Diffusion measurement from observed transverse beam echoes

We study the measurement of transverse diffusion through beam echoes. We revisit earlier observations of echoes in RHIC and apply an updated theoretical model to these measurements. We consider three possible models for the diffusion coefficient and show that only one is consistent with measured echo amplitudes and pulse widths. This model allows us to parameterize the diffusion coefficients as functions of bunch charge. We demonstrate that echoes can be used to measure diffusion much quicker than present methods and could be useful to a variety of hadron synchrotrons.


I. INTRODUCTION
Beam diffusion can lead to emittance growth, halo formation and particle loss. A standard method currently used to measure transverse diffusion requires scraping the beam with collimator jaws moved close to the beam, then retracting the jaws and waiting for the beam to diffuse to the outer position of the jaws [1][2][3][4][5]. This procedure is time consuming and the method is only applicable to storage rings where the beam circulates for times long enough to enable the measurement. Beam echoes were introduced into accelerator physics more than two decades ago [6,7] and then shown to be useful as a novel method to measure transverse diffusion [8]. A single echo observation can be done typically within a thousand turns with nonlinear tune spreads in the range 0.001 -0.01. Hence diffusion measurements with echoes would be considerably faster than the standard method and could also enable diffusion to be measured in synchrotrons where beams circulate for relatively short times.
Shortly after the introduction of the beam echo concept, longitudinal unbunched beam echoes were observed at the Fermilab Antiproton Accumulator [9] and then at the CERN SPS [10] The original motivation however had been to measure transverse diffusion from transverse echoes. In the year 2000, transverse bunched beam echoes were observed in the SPS with two consecutive dipole kicks [11] but no diffusion coefficients were extracted.
Later in 2004-2005 an extensive set of dedicated experiments was carried out at RHIC with dipole and quadrupole kicks [12] and these will be the focus in this paper. The existing model as applied to the data did not yield consistent values for the diffusion coefficients [13].
The next generation of intensity frontier hadron synchrotrons will require tight control of particle amplitude growth. At Fermilab the Integrable Optics Test Accelerator (IOTA) [14] ring is under construction where the novel concept of nonlinearly integrable lattices will be tested and could serve as a model for future synchrotrons. This ring offers the opportunity of testing a fast diffusion measurement technique which could help determine the degree of integrability (or stable motion) among different lattice models. With this motivation, we revisit the earlier RHIC measurements with an updated theoretical model to enable extraction of self-consistent diffusion coefficients. In Section II we describe the updated model, in Section III we apply this model to the RHIC data, in Section IV we consider beam related time scales and we summarize in Section V with lessons to be applied to future echo measurements.

