Analytic approximation of the head-tail phase difference from continuous transverse excitation for measuring chromaticity

We will explore a method for measuring chromaticity by continuously kicking the beam transversely. This is called the continuous head-tail method for measuring chromaticity. The complete analytic approximation in terms of trigonometric functions is derived for zero transverse emittance beam. A simple formula for calculating chromaticity from experimental data is also shown. Finally, the theory is compared with experimental data.


I. INTRODUCTION
Continuous chromaticity control up the ramp and through the squeeze will be vital for operating the Large Hadron Collider (CERN-LHC). It is especially important at the beginning of the ramp because of snapback where chromaticity swings are expected to drift at a rate of 0:33 s ÿ1 even with preprogrammed chromaticity correction tables [1]. Furthermore, the tolerance required for nominal 7 TeV head-on collisions operations is 1 unit. It is because of these tight tolerances that a chromaticity feedback system has been envisioned for the LHC. However, the best method for measuring chromaticity which has a measurement rate better than 1 Hz has not yet been pinned down. During the 2006 LHC Tune Feedback Review as part of the U.S. LHC Accelerator Research Program (US-LARP), it was pointed out that there are at least five different methods for doing this. One method, however, stirred up considerable interest: the continuous head-tail measurement technique.
To start off the discussion, we will look first at the traditional head-tail method [2]. In this method, the beam is kicked once transversely and, because of chromaticity, the head and the tail dephase differently. When the phase difference between the head and the tail is measured, it can be shown that, when the phases are maximally different, this difference is linearly related to the chromaticity. However, this technique, although proven to work at both CERN and Fermilab, suffers from at least three deficiencies: (i) It causes large emittance growth because of the large kicks. (ii) The chromaticity cannot be continuously measured because after a small number of kicks ( < 5), there is so much beam loss that the signal to noise (S/N) becomes really poor. (iii) It is not compatible with the phase locked loop (PLL) tune tracking method.
Some preliminary measurements at the Relativistic Heavy Ion Collider (BNL-RHIC) with a special detector called the [3] direct diode detector baseband tune (3D-BBQ) have shown a phase difference between the head and the tail which changed with chromaticity. Later work at the Super Proton Synchrotron (CERN-SPS) performed by the authors and the CERN beam instrumentation group also measured a phase difference in a few data sets, see Fig. 1. We believe that those that did not were due to saturation in the 3D-BBQ electronics because, although the 3D-BBQ electronics is extremely sensitive, it has been demonstrated at both RHIC and the Tevatron to be prone to saturation [4,5]. With these ambiguous results, it begged a theoretical understanding of whether the phase difference was real or not and thus this paper. If this method can be proven to work it has the following advantages: (i) There is already a continuous transverse kick from the tune tracker PLL. It has already been demonstrated at Fermilab, RHIC, and SPS that these small kicks 1 nrad do not blow up the emittance or cause beam lifetime problems. (ii) No extra modulations are required for the chromaticity measure- ment. For example, the traditional method for measuring chromaticity requires changing the RF frequency to change the momentum of the beam. It is compatible with the tune tracker PLL.
Therefore, the goal of this paper is to answer the following two questions: (i) Is there a phase difference between the head and tail when the bunch is continuously kicked transversely? (ii) Is there a formula which connects chromaticity to this phase difference if the first item is true?

II. THEORY
The way in which we will find the analytic solution for the phase difference between the head and the tail of the bunch is the following: (i) We write down the ordinary differential equation (o.d.e.) which describes the dynamics of a single particle. (ii) We transform to a rotating frame which simplifies the o.d.e. In this frame, we find that it is a simple harmonic oscillator with two forcing terms: one from the transverse kick of the tune tracker PLL and the other from the radio frequency (RF) (which is the source of the synchrotron frequency). (iii) The method of averaging is used to solve the o.d.e. for two cases: (a) when the transverse kick is on the betatron frequency and (b) when the transverse kick is close to the betatron frequency. (iv) The solution in the rotating frame is transformed back to the laboratory frame. (v) The phase difference between the head and tail is calculated. (vi) The particles in phase space are projected down from momentum space onto position space so as to reflect what is seen at the output of the beam position monitors.

