modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes

Modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes Z... Modeling the evaporation rate of cesium off tungsten based controlled porosity

Modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes

Z Pan and K L Jensen

Citation: AIP Advances 3, 042105 (2013); doi: 10.1063/1.4800700

View online: http://dx.doi.org/10.1063/1.4800700

View Table of Contents: http://aip.scitation.org/toc/adv/3/4

Published by the American Institute of Physics

Modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes

Z Pan1,2and K L Jensen2

1Physics Department University of Maryland, College Park, MD 20742, USA

2Code 6843, Naval Research Laboratory, Washington, DC 20375, USA

(Received 3 January 2013; accepted 25 March 2013; published online 4 April 2013)

The evaporation of cesium from a tungsten surface is modeled using an effective one-dimensional potential well representation of the binding energy The model accounts for both local and global interactions of cesium with the surface metal as well as with other cesium atoms The theory is compared with the data of Taylor and Langmuir

[Phys Rev 44, 423 (1933)] comparing evaporation rates to sub-monolayer surface

coverage of cesium, gives good agreement, and reproduces the nonlinear behavior of

evaporation with varying coverage and temperature Copyright 2013 Author(s) All

article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4800700]

The development of robust, long life, high efficiency photoemitters is critically needed for ap-plications demanding high brightness electron sources.1To that end, a cesium (Cs) based dispenser photocathode is under development.2 4The determination of an optimal in situ rejuvenation proce-dure to restore quantum efficiency degradation due to lost Cs motivates the modeling of evaporation and surface diffusion

High efficiency photocathodes generally rely on a thin coating of Cs to lower the work function and enhance emission.5However, Cs loss severely shortness the cathode lifetime and is a concern in high gradient accelerators A sintered wire controlled porosity dispenser (CPD) cathode can actively dispense fresh Cs to the surface during operation through a periodic array of pores leading from the surface down to a cesium reservoir6to replace Cs even as it is being lost.7Previously, the prediction

of the surface coverage during operation of a CPD was based on evaporation models using empirical fits and adjustable fitting parameters.8In this work, a model of evaporation without fitting parameters

is presented and shown to agree with experimental data Although applied to Cs evaporation from tungsten, the model can be extended to Cs evaporation off other substrates

Cesium atoms occupy well defined adsorption sites at the surface,9 modeled by a binding potential characteristic of the surface and the interactions present The evaporation rate is a product

of the frequency at which the Cs atom oscillates, the probability that it has sufficient energy to overcome the binding potential, and the local surface density of atoms Evaporation per unit area per unit time is then given by:

whereτ is the characteristic evaporation time, P is the probability that an atom can overcome the

binding potential and evaporate,σ is the number density per unit area of available binding sites, and

θ is the fraction of those sites which are occupied (aka the coverage).

An atom at an adsorption site can occupy discrete bound energy states If the cesium atoms are

in thermal equilibrium with the rest of the solid lattice, the probability of a particular energy state E

is proportional to e −E/k B T P is then given by:

P =



f r ee e −E/k B T

n

i=0e −E i /k B T+f r ee e −E/k B T (2)

042105-2 Z Pan and K L Jensen AIP Advances 3, 042105 (2013)

FIG 1 Schematic diagram of the effective potential well binding the cesium atom to a tungsten surface.

where k B is the Boltzmann constant and T is the temperature To apply Eq(2)to calculate P, the bound state energies E n and free state energies E freerequire determination by solving Schr¨odinger’s equation Ignoring surface roughness and contamination, the binding sites on a given crystal face are indistinguishable Since the motion of an atom along the surface plane does not contribute to

evaporation, the problem becomes an effective one dimensional binding potential in the ˆz direction

(perpendicular to the surface), as indicated in Fig 1 Although a realistic binding potential is undoubtedly more complex, the square well approximation gives good agreement to the evaporation

data The well width w is less than a nm (here, 5 Å, or the covalent diameter of a Cs atom), and is set in a larger bounded region of length L  w so that E freeis discrete as well

The well depth parameter V0 is separable into contributions arising from different sources:

(i) Coulomb contribution, V C, arising from the partial transfer of charge±F(θ)e between the cesium atoms and the surface (ii) a Van der Waals contribution, V L J, accounting for nearest neighbor cesium interactions which are approximated by the semi-empirical Lenard Jones potential10 (iii) a

Thermodynamic contribution, V μ, accounting for the change in surface free energy and entropy as

coverage is varied from 0 to 1 monolayer, and (iv) a Covalent contribution, V C V, due to the partial sharing of the valence band electrons between substrate and surface atoms.9Therefore, the potential

energy V0is given by

Each are discussed in turn The coulomb contribution to the well depth V C, is

VC = δ − F(θ) + F(θ)2V f +

4



i=0

N (i , θ)V i

where δ accounts for global interactions from charged cesium atoms farther away (because the

electrostatic force is long range, it would be in error to consider only nearest neighbor interactions),

