Storage Ring Design

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Joint US-CERN-Japan-Russia School on Particle Accelerators Course on Synchrotron Radiation and Free Electron Lasers Erice, Sicily, April 2011

Storage Ring Design

Part 1: Beam Dynamics with Synchrotron Radiation

Andy Wolski The Cockcroft Institute, and the University of Liverpool, UK

Brightness is a key ﬁgure of merit for SR sources

Storage Ring Design 1 Part 1: Beam Dynamics with SR High photon brightness needs low electron beam emittance

Storage Ring Design 2 Part 1: Beam Dynamics with SR

Storage Ring Design

1. Beam dynamics with synchrotron radiation Eﬀects of synchrotron radiation on particle motion. • The synchrotron radiation integrals. • Damping times of the beam emittances. • Quantum excitation and equilibrium emittances. • 2. Equilibrium emittance and storage ring lattice design Natural emittance in diﬀerent lattice styles. • Achromats and “quasi-achromats”. • 3. Chromaticity, dynamic aperture, beam lifetime Focusing errors. • Chromaticity and chromatic correction. • Dynamic aperture and Touschek lifetime. • 4. Beam instabilities Coupled bunch instabilities. • Single bunch instabilities. •

Storage Ring Design 3 Part 1: Beam Dynamics with SR Lecture 1 objectives: linear dynamics with synchrotron radiation

In this lecture, we shall:

deﬁne action-angle variables for describing symplectic • motion of a particle along a beam line;

discuss the eﬀect of synchrotron radiation on the (linear) • motion of particles in storage rings;

derive expressions for the damping times of the vertical, • horizontal, and longitudinal emittances;

discuss the eﬀects of quantum excitation, and derive • expressions for the equilibrium horizontal and longitudinal beam emittances in an electron storage ring.

Storage Ring Design 4 Part 1: Beam Dynamics with SR

Coordinate system

We work in a Cartesian coordinate system, with a reference trajectory that we deﬁne for our own convenience:

In general, the reference trajectory can be curved. At any point along the reference trajectory, the x and y coordinates are perpendicular to the reference trajectory. Storage Ring Design 5 Part 1: Beam Dynamics with SR Momenta

The transverse momenta are the canonical momenta , normalised by a reference momentum , P0: 1 dx px = γm + qA x , (1) P0 dt 1 dy py = γm + qA y , (2) P0 dt where m and q are the mass and charge of the particle, γ is the relativistic factor, and Ax and Ay are the transverse components of the vector potential.

We can choose the reference momentum for our own convenience; usually, we choose P0 to be equal to the nominal momentum for a particle moving along the beam line.

The transverse dynamics are described by giving the transverse coordinates and momenta as functions of s (the distance along the reference trajectory). Storage Ring Design 6 Part 1: Beam Dynamics with SR

Longitudinal coordinate

The longitudinal coordinate of a particle is deﬁned by:

z = β c(t t), (3) 0 0 − where β0 is the normalized velocity of a particle with the reference momentum P0; t0 is the time at which the reference particle is at a location s, and t is the time at which the particle of interest arrives at this location. z is approximately the distance along the reference trajectory that a particle is ahead of a reference particle travelling along the reference trajectory with momentum P0.

Storage Ring Design 7 Part 1: Beam Dynamics with SR Energy deviation

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 δ, which measures the diﬀerence between the energy of the particle and the energy of a particle with the reference momentum P0: E 1 1 γ δ = = 1 , (4) P0c − β0 β0 γ0 − ! where E is the energy of the particle, and β0 is the normalized velocity of a particle with the reference momentum P0.

Note that for a particle whose momentum is equal to the reference momentum, γ = γ0, and hence δ = 0.

Storage Ring Design 8 Part 1: Beam Dynamics with SR

Canonical variables

Using the deﬁnitions on the previous slides, the coordinates and momenta form canonical conjugate pairs :

(x, p x), (y, p y), (z, δ ). (5)

This means that if M represents the linear transfer matrix for a beam line consisting of some sequence of drifts, solenoids, dipoles, quadrupoles, RF cavities etc., so that: x x  px   px  y y   = M(s ; s )   (6)   1 0    py  ·  py       z   z       δ   δ   s=s1  s=s0     then, neglecting radiation from the particle , the matrix M is symplectic.

