Low-Emittance Storage Rings

# Low-Emittance Storage Rings

A. Wolski
###### Abstract

The effects of synchrotron radiation on particle motion in storage rings are discussed. In the absence of radiation, particle motion is symplectic, and the beam emittances are conserved. The inclusion of radiation effects in a classical approximation leads to emittance damping: expressions for the damping times are derived. Then, it is shown that quantum radiation effects lead to excitation of the beam emittances. General expressions for the equilibrium longitudinal and horizontal (natural) emittances are derived. The impact of lattice design on the natural emittance is discussed, with particular attention to the special cases of FODO, achromat, and TME style lattices. Finally, the effects of betatron coupling and vertical dispersion (generated by magnet alignment and lattice tuning errors) on the vertical emittance are considered.

Low-Emittance Storage Rings

A. Wolski

University of Liverpool, Liverpool, UK

## 1 Introduction

Beam emittance in a storage ring is an important parameter for characterising machine performance. In the case of a light source, for example, the brightness of the synchrotron radiation produced by a beam of electrons is directly dependent on the horizontal and vertical emittances of the beam and is one of the main figures of merit for users. Second generation light sources had natural emittances of order 100 nm. Over the years, significant improvements in lattice designs have been achieved (see Fig. 1), motivated largely by user requirements; third generation light sources now typically aim for natural emittances of just a few nanometres. In the case of colliders for high energy physics, one of the main figures of merit is the luminosity, which is a measure of the rate of particle collisions. Lower emittances allow smaller beam sizes at the interaction point, leading to higher particle density in the colliding bunches, and higher luminosity for the same total number of particles in the beam.

There are of course ways of improving the brightness of a light source and the luminosity of a collider without reducing the emittances: in both cases, for example, the beam current could be increased. However, beam currents are generally limited by collective effects such as impedance-driven instabilities, Touschek scattering, or (for colliders) beam-beam effects. Designing and operating a storage ring for maximum performance involves a good understanding and control of effects that impact the beam emittances.

In this note, we shall consider the emittance of electron (and positron) storage rings: because of synchrotron radiation effects, lepton storage rings are able to achieve very small emittances (of order 1 nm horizontal emittance, and less than 10 pm vertical emittance). We shall begin in Section 2 by reviewing some of the key features of beam dynamics in the absence of synchrotron radiation. In particular, an important property of the dynamics in such cases is that the particle motion is symplectic: this has the consequence that the beam emittances (which characterise the phase space volume occupied by the particles in a beam) are conserved as the beam moves around the storage ring. We shall then show that, in a classical approximation, radiation effects lead to damping of the emittances. We shall derive expressions for the exponential damping times. Then, we shall discuss how quantum effects of synchrotron radiation lead to excitation of the beam emittances. As a result, the emittances of beams in electron (or positron) storage rings reach equilibrium values determined by the beam energy and lattice design.

In Section 3 we shall apply the expression for the natural emittance derived in Section 2 to particular styles of lattice design. In particular, we shall consider FODO, double bend achromat, theoretical minimum emittance, and multi-bend achromat lattices. Double bend achromats are of particular interest for light sources, because they achieve low natural emittance (leading to high brightness) while providing long, dispersion-free (or low dispersion) straight sections that are ideal locations for insertion devices such as undulators or wigglers [1]. Insertion devices are useful for providing intense beams of synchrotron radiation with specific properties.

In a planar storage ring, the vertical emittance is dominated by alignment and tuning errors, rather than by the design of the lattice. In Section 4 we shall discuss how the vertical emittance is related to a range of errors, including steering errors, tilt errors on quadrupoles and vertical alignment errors on sextupoles. Betatron coupling and vertical dispersion are important features of the dynamics in this context, and both will be discussed. Optimisation of a lattice design for a low-emittance storage ring will generally involve simulations to characterise the sensitivity of the vertical emittance to different types of machine error. For this, techniques are needed for accurate computation of the equilibrium emittances from models in which different errors can be included. We shall consider three techniques that are widely used for emittance computation, discussing the envelope method in particular in some detail. Finally, we shall mention briefly some of the issues associated with operational tuning of a storage ring for low-emittance operation.

## 2 Beam dynamics with synchrotron radiation

Briefly, we shall proceed as follows. The symplectic motion of particles in an accelerator (i.e. motion neglecting synchrotron radiation and collective effects) is conveniently described using action-angle variables. We shall define these variables, and use them to review the key features of particle motion in synchrotron storage rings. We shall then include the effects of synchrotron radiation, initially in a classical approximation, leading to expressions for the energy lost per turn in a storage ring, and the damping times for the horizontal, vertical and longitudinal emittances. Finally, we shall discuss the effects of quantum excitation, and derive results for the equilibrium beam emittances. These results will be used in Section 3, where we consider how the equilibrium emittances are affected by the lattice design in a storage ring.

