May 16, 2006 - They have been calculated as a function of vessel radius for .... 5.00. 21. Aluminium. 3.62. 0.8. 3.33. 6.5. Gold. 5.93. 3.0. 2.00. 20. Iron ..... cm at room temperature and a number of valence electrons that is not well defined,.
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ASTEC-ID-040
Longitudinal Resistive Wall Wakefields for the ILC Positron Undulator Vessel DUNCAN SCOTT
MAGNETICS & RADIATION SOURCES GROUP: DARESBURY LABORATORY
ABSTRACT The DC, AC and anomalous skin effect longitudinal resistive wall wakefields of a round pipe have been computed for the ILC positron source undulator vessel. They have been calculated as a function of vessel radius for copper, aluminium, gold, iron and stainless steel materials and for the minimum, nominal and maximum ILC parameters. Gaussian, trapezium and ‘Linac Coherent Light Source-type’ charge distributions have been modelled as examples. To help assess the effects of cryogenic temperatures (the undulator is superconducting) the wakes have been calculated at 273K and 77K. For an undulator aperture of 5.6mm the increase in energy spread is 2% for gold at 273K with a Gaussian bunch profile and the minimum ILC parameters. The induced energy spread decreases with temperature and at 77K is 1.3% for gold and a Gaussian bunch.
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1. Introduction In the baseline design the positrons for the ILC are produced via pair production in a metallic target using a photon drive beam. The photons are produced using a helical undulator and the main ILC electron beam at 150GeV. For a polarised positron beam there needs to be approximately 200m of active undulator. The helical undulator is a super-conducting bi-filar helix design that has a circular vessel. To create the necessary on axis magnetic field in the helical undulator the aperture of the magnet will be less than 10mm in diameter. As the main electron beam is passing through such a long narrow gap vessel the wakefield effects of the vessel need to be assessed. A general introduction to resistive wall wakefields has already been given [1] and can be found elsewhere [2, 3] in this note the longitudinal wakefield effects are discussed. These change the energy of the beam and induce and energy spread within a bunch. In the minimal set of ILC parameters there is ~154ns between bunches and so only short range wakefields are considered. Resistive wall wakefields of a cylindrical pipe with finite conductivity have been calculated for various bunch charge distributions, vessel materials, vessel radius and the different ILC beam parameter sets. Relevant parameters for the beam and undulator are given in Table 1. The minimum vessel radius is ~2mm, but the current nominal vessel radius is ~2.8mm. Wakes have been calculated up to a 5mm radius. Parameter ILC Mode Rms Bunch Length
Symbol
Unit
σz
10-6 m
Number of electrons Beam Energy Nominal RMS Energy Spread Vessel Length Vessel radius (min) Vessel radius (nominal)
N E
1010 GeV %
L
m mm mm
σE E
Min 150 1
Value Nominal 300 2 150 0.05
Max 500 2
200 2 2.8
Table 1: undulator vessel and ILC beam parameters.
2. Conductivity and Impedance Models In the Drude-Sommerfield model of electron conductivity [4] conduction electrons are treated as an ideal gas obeying Maxwell-Boltzmann or Fermi-Dirac statistics respectively. The conductivity is related to the mean free time between collisions (or metal relaxation time), τ , of the conduction electrons moving in an applied field, or equivalently the mean free path between collisions, l , where:
l = v Fτ , and vF is the Fermi velocity for the particular metal.
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If the field the electrons are moving in is constant during this time then the conductivity effects are said to be DC. The DC conductivity of a metal, σ DC , is given by: ne 2τ σ DC = , me where n is the density of conduction electrons, e their charge and me the electron mass. In general the wakefields/impedances can be defined in terms of a characteristic distance, s0 , which is a function of the pipe radius, b , it’s DC conductivity, the speed of light, c , and the permittivity of free space, ε 0 : 1
⎛ 2cb 2ε 0 ⎞ 3 s0 = ⎜ ⎟ . ⎝ σ DC ⎠ 0 For a round metallic pipe the DC monopole longitudinal impedance, Z||,DC , is: −1
Z s Z||,0DC ( k ) = 0 02 2π b
⎡1 − isign ( ks0 ) iks ⎤ + 0⎥ , ⎢ 12 2 ⎥⎦ ks0 ⎢⎣
where Z 0 = ( cε 0 ) is the vacuum impedance. As the undulator is superconducting the wakefield effects at two temperatures, 273K and 77K have been looked at for a variety of different materials. Below 77K values for the conductivity depend upon many factors such as the purity of the material and grain size. Table 2 gives values for σ DC and τ for copper, aluminium, gold and iron [4]. The vessel could be made out of aluminium or stainless steel and then perhaps coated in a thin layer of gold or copper if necessary. Iron has been looked at as values for stainless steel are difficult to obtain. (In section 8 an estimation of the values for stainless steel type 316L has been made.) −1
σ DC
Unit Copper Aluminium Gold Iron
(10 ) Ω 7
−1
5.88 3.62 5.93 1.10
273K m −1
τ
σ DC
(10 ) s (10 ) Ω −14
2.7 0.8 3.0 0.23
8
−1
5.00 3.33 2.00 1.51
77K m −1
τ
(10 ) s −14
21 6.5 20 3.2
Table 2: DC conductivities and mean time between collisions for different materials at 273K and 77K.
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The DC conductivity model assumes that the applied field is constant between electron collisions. If the applied field changes significantly between the collisions then the conductivity is said to be AC and is given by [4]:
σ AC =
σ DC , 1 − iωτ
where ω is the angular frequency of the field. For short bunches the oscillation frequency of the fields from the beam can be shorter than the mean free time between collisions and so both effects need to be calculated. For AC conductivity a dimensionless metal relaxation time, Γ , can be defined which indicates the strength of the AC effects [5]: Γ=
τc s0
,
Table 3 gives vales of Γ for a 2mm radius vessel and the various materials considered. From the values of Γ it should be expected that the AC effects at 77K are will be more significant than at 273K for all materials. Γ , 273K 1.13 0.27 1.267 0.055
Copper Aluminium Gold Iron
Γ , 77K 18.0 4.88 12.67 1.84
Table 3: values for the dimensionless metal relaxation time at 273K and 77K for a 2mm radius vessel of different materials. The monopole AC conductivity impedance, Z||,0 AC , can be given in terms of Γ and s0 [5]:
Z
0 ||, AC
where tλ =
−1
Z s ⎡ t ik ⎤ ( k ) = 0 02 ⎢ λ 2 i 1 + tλ + sgn ( k ) 1 − tλ − ⎥ , 2π b ⎣ Γk 2⎦
(
ks0 Γ 1 + ( ks0 Γ )
2
)
.
Another effect that needs to be included is the anomalous skin effect (ASE). If the frequency of the field is not too high then the field will penetrate into a metal a distance δ 0 (the “classical” skin depth) given by [6]:
δ0 ( k ) =
2 Z 0σ DC k
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The derivation of the skin depth of a material assumes that the field in the metal varies little over a mean free path, δ 0 >> l . When this assumption is not valid and δ 0 ~ l the theory of the ASE must be used. For cases where δ 0