Storage Ring Design 9 Part 1: Beam Dynamics with SR Symplectic matrices

Mathematically, a matrix M is symplectic if is satisﬁes the relation: M T S M = S, (7) · · where S is the antisymmetric matrix: 0 1 0 0 0 0 10 0 0 0 0  −0 0 0 1 0 0  S = . (8)  0 0 10 0 0   0 0− 0 0 0 1     0 0 0 0 1 0   −   

Physically, symplectic matrices preserve areas in phase space. For example, in one degree of freedom:

Storage Ring Design 10 Part 1: Beam Dynamics with SR

Twiss parameters and the particle action

In an uncoupled periodic beam line, particles trace out ellipses in phase space with each pass through a periodic cell. The shape of the ellipse deﬁnes the Twiss parameters at the ob- servation point. The area of the ellipse deﬁnes the action Jx of the particle.

Storage Ring Design 11 Part 1: Beam Dynamics with SR Cartesian variables and action-angle variables

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: 2 2 2Jx = γxx + 2 αxxp x + βxpx. (9)

We also deﬁne the angle variable φx as follows: px tan φx = βx αx. (10) − x −

The action-angle variables provide an alternative to the Cartesian variables for describing the dynamics of a particle moving along a beam line. The advantage of action-angle variables is that, under symplectic transport, the action of a particle is constant.

It turns out that the action-angle variables form a canonically conjugate pair. Storage Ring Design 12 Part 1: Beam Dynamics with SR

Action and emittance

The action Jx is a variable that is used to describe the amplitude of the motion of a single particle. In terms of the action-angle variables, the Cartesian coordinate and momentum can be written:

2Jx x = 2βxJx cos φx, p x = (sin φx + αx cos φx) . (11) −s β q x

The emittance εx is the average amplitude of all particles in a bunch:

εx = Jx . (12) h i

With this relationship between the emittance and the average action, we can obtain (for uncoupled motion) the following relationships for the second-order moments of the particle distribution within the bunch: 2 2 x = βxεx, xp x = αxεx, p = γxεx. (13) h i h i − h xi Storage Ring Design 13 Part 1: Beam Dynamics with SR Action and radiation

So far, we have considered only symplectic transport, i.e. motion of a particle in the electromagnetic ﬁelds of drifts, dipoles, quadrupoles, 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 electromagnetic waves.

What impact will the radiation have on the motion of a particle?

In answering this question, we wil consider ﬁrst the case of uncoupled vertical motion: for a particle in a storage ring, this turns out to be the simplest case.

Storage Ring Design 14 Part 1: Beam Dynamics with SR

Radiation damping of vertical emittance

The radiation emitted by a relativistic particle has an opening angle of 1 /γ , where γ is the relativistic factor for the particle.

For an ultra-relativistic particle, γ 1, and we can assume that the radiation is emitted directly along the instantaneous direction of motion of the particle.

Storage Ring Design 15 Part 1: Beam Dynamics with SR Radiation damping of vertical emittance

The momentum of the particle after emitting radiation is: dp p0 = p dp p 1 , (14) − ≈ − P0! where dp is the momentum carried by the radiation, and we assume that: p P . (15) ≈ 0 Since there is no change in direction of the particle, we must have: dp py0 py 1 . (16) ≈ − P0!

Storage Ring Design 16 Part 1: Beam Dynamics with SR

Radiation damping of vertical emittance

After emission of radiation, the vertical momentum of the particle is: dp py0 py 1 . (17) ≈ − P0! Now we substitute this into the expression for the vertical betatron action (valid for uncoupled motion): 2 2 2Jy = γyy + 2 αyyp y + βypy , (18) to ﬁnd the change in the action resulting from the emission of radiation: 2 dp dJ y = αyyp y + βyp . (19) − y P 0 Then, we average over all particles in the beam, to ﬁnd: dp dJ y = dε y = εy , (20) h i − P0 where we have used: 2 2 yp y = αyεy, p = γyεy, and βyγy α = 1 . (21) h i − h y i − y Storage Ring Design 17 Part 1: Beam Dynamics with SR Radiation damping of vertical emittance

For a particle moving round a storage ring, we can integrate the loss in momentum around the ring, to ﬁnd the total change in momentum in one turn. The emittance is conserved under symplectic transport, so if the non-symplectic (radiation) eﬀects are slow, we can write:

dp dε y εy dp U0 dε y = εy ∴ = εy, (22) − P0 dt −T0 I P0 ≈ − E0T0 where T0 is the revolution period, and U0 is the energy loss in one turn. The approximation is valid for an ultra-relativistic particle, which has E pc . ≈

We deﬁne the damping time τy:

E0 τy = 2 T0, (23) U0 so the evolution of the emittance is: t εy(t) = εy(t) exp 2 . (24) − τy !

