An Estimation of the Nucleonic Charge of a Neutron Relative to that of a Proton Using the Ratios of Second Differences </HEAD> <STYLE> BODY {background:#AAFFDD; color:#000000; margin-left:0.2in; margin-right:0.2in;} H2 { text-align:center; font-size:20pt; color:#FF0000; } H4 {margin-left:2in; margin-right:2in; text-align:center; font-size:20pt; color:#000000; background:#FFFF00; padding:20px; border-style:outset; border-color:#FF0000; border-width:1px;} LI {font-size:18pt; text-indent:0.2in; text-weight:400; line-height:1.2em; background:#0077FF; border-style:outset; border-color: white; border-width:5px; padding:20px; } UL {background:#AAFFFF; border: 20px black; padding:20px; spacing:20px;} TH {font-size:20pt} TD {font-size:20pt; text-align:right spacing: 20px;} P {font-size:20pt; } </STYLE> <BODY BGCOLOR=#9ACD32 LINK=#000000 VLINK=#FF0000 ALINK=#FF0000> <center><table bgcolor=#00ffff frame=box border=9 cellpadding=15><th><B>San José State University</B></th></table></center> <P><CENTER><TABLE BGCOLOR=#00ffff FRAME=box BORDER=9 CELLPADDING=15> <TH><FONT SIZE="5" COLOR=#000000>applet-magic.com<br> <I>Thayer Watkins</I><br>Silicon Valley<BR>& Tornado Alley<br>USA</font></B></FONT></CENTER> </TH></TABLE></CENTER> <P><CENTER><TABLE BORDER=9 FRAME=box BGCOLOR=#000000 CELLPADDING=10><TR><TD> <CENTER><FONT SIZE="5" COLOR=#FFFFFF> An Estimation of the Nucleonic Charge<br> of a Neutron Relative to that of a Proton<br> Using the Ratios of Second Differences </FONT></CENTER> </TD></TR></TABLE> </CENTER> <b> <p>What is conventionally called the <i>nuclear strong force</i> is here called the <i>nucleonic force</i>. The reason for this change in terminology is that this force is not so strong in comparison with another type of force involved in the structure of nuclei. That other type of force is the force involved in the formation of spin pairs. That force, although strong, is exclusive in the sense that one neutron can form a spin pair with one and only one other neutron and with one and only one proton. The same holds true for a proton. <p>Each nucleon has a charge with respect to the nucleonic force. The force between two nucleons or aggregates of nucleons is of the form <h4> F = Hq<sub>1</sub>q<sub>2</sub>f(s)/s² </h4> <p>where H is a constant, q<sub>1</sub> and q<sub>2</sub> are the nucleonic charges, s is the distance between the particles and f(s) is a declining function of distance. Since f(s) is a declining function this means the nucleonic force drops off with distance faster than inverse distance-squared. <p>For now the repulsion between two protons is ignored. <p>The loss in potential energy involved in the formation of the system is then proportional q<sub>1</sub>q<sub>2</sub>U(s), where U(s) is an integral of f(z)/z² from ∞ down to s. <p>Let the nucleonic charge of the proton be taken to be 1 and denote the nucleonic charge of the neutron as q, where q can be a negative or positive number. The binding energy of a nucleus is closely related to the loss of potential energy involved. The three types of interactions through the nucleonic force are then <h4> neutron-neutron: I<sub>nn</sub> = Hq²U(s<sub>nn</sub>) <br> neutron-proton: I<sub>np</sub> = HqU(s<sub>np</sub>) <br> proton-proton: I<sub>pp</sub> = HU(s) </h4> <p>Thus, among other relationship, there is <h4> I<sub>nn</sub>/I<sub>pp</sub> = q²[U(s<sub>nn</sub>)/U(s<sub>pp</sub>)] </h4> <p>If s<sub>nn</sub> is equal to s<sub>pp</sub> then the ratio R=I<sub>nn</sub>/I<sub>pp</sub> gives an estimate of q². and √R gives an estimate of |q|. <p>The second differences in the binding energies of neutron