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pauli

New prescription 06/14/06

Pauli blocking was again applied as a weight (in SIMC) based on the neutron momentum and the correlated occupation number distribution (Fantoni et al).  The Pauli blocking weight is 1-n(k_n),  where k_n is the magnitude of the three-momentum of the recoiling neutron.  n(k_n) is the correlated occupation number distribution, which gives the occupancy of this momentum state. 

However, this time, deuterium was treated like the other nuclear targets.  The Fermi energy was set to 55 MeV (see Maieron et al PRC 65 025502).  The recoil of the spectator neutron was not considered, and the Pauli blocking weight was set to half that from nuclear targets (becuase only one spin state is blocked).  Pauli blocking in deuterium has almost no effect.

The Fermi momentum of the other targets were obtained from similar targets in
Whitney, Sick, et al (1974)  PRC 9 2230

targ   p_F(MeV)
12C    221
27Al   260
64Cu   265
197Au  265

Before Pauli blocking was applied
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

After Pauli blocking was applied
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)



Missing mass shifts 04/27/06

After the hydrogen missing mass peaks in SIMC were made to agree with the data (link), the missing mass shifts observed between SIMC and the data for nuclear targets are listed below.
Target
 Q2
(GeV2)
 Nucleon missing mass shift (GeV) Nuclear missing mass shift
No Pauli blocking With Pauli blocking No Pauli blocking With Pauli blocking
d 
d 
d 
d 
d 
c 
c 
c 
c 
c 
al      
al
al
al
al
cu      
cu
cu
cu
cu
au      
au
au
au
au
1.1
2.15
3
4
4.8
1.1
2.15
3
4
4.8
1.1
2.15
3
4
4.8
1.1
2.15
3
4
4.8
1.1
2.15
3
4
4.8
 -0.002
-0.002
-0.003
-0.002
-0.002
-0.01 
-0.0135
-0.012
-0.0145
-0.0118
-0.01 
-0.0135
-0.012
-0.0145
-0.0118
-0.01 
-0.0135
-0.012
-0.0145
-0.0118
-0.01 
-0.0135
-0.012
-0.0145
-0.0118
 -0.0025
 -0.004
 -0.004
 -0.004
 -0.004
 -0.024
 -0.02
 -0.011
 -0.0175
 -0.012
 -0.024
 -0.02
 -0.011
 -0.0175
 -0.012
 -0.024
 -0.02
 -0.011
 -0.0175
 -0.012
 -0.024
 -0.02
 -0.011
 -0.0175
 -0.012
 -0.002
-0.002
-0.003
-0.002
-0.002
-0.007
-0.014
-0.026
-0.02
-0.03
-0.007
-0.014
-0.026
-0.02   
-0.03
-0.007
-0.014
-0.026
-0.02
-0.03
-0.047
-0.057
-0.067
-0.06
-0.06
 -0.0025
-0.004
-0.004
-0.004
-0.004
-0.027
-0.023
-0.027
-0.027
-0.04
-0.027
-0.023
-0.027
-0.027
-0.04
-0.027
-0.023
-0.027
-0.027
-0.04
-0.07
-0.07
-0.07
-0.06
-0.06

The peaks after the missing mass shifts are included in the links below.

No Pauli blocking
Nucleon, mmx shift
 1H 2H C Al Cu Au
Nuclear, missmass shift
2H C Al Cu Au

With Pauli blocking
Nucleon, mmx shift
 1H 2H C Al Cu Au
Nuclear, missmass shift
2H C Al Cu Au

Results without Pauli blocking and without missing mass shifts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

Results with Pauli bocking and without missing mass shifts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

Results without Pauli blocking and with missing mass shifts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

Results with Pauli bocking and with missing mass shifts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

Deuterium momentum distribution 04/26/06 (rev. 04/27/06)

Plots of the momentum of the struck neutron for a deuterium target are in the links below.    The momentum is in the lab frame. 

The Pauli blocking weight is 0.5 when the relative momentum between the two neutrons in the final state in their CM frame multiplied by 1/2 is less than 243 MeV, and 1.0 otherwise.

Q^2 = 1.1   GeV^2
Q^2 = 2.15 GeV^2
Q^2 = 3.0   GeV^2
Q^2 = 4.0   GeV^2
Q^2 = 4.8   GeV^2

The ratio of the normalized yield with Pauli blocking over the normalized yield without Pauli blocking is at this link: 
Deuterium


Pauli blocking for deuterium 04/17/06

Pauli blocking was added for deuterium (there are two neutrons in the final state). 