Storage Ring Design 18 Part 1: Beam Dynamics with SR

Radiation damping of vertical emittance

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

But note that we made the assumption that the momentum of the particle was close to the reference momentum, i.e. p P . ≈ 0

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. But then, we should consider the change in momentum of a particle as it moves through an RF cavity.

Storage Ring Design 19 Part 1: Beam Dynamics with SR Radiation damping of vertical emittance

Fortunately, RF cavities are usually designed to provide a longitudinal electric ﬁeld, so that particles experience a change in longitudinal momentum as they pass through, without any change in transverse momentum.

This means that we do not have to consider explicitly the eﬀects of RF cavities on the emittance of the beam.

Storage Ring Design 20 Part 1: Beam Dynamics with SR

Synchrotron radiation energy loss

To complete our calculation of the the vertical damping time, we need to ﬁnd the energy lost by a particle through synchrotron radiation on each turn through the storage ring. We quote the (classical) result that the power radiated by a particle of charge e and energy E in a magnetic ﬁeld B is given by:

Cγ 3 2 2 2 Pγ = c e B E . (25) 2π Cγ is a physical constant given by: e2 = 8 846 10 5 m/GeV 3 (26) Cγ 2 4 . − . 3ε0(mc ) ≈ ×

Storage Ring Design 21 Part 1: Beam Dynamics with SR Synchrotron radiation energy loss

A charged particle with energy E in a magnetic ﬁeld B follows a circular trajectory with radius ρ, given by: p Bρ = . (27) q For an ultra-relativistic electron, E pc : ≈ E Bρ . (28) ≈ ec Hence, the synchrotron radiation power can be written: 4 Cγ E Pγ c . (29) ≈ 2π ρ2

Storage Ring Design 22 Part 1: Beam Dynamics with SR

Synchrotron radiation energy loss

For a particle with the reference energy, travelling at (close to) the speed of light along the reference trajectory, we can ﬁnd the energy loss by integrating the radiation power around the ring: ds U0 = Pγdt = Pγ . (30) I I c Using the previous expression for Pγ, we ﬁnd: C 1 = γ 4 (31) U0 E0 2 ds, 2π I ρ where ρ is the radius of curvature of the reference trajectory.

Note that for these expressions to be valid, we require that the reference trajectory be a real physical trajectory of a particle.

Storage Ring Design 23 Part 1: Beam Dynamics with SR The second synchrotron radiation integral

Following convention, we deﬁne the second synchrotron radiation integral , I2: 1 = (32) I2 2 ds. I ρ

In terms of I2, the energy loss per turn U0 is written:

Cγ 4 U0 = E I2. (33) 2π 0

Note that I2 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 defined, which are used to express in convenient form the effect of radiation on the dynamics of ultra-relativistic particles in a storage ring (or beam line). Storage Ring Design 24 Part 1: Beam Dynamics with SR

The ﬁrst synchrotron radiation integral

The first synchrotron radiation integral is not, however, directly related to the radiation effects. It is defined as: ηx I1 = ds, (34) I ρ where ηx is the horizontal dispersion.

The change in length of the closed orbit with respect to particle energy is expressed in terms of the momentum compaction factor, αp:

∆C 2 = αpδ + O(δ ). (35) C0 The momentum compaction factor can be written: 1 dC 1 1 1 αp = = ds = I1. (36) C dδ C ρ C 0 δ=0 0 I 0

Storage Ring Design 25 Part 1: Beam Dynamics with SR Damping of horizontal emittance

Analysis of radiation eﬀects on the vertical emittance was relatively straightforward. When we consider the horizontal emittance, there are three complications that we need to address:

The horizontal motion of a particle is often strongly • coupled to the longitudinal motion.

Where the reference trajectory is curved (usually, in • dipoles), the path length taken by a particle depends on the horizontal coordinate with respect to the reference trajectory.

Dipole magnets are sometimes built with a gradient, so • that the vertical ﬁeld seen by a particle in a dipole depends on the horizontal coordinate of the particle.

Storage Ring Design 26 Part 1: Beam Dynamics with SR

Horizontal-longitudinal coupling

Coupling between transverse and longitudinal planes in a beam line is usually represented by the dispersion, ηx. So, in terms of the horizontal dispersion and betatron action, the horizontal coordinate and momentum of a particle are given by:

x = 2βxJx cos φx + ηxδ (37) q 2Jx px = (sin φx + αx cos φx) + ηpx δ. (38) −s βx

Storage Ring Design 27 Part 1: Beam Dynamics with SR Horizontal-longitudinal coupling

When a particle emits radiation, we have to take into account:

the change in momentum of the particle; •

the change in coordinate x and momentum px, resulting • from the change in the energy deviation δ.

