Energy Calibration

The precession equation of motion for the spin $\vec{S}$ of an electron at rest in a magnetic field is the Larmor equation with the angular velocity $\vec \Omega = \vec{B}\cdot {ge}/{2{m_e}c}$. Using Lorentz transformations the last equation can be rewritten [6] for a highly relativistic ( $\frac{1}{\gamma}\ll 1$) electron moving in the electromagnetic field of an accelerator, which, when substituted into the Larmor equation is called the Thomas-BMT Equation [7]. An electron traveling along the design orbit will only see the guiding magnetic dipole field $\vec{B}_{\perp}$ as well as the accelerating electric field. Therefore the spin precession frequency in the particle's rest frame (i.e. in machine coordinates) is [8] [9]:

$$\vec{\Omega_{sp}} = \frac{e\vec{B}_\perp}{{m_e}c\gamma} \cdot a\gamma = \vec{\omega_0} \cdot \nu$$ (4)

where $a$ is the anomalous magnetic moment of the electron, $\vec{\omega_0} = {e\vec{B}_\perp}/{{m_e}c\gamma}$ the revolution frequency in the storage ring and $\nu = a\gamma$ is the spin tune ( $a\gamma = 5.45$ in the SLS at $E=2.4$ GeV). An approximated solution for the flat machine leads to $\vec{S}_z$ remaining constant while $\vec{S}_x, \vec{S}_y$ are statistically distributed among the electrons in the beam and cancel. The time-integration of the remaining component in the ensemble leads to a certain degree of polarization.

If a time-varying radial magnetic field is applied in resonance with the electron's spin revolutions the mean spin vector can be tilted into the horizontal plane which (in conjunction with spin diffusion) leads to zero polarization. Thus finding the resonant depolarization frequency is equivalent to finding the beam energy (eq. 4).

In the experimental setup a sinusoidal signal is fed into a vertical tune kicker magnet. The frequency of this signal is swept over pre-defined intervals. As soon as the sweep frequency hits the resonant depolarizing frequency and the beam gets depolarized, the product of beam current and lifetime drops and loss monitor (a pair of scintillators installed downstream of the in-vacuum undulator U24) coincidence signals rise due to an increased number of pairs of Touschek scattered electrons (see fig. 3).

Figure 3: The resonant depolarizing frequency is reached at 580 kHz.

Due to a sampling with $\omega_0$ there is the ambiguity that the resonance has to be distinguished from its mirror above the half-integer spin tune. This is done by a slight variation of the RF main frequency leading to a change in beam energy and thus to a shift of the resonance in the same direction and to a shift of the mirror in the opposite. The resonance also carries sidebands (the spin tune is modulated by synchrotron oscillations) which are equally distanced from the resonant frequency by multiples of the synchrotron tune $Q_s$; a variation of the RF voltage leading to a change in distance between main resonance and sidebands allows the identification of the unshifted main resonance (see fig. 4).

Figure 4: Dip patterns of two sweeps differing due to a $11\%$ increase of RF voltage. The main resonance and its sideband are separated by the synchrotron tune $Q_s=6.17 \cdot 10^{-3}$ respectively by $Q_s+\Delta Q_s=6.48 \cdot 10^{-3}$ (an increase of $Q_s$ by $\approx 5\%$).

Once these checks had been done the precise energy calibration was obtained with a fit using the Froissart-Stora formula for resonance crossing [10]. We specify the energy uncertainty with the half-FWHM of the fit, since the resonance uncertainty is independent of the signal generator driving the sweep. Applied to the data in fig. 5 this leads to an energy of $(2.4361\pm0.00024)$ GeV.

Figure 5: Resonance with fit according to the Froissart-Stora equation for resonance-crossing.

By decreasing the sweep dwell and the kicker power a higher accuracy can be reached. A series of measurements [1] has been performed finally leading to the energy calibration of $E=(2.4361 \pm 0.00018)$ GeV with a very low uncertainty of $\frac{\Delta E}{E}=4.5\cdot 10^{-5}$ which we expect to be able to under-run in upcoming measurements.

The energy calibration results are supported by independent measurements [11] of characteristic line spectra of undulator U24 revealing $E=(2.44\pm0.02)$ GeV which is in excellent agreement to the presented measurements.