II. ECHO PULSE WITH DIFFUSION
The basic beam echo generating mechanism is well known. If at some initial time the beam is kicked away from the central orbit, the beam centroid will decohere due to a nonlinear spread of frequencies. If subsequently a quadrupole kick is applied after the centroid response has decayed away, a diminished coherent response will reappear after a time interval equal to the delay between the dipole and quadrupole kicks. Figure 5 in Section III C shows an example of this echo formation during the measurements at RHIC.
Here we discuss the model to calculate the echo amplitude with diffusion using the same method and notation as in [15]. The phase space coordinates used x, p and action angle coordinates J, φ are related as The initial distribution is taken to be exponential in the action where J 0 = ε 0 , the initial rms emittance.
We first consider the dipole moment after a dipole kick and the general case where the dipole kicker is at a non-zero phase advance from the BPM location where the centroid is measured. Following the procedure in [15], the dipole moment after the dipole kick by an angle θ is where β K , β are the beta functions at the kicker and BPM respectively, Θ = ω ′ J 0 t with ω ′ ≡ dω/dJ the constant slope of the betatron angular frequency with action. This moment is independent of the phase advance from the kicker to the BPM. It differs from the expression in [15] only by the replacement of β by the geometric mean β G = β K β and β in the exponent replaced by β K . Following the dipole kick, the beam decoheres with the centroid amplitude decaying over a characteristic time τ D = 1/(ω ′ J 0 ), the decoherence time. At time τ ≫ τ D after the dipole kick, a single turn quadrupole kick is applied to generate the echoes, the first of which occurs around time 2τ. The echo amplitude and pulse shape is affected by the diffusive beam motion. We consider the density distribution to evolve according to the conventional form of the diffusion equation Here the diffusion coefficient D(J) has the usual dimension of [action 2 /time] and it differs from the definition of D(J) used in [8,15]. The treatment in [15] had developed the theory of the echo response to first order in the quadrupole kick strength. Since the experiments reported in [12] had observed a linear increase of the echo amplitude with quadrupole strength, this theory should suffice to discuss these experiments. We note that the theory developed earlier in [7] was nonlinear in this strength parameter. Using the method of [15], we find that the echo amplitude near time t > 2τ is where q is the dimensionless quadrupole kick strength defined as q = β Q / f , the ratio of the beta function at the quadrupole to its focal length and we defined t 3 1 = (t − τ) 3 + τ 3 . We consider the action dependent transverse angular frequency to be of the form ω(J) = ω β + ω ′ J where ω β is the angular betatron frequency and we consider the diffusion coefficient to be of the form where all coefficients D n have the same dimensions. The average dipole moment is given by where ω rev is the angular revolution frequency, Φ 0 = ω β (t − 2τ) and Φ 1 = ω ′ (t − 2τ).
Using ω ′ = (ω rev /ε)µ where µ = ν(ε) − ν β is the tune shift (from the bare tune ν β ) at an action equal to the emittance, it is convenient to define scaled diffusion coefficients d n as These coefficients d n have the dimension of time −3 . In the following we will consider specific cases of the above general form of D(J).
Different physical processes contribute to the diffusion coefficients D n . It is likely that space charge effects, beam-beam interactions (not present in the RHIC measurements discussed below) and intra-beam scattering all contribute to D 0 and higher order coefficients.
Early studies at the Tevatron at injection energy [16] with additional sextupoles as the driving nonlinearity had measured a constant D 0 term which varied with the proximity to a fifth order resonance. Measurements at the LHC at top energy during collisions showed that diffusion at the smallest amplitude measurable was finite [4], implying a non-zero D 0 .
A numerical simulation [17] showed that modulation diffusion leads to a constant diffusion term. Beam-gas scattering and noise in dipoles lead to a D 1 term while noise in quadrupoles leads to a D 2 term. There are likely other sources for these coefficients. Given that the beam is subject to multiple effects, the complete action dependence of the diffusion may be complex. Here we focus on the three simplest models with two diffusion coefficients that can be compared to measurements.
In the first case, we assume that the diffusion is of the form in this case, the dipole moment is given by The second case is the quadratic dependence model where The general time dependent form of the echo at time t = 2τ + ∆t where ∆t can have either sign is Here Erfc is the complementary error function.
The last case we consider is the linear and quadratic dependence In this case, the time dependent form of the echo at time t = 2τ + ∆t is The left plot in Fig. 1 shows the relative echo amplitude as a function of the diffusion coefficient D n for three values of n. In each case, only the single D n was non-zero.

A. Optimum tune shift and delay time
Analytical results for the optimum values of the tune shift and delay parameters that maximize the echo amplitude can be obtained for model 1 with diffusion coefficients . As a function of the time delay, this amplitude has a maximum at a delay τ = τ opt , such that the two coefficients can be related as It is understood that µ is held fixed at µ f ix while finding the optimum delay τ opt . Defining c τ = µ 2 f ix τ 3 opt and substituting this into the equation for the relative amplitude, we have for the maximum amplitude obtained at the delay τ opt This equation can be solved for d 0 and subsequently d 1 can be found. Positivity of d 1 requires that the solution for d 0 obey 3d 0 c τ ≤ 1.
Similarly, as a function of the tune shift, the amplitude has a maximum at µ = µ opt such that Here τ is held fixed at τ f ix while finding the optimum in µ. Defining c µ = µ 2 opt τ 3 f ix and again, substituting for d 1 , we can write the maximum relative amplitude at µ opt as Here d 1 ≥ 0 requires that the solution for d 0 obey 2d 0 c µ ≤ 1.
If both µ opt and τ opt are measured, then the diffusion coefficient d 0 can be found from equating the two expressions for d 1 which results in a quadratic equation for d 0 with the Once d 0 is determined, d 1 can be determined from either of Equations (18)  For the (d 0 , d 2 ) or (d 1 , d 2 ) models discussed here, the optimum values of the tune shift and delay parameters must be found numerically.

