Characterization of beam brightness

In particle accelerators, the average beam brightness \(\mathop{B}\limits^{-}\) is defined as the beam current, I, passing through a transverse phase-space volume \({{\mathscr{V}}}_{4}\) (ref. 51)

$$\mathop{B}\limits^{-}=\frac{I}{{{\mathscr{V}}}_{4}}$$ (1)

The normalized r.m.s. emittance is often used as an indicator of the phase-space volume occupied by the beam and is given by29

$${\varepsilon }_{\perp }=\frac{\sqrt[4]{|{V}|}}{{m}_{\mu }c}$$ (2)

where m μ is the muon mass and |V| is the determinant of the covariance matrix of the beam in the transverse phase space u = (x, p x , y, p y ). The covariance matrix has elements \({v}_{ij}=\langle {u}_{i}{u}_{j}\rangle -\langle {u}_{i}\rangle \langle {u}_{j}\rangle \). The distribution of individual particle amplitudes also describes the volume of the beam in phase space.

The amplitude is defined by30

$${A}_{\perp }={\varepsilon }_{\perp }{R}^{2}({\bf{u}},\langle {\bf{u}}\rangle )$$ (3)

where R2(u, v) is the square of the distance between two points, u and v, in the phase space, normalized to the covariance matrix:

$${R}^{2}({\bf{u}},{\bf{v}})={({\bf{u}}-{\bf{v}})}^{{\rm{T}}}{V}^{-1}({\bf{u}}-{\bf{v}})$$ (4)

The normalized r.m.s. emittance is proportional to the mean of the particle amplitude distribution. In the approximation that particles travel near the beam axis, and in the absence of cooling, the particle amplitudes and the normalized r.m.s. emittance are conserved quantities. If the beam is well described by a multivariate Gaussian distribution, then R2 is distributed according to a χ2 distribution with four degrees of freedom, so the amplitudes are distributed according to

$$f({A}_{\perp })=\frac{{A}_{\perp }}{4{\varepsilon }_{\perp }^{2}}\exp \left(\frac{-{A}_{\perp }}{2{\varepsilon }_{\perp }}\right)$$ (5)

The rate of change of the normalized transverse emittance as the beam passes through an absorber is given approximately by8,29,31

$$\frac{{\rm{d}}{\varepsilon }_{\perp }}{{\rm{d}}z}\approx -\frac{{\varepsilon }_{\perp }}{{\beta }^{2}{E}_{\mu }}|\frac{{\rm{d}}{E}_{\mu }}{{\rm{d}}z}|+\frac{{\beta }_{\perp }{(13.6{\rm{M}}{\rm{e}}{\rm{V}}{c}^{-1})}^{2}}{2{\beta }^{3}{E}_{\mu }{m}_{\mu }{X}_{0}}$$ (6)

where βc is the muon velocity, E μ is the muon energy, |dE μ /dz| is the mean energy loss per unit path length, X 0 is the radiation length of the absorber and β ⊥ is the transverse betatron function at the absorber29. The first term of this equation describes ‘cooling’ by ionization energy loss and the second term describes ‘heating’ by multiple Coulomb scattering. Equation (6) implies that there is an equilibrium emittance for which the emittance change is zero.

If the beam is well described by a multivariate Gaussian distribution both before and after cooling, then the downstream and upstream amplitude distributions f d(A ⊥ ) and f u(A ⊥ ) are related to the downstream and upstream emittances \({\varepsilon }_{\perp }^{{\rm{d}}}\) and \({\varepsilon }_{\perp }^{{\rm{u}}}\) by

$$\frac{{f}^{{\rm{d}}}({A}_{\perp })}{{f}^{{\rm{u}}}({A}_{\perp })}={\left(\frac{{\varepsilon }_{\perp }^{{\rm{u}}}}{{\varepsilon }_{\perp }^{{\rm{d}}}}\right)}^{2}\exp \left[-\frac{{A}_{\perp }}{2}\left(\frac{1}{{\varepsilon }_{\perp }^{{\rm{d}}}}-\frac{1}{{\varepsilon }_{\perp }^{{\rm{u}}}}\right)\right]$$ (7)

In the experiment described in this paper, many particles do not travel near the beam axis. These particles experience effects from optical aberrations, as well as geometrical effects such as scraping, in which high-amplitude particles outside the experiment’s aperture are removed from the beam. Scraping reduces the emittance of the ensemble and selectively removes those particles that scatter more than the rest of the ensemble. Optical aberrations and scraping introduce a bias in the change in r.m.s. emittance that occurs because of ionization cooling. In this work the distribution of amplitudes is studied. To expose the behaviour in the beam core, independently of aberrations affecting the beam tail, V and ε ⊥ are recalculated for each amplitude bin, including particles that are in lower-amplitude bins and excluding particles that are in higher-amplitude bins. This results in a distribution that, in the core of the beam, is independent of scraping effects and spherical aberrations.

