Re: Preferred orientation?

[Jonathan Wright]

Martin wrote:

PS For anyone interested in an explanation of texture vs. PO, see:
 
http://pd.chem.ucl.ac.uk/pdnn/inst1/texture1.htm

http://pd.chem.ucl.ac.uk/pdnn/inst1/texture2.htm


I disagree! The web page has confused "texture" with "granularity". See:

http://en.wikipedia.org/wiki/Texture_(crystalline)

If you need a way to separate students in an exam, you could pretend to 
believe that texture refers to solid samples, but preferred orientation 
is for free flowing powders. eg: texture is a static property of some 
object, preferred orientation is what you got once with a particular 
sample holder.


All the best,

Jon

Re[2]: Preferred orientation?

[Yaroslav Filinchuk, SNBL at ESRF]

Dear Martin,

What is called "texture" in the link you've provided actually is a
"bad powder average". Texture sounds to me as a rather general term,
while "preferred orientation" is its particular case. Texture is
the distribution of crystallographic orientations in a
polycrystalline sample, it does not need to be uniaxial like
"preferred orientation", and it does not need to be only due to a
more-or-less random distribution of intensities from a few
randomly oriented particles. The latter effect should be better
called "bad powder average".
IMHO, the link you've provided pretends to revise the terminology.
I wonder, what "texture analysis" means according to it...

Best regards,
 Yaroslav

http://filinchuk.com


===8<==Original message text===
Any data from anywhere will lead the gullible and unwary up a certain
creek without a particular implement and flat-plate is better than
transmission for doing so. Older saying still (ca. 1503-07), "thar be
Dragons".Martin PS For anyone interested in an explanation of texture
vs. PO, see:
http://pd.chem.ucl.ac.uk/pdnn/inst1/texture1.htm
http://pd.chem.ucl.ac.uk/pdnn/inst1/texture2.htm

PPS for anyone interested in the TV ad that came to mind during
this discussion, see: 
http://www.nationalarchives.gov.uk/films/1964to1979/filmpage_lonely.htmMartin

Subject: RE: Preferred orientation?Date: Fri, 16 May 2008 13:05:24 -0400From: 
[EMAIL PROTECTED]: [EMAIL PROTECTED]: rietveld_l@ill.fr




I'm not really arguing with you here as I prefer capillary data myself as it 
gives better data in many circumstances - however it can sometimes lead you up 
the garden path (another old saying!).  I suppose what it boils down to is that 
needles are a pain as they can orientate whatever you do them (reflection or 
transmission).  
 
The texture versus preferred orientation difference has some signficant blurry 
edges from a practical point of view.  
 
Anyway - I'm on holiday so I'm going to put my brain to sleep and go and do 
some gardening!
 
Pam




From: Martin [mailto:[EMAIL PROTECTED]: Fri 16/05/2008 11:47 AMTo: Whitfield, 
Pamela; [EMAIL PROTECTED]: RE: Preferred orientation?
Hi, happy RietveldersSome elements of confusion creeping in here. I think you 
said, Pam that transmission wont help much if it's wollastonite and what I'm 
saying is that it does and gave a pointer to a study that shows it. Indeed I 
don't claim transmission gets rid of PO either, but it does reduce it hugely 
which, if one reads back, is my claim here. The PO function in this case is 
merely to illustrate the point: 0.9 and a refined model vs. 1.6 and a bad 
refinement. The merits of various PO functions aren't important when it's the 
data that really count (or the counts that count, if you like). Why start out 
with bad data in the first place? As my old dad says, you can't make a silk 
purse out of a pigs ear. regards, Martin PS Just to clear up another possible 
point of confusion: large particles lead to texture effects, not preferred 
orientation.  


Subject: FW: Preferred orientation?Date: Fri, 16 May 2008 10:43:21 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr



 


I don't remember saying that reflection would work well with wollastonite only 
that a capillary won't get rid of the orientation (or at least that was my 
intention). As always this is going to vary from sample to sample, i.e. how 
large the particles are, aspect ratio of the particles, the diameter of the 
capillary, pure or mixture, etc.  Orientation with needles is going to be more 
of a problem in a 0.3mm versus a 0.8mm capillary.  Particle statistics are also 
a potential issue with samples like this.  If they are large enough to 
orientate badly then the crystallites are probably large and the capillary will 
do a better job with the statistics
 
The MD correction doesn't work very well for alot of these systems, SH is 
better as long as the correlations don't get out of hand (which they can quite 
easily).  However 0.9 is still significantly orientated so it doesn't get rid 
of it by any means.  With platey particles you can pretty much eliminate the 
preferential orientation with a capillary versus flat plate with significant 
impact of quantitative analysis results (paper published in Powder Diffraction 
a couple of years ago).
 
