Positronium in a Liquid Phase: Formation, Bubble State and Chemical Reactions

<table>Tao-Eldrup model of Ps atom in a liquid phase. It is assumed that Ps is confined in an infinite spherically symmetric potential well of radius <svg height="15.4" id="M334" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 23.8375 15.4" width="23.8375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
<g transform="matrix(.012,0,0,-.012,11.013,15.188)"><path d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196
q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" id="x221E"></path></g>
</svg>. <svg height="15.0875" id="M335" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 31.387501 15.0875" width="31.387501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,12.138)"><path d="M699 659l10 -27q-31 -17 -42 -39t-21 -74l-14 -69q-10 -53 -26 -82t-47 -53q-58 -45 -156 -48v-145q0 -58 14.5 -74t71.5 -20v-28h-256v28q57 4 71 20t14 74v145q-104 3 -163 49q-29 23 -45 54t-25 82l-12 72q-11 62 -20 77q-11 19 -38 31l10 27q74 -11 106 -50
q20 -24 30 -78l11 -60q16 -92 48.5 -128t97.5 -41v227q0 57 -13.5 73t-66.5 20v28h245v-28q-53 -4 -66.5 -20t-13.5 -73v-227q75 4 103.5 37.5t45.5 132.5l10 58q8 51 33 80q35 38 104 49z" id="x3A8"></path></g><g transform="matrix(.017,0,0,-.017,12.387,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,0,0,-.017,18.268,12.138)"><path d="M393 379q-9 -16 -28 -29q-15 -10 -23 -2q-19 19 -36 19q-21 0 -52 -38q-57 -72 -82 -126l-40 -197q-23 -3 -75 -18l-7 7q49 196 74 335q7 43 -2 43q-7 0 -30 -14.5t-47 -37.5l-16 23q37 42 82 73t67 31q41 0 15 -113l-11 -50h4q41 71 85 117t77 46q29 0 45 -26
q13 -21 0 -43z" id="x1D45F"></path></g><g transform="matrix(.017,0,0,-.017,25.442,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> is the center-of-mass wave function of Ps. <svg height="11.3125" id="M336" style="vertical-align:-0.1092pt" version="1.1" viewbox="0 0 11.075 11.3125" width="11.075" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg> is the free volume radius of the Ps bubble. <svg height="12.4375" id="M337" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 8.9125004 12.4375" width="8.9125004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,12.163)"><path d="M494 514l-10 -15q-56 76 -106.5 117t-110.5 41q-45 0 -45 -30q0 -28 109 -152q59 -68 86 -118.5t27 -107.5q0 -62 -29.5 -119.5t-86.5 -96.5q-64 -45 -138 -45q-76 0 -121.5 53t-45.5 136q0 74 40.5 135t98.5 95q43 26 87 42q-40 59 -57.5 92t-17.5 62q0 46 39 77.5
t90 31.5q54 0 97.5 -35.5t65 -78.5t28.5 -84zM359 234q0 84 -90 191q-47 -17 -74 -47q-81 -89 -81 -200q0 -71 32.5 -109.5t72.5 -38.5q47 0 80 33t46.5 77.5t13.5 93.5z" id="x1D6FF"></path></g>
</svg> is the penetration depth of the molecular electrons into the Ps bubble.</table>

Advances in Physical Chemistry

fig11

Figure 11

Figure 11: Positronium in a Liquid Phase: Formation, Bubble State and Chemical Reactions 