### 2.1 Symplectic motion

We work in a co-ordinate system based on a reference trajectory that we define for our own convenience (see Fig. 2). The distance along the reference trajectory is specified by the independent variable . For simplicity, in a planar storage ring, the reference trajectory is generally chosen to be a straight line (passing through the centres of all quadrupole and higher-order multipole magnets) everywhere except in the dipoles. In the dipoles, the reference trajectory follows the arc of a circle with radius , such that:

 Bρ=P0q, (1)

where is the dipole field, is the reference momentum (i.e. the momentum of particles for which the storage ring is designed) and is the particle charge. is the beam rigidity.

At any point along the reference trajectory, the position of a particle is specified by the and co-ordinates in a plane perpendicular to the reference trajectory. We follow the convention in which is the horizontal (transverse) co-ordinate, and is the vertical co-ordinate.

To describe the motion of a particle, we need to give the components of the momentum of a particle, as well as its co-ordinates. In the transverse directions (i.e. in a plane perpendicular to the reference trajectory) we use the canonical momenta [2] scaled by the reference momentum :

 px = 1P0(γmdxdt+qAx), (2) py = 1P0(γmdydt+qAy). (3)

Here, and are the mass and charge of the particle, is the relativistic factor for the particle, and and are the and components respectively of the electromagnetic vector potential. The transverse dynamics are described by giving the transverse co-ordinates and momenta as functions of (the distance along the reference trajectory).

To describe the longitudinal dynamics of a particle, we use a longitudinal co-ordinate defined by:

 z=β0c(t0−t), (4)

where is the normalised velocity of a particle with the reference momentum , is the time at which the reference particle is at a location , and is the time at which the particle of interest arrives at this location. Physically, the value of for a particle is approximately equal to the distance along the reference trajectory between the given particle and a reference particle travelling along the reference trajectory with momentum (see Fig. 3). A positive value for means that the given particle arrives at a particular location at an earlier time than the reference particle, i.e. the given particle is ahead of the reference particle.

The final dynamical variable needed to describe the motion of a particle is the energy of the particle. Rather than use the absolute energy or momentum, we use the energy deviation , which provides a measure of the difference between the energy of a particle and the energy of a particle with the reference momentum :

 δ=EP0c−1β0=1β0(γγ0−1). (5)

Here, is the relativistic factor for a particle with momentum equal to the reference momentum. A particle with momentum equal to the reference momentum has .

Using the above definitions the co-ordinates and momenta form canonical conjugate pairs:

 (x,px),(y,py),(z,δ). (6)

This means that (continuing to neglect radiation and collective effects) the equations of motion for particles in an accelerator beam line are given by Hamilton’s equations [2], with an appropriate Hamiltonian that describes the electromagnetic fields along the beam line. In a linear approximation, the change in the values of the variables when a particle moves along a beam line can be represented by a transfer matrix, :

 (7)

It is a general property of Hamilton’s equations that the transfer matrix is symplectic. Mathematically, this means that satisfies the relation:

 RTSR=S, (8)

where is the antisymmetric matrix:

 S=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010000−10000000010000−10000000010000−10⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (9)

The symplectic condition (8) imposes strong constraints on the dynamics. Physically, symplectic matrices preserve volumes in phase space (this result is sometimes expressed as Liouville’s theorem [2]). For example, for a linear transformation in one degree of freedom, a particular ellipse in phase space will be transformed to an ellipse with (in general) a different shape; but the area of the ellipse will remain the same. The number of invariants associated with a linear symplectic transformation is at least equal to the number of degrees of freedom in the system. Thus, for motion in three degrees of freedom, there are at least three invariants. For particles in a beam in an accelerator beam line, the invariants are associated with the emittances. If there is no coupling between the degrees of freedom (so that motion in any direction , or is independent of the motion in the other two directions) then we can associate an emittance with each of the three co-ordinates, i.e. there is a horizontal emittance, a vertical emittance and a longitudinal emittance. We shall give a more formal definition of the emittances shortly.

Consider a particle moving through a periodic beam line, without coupling (i.e. a beam line with no skew quadrupoles or solenoids). After each periodic cell, we can plot the horizontal co-ordinate and momentum as a point in the horizontal phase space. After passing through many cells, observing the particle always at the corresponding locations in successive cells, and assuming that the motion of the particle is stable, we find that the points trace out an ellipse in phase space. The shape of the ellipse defines the Courant–Snyder parameters [3] in the beam line at the observation point: see Fig. 4. The area of the ellipse is a measure of the amplitude of the oscillations. We define the horizontal action of the particle such that the area of the ellipse is equal to .

Applying simple geometry to the phase space ellipse, we find that the action (for uncoupled motion) is related to the Cartesian variables for the particle by:

 2Jx=γxx2+2αxxpx+βxp2x, (10)

where the Courant–Snyder parameters satisfy the relation:

 βxγx−α2x=1. (11)

We define the horizontal angle variable as follows:

 tanϕx=−βxpxx−αx. (12)

For a particle with a particular action (i.e. on an ellipse with a given area) the angle variable specifies the position of the particle around the ellipse. The action-angle variables [2] provide an alternative to the Cartesian variables for describing the dynamics. Although we have not shown that this is the case, the action-angle variables form a canonical conjugate pair: that is, the equations of motion expressed in terms of the action-angle variables can be derived from Hamilton’s equations, using an appropriate Hamiltonian (determined as before by the electromagnetic fields along the beam line). The advantage of using action-angle variables to describe particle motion in an accelerator is that, under symplectic transport (i.e. neglecting radiation and collective effects), the action of a particle is constant. We can of course define vertical and (in a synchrotron storage ring) longitudinal action-angle variables in the same way as we defined the horizontal action-angle variables.