I do have reflection data from 400 mesh wollastonite (albeit with MoKa from a 
high pressure gas cell) which can be fitted quite nicely with SH (lousy with 
MD).  The quantitative analysis results from the carbonation are good enough to 
extract a rate constant which suits me nicely.  The additional penetration of 
the MoKa should help with the stats in this case even if transparency becomes a 
problem - you can't win eh?  
 
Unfortunately it's quite difficult to completely decouple orientation, 
statistcs and microabsorption as they are all related to size.
 
Pam
 
  


From: Martin [mailto:[EMAIL PROTECTED]: Fri 16/05/2008 1

RE: Preferred orientation?

[Martin]

Any data from anywhere will lead the gullible and unwary up a certain creek 
without a particular implement and flat-plate is better than transmission for 
doing so. Older saying still (ca. 1503-07), "thar be Dragons".Martin PS For 
anyone interested in an explanation of texture vs. PO, see: 
http://pd.chem.ucl.ac.uk/pdnn/inst1/texture1.htmhttp://pd.chem.ucl.ac.uk/pdnn/inst1/texture2.htm
 PPS for anyone interested in the TV ad that came to mind during this 
discussion, see: 
http://www.nationalarchives.gov.uk/films/1964to1979/filmpage_lonely.htmMartin


Subject: RE: Preferred orientation?Date: Fri, 16 May 2008 13:05:24 -0400From: 
[EMAIL PROTECTED]: [EMAIL PROTECTED]: rietveld_l@ill.fr




I'm not really arguing with you here as I prefer capillary data myself as it 
gives better data in many circumstances - however it can sometimes lead you up 
the garden path (another old saying!).  I suppose what it boils down to is that 
needles are a pain as they can orientate whatever you do them (reflection or 
transmission).  
 
The texture versus preferred orientation difference has some signficant blurry 
edges from a practical point of view.  
 
Anyway - I'm on holiday so I'm going to put my brain to sleep and go and do 
some gardening!
 
Pam




From: Martin [mailto:[EMAIL PROTECTED]: Fri 16/05/2008 11:47 AMTo: Whitfield, 
Pamela; [EMAIL PROTECTED]: RE: Preferred orientation?
Hi, happy RietveldersSome elements of confusion creeping in here. I think you 
said, Pam that transmission wont help much if it's wollastonite and what I'm 
saying is that it does and gave a pointer to a study that shows it. Indeed I 
don't claim transmission gets rid of PO either, but it does reduce it hugely 
which, if one reads back, is my claim here. The PO function in this case is 
merely to illustrate the point: 0.9 and a refined model vs. 1.6 and a bad 
refinement. The merits of various PO functions aren't important when it's the 
data that really count (or the counts that count, if you like). Why start out 
with bad data in the first place? As my old dad says, you can't make a silk 
purse out of a pigs ear. regards, Martin PS Just to clear up another possible 
point of confusion: large particles lead to texture effects, not preferred 
orientation.  


Subject: FW: Preferred orientation?Date: Fri, 16 May 2008 10:43:21 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr



 


I don't remember saying that reflection would work well with wollastonite only 
that a capillary won't get rid of the orientation (or at least that was my 
intention). As always this is going to vary from sample to sample, i.e. how 
large the particles are, aspect ratio of the particles, the diameter of the 
capillary, pure or mixture, etc.  Orientation with needles is going to be more 
of a problem in a 0.3mm versus a 0.8mm capillary.  Particle statistics are also 
a potential issue with samples like this.  If they are large enough to 
orientate badly then the crystallites are probably large and the capillary will 
do a better job with the statistics
 
The MD correction doesn't work very well for alot of these systems, SH is 
better as long as the correlations don't get out of hand (which they can quite 
easily).  However 0.9 is still significantly orientated so it doesn't get rid 
of it by any means.  With platey particles you can pretty much eliminate the 
preferential orientation with a capillary versus flat plate with significant 
impact of quantitative analysis results (paper published in Powder Diffraction 
a couple of years ago).
 