−F(θ) + F(θ)2V f ensures charge neutrality upon evaporation, where  is the work function,

and V f is the ionization potential of cesium Finally, the last term is the weighted average over the

electrostatic potential V i

nn that a partially charged cesium atom on the surface experiences from “i” nearest neighbors, where N(i, θ) is the probability that a cesium atom will have i nearest neighbors

for a givenθ: N(i, θ) is given by the Bethe-Peierls approximation for nearest neighbor atoms.11The total number of nearest neighbors sites is dependent on the crystal structure of the substrate as well

as the crystal face cut of the surface For tungsten, the crystal structure is body centered cubic There are a total of 4 nearest neighbor sites for the [001] and [011] crystal face cuts of a tungsten surface Let±F(θ)e be the charge of the cesium atoms and their images, then V i

nnis given by

V nn i =−k (F(θ)e)2



k (F ( θ)e)2

a −k√(F ( θ)e)2

d2+ a2



(5)

where a is the nearest neighbor separation distance, d is the distance between the cesium atom and its corresponding image, e is the fundamental charge unit, and k is the Coulomb constant The charge

±F(θ)e can be derived from knowing the strength of the dipole moment M produced by each cesium atom at the surface The factor F( θ) is given by

F (θ) = M

ed = 4k (x s − x Cs ) G( θ)

where x s and x Csare the relative electronegativities of the substrate atom and the adsorbed cesium

atom, respectively, R is the covalent radius of the cesium atom, α is the polarizability of cesium, and

Gyftopolous structure factor G( θ) accounts for the electronegativity variation with coverage at the

surface.12

In Eq(4), the long range electrostatic interactions, represented byδ, are approximated by

δ ≈ 2πθ

a

where is the interaction energy between nearest neighbor cesium atoms, and β T = 1/k BT Eq(7)is obtained from representing the interactions as a sum over the average number of occupied adsorption sites for a givenθ, then approximating the summation by an integral The interactions between Cs

atoms far away is treated as a “screened” coulomb interaction e −k0r

where the damping factor k0is from the Debye-H¨uckel approximation13and is

βT k θσ e2

The second component to V0uses a Lenard Jones (LJ) potential, which is an empirical formula for the short range energy of interaction between atoms due to van der Waals dispersion forces as well as Pauli repulsion The LJ parameters are from Ref.14 The net contribution from the weighted average of nearest neighbor cesium interactions is given by

VL J =

4



i=0

where L is the LJ potential (e.g Eq 1.3 of Ref.10)

The third term in Eq(3)is the thermodynamic contribution V μto the well depth and is

V μ = k B T θ

∂(μ/k

B T )

∂ log θ



T ,A

(9)

The proportionality factor∂(μ/kT)/∂(log θ) is from the Darken equation relating the change in the

relative “order” of the surfaceθ  /  δθ2 to the chemical potential μ set up by the cesium atoms

on the surface.15Since systems naturally tend towards states of higher disorder, the effect of V μis

to lower the well depth V0as the coverage approaches a monolayer (θ → 1) For a 2D Langmuir

layer, the thermodynamic contribution V μis then

V μ = k B T



θ

1− θ



(10)

The final contribution to V C Varises from the covalent bond formed between the adsorbed cesium and the substrate due to the partial sharing of the valence band electrons The covalent contribution should be independent ofθ and T and will be approximated as a constant to be determined from

experimental data All terms in Eq(3)to evaluate V are now in place: with V in Eq(3)determined

042105-4 Z Pan and K L Jensen AIP Advances 3, 042105 (2013)

2 s]

Coverage

Evaporation of Cs off W

800 K

700 K

600 K

500 K

Taylor - Langmuir extrapolated data

as a function ofθ and T, Eq(2)is applied to determine P Combined with τ and σ, Eq(1)then gives the evaporation rate per unit area per unit time

The [001] crystal face of tungsten was used andσ taken to be σ = 4 × 1014 sites/cm2as per Ref.12 Forτ, it is assumed that a bound cesium atom will undergo random energy changes from

collisions with the well walls for each period, making 1/τ the frequency of oscillation averaged

over all available energy states Assuming Boltzmann statistics, the average cesium frequency of oscillation is approximated as

ν =

n

i=0 2E π i e −E i /k B T

n

where E i/2π is the quantum mechanical oscillation frequency in a bound state with energy Ei Fig.2shows the comparison between the evaporation rates calculated using the theory with ex-perimental data (and its extrapolation by Taylor and Langmuir) for cesium evaporation off tungsten.16

It is seen that the theory performs well in capturing the qualitative behavior of evaporation over

a wide range of coverages and temperatures It can also be noticed that the evaporation rate can vary over several orders of magnitude as coverage at the surface is varied from 0 to 1 for the range

of temperatures considered Quantitatively, the evaporation values predicted by the model agree to within 10% for the range of data presented in the figure At full monolayer cesium coverages, the model has a singularity and diverges to infinity due to the assumptions in the model However, as shown in Fig.2, good agreement for coverages close to a monolayer up to 0.98 are found

In conclusion, a model for cesium evaporation off tungsten was developed as part of a program

to optimize and predict the performance of CPD photocathodes The model captures the nonlinear dependence of evaporation on surface coverage of cesium and the temperature even using the simplifying assumption of a flat well to model the bound states No adjustable parameters for the

physical model for the evaporation process were required The methodology can be extended to other substrates as well

ACKNOWLEDGMENTS

We thank the Office of Naval Research and the Joint Technology Office for their support, and E.

Montgomery and D Feldman for useful discussions

Energy 3, 66 (2008).

Wan, Nucl Instr and Meth A622, 685 (2010).