ch24_p76 - 76. We note that for two points on a circle,...

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76. We note that for two points on a circle, separated by angle θ (in radians), the direct- line distance between them is r = 2 R sin( /2). Using this fact, distinguishing between the cases where N = odd and N = even, and counting the pair-wise interactions very carefully, we arrive at the following results for the total potential energies. We use k = 14 π 0 ε . For configuration 1 (where all N electrons are on the circle), we have () 1 1 22 1, even 1, odd 11 1 , 2 sin 2 2 2 sin 2 NN jj Nke Nke UU Rj θθ == §· § · ¨¸ ¨ ¸ =+ = ¨ ¸ ¨ ¸ ©¹ © ¹ ¦¦ where = 2 π N . For configuration 2, we find U Nk e U e N j N N j N 2 2 1 2 1 2 2 1 3 2 1 2 1 2 2 1 2 1 2 5 2 , , sin = = = = = + F H G G I K J J = + F H G G I K J J ¦ ¦ even odd b g b g b g b g where ′ = 2 1 π N . The results are all of the form U ke R 1 2 2 or2 a pure number. × In our table, below, we have the results for those “pure numbers” as they depend on N and on which configuration we are considering. The values listed in the U rows are the potential energies divided by ke 2 /2 R . N 4 5 6 7 8 9 10 11 12 13 14 15 U 1 3.83 6.88 10.96 16.13 22.44 29.92 38.62 48.58 59.81 72.35 86.22 101.5 U 2 4.73 7.83 11.88 16.96 23.13 30.44 39.92
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ch24_p76 - 76. We note that for two points on a circle,...

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