When we analysed the vertical motion, we ignored the second eﬀect, because we assumed that the vertical dispersion was zero.

Storage Ring Design 28 Part 1: Beam Dynamics with SR

Damping of horizontal emittance

Taking all the above eﬀects into account, we can proceed along the same lines as for the analysis of the vertical emittance. That is:

Write down the changes in coordinate x and momentum px resulting • from an emission of radiation with momentum dp (taking into account the additional eﬀects of dispersion).

Substitute expressions for the new coordinate and momentum into the • expression for the horizontal betatron action, to ﬁnd the change in action resulting from the radiation emission.

Average over all particles in the beam, to ﬁnd the change in the • emittance resulting from radiation emission from each particle.

Integrate around the ring (taking account of changes in path length • and ﬁeld strength with x in the bends) to ﬁnd the change in emittance over one turn.

The algebra gets somewhat cumbersome, and is not especially enlightening. See Appendix A for more details. Here, we just quote the result...

Storage Ring Design 29 Part 1: Beam Dynamics with SR Damping of horizontal emittance

The horizontal emittance decays exponentially:

dε x 2 = εx, (39) dt −τx where the horizontal damping time is given by:

2 E0 τx = T0. (40) jx U0

The horizontal damping partition number jx is given by:

I4 jx = 1 , (41) − I2 where the fourth synchrotron radiation integral is given by: η 1 e ∂B = x + 2 = y (42) I4 2 k1 ds, k 1 . I ρ ρ ! P0 ∂x

Storage Ring Design 30 Part 1: Beam Dynamics with SR

Damping of synchrotron oscillations

So far we have considered only the eﬀects of synchrotron radiation on the transverse motion. There are also eﬀects on the longitudinal motion.

Generally, synchrotron oscillations are treated diﬀerently from betatron oscillations, because the synchrotron tune in a storage ring is typically much less than 1, while the betatron tunes are typically much greater than 1.

Storage Ring Design 31 Part 1: Beam Dynamics with SR Damping of synchrotron oscillations

To ﬁnd the eﬀects of radiation on synchrotron motion, we proceed as follows:

We write down the equations of motion (for the variables z • and δ) for a particle performing synchrotron motion, including the radiation energy loss.

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.

Solving the equations of motion gives synchrotron • oscillations (as expected) with amplitude that decays exponentially.

Storage Ring Design 32 Part 1: Beam Dynamics with SR

Damping of synchrotron oscillations

The change in energy deviation δ and longitudinal coordinate z for a particle in one turn around a storage ring are given by:

eV RF ωRF z U ∆δ = sin φs , (43) E0 − c − E0 ∆z = αpC δ, (44) − 0 where VRF is the RF voltage, and ωRF the RF frequency, E0 is the reference energy of the beam, φs is the nominal RF phase, and U (which may be diﬀerent from U0) is the energy lost by the particle through synchrotron radiation.

If the revolution period is T0, then we can write the longitudinal equations of motion for the particle:

dδ eV RF ωRF z U = sin φs , (45) dt E0T0 − c − E0t0 dz = αpcδ. (46) dt −

Storage Ring Design 33 Part 1: Beam Dynamics with SR Damping of synchrotron oscillations

Let us assume that z is small compared to the RF wavelength, i.e. ω z/c 1. RF Also, the energy loss per turn is a function of the energy of the particle (particles with higher energy radiate higher synchrotron radiation power), so we can write (to ﬁrst order in the energy deviation): dU dU U = U0 + ∆E = U0 + E0δ . (47) dE E=E0 dE E=E0

Further, we assume that the RF phase φs is set so that for z = δ = 0, the RF cavity restores exactly the amount of energy lost by synchrotron radiation. The equations of motion then become: dδ eV RF ωRF 1 dU = cos( φs) z δ , (48) dt −E0T0 c − T0 dE E=E0

dz = αpcδ. (49) dt −

Storage Ring Design 34 Part 1: Beam Dynamics with SR

Damping of synchrotron oscillations

Combining these equations gives: d2δ dδ + 2 α + ω2δ = 0 . (50) dt 2 E dt s

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

2 eV RF ωRF ωs = cos( φs) αp, (51) − E0 T0 1 dU αE = . (52) 2T0 dE E=E0

Storage Ring Design 35 Part 1: Beam Dynamics with SR Damping of synchrotron oscillations

If α ωs, the energy deviation and longitudinal coordinate E damp as:

δ(t) = δˆexp( α t) sin( ωst θ ), (53) − E − 0

αpc z(t) = δˆexp( αEt) cos( ωst θ0). (54) ωs − − where δˆ is a constant (the amplitude of the oscillation at t = 0).

