NATIONAL NUCLEAR DATA CENTER
Bldg. 197DBrookhaven National Laboratory
P. O. Box 5000
Upton, NY 11973-5000, U.S.A.
(Internet) nndc@bnl.gov

Telephone: (631)344-2902
FAX: (631)344-2806

Memo CP-C/271 (Revised)
DATE:

8 September 2000

TO:

Distribution

FROM:

V. McLane

SUBJECT:

Polarization LEXFOR entry

Attached is the proposed new version of the LEXFOR entry on polarization. If
accepted, it will involve some new codes and changes to already existing codes. I
have checked our library, and there exists only one entry outside of our area for which
updates will be needed.
A memo on the dictionary updates will be sent shortly, after everyone has had achance to look at the LEXFOR entry.

Distribution:
M. Chiba, Sapporo
F. E. Chukreev, CAJaD
S. Dunaeva, Sarov
O. Gritzay, KINR
K. Kato, JCPDG
M. Kellett, NEADB
V. N. Manokhin, CJD

S. Maev, CJD
O. Schwerer, NDS
S. Takács, ATOMKI
F. T. Tárkányi, ATOMKI
V. Varlamov, CDFE
Zhuang Youxiang, CNDC
NNDC FilePolarization 1If the spins of a assembly of particles, either stationary or in a beam, are oriented in all
directions with equal probability, the assembly is said to be unpolarized. If the spins arepredominantly oriented in one direction, the assembly is said to be polarized.
Under the influence of a spin-orbit force upon scattering from a target, an unpolarized beamof particles becomes polarized.
P(θ)θθ

The inverse of this situation is the asymmetric scattering of a polarized beam of particles.
The degree of polarization of such a beam may be determined by measuring the leftright asymmetry upon scattering from a target nucleus which plays the role of an analyzer.
PbleftθθrightFor elastic scattering from a spin-0 nucleus: the principle of Polarization-Asymmetry Equality
states that, for time-reversal invariant reactions, the polarization induced in a previously
completely unpolarized beam is identically equal to the asymmetry ensuing from thescattering of a perfectly polarized beam under the same conditions.

Conventions for quantities given in Cartesian coordinates
Basel Convention for spin-½ particles 2
In nuclear reactions, the positive polarization of spin-½ particles is taken in the direction of the vector product k in × k out where k in and k out are the circular wave vectors of the incomingand outgoing particles.

1
2

See [1970Ma].
See [1960Hu].

Madison Convention for spin-1 particles; quantities given in Cartesian coordinates 3
1. Polarization (P), the state of spin orientation of an assembly of particles, is referred to aright-handed coordinate system in which the positive z-axis is along the direction of 
momentum of the incident particles, and the positive y-axis along k in × k out(perpendicular to the beam direction in the reaction plane) for the nuclear reaction which
the polarized particles initiate, or from which they emerge.
2. Analyzing power (A) is referred to a right-handed coordinate system in which the positive z-axis is along the direction of the incident beam, and the positive y-axis along k in × k out
for the reaction.
The cross section for a reaction initiated by polarized particles with spin-1 may be written
in terms of Cartesian tensors as:σ =σ 0 1 + 23 p y Ay + 12 pzz Azz + 32 pxz Axz + 16 ( pxx − p yy )( Axx − Ayy ) 
3. In the expression for a nuclear reaction A(b,c)D, an arrow placed over the symbol denotesa particle in a polarized state.
 Example:
A(b, c) D
Additional conventions
1. The following subscripts are defined:b beamt target↑ spin up↓ spin down
i ith component for beam
k kth component for scattered particle
j jth component for target
p pth component for recoil nucleus
N=n=y normal to the scattering plane
L=l=z longitudinal, i.e., parallel to incident particle momentum in scatteringplane
S=s=x N�L (sideways), i.e., perpendicular to particle momentum inscattering plane
SLaboratory system

LCenter-of-mass system

slc

SLb

Lscl

Sb

A

A
DDSl

Lls



N and n are identical with direction out of page for all particles.

3See [1971Sa].

sOccasionally, a 4-index notation is used; the order of the subscript in a 4-index notation is(i,j;k,p). A zero in a field denotes unpolarized or polarization not measured.
2. Y is the normalized yield.
3. σ 0 is the differential cross section for an unpolarized (spin-averaged) beam.
Quantities given in Cartesian coordinates and compiled in EXFOR
Polarization:refers to the degree of polarization of a single particle or ensemble.
P s (θ) is the polarization of the outgoing particle beam as a function of angle.