The expressions for the action (10) and the angle (12) can be inverted, to give expressions for the Cartesian co-ordinate and momentum in terms of and :

 x = √2βxJxcosϕx, (13) px = −√2Jxβx(sinϕx+αxcosϕx). (14)

The emittance of a bunch of particles can be defined as the average action of all particles in the bunch:

 εx=⟨Jx⟩. (15)

For uncoupled motion, and assuming that the angle variables of different particles are uncorrelated, it follows from (13) and (14) that the second order moments of the particle distribution are related to the Courant–Snyder parameters and the emittance:

 ⟨x2⟩ = βxεx, (16) ⟨xpx⟩ = −αxεx, (17) ⟨p2x⟩ = γxεx. (18)

Using (11), we then find that the emittance can be expressed in terms of the second order moments as:

 ε2x=⟨x2⟩⟨p2x⟩−⟨xpx⟩2. (19)

However, we stress that this relation holds only for uncoupled motion. The expression for the emittance (15) can be generalised without too much difficulty to coupled motion (see, for example [4]), leading to normal mode emittances that are conserved under symplectic transport even where coupling is present. However, the expression for the emittance (19) is less easily generalised to include coupling, and an emittance that is defined by (19) will, in general, not be constant in a beam line where there is coupling.

### 2.2 Vertical damping by synchrotron radiation

So far, we have considered only symplectic transport, i.e. motion of a particle in drift spaces or in the electromagnetic fields of dipoles, quadrupoles, rf cavities etc. without any radiation. However, we know that a charged particle moving through an electromagnetic field will (in general) undergo acceleration, and a charged particle undergoing acceleration will radiate energy in the form of electromagnetic waves. We now address the question of the impact that this radiation will have on the motion of a particle in a synchrotron storage ring. We shall consider first the case of uncoupled vertical motion: for a particle in a storage ring, this turns out to be the simplest case. Since we are primarily interested in the dynamics of the particles generating the radiation, we quote a number of results regarding the properties of the radiation itself (rather than derive these results from first principles).

The first result that we quote for the properties of synchrotron radiation, is that radiation from a relativistic charged particle is emitted within a cone of opening angle of , where is the relativistic factor for the particle [5]. The axis of the cone is tangent to the trajectory of the particle at the point where the radiation is emitted. For an ultra-relativistic particle, , and we can assume that the radiation is emitted directly along the instantaneous direction of motion of the particle.

Consider a particle with initial momentum , that emits radiation carrying momentum . The momentum of the particle after emitting radiation is:

 P′=P−dP≈P(1−dPP0). (20)

Since there is no change in direction of the particle, the vertical component of the momentum must scale in the same way as the total momentum of the particle:

 p′y≈py(1−dPP0). (21)

Now we substitute this into the expression for the vertical betatron action (valid for uncoupled motion):

 2Jy=γyy2+2αyypy+βyp2y, (22)

to find the change in the action resulting from the emission of radiation:

 dJy=−(αyypy+βyp2y)dPP0. (23)

Note that in (23) we neglect a term that is second order in . This term vanishes in the classical approximation when we consider the emission of an infinitesimal amount of radiation in an infinitesimal time interval ; however, we shall see later that including quantum effects, the second order term will lead to excitation of the action. Retaining for the present only the first order term in , averaging (23) over all particles in the beam gives:

 ⟨dJy⟩=dεy=−εydPP0, (24)

where we have used:

 ⟨ypy⟩ = −αyεy, (25) ⟨p2y⟩ = γyεy, (26)

and:

 βyγy−α2y=1. (27)

The emittance is conserved under symplectic transport, so if the effects of radiation are ‘slow’ (i.e. the rate of change of energy from radiation is small compared to the total energy of a particle divided by the revolution period), then for a particle in a storage ring we can average the momentum loss around the ring. From (24):

 dεydt=−εyT0∮dPP0≈−U0E0T0εy=−2τyεy, (28)

where is the revolution period, and is the energy lost through synchrotron radiation in one turn. The approximation is valid for an ultra-relativistic particle, which has . The damping time is defined by:

 τy=2E0U0T0. (29)

The evolution of the emittance is given by:

 εy(t)=εy(t=0)exp(−2tτy). (30)

Typically, in an electron storage ring, the damping time is of order several tens of milliseconds, while the revolution period is of the order of a microsecond. In such a case, radiation effects are indeed slow compared to the revolution frequency.