spin pairs and proton spin pairs were computed and used as estimates of I<sub>nn</sub>T and I<sub>pp</sub>, respectively. Their ratios were computed and the square roots taken to give estimates of |q|. There were 450 separate values for √R computed. The average of these 450 values is 0.63, only about 5.3 percent lower than value of 0.67=2/3 found by other methods. The crucial matter is that q is less than unity in magnitude. <h2>The Effect of the Electrostatic<br> Repulsion between Two Protons</h4> <p>The force between two protons is <h4> F = Hf(s)/s² + Ke²/s² </h4> <p>where K is a constant and e is the electrostatic charge of a proton. <p>The above force can be put into the form <h4> F = Hf(s)/s²[1 + Ke²/(Hf(s))] </h4> <p>Let Ke²/(Hf(s)) be denoted as D(s). Then the interaction between two protons can be treated as though it is an interaction between two particles each with a nucleonic charge of (1+D(s))<sup>½</sup>. The same would hold true for the force between two proton spin pairs. <p> The ratio R previously computed would be equal to <h4> R = q²/(1+D(s)) <br>and hence<br> √R = q/(1+D(s))<sup>½</sup> </h4> <p>To see how this would affect the result consider the case in which f(s)=exp(−s/s<sub>0</sub>), which may well be the actual value of f(s) for nuclei. <p>Then <h4> D(s) = [Ke²/H]exp(s/s<sub>0</sub>)] </h4> <p>Then √R could take the following form. <p><center><img src="chargeratios2a.gif"></center> <p>For each number of proton spin pairs the estimates of q were averaged. The plot of those averages is shown below. <p><center><img src="chargeratios2c.gif"></center> <hr> <p>Appendix <p><center> <TABLE BORDER=9 FRAME=box BGCOLOR=#ffffaa CELLPADDING=10><th colspan=11 bgcolor=#FFFFFF><b> The Square Roots of the Ratios of the Second Difference in Binding Energy<br> with respect to the Number of Neutron Pairs the Number of Proton Pairs</b></th> <tr><td> </td><td colspan=10>Number of Proton Spin Pairs </td></tr> <tr><td>Number<br>of Neutron<br>Spin Pairs </td><td> 3 </td><td> 4 </td><td> 5 </td><td> 6 </td><td> 7 </td><td> 8 </td><td> 9 </td><td> 10 </td></tr> <tr><td> 3 </td><td> 0.76097678 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 4 </td><td> 3.342440919 </td><td> 0.888873603 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 5 </td><td> </td><td> </td><td> 0.878239049 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 6 </td><td> </td><td> </td><td> 1.105435604 </td><td> 0.86545272 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 7 </td><td> </td><td> </td><td> 0.4840778 </td><td> 1.506199686 </td><td> 0.843728527 </td><td> 0.542130869 </td><td> </td><td> </td></tr> <tr><td> 8 </td><td> </td><td> </td><td> 0.533766449 </td><td> 1.063771328 </td><td> 1.374851088 </td><td> 0.888299998 </td><td> 0.80928372 </td><td> </td></tr> <tr><td> 9 </td><td> </td><td> </td><td> </td><td> 1.148258972 </td><td> 0.81471709 </td><td> 0.927069005 </td><td> 0.884754145 </td><td> 0.687026576 </td></tr> <tr><td> 10 </td><td> </td><td> </td><td> </td><td> </td><td> 0.989685838 </td><td> 0.613669826 </td><td> 1.052091321 </td><td> 0.850028537 </td></tr> <tr><td> 11 </td><td> </td><td> </td><td> </td><td> </td><td> 0.659535677 </td><td> 0.842676303 </td><td> 0.813977872 </td><td> 1.394669167 </td></tr> <tr><td> 12 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.196923806 </td><td> 0.353210955 </td><td> 0.414078855 </td></tr> <tr><td> 13 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.452368543 </td><td> 0.549081208 </td></tr> <tr><td> 14 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.230450308 </td><td> 0.367137041 </td></tr> </table></center> <p><center> <TABLE BORDER=9 FRAME=box BGCOLOR=#ffffaa CELLPADDING=10><th colspan=11 bgcolor=#FFFFFF><b> The Square Roots of the Ratios of the Second Difference in Binding Energy<br> with respect to the Number of Neutron Pairs the Number of Proton Pairs</b></th> <tr><td></td><td colspan=10>Number of Proton Spin Pairs </td></tr> <tr><td>Number<br>of Neutron<br>Spin Pairs </td><td> 11 </td><td> 12 </td><td> 13 </td><td> 14 </td><td> 15 </td><td> 16 </td><td> 17 </td><td> 18 </td><td> 19 </td><td> 20 </td></tr> <tr><td> 10 </td><td> 0.412240303 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 11 </td><td> 0.92383741 </td><td> 0.715591185 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 12 </td><td> 1.154897974 </td><td> 0.807742496 </td><td> 0.593964561 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 13 </td><td> 0.633474302 </td><td> 1.283470596 </td><td> 0.82496589 </td><td> 0.561499917 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 14 </td><td> 0.442030824 </td><td> 0.841113184 </td><td> 1.372386581 </td><td> 0.762629328 </td><td> 0.544042857 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 15 </td><td> 0.873630372 </td><td> 1.224955057 </td><td> 0.969849017 </td><td> 1.437655259 </td><td> 0.904114113 </td><td> 0.910376551 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 16 </td><td> 0.530033474 </td><td> 0.924241886 </td><td> 1.025375303 </td><td> 0.676452735 </td><td> 0.934241329 </td><td> 0.816922913 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 17 </td><td> 0.677656808 </td><td> 0.79637092 </td><td> 0.828629402 </td><td> 0.689820564 </td><td> 0.595038985 </td><td> 1.169928848 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 18 </td><td> </td><td> 0.756477405 </td><td> 0.75235399 </td><td> 0.701587149 </td><td> 0.549998082 </td><td> 0.702978671 </td><td> 1.223239667 </td><td> </td><td> </td><td> </td></tr> <tr><td> 19 </td><td> </td><td> </td><td> 0.540164584 </td><td> 0.691604293 </td><td> 0.506387829 </td><td> 0.720788966 </td><td> 0.753661371 </td><td> 1.293016368 </td><td> </td><td> </td></tr> <tr><td> 20 </td><td> </td><td> </td><td> </td><td> 0.61742568 </td><td> 0.468351381 </td><td> 0.71505808 </td><td> 0.694461522 </td><td> 0.654405692 </td><td> 1.393298033 </td><td> </td></tr> <tr><td> 21 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.598265291 </td><td> 0.607607441 </td><td> 0.747394159 </td><td> 0.76926557 </td><td> </td></tr> <tr><td> 22 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.585690218 </td><td> 0.700897889 </td><td> 0.719943941 </td><td> 0.731925055 </td></tr> <tr><td> 23 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.514837928 </td><td> 0.581624497 </td><td> 0.620911323 </td><td> 0.598849472 </td></tr> <tr><td> 24 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.469354166 </td><td> 0.50773752 </td><td> 0.525408588 </td><td> 0.402415086 </td></tr> <tr><td> 25 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.647072007 </td><td> 0.578692723 </td><td> 0.424249904 </td><td> 0.422447179 </td></tr> <tr><td> 26 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.933029127 </td><td> 1.223358315 </td><td> 1.361217392 </td><td> 1.208345115 </td></tr> <tr><td> 27 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.644499098 </td><td> 0.58584714 </td><td> 0.498485136 </td></tr> <tr><td> 28 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.64040202 </td><td> 0.42935013 </td></tr> <tr><td> 29 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.933654393 </td><td> 0.820864864 </td></tr> <tr><td> 30 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.374312557 </td></tr> </table></center> <p><center> <TABLE BORDER=9 FRAME=box BGCOLOR=#ffffaa CELLPADDING=10><th colspan=11 bgcolor=#FFFFFF><b> The Square Roots of the Ratios of the Second Difference in Binding Energy<br> with respect to the Number of Neutron Pairs the Number of Proton Pairs</b></th> <tr><td> </td><td colspan=10>Number of Proton Spin Pairs </td></tr> <tr><td>Number<br>of Neutron<br>Spin Pairs </td><td> 21 </td><td> 22 </td><td> 23 </td><td> 24 </td><td> 25 </td><td> 26 </td><td> 27 </td><td> 28 </td><td> 29 </td><td> 30 </td></tr> <tr><td> 21 </td><td> 0.507916059 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 22 </td><td> 0.526087363 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 23 </td><td> 0.543366894 </td><td> 0.574665585 </td><td> 0.703825526 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 24 </td><td> 1.403022198 </td><td> 1.431831591 </td><td> 1.383181073 </td><td> 1.424973006 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 25 </td><td> 0.653526139 </td><td> 0.804121575 </td><td> 0.702760878 </td><td> 0.784844117 </td><td> 0.724750047 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 26 </td><td> 0.519415807 </td><td> 0.860628574 </td><td> 0.715428039 </td><td> 0.433324891 </td><td> 0.598188879 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 27 </td><td> 0.604707175 </td><td> 0.747549933 </td><td> 0.686488254 </td><td> 0.749161197 </td><td> 0.672789893 </td><td> 0.480088417 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 28 </td><td> 0.481487527 </td><td> 0.604734382 </td><td> 0.591094882 </td><td> 0.648787002 </td><td> 0.519677858 </td><td> 0.560152714 </td><td> 0.838069443 </td><td> </td><td> </td><td> </td></tr> <tr><td> 29 </td><td> 0.452524206 </td><td> 0.540652136 </td><td> 0.566721364 </td><td> 0.65713604 </td><td> 0.632267198 </td><td> 0.247860444 </td><td> 0.675233572 </td><td> 0.727392967 </td><td> </td><td> </td></tr> <tr><td> 30 </td><td> 0.497582581 </td><td> 0.603779628 </td><td> 0.540834407 </td><td> 0.59201877 </td><td> 0.640263921 </td><td> 0.494903223 </td><td> 0.699628804 </td><td> 0.784464541 </td><td> </td><td> </td></tr> <tr><td> 31 </td><td> 0.507819985 </td><td> 0.579309934 </td><td> 0.552706907 </td><td> 0.560814517 </td><td> 0.653852381 </td><td> 0.428225768 </td><td> 0.59057739 </td><td> 0.928762684 </td><td> 0.73073975 </td><td> </td></tr> <tr><td> 32 </td><td> </td><td> 0.50481552 </td><td> 0.548085867 </td><td> 0.552130407 </td><td> 0.605157522 </td><td> 0.376063673 </td><td> 0.612890298 </td><td> 0.599590316 </td><td> 0.657382779 </td><td> </td></tr> <tr><td> 33 </td><td> </td><td> </td><td> 0.499818149 </td><td> 0.54037939 </td><td> 0.537527832 </td><td> 0.386554525 </td><td> 0.689687399 </td><td> 0.719045664 </td><td> 0.716014739 </td><td> 0.698430296 </td></tr> <tr><td> 34 </td><td> </td><td> </td><td> </td><td> 0.617558274 </td><td> 0.488948361 </td><td> 0.358790829 </td><td> 0.704166004 </td><td> 0.591968447 </td><td> 0.648355085 </td><td> 0.675394799 </td></tr> <tr><td> 35 </td><td> </td><td> </td><td> </td><td> 0.540467634 </td><td> 0.503261464 </td><td> 0.337698391 </td><td> 0.636742579 </td><td> 0.605594819 </td><td> 0.673140394 </td><td> 0.771308091 </td></tr> <tr><td> 36 </td><td> </td><td> </td><td> </td><td> </td><td> 0.502108929 </td><td> 0.313738001 </td><td> 0.565264856 </td><td> 0.545296545 </td><td> 0.540714116 </td><td> 0.644465254 </td></tr> <tr><td> 37 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.283926936 </td><td> 0.54632173 </td><td> 0.538543285 </td><td> 0.498473996 </td><td> 0.5 </td></tr> <tr><td> 38 