σ P = σ 0 P(θ )whereREACTION Coding: ,POL/DA
Asymmetry: Asymmetry is the relative difference between:a.) the number of particles scattered to the right and to the left in the reaction plane.
ε = ( L − R) ( L + R) = Pb Ay
where ε =Pb =Ay =L,R =asymmetry
incident beam polarization
analyzing power of target
intensity of particles scattered left and right in the same plane under thesame angle.
REACTION coding: POL/DA,,ASYb.) the angular distribution for incident projectiles with their polarization vectors
perpendicular and parallel to the reaction plane.

ε=( dσ ⊥ − dσ || ) ( dσ || + dσ ⊥ )Where refers to the parallel component,⊥ refers to perpendicular component.

REACTION coding: ,POL/DA,,ASY/PP
If data are given as (dσ || − dσ ⊥ ) /(dσ || + dσ ⊥ ) , the negative of the value given is coded.
c.) the angular distribution for forward and backward scattered particles.

ε =−( dσ F dσ B ) ( dσ F + dσ B )REACTION Coding: ,POL/DA,,ASY/FB
The angles are given as ANG1 and ANG2.

Analyzing Power: the relative difference in the cross sections for corresponding spin up vs.
spin down.
Vector analyzing power (A i or A i,0;0,0 and A j or A 0 , j;0,0 ):Coding: Modifier ANA in SF8.
Example:,POL/DA,,ANAAnalyzing power, A n , incident projectile spin normal toscattering plane.

Tensor analyzing power (A ij or A i,j,0,0 ): initial spin state correlation parameters. 4εA=
A=yyNNPb Pt

Coding: Modifier A in SF8, branch code in SF5 giving components of beam and target spin
Example:NN,POL/DA,,ANA

Tensor analyzing power, A NN , incident projectile and target
spins normal to scattering plane.

Spin-Correlation Parameter (C kp or C 0,0,k,p ): final spin state correlation parameters.
For spins in the scattering plane:Ckpdσ / d Ω↑↑ + dσ / d Ω↓↓ − dσ / d Ω↑↓ − dσ / d Ω↓↑
1 Y↑↑ + Y↓↓ − Y↑↓ − Y↓↑=dσ / d Ω↑↑ + dσ / d Ω↓↓ + dσ / d Ω↑↓ + dσ / d Ω↓↑ pb Pt Y↑↑ + Y↓↓ + Y↑↓ + Y↓↑
LU

PkA
LDRUA

Pp

where:
L = left
R = right
D = down
U = up

RDFor spins normal to the scattering plane:
CNN =1 ( LL)( RL) + ( LR )( RR ) − ( LL)( RR ) − ( LR )( RL)
pb Pt ( LL)( RL) + ( LR )( RR ) + ( LL)( RR ) + ( LR )( RL)
LLPkALRRLPpARR4

May be referred to in the literature as spin correlation parameter.

where:L = left
R = right

REACTION Coding: Modifier C in SF8, branch code in SF5 giving components of beamand target spin.
Example: LL,POL/DA,,CC LL , spin correlation function, outgoing particle andrecoil nucleus spins parallel to beam direction inscattering plane

If time reversal holds: C NN = A NN
Spin-Rotation-Depolarization Parameters (D ik or D i,0,k,0 for incident beam and D jp or
D 0,j,0,p for target)
The spin-rotation parameter is a measure of the rotation of the spin of the scattered beam.
The three orthogonal components of the outgoing particle polarization are related to thecomponents of the incident particle polarization through the set of spin-rotation parameters.
These parameters are related to the Wolfensein [1956Wo] parameters as follows:
Wolfenstein
Parameter
D NN =

D

D SS =

R

D LS =R′

D SL =A

D LL =

A′

Representation⊗ signifies normal to the scattering plane

D NN is also known as the Spin-Depolarization Parameter
REACTION Coding: Modifier D in SF8, branch code in SF5 giving components of beamand target spin.
Example:
LL,POL/DA,,D

Spin-Rotation Parameter D LL , beam and scattered particle spins
parallel to beam direction in scattering plane.

LL,POL/DA,*,D

Spin-Rotation Parameter D LL , target and recoil nucleus spins
parallel to beam direction in scattering plane.

