 not so Frequently Asked Questions  update 2005/11/19


Deformed Nucleus (Legendre Expansion)[ver.4] ONLY ! The spherical harmonics Y[lm](theta,phi) is reduced to a simple Legendre function p[l](cos(theta)) scaled by a constant when m=0, which is independent of phi. The following equations and figure are Y[l](theta) for several lvalues. Using this Y[l](theta) with l=lambda=even terms, shape of a deformed nucleus can be expanded as follows: where the beta a parameter of deformation. If beta=0, the nucleus is spherical. The 3dim. shape given by this equation is shown with gnuplot. As it is already shown in the previous section, we express the (x,y,z) coordinate with the angles u,v and radius r. x = R(theta)*cos(u)*cos(v) y = R(theta)*sin(u)*cos(v) z = R(theta)*sin(v) where the theta is the angle measured from the Zaxis, so that the relation between theta and v is theta = pi/2v. To draw the surface, the parameters u,v are varied from 0 to 360 deg. In the case of beta_2 = 0.3, beta_4 = 0.1, and R_0 = 1: gnuplot> set parametric dummy variable is t for curves, u/v for surfaces gnuplot> set angle degree gnuplot> set urange [0:360] gnuplot> set vrange [0:360] gnuplot> set isosample 16,16 gnuplot> set ticslevel 0 gnuplot> set view 75,25 gnuplot> set size 0.7,1.0 gnuplot> set xrange [2:2] gnuplot> set yrange [2:2] gnuplot> set zrange [2:2] gnuplot> set urange [0:360] gnuplot> set vrange [0:360] gnuplot> y0(t)=1.0 gnuplot> y2(t)=sqrt(5.0/(4*pi))*( 3.0*cos(t)**2  1.0 )/2.0 gnuplot> y4(t)=sqrt(9.0/(4*pi))*(35.0*cos(t)**4  30*cos(t)**2 +3.0)/8.0 gnuplot> b2=0.3 gnuplot> b4=0.1 gnuplot> r(t) = 1 + b2*y2(0.5*pit) + b4*y4(0.5*pit) gnuplot> fx(u,v)=cos(u)*cos(v) gnuplot> fy(u,v)=sin(u)*cos(v) gnuplot> fz(v)=sin(v) gnuplot> set pm3d gnuplot> splot r(v)*fx(u,v),r(v)*fy(u,v),r(v)*fz(v) with lines The deformation parameters, beta_2 and beta_4 can be positive or negative. Here are some examples for some combinations of beta_2 and beta_4. The beta_2 parameters are taken to be 0.4 or 0.4, and for each beta_2, we changed the beta_4 value from 0.2 to 0.2. When beta_2 is positive the shape of nucleus is prolate, while it becomes oblate if beta_2 is negative. gnuplot> set border 0 gnuplot> unset xtics gnuplot> unset ytics gnuplot> unset ztics gnuplot> unset colorbox gnuplot> b2 = 0.4 ; b4 = 0.2 ; replot ; pause 1 gnuplot> b2 = 0.4 ; b4 = 0.0 ; replot ; pause 1 gnuplot> b2 = 0.4 ; b4 = 0.2 ; replot ; pause 1 gnuplot> b2 = 0.4 ; b4 = 0.2 ; replot ; pause 1 gnuplot> b2 = 0.4 ; b4 = 0.0 ; replot ; pause 1 gnuplot> b2 = 0.4 ; b4 = 0.2 ; replot ; pause 1