B. Echo pulse width
In addition to the amplitude, the echo can also be characterized by the echo pulse width, e.g the full width at half maximum (FWHM) can be chosen as a width measure.
For the model D(J) = D 0 + D 1 (J/J 0 ), the FWHM can be found analytically from Eq.
(11). We define a variable D up which depends on a upper limit to the pulse full width (∆t) up FW and other parameters as follows For example, with an upper limit to the pulse width of a 100 turns, we have D up = 2.6 × 10 −12 m 2 /s. For pulse widths ∆t FW HM < (∆t) up FW such that (D 0 /D up , D 1 /D up ) ≪ 1, we can keep terms to first order in D 0 /D up , D 1 /D up , and we find for the FWHM As we see later, we have typically (D 0 /D up , D 1 /D up ) ≈ 0.1, so the above assumption is satisfied for pulse widths up to a 100 turns or somewhat larger. We find that the FWHM increases with increasing D 1 but very slowly with D 0 as seen in Fig. 2. When there is no diffusion, we have for the minimum FWHM In units of turns, this theoretical minimum FWHM depends only on the tune shift coefficient µ. This value when compared with measured FWHM values can set limits on the tune shift parameter, as will be seen later.
For the other models with either (D 0 , D 2 ) or (D 1 , D 2 ), the time dependent pulse shape and hence the FWHM must be found numerically. From this pulse shape, the upper envelope is found numerically as an interpolating function and the FWHM then calculated from this envelope function.

III. ANALYSIS OF RHIC DATA WITH AU IONS
We briefly discuss the experimental procedure here, more details can be found in [12].
The echo experiments were first done with Au ions, later with Cu ions and also with protons, all at injection energy. A special purpose quadrupole kicker was used with a rise time   Table I shows some of the relevant parameters for the Au ions [12].

A. Emittance growth and rescaling tune shift
In evaluating the tune shift parameter µ for calculating echo amplitudes, it is important to use the emittance following the dipole kick. The rms emittance is given by The ensemble averages are calculated using the distribution function at time t after the dipole kick which can be written in the notation of [15] as and the averages are found from e.g. J cos 2 φ = dJdφ J cos 2 φ ψ 2 (J, φ ,t) etc. It can be shown this leads to an rms emittance given by At times t ≫ τ D , the term A 2 → 0 and we can approximate where ε 0 = J 0 is the initial emittance, ∆x = β K β θ is the change in beam position at the BPM and σ 0 = β ε 0 is the initial beam size at the BPM. The last expression in Eq.
(29) has the same form as in [18]. Thus a kick to a 3σ amplitude results in an emittance which is 5.5 times larger than the initial emittance. We will take this as an average estimate for the emittance following the dipole kick. By definition, the tune shift parameter µ increases linearly with emittance and hence µ increases from its nominal value of 0.0014 to 0.0077 following the dipole kick. Without this rescaling, the model cannot agree with the experimental results, as seen in the earlier analysis [12,13].

B. Diffusion coefficients from optimum tune shift and delay
The theory predicts that the maximum echo amplitude, which occurs close to the time 2τ after the dipole kick, grows indefinitely with the product µτ in the absence of diffusion. In the presence of any diffusion, the echo amplitude grows more slowly, reaches a maximum and then decreases as either µ or τ is increased. In each case, the irreversible particle motion caused by the presence of diffusion reduces the amplitude of the recohering signal at the time of the echo. Here we will apply the formulas developed in Section II A to extract diffusion coefficients from measurements of the optimal tune shift and optimal delay.
We discuss first the analysis of the nonlinear tune shift scan done on March 11, 2004. For the (D 0 , D 1 ) model, we start by solving Eq. (18) and Eq. (19) for the coefficients from the echo amplitude and the value of the optimum delay τ opt . Again, better fits to the data are obtained by a least square minimization which is also the procedure for the other two models. Table II shows the best fit values with this delay scan. Compared to the values from the tune shift scan, the coefficients for the same model are within a factor of two from this delay scan. Some of the variation in the values between the scans can be due to different beam conditions on the two days such as bunch intensities and machine tunes.
However the uncertainties associated with these values are large since there were too few data points. Fig. 4 shows the comparison of the fitted models with the data. Again all three