To ﬁnd the damping constant αE, we need to know how the energy loss per turn U depends on the energy deviation δ...

Storage Ring Design 36 Part 1: Beam Dynamics with SR

Damping of synchrotron oscillations

We can ﬁnd the total energy lost by integrating over one revolution period:

U = Pγ dt. (55) I 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.

dC dt = (56) c

x ηxδ dC = 1 + ds = 1 + ds ρ! ρ ! (57) 1 ηxδ U = Pγ 1 + ds. (58) c I ρ !

Storage Ring Design 37 Part 1: Beam Dynamics with SR Damping of synchrotron oscillations

With the energy loss per turn given by:

1 ηx U = Pγ 1 + δ ds, (59) c I ρ ! and the synchrotron radiation power given by: 4 Cγ 3 2 2 2 Cγ E Pγ = c e B E = c , (60) 2π 2π ρ2 we ﬁnd, after some algebra:

dU U0 = jE , (61) dE E=E0 E0

where: Cγ 4 I4 U0 = E0I2, j E = 2 + . (62) 2π i2 I2 and I4 are the same synchrotron radiation integrals that we saw before, in Eqs. (32) and (42).

Storage Ring Design 38 Part 1: Beam Dynamics with SR

Damping of synchrotron oscillations

Finally, we can write the longitudinal damping time:

1 2 E0 τz = = T0. (63) αE jz U0

U0 is the energy loss per turn for a particle with the reference energy E0, following the reference trajectory. It is given by:

Cγ 4 U0 = E I2. (64) 2π 0 jz is the longitudinal damping partition number, given by:

I4 jz = 2 + . (65) I2

Storage Ring Design 39 Part 1: Beam Dynamics with SR Damping of synchrotron oscillations

The longitudinal emittance can be given by a similar expression to the horizontal and vertical emittance:

2 2 2 εz = z δ zδ . (66) qh ih i − h i

Since the amplitudes of the synchrotron oscillations decay with time constant τz, the damping of the longitudinal emittance can be written: t εz(t) = εz(0) exp 2 . (67) − τz

Storage Ring Design 40 Part 1: Beam Dynamics with SR

Summary: synchrotron radiation damping

The energy loss per turn is given by:

Cγ 4 5 3 U0 = E I2, C γ 8.846 10 − m/GeV . (68) 2π 0 ≈ ×

The radiation damping times are given by:

2 E0 2 E0 2 E0 τx = T0, τ y = T0, τ z = T0. (69) jx U0 jy U0 jz U0

The damping partition numbers are:

I4 I4 jx = 1 , j y = 1 , j z = 2 + . (70) − I2 I2

The second and fourth synchrotron radiation integrals are: 1 η 1 = = x + 2 (71) I2 2 ds, I 4 2 k1 ds. I ρ I ρ ρ !

Storage Ring Design 41 Part 1: Beam Dynamics with SR Quantum excitation

If radiation were a purely classical process, the emittances would damp to nearly zero.

However radiation is emitted in discrete units (photons), which induces some “noise” on the beam. The eﬀect of the noise is to increase the emittance.

The beam eventually reaches an equilibrium distribution determined by a balance between the radiation damping and the quantum excitation.

Storage Ring Design 42 Part 1: Beam Dynamics with SR

Quantum excitation of horizontal emittance

By considering the change in the phase-space variables when a particle emits radiation carrying momentum dp, we ﬁnd that the associated change in the betatron action is: 2 dp dp dJ x = w1 + w2 , (72) − P0 P0! where w1 and w2 are functions of the Twiss parameters, the dispersion, and the phase-space variables (see Appendix A).

The time evolution of the action can then be written: dJ x 1 dp dp dp = w1 + w2 2 . (73) dt − P0 dt P0 dt

In the classical approximation, we can take dp 0 in the limit → of small time interval, dt 0. In this approximation, the second → term on the right hand side in the above equation vanishes, and we are left only with damping. But since radiation is quantized, it makes no real sense to take dp 0... → Storage Ring Design 43 Part 1: Beam Dynamics with SR Quantum excitation of horizontal emittance

To take account of the quantization of synchrotron radiation, we write the time-evolution of the action as: dJ x 1 dp dp dp = w1 + w2 2 , (74) dt − P0 dt P0 dt dJ u u2 ∴ x = ˙ + ˙ (75) w1N h i w2N h 2 i2, dt − P0c P0 c where u is the photon energy, and N˙ is the number of photons emitted per unit time.

