Effects Limiting High-Gradient Operation of Metallic Accelerating

Structures

G. S. Nusinovich, D. G. Kashyn,

A. C. Keser, O. V. Sinitsyn, T. M. Antonsen, Jr., and K. Jensen (NRL)

US High Gradient Research Collaboration Workshop,February 9-10, 2011, SLAC, Menlo Park, CA

Focus of UMd Efforts

• UMd theoretical efforts are focused on studying various processes limiting high-gradient operation of room temperature structures:

- multipactor in DLA structures,

- heating of micro-cracks by RF magnetic fields,

- processes in microprotrusions.

What was done and what will be presented here?

• Progress in studies of multipactor in DLA structures will be presented by O. Sinitsyn.

• Studies of micro-crack heating by RF magnetic fields were reported at two previous workshops and described in the paper:

W. Zhu, J. Mizrahi, T. M. Antonsen, Jr. and G. S. Nusinovich, “Temperature rise and stress induced by microcracks in accelerating structures”, Phys. Rev. ST – A&B, 13, 121003 (2010).

• Below we focus on processes in micro-protrusions.

Possible origin of microprotrusions

• It seems possible that microprotrusions originate from microcracks on the surface of an initially smooth structure.

• First, the surface heating in the region of high RF magnetic field causes cracking; some experimental data indicate that this happens when the temperature rise in an RF pulse exceeds 100oC and this agrees with some theoretical results (Nezhevenko - Kovalenko).

• Next, sharp rims of the cracks magnify local RF electric field that causes the field emission of the dark current which is a precursor of the RF breakdown.

• This scenario may take place in the regions where not only RF magnetic field is high, but also RF electric field is strong enough. (SLAC experiments confirm this point.)

Thermal processes in microprotrusions

• Joule heating

• Ion bombardment

• Role of the Nottingham effect

• Possible role of the Thomson effect

D. Kashyn

Joule heating of microprotrusions

• Size of microprotrusions is much smaller than the wavelength of the RF radiation.

• The field in the vicinity of such protrusion and inside it (final conductivity – non-zero skin depth) can be described by the Laplace equation instead of the Helmholtz equation.

022 k 02 Point Charge Models – monopole model,- dipole model

Point Charge Models

• Monopole model: charges are assembled into the line such that the n-th charge is located on the apex of n-1-th sphere. This assembly approximates the protrusion. The electric field can be determined analytically if the ratio for all n. baa nn /1

All charges of the same sign – monopole model. Zero equipotential line corresponds to the protrusion surface. Conclusion: Monopole PCM allows one to mimic the profile of a smooth protrusion, however, such profiles are not realistic.

Point Charge Models

• Dipole PCM: to obtain more realistic protrusions the set of image charges should be added to the original assembly.

n

j j

jn

j j

jn

zza

zza

a

zaFzV

0220

0220

000

)()(),(

Potential in the vicinity of protrusion

F0 is the background field, a0 is the scaling factor, and

are the magnitudes of the virtual charges.j

DPCM yields a sharper profile of the asperity.

What do we solve?We solve the Laplace’s equation in the region

and where describes the shape of the protrusion.

max0 rr )(0 rzz s )(rzs

Laplace equation is the subject to boundary conditions: at

and at

02

0)(r

xn

)(rzs

0z

We use relaxation scheme to solve the Laplace equation.

In order to specify the boundary conditions on the we need to introduce the quantity:

which is the flux through the boundary of the protrusion. The protrusion crosses the grid at some points. Associated with each crossed grid there is a numerical flux Qk. Values of the potential are adjusted in such a way that numerical flux matches the real flux.