Note that we made the assumption that the momentum of the particle was close to the reference momentum, i.e. . If the particle continues to radiate without any restoration of energy, we will reach a point where this assumption is no longer valid. However, electron storage rings contain rf cavities to restore the energy lost through synchrotron radiation. For a thorough analysis of synchrotron radiation effects on the vertical motion (at least, with a classical model for the radiation), we should consider the change in momentum of a particle as it moves through an rf cavity. However, in general, rf cavities are designed to provide a longitudinal electric field. This means that particles experience a change in longitudinal momentum as they pass through a cavity, without any change in transverse momentum. In other words, the vertical momentum of a particle will remain constant as the particle moves through an rf cavity, which will therefore have no effect on the emittance of the beam.

To complete our calculation of the vertical damping time, we need to find the energy lost by a particle through synchrotron radiation on each turn through the storage ring. At this point, we quote a second result from the theory of synchrotron radiation: the radiation power from a relativistic particle following a circular trajectory of radius is given by Liénard’s formula [5]:

 Pγ=q2c6πϵ0β4γ4ρ2=Cγc2πc4P4ρ2=Cγ2πc5q2B2P2≈Cγ2πc3q2B2E2, (31)

where the particle has charge , velocity , energy and momentum . The particle travels on a path with radius in a magnetic field of strength . The approximation in the final expression of (31) is valid for ultra-relativistic particles, . is the permittivity of free space, and is a physical constant given by:

 Cγ=q23ϵ0(mc2)4. (32)

For electrons, . Note that the radiation power has a very strong scaling with the particle mass: the larger the mass of the particle, the smaller the amount of radiation emitted. In proton storage rings, except at extremely high energy, synchrotron radiation effects are generally negligible. For a particle with the reference energy, travelling close to the speed of light along the reference trajectory, we can find the energy loss by integrating the radiation power around the ring:

 U0=∮Pγdt≈∮Pγdsc. (33)

Using the expression (31) for , we find:

 U0≈Cγ2πE40∮1ρ2ds, (34)

where is the radius of curvature of the particle trajectory, and we assume that the particle energy is equal to the reference energy . For convenience, we assume that the closed orbit is the same as the reference trajectory for a particle with the reference momentum.

Following convention, we define the second synchrotron radiation integral, [6]:

 I2=∮1ρ2ds. (35)

In the ultra-relativistic limit, the energy loss per turn is written in terms of as:

 U0=Cγ2πE40I2. (36)

Note that is a property of the lattice (actually, a property of the reference trajectory), and does not depend on the properties of the beam. Conventionally, there are five synchrotron radiation integrals used to express the effects of synchrotron radiation on the dynamics of ultra-relativistic particles in an accelerator. The first synchrotron radiation integral is not, however, directly related to the radiation effects. It is defined as:

 I1=∮ηxρds, (37)

where is the horizontal dispersion. is related to the momentum compaction factor , which plays an important role in the longitudinal dynamics, and describes the change in the length of the closed orbit with respect to particle energy:

 ΔCC0=αpδ+O(δ2). (38)

The length of the closed orbit changes with energy because of dispersion in regions where the reference trajectory has some curvature (see Fig. 5):

 dC=(ρ+x)dθ=(1+xρ)ds. (39)

If , then:

 dC=(1+ηxδρ)ds. (40)

The momentum compaction factor can be written:

 αp=1C0dCdδ∣∣∣δ=0=1C0∮ηxρds=I1C0. (41)

### 2.3 Horizontal damping

Analysis of the effect of synchrotron radiation on the vertical emittance was relatively straightforward. When we consider the horizontal emittance, there are three complications that we need to address. First, the horizontal motion of a particle is often strongly coupled to the longitudinal motion. We cannot treat the horizontal motion without also considering (to some extent) the longitudinal motion. Second, where the reference trajectory is curved (usually, in dipoles), the length of the path taken by a particle depends on the horizontal co-ordinate with respect to the reference trajectory. This can be a significant effect since dipoles inevitably generate dispersion (a variation of the orbit with respect to changes in particle energy), so the length of the path taken by a particle through a dipole will depend on its energy. Finally, dipole magnets are sometimes built with a gradient, in which case the vertical field seen by a particle in a dipole will depend on the horizontal co-ordinate of the particle.

Coupling between transverse and longitudinal planes in a beam line is usually represented by the dispersion, and , defined by:

 ηx = dxcodδ∣∣∣δ=0, (42) ηpx = dpx,codδ∣∣∣δ=0, (43)

where and are the co-ordinate and momentum for a particle with energy deviation on a closed orbit. We use the horizontal action-angle variables and to describe the horizontal betatron oscillations of a particle with respect to the dispersive closed orbit, i.e. the closed orbit for a particle with energy deviation . In terms of the horizontal dispersion and betatron action, the horizontal co-ordinate and momentum of a particle are given by:

 x = √2βxJxcosϕx+ηxδ, (44) px = −√2Jxβx(sinϕx+αxcosϕx)+ηpxδ. (45)

When a particle emits radiation, we have to take into account both the change in momentum of the particle, and the change in co-ordinate and momentum with respect to the new (dispersive) closed orbit. Note that when we analysed the vertical motion, we assumed that there was no vertical dispersion. This is the case in an ideal, planar storage ring, but as we shall discuss later, alignment errors on the magnets can lead to the generation of some vertical dispersion that depends on the errors, the effects of which cannot always be neglected.