A. The differential equation
We can write down the transverse equation of motion for a single particle as Here X is the transverse position of the particles, s is the longitudinal coordinate, and ! Q is the betatron frequency and c is the speed of light. However, if the particle resides in an RF bucket, we must consider its longitudinal motion inside the bucket and so the equation of motion becomes Here z defines the longitudinal position relative to the center of the RF bucket and is the relative momentum difference from the ''on momentum'' particle. If we expand the betatron frequency to first order in , we obtain where ! 0 is the revolution frequency, Q ! Q =! 0 is the betatron tune, and is the linear chromaticity. Therefore Eq. (2) becomes d 2 Xs; ; z ds 2 ! 2 0 Q 2 =c 2 Xs; ; z 0: We will approximate the longitudinal motion inside the RF bucket with s ÿ ! s c r sin ! s s c zs r cos ! s s c ; where ! s is the synchrotron frequency, r; is the position of the particle at s=c 0, and is the slip factor. We can write down a linear transformation with maps z0; 0 z 0 ; 0 to zs; s: We can also think of the map given by Eq. (6) as a frame that is rotating at the synchrotron frequency. Thus in this frame, Eq. (4) becomes where 0 z 0 =c and we have mapped X ! x to remind us that we are in the rotating frame. Next, let us change variables to use turns n rather than s, i.e.
where R is the radius of the accelerator. In this variable, Eq. (7) becomes where s=c 2n=! 0 , Q s ! s =! 0 is the synchrotron tune.
If the weak sinusoidal kick from the tune tracker PLL is given by cos2Q k n where 1, Q k is the frequency of the kick in tune units and 1 has the same dimensions as x to keep the dimensions of the left-hand side and right-hand side (rhs) of the ordinary differential equation correct, then Eq. (9) with this force is x 2Q 2 0 cos2Q s n ÿ ! s 0 sin2Q s n 2 x cos2Q k n; Let us define the following new variables so that it is easier to lug Eq. (10) around: Thus Eq. (10) becomes It is clear that (12) is Hill's equation with an external periodic forcing because Wn Wn 2= s .

B. The single particle solution
The approach to solving Eq. (12) is to find the solution to the homogenous problem, then solve for the first order solutions with the method of variation of parameters and, finally, obtain the single particle solution by the method of averages.

Homogeneous solution
The o.d.e. from Eq. (12) with the rhs set to zero is y h Wn 2 y h 0; where we have introduced y h to be the solution of this equation. An approximate solution of Eq. (13) is [6] where 0 2 C is a constant determined from initial conditions. Therefore, the homogeneous solution is where 01 , 02 2 R are constants to be derived from initial conditions and 2 2 q ' arctan :

Variation of parameters
The two relevant equations which can be written down from Eq. (15) which are prescribed by the variation of parameters method are y z 1 ny 1 n z 2 ny 2 n _ y z 1 n _ y 1 n z 2 n _ y 2 n; where z 1 and z 2 are the equations to be solved from initial conditions. The first order differential equations which when integrated give us z 1 and z 2 are if we assume that = Q 1. To integrate the above, we will apply the method of averaging.

Averaging
Notice that the rhs of Eq. (18) is not periodic but almost periodic because of the three tunes k , Q , and s . The nonlinear averaging theorems described by Hoppensteadt [7] can be used to approximately integrate these two first order differential equations.
The definition of averaging of a function ft 1 ; t 2 ; . . . with respect to (w.r.t.) t 1 in this theorem is where f n are the Fourier coefficients for the expansion of ft 1 ; t 2 ; . . . in t 1 and ! n . In this context, n Q and n s are treated as independent variables when we perform averaging. Furthermore, we will find solutions for only j k ÿ Q j 1 because the tune tracker PLL is always assumed to lock onto the betatron tune or close to it.
Suppose the tune tracker PLL locks close or onto Q , i.e. j k ÿ Q j 1. Let us write When we substitute this into Eq. (18) we get if we assume that = s 1. Equation (22) is actually integrable and so we get As expected, as Q ! 0, cosn Q = Q ! 1 in the z 1 equation. This singularity will be removed when the initial conditions are included in the full solution [8]. The secular term arises when Q ! 0, sinn Q = Q ! n, which is exactly what we expect when the excitation is exactly on the betatron tune. Furthermore, we notice that there are extra singularities which occur when Q s .
Carrying on, we can calculate the particular integral from Eq. (17) and therefore, the complete solution to Eq. (10) when k ÿ Q Q is (24)

C. Laboratory frame
All the calculations that we have done so far are in the frame that follows the single particle. In the laboratory frame, we measure the transverse position of the beam once a turn at the longitudinal positions B while we continuously kick the beam. To arrive at an analytic solution for this paper, we will make the assumption that the beam is matched to the RF bucket, i.e., in the ; plane the particle distribution is stationary [9]. This means that the particles are dense on the contour r B =c, see where yn comes from Eq. (24). For the remainder of this paper, we will use the yn from Eq. (24). It will be more and so with the above relationship, we can writeŶ B n as It is interesting to note that the time dependent synchrotron term n s is now embedded in 01 and 02 and removed from the terms cosn Q , sinn Q , cosn Q Q , and sinn Q Q . The reason why there can be any phase difference between the head and the tail originates from here. This means that, if we had ignored the initial conditions, we would have obtained an incorrect solution.

