Ph.D. Thesis - Physics
Ph.D. Thesis - Physics
Ph.D. Thesis - Physics
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In[1]:= <br />
File: crystal_shape_6uraniborg.nb<br />
Author: Kenan Diab<br />
Calculates the structure of ion<br />
crystals in a trap using the three secular frequencies wx,<br />
wy, and wz and the number of ions n as inputs. The variables z3coeff<br />
and z4coeff are optional anharmonicity terms.<br />
<br />
everything is done is cgs units. to simplify the coulomb<br />
contribution to the energy, we divide everything by e^2, introducing<br />
the factor of 0.00131 to the harmonic part. <br />
wx 2Pi.15010^6;<br />
wy 2Pi.15010^6;<br />
wz 2Pi1.310^6;<br />
z3coeff 0; 6.7837210^10; <br />
z4coeff 0;6.5239610^11; <br />
n 13; number of ions <br />
r is the array of ion coordinates. r3i1, r3i2,<br />
r3i3 are the x,y,z coordinates of the i'th ion. <br />
rArraySubscripta,1 &,3 n;<br />
mathematica uses a relatively efficient simplex method <br />
ans NMinimize0.00063060.5wx^2Sumr3 i1^2,i, 0, n1<br />
wy^2Sumr3 i2^2,i, 0, n1wz^2Sumr3 i3^2,i, 0, n1<br />
z3coeffSumr3 i3^3,i, 0, n1 z4coeffSumr3 i3^4,i, 0, n1<br />
SumSqrt1r3 i1r3 j1^2<br />
r3 i2r3 j2^2r3 i3r3 j3^2,<br />
i, 0, n1,j, i1, n1, r, WorkingPrecision15<br />
plot points 3D <br />
s r. Extractans, 2;<br />
points ;<br />
Fori 0, i n, i, points Appendpoints,s3 i1, s3 i2, s3 i3;<br />
ListPlot3Dpoints10 000, AspectRatioAutomatic, AxesLabel "x Μ m", "y Μ m"<br />
plot points 2D, dropping z coordinate <br />
s r. Extractans, 2;<br />
points ;<br />
Fori 0, i n, i, points Appendpoints,s3 i1, s3 i2;<br />
ListPlotpoints10 000, AspectRatio Automatic,<br />
PlotStyleRed, Thick, PointSizeLarge,<br />
AxesLabel "x Μ m", "y Μ m", PlotRange50, 50,50, 50<br />
calculate max distance between y vals <br />
yvals ;<br />
Fori 0, in, i, yvals Appendyvals, s3 i2;<br />
dist MaxyvalsMinyvals<br />
convert energy to electron volts <br />
energy ans11.43965000000000000010^7<br />
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