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.251259706 </td><td> 0.519362575 </td><td> 0.518504836 </td><td> 0.570757487 </td><td> 0.496897008 </td></tr> <tr><td> 39 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.267702985 </td><td> 0.559279924 </td><td> 0.507942408 </td><td> 0.528009347 </td><td> 0.700542471 </td></tr> <tr><td> 40 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 1.485077286 </td><td> 1.313396314 </td><td> 1.463747054 </td><td> 1.339896129 </td></tr> <tr><td> 41 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.588181474 </td><td> 0.452605515 </td><td> 0.500561987 </td><td> 0.541352718 </td></tr> <tr><td> 42 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.536957095 </td><td> 0.504894775 </td><td> 0.55102272 </td></tr> <tr><td> 43 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.546410645 </td><td> 0.527523026 </td><td> 0.316304548 </td></tr> <tr><td> 44 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.24743583 </td><td> #NUM! </td></tr> <tr><td> 45 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.657813723 </td></tr> <tr><td> 46 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.588348405 </td></tr> </table><br> #NUM! means an estimate of |q| cannot be computed because R is negative. </center> <p><center> <TABLE BORDER=9 FRAME=box BGCOLOR=#ffffaa CELLPADDING=10><th colspan=11 bgcolor=#FFFFFF><b> The Square Roots of the Ratios of the Second Difference in Binding Energy<br> with respect to the Number of Neutron Pairs the Number of Proton Pairs</b></th> <tr><td> </td><td colspan=10>Number of Proton Spin Pairs </td></tr> <tr><td> Number<br>of Neutron<br>Spin Pairs </td><td> 31 </td><td> 32 </td><td> 33 </td><td> 34 </td><td> 35 </td><td> 36 </td><td> 37 </td><td> 38 </td><td> 39 </td><td> 40 </td></tr> <tr><td> 36 </td><td> 0.625543242 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 37 </td><td> 0.657951695 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 38 </td><td> 0.622841099 </td><td> 0.764922469 </td><td> #NUM! </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 39 </td><td> 0.350792193 </td><td> 0.313265745 </td><td> 0.369989918 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 40 </td><td> 0.525872773 </td><td> 0.41872275 </td><td> 0.625719301 </td><td> 0.645497224 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 41 </td><td> 0.548460231 </td><td> 0.688771611 </td><td> 0.560068108 </td><td> 0.765031867 </td><td> 0.729756383 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 42 </td><td> 1.417086523 </td><td> 1.422962647 </td><td> 1.281415268 </td><td> 1.404804186 </td><td> 1.518785457 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 43 </td><td> 0.574849101 </td><td> 0.588858133 </td><td> 0.57581346 </td><td> 0.444570892 </td><td> 0.506911545 </td><td> 0.504793677 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 44 </td><td> 0.555758036 </td><td> 0.524548287 </td><td> 0.520244007 </td><td> 0.519932371 </td><td> 0.427109825 </td><td> 0.561400873 </td><td> 0.545942788 </td><td> </td><td> </td><td> </td></tr> <tr><td> 45 </td><td> 0.037062466 </td><td> #NUM! </td><td> 0.255757238 </td><td> 0.548032518 </td><td> 0.583595369 </td><td> 0.455573452 </td><td> 0.55365392 </td><td> 0.511271138 </td><td> </td><td> </td></tr> <tr><td> 46 </td><td> 0.081549571 </td><td> 0.275599449 </td><td> 0.337084671 </td><td> 0.341532187 </td><td> 0.495384099 </td><td> 0.603022689 </td><td> 0.45142684 </td><td> 0.561996995 </td><td> </td><td> </td></tr> <tr><td> 47 </td><td> 0.532878539 </td><td> 0.535259327 </td><td> 0.512836087 </td><td> 0.455055013 </td><td> 0.518063376 </td><td> 0.556003404 </td><td> 0.655656204 </td><td> 0.45692449 </td><td> 0.598837672 </td><td> </td></tr> <tr><td> 48 </td><td> 0.636366115 </td><td> 0.645169478 </td><td> 0.550603121 </td><td> 0.524163595 </td><td> 0.510593657 </td><td> 0.510044784 </td><td> 0.54238798 </td><td> 0.590778815 </td><td> 0.453180022 </td><td> </td></tr> <tr><td> 49 </td><td> 0.516913919 </td><td> 0.626856658 </td><td> 0.571573996 </td><td> 0.479032505 </td><td> 0.512368607 </td><td> 0.55171661 </td><td> 0.632291322 </td><td> 0.543384093 </td><td> 0.577073959 </td><td> </td></tr> <tr><td> 50 </td><td> </td><td> 0.573451091 </td><td> 0.728615823 </td><td> 0.616206252 </td><td> 0.547861274 </td><td> 0.582290112 </td><td> 0.46778027 </td><td> 0.606761241 </td><td> 0.527798663 </td><td> 0.655295174 </td></tr> <tr><td> 51 </td><td> </td><td> </td><td> 0.586979944 </td><td> 0.633193012 </td><td> 0.528279282 </td><td> 0.529430928 </td><td> 0.657315505 </td><td> 0.507599036 </td><td> 0.601867841 </td><td> 0.591877075 </td></tr> <tr><td> 52 </td><td> </td><td> </td><td> </td><td> 0.57644705 </td><td> 0.553297397 </td><td> 0.507540317 </td><td> 0.645679029 </td><td> 0.541048913 </td><td> 0.481812056 </td><td> 0.635394923 </td></tr> <tr><td> 53 </td><td> </td><td> </td><td> </td><td> </td><td> 0.627007924 </td><td> 0.572005833 </td><td> 0.562138773 </td><td> 0.578227035 </td><td> 0.510384209 </td><td> 0.474798406 </td></tr> <tr><td> 54 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.397866324 </td><td> 0.607030141 </td><td> 0.42435068 </td><td> 0.586032715 </td><td> 0.680599191 </td></tr> <tr><td> 55 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.758776861 </td><td> 0.64252637 </td><td> 0.398296657 </td><td> 0.68102254 </td></tr> <tr><td> 56 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.504170214 </td><td> 0.467378458 </td><td> 0.605568891 </td></tr> <tr><td> 57 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.500551086 </td><td> 0.465795816 </td><td> 0.566484462 </td></tr> <tr><td> 58 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.426123116 </td><td> 0.550752102 </td></tr> <tr><td> 59 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.494108971 </td></tr> <tr><td> 60 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.533532688 </td></tr> </table><br> #NUM! means an estimate of |q| cannot be computed because R is negative. </center> <p><center> <TABLE BORDER=9 FRAME=box BGCOLOR=#ffffaa CELLPADDING=10><th colspan=11 bgcolor=#FFFFFF><b> The Square Roots of the Ratios of the Second Difference in Binding Energy<br> with respect to the Number of Neutron Pairs the Number of Proton Pairs</b></th> <tr><td> </td><td colspan=10>Number of Proton Spin Pairs </td></tr> <tr><td>Number<br>of Neutron<br>Spin Pairs </td><td> 41 </td><td> 42 </td><td> 43 </td><td> 44 </td><td> 45 </td><td> 46 </td><td> 47 </td><td> 48 </td><td> 49 </td><td> 50 </td><td> 51 </td><td> 52 </td><td> 53 </td></tr> <tr><td> 53 </td><td> 0.53311399 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 54 </td><td> 0.403844189 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 55 </td><td> 0.415568729 </td><td> 0.330969237 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 56 </td><td> 0.474814687 </td><td> 0.385066049 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 57 </td><td> 0.420285892 </td><td> 0.429842985 </td><td> 0.503154605 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 58 </td><td> 0.422125093 </td><td> 0.394220045 </td><td> 0.610467696 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 59 </td><td> 0.416666667 </td><td> 0.389140301 </td><td> 