C. Diffusion coefficients from the echo amplitude and the FWHM
The above analysis has shown that all three models are viable candidates in describing the data dependence on either the tune shift or the delay. We now use turn by turn (TBT) data to fit both the echo amplitude and the echo pulse width with each model. Ten such data sets could be retrieved from the 2004 measurements. In this TBT set, the initial dipole kick and bunch charge varied but the other parameters including the quadrupole kick strength, tunes, delay and octupole strengths were kept constant. Figures 5 and 6 show two examples from this set, one with a clean echo pulse and the other where the beam centroid takes a longer time to decohere after the initial kick and the echo pulse is also much wider. Some of the more distorted signals could be due to oscillations from off-axis injection and could partly be due to a fourth order resonance and slightly higher bunch charge. For each data set, an interpolating function was found to fit the upper envelope of the echo pulse and the FWHM was extracted from this interpolating function. Using the value of the rescaled tune shift parameter µ = 0.077, the minimum theoretical value of the FWHM without We now solve for two diffusion coefficients using the relative echo amplitude and the FWHM. For the (D 0 , D 1 ) model, the FWHM can be found analytically, as shown in Eq.
(24). The d 0 coefficient can be written as a function of the echo amplitude and d 1 using the echo amplitude equation Eq.(11) as where x amp rel = x amp /(β K θ ) is the relative echo amplitude in terms of the dipole kick and N delay = τ/T rev is the delay in units of turns. The positivity of d 0 implies an upper limit to d 1 as The value of d 1 can be found by numerically solving the equation Eq. (24) for the FWHM with d 0 substituted from Eq. (31). We find that this (D 0 , D 1 ) model yields positive d 0 coefficients in only four of the ten cases. We conclude therefore that the D 0 , D 1 model is not well suited for this data.
With the (D 0 , D 2 ) model,the d 0 coefficient can again be found analytically as a function of the echo amplitude and d 1 using where H 02 is defined in Eq. (13). We find again that no solutions with positive D 0 can be found in all cases with FWHM > 70 turns. Even in other cases where the solutions can be found, the values of D 2 are significantly larger than the values found in the previous sections, hence appear to be in a disconnected region of the parameter space. Since D 0 has little impact on the FWHM (see Fig. 2 It turns out to be easier to do a least squared minimization to find the solution. Here we define the χ 2 function as where ampl(d 1 , d 2 ) and fwhm (d 1 , d 2 ) are the amplitude function (from Eq. (16) ) and the FWHM function defined numerically and σ ampl = 0.05 and σ f whm = 2 are the estimated uncertainties in the two data variables. This least squares method turns out to be efficient and leads to positive solutions for d 1 , d 2 in all cases. Table III shows the values of the diffusion coefficients in these cases. We observe that these values are close to the values of D 1 found from the optimal tune shift and delay measurements shown in Table II tune shift scan methods for the amplitude are less sensitive to D 2 and also from the larger number of data points in the FWHM analysis, we expect the values in Table III to be more accurate. In most cases, the D 1 coefficient is an order of magnitude greater than D 2 . The single exception (row 2 of this table) corresponds to the outlier point labeled 1 in Fig. 7.
As a function of charge, D 1 increases while D 2 appears to be independent of the charge.

D. Diffusion dependence on bunch charge
We focus now on the (D 1 , D 2 ) model which is the only one of those studied that can de-  Fig. 9 shows the measured echo amplitudes (in red) as a function of the number of particles per bunch and also the calculated amplitude (in blue) from these fits for (D 1 , D 2 ). The measured echo amplitude decreases with increasing charge, and this trend is well reproduced by the theoretical amplitude function. This is a consistency check and is to be expected, since the linear fit for d 1 and constant for d 2 were obtained from the data set. The comparison in Fig. 9 shows that we can parameterize the diffusion coefficients as diffusion for heavy ions such as Au in RHIC, at injection energy. The incoherent space charge and IBS induced diffusion and emittance growth depends linearly on the charge and our analysis confirms that the leading diffusion coefficient D 1 increases linearly with charge. The coefficient D 2 is likely to be determined by diffusion from single particle nonlinear dynamics processes.
In the above analysis we have neglected the effect of wakefields on the echo formation.
Their impact on the calculations above is not likely to be significant. As seen in Figures 5   and 6 and generally true for the available turn by turn data, the centroid response after the dipole kick is cleaner and the relative echo amplitude is larger with the larger amplitude kick. This would likely not be the case if the effects of the transverse wake were significant.
Instead, effects due to injection oscillations and fourth order resonance which shows up at intermediate amplitudes are the likely reason for the response seen in Fig. 6. In addition, the effect of wake fields would be visible in a change in the decoherence time with intensity.
An analysis of the intensity scan data shows no correlation between the decoherence time and the bunch intensity.

