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 P = 2 3rec γ3β4E ρ2 , Instantaneous radiated power re = (Ze)2 4πǫ0mc2 , Classical electromagnetic radius ∆t = L βc , Time needed to travel through an arc of length L ∆E = P ∆t = P L βc Average total energy loss = 2 3rec γ3β4E ρ2 L βc = 2 3re γ3β3E ρ2 L.

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One can write the energy loss as a function of the arc length L, and of the bending radius ρ: ∆E (L, ρ) = 2 3reγ3β3E L ρ2 ,

• r, as a function of the instantaneous kick θ = L

ρ:

∆E (L, θ) = 2 3reγ3β3E θ2 L .

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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 3reγ3 θ2 L . re 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,θ2 = θ2

x + θ2 y:

rfac = 1 3aradγ3 θ2

x + θ2 y

L

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Occasionally, such expressions are written using the instantaneous curvatures hx and hy, hx = θx L , hy = θy L , so that in the code rfac can be found written either ways: rfac = 1 3aradγ3 θ2

x + θ2 y

L = 1 3aradγ3 h2

x + h2 y

• L

(1)

pt = orbit(6)

• rbit(2) = orbit(2) - rfac * (one + pt) * orbit(2)
• rbit(4) = orbit(4) - rfac * (one + pt) * orbit(4)
• rbit(6) = orbit(6) - rfac * (one + pt) ** 2

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TWISS

◮ tmbend:

rfac = 1 3 aradγ3 h2

x + h2 y

• L (1 + hx) (1 − tan (E1,2) x)
• L1,2

half effects at the two ends, where arad = classical electromagnetic radius h = inverse of the nominal curvature radius L1 = L 2 (1 + hx) (1 − tan (E1) x) = path length for the first 1/2 magnet L2 = L 2 (1 + hx) (1 − tan (E2) x) = path length for the second 1/2 magnet with instantaneous curvature at entrance and exit: hx = (h + ∆h) +k1

• x − h y2

2

• +k1sy

+ k2

2

• x2 − y2

, hy = −k1y +k1sx −k2xy. (2) k0, k1, k1s, k2, k2s 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.

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TWISS

◮ tmcorr:

rfac = 1 3 aradγ3 θ2

x + θ2 y

L half effect at the two ends

◮ tmquad: from the definition of quadrupole kick

θ2

x + θ2 y = (k1L)2 x2 + (−k1L)2 y2 = (k1L)2

x2 + y2 it derives that rfac = 1 3 aradγ3 θ2

x + θ2 y

L = 1 3 aradγ3k2

1L

• x2 + y2

half effect at the two ends

◮ tmsext: from the definition of sextupole kick:

θx = k2 2

• x2 − y2

θy = k2xy θ2

x + θ2 y =

k2 2

• x2 − y22

+ (k2xy)2 = k2

2

4

• x2 + y22

(and k2 has the units of m−3) it derives that rfac = 1 3 aradγ3 k2

2

4

• x2 + y22

L

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TWISS

◮ tmmult, tmoct, tmrfmult, tmcrab follow the scheme:

rfac = 1 3 aradγ3 θ2

x + θ2 y

Lrad . where Lrad 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.

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TRACK

Subroutines in Track usually define a variable curv =

• θ2

x + θ2 y

L which is used as an input parameter to trphot();

◮ ttmult:

◮ for each particle, at entrance and at exit ◮ defines const = 1

3 aradγ3, uses elrad

◮ defines

curv = sqrt((dipr + dxt(jtrk))**2 + (dipi + dyt(jtrk))**2) / elrad

◮ then

if (!quantum)

rfac = const · curv2 · elrad

else trphot(el,curv,rfac,deltas);

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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.

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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

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Potential issue in TRACK

◮ Issues in Track

The subroutine trphot is capable of computing the emission of many photons, but it is

• nly 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, nγ = 5αf 2 √ 3 βrelγrelθ⊥, with αf as the fine-structure constant, αf =

1 4πǫ0 e2 c , and then slices the element in

nslices = 10 · nγ + 3 slices.

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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.

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