The nucleon-nucleon center-of-mass frame is ideal for deriving the relations between spin observables and the interaction matrix, but it is not always a useful way to conduct experiments. Intermediate energy nuclear physics experiments are typically fixed target experiments, so it is necessary to convert the expressions for to be in terms of laboratory frame polarization transfer observables.
The laboratory frame will be defined as:
The axes are typically referred to as longitudinal , normal , and sideways . and represent right-handed Cartesian coordinate frames [BBW82]. In terms of the laboratory frame coordinates equation can be expressed as [Che93]
The analyzing power associated with the out-of-plane component is still referred to as by convention. Therefore, the laboratory frame polarization observables can be obtained by measuring both the initial and final polarization.
In order to determine the Bleszynski, et al. observables, , [BBW82] it is necessary to express the laboratory frame observables (i.e. ) in terms of the center-of-mass spin observables (i.e. ). This requires a transformation of the reference frame which includes a relativistic effect on the spin rotation. This transformation is presented in detail by [IcK92].
Figure: Scattering kinematics in the center-of-mass and laboratory reference frames including the relativistic effect on the spin rotation explicitly [IcK92].
First of all, the unit vector normal to the scattering plane will remain unchanged , so . For the in-plane components the final state coordinates, and , will be rotated by an angle, , which is the relativistic spin rotation as is shown in figure . is given by
where is the velocity of the particle, is the velocity of the c.m. frame with respect to the laboratory frame, , and and are the scattering angles in the c.m. and laboratory frames, respectively. If we take to be the angle between the incident beam direction, , and the unit vector as defined in equation (see fig. ) then the transformations look like
Given these expressions between the coordinates the relations between in-plane spin observables is
where . Using equation , equations to can be rewritten as [Lut93]
Note: It is also possible to define as [Che93]: