V.S. Morozov 4 February 2004
Dear Colleagues,
We have been discussing how to extract the vector and tensor polarizations from the measured
asymmetries. We found that there was some confusion about the coordinate system and the
notation; note that our recent IUCF vertically polarized deuteron PRL, probably unwisely, used P z
and Pzz rather than Py and Pyy. We now have some suggestions that might eliminate this confusion:
1. We propose using the Madison Convention coordinate system [1], which is a right-handed
system with: the positive z-axis along the beam momentum k in ; and the positive y-axis along
k in  k fs where k fs is momentum of the forward scattered particle. For example, if the reaction
plane is horizontal, then the x-axis is radial, the y-axis is vertical, and the z-axis is longitudinal.
2. The vector and tensor polarizations should be called PV and PT, respectively. They should have
components: Px & Pxx; Py & Pyy ; and Pz & Pzz for beam polarization directions pointing along the
x, y, and z axes, respectively.
3. We propose changing the notation for the measured asymmetries from 1 and  5 to:
LR
(L  R )  (T  B)
V 
and  T  4
(1)
LR
LRTB
where L, R, T, and B are the number of forward scattered events into the left, right, top, and
bottom quadrants, respectively.
4. The detector efficiencies for forward scattering into the left, right, top, and bottom quadrants
should be called L , R , T , and B , respectively.
We tried to derive the equations relating  V and T to the vector and tensor polarizations and the
effective analyzing powers. We used the differential cross section for a reaction initiated by a beam
of vertically upward polarized spin-1 particles [1]:
d d 0
3
1
2
1
(2)

[1  Py A y  Pzz A zz  Pxz A xz  (Pxx  Py y )(A xx  A y y )] ,
d d
2
2
3
6
where d 0 / d is the unpolarized differential cross section, and Ay, Axx , Ayy , Azz , and Axz are
analyzing powers.
For the forward scattering of a vertically upward polarized deuteron into COSY’s horizontal plane,
the EDDA detector counts the recoil protons, L’ and R’, going into its left and right quadrants,
respectively. These, respectively, correspond to right (R) and left (L) forward scattering of the
deuterons in Eq. (2). Now notice that L = R’ and R = L’; moreover, P y = PV; Pyy = PT; Pxz = 0; and
Pxx = Pzz = -½ PT [2, 1]; thus, using Eq. (2) and the relation Axx + Axx + Axx = 0, we obtain the
number of detected L and R events:
1
 3

L  R '  c L 1  PV A y  PT A y y  ,
2
 2

1
 3

R  L'  c R 1  PV A y  PT A y y  ,
(3)
2
 2

where c is a constant, while PV and PT are the vector and tensor polarizations along the upward
vertical axis perpendicular to COSY’s horizontal plane. The Ay and Ayy are the “effective”
analyzing powers. NOTE: Were these obtained by averaging over the  for each EDDA
quadrant, while using the Ay, Ayy, and d 0 / d values obtained by interpolating published data at
1042 [2] and 2430 MeV/c [3]?
VECTOR ASYMMETRY
Using Eq. (1), the vector asymmetry  V is:
 L
   3

1

 1  1  L  PV A y   L  1 PT A y y

  R  2
 R
2
.
(4)
V   R
 L   L
3
 L  1
1 
  
 PT A y y
 1 PV A y  1 
 R   R
2
 R  2
In the limit that L / R  1 , this becomes:
3 PV A y
.
(5)
V 
1
2
1  PT A y y
2
During the December 2003 run Dr. Rohdjess prepared two papers [4] on the calibration of scalers,
which appeared to give a slightly different αV:
3 PV A y
 1   V 
.
(6)
1
2
1  PT A zz
4
In these papers [4], he also obtained the average relative scaler efficiencies from a 2 fit; these were:
R ' / L'  L / R  1.0396  0.0006 ,
T ' / L'  B / R  0.9508  0.0023 ,
(7)
B' / L'  T / R  0.9500  0.0023 .
Substituting these efficiencies into Eq. (4) gives:
3
1
0.0396  2.0396  PV A y  0.0396  PT A y y
2
2
V 
.
(8)
3
1
2.0396  0.0396  PV A y  2.0396  PT A y y
2
2
TENSOR ASYMMETRY
Using Eq. (2) and Refs. [1, 2], for the forward scattering of vertically polarized deuterons into the
vertical reaction plane, gives:
 1

T  B'  c T 1  PT A xx  ,
 2

 1

B  T'  c B 1  PT A xx  .
(9)
 2

Using Eqs. (1), (3), and (9), the tensor asymmetry is then:
  
1
 L T B   L
3

 
1 
  


 1 PV A y  1  L A y y   T  B A xx  PT
 R R R   R
2
 R R 
  R 
 2 . (10)
T  4
  
1
 L T B   L
3

 
1 
  


 1 PV A y  1  L A y y   T  B A xx  PT
 R R R   R
2
 R R 
  R 
2
Note that A xx  A y y  A zz  0 [2]; thus, when L / R  T / R  B / R  1 , Eq. (10) becomes:
[A y y  A xx ]PT
[A  A y y ]PT
.
(11)
  xx
1
1
1  A y y  A xx PT
1  A zz PT
4
4
The Dec 03 papers [4] gave the below Eq. (12), which differs from Eq. (11) by a minus sign
[A xx  A y y ]PT
.
(12)
5  T 
1
1  A zz PT
4
Substituting the efficiencies from Eq. (7) into Eq. (10) gives
3
1
0.1388  0.0396  PV A y  2.0396  A y y  1.9008  A xx  PT
2
2 .
(13)
T  4
3
1
3.9404  0.0396  PV A y  2.0396  A y y  1.9008  A xx  PT
2
2
It seems that there are some inconsistencies between Eqs. (6), (12) and Eqs. (5), (11), which could
be due to the confusion about the coordinate system; this should be discussed. Also notice from
Eqs. (8) and (13), that by including the efficiencies given by Dr. Rohdjess [4], one may explain at
least part of the ε1 and ε5 offsets observed during the February and December 2003 deuteron runs.
T 
ASYMMETRY  2 (was  2 )
The asymmetry  2 is defined as:
TB
2 
TB
Using Eqs. (14) and (9), this asymmetry can written as:
 T  B   T B  1

  
 PT A xx






R 
R  2
 R
.
2   R
 T B   T B  1

  
 PT A xx


 R R   R R  2
In the limit that T / R  B / R  1, this becomes:
2  0 .
Page 16 of the February 2003 run data book gives
2  2  0 .
Substituting the efficiencies from Eq. (7) into Eq. (15) gives
1
 0.0008  0.0008  PT A xx
2
2 
.
1
1.9008  1.9008  PT A xx
2
Assuming PT = 0.541, Axx = - 0.0540 would give 2 = - 0.000421.
(14)
(15)
(16)
(17)
(18)
ASYMMETRY  3 (was  3 )
The asymmetry  3 is defined as:
LR
3  8
LRTB
Using Eqs. (19), (3), and (9), this asymmetry can written as:
(19)
 L
   3


 1  1  L  PV A y   L
 R
  R  2
 R
3  8
 
 L T B   L
3
1 
  


 1 PV A y  1  L
 R R R   R
2
  R
1
 1 PT A y y
2



A y y   T  B

 R R
1

A xx  PT

2
Using A xx  A y y  A zz  0 , in the limit that L / R  T / R  B / R  1 , this becomes:
PV A y
.
1
4  [A y y  A xx ]PT
1  A zz PT
4
Page 16 of the February 2003 run data book gives
PV A y
.
 3   3  6
1
1  (A zz  A xx  A y y )PT
4
Using A xx  A zz  A y y we obtain
3  8
3PV A y
6
PV A y
.
1
1  A y yPT
2
Substituting the efficiencies from Eq. (7) into Eq. (20) gives
3
1
0.0396  2.0396  PV A y  0.0396  PT A y y
2
2
.
3  8
3
1
3.9404  0.0396  PV A y  [2.0396  A y y  1.9008  A xx ] PT
2
2
3  6
References
1.
2.
3.
4.
Madison Convention, Proceedings of the 3rd International Symposium on Polarization Phenomena in
Nuclear Physics, Madison, 1970, edited by H.H. Barschall and W. Haeberli (University of Wisconsin
Press, Madison, WS, 1971), p. xxv.
K. Sekiguchi et al., Phys. Rev. C 65, 034003 (2002).
M. Haji-Saied et al., Phys. Rev. C 36, 2010 (1987).
H. Rohdjess, “Calibration of Scalers” papers, 9 and 10 Dec 2003.
. (20)
(21)
(22)
(23)
(24)