0.583459966 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 60 </td><td> 0.380104585 </td><td> 0.380627379 </td><td> 0.618195442 </td><td> 0.622045617 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 61 </td><td> 0.417186128 </td><td> 0.333368883 </td><td> 0.629800004 </td><td> 0.636367878 </td><td> #NUM! </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 62 </td><td> 0.407566854 </td><td> 0.323165947 </td><td> 0.585838456 </td><td> 0.641718672 </td><td> 1.126095454 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 63 </td><td> </td><td> 0.348807492 </td><td> 0.77313118 </td><td> 0.783894618 </td><td> 0.830168583 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 64 </td><td> </td><td> 0.820065761 </td><td> 1.417765556 </td><td> 1.363105376 </td><td> 1.287119167 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 65 </td><td> </td><td> </td><td> 0.379447811 </td><td> 0.315185202 </td><td> 0.378080895 </td><td> 0.294609559 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 66 </td><td> </td><td> </td><td> 0.514456759 </td><td> 0.403579558 </td><td> 0.356372091 </td><td> 0.297670279 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 67 </td><td> </td><td> </td><td> </td><td> 0.541653422 </td><td> 0.366302082 </td><td> 0.361556208 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 68 </td><td> </td><td> </td><td> </td><td> </td><td> 0.454740049 </td><td> 0.393643182 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 69 </td><td> </td><td> </td><td> </td><td> </td><td> 0.456510161 </td><td> 0.535136878 </td><td> 0.544952163 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 70 </td><td> </td><td> </td><td> </td><td> </td><td> 0.547664265 </td><td> 0.57994199 </td><td> 0.525841477 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 71 </td><td> </td><td> </td><td> </td><td> </td><td> 0.518813713 </td><td> 0.547930415 </td><td> 0.528889101 </td><td> 0.478981809 </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 72 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.629075377 </td><td> 0.614020657 </td><td> 0.669416416 </td><td> 0.483050923 </td><td> </td><td> </td><td> </td><td> </td></tr> <tr><td> 73 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.574516485 </td><td> 0.616382144 </td><td> 0.637922793 </td><td> 0.648197216 </td><td> 0.554910164 </td><td> </td><td> </td><td> </td></tr> <tr><td> 74 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.61591384 </td><td> 0.571348091 </td><td> 0.500603944 </td><td> 0.535849027 </td><td> </td><td> </td><td> </td></tr> <tr><td> 75 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.563233571 </td><td> 0.542098077 </td><td> 0.421927449 </td><td> </td><td> </td><td> </td></tr> <tr><td> 76 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.663485587 </td><td> 0.554291084 </td><td> </td><td> </td><td> </td></tr> <tr><td> 77 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.857840551 </td><td> #NUM! </td><td> </td><td> </td></tr> <tr><td> 78 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.5137803 </td><td> #NUM! </td><td> 0.250036629 </td><td> </td></tr> <tr><td> 79 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> #NUM! </td><td> 0.209369569 </td><td> </td></tr> <tr><td> 80 </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> 0.415604707 </td></tr> </table><br> #NUM! means an estimate of |q| cannot be computed because R is negative. </center> <p>(To be continued.) <p> <P><hr> <p> <CENTER><TABLE BORDER=9 FRAME=box BGCOLOR=#00aa00 CELLPADDING=20><TD> <CENTER><A HREF="index.html">HOME PAGE OF 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