Taking all the above effects into account for the horizontal motion, we can proceed along the same lines as for the analysis of the vertical emittance. That is, we first write down the changes in co-ordinate and momentum resulting from an emission of radiation with momentum (taking into account the additional effects of dispersion). Then, we substitute expressions for the new co-ordinate and momentum into the expression for the horizontal betatron action, to find the change in action resulting from the radiation emission. Averaging over all particles in the beam gives the change in the emittance that results from radiation emission from each particle in the beam. Finally, we integrate around the ring (taking account of changes in path length and field strength with the horizontal position in the bends) to find the change in emittance over one turn.

Filling in the steps in this calculation, we proceed as follows. First, we note that, in the presence of dispersion, the action is written:

 2Jx=γx~x2+2αx~x~px+βx~p2x, (46)

where and are the horizontal co-ordinate and momentum with respect to the dispersive closed orbit:

 ~x = x−ηxδ, (47) ~px = px−ηpxδ. (48)

After emission of radiation carrying momentum , the variables change by:

 δ ↦ δ−dPP0, (49) ~x ↦ ~x+ηxdPP0, (50) ~px ↦ ~px(1−dPP0)+ηpxdPP0. (51)

We write the resulting change in the action as:

 Jx↦Jx+dJx. (52)

The change in the horizontal action is:

 dJx=−w1dPP0+w2(dPP0)2, (53)

where, in the limit :

 w1=αxxpx+βxp2x−ηx(γxx+αxpx)−ηpx(αxx+βxpx), (54)

and:

 w2=12(γxη2x+2αxηxηpx+βxη2px)−(αxηx+βxηpx)px+12βxp2x. (55)

Treating radiation as a classical phenomenon, we can take the limit in the limit of small time interval, . In this approximation, the term that is second order in vanishes, and we can write for the rate of change of the action:

 dJxdt=−w11P0dPdt≈−w1PγP0c, (56)

where is the rate of energy loss of the particle through synchrotron radiation (31). To find the average rate of change of horizontal action, we integrate over one revolution period:

 dJxdt=−1T0∮w1PγP0cdt. (57)

It is more convenient, given a particular lattice design, to integrate over the circumference of the ring, rather than over one revolution period. However, we have to be careful changing the variable of integration (from time to distance ) where the reference trajectory is curved:

 dt=dCc=(1+xρ)dsc. (58)

So:

 dJxdt=−1T0P0c2∮w1Pγ(1+xρ)ds, (59)

where the rate of energy loss is given by (31).

We have to take into account the fact that in general, the field strength in a dipole can vary with position. To first order in we can write:

 B=B0+x∂B∂x. (60)

Substituting (60) into (31), and with the use of (54), we find (after some algebra) that, averaging over all particles in the beam:

 ∮⟨w1Pγ(1+xρ)⟩ds=cU0(1−I4I2)εx, (61)

where the energy loss per turn is given by (36), the second synchrotron radiation integral is given by (35), and the fourth synchrotron radiation integral is :

 I4=∮ηxρ(1ρ2+2k1)ds. (62)

 k1=qP0∂By∂x. (63)

Note that in (62), the dispersion and quadrupole gradient contribute to the integral only in the dipoles: in other parts of the ring, where the beam follows a straight path, the curvature is zero.

Averaging (59) over all particles in the beam and combining with (61) we have:

 dεxdt=−1T0U0E0(1−I4I2)εx. (64)

Defining the horizontal damping time :

 τx=2jxE0U0T0, (65)

where:

 jx=1−I4I2, (66)

the evolution of the horizontal emittance can be written:

 dεxdt=−2τxεx. (67)

The quantity is called the horizontal damping partition number. For most synchrotron storage ring lattices, if there is no gradient in the dipoles then is very close to 1. From (67) the horizontal emittance decays exponentially:

 εx(t)=εx(t=0)exp(−2tτx). (68)

### 2.4 Longitudinal damping

So far we have considered only the effects of synchrotron radiation on the transverse motion, but there are also effects on the longitudinal motion. Generally, synchrotron oscillations are treated differently from betatron oscillations because in one revolution of a typical storage ring, particles complete many betatron oscillations but only a fraction of a synchrotron oscillation. In other words, the betatron tunes are , but the synchrotron tune is . To find the effects of radiation on synchrotron motion, we proceed as follows. We first write down the equations of motion (for the dynamical variables and ) for a particle performing synchrotron motion, including the radiation energy loss. Then, we express the energy loss per turn as a function of the energy deviation of the particle. This introduces a damping term into the equations of motion. Finally, solving the equations of motion gives synchrotron oscillations (as expected) with amplitude that decays exponentially.