Changing perspective
To calculate the phase difference between the head and tail, we will first change perspective from that of a fixed betatron tune Q and variable kick tune k to that of a fixed k and variable Q because ultimately this is what is done in our experiment when we use the tune tracker PLL: we measure phase w.r.t. the frequency of the kick and not the betatron frequency. In this perspective, we have from Eq. (20), and thusŶ B n is transformed to this perspective with a trivial change of variables, where we just replace Q with k ÿ Q to become where we have added k to the subscript ofŶ k;B n to remind ourselves that we are in the perspective of the kicker and the new functions k;01 and k;02 are 01 and 02 with Q ! k ÿ Q i.e.
When we look at Eqs. (31) and (33)-which are rather complicated -very carefully, we can extract out two different cases. The first is when j Q j s 1 or Q 0, i.e., the kick is very close to the betatron tune but not on a synchrotron line or on the betatron tune. The other case is when j Q j s 1, i.e. when the kick is close to or on a synchrotron line. The second case will not be considered in this paper but has been discussed elsewhere [11].

Solution when j Q j s 1 or Q 0
In this case, the dominant terms in Eq. (31) are those with coefficients which contain 1= Q . Picking these out, we have and the string ofŶ subscripts has increased with the addition of Q which reminds us that we are only looking at terms which contain 1= Q .

Averaged transverse position
To obtain an averaged transverse positionX, we have to integrateŶ over [12], see Fig. 3. Note that the distribution is time independent because we have assumed that the ; distribution is matched to the bucket and thus stationary. Let B ; be the density of particles in the ; plane which is normalized, i.e.
Therefore, the fraction of particles in the small segment is d which gives us the weight for calculating the averageX, i.e.
Thus, the averaged transverse positionX at B is which is independent of . If we define to be the phase w.r.t. the kick cosn k , then it is easy to show that arctan cos 3 2 because cot tan=2 ÿ and the ÿ is from the ''ÿ'' sign in front of .  which is astonishingly simple for all the work that we have done. However, we must again remind ourselves that Eq. (39) is only valid if Q s 1 or Q 0. Note: The head-tail phase difference has also been derived by Fartoukh [13] using a different method and agrees with our answer. We have also used numerical calculations to verify this formula as well as Eq. (29). The numerical results agree very well with the analytic solutions.

III. EXPERIMENT
In this experiment, the basic idea is for us to continuously kick the beam vertically with the AC-dipole. The phase difference between the head and tail due to this kick is then measured and plotted for different chromaticities.
The advantage of doing it this way is because the ACdipole kicks are big enough to be seen directly with an oscilloscope and so do not require any special high gain electronics. Furthermore, this method is independent of the tune tracker PLL electronics which gave ambiguous results (discussed in the Introduction). Unfortunately, the disadvantage is that the transverse emittance will blow up because the AC-dipole kicks are very close to the betatron tune.
Practically, we must first calibrate the knob T:CYINJ which we will use to change the vertical chromaticity. We do this by measuring the vertical chromaticity v for different T:CYINJ settings. The reason for using uncoalesced [14] rather than coalesced [15] beam is because it is much easier to measure chromaticity with uncoalesced beam with the traditional change of RF frequency method than with coalesced beam. The fit that we have found is From this fit, we find that the error in the value of v when we are given T:CYINJ is approximately 0:65 units at a confidence level of 95% for the range of T:CYINJ that we will be setting. See Fig. 4.
The experimental setup is shown in Fig. 5 [16]. We inject one coalesced bunch, which contains about 330 10 9 protons, to be used for the entire experiment. We set the chromaticity with T:CYINJ and then we kick the bunch transversely with the AC-dipole for a short period of time for each head-tail phase measurement. The reason for turning off the AC-dipole is to keep the vertical emittance growth to a minimum. The exact ramp waveform of the AC-dipole is shown in Fig. 5. The frequency of the kick is set to 0:6 10 ÿ3 tune units <Q s 1:7 10 ÿ3 below the vertical betatron tune which is at 0.5776. The transverse position of the head and the tail are measured with a fast sampling oscilloscope and the phase difference is calculated off-line. After each change of chromaticity, we move the vertical tune if necessary to ensure that the betatron tune remains at 0.5776. Despite having set the AC-dipole kick to its minimum value, the vertical emittance measured with the flying wires system, grows from 17 mm mrad to 36 mm mrad during the experiment. The sigma bunch length B did not change during this time and is 2.9 ns. Figure 6 shows an example of the head-tail phase difference which we have analyzed. In this example v 4 and B 0:8 ns. From here it is unambiguous that there is a phase difference between the head w.r.t. tail.
The measured phase difference for three values of B 0:4 ns, 0.8 ns, and 1.2 ns versus v are shown in Fig. 7. The red line is the expected as a function of v from Eq. (39) where we have used the Tevatron parameters ! 0 2 47:71 10 3 s ÿ1 and 0:0028. To check how good our theory fits to the data despite the nonzero vertical emittance, we have calculated 2 and the reduced 2 for these three cases which are summarized in Table I. For all three B cases, the 2 and the reduced 2 results tell us that the phase difference data does not match Eq. (39) very well and, in fact, this is rather obvious by eye from Fig. 7.
We suspect that there are nonlinear components in that are not taken care of in the theory. If we assume that the lowest power nonlinear term is quadratic in , then we can try to fit the data with this added 2 term, where a and b are fit parameters. The blue curves in Fig. 7 show the quadratic fit and Table II shows the fit parameters. It is clear from the plots that the quadratic fits are rather good despite the wildly different reduced 2 for each case.

