II-9. Instrumentation

RIKEN Accel. Prog. Rep. 55 (2022)

RIKEN Accel. Prog. Rep. 55 (2022)

Compensation for energy-spread growth in RUNBA
M. Wakasugi,∗1,∗2 Y. Abe,∗2 R. Ogawara,∗1,∗2 T. Ohnishi,∗2 H. Tongu,∗1 K. Tsukada,∗1 and Y. Yamaguchi∗2
A small heavy-ion storage ring, RUNBA,1) is underconstruction at E21. It will be used to develop a beam
recycling technique, which will be a novel tool for the nuclear reaction study of rare radioactive isotopes (RIs). Afew RIs, less than 10 in number, accumulate in RUNBA,
which is equipped with an internal target, and hit thetarget many times until a nuclear reaction occurs. Thisis the beam recycling technique, which is established bycompensating the energy loss U , energy straggling, andangular struggling given at a target. The energy loss
can be compensated for by a radio-frequency (RF) cavity. Specially designed devices are, however, required
to compensate for, turn by turn, the growth in energy
spread and emittance due to the straggling. We will install an energy dispersion corrector (EDC) in RUNBA tocompensate for energy-spread growth. Assuming a thinfoil internal target, we form a sinusoidal single-wave signal from the timing signal created when the beam passes
through the target. This correction signal is applied toan acceleration gap in EDC in synchronization with thebeam arrival. In this report, we analytically show therequired characteristics of EDC.
The invariant of the longitudinal particle motion in(δφ, δ) phase space is given as2Q(δφ, δ) =1 ηωrfqVrf ηωrf
δ2 +22βAβ 2 ET{cos(φs +δφ)−cos(φs )+δφ sin(φs )},(1)

where E is the energy of ions, δ = δE/E, φs a synchronous phase; A the mass number, q the charge, β thevelocity, T the revolution time, η the slipping factor, andVrf and ωrf are the amplitude and angular frequency ofRF, respectively. The RF phase is shifted by Δφt andΔφe due to the energy straggling Δt at the target andthe energy correction Δe given at EDC, respectively.
Assuming Δe is proportional to a time-of-ﬂight diﬀerence in the design value from the target to EDC, it isexpressed with a proportional coeﬃcient λe asΔe = Kte (λe ) δ + Δt + Kδ (λe )δt ,(2)
2+1 Uqλee ηte Tte
where Kte (λe ) = qλAβ
1 + γγ 2 −12EE ; Kδ (λe ) = AE ;
ηte and Tte are the partial slipping factor and time ofﬁght between the target and EDC, respectively; and δtis the uncertainty in timing measurement. Since Δt andδt are stochastic variables, their probability density distributions are assumed to be Gaussian and denoted byψ(Δt ) and θ(δt ), with standard deviations of σt and στ ,respectively. The averaged time derivative of Q(δφ, δ)is calculated as dQ 
T = Q(δφ + Δφt + Δφe , δ + Δt + Δe )dt∗1∗2Fig. 1. (a) Time derivative of the invariant Q at λe = 40 V/ns
calculated using Eq. (3) and (b) Γ (λe ) calculated usingEq. (4) for several cases of uncertainty στ .

− Q(δφ, δ) ψ(Δt )dΔt θ(δt )dδt .

(3)Since this is a function of the position in phase space(δφ, δ) and the coeﬃcient λe , we take its average withina separatrix as dQ 1T dδφdδ.
(4)
Γ (λe ) =S S dt
Γ is a function of λe only. If Γ takes negative values,
the invariant Q(δφ, δ) decreases with time on average
and, consequently, the beam continues to circulate thering. Here we omit the explicit expression of Eqs. (3)and (4) because of the limited text space.
We assume in the following calculation that the beamin RUNBA is a 10 MeV/nucleon 12 C6+ beam and thetarget is a thin 12 C foil with a thickness of 1018 /cm2 .
Figure 1(a) shows the time-derivative distribution in longitudinal phase space calculated using Eq. (3) at λe =40 V/ns. This has a quadric-like surface and is positivein the small δ region but negative in the large δ region.
The averaged time derivative calculated using Eq. (4) isshown in Fig. 1(b) for several cases of the uncertaintyστ . λe has two solutions for each case: one indicates theminimum required correction, and the other indicatesthe maximum limit to avoid the invariant growth due toovercorrection. Although the λe value can be chosen in
between the two solutions, the ﬂexibility is reduced as
uncertainty increases. Practically, the smallest possible
value is the most convenient because we can reduce thesignal power for EDC. The inset in Fig. 1(b) shows an
expanded view around the lower limit of λe . We foundthat this is insensitive to the uncertainty and the minimum required correction coeﬃcient is λe ∼40 V/ns.
Reference1) M. Wakasugi et al., RIKEN Accel. Prog. Rep. 54, 87(2021).

Institute for Chemical Research, Kyoto University
RIKEN Nishina Center- 79 -1