The changes in energy deviation and longitudinal co-ordinate for a particle in one turn around a storage ring are given by:

 Δδ = qVrfE0sin(ϕs−ωrfzc)−UE0, (69) Δz = −αpC0δ, (70)

where is the rf voltage, the rf frequency, is the reference energy of the beam, is the nominal rf phase, and (which may be different from ) is the energy lost by the particle through synchrotron radiation. Strictly speaking, since the longitudinal co-ordinate is a measure of the time at which a particle arrives at a particular location in the ring, changes in with respect to energy should be written in terms of the phase slip factor , which describes the change in revolution period with respect to changes in energy, rather than in terms of the momentum compaction factor . The phase slip factor and the momentum compaction factor are related by (see, for example [7]):

 ηp=αp−1γ20, (71)

where is the relativistic factor for a particle with the reference momentum. But for a storage ring operating a long way above transition (which is the situation we shall assume here) , so . It is slightly more convenient to work with the momentum compaction factor, since this depends (essentially) on just the geometry of the lattice and the optical functions (in particular, the dispersion); whereas the phase slip factor depends also on the beam energy.

If the revolution period in the storage ring is , then we can write the longitudinal equations of motion for the particle:

 dδdt = qVrfE0T0sin(ϕs−ωrfzc)−UE0T0, (72) dzdt = −αpcδ. (73)

To solve these equations, we have to make some assumptions. First, we assume that is small compared to the rf wavelength:

 ωrf|z|c≪1. (74)

The synchrotron radiation power produced by a particle depends on the energy of the particle. We assume that the energy deviation is small, , so we can work to first order in :

 U=U0+ΔEdUdE∣∣∣E=E0=U0+E0δdUdE∣∣∣E=E0. (75)

Finally, we assume that the rf phase is set so that for , the rf cavity restores exactly the amount of energy lost by synchrotron radiation. With these assumptions, the equations of motion become:

 dδdt = −qVrfE0T0cos(ϕs)ωrfcz−1T0δdUdE∣∣∣E=E0, (76) dzdt = −αpcδ. (77)

Taking the derivative of (76) with respect to , and substituting for from (77) gives:

 d2δdt2+2αEdδdt+ω2sδ=0. (78)

This is the equation for a damped harmonic oscillator, with frequency and damping constant given by:

 ω2s = −qVrfE0cos(ϕs)ωrfT0αp, αE = 12T0dUdE∣∣∣E=E0. (80)

If , the energy deviation and longitudinal co-ordinate damp as:

 δ(t) = δ0exp(−αEt)sin(ωst−θ0), z(t) = αpcωsδ0exp(−αEt)cos(ωst−θ0). (82)

where is a constant (the amplitude of the oscillation in at ), and is a fixed phase (the phase of the oscillation at ).

To find an explicit expression for the damping constant , we need to know how the energy loss per turn depends on the energy deviation . The total energy lost per turn by a particle is found by integrating the synchrotron radiation power over one revolution period:

 U=∮Pγdt. (83)

To convert this to an integral over the circumference, we should recall that the path length depends on the energy deviation; so a particle with a higher energy takes longer to travel around the lattice:

 dt=dCc=1c(1+xρ)ds=1c(1+ηxδρ)ds. (84)

Therefore, the radiation energy loss per turn is:

 U=1c∮Pγ(1+ηxδρ)ds. (85)

Using (31), we find after some algebra:

 dUdE∣∣∣E=E0=jzU0E0, (86)

where is given by (36), and the longitudinal damping partition number is:

 jz=2+I4I2. (87)

and are the same synchrotron radiation integrals that we saw before, in (35) and (62). Finally, we can write the longitudinal damping time:

 τz=1αE=2jzE0U0T0. (88)

Neglecting coupling, the longitudinal emittance can be given by a similar expression to the horizontal and vertical emittance:

 εz=√⟨z2⟩⟨δ2⟩−⟨zδ⟩2. (89)

Even where dispersion is present, so that the horizontal and longitudinal motion are coupled, the expression (89) can provide a useful definition of the longitudinal emittance, since the longitudinal variables usually have a much weaker dependence on the transverse variables, than the transverse variables have on the longitudinal. Since the amplitudes of the synchrotron oscillations decay with time constant , the damping of the longitudinal emittance can be written:

 εz(t)=εz(t=0)exp(−2tτz). (90)

It is worth commenting on the fact that the horizontal, vertical and longitudinal emittances are all damped by synchrotron radiation with exponential damping times that depend on the beam energy and the rate at which particles lose energy through synchrotron radiation. In the case of the horizontal and longitudinal emittances, there is an additional factor in the expressions for the damping times that depends on details of the lattice, or, more precisely, on the properties of the dipoles. The additional factors are given by the damping partition numbers and . From (66) and (87), we see that:

 jx+jz=3. (91)

In general, there can also be a vertical damping partition number , although in the simple case we have considered here (of a perfectly planar storage ring) . A more general analysis would lead to the result:

 jx+jy+jz=4, (92)

which is known as the Robinson damping theorem [8]. The significance of this result is that while it is possible (for example, by changing the field gradient in the dipoles) to ‘shift’ the radiation damping between the different degrees of freedom, the overall amount of damping is fixed. In a planar storage ring, for example, one can reduce the horizontal damping time, but only at the expense of increasing the longitudinal damping time.

