# Presentation: 897110

## Longitudinal Dynamics in Storage Rings

**Longitudinal Dynamics in Storage Rings**
**David Robin**

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## Particle Storage Rings

- Particle bunches

- Optics elements

- quadrupoles

- dipoles

- RF cavity

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**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## Particle Storage Rings

**Length and Momentum**
**Lo ******** Revolution Length of the Reference Particle**
**Po ******** Momentum of the Reference Particle**

- Reference Particle

******L/Lo, ********P/Po**

- Particle

******L**

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**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## At a given instant t:

**Consider two particles with different momentum on parallel trajectories:**

** ****This path length dependence on momentum applies everywhere,**
**also in straight trajectories.**

**At a given instant *****t*****:**

**But****:**

** ****The effect quickly vanishes for relativistic particles.**

** ****Higher momentum particles precede the ones with lower momentum.**

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## In the example (sector bending magnet) L > L0 so that  > 0

**In the example (sector bending magnet) *****L***** > *****L*****0**** so that ********** > 0**
**Higher energy particles will leave the magnet later.**

**where********** ****is constant**

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## Momentum compaction, , is the change in the closed orbit length as a function of momentum.

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**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## Particle Storage Rings

**Gain Energy in the RF Cavity**

- dipoles

- RF cavity

- 7

**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

##  We define as V  the voltage gain for the particle. 

** ****We define as *****V*****  the voltage gain**** for the particle. **
***V***** depends only on the particle trajectory and includes the contribution of every electric field present in the area (RF fields, space charge fields, fields due to the interaction with the vacuum chamber, ...)**

** ****The particle can also experience ****energy variations *****U*****(*****E*****) that depend also on its energy****, as for the case of the radiation emitted by a particle under acceleration (synchrotron radiation when the acceleration is transverse).**

** ****The energy gain for a particle that moves from ****A**** to ****B**** is given by:**

** ****The total energy variation will be given by the sum of the two terms:**

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## The energy variation for the reference particle is given by:

**The energy variation for the reference particle is given by:**

**For particle with energy *****E = E*****0***** + ************E***** and orbit position *****s = s*****0***** + ************s*****:**

**Where the last expression holds for the case where**
*********s << L*****0**** (reference orbit length) and ***********E << E*****0****.**

**In this approximation we can express the average rate of change of the energy respect to the reference particle energy by:**

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## Finally, by defining the quantities:

**Finally, by defining the quantities:**

**We obtain the equations of motion for the longitudinal plane:**

**We will study the case of storage rings where *****dV*****/*****ds***** is mainly due to the RF system used for restoring the energy lost per turn by the beam.**

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## This expression is the well known damped harmonic oscillator equation, which has the general solution:

**This expression is the well known damped harmonic oscillator equation, which has the general solution:**

**The stable solution represents an oscillation with frequency ****2************** and with exponentially decreasing amplitude.**

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##  The damping time 1/D (~ ms for e-, ~ 13 hours LHC at 7 TeV) is usually much larger than the period of the longitudinal oscillations  1/2(~s). This implies that the damping term can be neglected when calculating the particle motion for t << 1/D :

** ****The case of damped oscillations is exactly what we want for storing particles in a storage ring.**

** ****The synchrotron radiation (SR) emitted when particles are on a curved trajectory satisfies the condition. The SR power scales as: **

** ****Typically, synchrotron radiation damping is very efficient in electron storage rings and negligible in proton machines.**

** ****The ****damping time**** ****1/***********D***** (~ ms for e****-****, ~ 13 hours LHC at 7 TeV) is usually much larger than the period of the longitudinal oscillations  ****1/2**************(~********s). This implies that the damping term can be neglected when calculating the particle motion for *****t *****<< ****1/***********D***** :**

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## Let’s consider a storage ring with reference trajectory of length L0:

**Let’s consider a storage ring with reference trajectory of length *****L*****0****:**

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## For our storage ring:

**For our storage ring:**

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## If

**If**

**Additionally:**

**Synchrotron Oscillations**

**A different set of variables:**

**For***********s << L*****0**** and ***********E << E*****0****.**

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## Particle Storage Rings

- Reference Particle

- Particle

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**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## We just found:

**We just found:**

**This equation represents an ellipse in the**
**longitudinal phase space {**********, **********}**

**With damping:**

**In rings with negligible synchrotron radiation (or with negligible non-Hamiltonian forces, the longitudinal emittance is conserved.**

**This is the case for heavy ion and for most proton machines. **

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## Two synchronous phases  one stable one unstable

**Two synchronous phases ******** one stable one unstable**

**But**

**For positive charge particles:**

**For negative charge particles all the phases are shifted by **********.**

**Crossing the transition energy during energy ramping requires a phase jump of ~ ********

**We define as *****transition energy***** the energy at which ********** changes sign.**

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## So far we have used the small oscillation approximation where:

