Synchrotron Radiation in MAD-X Andrea Latina A. Latina (CERN) - - PowerPoint PPT Presentation

Synchrotron Radiation in MAD-X Andrea Latina A. Latina (CERN) - Mar 3, 2017

Physics recap The average energy loss by a radiating particle is: ∆ E = P ∆ t , with 3 r e c γ 3 β 4 E P = 2 , Instantaneous radiated power ρ 2 ( Ze ) 2 r e = 4 πǫ 0 mc 2 , Classical electromagnetic radius ∆ t = L β c , Time needed to travel through an arc of length L ∆ E = P ∆ t = P L Average total energy loss β c 3 r e c γ 3 β 4 E γ 3 β 3 E = 2 β c = 2 L L . 3 r e ρ 2 ρ 2 2/13 A. Latina - Synchrotron Radiation in MAD-X

One can write the energy loss as a function of the arc length L , and of the bending radius ρ : ∆ E ( L , ρ ) = 2 3 r e γ 3 β 3 E L ρ 2 , or, as a function of the instantaneous kick θ = L ρ : 3 r e γ 3 β 3 E θ 2 ∆ E ( L , θ ) = 2 L . 3/13 A. Latina - Synchrotron Radiation in MAD-X

In MAD-X In MAD-X this energy loss is computed with the assumption that β ≈ 1, splitting the magnets in two halves, and normalising to the beam energy, E . In the code such a normalised energy loss is typically called rfac , and reads: rfac = ∆ E ( L / 2 , θ/ 2 ) E = 1 3 r e γ 3 θ 2 L . r e is replaced by arad , a user-defined parameter characterising the classical electromagnetic radius of the particle being considered (see formula above for a particle of arbitrary mass m and charge state Z ). The total kick is separated in its horizontal and vertical components, θ 2 = θ 2 x + θ 2 y : 3 arad γ 3 θ 2 x + θ 2 rfac = 1 y L 4/13 A. Latina - Synchrotron Radiation in MAD-X

Occasionally, such expressions are written using the instantaneous curvatures h x and h y , h x = θ x h y = θ y L , L , so that in the code rfac can be found written either ways: 3 arad γ 3 θ 2 x + θ 2 = 1 y rfac L (1) = 1 3 arad γ 3 � h 2 x + h 2 � L y pt = orbit(6) orbit(2) = orbit(2) - rfac * (one + pt) * orbit(2) orbit(4) = orbit(4) - rfac * (one + pt) * orbit(4) orbit(6) = orbit(6) - rfac * (one + pt) ** 2 5/13 A. Latina - Synchrotron Radiation in MAD-X

TWISS ◮ tmbend : rfac = 1 3 arad γ 3 � � h 2 x + h 2 L ( 1 + hx ) ( 1 − tan ( E 1 , 2 ) x ) y � �� � L 1 , 2 half effects at the two ends, where arad = classical electromagnetic radius h = inverse of the nominal curvature radius L 1 = L 2 ( 1 + hx ) ( 1 − tan ( E 1 ) x ) = path length for the first 1/2 magnet L 2 = L 2 ( 1 + hx ) ( 1 − tan ( E 2 ) x ) = path length for the second 1/2 magnet with instantaneous curvature at entrance and exit: � � x − h y 2 � x 2 − y 2 � + k 2 = ( h + ∆ h ) + k 1 + k 1 s y , h x 2 2 (2) h y = − k 1 y + k 1 s x − k 2 xy . k 0 , k 1 , k 1 s , k 2 , k 2 s are divided by ( 1 + δ ) to keep into account the energy deviation of the particle. These equations are identical to those implemented in MAD-8. 6/13 A. Latina - Synchrotron Radiation in MAD-X