In a typical storage ring, the dispersion in the dipoles is small compared to the bending radius of the dipoles, that is:

 ηxρ≪1. (93)

Then, if there is no quadrupole component in the dipoles (so that in the dipoles), comparing (35) and (62) leads to:

 I4I2≪1, (94)

in which case:

 jx ≈ 1, (95) jz ≈ 2. (96)

The horizontal damping time is approximately equal to the vertical damping time; the longitudinal damping time is about half the vertical damping time. Typical values for the damping times in medium energy synchrotron light sources are some tens of milliseconds, or a few thousand turns.

### 2.5 Quantum excitation

So far, we have assumed a purely classical model for the radiation, in which energy can be radiated in arbitrarily small amounts. From the expressions for the evolutions of the emittances (30), (68) and (90), we see that if radiation was a purely classical process, the emittances would damp towards zero. However, quantum effects mean that radiation is emitted in discrete units (photons). As we shall see, this induces some ‘noise’ on the beam, known as quantum excitation, the effect of which is to increase the emittance. The beam in an electron (or positron) storage ring will eventually reach an equilibrium distribution determined by a balance between the radiation damping and the quantum excitation. In the remainder of this section, we shall derive expressions for the rate of quantum excitation and for the equilibrium emittances in an electron storage ring.

In deriving the equation of motion (59) for the action of a particle emitting synchrotron radiation, we made the (classical) approximation that in a time interval , the momentum of the radiation emitted goes to zero as goes to zero. In reality, emission of radiation is quantized, so we are prevented from taking the limit . The equation of motion for the action (56) should then be written:

 dJxdt=−w1P0c∫∞0˙N(u)udu+w2P20c2∫∞0˙N(u)u2du, (97)

where is the number of photons emitted per unit time in the energy range from to . The first term on the right hand side of (97) just gives the same radiation damping as in the classical approximation; the second term is an excitation term that we previously neglected.

To find an explicit expression for the rate of change of the action in terms of the beam and lattice parameters, we need to find expressions for the integrals and . The required expressions can be found from the spectral distribution of synchrotron radiation from a dipole magnet, which is another result that we quote from synchrotron radiation theory. The spectral distribution of radiation from a dipole magnet is given by [5]:

 dPdϑ=9√38πPγϑ∫∞ϑK5/3(x)dx, (98)

where is the energy radiated per unit time per unit frequency range, and is the radiation frequency divided by the critical frequency :

 ωc=32γ3cρ. (99)

is the total energy radiated per unit time (31), and is a modified Bessel function. Since the energy of a photon of frequency is , it follows that:

 ˙N(u)du=1ℏωdPdϑdϑ. (100)

Using (98) and (100), we find:

 ∫∞0˙N(u)udu=Pγ, (101)

and:

 ∫∞0˙N(u)u2du=2Cqγ2E0ρPγ. (102)

is a constant given by:

 Cq=5532√3ℏmc. (103)

For electrons (or positrons) .

The next step is to substitute for the integrals in (97) from (101) and (102), substitute for and from (54) and (55), and average over the circumference of the ring. This gives an expression for the evolution of the horizontal action (for and ):

 dεxdt=−2τxεx+2jxτxCqγ2I5I2, (104)

where the fifth synchrotron radiation integral is given by:

 I5=∮Hx|ρ3|ds. (105)

The function () is given by:

 Hx=γxη2x+2αxηxηpx+βxη2px. (106)

The damping time and horizontal damping partition number are given, as before, by (65) and (66). Note that the excitation term is independent of the emittance: the quantum excitation does not simply modify the damping time, but leads to a non-zero equilibrium emittance. The equilibrium emittance is determined by the condition:

 dεxdt∣∣∣εx=ε0=0, (107)

and is given by:

 ε0=Cqγ2I5jxI2. (108)

Note that is determined by the beam energy, the lattice functions (Courant–Snyder parameters and dispersion) in the dipoles, and the bending radius in the dipoles. We shall discuss how the design of the lattice affects the value of (and hence, the equilibrium horizontal emittance) in Section 3. The equilibrium horizontal emittance (108) determined by radiation is sometimes called the natural emittance of the lattice, since it includes only the most fundamental effects that contribute to the emittance: radiation damping and quantum excitation. Other phenomena (such as impedance or scattering effects) can lead to some increase in the equilibrium emittance actually achieved in a storage ring, compared to the natural emittance. Typically, third generation synchrotron light sources have natural emittances of order of a few nanometres. With beta functions of a few metres, this implies horizontal beam sizes of tens of microns (in the absence of dispersion).

In many storage rings, the vertical dispersion in the absence of alignment, steering and coupling errors is zero, so that . However, the equilibrium vertical emittance is larger than zero, because the vertical opening angle of the radiation excites some vertical betatron oscillations. The fundamental lower limit on the vertical emittance, from the opening angle of the synchrotron radiation, is given by [9]:

 εy=1355CqjyI2∮βy|ρ3|ds. (109)

In most storage rings, this is an extremely small value, typically four orders of magnitude smaller than the natural (horizontal) emittance. In practice, the vertical emittance is dominated by magnet alignment errors. Storage rings typically operate with a vertical emittance that is of order 1% of the horizontal emittance, but many can achieve emittance ratios somewhat smaller than this. We shall discuss the vertical emittance in more detail in Section 4.