A. Sources of the nonlinearity
Equation (39) has been derived by assuming that the RF bucket is linear and that the transverse emittance is zero. The effects of nonlinearity from transverse impedance,  second order chromaticity as well as amplitude driven effects have not been included. A full accounting of all these effects is beyond the scope of this paper and would require a complete numerical simulation where the beam is driven resonantly. However, some preliminary insight can be gained by considering a single kick computer simulation which includes the effects of transverse impedance from short range resistive wall wakefields, linear and second order chromaticity. These simulations seem to indicate that neither second order chromaticity by itself nor resistive wall impedance by itself can account for the observed phase structure. This is because the phase effects of 2nd order chromaticity by itself cancel when the phase difference between the head and the tail is calculated. The effect of the impedance of the beam pipe can introduce a nonlinear phase response which can be seen in Fig. 8. It is clear from this figure that the nonlinear effect becomes more pronounced as the effective impedance Z eff is increased from zero to 1:5 M=m at 100 MHz. Note: 1:5 M=m at 100 MHz is a realistic value for the Tevatron from previous analysis. However, resistive wall impedance by itself fails to fully capture the observed phase response [17]. When both 2nd order and resistive wall impedance effects are included, a complex interplay between these effects is observed which can be seen in Fig. 9. In this simulation three different values of second order chromaticity are used to illustrate the behavior of the head-tail phase difference as a function of linear chromaticity. For reference, the Tevatron 2nd order chromaticity is 1000. Notice that, as the second order chromaticity is increased, the maximum phase difference moves towards higher chromaticity and the curvature of the phase difference before the maxima takes on a similar look to the measured data shown in Fig. 7. However, it must be emphasized that these simulations have been performed with a single kick instead of a resonantly driven system and so a direct comparison cannot be made between these two results.

IV. CONCLUSION
We have found an analytic approximation for the weakly forced Hill's equation which describes the transverse motion of the beam in the presence of chromaticity for a single particle. We have checked that our analytic approximation matches very closely to the numerically integrated solution. From here, we have derived an analytic approximation which shows that there is a phase difference between the head and tail for a zero transverse emittance but nonzero longitudinal bunch length. A simple formula for this phase difference has been derived when the kick tune is very close to the betatron tune.
For the experiment, we have used the AC-dipole to excite the beam transversely and then measured the headtail phase of the bunch. We have shown that there is a phase difference between the head and tail and thus have removed any lingering doubts as to whether the effect is real or not. From the measured data, we have found that there are nonlinear terms which are not taken care of by the theory. There are several possible sources for these terms, with short range wakefields coupling with second order chromaticity being the best candidate. The exact nature of this coupling will have to be resolved with computer simulations in the future. order chromaticity on the measured phase difference between the head slice ( 0:4 ns from bucket center) and the tail slice ( ÿ 0:4 ns from bucket center) for a single kick. In the presence of wakefields, 2nd order chromaticity can now contribute to the phase response, and as it increases, the maximum phase difference shifts to higher chromaticity. The curvature of the phase difference before the peak has a similar flavor to the measured phase difference shown in Fig. 7.   FIG. 8. (Color) The effect of resistive wall wakefield on the measured phase difference between the head slice ( 0:4 ns from bucket center) and the tail slice ( ÿ 0:4 ns from bucket center) for a single kick.