Hi Xiujun
Topas implements a normalized symmetrized sperical harmonics function, see
Jarvine
J. Appl. Cryst. (1993). 26, 525-531
http://scripts.iucr.org/cgi-bin/paper?S0021889893001219
The expansion is simply a series that is a function hkl values.
The series is normalized such that the maximum value of each component is 1.
The normalized components are:
Y00 = 1
Y20 = (3.0 Cos(t)^2 - 1.0)* 0.5
Y21p = (Cos(p)*Cos(t)*Sin(t))* 2
Y21m = (Sin(p)*Cos(t)*Sin(t))* 2
Y22p = (Cos(2*p)*Sin(t)^2)
Y22m = (Sin(2*p)*Sin(t)^2)
Y40 = (3 - 30*Cos(t)^2 + 35*Cos(t)^4) *.1250000000
Y41p = (Cos(p)*Cos(t)*(7*Cos(t)^2-3)*Sin(t)) *.9469461818
Y41m = (Sin(p)*Cos(t)*(7*Cos(t)^2-3)*Sin(t)) *.9469461818
Y42p = (Cos(2*p)*(-1 + 7*Cos(t)^2)*Sin(t)^2) *.7777777778
Y42m = (Sin(2*p)*(-1 + 7*Cos(t)^2)*Sin(t)^2) *.7777777778
Y43p = (Cos(3*p)*Cos(t)*Sin(t)^3) *3.0792014358
Y43m = (Sin(3*p)*Cos(t)*Sin(t)^3) *3.0792014358
Y44p = (Cos(4*p)*Sin(t)^4)
Y44m = (Sin(4*p)*Sin(t)^4)
Y60 = (-5 + 105*Cos(t)^2 - 315*Cos(t)^4 + 231*Cos(t)^6) *.62500.0000
Y61p = (Cos(p)*(-5 + 30*Cos(t)^2 - 33*Cos(t)^4)*Sin(t)*Cos(t)) *.6913999628
Y61m = (Sin(p)*(-5 + 30*Cos(t)^2 - 33*Cos(t)^4)*Sin(t)*Cos(t)) *.6913999628
Y62p = (Cos(2*p)*(1 - 18*Cos(t)^2 + 33*Cos(t)^4)*Sin(t)^2) *.6454926483
Y62m = (Sin(2*p)*(1 - 18*Cos(t)^2 + 33*Cos(t)^4)*Sin(t)^2) *.6454926483
Y63p = (Cos(3*p)*(3- 11*Cos(t)^2)*Cos(t)*Sin(t)^3) *1.4168477165
Y63m = (Sin(3*p)*(3- 11*Cos(t)^2)*Cos(t)*Sin(t)^3) *1.4168477165
Y64p = (Cos(4*p)*(-1 + 11*Cos(t)^2)*Sin(t)^4) *.8167500000
Y64m = (Sin(4*p)*(-1 + 11*Cos(t)^2)*Sin(t)^4) *.8167500000
Y65p = (Cos(5*p)*Cos(t)*Sin(t)^5) *3.8639254683
Y65m = (Sin(5*p)*Cos(t)*Sin(t)^5) *3.8639254683
Y66p = (Cos(6*p)*Sin(t)^6)
Y66m = (Cos(6*p)*Sin(t)^6)
Y80 = (35 - 1260*Cos(t)^2 + 6930*Cos(t)^4 - 12012*Cos(t)^6 +
6435*Cos(t)^8)* .0078125000
Y81p = (Cos(p)*(35*Cos(t) - 385*Cos(t)^3 + 1001*Cos(t)^5 -
715*Cos(t)^7)*Sin(t))* .1134799545
Y81m = (Sin(p)*(35*Cos(t) - 385*Cos(t)^3 + 1001*Cos(t)^5 -
715*Cos(t)^7)*Sin(t))* .1134799545
Y82p = (Cos(2*p)*(-1 + 33*Cos(t)^2 - 143*Cos(t)^4 + 143*Cos(t)^6)*Sin(t)^2)*
.5637178511
Y82m = (Sin(2*p)*(-1 + 33*Cos(t)^2 - 143*Cos(t)^4 + 143*Cos(t)^6)*Sin(t)^2)*
.5637178512
Y83p = (Cos(3*p)*(-3*Cos(t) + 26*Cos(t)^3 - 39*Cos(t)^5)*Sin(t)^3)*
1.6913068375
Y83m = (Sin(3*p)*(-3*Cos(t) + 26*Cos(t)^3 - 39*Cos(t)^5)*Sin(t)^3)*
1.6913068375
Y84p = (Cos(4*p)*(1 - 26*Cos(t)^2 + 65*Cos(t)^4)*Sin(t)^4)* .7011002983
Y84m = (Sin(4*p)*(1 - 26*Cos(t)^2 + 65*Cos(t)^4)*Sin(t)^4)* .7011002983
Y85p = (Cos(5*p)*(Cos(t) - 5*Cos(t)^3)*Sin(t)^5)* 5.2833000817
