P25_086.PDF

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Chapter 25 - 25.86
86. 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( θ/ 2). Using this fact, distinguishing between the cases where N =oddand N =even,
and counting the pair-wise interactions very carefully, we arrive at the following results for the total
potential energies. We use k =1 / 4 πε 0 . For configuration 1 (where all N electrons are on the circle), we
have
Nke 2
2 R
1
sin( jθ/ 2) + 1
1
U 1 ,N =even =
2
j =1
N
−
1
Nke 2
2 R
2
1
sin( jθ/ 2)
U 1 ,N =odd =
j =1
where θ = 2 N . For configuration 2, we find
( N
−
1) ke 2
2 R
N
2 − 1
1
sin( jθ / 2) +2
U 2 ,N =even =
j =1
N
−
3
( N
−
1) ke 2
2 R
2
sin( jθ / 2) + 5
1
U 2 ,N =odd =
2
j =1
where θ =
2 π
N− 1 . The results are all of the form
U 1or2 = ke 2
2 R ×
a pure number .
In our table, below, we have the results for those “pure numbers” as they depend on N andonwhich
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 48.62 59.58 71.81 85.35 100.2
We see that the potential energy for configuration 2 is greater than that for configuration 1 for N< 12,
but for N
≥
12 it is configuration 1 that has the greatest potential energy.
(a) Configuration 1 has the smallest U for 2
≤
N
≤
11, and configuration 2 has the smallest U for
15.
(b) N = 12 is the smallest value such that U 2 <U 1 .
(c) For N = 12, configuration 2 consists of 11 electrons distributed at equal distances around the circle,
and one electron at the center. A specific electron e 0 on the circle is R distance from the one in the
center, and is
≤
N
≤
r =2 R sin π
11
≈
0 . 56 R
distance away from its nearest neighbors on the circle (of which there are two – one on each side).
Beyond the nearest neighbors, the next nearest electron on the circle is
r =2 R sin 2 π
11
≈
1 . 1 R
distance away from e 0 . Thus, we see that there are only two electrons closer to e 0 than the one in
the center.
N
2 −
12
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