVrF+/nuLiYsLltXHjRqZPn05ed3c3uVyOq0lCuKj+/n6y2SxlZWXknTlzhsOHDxMuj/7+frLZLGVlZeSdOXOGw4cPM1RfXx+nTp1i1qxZDPXCCy/Q0tLCPffcw6pVqygqKiKMvLvuuovu7m4OHz7MUC0tLYwePZrKykrCyOrv7yebzVJWVkbemTNnOHz4MEP19fVx6tQpZs2axVAvvPACLS0t3HPPPaxatYqioiLC5dXf3082m6WsrIy8M2fOcPjwYa4mCeGiNmzYQFVVFYOKioro7u4mXB4bNmygqqqKQUVFRXR3dzNUe3s7kydPpqSkhEHHjx9n3bp15J0+fZqGhgYaGhpoaGigoaGBhoYGNm3aRLj06urqGDt2LHV1dbS3t3P48GFWrlzJe++9x8MPP8yYMWMII2vDhg1UVVUxqKioiO7uboZqb29n8uTJlJSUMOj48eOsW7eOvNOnT9PQ0EBDQwMNDQ00NDTQ0NDApk2bCCNrw4YNVFVVMaioqIju7m6uJgnha+VyOZqbm6murmZQeXk5vb29hJGXy+Vobm6murqaQeXl5fT29jJUR0cHlZWVDNXZ2ckXX3xBXmtrK62trbS2ttLa2kprayutra309vYSLr2JEyfy0ksvkVdXV8eCBQvYuHEjjzzyCEuWLCGMrFwuR3NzM9XV1QwqLy+nt7eXoTo6OqisrGSozs5OvvjiC/JaW1tpbW2ltbWV1tZWWltbaW1tpbe3lzBycrkczc3NVFdXM6i8vJze3l6uJgnhayVJwtatW6msrGTQokWL2L59O2HkJUnC1q1bqaysZNCiRYvYvn07Qy1dupRUKsVQCxcuxDa2sY1tbGMb29hm5cqVhJFRXl7Ojh072LZtG6+++ir79u1j6dKlhJGXJAlbt26lsrKSQYsWLWL79u0MtXTpUlKpFEMtXLgQ29jGNraxjW1sY5uVK1cSRk6SJGzdupXKykoGLVq0iO3bt3M1SQgXNWnSJIYrLS0lXB6TJk1iuNLSUoaaOXMmEyZMIFz5ysrKmDlzJkVFRYTLZ9KkSQxXWlrKUDNnzmTChAmEK9+kSZMYrrS0lKtJQgghhBBCgUkIIYQQQigwCSGEEEIIBSYhhBBCCKHAJIQQQgghFJiEEEIIIYQCkxBCCCGEUGASQgghhBAKTEIIIVwhMpkMR48eJYQQLiYhhBCuEJs2bWLixImEEMLFJIQQwhXiW9/6FkmSEEIIF5MQQghXgIGBAa677jpCCOHfkRBCCFeAzs5OfvzjHxNCCP+OhBBCGEHHjx+nvb2dXC7HUJ2dnUyfPp28zz//nHQ6TQghXEhCCCGMkJ6eHl5//XUymQw1NTUMlyQJec8++yz33nsvBw4cIIQQvkpCCCGMgFwuR2trK0uXLuUHP/gB2WyW06dPkzcwMMCECRMY9NOf/pS8jz/+mBBC+CoJIYQwAtrb25k7dy55bW1t3HjjjVx77bXkdXZ2Ul5ezqA5c+Ywe/ZsJk2aRAghfJWEEEIYAdOmTePOO+/k3LlzbN26lZqaGgZ1dHQwa9YshpowYQK33HILIYTwVRJCCGEEFBcXk/fuu+9y+vRpqqurGZQkCUmSMCiXy3HdddcRQggXkhBCCCNoz549fOc73+Gmm24ib2BggAkTJjBUc3MzCxcuJIQQLiQhhBBG0BdffMEtt9zCoI6ODsrLyxlkm3/9619MmTKFEEK4kIQQQhhBP/nJT9i7dy9Hjx4lr7Ozk1mzZpG3d+9e3nrrLZYuXUoIIXydhBBCGEF33XUXv//973nooYdYt24dR48e5S9/+QtPPfUU/f39/Pa3vyWEEC4mIYQQRtivfvUrtmzZwrx58ygpKeF73/seTz31FL/85S8JIYR/R0IIIVwm/f39/OxnP0MSIYTwn0gIIYTLpLOzk4qKCkII4T+VEEIIl0kul6OoqIgQQvhPJYQQwmVy3333EUII/43/Bz7kwY7MRpk7AAAAAElFTkSuQmCC">
</div></div></div></div></div></div></div></div><div class = 'S8'><span>When looking very carefully it is possible to note that the final distribution of the electrons that start with </span><span texencoding="\eta=0" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36" height="18" /></span><span> is symmetric; half the electrons gains energy (</span><span texencoding="\dot\psi" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="13" height="20" /></span><span> above the green line) and the other half loses energy (</span><span texencoding="\dot\psi" style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAoCAYAAADg+OpoAAACYElEQVRYR+3WSaiOYRQH8N/NkKHIUErKlJJEZGsqGbKQYYmkJJKFsZCFsRQliYUSFlaGJREybMgQSTJtiBIhRRk7db7b2+2997vfd7tX6Z7V973vOc//Of/nf/7P26CDoqGDcHQC1c10J3X/N3UjcRYfMBffWmq3LWJYh/25+BxcaC+goTiNT1jYnh3VJIy2UNcJVBMDdSf/szMKmR7BH5zEpvxd7GQVJuJG5hTfTUPM1wTcxGL8ioSyjqbjSguD+AaDsRzHCyixgcM5T7HG7eIOyoC64yu6YTt2FgrG4lH+H4IAjRiGZ+iKFTjW9DCbO6P3GIgt2FsoqtjOYwRoJQ5hDZ5jVJliyoB6Z0eRPx/nC4XhZ7NwAOvzeTDwEVHXlIHG0jKgmbiYGaPxNH/3yAV7YnYhZyquZU6I4EFrO9qHjfiBXviZhTNwCd/Rv2CicYbb8Dmf/24t0B1MwhOMKRRVNhBg0XUl7qWcz2BRGUg8K6MubL9vXmoxV5UISsZjQ+EeCrW9yoSlOFULUFAWMo0ONmfhILzNjY0rSPwg1uIdhietpVhlHb3AiHSI1Vm1JF0gwGJYI0Io9xEiabGb5qjbnfPzMKmKvKAk7OQElqEfrub7uM6DzhajrKNwhMuYkvMS3he+FfTFwL7GjqRpF0IEVaM5Z+iSsxLqizNZkOa6By9ztm5VXb2Q0JprYiti53dT9rWs35jbGqDrmIzoJkDrimpAffIDMeQeZxZ3UF1RDShMNb5Gv2BAwY5qBqsGdBQrcS4FUTNApaAa0Ly8l8LPYjjrjmpAdS/ctLDDgP4CU2NxKfuIXAUAAAAASUVORK5CYII=" width="13" height="20" /></span><span>below the green line), such that the net effect is zero. On the other hand, the final positions of the electrons that start with </span><span texencoding="\eta=\dot\psi=0.4" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="76" height="20" /></span><span> show a slight imbalance between those that gain and those that lose energy.