After pfermi has been generated (pfermi is actually P_m from the spectral function) the momentum of the two neutrons can be calculated.  The spectator neutron has momentum given by -pfermi. 

The relative momentum between the two neutrons in the CM frame was already in the SIMC ntuple (calculations).

The cut-off for the relative momentum was based on the radius of the deuteron and the uncertainty principle, and was 243 MeV  (calculations).  If the relative momentum was below this cut-off, the event weight was multiplied by 0.5.

Deuterium, all Q2, missmass comparison
No Pauli, no Jost
With Pauli, no Jost
With Pauli, with Jost (del=1.2)

Deuterium transparency,  Nucleon mmx cut only, no shifts in mmx applied.
No Pauli, no Jost
NUCLEON mmx cuts 2H Transp.

With Pauli, no Jost
NUCLEON mmx cuts 2H Transp.

With Pauli, with Jost (del=1.2)
NUCLEON mmx cuts 2H Transp.


Effect of correlations on missing mass 04/16/06

The plots below are the nuclear missing mass (missmass) for carbon.  The first plot has no Pauli blocking, the second has the ideal Fermi gas model of Pauli blocking, while the third has the correlated Pauli blocking described by Fantoni et al.

carbon_missmass_nopauli.gif
carbon_missmass_simple.gif
carbon_missmass_correlated.gif

The SIMC plots have an arbitrary normalization based on the area under the curve and a normalization determined by eye. 

Pauli blocking makes the shift between SIMC and data larger, especially at the lowest Q2.  The correlated Pauli blocking produces a low energy tail in the missing mass that helps match SIMC with the data, but the match is still worse compared to the plots without Pauli blocking.

New occupation number (Fantoni et al.) 04/09/06 (rev. 04/10/06)

The occupation number, n_2(k), in Tables 2 and 3 of Fantoni and Pandharipande Nucl. Phys. A 427 (1984) 473 has rounded edges.  The values in the tables were parameterized and used to weight events using a UWFUNC (pbweight.f).  The plot below shows the Fermi distribtion (n_2 vs. k/k_F) from the table and my parameterization to these values.

The parameterization is:
         if ((k/k_F)<0.223) then
            n=0.866
         elseif ((k/k_F)<1.0) then
            n=0.85+0.0909*(k/k_F)-0.134*(k/k_F)**2
         elseif ((k/k_F)<2.14) then
            n= 0.595-0.967*(k/k_F)+0.536*(k/k_F)**2-0.0998*(k/k_F)**3
         else
            n=0.0
         endif
Just to check, this is a plot of the occupation number vs. k/k_F from the SIMC ntuple.

Table of results WITHOUT Pauli blocking
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

Table of results with the "old" ideal Fermi gas model of the Fermi distribution
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)

Table of results with the "new" Fantoni et al. Fermi distribution
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield

2H C Al Cu
 Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)
Missmass cuts by eye
 norm yield

2H C Al Cu Au Transp. A/2H A/C A/Cu Alpha Alpha (H_to_Al) Alpha (Al_to_Au)




Cuts on hsdelta and ssdelta repeated 04/07/06

Similar to 04/02/06 below, tight cuts were again placed on ssdelta and hsdelta to select regions of phase space that are less affected by Pauli blocking.   These results actually use the occupation number with rounded edges from Fantoni et al, described above 04_09_06.

Results without Pauli blocking (and tight hsdelta and ssdelta cuts)
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)

Results with Pauli blocking (and tight hsdelta and ssdelta cuts)
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)









Pauli blocking correction 04/05/06

I found a mistake in my code in event.f.  The calcuation of the neutron momentum was performed too early in complete_ev.  After moving it lower in the subroutine, things started to make more sense.
Pauli blocking vs. Pfermi using event.f to reject events (using the return command)
below the Fermi momentum

 Pauli blocking vs. Pfermi using the UWFUNC and setting
weight=0 for kn/k_F<1
These two distributions now have the same shape.  The plot on the left seems to have a larger statistical error, but in fact, it has more events.  The reason that it seems to have a larger statistical error is because the pfermi hist before Pauli blocking used a different random number pattern in SIMC compared to the hist after Pauli blocking (because some events are thrown away).  The histograms at the top are normalized so that the area under the histogram is the normalized yield. 

Now that the results are repeatable, there is still the issue of the strange dependence of Pauli blocking vs. Pfermi (the bottom plots in the figures above).

I wrote to the ntuple a new variable          qdotpfer  = (unit three vector q) dot (unit three vector pfer)
I then used cuts to select
The negative slope at negative pfermi of the bottom plots in the figures above appears to be due to transverse events.

