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 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 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 Two other corrections to the angular velocity of spin precession caused by the longitudinal magnetic field and The corrected values of L$_{0}$ also coincide for muons and protons because particles with equal momenta move in the same field. Evidently, the correction to the local value of ω ( a ) z is given by 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. Evidently, the spin-isochronous ring ( α = 1) is not isochronous in the usual sense, i.e., the beam revolution frequency depends on the momentum. where T is the revolution period, the definition of α can be brought to the usual form: Since where C is the orbit circumference. As a result, the mo- mentum compaction factor is and the following relation takes place: ω ( a ) z = − e m aB$_{z}$. (2) d Ω ( a ) z dp = 0 . (3) n = − R$_{0}$ B$_{0}$ ( ∂B$_{z}$ ∂ρ ) ρ = R 0 , ν ( b ) x = √$_{1}$ − n, ν ( b ) z = √$_{n,}$ Ω ( a ) = ω ( a ) z πρ πρ + L = − πeaρB$_{z}$ ( ρ ) m ( πρ + L ) , (4) a$_{µ}$ = g$_{p}$ − 2 2 m$_{µ}$ m$_{p}$ Ω ( a ) µ Ω ( a ) p , (5) d Ω ( a ) z dp = d Ω ( a ) z dρ ( dp dρ ) − 1 , dp dρ = (1 − n ) eB$_{z}$ ( ρ ) . (6) L = L$_{0}$ = n 1 − n πR$_{0}$, (7) R$_{0}$ = p$_{0}$ | e | B$_{0}$ , Ω ( a ) = Ω ( a ) 0 = − (1 − n ) eaB$_{0}$ m , (8) δω ( a ) z /ω ( a ) z = δB$_{z}$/B$_{z}$. Δ T T$_{0}$ = ( α − 1 γ 2 0 ) Δ p p$_{0}$ . Δ C C$_{0}$ = 1 γ 2 0 Δ p p$_{0}$ + Δ T T$_{0}$ , α = Δ C/C$_{0}$ Δ p/p$_{0}$ = 1 . (9) Δ C C$_{0}$ = Δ p p$_{0}$ = (1 − n ) x R$_{0}$ , FIG. 1: The storage ring. b/2 R GLYPH<c114> 0 L p=p p>p 0 0 2