Extension of Busch's Theorem to Particle Beams

In 1926, H. Busch formulated a theorem for one single charged particle moving along a region with a longitudinal magnetic field [H. Busch, Berechnung der Bahn von Kathodenstrahlen in axial symmetrischen electromagnetischen Felde, Z. Phys. 81 (5) p. 974, (1926)]. The theorem relates particle angular momentum to the amount of field lines being enclosed by the particle cyclotron motion. This paper extends the theorem to many particles forming a beam without cylindrical symmetry. A quantity being preserved is derived, which represents the sum of difference of eigen-emittances, magnetic flux through the beam area, and beam rms-vorticity multiplied by the magnetic flux. Tracking simulations and analytical calculations using the generalized Courant-Snyder formalism confirm the validity of the extended theorem. The new theorem has been applied for fast modelling of experiments with electron and ion beams on transverse emittance re-partitioning conducted at FERMILAB and at GSI.


BUSCH'S THEOREM FOR SINGLE PARTICLE
In 1926, H. Busch applied the preservation of angular momentum for systems with cylindrical symmetry to a charged particle moving inside a region with magnetic field ì B [1]. Using conjugated momenta, the magnetic field strength is intrinsically included into the equations of motion. In linear systems, the normalized conjugated momenta p x and p y are related to the derivatives of the particle position coordinates (x, y) w.r.t. the main longitudinal direction of motion ì s through where ì A is the magnetic vector potential with ì B = ì ∇× ì A, B s is the longitudinal component of the magnetic field, and (Bρ) is the particle rigidity, i.e., its momentum per charge p/(qe), with p as total momentum, q as charge number, and e as elementary charge. Busch's theorem states that the canonical angular momentuml = xp y − yp x is a constant of motion that is written in cylindrical coordinates as A general formulation of Eq. (3) has been derived in [2], which is regarded as the generalized Busch theorem i.e., the path integral around a stream of possible particle trajectories along a closed contour C, plus the magnetic flux through the area enclosed by C is an invariant of the motion. Figure 1 illustrates the first part of the left hand side of Eq. (4). Busch's theorem of Eq. (3) is the special case of this generalized form for C being a circle of radius r. It is emphasized that Eq. (4) is only valid for laminar motion.

TRANSVERSE EIGEN EMITTANCES
The transverse eigen emittances were introduced by Dragt in 1992 [4] as the two transverse rms-emittances the beam acquires after all inter-plane correlations have been removed by appropriate linear beam line elements. They are calculated from the 2 nd -moments beam matrix together with as [5] The product of the two eigen emittances is equal to the 4d rms-emittanceε 4d being in turn equal to the square root of the determinant ofC. By definitionε 1/2/4d are preserved by linear beam line elements as drifts, quadrupoles, dipoles, and solenoids even if they are tilted.

PRESERVATION OF EIGEN EMITTANCES
Busch's theorem was obtained from expressing preservation of axial angular momentum using conjugate momenta. Its extension to beams is achieved in analogue way by ex- TUPO078 pressing the preservation of the sum using conjugate momenta as shown in detail in [3]. Substitution of the conjugate momenta by Eqs. 1 and 2 leads to the following expression including just laboratory coordinates that can be applied directly to emittance manipulation scenarios where A := x 2 y 2 − x y 2 is the rms-area of the beam divided by π. The last summand (abbreviated as 2 B s (Bρ) W A ) is invariant under rotation around the beam axis and it vanishes for uncorrelated beams. As shown in [6] Eq. (9) is equivalent to Emittance reduction through acceleration can be included by multiplying Eqs. (1) and (2) initially by p = mγ βc, where β is the longitudinal particle velocity normalized to the velocity of light c. The extension of Busch's theorem to beams including acceleration is where ψ is the magnetic flux through the beam rms-area A. This expression can be further compressed by using the mechanical canonical momentum ì P c = ì P + eq ì A with ì P := mγì v and the canonical vorticity ì

FLAT BEAM EXPERIMENT AT FERMILAB
At FERMILAB's NICADD photoinjector, flat electron beams were formed by first producing the beams at the surface of a photo cathode placed inside an rf-gun to which longitudinal magnetic field B s = B 0 was imposed [7]. The schematic beam line is sketched in Fig. 2. Along the subsequent region with B s = 0, the beam was accelerated to 16 MeV. Finally, correlations initially imposed by the magnetic exit fringe field of the rf-gun were removed by three skew quadrupole magnets. Equation (11) equalizes the situation at the cathode surface at the left-hand side to the situation of the finally flat beam on the right-hand side (q = 1) where A 0 is the beam rms-area at the cathode surface. The authors of [7] used the definitions [8] (ε u n ) 2 := ε n f 1 · ε n f 2 (14) resulting in of which only the upper sign gives a meaningful positive result. Re-plugging this expression for ε n f 1 into Eq. (13) leads to being identical to their original expression (Eq. (1) of [7]).

TRANSVERSE EMITTANCE TRANSFER EXPERIMENT AT GSI
At GSI, the EMittance Transfer EXperiment (EMTEX) transferred emittance from one transverse plane into the other one by passing the beam through a short solenoid [5,9]. The schematic EMTEX beam line is depicted in Fig. 3. In the solenoid center, the ions charge state, i.e., their rigidity was changed by placing a thin carbon foil therein from 14 N 3+ to 14 N 7+ . Charge state stripping is a standard procedure used at several laboratories that deliver heavy or intermediate mass ions [10]. In front of the solenoid, the beam had no inter-plane correlations, and thus, the difference of rms-emittances was equal to the difference of eigenemittances (mod. sign). Since the solenoid was short, the beam area at the foil A := A f can be approximated as constant during the beam transit through the solenoid. Table 1 lists the relevant parameters of the experiment. Equation (9) Figure 3: Fig. 7 from [3]: schematic beam line of EMTEX at GSI for transverse emittance transfer [9].
relates the beam parameters in front of the solenoid (B s = 0, no correlations → W A = 0, ε 10 = ε x,3+ , ε 20 = ε y,3+ ) to those in front of the foil in the center of the short solenoid: where the index f refers to the location of the foil. The entrance fringe field of the solenoid causes the rms-vorticity leading to Using the initial beam parameters of the experiment from Table 1, A f = ε x β x ε y β y = 4.166 mm 2 , and the identity 1 mm mrad = 1 µm gives Equation (9) is re-used to relate the beam parameters just behind the foil but still at the center of the solenoid to those at the exit of the beam line, where B s = 0 and the beam correlations have been removed again. Angular scattering in the foil is neglected. As the beam changed rigidity in the foil, (Bρ) 3+ must be properly replaced by (Bρ) 7+ . However, second beam moments are not changed by the foil, i.e., W A = W A f , right in front and right behind the foil. Accordingly, which by using Eq. (19) and plugging in the values delivers |ε x,7+ − ε y,7+ | = 2.2 mm mrad (23) 1.040 mm mrad ε y,3+ 0.825 mm mrad β x, 3+ 3.461 m β y, 3+ 5.845 m q in/out 3+ / 7+ (Bρ) 3+/7+ 2.278 / 0.976 Tm B 0 0.9 T fitting well the measured value of 2.0 mm mrad (see Fig. 2 of [9]).