The spin structure of the reaction is given by , where A and B represent the spin of the target's initial and final states, respectively. The derivations in this section are made for the case of a spinless target, but the resulting relations hold for the more general case. The complete derivation for arbitrary projectile and target spin can be found in [Wol56]. Details on the method used here can be found in [Ohl72].
In non-relativistic quantum mechanical scattering one can define a transformation matrix M which contains all the spin related information in a reaction. The outgoing nucleon spinor is related to the incident nucleon spinor linearly by the relationship
The expectation value of an observable corresponding to a given Hermitian operator, , is
for some given . That spinor represents a single particle of definite polarization. It is possible to define a density matrix, , such that
where means the trace of the argument.
If we want to work with a large ensemble of particles, such as in an experiment, it is necessary to find the ensemble average of the expectation value, , defined as
where N is the total number of particles. Working with the ensemble average of the expectation value is necessary because the spinors describing the individual particles will not likely be identical. That is, the polarization of the individual particles may vary from one particle to the next. In this case it is still possible to define a density matrix:
For the remainder of this chapter the ``ensemble'' will be understood and the overline notation will be not be used.
The average polarization of an ensemble can be found by applying the standard Pauli matrices:
as the Hermitian operators. Therefore,
The density matrix can be rewritten in terms of the polarization by realizing that the density matrix, being Hermitian itself, can be represented in terms of the set I, , , :
where j = 0,x,y,z and is the unit matrix
Using the identity , the coefficients, , are
The density matrix can then be written as:
assuming the density matrix is normalized such that
Given the definition in equation the relationship between the initial and final spinors shown in equation can be rewritten as
If is normalized to unity the differential cross section, I, for a polarized beam is given by
If the beam is unpolarized,
then the expression reduces to the standard definition for unpolarized cross section:
Using the definition for (eqn. ) in equation one finds
which, in turn, yields
where the analyzing power, , is defined as
If we wish to also consider the polarization of the scattered particles it is useful to note that is normalized to unity so that
where the prime indicates the outgoing particle. Therefore,
Once again using eqn. this becomes
where is the component of the polarization that scattered particle would have if the beam were unpolarized (called induced polarization), and is the polarization transfer coefficient which relates the initial polarization component to the final polarization component. and are defined as:
In their fully expanded forms the expressions for cross section and polarization are:
which is the most general form of the expressions allowed by conservation of angular momentum despite the fact that we initially made the assumption of a spinless target. The next step in the derivation is to consider the restriction imposed on the polarization observables imposed by parity conservation and time reversal invariance.
Under a parity transformation all vector quantities should reverse sign while pseudovectors, such as a polarization, should be unaffected. Therefore the initial and final momentum vectors, and respectively, being vector quantities will change sign. Assume the particle motion is defined as being along the z-axis, and the scattering plane is defined by the x- and z-axes. Under a parity transformation the in-plane coordinates, being defined by linear combinations of and , will rotate . The y-axis will be perpendicular to the scattering plane (in the direction) and will not change sign under this transformation. The polarization transfer coefficients which relate polarization vectors in the scattering plane will be allowed by parity conservation because after the parity transformation the reaction will look exactly the same as before. Actually, to be precise, the reaction will look as though both the ingoing and outgoing polarization vectors are reversed, but the linear nature of equations to means that the two cases are equivalent. Therefore, , , , and are allowed by conservation of parity. On the other hand, if one of the polarization vectors is out of the plane (y-axis) then that polarization vector will not change since the parity transform is closely related to a rotation around the y-axis. Because the in-plane polarization vector changes sign, as described above, then those coefficients must obey for parity to be conserved, and so those terms must be identically zero. Finally, because the parity transformation doesn't affect the out-of-plane polarization vectors at all will be allowed. For similar reasons and are the only components of analyzing power and induced polarization that are non-zero. Therefore equations to reduce to:
Recasting these equations into a single matrix equation defining the resultant polarization vector: