# [Numpy-discussion] Even Sphere Volume

Fri Jul 10 03:04:29 CDT 2009

```Here is a different one, based on the ZCW  (Zaremba, Conroy,
Wolfsberg) algorithm used for powder spectra to equally distribute
points over a sphere surface:
This paper gives an nice overview (I think I took the actual algorithm
from there):

Computer simulations in solid-state NMR. III. Powder averaging
M. Edén
Concepts in Magnetic Resonance Part A  18A  24-55  (2003)
http://dx.doi.org/10.1002/cmr.a.10065

############# Fibonacci number generator ##################
def fib(n):
"""
From http://en.literateprograms.org/Fibonacci_numbers_(Python)
Returns (n+2)-th and n-th fibonacci number
"""
if n > 90:
raise ValueError,"Can't represent higher numbers!"
start = N.array([[1,1],[1,0]],dtype='int64')
temp = start[:]
for i in xrange(n):
temp = N.dot(start,temp)
return temp[0,0],temp[0,1],temp[1,1]

############# ZCW Angles ##########
def zcw_angles(m):
"""
Getting N=fibonacchi(m) numbers of ZCW angles

"""

samples, fib_1, fib_2 = fib(m)
#fib_2 = fib_1
print "Using %i Samples"%samples

js = N.arange(samples, dtype='Float64')/samples

c_full = (1.,2.,1.)
c_hemi = (-1.,1.,1.)
c_oct  = (2.,1.,8.)

c = c_full

j_samples = fib_2*js
# alpha
phi = 2*N.pi/c[2] * N.mod(j_samples,1.0)
# beta
theta = N.arccos(c[0]*(c[1]*N.mod(js,1.0) - 1))
# weights
weight = N.ones(samples)/samples
return phi,theta,weight

Here is a interesting one, REPULSION:

REPULSION, A Novel Approach to Efficient Powder Averaging in Solid-
State NMR
M. Bak and N. C. Nielsen
Journal of Magnetic Resonance  125  132--139  (1997)
http://www.sciencedirect.com/science/article/B6WJX-45N4XS2-J/1/58bb5f6971e8a76ab01a17e2794f98b7
A novel approach to efficient powder averaging in magnetic resonance
is presented. The method relies on a simple numerical procedure which
based on a random set of crystallite orientations through simulation
of fictive intercrystallite repulsive forces iteratively determines a
set of orientations uniformly distributed over the unit sphere. The so-
called REPULSION partition scheme is compared to earlier methods with
respect to the distribution of crystallite orientations, solid angles,
and powder averaging efficiency. It is demonstrated that powder
averaging using REPULSION converges faster than previous methods with
respect to the number of crystallite orientations involved in the
averaging. This feature renders REPULSION particularly attractive for
calculation of magic-angle-spinning solid-state NMR spectra using a
minimum of crystallite orientations. For numerical simulation of
powder spectra, the reduced number of required crystallite
orientations translates into shorter computation times and simulations
less prone to systematic errors induced by finite sets of nonuniformly
distributed crystallite orientations.

--
"A program that produces incorrect results twice as fast is infinitely
slower." -- John Osterhout

```