</span></div><h4 class = 'S9'><span>Small-signal gain</span></h4><div class = 'S0'><span>In order to explore this imbalance we select a larger range of starting energies </span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: normal; color: rgb(0, 0, 0);">η</span><span> (green) and calculate the average value of </span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: normal; color: rgb(0, 0, 0);">η</span><span> at the final (red) positions, which will be proportional to the averaged energy loss of the electrons. Since energy is conserved this energy increases the energy of the laser and is therefore normally called </span><span style=' font-family: monospace;'>gain</span><span>. </span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span >eta=-5.002:0.2:5; </span><span style="color: rgb(2, 128, 9);">% range of etas to explore</span></span></div></div><div class="inlineWrapper"><div class = 'S7'><span style="white-space: pre"><span >gain=zeros(1,length(eta)); </span></span></div></div></div><div class = 'S8'><span>We initialize the figure so we can watch the points added as they are calculated.</span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span >figure; hold </span><span style="color: rgb(170, 4, 249);">on</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >xlim([-5,5]); ylim([-7e-3,7e-3])</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >xlabel(</span><span style="color: rgb(170, 4, 249);">'\eta_0 = (\gamma_0-\gamma_r)/\gamma_r'</span><span >); </span></span></div></div><div class="inlineWrapper"><div class = 'S7'><span style="white-space: pre"><span >ylabel(</span><span style="color: rgb(170, 4, 249);">'<\eta_f> - \eta_0'</span><span >)</span></span></div></div></div><div class = 'S8'><span>In the following loop we scan the range of </span><span style=' font-family: monospace;'>eta</span><span> and then over the starting phases phi. Inside these loops we use pendulumtracker() to move the electrons to their final position, where we use their energy, available as x(2), to contribute to the gain. Whenever the loop over the phases completes, we plot an asterisk with the initial </span><span style=' font-family: monospace;'>eta</span><span> as the horizontal coordinate and the </span><span style=' font-family: monospace;'>gain</span><span> as the vertical coordinate.</span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">for </span><span >m=1:length(eta)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > </span><span style="color: rgb(14, 0, 255);">for </span><span >k=1:length(phi)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > x=pendulumtracker([phi(k),eta(m)],Omega,dt);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > gain(m)=gain(m)+x(2);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > </span><span style="color: rgb(14, 0, 255);">end</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > gain(m)=eta(m)-gain(m)/length(phi);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > plot(eta(m),gain(m),</span><span style="color: rgb(170, 4, 249);">'k*'</span><span >);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > pause(0.001);</span></span></div></div><div class="inlineWrapper outputs"><div class = 'S4'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">end</span></span></div><div class = 'S5'><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="2BC1DF0A" data-scroll-top="null" data-scroll-left="null" data-testid="output_2" style="width: 1405px;"><div class="figureElement"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="uniqName_333_407" widgetid="uniqName_333_407" style="width: 100%; height: auto;"><div class="ImageView" id="uniqName_333_409" widgetid="uniqName_333_409" style="width: 100%; height: auto;">