Solving the equation gives the potential distribution inside of the protrusion.

rdrzrrQr

s 0

2' )(12)(

)(rzs

Amplification factor and field inside the protrusion

1

12

1

20

1

12

1

20

0 )()(1

),0(1)0( 1

n

j jn

jn

j jn

jzzn

n zza

zza

z

zV

Fr n

For the DPCM the amplification factor is given by

b=0.9 n=40, background field 300 MV/m, amplification factor 45

The field inside the protrusion is rather small, however, given the high conductivity of metals this field leads to current densities on the order of 0.1-1 A/micron2

Protrusion heating

C

EJTD

t

T

2

After the field and the current density inside the protrusion are determined, the volume power density can be calculated to solve the heat equation with proper boundary conditions:

0T at z=zmin and at 0

nx

T )(rzsElectron current density at the protrusion surface is equal to the Fowler-Nordheim current density:

F

vb

t

Faj FNFNFN

2/3

2

2

exp

teEeEF RF sin0where is the periodic RF field and aFN and bFN are constants equal to 1.37 A/eV and 6.83 /(eV1/2nm), respectively.

The difference between RF and DC heating is characterized by the following ratio (cf. A. Grudiev, S. Calatroni and W. Wuench, PRST-A&B 2009):

0 00

2/3

094

20

2

1sin

1106618.13expsin

2

1d

yv

yv

Fyv

j

j

FN

1D Model of heat propagation

In order to verify our calculation we used a simplified model. We assumed that a uniform electric current flows through the protrusion. The current density is given by:

where is the area of the last charge in the PCM assembly and is

the area of the corresponding layer. Under these assumption the following equation should be solved:

22

za

aJzJ apexFN

apexa za

pCEJ

dSz

TzAD

zzAT

T)()(

To avoid any influence of the base, we imposed the boundary condition for the zero temperature rise deeply enough (at z=-10a0 ) inside the base.

A(z) is the area of the corresponding layer

Comparison of 1D and accurate models

The comparison shows that the temperature rise is slightly smaller for the 1D model than for the accurate one.

20 ns pulse, b=0.9, background field is 300MV/m

Accurate model predicts a little higher temperature rise than a 1D model.

Results of simulations

The scaling parameter a0 affects the temperature rise significantly.

Increasing the height-to-base ratio decreases the time required to melt the protrusion

copper, b=0.9, background field is 300MV/m

Conclusions from studying the joule heating of microprotrusions

A theory describing the field distribution inside micro-protrusions is developed. A simplified model is also created to test the predictions of the theory.

The theory allows one to accurately analyze the protrusion heating during RF pulses.

It was found that it is possible to achieve melting for the certain geometries of the protrusions with pulse length in the range 100-600 ns.

In order to achieve melting amplification factors should be greater than 50.

Nusinovich

Ion bombardment

• Many effects on the metallic surfaces and, in particular, on protrusions were attributed by some authors to ion bombardment.

• At the same time, people developing field emitters know that operation of such emitters in RF fields is more stable than in DC fields.

• This difference can be explained by the fact that in the vicinity of a microprotrusion the ions experience the ponderomotive force pushing them away from the protrusion surface.

Ion motion

• Analytical theory (Kapitsa’s method)• This method is based on separation of slow

motion caused by the ponderomotive force and fast, but small RF oscillations:

• Equation for slow motion can be written as

• The RF field near the apex of small protrusion has spatial distribution quite similar to the spherically symmetrical distribution of the field of a small sphere. This allows one to describe the potential well by

rRr ~

R

U

dt

Rdmi

2

2

4

02

20 1

4

R

r

m

EeU

i

i

Such potential well pushes charges away from the center!

Ion motion

• Only ions with large enough initial velocities directed toward protrusion can reach it:r0 – apex radius,R0 – ion initial coordinate

Critical normalized RF field as the function of R0/r0

2

0

0

2

0

00*0,0,

22

1

R

r

R

r

cm

Ee

i

iii

Ion motion (PIC simulations - WARP)

t=0 t=T t=2T t=3TEvolution of ion distribution (initial thermal spread in velocities).The velocity of propagation of the ion region boundary agrees with estimatesbased on Kapitsa’s method.

Phase space in r and zafter 4 RF periods

Ion motion: effect of the dark current

• Effect of the field emitted electron dark current on the ion motion.

• This effect can be significant when ions are located close to the dark current and far from the apex.

• To describe this effect one should add to the equation for ion motion averaged over the RF period the radial force due to the presence of the electron dark current (averaged over the RF period).

