For
ultrarelativistic particles with fixed
magnet settings,
A(δ)=A0(1−δ) and
B(δ)=B0(1−δ). Hence the half-trace is
C(δ)=C0+δ[d(A0+B0)−d2A0B0]+O(δ2),C0=1−d(A0+B0)+21d2A0B0.
Off-
energy motion is stable when
∣C(δ)∣<1, separately in each transverse
plane. Where
sinμ0=0, its
first-order
phase shift is
Δμ=−sinμ0d(A0+B0)−d2A0B0δ.
For equal opposite strengths,
μ(δ)=μ0−2tan(μ0/2)δ+O(δ2).
Thus positive
momentum error reduces the focusing and phase advance; negative error moves toward the overfocused unstable boundary. The unexpanded symmetric stability condition is
∣f0∣(1+δ)>d/2, with
1+δ>0, within the ultrarelativistic
energy scaling. Its natural
chromaticity per
cell is
d(μ/2π)/dδ=−tan(μ0/2)/π. Away from the
ultrarelativistic limit replace the displayed
first-order
energy error by
δ/β02. Linearizing at
a stability boundary is singular and needs separate
treatment.