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">
</div></div></div></div></div></div></div></div><div class = 'S8'><span>This curve is known as the small-signal FEL gain curve, which can also be derived analytically with the result shown in Equation 10.31. </span></div><div class = 'S0'><span></span></div><h2 class = 'S10'><span>Appendix</span></h2><div class = 'S0'><span>The function</span><span style=' font-family: monospace;'> pendulumtracker()</span><span> receives the phase-space coordinates </span><span style=' font-family: monospace;'>x</span><span> at the start, the small-amplitude oscillation frequency </span><span style=' font-family: monospace;'>omega</span><span>, and the integration time </span><span style=' font-family: monospace;'>dt</span><span> as input and returns the phase-space coordinates </span><span style=' font-family: monospace;'>xout</span><span>. Internally, it integrates the equations of motion for a mathematical pendulum in closed form using Jacobi elliptic functions. This is much faster than numerically integration, expecially for extremely large times, such as thousands or even millions of synchrotron periods. The coding closely follows Section 5.4, especially Equations 5.50 to 5.54.</span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S2'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">function </span><span >xout=pendulumtracker(x,omega,dt)</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >k2=(0.5*x(2)/omega)^2+sin(0.5*x(1))^2; </span><span style="color: rgb(2, 128, 9);">% just after eq. 5.45</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >k=sqrt(k2);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">if </span><span >(x(1)>pi), x(1)=x(1)-2*pi; </span><span style="color: rgb(14, 0, 255);">end</span><span > </span><span style="color: rgb(2, 128, 9);">% map back into range [-pi,pi]</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">if </span><span >(x(1)<-pi), x(1)=x(1)+2*pi; </span><span style="color: rgb(14, 0, 255);">end</span><span > </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >s=1; </span><span style="color: rgb(14, 0, 255);">if </span><span >(x(1)<0), s=-s; x(1)=-x(1); </span><span style="color: rgb(14, 0, 255);">end</span><span > </span><span style="color: rgb(2, 128, 9);">% keep track of quadrant</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span >s1=1; </span><span style="color: rgb(14, 0, 255);">if </span><span >(x(2)<0), s1=-s1; </span><span style="color: rgb(14, 0, 255);">end</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">if </span><span >(k>1) </span><span style="color: rgb(2, 128, 9);">% outside the separatrix</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > kelf=ellipke(1/k2); </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > trev=2*kelf/(k*omega);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > t0=mod(dt,trev);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > tmp=s1*k*omega*t0+s*elliptic12(0.5*x(1),1/k2); </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > [sn,cn,dn]=ellipj(tmp,1/k2); </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > </span><span