I do have reflection data from 400 mesh wollastonite (albeit with MoKa from a 
high pressure gas cell) which can be fitted quite nicely with SH (lousy with 
MD).  The quantitative analysis results from the carbonation are good enough to 
extract a rate constant which suits me nicely.  The additional penetration of 
the MoKa should help with the stats in this case even if transparency becomes a 
problem - you can't win eh?  
 
Unfortunately it's quite difficult to completely decouple orientation, 
statistcs and microabsorption as they are all related to size.
 
Pam
 
  


From: Martin [mailto:[EMAIL PROTECTED]: Fri 16/05/2008 10:04 AMTo: Whitfield, 
Pamela; [EMAIL PROTECTED]: RE: Preferred orientation?
In fact I think you might find it helps quite a bit. Have a look at: 
http://img.chem.ucl.ac.uk/www/vickers/po/po.htm Martin 

Subject: RE: Preferred orientation?Date: Thu, 8 May 2008 10:55:12 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr






I do that myself but it doesn’t always help much if you’ve got something like 
wollastonite! J
 


From: Martin [mailto:[EMAIL PROTECTED] Sent: May 8, 2008 10:51 AMTo: [EMAIL 
PROTECTED]: RE: Preferred orientation?
 
Forget all that long winded stuff. Just collect the data on capillary 
transmission geometry and avoid all (well, most of) the fuss. Martin Vickers



Get fish-slapping on Messenger! Play Now

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RE: Preferred orientation?

[Whitfield, Pamela]

I'm not really arguing with you here as I prefer capillary data myself as it 
gives better data in many circumstances - however it can sometimes lead you up 
the garden path (another old saying!).  I suppose what it boils down to is that 
needles are a pain as they can orientate whatever you do them (reflection or 
transmission).  
 
The texture versus preferred orientation difference has some signficant blurry 
edges from a practical point of view.  
 
Anyway - I'm on holiday so I'm going to put my brain to sleep and go and do 
some gardening!
 
Pam



From: Martin [mailto:[EMAIL PROTECTED]
Sent: Fri 16/05/2008 11:47 AM
To: Whitfield, Pamela; rietveld_l@ill.fr
Subject: RE: Preferred orientation?


Hi, happy Rietvelders

Some elements of confusion creeping in here. I think you said, Pam that 
transmission wont help much if it's wollastonite and what I'm saying is that it 
does and gave a pointer to a study that shows it. Indeed I don't claim 
transmission gets rid of PO either, but it does reduce it hugely which, if one 
reads back, is my claim here. The PO function in this case is merely to 
illustrate the point: 0.9 and a refined model vs. 1.6 and a bad refinement. The 
merits of various PO functions aren't important when it's the data that really 
count (or the counts that count, if you like). Why start out with bad data in 
the first place? As my old dad says, you can't make a silk purse out of a pigs 
ear.
 
regards, Martin 
PS Just to clear up another possible point of confusion: large particles lead 
to texture effects, not preferred orientation.  








Subject: FW: Preferred orientation?
Date: Fri, 16 May 2008 10:43:21 -0400
From: [EMAIL PROTECTED]
To: rietveld_l@ill.fr


 
I don't remember saying that reflection would work well with 
wollastonite only that a capillary won't get rid of the orientation (or at 
least that was my intention). As always this is going to vary from sample to 
sample, i.e. how large the particles are, aspect ratio of the particles, the 
diameter of the capillary, pure or mixture, etc.  Orientation with needles is 
going to be more of a problem in a 0.3mm versus a 0.8mm capillary.  Particle 
statistics are also a potential issue with samples like this.  If they are 
large enough to orientate badly then the crystallites are probably large and 
the capillary will do a better job with the statistics
 
The MD correction doesn't work very well for alot of these systems, SH 
is better as long as the correlations don't get out of hand (which they can 
quite easily).  However 0.9 is still significantly orientated so it doesn't get 
rid of it by any means.  With platey particles you can pretty much eliminate 
the preferential orientation with a capillary versus flat plate with 
significant impact of quantitative analysis results (paper published in Powder 
Diffraction a couple of years ago).
 