**So far we have used the *****small oscillation approximation***** where:**

**In the more general case of larger phase oscillations:**

** ****Stable and unstable orbits exist. The two regions are separated by a special trajectory called *****separatrix***

** ****Larger amplitude orbits have smaller synchrotron frequencies**

**And by Numerical integration:**

** ****For larger amplitudes, trajectories in the phase space are not ellipsis anymore.**

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## The RF bucket is the area of the longitudinal phase space where a particle orbit is stable

**The RF bucket is the area of the longitudinal phase space where a particle orbit is stable**

**The *****momentum acceptance***** is defined as the maximum momentum that a particle on a stable orbit can have.**

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##  In electron storage rings, the statistical emission of synchrotron radiation photons generates gaussian bunches.

** ****In electron storage rings, the statistical emission of synchrotron radiation photons generates gaussian bunches.**

** ****The over voltage *****Q***** is usually large so that the core of the bunch “lives” in the small oscillation region of the bucket. The equation of motion in the phase space are elliptical:**

** ****If ***********p*****/*****p*****0**** is the *****rms relative momentum spread***** of the gaussian distribution, then the *****rms bunch length***** is given by:**

** ****In the case of heavy ions and of most of protons machines, the whole RF bucket is usually filled with particles. The bunch length *****l***** is then proportional to the difference between the two extreme phases of the separatrix:**

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##  A charged particle when accelerated radiates.

** ****A charged particle when accelerated radiates.**

** ****In high energy storage rings transverse acceleration induces significant radiation (synchrotron radiation) while longitudinal acceleration generates negligible radiation (****1/**********2****).**

** ****Synchrotron radiation plays a major role in the dynamics of an electron storage ring**

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##  For relativistic electrons:

** ****For relativistic electrons:**

** ****In the case of dipole magnets with constant radius **********(*****iso-magnetic***** case):**

** ****The average radiated power is given by: **

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## By performing the calculation one obtains:

**By performing the calculation one obtains:**

**Where *****D***** depends on the lattice parameters.**
**For the *****iso-magnetic separate function***** case:**

**Analogously, for the transverse plane:**

**and**

**Sometimes the *****partition numbers***** are used:**

**with**

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##  We saw that synchrotron radiation induces damping in all the planes.

** ****We saw that synchrotron radiation induces damping in all the planes.**

** ****Because of that, one would expect that all the particles should collapse in a single point.**

** ****This **_**does not**_** happen because of the ****quantum nature of synchrotron radiation****.**

** ****In fact, photons are randomly emitted in quanta of discrete energy and every time a photon is emitted the parent electron undergoes to a “jump” in energy.**

** ****Such a process perturbs the electron trajectories exciting oscillations in all the planes.**

** ****These oscillations grow until reaching **_**equilibrium**_** when balanced by the radiation damping.**

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##  At equilibrium the momentum spread is given by:

** ****At equilibrium the momentum spread is given by:**

** ****For the horizontal emittance at equilibrium:**

**where:**

** ****In the vertical plane, when no vertical bend is present, the synchrotron radiation contribution to the equilibrium emittance is very small and the vertical emittance is defined by machine imperfections and nonlinearities that couple the horizontal and vertical planes:**

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## At this point we have discussed the motion of a particle in an accelerator for all the planes.

**At this point we have discussed the motion of a particle in an accelerator for all the planes.**

**Damping:**** several ms for electrons, ~ infinity for heavier particles**

**It can be helpful remarking the time scale for the different phenomena governing the particle dynamics.**

**Betatron oscillations:**** ~ tens of ns**

**Synchrotron oscillations:**** ~ tens of ********s**

**Revolution period:**** ~ hundreds of ns to ********s**

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## 7. Calculate the synchrotron frequency and tune for the ALS when the ring is operating in the following configuration: RF = 500 MHz, harmonic number = 328, E = 1.9 GeV, momentum compaction = 0.00137, energy lost per turn = 279 keV, peak RF voltage = 1.3 MV.

**7. Calculate the synchrotron frequency and tune for the ALS when the ring is operating in the following configuration: RF = 500 MHz, harmonic number = 328, E = 1.9 GeV, momentum compaction = 0.00137, energy lost per turn = 279 keV, peak RF voltage = 1.3 MV.**

**6. Calculate the ratio between the synchrotron radiation power radiated by a particle in the Large Hadron Collider (LHC), the proton collider at CERN, and the one radiated by a particle in the Advanced Light Source (ALS), the electron storage ring in Berkeley. The magnet bending radius is ~2810 m and ~5 m and the particle energy is 7000 GeV and 1.9 GeV for the LHC and the ALS respectively. (Remember that the electron mass is 9.1095 10****-31**** Kg while the proton one is 1.6726 10****-27**** Kg) **

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