content_image/395214.jpg content_image/395217.jpg content_image/395219.jpg content_image/395215.jpg content_image/395212.jpg content_image/395213.jpg content_image/395216.jpg content_image/395218.jpg content_image/395226.jpg content_image/395224.jpg content_image/395223.jpg content_image/395222.jpg content_image/395221.jpg content_image/395220.jpg
field is defined by 

Let Ω ( a ) denotes the average value of ω ( a $^{)}$. The spin coherence is kept when 

For a storage ring with a noncontinuous field, the quan- tity B$_{z}$ should be averaged. 

This condition defines a spin-isochronous ring, i.e., the spin precession frequency is independent of the momen- tum at the first order. 

Condition (3) can be satisfied for ordinary storage rings with magnets creating nonuniform field (Fig. 1). Beam direction is normal to the magnet faces and there is not edge focusing. The number of bending sections can be different. If the field created by the magnets is given by B$_{z}$ ( ρ ) = const · ρ − $^{n}$, the field index and betatron tunes into bending sections are equal to 

<img src='content_image/395214.jpg'>

where B$_{0}$ ≡ B$_{z}$ ( R$_{0}$ ) , x = ρ − R$_{0}$ , and R$_{0}$ is the ring radius. Average angular frequency of spin precession is given by 

where L is a half of the total length of the straight sec- tions (Fig. 1). The muon anomaly is equal to 

where the fundamental constants g$_{p}$ and m$_{µ}$/m$_{p}$ are mea- sured with a high precision. The magnetic field is the same for muons and protons when they move on the same trajectory. In this case, their momenta coincide. 

When the momentum increases ( p > p$_{0}$ ), the magnetic field becomes weaker, but the time of flight in the mag- netic field becomes longer. The spin precession is defined by 

<img src='content_image/395217.jpg'>

Condition (3) leads to d Ω ( a ) z /dρ = 0 and is satisfied when 

where R$_{0}$ corresponds to p$_{0}$ and B$_{0}$ . In this case 

<img src='content_image/395219.jpg'>

<img src='content_image/395215.jpg'>

<img src='content_image/395212.jpg'>

<img src='content_image/395213.jpg'>

<img src='content_image/395216.jpg'>

<img src='content_image/395218.jpg'>

<img src='content_image/395226.jpg'>

and the following relation takes place: 

where C is the orbit circumference. As a result, the mo- mentum compaction factor is 

Since 

where T is the revolution period, the definition of α can be brought to the usual form: 

Evidently, the spin-isochronous ring ( α = 1) is not isochronous in the usual sense, i.e., the beam revolution frequency depends on the momentum. 

Eq. (7) is not exact because it does not include a correction for the fringe field. This field also contributes to the average field, but it is independent of ρ . The fringe field is important only near the magnet edges and causes the correction to L$_{0}$ of order of the ratio of the magnet gap to the ring radius ( ∼ 10 − $^{2}$). This correction depends on the number of the straight sections and can be analytically and numerically calculated because the magnet field is known with a needed accuracy. ( a ) 

Evidently, the correction to the local value of ω z is given by 

The corrected values of L$_{0}$ also coincide for muons and protons because particles with equal momenta move in the same field. 

Two other corrections to the angular velocity of spin precession caused by the longitudinal magnetic field and 

<img src='content_image/395224.jpg'>

<img src='content_image/395223.jpg'>

<img src='content_image/395222.jpg'>

<img src='content_image/395221.jpg'>

<img src='content_image/395220.jpg'>