I do have reflection data from 400 mesh wollastonite (albeit with MoKa 
from a high pressure gas cell) which can be fitted quite nicely with SH (lousy 
with MD).  The quantitative analysis results from the carbonation are good 
enough to extract a rate constant which suits me nicely.  The additional 
penetration of the MoKa should help with the stats in this case even if 
transparency becomes a problem - you can't win eh?  
 
Unfortunately it's quite difficult to completely decouple orientation, 
statistcs and microabsorption as they are all related to size.
 
Pam
 
  



From: Martin [mailto:[EMAIL PROTECTED]
Sent: Fri 16/05/2008 10:04 AM
To: Whitfield, Pamela; rietveld_l@ill.fr
Subject: RE: Preferred orientation?


In fact I think you might find it helps quite a bit. Have a look at:
 
http://img.chem.ucl.ac.uk/www/vickers/po/po.htm

 
Martin 



Subject: RE: Preferred orientation?
Date: Thu, 8 May 2008 10:55:12 -0400
From: [EMAIL PROTECTED]
To: rietveld_l@ill.fr




I do that myself but it doesn't always help much if you've got 
something like wollastonite! J

 

From: Martin [mailto:[EMAIL PROTECTED] 
Sent: May 8, 2008 10:51 AM
To: rietveld_l@ill.fr
Subject: RE: Preferred orientation?

 

Forget all that long winded stuff. Just collect the data on 
capillary transmission geometry and avoid all (well, most of) the fuss.
 
Martin Vickers







Get fish-slapping on Messenger! Play Now 

RE: Preferred orientation?

[Martin]

 
 
er.. - sorry - making things worse here - quite right, large particles can lead 
to PO. I meant crystallites . 'umble apologies.
 
M
 
 


Subject: RE: Preferred orientation?From: [EMAIL PROTECTED]: [EMAIL PROTECTED]; 
[EMAIL PROTECTED]: Fri, 16 May 2008 15:47:31 +


Hi, happy RietveldersSome elements of confusion creeping in here. I think you 
said, Pam that transmission wont help much if it's wollastonite and what I'm 
saying is that it does and gave a pointer to a study that shows it. Indeed I 
don't claim transmission gets rid of PO either, but it does reduce it hugely 
which, if one reads back, is my claim here. The PO function in this case is 
merely to illustrate the point: 0.9 and a refined model vs. 1.6 and a bad 
refinement. The merits of various PO functions aren't important when it's the 
data that really count (or the counts that count, if you like). Why start out 
with bad data in the first place? As my old dad says, you can't make a silk 
purse out of a pigs ear. regards, Martin PS Just to clear up another possible 
point of confusion: large particles lead to texture effects, not preferred 
orientation.  


Subject: FW: Preferred orientation?Date: Fri, 16 May 2008 10:43:21 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr



 


I don't remember saying that reflection would work well with wollastonite only 
that a capillary won't get rid of the orientation (or at least that was my 
intention). As always this is going to vary from sample to sample, i.e. how 
large the particles are, aspect ratio of the particles, the diameter of the 
capillary, pure or mixture, etc.  Orientation with needles is going to be more 
of a problem in a 0.3mm versus a 0.8mm capillary.  Particle statistics are also 
a potential issue with samples like this.  If they are large enough to 
orientate badly then the crystallites are probably large and the capillary will 
do a better job with the statistics
 
The MD correction doesn't work very well for alot of these systems, SH is 
better as long as the correlations don't get out of hand (which they can quite 
easily).  However 0.9 is still significantly orientated so it doesn't get rid 
of it by any means.  With platey particles you can pretty much eliminate the 
preferential orientation with a capillary versus flat plate with significant 
impact of quantitative analysis results (paper published in Powder Diffraction 
a couple of years ago).
 
I do have reflection data from 400 mesh wollastonite (albeit with MoKa from a 
high pressure gas cell) which can be fitted quite nicely with SH (lousy with 
MD).  The quantitative analysis results from the carbonation are good enough to 
extract a rate constant which suits me nicely.  The additional penetration of 
the MoKa should help with the stats in this case even if transparency becomes a 
problem - you can't win eh?  
 