style="color: rgb(14, 0, 255);">if </span><span >(abs(tmp) > kelf), sn=-sn; </span><span style="color: rgb(14, 0, 255);">end</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > xout(1)=2*asin(sn); </span><span style="color: rgb(2, 128, 9);">% eq. 5.52</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > xout(2)=2*s1*omega*k*dn;</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">else</span><span > </span><span style="color: rgb(2, 128, 9);">% inside the separatrix</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > trev=4*ellipke(k2)/omega; </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > t0=mod(dt,trev);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > z0=asin(min(1,sin(0.5*x(1))/k));</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > tmp=s1*omega*t0+s*elliptic12(z0,k2); </span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > [sn,cn,dn]=ellipj(tmp,k2);</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > xout(1)=2*asin(k*sn); </span><span style="color: rgb(2, 128, 9);">% eq. 5.54</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span > xout(2)=2*s1*omega*k*cn;</span></span></div></div><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">end</span></span></div></div><div class="inlineWrapper"><div class = 'S7'><span style="white-space: pre"><span style="color: rgb(14, 0, 255);">end</span></span></div></div></div>
<br>
<!--
##### SOURCE BEGIN #####
%%
% Companion software for "Volker Ziemann, _Hands-on Accelerator physics using
% MATLAB, CRCPress, 2019_" (https://www.crcpress.com/9781138589940)
%% Phase space of a small-signal free-electron laser (Section 10.3)
% Volker Ziemann, 211124, CC-BY-SA-4.0
%
% *Important:* requires the elliptic package from <https://github.com/moiseevigor/elliptic
% https://github.com/moiseevigor/elliptic> located in a subdirectory below the
% present one (for the fast evaluation of elliptic functions).
%
% The dynamics of an electron moving in an undulator magnet and, at the same
% time, exposed to the field of an external laser can be described by a mathematical
% pendulum, given by Equation 10.30, which reads $\ddot\psi+\Omega^2\sin\psi=0$.
% Here $\eta=\dot\psi$ describes the energy difference with respect to the energy
% at which the electron motion and laser are synchronous, which is is encoded
% in the FEL resonance condition in Equation 10.28. Since the electron bunches
% are much longer than the laser wavelength, we can assume that the electrons
% are uniformly distributed in phase and follow these uniformly distributed electrons
% for some time and calculate where they are at the end of the FEL.
%
% The intensity of the laser and the parameters of the undulator are encoded
% in the oscillation frequency $\Omega$ and is given in Equation 10.30. We set
% the duration of the FEL process to 1/10 of the full oscillation period and then
% define a range of starting phase $\phi$.
clear
addpath ./elliptic
Omega=0.25 % Slider to set Omega
Ts=2*pi/Omega;
dt=0.1*Ts; % time step
phi=-pi:2*pi/201:pi;
%%
% We define the separatrix, which separates the periodic from the non-periodic
% trajectories, and plot both its upper and the lower branch. Then we scale the
% axes and annotate the plot.
separatrix=2*Omega*cos(0.5*phi); % eq. 5.47
clf
plot(phi,separatrix,'k',phi,-separatrix,'k')
axis([-pi,pi,-0.85,0.85]);
set(gca,'xtick',[-pi,-pi/2,0,pi/2,pi],'fontsize',14, ...
