Re: Help: General spherical harmonics
Dear all,
I too have posed style similar questions to Arnt Kern at Bruker.
He pointed me to amongst other sources Wikipedia for a general
background in this topic.
However Im still keen to get a better understanding of
use/parameterisation within Topas and I too welcome any contributions to
this subject.
Dr Tim Hyde
Principal Scientist
Johnson Matthey Technology Centre
Blounts Court
Sonning Common
Reading RG4 9NH
Tel: +44 (0) 118 924 2152
Fax: +44 (0) 118 924 2254
email [EMAIL PROTECTED]
[EMAIL PROTECTED] 19/04/08 00:40
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.
k41sh_c41 0.36706`
k61sh_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
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Re: Help: General spherical harmonics
Hi Xiujun Tim,
while I am not familiar with the texture analysis offered in Topas, I can
offer a few hopefully useful hints.
- The texture index is a number that is 1 for a perfectly random sample and
infinity for a single crystal. A weak texture would have an index between say
1 and 1.5, a moderate texture between say 1.5 and 3. Above that one can say
the sample has a strong texture. I am not aware of a standard for this, so
these numbers might be somewhat biased with my personal judgement.
- The texture index condenses the whole ODF into a single number, so it's
value is fairly limited.
- Before you attempt to do combined texture and structure refinements, you
should establish that your pole figure coverage in your given instrument
setup is sufficient. Any texture analysis will refine to some numbers, but if
there is not enough pole figure coverage they will be meaningless, flawing
your structure analysis.
- If you don't use an image plate, you will most certainly not have enough
coverage from a single sample orientation. Even with an image plate, you
might need multiple sample orientations.
- To establish whether the texture analysis works, you could measure the
texture of household tin/aluminum foil and make sure you see a fcc rolling
texture. You would have to analyze the data without any symmetry using
something like a 10th or 12th order spherical harmonics in the texture
analysis. If you can reproduce the rolling texture, your coverage is probably
sufficient for texture analysis and hence for a combined texture crystal
structure refinement of your actual sample. If not, you might have to do a
combined refinement against multiple patterns taken in different sample
orientations.
- An excellent book on the subject is Kocks/Wenk/Tome, Texture and
Anisotropy, it has among a lot of other valuable information a few words on
the texture index and pretty much an atlas of possible textures for various
materials and processing conditions to help you judging whether the texture
you see makes sense.
- To my knowledge the only Rietveld software supporting the more powerful WIMV
and E-WIMV algorithms for texture (and also a somewhat improved exponential
spherical harmonics algorithm) is MAUD, freely available. Not sure what
flavors Topas supports, but it probably doesn't hurt to try different
programs. There are tutorials available on the MAUD website.
Hope this helps,
Sven
On Friday 18 April 2008 17:40:21 [EMAIL PROTECTED] wrote:
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.
k41sh_c41 0.36706`
k61sh_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 its 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
RE: Help: General spherical harmonics
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) *.125000
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) *.78
Y42m = (Sin(2*p)*(-1 + 7*Cos(t)^2)*Sin(t)^2) *.78
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.
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) *.816750
Y64m = (Sin(4*p)*(-1 + 11*Cos(t)^2)*Sin(t)^4) *.816750
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.
k41sh_c41 0.36706`
k61sh_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