Unfortunately it's quite difficult to completely decouple orientation, 
statistcs and microabsorption as they are all related to size.
 
Pam
 
  


From: Martin [mailto:[EMAIL PROTECTED]: Fri 16/05/2008 10:04 AMTo: Whitfield, 
Pamela; [EMAIL PROTECTED]: RE: Preferred orientation?
In fact I think you might find it helps quite a bit. Have a look at: 
http://img.chem.ucl.ac.uk/www/vickers/po/po.htm Martin 

Subject: RE: Preferred orientation?Date: Thu, 8 May 2008 10:55:12 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr






I do that myself but it doesn’t always help much if you’ve got something like 
wollastonite! J
 


From: Martin [mailto:[EMAIL PROTECTED] Sent: May 8, 2008 10:51 AMTo: [EMAIL 
PROTECTED]: RE: Preferred orientation?
 
Forget all that long winded stuff. Just collect the data on capillary 
transmission geometry and avoid all (well, most of) the fuss. Martin Vickers



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RE: Preferred orientation?

[Martin]

Hi, happy Rietvelders
Some elements of confusion creeping in here. I think you said, Pam that 
transmission wont help much if it's wollastonite and what I'm saying is that it 
does and gave a pointer to a study that shows it. Indeed I don't claim 
transmission gets rid of PO either, but it does reduce it hugely which, if one 
reads back, is my claim here. The PO function in this case is merely to 
illustrate the point: 0.9 and a refined model vs. 1.6 and a bad refinement. The 
merits of various PO functions aren't important when it's the data that really 
count (or the counts that count, if you like). Why start out with bad data in 
the first place? As my old dad says, you can't make a silk purse out of a pigs 
ear. regards, Martin PS Just to clear up another possible point of confusion: 
large particles lead to texture effects, not preferred orientation.  


Subject: FW: Preferred orientation?Date: Fri, 16 May 2008 10:43:21 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr




 


I don't remember saying that reflection would work well with wollastonite only 
that a capillary won't get rid of the orientation (or at least that was my 
intention). As always this is going to vary from sample to sample, i.e. how 
large the particles are, aspect ratio of the particles, the diameter of the 
capillary, pure or mixture, etc.  Orientation with needles is going to be more 
of a problem in a 0.3mm versus a 0.8mm capillary.  Particle statistics are also 
a potential issue with samples like this.  If they are large enough to 
orientate badly then the crystallites are probably large and the capillary will 
do a better job with the statistics
 
The MD correction doesn't work very well for alot of these systems, SH is 
better as long as the correlations don't get out of hand (which they can quite 
easily).  However 0.9 is still significantly orientated so it doesn't get rid 
of it by any means.  With platey particles you can pretty much eliminate the 
preferential orientation with a capillary versus flat plate with significant 
impact of quantitative analysis results (paper published in Powder Diffraction 
a couple of years ago).
 
I do have reflection data from 400 mesh wollastonite (albeit with MoKa from a 
high pressure gas cell) which can be fitted quite nicely with SH (lousy with 
MD).  The quantitative analysis results from the carbonation are good enough to 
extract a rate constant which suits me nicely.  The additional penetration of 
the MoKa should help with the stats in this case even if transparency becomes a 
problem - you can't win eh?  
 
Unfortunately it's quite difficult to completely decouple orientation, 
statistcs and microabsorption as they are all related to size.
 
Pam
 
  


From: Martin [mailto:[EMAIL PROTECTED]: Fri 16/05/2008 10:04 AMTo: Whitfield, 
Pamela; [EMAIL PROTECTED]: RE: Preferred orientation?
In fact I think you might find it helps quite a bit. Have a look at: 
http://img.chem.ucl.ac.uk/www/vickers/po/po.htm Martin 

Subject: RE: Preferred orientation?Date: Thu, 8 May 2008 10:55:12 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr






I do that myself but it doesn’t always help much if you’ve got something like 
wollastonite! J
 


From: Martin [mailto:[EMAIL PROTECTED] Sent: May 8, 2008 10:51 AMTo: [EMAIL 
PROTECTED]: RE: Preferred orientation?
 