Quantum effects excite longitudinal emittance as well as transverse emittance. Consider a particle with longitudinal co-ordinate and energy deviation , which emits a photon of energy (see Fig. 6). The co-ordinate and energy deviation after emission of the photon are given by:

 δ′ = δ′0sinθ′=δ0sinθ−uE0, (110) z′ = αpcωsδ′0cosθ′=αpcωsδ0cosθ. (111)

Therefore:

 δ′20=δ20−2δ0uE0sinθ+u2E20. (112)

Averaging over the bunch gives:

 Δσ2δ=⟨u2⟩2E20, (113)

where:

 σ2δ=⟨δ2⟩=12⟨δ20⟩. (114)

 dσ2δdt=12E201C0∮dC∫∞0du˙N(u)u2−2τzσ2δ, (115)

where we have averaged the radiation effects around the ring by integrating over the circumference. Using (102) for , we find:

 dσ2δdt=Cqγ22jzτzI3I2−2τzσ2δ. (116)

The equilibrium energy spread is given by :

 σ2δ0=Cqγ2I3jzI2, (117)

where the third synchrotron radiation integral is defined:

 I3=∮1|ρ3|ds. (118)

The equilibrium energy spread determined by radiation effects is often referred to as the natural energy spread, since collective effects can often lead to an increase in the energy spread with increasing bunch charge. Note that the natural energy spread is determined essentially by the beam energy and by the bending radii of the dipoles; rather counterintuitively, it does not depend on the rf parameters (either the voltage or the frequency). On the other hand, the bunch length does have a dependence on the rf. The ratio of the bunch length to the energy spread in a matched distribution (i.e. a distribution that is unchanged after one complete revolution around the ring) can be determined from the shape of the ellipse in longitudinal phase space followed by a particle obeying the longitudinal equations of motion (72) and (73). Neglecting radiation effects (which can be assumed to be small) the result is:

 σz=αpcωsσδ. (119)

We can increase the synchrotron frequency , and hence reduce the bunch length, by increasing the rf voltage, or by increasing the rf frequency.

### 2.6 Summary of radiation damping and quantum excitation

To summarise, including the effects of radiation damping and quantum excitation, the emittances (in each of the three degrees of freedom) evolve with time as:

 ε(t)=ε(t=0)exp(−2tτ)+ε(t=∞)[1−exp(−2tτ)], (120)

where is the initial emittance (for example, of a beam as it is injected into the storage ring), and is the equilibrium emittance determined by the balance between radiation damping and quantum excitation. The damping times are given by:

 jxτx=jyτy=jzτz=2E0U0T0, (121)

where the damping partition numbers are given by:

 jx=1−I4I2,jy=1,jz=2+I4I2. (122)

The energy loss per turn is given by:

 U0=Cγ2πE40I2, (123)

where for electrons (or positrons) The natural emittance is:

 ε0=Cqγ2I5jxI2, (124)

where for electrons (or positrons) The natural rms energy spread and bunch length are given by:

 σ2δ = Cqγ2I3jzI2, (125) σz = αpcωsσδ. (126)

The momentum compaction factor is:

 αp=I1C0. (127)

The synchrotron frequency and synchronous phase are given by:

 ω2s = −qVrfE0ωrfT0αpcosϕs, (128) sinϕs = U0qVrf. (129)

Finally, the synchrotron radiation integrals are:

 I1 = ∮ηxρds, (130) I2 = ∮1ρ2ds, (131) I3 = ∮1|ρ|3ds, (132) I4 = ∮ηxρ(1ρ2+2k1)ds,k1=eP0∂By∂x, (133) I5 = ∮Hx|ρ|3ds,Hx=γxη2x+2αxηxηpx+βxη2px. (134)

## 3 Equilibrium emittance and storage ring lattice design

In this section, we shall derive expressions for the natural emittance in four types of lattices: FODO, double bend achromat (DBA), multi-bend achromat (including the triple bend achromat) and theoretical minimum emittance (TME) lattices. We shall also consider how the emittance of an achromat may be reduced by ‘detuning’ the lattice from the strict achromat conditions.

Recall that the natural emittance in a storage ring is given by (108):

 ε0=Cqγ2I5jxI2, (135)

where is a physical constant, is the relativistic factor, is the horizontal damping partition number, and and are synchrotron radiation integrals. Note that , and are all fixed by the layout of the lattice and the optics, and are independent of the beam energy. In most storage rings, if the bends have no quadrupole component, the damping partition number . In that case, to find the natural emittance we just need to evaluate the two synchrotron radiation integrals and . If we know the strength and length of all the dipoles in the lattice, it is straightforward to calculate . For example, if all the bends are identical, then in a complete ring (total bending angle = 2):

 I2=∮1ρ2ds=∮B(Bρ)dsρ=