Particle beam focussing and other confined physical systems often obey the equation ¨x = kx, where k < 0 implies stable motion. A constant negative value of k yields simple harmonic motion (constant focussing), while positive values give unstable (defocussing) motion. Alternating gradient focussing is the surprising principle that if k(t) is varied periodically, the overall motion can sometimes be stable even if the average value of k is zero or negative. It is used where focussing in one direction implies defocussing in one or more other directions, such as with electromagnetic fields in free space that have to satisfy ∇ · E = 0 and similar.

43 PARTICLE ACCELERATORS↗

DOE OSTI2024

Benchmark Midplane Field for Medical FFA

This is a lattice cell for a fixed-field medical accelerator, transmitting protons from 10–250 MeV. The goal is to have fixed tunes throughout this energy range. The cell is based on the lattice Feb7 2024 ProtonAccelerator.f90 from Dejan Trbojevic and had the magnet field profiles re-optimised to attain fixed tunes with realistic magnet fringe fields.

43 PARTICLE ACCELERATORS↗

DOE OSTI2023

Higher-Order Corrections to Optimisers based on Newton's Method

The Newton, Gauss–Newton and Levenberg–Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some directions than others and the space of possible improvement in sum of squares becomes a long narrow ellipsoid in the linear model. This means that even a small amount of nonlinearity in the problem parameters can cause a proposed point far down the long axis of the ellipsoid to fall outside of the actual curved valley of improved values, even though it is quite nearby. This paper presents a differential equation that ‘follows’ these valleys, based on the technique of geodesic acceleration, which itself provides a 2 nd order improvement to the Levenberg–Marquardt iteration step. Higher derivatives of this equation are computed that allow n th order improvements to the optimisation methods to be derived. These higher-order accelerated methods up to 4 th order are tested numerically and shown to provide substantial reduction of both number of steps and computation time.

43 PARTICLE ACCELERATORS↗

DOE OSTI2023

Integrals of Polynomial Functions over Spheres and Balls

Abstract not provided.

43 PARTICLE ACCELERATORS↗

DOE OSTI2022

Paraxial, Thin-Lens Analysis of Fixed-Tune, Non-Scaling FFAs with Two Magnets per Cell

Fixed-field accelerators (FFAs) have closed orbits that change as a function of beam momentum. It is sometimes useful to avoid various resonances by keeping the tunes constant as these orbits change. A well-known example is the scaling FFA where the entire beam orbit and optics are geometrically scaled as a function of momentum. However, this is stricter than necessary: cells with three or more lenses can have fixed tunes even when the focussing strengths in the lenses change. Recently, Dejan Trbojevic has found a pair of nonlinear magnets that produce fixed tunes and fixed beta functions while not being a scaling FFA. This improves on a more approximate solution in. Notably a scaling FFA would require one magnet to be entirely reverse-bending, whereas the Trbojevic solution does not do this and instead resembles an intermediate point between the traditional scaling field profiles and the nonscaling profiles centred around a momentum with equal and positive fields. This suggests there are at least three levels of stringency that one can apply to a fixed-tune FFA design: 1. Fixed cell tunes (in both planes) as a function of momentum; 2. Fixed optics (beta functions) as a function of momentum; 3. Similarity of all orbits via a scaling symmetry law ↔ traditional scaling FFA. This note studies the interesting case #2 above (#3 being fully characterised by the orbit at a single energy) in the simplest possible example: a cell of two thin lenses in the small angle (paraxial) approximation.

43 PARTICLE ACCELERATORS↗

DOE OSTI2021

Integrated Fields of Permanent Magnet Dipole Trims

To adjust the integrated dipole field of a magnet (through the longitudinal z axis), pairs of small permanent magnets may be added at the end, symmetrically above and below the axis. If these are sufficiently small, they may be approximated by magnetic dipole point sources.

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

The Connection of Adiabaticity to the Fourier Transform

This is probably the first thing I’d suggest anyone learns about adiabatic transitions because it tells you how a linear system reacts to an arbitrary forcing function. Not all systems are linear but they are usually locally linear near their stable fixed point. I think Scott Berg pointed out the relation to the Fourier transform to me and there’s probably literature about this under generic dynamical systems.