'xticklabels',{'-\pi','-\pi/2','0','\pi/2','\pi'})
xlabel('\psi');
ylabel('$$\dot\psi$$','interpreter','latex')
hold on;
%%
% Now we define a range of initial starting values for $\eta=\dot\psi$ and loop
% over the two values. Inside that loop we loop over the uniformly distributed
% starting phases of the electron that we mark with a green dot and use |pendulumtracker()|,
% defined in the Appendix, to find the final position after time |dt|, which we
% mark by a red dot. In this way we follow where each electron starts and ends
% it journey through the FEL.
eta2=0.4; % Slider for the second value of eta
eta0=[0,eta2];
for m=1:length(eta0)
for k=1:length(phi)
plot(phi(k),eta0(m),'g.')
x=pendulumtracker([phi(k),eta0(m)],Omega,dt);
plot(x(1),x(2),'r.')
pause(0.001)
end
end
%%
% When looking very carefully it is possible to note that the final distribution
% of the electrons that start with $\eta=0$ is symmetric; half the electrons gains
% energy ($\dot\psi$ above the green line) and the other half loses energy ($\dot\psi$below
% the green line), such that the net effect is zero. On the other hand, the final
% positions of the electrons that start with $\eta=\dot\psi=0.4$ show a slight
% imbalance between those that gain and those that lose energy.
% Small-signal gain
% In order to explore this imbalance we select a larger range of starting energies
% $\eta$ (green) and calculate the average value of $\eta$ at the final (red)
% positions, which will be proportional to the averaged energy loss of the electrons.
% Since energy is conserved this energy increases the energy of the laser and
% is therefore normally called |gain|.
eta=-5.002:0.2:5; % range of etas to explore
gain=zeros(1,length(eta));
%%
% We initialize the figure so we can watch the points added as they are calculated.
figure; hold on
xlim([-5,5]); ylim([-7e-3,7e-3])
xlabel('\eta_0 = (\gamma_0-\gamma_r)/\gamma_r');
ylabel('<\eta_f> - \eta_0')
%%
% In the following loop we scan the range of |eta| and then over the starting
% phases phi. Inside these loops we use pendulumtracker() to move the electrons
% to their final position, where we use their energy, available as x(2), to contribute
% to the gain. Whenever the loop over the phases completes, we plot an asterisk
% with the initial |eta| as the horizontal coordinate and the |gain| as the vertical
% coordinate.
for m=1:length(eta)
for k=1:length(phi)
x=pendulumtracker([phi(k),eta(m)],Omega,dt);
gain(m)=gain(m)+x(2);
end
gain(m)=eta(m)-gain(m)/length(phi);
plot(eta(m),gain(m),'k*');
pause(0.001);
end
%%
% This curve is known as the small-signal FEL gain curve, which can also be
% derived analytically with the result shown in Equation 10.31.
%
%
%% Appendix
% The function |pendulumtracker()| receives the phase-space coordinates |x|
% at the start, the small-amplitude oscillation frequency |omega|, and the integration
% time |dt| as input and returns the phase-space coordinates |xout|. Internally,
% it integrates the equations of motion for a mathematical pendulum in closed
% form using Jacobi elliptic functions. This is much faster than numerically integration,
% expecially for extremely large times, such as thousands or even millions of
% synchrotron periods. The coding closely follows Section 5.4, especially Equations
% 5.50 to 5.54.
function xout=pendulumtracker(x,omega,dt)
k2=(0.5*x(2)/omega)^2+sin(0.5*x(1))^2; % just after eq. 5.45
k=sqrt(k2);
if (x(1)>pi), x(1)=x(1)-2*pi; end % map back into range [-pi,pi]
if (x(1)<-pi), x(1)=x(1)+2*pi; end
s=1; if (x(1)<0), s=-s; x(1)=-x(1); end % keep track of quadrant
s1=1; if (x(2)<0), s1=-s1; end
if (k>1) % outside the separatrix
kelf=ellipke(1/k2);
trev=2*kelf/(k*omega);
t0=mod(dt,trev);
tmp=s1*k*omega*t0+s*elliptic12(0.5*x(1),1/k2);
[sn,cn,dn]=ellipj(tmp,1/k2);
if (abs(tmp) > kelf), sn=-sn; end
xout(1)=2*asin(sn); % eq. 5.52
xout(2)=2*s1*omega*k*dn;
else % inside the separatrix
trev=4*ellipke(k2)/omega;
t0=mod(dt,trev);
z0=asin(min(1,sin(0.5*x(1))/k));
tmp=s1*omega*t0+s*elliptic12(z0,k2);
[sn,cn,dn]=ellipj(tmp,k2);
xout(1)=2*asin(k*sn); % eq. 5.54
xout(2)=2*s1*omega*k*cn;
end
end
##### SOURCE END #####
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