Forget all that long winded stuff. Just collect the data on capillary 
transmission geometry and avoid all (well, most of) the fuss. Martin Vickers



Get fish-slapping on Messenger! Play Now

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http://clk.atdmt.com/UKM/go/msnnkmgl001009ukm/direct/01/

FW: Preferred orientation?

[Whitfield, Pamela]

 
I don't remember saying that reflection would work well with wollastonite only 
that a capillary won't get rid of the orientation (or at least that was my 
intention). As always this is going to vary from sample to sample, i.e. how 
large the particles are, aspect ratio of the particles, the diameter of the 
capillary, pure or mixture, etc.  Orientation with needles is going to be more 
of a problem in a 0.3mm versus a 0.8mm capillary.  Particle statistics are also 
a potential issue with samples like this.  If they are large enough to 
orientate badly then the crystallites are probably large and the capillary will 
do a better job with the statistics
 
The MD correction doesn't work very well for alot of these systems, SH is 
better as long as the correlations don't get out of hand (which they can quite 
easily).  However 0.9 is still significantly orientated so it doesn't get rid 
of it by any means.  With platey particles you can pretty much eliminate the 
preferential orientation with a capillary versus flat plate with significant 
impact of quantitative analysis results (paper published in Powder Diffraction 
a couple of years ago).
 
I do have reflection data from 400 mesh wollastonite (albeit with MoKa from a 
high pressure gas cell) which can be fitted quite nicely with SH (lousy with 
MD).  The quantitative analysis results from the carbonation are good enough to 
extract a rate constant which suits me nicely.  The additional penetration of 
the MoKa should help with the stats in this case even if transparency becomes a 
problem - you can't win eh?  
 
Unfortunately it's quite difficult to completely decouple orientation, 
statistcs and microabsorption as they are all related to size.
 
Pam
 
  



From: Martin [mailto:[EMAIL PROTECTED]
Sent: Fri 16/05/2008 10:04 AM
To: Whitfield, Pamela; rietveld_l@ill.fr
Subject: RE: Preferred orientation?


In fact I think you might find it helps quite a bit. Have a look at:
 
http://img.chem.ucl.ac.uk/www/vickers/po/po.htm

 
Martin 



Subject: RE: Preferred orientation?
Date: Thu, 8 May 2008 10:55:12 -0400
From: [EMAIL PROTECTED]
To: rietveld_l@ill.fr




I do that myself but it doesn't always help much if you've got 
something like wollastonite! J

 

From: Martin [mailto:[EMAIL PROTECTED] 
Sent: May 8, 2008 10:51 AM
To: rietveld_l@ill.fr
Subject: RE: Preferred orientation?

 

Forget all that long winded stuff. Just collect the data on capillary 
transmission geometry and avoid all (well, most of) the fuss.
 
Martin Vickers







Get fish-slapping on Messenger! Play Now 
 




Get fish-slapping on Messenger! Play Now 
  

RE: Preferred orientation?

[Martin]

In fact I think you might find it helps quite a bit. Have a look at:
 
http://img.chem.ucl.ac.uk/www/vickers/po/po.htm
 
Martin 

Subject: RE: Preferred orientation?Date: Thu, 8 May 2008 10:55:12 -0400From: 
[EMAIL PROTECTED]: rietveld_l@ill.fr







I do that myself but it doesn’t always help much if you’ve got something like 
wollastonite! J
 


From: Martin [mailto:[EMAIL PROTECTED] Sent: May 8, 2008 10:51 AMTo: [EMAIL 
PROTECTED]: RE: Preferred orientation?
 
Forget all that long winded stuff. Just collect the data on capillary 
transmission geometry and avoid all (well, most of) the fuss. Martin Vickers



Get fish-slapping on Messenger! Play Now
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Re: burgers vector

[Vincent Favre-Nicolin]

   Dear Muruges,

> I need a burgers vector for BCC, FCC, HCP systems, and how to calculate the
> burgers vector for other systems also.
>
> kindly give me the name of books, literature reference, tutorial, where
> available, it will more help full for me in research,

  I would *strongly* suggest spending some time on crystallography books in your
local university/laboratory library, where this will be explained at length.

  You can also find some starting information on the web, e.g. from the
University of Kiel:
http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html

regards,
Vincent Favre-Nicolin