Pauli blocking with UWFUNC 04/04/06 (rev. 04/05/06)

I changed event.f so that Pauli blocking does not block any events in SIMC, however, the ratio k_n/k_F is now written to the ntuple.  I then made a UWFUNC that will weight events based on the occupation number:
n(k_n/k_F)

For an ideal Fermi gas model,
n(k_n/k_F) = 1   for    k_n/k_F < 1
n(k_n/k_F) = 0   for    k_n/k_F > 1

I was not able to reproduce the previous results by setting the pauli weight = 0 when k_n/k_F < 1, and 1 otherwise.
Pauli blocking vs. Pfermi using event.f to reject events below the Fermi momentum

 Pauli blocking vs. Pfermi using the UWFUNC and setting weight=0 for kn/k_F<1

In fact, if I have:
case (1), where the return statement is used in event.f to block events with k<k_F
case (2), where the return statement is commented out, and an ntuple variable is set to 0 when k<k_F, and 1 otherwise

In the plots below Red is case (1) and Black is case (2).  Both cases have Pauli blocking, but applied differently.

This suggests that pfermi IS generated differently when the return statement is used in event.f.

Cuts on hsdelta and ssdelta 04/02/06

Tight cuts were placed on hsdelta and ssdelta and the data was analyzed with Pauli blocking and without Pauli blocking.  The goal was to select regions of phase space that were less dependent on the model of Pauli blocking, and hopefully determine if our model of Pauli blocking is reasonable. 

My conclusion is that the new transparency results display the same upward trend vs. Q2 as the analysis without the tight delta cuts and without Pauli blocking.  It seems that there are problems in our model of Pauli blocking.

In the analysis below, only NUCLEON missing mass cuts were used, because the other missing mass cuts did not produce enough events.

No Pauli blocking, and with tight delta cuts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)

With Pauli blocking, and with tight delta cuts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)

No Pauli blocking, and no tight delta cuts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)

With Pauli blocking, and no tight delta cuts
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)


The position of the delta cuts were determined setting-by-setting so as to provide enough statistics, but reduce the effects of Pauli blocking.  The positions of the delta cuts are here.

The reduction in the effects of Pauli blocking can be seen by comparing the plots below. 
The plots contain SIMC data without Pauli blocking (black) and with Pauli blocking (red).
Compare the top plots on the LHS with the RHS.  The peak at negative pfermi is reduced by the delta cuts.
Compare the bottom plots on the LHS with the RHS.  The histogram is slightly closer to 1 (ie. no influence by Pauli blocking).
The overall statistics is reduced by about 1/3.
Setting 1acarbon, without delta cuts
 Setting 1acarbon, with delta cuts

Histograms from all of the carbon settings without delta cuts and with delta cuts.

One particular histogram that is interesting is that from 4acarbon (Q2=1.1 GeV2).  It is strange that the Red hist/Black hist changes to negative slope at negative pfermi.

Pfermi distributions 03/30/06


pfermi comes from the spectral function P_m
pfermi = abs(P_m) * abs( P_m . q ) / P_m . q
where P_m and q are three vectors.

The histograms below were normalized by Weight*normfac, so that the sum of the events in each histogram is the normalized yield.  The data comes from SIMC.

Pauli blocking 03/28/06


The interior nuclear densities were used to calculate the Fermi energy using the equation
E_F = hbar2/2M x (3/2pi2 rho)^(2/3)
The interior nuclear densities were estimated from the figure below in the paper by Hofstadter (1963).
Nuclei
 rho (C/cm^3)
 rho (nucleons/fm^-3)
 E_F (MeV)
 P_F (MeV)
12C
27Al
63Cu
197Au
 1.25
1.35
1.25
1.07
 0.156
0.176
0.172
0.167
 36.2
39.2
38.6
37.9
 261
271
269
267
Note that rho (nucleons/fm^-3) was calculated using the ratio of protons to neutrons, which is 1:1.49 for Au.

Dipangkar suggested to use the recoiling neutron kinetic energy and require this to be above the Fermi energy, where
(Tn + Mn)2 =Pn2 + Mn2

The transparency results at low Q2 shifted up slightly from the ones at the bottom of the next entry below.  (NB only the nucleon mmx cuts have been shifted properly)
NUCLEON mmx cuts norm yield iteration
 1H 2H C
 Al
 Cu
 Au Transp.
 		A/2H
 A/C
 A/Cu
 Alpha
 Alpha (H_to_Al)
 Alpha (Al_to_Au)
NUCLEAR missmass cuts norm yield