IV. MEAN ESCAPE TIME
One useful time scale that can be extracted from the diffusion coefficients is the mean escape time t esc associated with probabilistic processes [20]. This time, also known as the mean first passage time, is the mean time taken (averaging over many realizations of the process) for a particle to escape from a certain region defined by a boundary. It was shown in [21] that in the case that D(J) = D 1 (J/J 0 ), the time dependent density distribution solution ψ(J,t) to the diffusion equation leads to a beam lifetime t L which is close to the escape time t esc estimate. Defining t L = −N(t)/(dN/dt) where N(t) = ψ(J,t)dJ is the particle number, it was shown that where J A is the action at the absorbing boundary. We will assume that the mean escape time is also a useful beam relevant time scale when D(J) = D 1 (J/J 0 ) + D 2 (J/J 0 ) 2 . The mean escape time from an action J to an absorbing boundary at action J A is given by This is the mean escape time for particles initially at a single action J to reach the aperture at action J A due to diffusion. A parameter describing the escape time for the beam can be obtained by averaging this over the initial beam distribution ψ 0 (J), which yields where Γ(0, z) is the incomplete Gamma function and we have assumed D 2 = 0. The dimensionless amplifying factor A F , defined by the terms in square brackets, depends only the ratios D 1 /D 2 , J A /J 0 , Figure 10 shows the dependence of the dimensionless terms on  Table   III, we have t esc ≈ 1s. While this time is extremely short, it corresponds to the lifetime of a beam at large amplitudes and not to a beam circulating on the nominal closed orbit.
Observations in RHIC did show that lifetimes of kicked beams were significantly smaller compared to that for beams not kicked. However the early losses of the kicked beams were dominated by scraping at aperture restrictions, so there is no straightforward way to determine the contribution of diffusion to those lifetimes. Nevertheless, the diffusion coefficients and the associated time scales should be useful for relative measures of beam growth and particle loss. As an example, it could be useful in IOTA to quickly distinguish between lattices with different degrees of integrability. If echoes can be generated by small amplitude kicks, then the calculated diffusion coefficients and the time scales would be more representative of beam behavior under nominal conditions. Determining the diffusion coefficients may require different parameterizations of D(J) at small and large amplitudes, as seen for example in [16].

V. SUMMARY
In this article, we revisited earlier observations of transverse beam echoes in RHIC to extract diffusion coefficients from those measurements. We considered three models for the action dependence of the diffusion coefficients: D(J) = D 0 + D 1 (J/J 0 ), D(J) = D 0 + D 2 (J/J 0 ) 2 , and D(J) = D 1 (J/J 0 ) + D 2 (J/J 0 ) 2 . All three models were found to adequately describe the echo amplitudes measured during scans of the nonlinear tune shift and the delay between the dipole and quadrupole kicks. Next, turn by turn data was used to extract both the amplitude and the FWHM of the pulse width. Here both models with D 0 do not describe the data with larger pulse widths, so the only model that successfully describes both the amplitude and the FWHM data is the (D 1 , D 2 ) model. We find that D 1 is an order of magnitude larger than D 2 in most cases; it increases linearly with the charge while D 2 is nearly independent of the charge. Using these charge dependencies, the (D 1 , D 2 ) model also adequately describes another set of data where the echo amplitudes were measured as a function of charge.
These results show that transverse echoes can indeed be used to measure transverse beam diffusion in existing and future hadron synchrotrons, We make some observations on requirements for future measurements. The diffusion measurements require good control of several machine and beam parameters such as the initial dipole kick, the quadrupole kick, machine nonlinearity, tunes and beam emittance, to name the most important. Injection oscillations can strongly influence the echo amplitude and pulse shape, so these need to be controlled to the extent possible. Alternatively if available, a fast dipole kicker in the ring would be preferable to initiate the echo. In such a case, a transverse damper can damp initial oscillations and then be turned off before the dipole kicker is used. While the echo amplitude variation with scans of the tune shift and time delay are useful, detailed analysis of the turn by turn data yields more information. As an example of this, we found that the FWHM scales quadratically with the charge and therefore is more sensitive to intensity changes than the echo amplitude. The proximity of resonances can also spoil echoes so the tunes and the dipole kick amplitudes need to be chosen carefully as well.