# Spin Rotators.

A spin rotation from vertical direction in the arcs to longitudinal one
in the IPs will be performed in two steps: at first, by a solenoidal
spin rotator to horizontal plane and then by dipoles. The
![](media/image1.wmf) spin rotator consists of two superconducting
solenoids, each 3 m long, and with the field of about 8 T. Between
solenoids is placed a focusing structure, which cancels the betatron
coupling and also creates the spin transparency. On the opposite side of
the interaction straight, spin is restored to vertical direction by the
negative spin rotator. As a result, the spin tune is undisturbed by the
interaction region insertion, and the polarization behavior is mainly
the same as without the spin rotators.

The equilibrium polarization direction (vector![](media/image2.wmf)) is
vertical in the main part of the ring and therefore one can expect a
relatively low depolarization rate of the electron beam. Moreover, the
Sokolov-Ternov polarization mechanism should provide a high enough beam
polarization.

Still there are some requirements, which should be satisfied by the
insertion optics, where the spin vector ![](media/image3.wmf) lies in
the horizontal plane. To minimize the negative effect from a spin
perturbation ![](media/image4.wmf) over the whole straight section, we
should fulfill the so called spin transparency condition, namely, the
integral of the perturbation through the insertion
azimuth![](media/image5.wmf):

![](media/image6.wmf)

should be made zero or, at least, small. Here ![](media/image7.wmf) is a
complex vector, which is composed from the unity vectors
![](media/image8.wmf) and ![](media/image9.wmf), which in turn are the
two orthogonal solutions of the spin motion equation for the equilibrium
particle. The spin perturbation components are:

![](media/image10.wmf)

where ![](media/image11.wmf) is the dimensionless spin tune,
![](media/image12.wmf) and ![](media/image13.wmf) are the second
derivatives of the vertical or horizontal displacements over the azimuth
![](media/image14.wmf), ![](media/image15.wmf) are respectively the
normalized horizontal, longitudinal or vertical magnetic fields.

Careful analysis shows that the spin transparency condition for the
interaction region straight section, which contains the discussed above
spin rotators, can be fulfilled, and a small decrease of the equilibrium
polarization degree is caused only by bending magnets in the straight
between two spin rotators.

![](media/image16.png){width="5.501388888888889in"
height="1.4222222222222223in"}

Fig. 1 Scheme of the solenoidal spin rotator in the electron ring.

We found a scheme of the focusing structure (see Fig.1), that contains
only regular quadrupoles inside the solenoidal spin rotator and cancels
the betatron coupling as well as creates the spin transparency. A
transfer matrices of a full insertion (from the first solenoid edge to
the second solenoid edge):

![](media/image17.wmf)

Here ![](media/image18.wmf) is a curvature radius in the solenoidal
field ![](media/image19.wmf):

![](media/image20.wmf)

Table 1 Main parameters of the rotator insertion.

  ----------------------------------- -----------------------------------
  **Parameter**                       **Value**

  Solenoid length                     3 m

  Solenoidal field                    8.7 T

  Quadrupole length                   0.4 m

  Maximal quad's gradient             28 T/m
  ----------------------------------- -----------------------------------

In our proposal the condition on transparency is fulfilled, thus, all
spin rotators as also the horizontal wiggle between them are spin
transparent against the betatron deviations. An inevitable spin-orbit
coupling for the energy-off particles is excited by the first spin
rotator and then compensated by the second one. In result the spin-orbit
coupling vector ![](media/image21.wmf) is exactly equal to zero in arcs.

![](media/image22.png){width="4.954166666666667in"
height="3.497916666666667in"}

Fig. 2 Spin-orbit coupling along the ring.

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