In Appendix B, we show that this leads to the equation for the evolution of the emittance, including both radiation damping and quantum excitation:

dε x 2 2 2I5 = εx + Cqγ , (76) dt −τx jxτx I2

Storage Ring Design 44 Part 1: Beam Dynamics with SR

Quantum excitation of horizontal emittance

The ﬁfth synchrotron radiation integral I5 is given by:

I5 = H ds, (77) ρ3 I | | where the “curly-H” function is deﬁned: H 2 2 = γxη + 2 αxηxηpx + βxη . (78) H x px

The “quantum constant” Cq is given by: ~ 55 13 Cq = 3.832 10 − m. (79) 32 √3mc ≈ ×

Storage Ring Design 45 Part 1: Beam Dynamics with SR Equilibrium horizontal emittance

Using Eq. (76) we see that there is an equilibrium horizontal emittance ε0, for which the damping and excitation rates are equal:

dε x 2 2 2I5 = 0 , ∴ ε0 = Cqγ . (80) dt εx=ε0 τx jxτx I2

The equilibrium horizontal emittance is given by: 2 γ I5 ε0 = Cq . (81) jx I2

Note that ε0 is determined by the beam energy, the lattice functions (Twiss parameters and dispersion) in the dipoles, and the bending radius in the dipoles.

Storage Ring Design 46 Part 1: Beam Dynamics with SR

Natural emittance

ε0 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.

Typically, third generation synchrotron light sources have natural emittances of order a few nanometres. With beta functions of a few metres, this implies horizontal beam sizes of tens of microns (in the absence of dispersion).

As the current is increased, interactions between particles in a bunch can increase the emittance above the natural emittance.

Storage Ring Design 47 Part 1: Beam Dynamics with SR Quantum excitation of vertical emittance

In many storage rings, the vertical dispersion in the absence of alignment, steering and coupling errors is zero, so y = 0. H 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 ∗:

13 Cq βy ε = ds. (82) y 3 55 jyI2 ρ I | | 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.

∗T. Raubenheimer, SLAC Report 387, p.19 (1991).

Storage Ring Design 48 Part 1: Beam Dynamics with SR

Quantum excitation of longitudinal oscillations

Quantum eﬀects excite longitudinal emittance as well as transverse emittance. Consider a particle with longitudinal coordinate z and energy deviation δ, which emits a photon of energy u.

u δ0 = δˆ0 sin θ0 = δˆsin θ . (83) − E0 αpc αpc z0 = δˆ0 cos θ0 = δˆcos θ. ωs ωs (84) u u2 ∴ ˆ 2 ˆ2 ˆ δ0 = δ 2δ sin θ + 2. (85) − E0 E0

Averaging over the bunch gives: 2 1 2 u 2 ˆ2 ∆σδ = h 2i where σδ = δ . (86) 2E0 2h i

Storage Ring Design 49 Part 1: Beam Dynamics with SR Quantum excitation of longitudinal oscillations

Including the eﬀects of radiation damping, the evolution of the energy spread is: dσ 2 1 2 δ ˙ 2 2 = 2 N u ds σδ . (87) dt 2E0C0 I h i − τz

Using Eq. (132) from Appendix B for N˙ u2 , we ﬁnd: h i 2 dσ δ 2 2 I3 2 2 = Cqγ σδ . (88) dt jzτz I2 − τz

2 We ﬁnd the equilibrium energy spread from dσ δ /dt = 0:

2 2 I3 σδ0 = Cqγ . (89) jzI2

The third synchrotron radiation integral I3 is given by: 1 I3 = ds. (90) ρ3 I | | Storage Ring Design 50 Part 1: Beam Dynamics with SR

Natural energy spread

The equilibrium energy spread determined by radiation eﬀects is: 2 2 I3 σδ0 = Cqγ . (91) jzI2

This 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.

The natural energy spread is determined essentially by the beam energy and by the bending radii of the dipoles. Note that the natural energy spread does not depend on the RF parameters (either voltage or frequency) .

Storage Ring Design 51 Part 1: Beam Dynamics with SR Natural bunch length

The bunch length σz in a matched distribution with energy spread σδ is: αpc σz = σδ. (92) ωs

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

Note: in a matched distribution, the shape of the distribution in phase space is the same as the path mapped out by a particle in phase space when observed on successive turns. Neglecting radiation eﬀects, a matched distribution stays the same on successive turns of the bunch around the ring.