The change in phase-space density provides a direct measurement of the cooling effect. The k-nearest-neighbour algorithm provides a robust non-parametric estimator of the phase-space density of the muon ensemble32,34,52. The separation of pairs of muons is characterized by the normalized squared distance, \({R}_{ij}^{2}({{\bf{u}}}_{i},{{\bf{u}}}_{j})\), between muons with positions u i and u j . A volume \({{\mathscr{V}}}_{ik}\) is associated with each particle, which corresponds to the hypersphere that is centred on u i and intersects the kth nearest particle (that is, the particle that has the kth smallest R ij ). The density, ρ i , associated with the ith particle is estimated by

$${\rho }_{i}({{\bf{u}}}_{i})=\frac{k}{n{|V|}^{1/2}}\frac{1}{{{\mathscr{V}}}_{ik}}=\frac{2k}{n{{\rm{\pi }}}^{2}{|V|}^{1/2}}\frac{1}{{R}_{ik}^{4}}$$ (8)

where n is the number of particles in the ensemble. An optimal value for k is used, \(k={n}^{4/(4+d)}=\sqrt{n}\), with phase-space dimension d = 4 (ref. 32).

Data taking and reconstruction

Data were buffered in the front-end electronics and read out after each target actuation. Data storage was triggered by a coincidence of signals in the photomultiplier tubes (PMTs) serving a single scintillator slab in the upstream TOF station closest to the cooling channel (TOF1). The data recorded in response to a particular trigger are referred to as a ‘particle event’.

Each TOF station was composed of a number of scintillator slabs that were read out using a pair of PMTs, one mounted at each end of each slab. The reconstruction of the data began with the search for coincidences in the signals from the two PMTs serving any one slab in a TOF plane. Such coincidences are referred to as ‘slab hits’. ‘Space points’ were then formed from the intersection of slab hits in the x and y projections of each TOF station separately. The position and time at which a particle giving rise to the space point crossed the TOF station were then calculated using the slab position and the times measured in each of the PMTs. The relative timing of the two upstream TOF stations (TOF0 and TOF1) was calibrated relative to the measured time taken for electrons to pass between the two TOF detectors, on the assumption that they travelled at the speed of light.

Signals in the tracker readout were collected to reconstruct the helical trajectories (‘tracks’) of charged particles in the upstream and downstream trackers (TKU and TKD, respectively). Multiple Coulomb scattering introduced significant uncertainties in the reconstruction of the helical trajectory of tracks with a bending radius of less than 5 mm. For this class of track, the momentum was deduced by combining the tracker measurement with the measurements from nearby detectors. The track-fitting quality was characterized by the χ2 per degree of freedom

$${\chi }_{{\rm{df}}}^{2}=\frac{1}{n}\sum _{i}\frac{{\rm{\delta }}{x}_{i}^{2}}{{\sigma }_{i}^{2}}$$ (9)

where δx i is the distance between the fitted track and the measured signal in the ith tracker plane, σ i is the resolution of the position measurement in the tracker planes and n is the number of planes that had a signal used in the track reconstruction. Further details of the reconstruction and simulation may be found in ref. 50.

Beam selection

Measurements made in the instrumentation upstream of the absorber were used to select the input beam. The input beam (the upstream sample) was composed of events that satisfied the following criteria:

Exactly one space point was found in TOF0 and TOF1 and exactly one track in TKU.

The track in TKU had \({\chi }_{{\rm{df}}}^{2} < 8\) and was contained within the 150-mm fiducial radius over the full length of TKU.

The track in TKU had a reconstructed momentum in the range 135–145 MeV c −1 , corresponding to the momentum acceptance of the cooling cell.

The time-of-flight between TOF0 and TOF1 was consistent with that of a muon, given the momentum measured in TKU.

The radius at which the track in TKU passed through the diffuser was smaller than the diffuser aperture.

The beam emerging from the cooling cell (the downstream sample) was characterized using the subset of the upstream sample that satisfied the following criteria:

Exactly one track was found in TKD.

The track in TKD had \({\chi }_{{\rm{df}}}^{2} < 8\) and was contained within the 150-mm fiducial radius of TKD over the full length of the tracker.

The same sample-selection criteria were used to select events from the simulation of the experiment, which included a reconstruction of the electronics signals expected for the simulated particles.

Calculation of amplitudes

The amplitude distributions obtained from the upstream and downstream samples were corrected for the effects of the detector efficiency and resolution and for the migration of events between amplitude bins. The corrected number of events in a bin, \({N}_{i}^{{\rm{corr}}}\), was calculated from the raw number of events, \({N}_{j}^{{\rm{raw}}}\), using

$${N}_{i}^{{\rm{corr}}}={E}_{i}\sum _{j}{S}_{ij}{N}_{j}^{{\rm{raw}}}$$ (10)

where E i is the efficiency correction factor and S ij accounts for the detector resolution and event migration. E i and S ij were estimated from the simulation of the experiment. The uncorrected and corrected amplitude distributions for a particular configuration are shown in Extended Data Fig. 1. The correction is small relative to the ionization cooling effect, which is clear even in the uncorrected distributions.

It can be seen from equation (7) that in the limit of small amplitudes, and in the approximation that the beam is normally distributed in the phase-space variables, the ratio of the number of muons is equal to the ratio of the square of the emittances,

$$\mathop{\mathrm{lim}}\limits_{{A}_{\perp }\to 0}\frac{{f}^{{\rm{d}}}({A}_{\perp })}{{f}^{{\rm{u}}}({A}_{\perp })}={\left(\frac{{\varepsilon }_{\perp }^{{\rm{u}}}}{{\varepsilon }_{\perp }^{{\rm{d}}}}\right)}^{2}$$ (11)

The ratio of f d to f u in the lowest-amplitude bin of Fig. 3, which is an approximation to this ratio, is listed in Extended Data Table 1.