Beam emittance calculation

The analytic formula of Bassetti and Erskine (3.6) requires the beam to be transversely bi-normally distributed [BE80]. Since σ_x is determined by Eq. (3.5) as the sum of two independent random distributions with rms values²ϵxand (∆p/p0)rms, coming from the betatron motion and the disper-sive motion, both of these two independent distributions must be normally distributed according to the following Theorem:

Theorem 3.1.1(Cramér’s decomposition theorem [Cra36]). Let a random variableZ be normally distributed and written as the sum Z =X+Y of two independent random variables which are not constant. Then both X andY must be normally distributed.

2In Ref. [Tit19] we present in detail the relation ofϵx to the rms value of a random distribution. The sum of both motions is a result of writing the solution of Eq. (1.24) in the form of a general solution of the homogeneous equation and of a particular solution of the energy-dependent inhomogeneous equation.

This is the converse of the well-known fact that the probability distribution of the sum of two normally distributed random variables is again normally distributed. Therefore, the analytical space charge models using Bassetti and Erskines formula together with Eq. (3.5) always make the indirect assumption that the distributions for the betatron motion and the dispersive motion, producing the two rms quantities on the right-hand side of Eq. (3.5), are normally distributed. As a consequence we require a form factor to modify the rms values in the case that our measured profiles are significantly different from a Gaussian shape.

In our situation of the PS experiment, the distribution in the longitudinal plane is nearly cir-cular (see Fig. 4.16 later): The bucket is not completely filled, hence the beam is not near the separatrix. As a result, the energy-profiles and the longitudinal profiles are in very good agreement to each other, see Fig. 3.1. Therefore we can equally well take the longitudinal profiles as reference to determine the form factor. The computation of this factor is described in Fig. 3.2, where we successively increased a fit region and computed the differences to the goal – until we arrived at a minimum.

Figure 3.1: Energy-profile (orange) versus a longitudinal profile (blue) in the PS, where we scaled the longitudinal profilen˜z to compare it with the energy profile, using the dispersion at the horizontal wirescanner. The energy profile was determined by a tomoscope application, see Chapter4for the reference.

The line density λ in a typical scenario with a bunched beam is not constant. MAD-X takes this into account by assuming a longitudinal Gaussian shape:

λ(z)∝exp (︄

−1

2(z− ⟨z⟩)² σ_z²

)︄

. (3.7)

The longitudinal beam size σ_z must be given by the user in the MAD-XRUN command in form of the variablesigma_z.

Of particular importance is that prior to the tracking, the user can set the MAD-X option

emittance_updateintrue orfalseto set the code to track in eitheradaptive or infrozen mode.

The adaptive mode determines, as often as every turn, the beam emittance and updates the beam-beam elements accordingly [KS13]. This computation is required in particular for the adaptive mode, but can also be printed to file in form of the variables EX_RMSand EY_RMS. Let us describe the computation of these emittances (for MAD-X version 5.02.07, which we were using here).

In a first step, MAD-X determines the twiss parameters α_x,y, β_x,y and γ_x,y, the coordinates x_co, (px)co, yco and (py)co of the closed orbit, the momentum deviationηˆ := ∆p/p0 and the dispersion

3.1. MAD-X 77

Figure 3.2: Determination of the form-factor for longitudinal parabolic profiles in the PS due to Thm. 3.1.1. Left: A given profile (blue) is fitted by a Gaussian curve (green) within the region indicated by the black dashed vertical lines. These lines correspond to a scaling of the rms value (indicated by the red dashed vertical lines) of the given measured profile by a cutting factor. For comparison a fitted parabolic profile is also shown (orange). Right: Due to the importance of the space charge tune shift in the region of maximal intensity, the fit difference to the measured profiles were weighted by the intensity. The plot shows the weighted error in dependency on various cutting factors. We have choosen the first minimum for this factor, indicated by the red vertical line.

termsD_x,D_px,D_y and D_py at the respective position of the current space charge node. Then the actionsJ_x and J_y of every particle, having transverse coordinates x,p_x, y and p_y, are determined by inserting the coordinates into the respective expressions for the phase space ellipses:

∀a∈ {x, px, y, py}: ad:=a−aco−Daηˆ, (3.8a) 2Jx(x, px) =γ_xx²_d+ 2αxx_d(px)d+β_x(px)²_d, (3.8b) 2Jy(y, py) =γyy_d²+ 2αyyd(py)d+βy(py)²_d. (3.8c) Note that the closed-orbita_coabove consists of its own dispersive and betatron part. As in Bassetti and Erskine’s work, one now assumes that the particle distribution is having a bi-Gaussian form with respect to the (x, px) and (y, py)-planes. Following the considerations in Ref. [Ale13], the density distributions for each direction is treated independently and hence one can focus, without loss of generality, on thex-direction:

f(x, px) = 1

2πϵ^CS_x exp^(︃−J_x(x, px) ϵ^CS_x

)︃

. (3.9)

Hereby, the emittance ϵ^CS_x can be understood as a volume measure so that the integral over the entire (x, px)-region is equal to the total number of particles: Using the transformation to angle-action variables (x, px)↦→(φx, J_x),f can be pulled back to a density distribution on Floquet-space, where the number of particles inside the region [0,2π[×[0, J˜_x[ is given by the cumulative distribution function

F(J˜_x) :=^{∫︂ 2π}

0

∫︂ J˜_x 0

f(φx, Jx)dJxdφx= 1−exp(−J˜_x/ϵ^SC_x ). (3.10)

On the other hand, the particles of the simulation correspond to a discrete distribution h with where Jx^(k) denotes the action of the particle k and θ the Heaviside-function. If these actions are ordered from smallest to largest values, the cumulative function G evaluated at Jx^(l) can be simplified toG(Jx^(l)) =l/N. Since we have only N particles in the simulation, we can not equate G(Jx^(l)) withF(Jx^(l)) directly (for example,F(Jx^(N)) would have to be 1). Instead, we can consider ϵ^CS_x as being dependent on the particle number land obtain

l/N= 1−exp(−J_x^(l)/ϵ^CS,(l)_x ), (3.12) In MAD-X version 5.02.07 used in this work, this averaging procedure is implemented in a slightly different manner via the harmonic mean:

in which a parameterα∈]0,1[ was introduced to deal with the casel=N. For further details we refer the reader to Ref. [Ale13].