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

Permanent magnets for the return loop of the Cornell-Brookhaven energy recovery linac test accelerator

214 neodymium permanent magnets have been manufactured for the return loop of the CBETA multi-turn ERL at Cornell University. There are 5 types of quadrupole and combined-function gradient magnets using a variant of the circular Halbach design. These are made out of NdFeB material and glued into an aluminium housing with water channels for temperature stabilisation. The NdFeB wedges and magnet construction were done by outside companies, while the final “tuning” using inserts containing 64 iron rods per magnet was done at BNL over a period of about 6 months. Average relative field errors of 2.2×10 -4 were achieved on the beam region. The magnet strengths vary by type but are of order 10 T/m for quadrupole component and up to 0.3 T for the dipole. This paper reports on the field quality and timeline achieved in this production process

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

Smoothed Retarded Potentials and Fields

This note will follow the derivation of the Liénard–Wiechert potentials for a moving charge, which give the entire electromagnetic field generated by that charge including time delays, while concurrently deriving a version for a smoothed charge. The smoothed charge replaces the delta function charge density ρ ∝ δ 3 (x) by a spherically symmetric function f(|x|), so instead of a potential φ ∝ 1/ |x| for a charge at rest, the potential will be the 3D convolution of f with 1/ |x|. The smoothing will be defined in the ‘laboratory frame’ rather than the rest frame of the particle. If a spherically-symmetric distribution is required in a different rest frame, for example the averaged rest frame of a whole particle beam, the particle histories may be Lorentz boosted and the resulting fields Lorentz boosted back. Another motivation for using the lab frame is that if a beam is uniformly accelerated by an electric field, it maintains its lab frame size, rather than shrinking as a rigid object would (if the beam enters an electric field from the side, the particles maintain their time separation and the length actually increases with the beam velocity). Properly defining a charge distribution of fixed size in the changing reference frame of an accelerating particle is somewhat complicated and relates to the concept of Born rigidity.

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

Integrals of Distorted Gaussian Functions

Series solutions involving Gaussian-smoothed sources may involve integrating a slightly-distorted Gaussian function, which can be expressed as a Gaussian multiplied by a polynomial or series. The 1D Gaussian function centred at $x = 0$ is $f_{1,σ}(x)$ = $\frac{1} {σ\sqrt{2π}}$ $e^{-x^{2}/2σ^{2}}$ and obeys $\int$$^{\infty}_{-\infty}$ $f_{1,σ}(x) dx =$

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

FETS-VFFA Cell using Contour-Based Magnets

A pure-conductor winding arrangement has been found that approximates the optics of the design given in (Figure 1). The tunes are not the same but are reasonably flat and exist through the entire 3–12 MeV energy range, so could conceivably be fine-tuned using the vertical winding density function in the next section. The lattice consists of two magnets: a reverse bending Bd magnet followed by a normal bending Bf magnet. These magnets are both sectors whose midpoints are displaced with respect to the midpoints of the sides of a 20-sided polygon with side length 1.25 m. The ring is said to have 10 cells (2 polygon sides per cell) that are 2.5 m long. The displacements are 17 cm inwards for Bd and 17 cm outwards for Bf.

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

Magnetic Potential of a Finite Wire

The wire segment in question travels in a straight line from position a to b and carries current I in the direction towards b . The magnetic field of this segment alone will not be Maxwellian because the current does not satisfy the continuity equation. However, the field sum of a loop of such wires will be.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Magnetic Field of a Winding Sheet defined by a Contour Function

Abstract not provided.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Computation of magnetic fields from field components on a plane grid

An algorithm is presented to calculate the field components of the magnetic field [B x (x, y, z), B y (x, y, z), B z (x, y, z)] at a point (x, y, z) in space, from the knowledge of the components [B x (x, y) = 0, z), B y (x, y = 0, z), B z (x, y = 0,z)] on a “reference plane”, which is normal to the y-axis at . The algorithm, which is a general one and is not restricted to fields with mid-plane symmetry is based on the Maclaurin series expansion of the magnetic field components at any point in space in terms of the distance (y) of the point from the reference plane. The coefficients of the Maclaurin series expansion are expressed in terms of the on-plane field components and their partial derivatives with respect to spatial coordinates (). The field components are usually generated from magnetic field measurements on a rectangular grid on the plane. This algorithm was employed in 1986 in the RAYTRACE computer code to help calculate the optical properties of magnets and of the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory (BNL). Here, a general mathematical formulation of this algorithm based on the differential algebraic method was presented by Makino in 2011. This paper presents the step by step derivation of the algorithm and provides the necessary formulas to be introduced by the reader in any computer code which requires the field components generated by magnetic devices. In addition provides an example of the use of the algorithm and its limitations as applied to a Halbach type magnet with or without median plane symmetry.

43 PARTICLE ACCELERATORS↗