Effect of impedance and higher order chromaticity on the measurement of linear chromaticity

The combined effect of impedance and higher order chromaticity can act on the beam in a nontrivial manner which can cause a tune shift which depends on the relative momenta with respect to the 'on momentum' particle ({Delta}p/p). Experimentally, this tune shift affects the measurement of the linear chromaticity which is traditionally measured with a change of {Delta}p/p. The theory behind this effect will be derived in this paper. Computer simulations and experimental data from the Tevatron will be used to support the theory.


I. INTRODUCTION
The control of chromaticity in modern high energy colliders such as the LHC (Large Hadron Collider), Tevatron and RHIC (Relativistic Heavy Ion Collider) is of critical importance for maintaining beam stability and in maximizing the beam lifetime both during acceleration and at HEP (high energy physics).
In order to deliver as many collisions as possible to the experiments, losses have to be reduced as much as possible. One significant source of continuous particle loss is related to the head-tail instabilities driven by wakefields. These instabilities can be controlled by increasing the betatron tune spread, and thus Landau damping, with large chromaticities.
However, when the chromaticity is too large, the beam's betatron tune footprint can cover more resonances and thus decrease its lifetime. Clearly an understanding of the true chromaticity of the machine is critical for optimizing the integrated luminosity delivered to the experiments.
One major motivation for this paper comes from the observation that the linear chromaticity measured using the "traditional" method for uncoalesced [10] and coalesced [11] proton beam yields consistently different results by ∼ 1 unit especially in the vertical plane, i.e. the measured linear chromaticity has a dependence on bunch structure. (The traditional method referred here is the method where the RF frequency is varied, and thus the relative momenta w.r.t. the "on momentum" particle ∆p/p, and the linear chromaticity is measured from the betatron tune excursions).
In this paper we will show that the combined effect of higher order chromaticity and resistive wall impedance will cause a betatron tune shift which depends on ∆p/p. This means that the linear chromaticity when measured with the traditional method will yield an incorrect result because the betatron tune does not shift as much as expected for coalesced protons.

II. THE COMBINED EFFECT OF CHROMATICITY AND IMPEDANCE ON THE COLLECTIVE FREQUENCY
In the simplest case a single particle's transverse motion can be characterized by Here Y is the transverse position of the particle and t is the time coordinate and Qω 0 is the angular betatron frequency where ω 0 is the angular revolution frequency and Q is the betatron tune. The solution of this differential equation will give us the transverse harmonic motion of a particle oscillating at the betatron angular frequency ω β = Qω 0 .
If we now add in the effect of wakefields to the transverse motion, Eq. 1 becomes the forced simple harmonic oscillator equation [2] For a broad-band impedance, the transverse force F ⊥ is where R is the mean radius of the accelerator, q is the electronic charge, β is the relative speed of the particle w.r.t. the speed of light c.
If we assume that the solutions of Eq. 3 take the form: where n is the revolution harmonic, θ is the angle along the closed orbit of the accelerator, Y k is the amplitude of the motion for the kth particle and Ω c is the collective oscillation angular frequency of the particles, then when we substitute it into Eq. 2, it becomes Now when we divide Eq. 5 with the term on the rhs and then integrate both sides over the transverse beam distribution ρ(δ), the lhs becomes ρ(δ)y k dδ = y . Here, we have defined δ ≡ ∆p/p. The result is the dispersion relation where γmc 2 is the total energy of each particle.
If we expand the Q and ω 0 in terms of δ we obtain.
Here ξ = dQ/dδ is called the linear chromaticity, ξ ′ the second order chromaticity and η is the phase slip parameter. Linearizing the denominator and keeping only first order term in It is customary to define new variables V and U which are proportional to the real and imaginary parts of the impedance Now we can write the dispersion relation in a more compact form, where we have defined If we consider a Gaussian distribution, where δ 0 is the collective mean relative momentum, then Eq. 10 can be transformed into, and Eq. 13 has a known solution in the form of the complex error function erfc(z) = 1 − erf(z), We can calculate u and v for the Tevatron with Eq. 16 from its measured parameters.
In Fig. 1, we plot u versus v for three different δ 0 offsets which are typically used for chromaticity measurements in the Tevatron, If we now consider the effects of second order chromaticity, we can expand Eq. 11 to second order in δ to get, Now Eq. 13 becomes, where we have defined, The denominator in Eq. 17 can be factored and recast as, Using Mathematica[6] this integral can be solved, yielding When we plot u versus v for the same Im(Ω c ) = −0.031σ ω in Fig. 3, we can see a clear δ 0 dependence in the curves. Typically 2nd order chromaticity at injection in the Tevatron has been found to be between ±1000 to ±5000 units in both planes (see Fig. 2). The second order chromaticity has introduced an additional δ 0 dependence apart from the normal first and second order chromaticity effects.
This shift will impact the measured chromaticity when the traditional method is used.
This effect has been postulated in our previous paper [1] and is due to the mixing of the wakefield and the higher order chromaticity. at the Tevatron injection energy. These measurements are shown in Figure 4 and in Table. I. There is a consistent though varied depression in the chromaticity measured as one goes from uncoalesced to coalesced protons.  the "zero" crossings of the phase response matter. [7] Fig. 6 and 7 show the typical phase of the BTF measurements for three different δ 0 changes. The "×"'s mark the zero crossings where the PLL can lock to -the phase offset in the PLL electronics is chosen so that it locks to a point which is symmetric about the central dip. We define this point to be the