Y85m = (Sin(5*p)*(Cos(t) - 5*Cos(t)^3)*Sin(t)^5)* 5.2833000775
Y86p = (Cos(6*p)*(-1 + 15*Cos(t)^2)*Sin(t)^6)* .8329862557
Y86m = (Sin(6*p)*(-1 + 15*Cos(t)^2)*Sin(t)^6)* .8329862557
Y87p = (Cos(7*p)*Cos(t)*Sin(t)^7)* 4.5135349314
Y87m = (Sin(7*p)*Cos(t)*Sin(t)^7)* 4.5135349313
Y88p = (Cos(8*p)*Sin(t)^8)
Y88m = (Sin(8*p)*Sin(t)^8)
where
t = theta
p = phi
theta and phi are the sperical coordinates of the normal to the hkl plane.
These components were obtained from Mathematica and mormalized using Topas.
The user determines how the series is used. In the case of correcting for
texture as per Jarvine then the
intensities of the reflections are multiplied by the series value. This is
accomplished bye first defining a series:
str...
spherical_harmonics_hkl sh sh_order 8
and then scaling the peak intensities, or,
scale_pks = sh;
after refinement the INP file is updated with the coefficients.
The macro PO_Spherical_Harmonics, as you have defined, can also be used.
Typically the C00 coeffecient is not refined as its series component Y00 is
simply 1 and is 100% correlated with the scale parameter.
You could output the series values as a function of hkl as follows:
scale_pks = sh;
phase_out sh.txt load out_record out_fmt out_eqn {
"%4.0f" = H;
"%4.0f" = K;
"%4.0f" = L;
" %9g\n" = sh;
}
Cheers
Alan
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
Sent: Saturday, 19 April 2008 9:40 AM
To: rietveld_l@ill.fr
Subject: Help: General spherical harmonics
Dear all,
Now i am using the Topas Academic software to do the refinement of my sample
which has stronger preferred orientations in some directions.
In the program, i use the general spherical harmonics function to correlate
the effect, as shown as below,
'Preferred Orientation using Spherical Harmonics
PO_Spherical_Harmonics(sh, 6 load sh_Cij_prm {
k00 !sh_c00 1.0000
k41 sh_c41 0.36706`
k61 sh_c61 -0.30246`
} )
And I see the literature, texture index J is used to evaluate the extent of
PO by the equation shown in attachment ( I don't how to put the equation
here).
But I am not sure what the l means and it's not easy to find the detailed
calculation in the literature. So I am wondering could someone of you give
me some advice of the meaning of parameters m, n, l and in my case. Is the l
is equal to 4 and 6?
Thank you very much for all your help and time.
Xiujun Li
Master Student
Advanced Materials and Processing Laboratory Chemical and Materials
Engineering University of Alberta Edmonton, Alberta, Canada T6G 2G6
Phone: 1-780-492-0701