TWISS ◮ tmcorr : 3 arad γ 3 θ 2 x + θ 2 rfac = 1 y half effect at the two ends L ◮ tmquad : from the definition of quadrupole kick y = ( k 1 L ) 2 x 2 + ( − k 1 L ) 2 y 2 = ( k 1 L ) 2 � x 2 + y 2 � θ 2 x + θ 2 it derives that 3 arad γ 3 θ 2 x + θ 2 rfac = 1 = 1 � x 2 + y 2 � y 3 arad γ 3 k 2 half effect at the two ends 1 L L ◮ tmsext : from the definition of sextupole kick: θ x = k 2 � x 2 − y 2 � 2 θ y = k 2 xy � k 2 x 2 − y 2 �� 2 + ( k 2 xy ) 2 = k 2 � � x 2 + y 2 � 2 θ 2 x + θ 2 2 y = 2 4 (and k 2 has the units of m − 3 ) it derives that � x 2 + y 2 � 2 3 arad γ 3 k 2 rfac = 1 2 4 L 7/13 A. Latina - Synchrotron Radiation in MAD-X

TWISS ◮ tmmult , tmoct , tmrfmult , tmcrab follow the scheme: 3 arad γ 3 θ 2 x + θ 2 rfac = 1 y . L rad where L rad is “a fictitious length, originally only used to compute synchrotron radiation effects. A non-zero LRAD in conjunction with OPTION, THIN FOC=true takes into account the weak focussing of bending magnets.” ◮ tmdpdg and tmbb do not implement radiation. 8/13 A. Latina - Synchrotron Radiation in MAD-X

TRACK Subroutines in Track usually define a variable � θ 2 x + θ 2 y curv = L which is used as an input parameter to trphot(); ◮ ttmult : ◮ for each particle, at entrance and at exit ◮ defines const = 1 3 arad γ 3 , uses elrad ◮ defines curv = sqrt((dipr + dxt(jtrk))**2 + (dipi + dyt(jtrk))**2) / elrad ◮ then if (!quantum) rfac = const · curv 2 · elrad else trphot(el,curv,rfac,deltas); 9/13 A. Latina - Synchrotron Radiation in MAD-X

TRACK ◮ ttcorr : computes curv = sqrt(dpx**2 + dpy**2) / el for trphot(), and rfac = arad * gammas**3 * (dpx**2 + dpy**2) / (three * el) ◮ A significant limitation of this implementation (an error?) is that the synchrotron radiation effects of tmcorr are computed for the first particle, and then applied to all particles equally. ◮ ttrfmult, tttquad, tttdipole : implement radiation effects just like above ◮ trphot : computes the average number of photons emitted, given a bending curvature, and then returns the sum of the energy losses due to each photons ◮ This concentrates the effect of all photons in one location, not considering the deflection due to the energy lost with each photon. 10/13 A. Latina - Synchrotron Radiation in MAD-X

EMIT The module calculates the effects of radiation damping. It computes the damping matrix (eigenvalues), and propagates it through the lattice, invoking the Twiss routines in order to extract the first-order map of each element. For each element, the misalignment are applied, then the effects of the radiation and of the damping are computed with the routine emdamp (), which computes the matrix of eigenvalues (at element entrance and at element exit) using the scheme illustrated below. The effect of radiation damping is computed for the elements: code_rbend , code_sbend , code_quadrupole , code_sextupole , code_octupole , code_multipole , code_rfcavity , code_hkicker , code_kicker , code_vkicker , code_tkicker . Note: RF-Multipole is missing 11/13 A. Latina - Synchrotron Radiation in MAD-X

Potential issue in TRACK ◮ Issues in Track The subroutine trphot is capable of computing the emission of many photons, but it is only called at the entrance and at the exit of each element. This corresponds to assuming that no photons are emitted inside the element, and therefore that the energy is constant within the element, which is of course incorrect. This might be overcome bu using many slicies. One could follow the approach used in PLACET where, given a certain instantaneous kick, θ ⊥ , one computes the average number of photons emitted, n γ = 5 α f β rel γ rel θ ⊥ , √ 2 3 e 2 1 with α f as the fine-structure constant, α f = � c , and then slices the element in 4 πǫ 0 n slices = 10 · n γ + 3 slices. 12/13 A. Latina - Synchrotron Radiation in MAD-X

Potential improvements ◮ Abandon the assumption β ≈ 1 ◮ Enrich MAD-X dictionary with pre-defined values for arad (better: improve the BEAM command, to compute arad internally) ◮ Suggest a split of the radiating elements in more than two parts, and apply the radiation effects to each slice. ◮ Implement radiation damping effects of RF-Multipoles in Emit. 13/13 A. Latina - Synchrotron Radiation in MAD-X