[Frost] Dilution method

[Michael Dugger]

Hi,

There are two common methods of dealing with the bound nucleons:
1) Subtraction method
2) Dilution method.

Either method should produce final results that are consistent with one 
another. I want to make sure that we are all in agreement as to the use of 
the dilution method. (the subtraction method is very simple and probably 
does not need much in the way of explanation.)

>From my discussion with Ken, I understand that (please correct me if I am 
wrong) the dilution method can be summarized through the equation:

O_free = [O_butanol*N_butanol - N_bound*O_carbon]/N_free,

where
O_free = Observable on the free proton
O_butanol = Observable on the butanol
O_carbon = Observable on the carbon
N_free = Number of events on free proton
N_butanol = Number of events of interest on butanol
N_bound = Number of events of interest on bound nucleons within butanol

If we introduce the dilution factor D = N_free/N_butanol, then we can 
rewrite O_free as

O_free = O_butanol/D - O_carbon*(1-D)/D

There are two observables and a dilution factor that make up the 
observable for free protons.

For the special case where O_carbon = 0, we get the nice result that
O_free = O_butanol/D.

The scale factor can be connected to the dilution factor through the 
equation
D = 1 - sf*N_carbon/N_butanol,
as Sung does.

It is important to note that for the Sigma, Cz and Cx observables O_carbon 
is not expected to be = 0.
This means that for these observables you must use the full equation:
O_free = O_butanol/D - O_carbon*(1-D)/D

Liam: It looks like it is possible for an incorrect dilution factor to 
cause a sign change in observables. Perhaps you can vary the dilution 
factors and see if you can get Cz and Cz observables to better agree with 
the g1c data.

Take care,
Michael