Storage Ring Design 52 Part 1: Beam Dynamics with SR

Summary: radiation damping

Including the eﬀects of radiation damping and quantum excitation, the emittances vary as: t t ε(t) = ε(0) exp 2 + ε( ) 1 exp 2 . (93) − τ ∞ − − τ

The damping times are given by:

E0 jxτx = jyτy = jzτz = 2 T0. (94) U0

The damping partition numbers are given by:

I4 I4 jx = 1 , j y = 1 , j z = 2 + . (95) − I2 I2

The energy loss per turn is given by:

Cγ 4 5 3 U0 = E I2, C γ = 9 .846 10 − m/GeV . (96) 2π 0 ×

Storage Ring Design 53 Part 1: Beam Dynamics with SR Summary: equilibrium beam sizes

The natural emittance is:

2 I5 13 ε0 = Cqγ , C q = 3 .832 10 − m. (97) jxI2 ×

The natural energy spread and bunch length are given by:

2 2 I3 αpc σδ = Cqγ , σ z = σδ. (98) jzI2 ωs

The momentum compaction factor is:

I1 αp = . (99) C0

The synchrotron frequency and synchronous phase are given by:

2 eV RF ωRF U0 ωs = αp cos φs, sin φs = . (100) − E0 T0 eV RF

Storage Ring Design 54 Part 1: Beam Dynamics with SR

Summary: synchrotron radiation integrals

The synchrotron radiation integrals are: ηx I1 = ds, (101) I ρ 1 = (102) I2 2 ds, I ρ 1 I3 = ds, (103) ρ 3 I | | η 1 e ∂B = x + 2 = y (104) I4 2 k1 ds, k 1 , I ρ ρ ! P0 ∂x x 2 2 I5 = H ds, x = γxη + 2 αxηxηpx + βxη . (105) ρ 3 H x px I | |

Storage Ring Design 55 Part 1: Beam Dynamics with SR Appendix A: Damping of horizontal emittance

In this Appendix, we derive the expression for radiation damping of the horizontal emittance: dε x 2 = εx, (106) dt −τx where: 2 E0 I4 τx = T0, j x = 1 . (107) jx U0 − I2

To derive these formulae, we proceed as follows:

1. We ﬁnd an expression for the change of horizontal action of a single particle when emitting radiation with momentum dp .

2. We integrate around the ring to ﬁnd the change in action per revolution period.

3. We average the action over all the particles in the bunch, to ﬁnd the change in emittance per revolution period.

Storage Ring Design 56 Part 1: Beam Dynamics with SR

Appendix A: Damping of horizontal emittance

To begin, we note that, in the presence of dispersion, the action Jx is written: 2 2 2Jx = γxx˜ + 2 αxx˜p˜x + βxp˜x, (108) where:

x˜ = x ηxδ, andp ˜ x = px ηpx δ. (109) − − After emission of radiation carrying momentum dp , the variables change by: dp dp dp dp δ δ , x˜ x˜ + ηx , p˜x p˜x 1 + ηpx (1 δ) . (110) 7→ − P 7→ P 7→ − P − P 0 0 0 0 We write the resulting change in the action as:

Jx Jx + dJ x. (111) 7→

Storage Ring Design 57 Part 1: Beam Dynamics with SR Appendix A: Damping of horizontal emittance

The change in the horizontal action is: 2 dp dp dJ x 1 dp dp dp dJ x = w1 + w2 ∴ = w1 + w2 , (112) − P P dt − P dt P 2 dt 0 0 0 0 where, in the limit δ 0: → 2 w1 = αxxp x + βxp ηx(γxx + αxpx) ηpx (αxx + βxpx), (113) x − − and:

1 2 2 1 2 w2 = γxη + 2 αxηxηpx + βxη (αxηx + βxηpx ) px + βxp . (114) 2 x px − 2 x " Treating radiation as a classical phenomenon, we can take the limit dp 0 in the limit of small time interval, dt 0. In this approximation: → → dJ x 1 dp Pγ w1 w1 , (115) dt ≈ − P0 dt ≈ − P0c where Pγ is the rate of energy loss of the particle through synchrotron radiation.

Storage Ring Design 58 Part 1: Beam Dynamics with SR

Appendix A: Damping of horizontal emittance

To ﬁnd the average rate of change of horizontal action, we integrate over one revolution period: dJ x 1 Pγ = w1 dt. (116) dt −T P c 0 I 0

We have to be careful changing the variable of integration where the reference trajectory is curved: dC x ds dt = = 1 + . (117) c ρ c So:

dJ x 1 x = w1Pγ 1 + ds, (118) dt −T P c2 ρ 0 0 I where the rate of energy loss is:

Cγ 3 2 2 2 Pγ = c e B E . (119) 2π

Storage Ring Design 59 Part 1: Beam Dynamics with SR Appendix A: Damping of horizontal emittance

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

∂B y B = B0 + x . (120) ∂x Substituting Eq. (120) into (119), and with the use of (113), we ﬁnd (after some algebra!) that, averaging over all particles in the beam:

x I4 w1Pγ 1 + ds = cU 0 1 εx, (121) ρ − I I 2 where:

Cγ 4 1 ηx 1 U0 = cE I2, I 2 = ds, I 4 = + 2 k1 ds, (122) 2π 0 ρ2 ρ ρ2 I I and k1 is the normalised quadrupole gradient in the dipole ﬁeld:

e ∂B y k1 = . (123) P0 ∂x

Storage Ring Design 60 Part 1: Beam Dynamics with SR

Appendix A: Damping of horizontal emittance

Combining Eqs. (118) and (121) we have:

dε x 1 U0 I4 = 1 εx. (124) dt −T E − I 0 0 2

Deﬁning the horizontal damping time τx:

2 E0 I4 τx = T0, j x = 1 , (125) jx U0 − I2 the evolution of the horizontal emittance can be written:

dε x 2 = εx. (126) dt −τx

The quantity jx is called the horizontal damping partition number . For most synchrotron storage ring lattices, if there is no gradient in the dipoles then jx is very close to 1.

Storage Ring Design 61 Part 1: Beam Dynamics with SR Appendix B: Quantum excitation of horizontal emittance

In deriving the equation of motion (118) for the action of a particle emitting synchrotron radiation, we made the classical approximation that in a time interval dt , the momentum dp of the radiation emitted goes to zero as dt goes to zero.

In reality, emission of radiation is quantized, so writing “ dp 0” actually makes no sense. →

Taking into account the quantization of radiation, the equation of motion for the action (112) should be written:

2 2 dp dp dJ x u u dJ x = w1 + w2 ∴ = w1N˙ h i + w2N˙ h i , (127) − P P dt − P c P 2c2 0 0 0 0 where N˙ is the number of photons emitted per unit time.

The ﬁrst term on the right hand side of Eq. (127) just gives the same radiation damping as in the classical approximation. The second term on the right hand side of Eq. (127) is an excitation term that we previously neglected...

Storage Ring Design 62 Part 1: Beam Dynamics with SR

Appendix B: Quantum excitation of horizontal emittance

Averaging around the circumference of the ring, the quantum excitation term can be written: u2 1 u2 w2N˙ h i w2N˙ h i ds. (128) P 2c2 ≈ C P 2c2 0 0 I 0

Using Eq. (114) for w2, we ﬁnd that (for x ηx and px ηpx ) the excitation term can be written: 2 u 1 2 w2N˙ h i xN˙ u ds, (129) P 2c2 ≈ 2E2C H h i 0 0 0 I where the “curly-H” function x is given by: H 2 2 x = γxη + 2 αxηxηpx + βxη . (130) H x px

Storage Ring Design 63 Part 1: Beam Dynamics with SR Appendix B: Quantum excitation of horizontal emittance

Including both (classical) damping and (quantum) excitation terms, and averaging over all particles in the bunch, we ﬁnd that the horizontal emittance evolves as:

dε x 2 1 2 = εx + N˙ u x ds. (131) dt −τ 2E2C h iH x 0 0 I We quote the result (from quantum radiation theory):

2 2 Pγ N˙ u = 2 Cqγ E0 , (132) h i ρ where the “quantum constant” Cq is: ~ 55 13 Cq = 3.832 10 − m. (133) 32 √3 mc ≈ ×

Storage Ring Design 64 Part 1: Beam Dynamics with SR

Appendix B: Quantum excitation of horizontal emittance

Using Eq. (132), and Eq. (119) for Pγ, and the results:

E0 Cγ 4 jxτx = 2 T0, U 0 = cE 0I2, (134) U0 2π we ﬁnd that Eq. (131) for the evolution of the emittance can be written:

dε x 2 2 2 I5 = εx + Cqγ (135) dt −τx jxτx I2 where the ﬁfth synchrotron radiation integral I5 is given by:

x I5 = H ds. (136) ρ3 I | | Note that the excitation term is independent of the emittance: it does not simply modify the damping time, but leads to a non-zero equilibrium emittance.

Storage Ring Design 65 Part 1: Beam Dynamics with SR