Spin-flip probability (Snn or Sflip)
REACTION coding:

NN,POL/DA,,SF

The Spin-Flip Probability is related to the spin depolarization parameter by:
1 (1 − DS=2nnNN ).
Spin-Rotation Function (SRF or Q): gives the coupling between the longitudinalcomponent of the beam polarization and the component of the outgoing particle spin
perpendicular to the beam direction.
Q=1 − P 2 sin β

where β is the angle through which the projection of the polarization upon the scattering
plane is rotated.
For spin-0 particles:
Q = A cos θ lab + R sin θ lab = DSL cos θ lab + DSS cos θ lab
REACTION coding: POL/DA,,SRF
Spin-Transfer Parameter (Kip for incident beam and Kjk for target)
P(θ)

PbCoding Example:
NN,POL/DA,,KLL,POL/DA,*,K

K NN , spin transfer parameter, incoming particle and recoil nucleus
spins normal to scattering plane
K NN , spin transfer parameter, target nucleus and outgoing particle
spins longitudinal in scattering plane

The spin-transfer parameter is related to the spin-rotation parameter by:.

D=K ij (π − θ )ij (θ )

Conventions for quantities given in Spherical Coordinates
The following subscripts are defined:k refers to the rank
q refers to the projection
Madison convention for spin-1 particles quantities given in spherical coordinates 5
For particles of spin ≥ 1, the polarization may be described in terms of tensor quantities ofinteger rank k ≥ 2i: 0 ≤ |q| ≤ k.
The differential cross section for a reaction initiated by a beam with tensor polarization
components t kq is(σ = σ 0 ∑ k ,q tkqTkq*Where)

T kq = analyzing power components.

For spin-1 particles, if parity is conserved, T 10 =0, T 11 is pure imaginary, and T 2q is pure real;
the cross section may be written asσ =σ 0 [1 + 2iT11 Re(t11 ) + T20t20 + 2T21 Re(t21 ) + 2T22 Re(t22 ) ]Quantities given in Cartesian coordinates and compiled in EXFOR
Analyzing Power (T kq )Vector analyzing power (iT 11 )Coding:quantity code VAP in SF6

Example: POL/DA,,VAP
Tensor analyzing power (T 20 , T 21 , T 22 )Coding:quantity code TAP in SF6; k and q are specified under the branch field using
the codes 20, 21, and 22.

Example: 20,POL/DA,,TAP

5

See [1971Sa].

Associated Information
Polarized beam specitifcation
For a polarized incident neutron beam, enter the code POLNS under the keyword INC-SOURCE;
for a polarized ion beam, enter POLIS. If the incident source is known, it should follow in thesame set of parenthesis.
Example; INC-SOURCE(POLNS,D-T)

Atomic beam and Lamb-shift sources are entered using the codes ATOMI and LAMB,respectively.
Polarized target specification
For a polarized target, enter the code POLTR under the keyword INC-SOURCE.
The polarization of the incident beam and target should be given in the data table, if known,
using the data headings POL-BM and POL-TR, respectively. The numerical uncertainties areentered using the headings POL-BM-ERR and POL-TR-ERR; an explanation may be given in
free text under ERR-ANALYS (see EXFOR 8.10).

REFERENCES
1956Wo L. Wolfenstein, Polarization of fast nucleons, Ann. Rev. Nucl. Sci. 6, (1956) 43-47.
1960Hu Proceedings of the International Symposium on Polarization Phenomena ofNucleons, P.Huber and K. P. Meyer, eds., Helv. Phys.Acta, Suppl. VI (1960) end ofvolume.
1970Ma P. Marmier and E. Sheldon, Physics of Nuclei and Particles, (Academic Press, NY,1970) Vol. II, Ch. 13.
1971Sa G. R. Satchler et al., Polarization Phenomena in Nuclear Reactions (University ofWisconsin Press, Madison, WI, 1971) pp. xxv-xxix; also S. E. Darden, pp. 39-56.
For more information, see also the following references.
T. A. Welton, The theory of polarization in reactions and scattering, Fast Neutron Physics,
Part II (Interscience Publishers, 1963) 1317-1377.
G. G. Ohlsen, Polarization transfer and spin correlation experiments in nuclear physics, Rev.
Prog. Phys. 35,(1972) pp. 717-801.
J. Ashkin et al., Convention for spin parameters in high energy scattering experiments,
Higher Energy Polarized Proton Beams (American Institute of Physics, 1978) pp.
142-146.
D. Crabb, Spin correlation measurements in nucleon-nucleon scattering at high energy,Polarization Phenomena in Nuclear Physics - 1980, (American Institute of Physics,1981) pp. 11-30.