• This force is equal to the ponderomotive force for ions when their radial departure from the dark current obeys the condition:

20

2

5

,

,930

r

R

m

MECr is

crion

Example: C=0.01, work function – 5eV, Mi/m=2000, wavelength – 3 cm, r0=10nmR=1, 2 and 3 micron – r^ion,cr=0.04, 1.4 and 10.8 microns, respectively.The region where ions are attracted radially to the dark current rapidly increaseswith the departure from the apex of microprotrusion. Keser

Nottingham Effect

• Above, the thermal conductivity equation

C

EJTD

t

T

2

was supplemented by the boundary condition 0

nx

T

ssss

n

TETe

TEj

x

T,

),(

More accurate temperature distribution can be obtained when the boundary condition takes into account the Nottingham effect

Here, we introduced an average value of the energy evolved at the surface per emitted electron: ss TE ,

(NB: practically all parameters in the equation above are temperature dependent)

Nottingham Effect

N. E. is a quantum mechanical effect that contributes to the thermal balance in field electron emitters.

Joule heating is dominated by this effect at the emitter tip.

N. E. was extensively studied in the realm of field electron emitters, because it influences the temperature rise and, hence, the emitter operation.

Theory The tunneling electrons are mostly from energy levels

below the Fermi level. They are replaced by more energetic electrons from the conduction band. As a result, emitter tip heats up. At high current densities, the energy associated with the Nottingham effect can exceed the energy due to Joule dissipation.

Ref. 1. G. Fursey, “Field Emission in Vacuum Microelectronics” (Kluwer Academic/ Plenum, New York, 2005).

Heating-Cooling Similarly, if the average energy of tunneling electrons

exceed the Fermi level, the surface starts to cool down! Since the electron supply function evolves with

temperature; there is a certain inversion temperature T* for which

T< T* implies heating T> T* implies cooling

Supply Function (βT = 1/kBT)

Transmission Probability

f E ln 1exp T EF E

D E 1 / 1exp F Eo E

Heating - Cooling

EE

A BEF

…

Below T*U replacement e- gives up energy (heating)above T* U replacement e- takes up energy (cooling)

T< T* ⃰ T> T* ⃰

The inversion temperature depends on the surface field gradient and on the work function.

Average energy term

– <ε> depends on transmission probability & supply function - but can be parameterized by

– It is often calculated using Fowler Nordheim (FN) equation for field emission (heating was the concern)

– BUT: at high temperatures, FN equation gives wrong result, and General Thermal Field equation should be used

– Nottingham heating at high T for dark current needs to be reconsidered

T / F

E EF E EF D E f E dED E f E dE

red region: heatingblue region: cooling

Evolution of temperature profiles with time

Fursey , p. 46

Profiles for the case of N. cooling

Significance for Us Temperature gradients are on the order of 10^8 K/cm Tangential stresses may exceed 2 10^9 Pa The resulting thermo-elastic stress can destroy the

emitter tip before melting point is reached. There is a certain controversy about realizing

such conditions: L. M. Baskin, D. V. Glazanov and G. N. Fursey “Influence of thermoelastic stresses on the destruction of field-emission cathode points and the transition to explosive emission”, Sov. Phys. Tech. Phys. (1989) versus M. G. Ancona, “Thermochanical analysis of failure of metal field emitters”, J. Vac. Sci. Technol., (1995)

This issue deserves a more detailed study

Nusinovich

Possible role of the Thomson effect

• Thomson effect is the thermoelectric effect that occurs in conductors with a non-uniform temperature.

• The electric field in such conductors is equal to

Tj

E

The Thomson coefficient does not exceedK

V

e

kB 41029.12

3

Then, the heat propagation equation can be written as:

TEJC

TDt

T

p

1)(

)( 2

Possible role of the Thomson effect

• The Thomson effect can play an important role when

Ez

T

Our simulations of the RF field penetrating into protrusions show that the RF electric field there (in copper) is about 3 kV/m.Hence, this effect becomes significant when the temperature gradient is on the order of 23 K/micron or above.In micro-protrusions of the height on the order of 10 microns, the non-uniform temperature rise in short pulses with high gradient of the RF field can easily lead to the fulfillment of such conditions.

Acknowledgments: this work is supported by the Office of HEP DOE.