# Copper coating specification for the RHIC arcs Page: 3 of 8

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2

where r measures the depth of the material, d is the thickness of the coating, and we have assumed diffuse reflection

of the electrons at the interface as verified by measurements [81. The time dependence is exp(-iwt) and the kernel

function is

k() = Je - Isa (- )ds,

1

where a = 1 - iwt/vferni and the Fermi velocity is vfermi = 1.57 x 106m/s for copper. Note that

dxk (x) = 4l/3a,

J dxk,(z

so that if the electric field is constant over many mean free paths equation (2) reduces to

dr2 + k2E = -ipicwJ(r), (3)

with J= o-E/a where o- = 1/p is the low frequency conductivity. This result was in fact used to obtain the coefficient

in (2).

The surface impedance is Z8 = E/H where H = B/pu for our nonmagnetic materials and the fields are evaluated

at the surface of the material. Faraday's law in cylindrical coordinates gives

E _ E=wpoH.

&z &r

For beam generated fields with J = o-E in the wall of a cylindrical pipe, the second term on the left is much bigger

than the first [9] giving

Z, -ikZo ,

z

where E' = aEz/r evaluated at the surface and Re(Z ) = R,(w) in equation (1). This is the same result one gets

for the reflection of normally incident light which allows us to use the same expressions for surface impedance in both

cases. For d - cx equation (2) can be solved in closed form using Laplace transforms [10} and expressions for the

surface impedance may be obtained. One finds

E'z a X(t)

=-- in {1 + t2 dt, (4)

Ez Qr

0

where the fields are evaluated at the surface of the metal, ( = 32/232a3, 5 = 2p/kZo is the skin depth and

x(t) = 2t-3[(1 + t2) arctan(t) - t] = 4/3 + 4t2/15 + 0(t4). While equation (2) holds for diffuse reflection of electrons

of the metal surface it is possible to solve the kinetic equations assuming electrons undergo perfect reflection at the

interface. In this case

Ez _ 22 dt (5)

E' ,ra t2+x(t)

In equations (4) and (5) r is zero at the interface between vacuum and copper and increases with depth in the copper.

This produces a sign change in the derivatives which is simple but needs to be looked after.

When the coating thickness is finite one may solve equation(2) numerically. To do this first take the dimensionless

variable u = r/Q. Next we take u = 0 to be at the interface between the copper and the stainless steel. Equation (2)

becomes

AdI

2+ k242 E = -i A dur E~us )k (u - u1) (6)

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### Reference the current page of this Report.

Blaskiewicz, M. Copper coating specification for the RHIC arcs, report, December 1, 2010; United States. (digital.library.unt.edu/ark:/67531/metadc846206/m1/3/: accessed February 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.