Synchrotrons and beamlines

These measurements will sometimes require a dependence on light po- larisation or a tuneable photon energy, which necessitates the use of syn- chrotron light. An accessible introduction to synchrotron radiation and some applications can be found in [37]. The following section summarises some

50 Chapter 3. Methods

of the key aspects from this reference, relating to the work carried out in the thesis.

The basic physics underpinning synchrotron operation is that acceler- ated charged particles emit electromagnetic radiation. Synchrotrons then provide a closed loop for the electron to follow under vacuum. Partly the acceleration is a centripetal acceleration provided by bending magnets. The acceleration is given straightforwardly in this case by the Lorentz equation and the power radiation by relativistic electrons on a circular orbit is given by Schwinger’s formula. Together, these provide a loss in energy within a complete orbit around the synchrotron of [37]:

∆Ee = 4π 3 e2 R E mc2 4 (3.22)

whereE,mandeare the electron energy, rest mass and charge respectively, and R is the radius of the orbit (or synchrotron ring radius). This loss in energy is compensated by a radio frequency cavity which stabilises the ring energy, as well as quadrupolar magnets for stabilising the trajectory. Addi- tionally, at the relativistic speeds of the electrons in their circular path, the emitted synchrotron light is highly collimated in the propagation direction. This is given by the so-called vertical half-opening angle (fig.3.9a), which is

ψ =γ−1 ≈mc2/E ≈1/(1957×E[in GeV]), which for a3GeV storage ring would give an angle ofψ≈0.01o[37].

The energy spectrum produced by bending magnets is very broad and not so easily tuneable and the polarisation is not so easily changed (being mostly linearly horizontal polarised). A popular way to easily tune the en- ergy and polarisation of the light is through an undulator (or a wiggler, which is similar in principle). This is a device which is also placed in the straight sections of the storage ring. It typically consists of a periodic array of permanent or electromagnets (fig. 3.9b,c). The magnetic force experi- enced by the electrons cause them to follow an oscillatory path through the undulator producing highly collimated synchrotron radiation with each os- cillation (fig. 3.9a). The sharper bends from the oscillations shifts the peak

3.3. Practical considerations 51 e- a b gap phase

c _{linear vertical polarisation}

linear horizontal polarisation

FIGURE3.9: a) Definitions of physical quantities relating to the trajectory of electrons through the undulator. An un- dulator (as opposed to a wiggler), has a ‘wiggling angle’

θ∼γ−1_{. b,c) Schematics of an APPLE II type undulator de-}

sign allowing for variable polarisation of light. Alignment of poles producing a vertical field leads to horizontal os- cillations and linearly horizontal polarised light (b). Phase offsetting poles forcing in-plane magnetic field lines creates vertical oscillations and therefore linearly vertical polarised

light (c). a) Adapted from [37], b,c) adapted from [38].

energy of the emittance to higher energies, since these require greater ac- celeration. The coherent superposition of synchrotron radiation from the oscillations then provides a greater intensity of radiation (so the intensity is additionally proportional to the length of the undulator). The amplitude of the oscillations can be easily changed (in some undulators) by changing the size of the gap between the magnetic arrays (fig.3.9b,c), therefore changing the peak in photon energy. A smaller gap, increases the magnetic force expe- rienced by the electrons, providing greater amplitude oscillations, yielding higher energy photons. The wavelength of the light emitted by the undula- tor is given by the following [37]:

λ= λu 2γ2 1 +K 2 2 +γ 2_θ2 (3.23)

whereλu is the period of an oscillation through the undulator given by the

distance between magnetic poles (fig. 3.9b),K = _2πmce λuB, for a magnetic

field strengthB, andθis the angle of the emitted radiation to the undulator axis (fig. 3.9a). As well as the fundamental wavelength, higher harmonics

52 Chapter 3. Methods

in energy (shorter wavelengths) are emitted withλn=λ/n.

In addition to the tuneable photon energy, the polarisation vector of the emitted light can be tuned. The polarisation vector is governed by the direc- tion of the oscillations in the trajectory of the electrons through the undula- tor. The direction of the oscillations through the undulator is defined by the magnetic field set up by the periodic magnetic arrays. This can be achieved in different ways but an example of a modern design of undulator is the AP- PLE II type undulator shown in figure3.9b,c. Linearly horizontal polarised light is created when the electrons oscillate in-plane (horizontally), requir- ing magnetic field lines out-of-plane (vertically). This is achieved by align- ing poles such that out-of-plane field lines are aligned vertically across the gap (fig. 3.9b). Conversely, linearly vertical polarised light requires in-plane magnetic field lines. This is achieved by offsetting the adjacent magnetic arrays (a ‘phase’ shift). It is required that the out-of-plane poles are op- posite across the gap, since magnetic field lines cannot cross and this then forces the magnetic field to be mostly in-plane (fig.3.9c). Instead of shifting the magnetic arrays such that two out-of-plane magnetic poles are aligned laterally, it is possible to adjacently align an in-plane and an out-of-plane array. This creates a magnetic field which forces the electrons to follow a circular trajectory through the undulator, which is effectively a combination of linearly horizontal and vertical polarised light. The handedness of the trajectory through the undulator then defines the polarisation vector of the light to be circularly left or circularly right polarised.