B. Difference in Tune shift between Coalesced and Uncoalesced Protons
Careful analysis of the BTF phase response shows that the tune measurements are different between uncoalesced and coalesced beam because of its dependence on δ 0 . This results in a difference in the measured chromaticity between these two bunch structures where we find that the chromaticity of the coalesced beam is always smaller than for uncoalesced beam.  Tables II and III. In the analysis of the data, we calculate the zero crossings using linear interpolation and  For example, when we consider measurements taken with sextupole magnet setting CYINJ = 27 (see Table. II), we obtain the measured phase response shown in Fig. 6 for uncoalesced protons and Fig. 7 for coalesced protons. In both cases, when the RF frequency is shifted by ∆f = ±40Hz, we expect δ 0 = ±2.69 × 10 −4 for f RF = 53.104 MHz and η = 0.0028 by using the formula ∆f /f RF = ηδ 0 , Let us look at the data in Table II. The tune difference between uncoalesced and coalesced beam (∆) for δ 0 = 0 is −2.0 × 10 −4 . This difference is close to what we expect from the coherent tune shift caused by the resistive wall impedance [3,4]. However, when we compare ∆ for δ 0 = ±2.69×10 −4 , we see that the difference is large enough to alter the measured linear chromaticity with the traditional method. This is the reason why the linear chromaticity for coalesced beam can be underestimated because the measurements of the zero phase crossings have been shifted from where we expect them to be. In this case we see that the measured chromaticity of the coalesced beam smaller than the coalesced beam by 0.59 units.
We repeated the experiment for chromaticity sextupole setting CXINJ = 33 units and we can see from Table III that ∆ shows the same type of tune differences between uncoalesced and coalesced beam for different δ 0 values. And again, the measured linear chromaticity for coalesced beam is smaller than uncoalesced beam by 0.87 units.
The data clearly shows that ∆ has a δ 0 dependence. Naïvely we might expect that the effect of impedance will shift the coherent tune uniformly with little or no dependence on δ 0 , i.e. the coherent betatron tune shift is effectively decoupled from δ 0 . We can explain this observation with our calculation in section II where we have shown that δ 0 is coupled to the coherent tune shift when second order chromaticity is included. We can use Eq. 21 to generate Table. IV to compare the theory to the experimental data. The results show that the predictions of our simple model match the characteristics of the experiment surprisingly well. The theory shows that there is an asymmetry in ∆ -larger shift for positive δ 0 shift than negative -and the size of ∆ for the different δ 0 's are comparable with the experimental data shown in Tables II and III.   TABLE IV: Table of  The simulation was setup to model the BTF measurement as closely as possible. The whole simulation consisted of 10 frequency sweeps of the kicker back and forth across the resonant betatron tune. Each sweep totalled about 19 × 10 3 turns. The average beam position at the pickup was then recorded for each turn. The simulation was further enabled to model an offset in energy due to a change in frequency that is done during a chromaticity measurement. The resistive wall wakefield was applied using a simple 1/ √ r model with the effects lumped into a single location in the ring. Linear and higher order chromaticity was modeled using kicks distributed around the ring. Our simulations typically ran between 3 × 10 4 to 3 × 10 5 particles.
Although part of the experiment was done using four coalesced bunches to improve the signal to noise ratio of the measurement, the primary transverse wake field effect from the resistive wall goes as z −1/2 and since the rms bunch size is 78 cm and the inter-bunch spacing is 21 buckets or 118 m, the inter-bunch effects are about a factor of 12 smaller than the intra-bunch effects. For this reason we will neglect the long range wakefield effects. We will benchmark BBSIMc against the BTF measurements to see whether we can reproduce it.
For uncoalesced protons we are able to reproduce fairly accurately the phase of the BTF. For coalesced protons, when we include resistive wall wakefields in the BBSIMc model together with second order chromaticity, we find that while the phase of the simulated BTFs begins to deviate from the uncoalesced simulations it is not an exact match to the measured BTFs. This is in part due to the fact that our model has only simulated chromatic effects to 2nd order, and recent evidence [5] suggests that the distribution in δ due to the coalescing process yields a persistent longitudinal dipole mode which distorts the longitudinal distribution unlike the smooth gaussian distribution that we have assumed. Fig. 9 shows the BBSIMc results overlaid on the measured BTFs.
From Fig. 10 we can obtain a better sense of the magnitude of this effect and its response to intensity and positive and negative 2nd order chromaticity values. We see that generally, as was observed in experiment, that coalesced measurements produce lower linear chromaticity values than uncoalesced. However the correlations between linear chromaticity and intensity and 2nd order chromaticity appears non-trivial (not simply linear).

V. CONCLUSION
We have shown that the naïve expectation that the coherent betatron tune shifts from δ 0 changes are independent of bunch structure is false. In fact, our experiments show that the amount of coherent tune shift is strongly dependent on bunch structure and therefore wakefields. When we include higher order chromaticity, we find that the net coherent tune shift has a significant dependence on ∆p/p. We have demonstrated this by using simple