^IMIZJL

UM-P-90/55

The Fall of Charged Particles Under Gravity:

A Study of Experimental Problems

T.W.Darling, F.Rossi, G.I.Opat and G.F.Moorhead

School of Physics University of Melbourne Parkville, Victoria 3052

Australia

Abstract

There are currently several proposals to study the motion of antiprotons,

negative hydrogen ions, positrons and electrons under gravity. The motions of

such charged particles are affected by residual gas, radiation, electric and

magnetic fields, as well as gravity. The electric Fields are particularly sensitive to

the state of the "shielding" container. In this paper we review, and extend where

necessary, the physics of these extraneous influences on the motion of charged

particles under gravity. The effects considered include: residual gas scattering,

wall potentials due to patches, stress, thermal gradients, and contamination states,

and image-charge induced dissipation.

2

1. Introduction

Experiments to determine the gravitational acceleration of charged

particles have been based on the following idea. A short burst of particles with a

range of initial velocities travels under gravity in a vertical evacuated metal tube,

known as a "drift Oibe", which also acts as an electrical shield. To ensure that the

particles only move in a vertical direction, an axial guiding magnetic field is

provided (Section 2.d). Fig. 1 illustrates how the arrival time depends on initial tpward velocity in an ideal experiment. Particles with initial speed v < v do not

re?ch the top. The critical velocity v is given by

v c = \2Ha , (1.1)

v

where a is the downward acceleration and H is the detector he.'ght. Particles with vertical speeds v > v arrive at the detector with a spectium

of times less than the critical time t , where

t c = V2H/a . (1.2)

By determining the upper cut-off of the arrival time spectrum, t a may be

determined.

Actual experiments of this kind include the electron fall experiments of

Witteborn and Fairbank (WF) (1967,1977) and of Lockhart, Witteborn and Fairbank

(LWF) (1977), and a proposed antiproton fall experiment (Beverini, 1986; Brown,

1986; Goldman and Nicto, 1981; Goldman, Hughes and Nieto, 1987). In the WF and

LWF experiments, low energy ( « leV) electrons were projected upwards into a -1 2 0.9m tall, 5 cm diameter, drift tube with pressure less then 10 torr, at 4.2K. and

with a typical guide magnetic field of magnitude 20G provided by a

superconducting solenoid.

3

The result of the WF experiments was that the acceleration of the electrons

was 0 ± 0.09 g. This is in agreement with the prediction of Schiff and Barnhill

(1966) (Section 5) that the gravitational force is cancelled for electrons by the

electric field induced by gravity within the drift tube metal. This is the subject of

some controversy since other theoretical work has predicted much larger

gravitationally induced electric forces (DMRT, 1928) (Section 5). Non-uniform

forces, not accounted for in equations 1.1 and 1.2, may also be expected to occur.

These might arise through collisions with other beam particles and residual gas

molecules (Section 3), and from the non-uniform electrostatic forces provided by

the space-varying electric potentials of the surface patches (Section 4). In the

LWF experiment, evidence for a curious temperature dependent "surface shielding

effect" has been reported Above 4.5 K the observed forces were consistent with

those now expected theoretically, however the low-temperature null result of the

WF experiments has yet to be fully explained.

The original intent of the WF experiment was to continue the drift tube

studies with positrons to measure the gravitational acceleration of antimatter.

This idea is being pursued by a consortium which intends to use the slow

antiprotons from the LEAR. facilities at CERN in an experiment similar to WF.

Since the interpretation of the results for antimatter are of fundamental

importance, we have reviewed (see in particular, Witteborn and Fairbank, 1977;

Beverini et al, 1986) and extended where necessary the analysis of the forces

experienced by a charged particle in a drift tube.

2. Interactions of a charged particle

A charged particle is coupled to its environment by:

(a) the gravitational field through its mass, m

(b) the electric field through its charge, q

(c) electric field gradients through its polarizability, a

(d) the magnetic field through its charge and velocity, V

4

(e) magnetic field gradients through its magnetic moment, u.

and diamagnctic polarisability. a

(f) radiation through the photon cross-section, a

(g) residual gas scattering through the cross-section, a

We believe that (a)-(g) exhaustively lists all the "handles" by which charged

particles are coupled to the r temal world. In the rest of this and subsequent

sections we shall investigate each of these couplings in tum.

( a ) G r a v i t a t i o n a l : The term "gravitational" in the above list refers to non-

electromagnetic interactions of adequate range. In the absence of non-standard

phenomena, the gravitational force on a non-relativistic particle is given by

F g = mg , (2.1)

where g is the gravitational field. In terms of the gravitational potential \y,

g is given by

g = - V V . (2.2)

( b ) E l ec t r i ca l : Unwanted electrical interactions are by far the most serious

problem in free-falling charge experiments. The magnitude of the electric leld

which just balances gravity is given by

E = mg/c = (mc2/c) (g/c 2) (2.3)

= 0.56 x 1 0 ' 1 0 V/m (electrons), and

= 1.02 x 10"7 V/m (protons).

Electric fields may arise from the following static effects involving the apparatus

and the shield surrounding the experimental space:

(;) Contact potential patches and surface contamination (Section 4).

( i i) Gravitationaily induced stress gradients leading to chemical potential

gradients, (Dessler, Michel, Rorschach, and Trammel, 1968) (Section 5).

(iii) Gravitationally induced electric compensation fields required to support

the electrons in the apparatus shield, (Schiff and Barnhill, 1966)

(Section 5).

5

(iv) Fields induced by thermal gradients (Section 6).

(v) Image charge attraction (Section 7).

Electric fields also may arise by the dynamic effect of dissipative forces due to

image charge motion (Section 7).

( c ) Electric field gradients: Electric field gradients can in principle exert

additional forces on polarisable systems. The only significantly polarisable

system under consideration is the negative hydrogen ion, for which we give an

order-of-magnitude force estimate. Wc start with the force equation,

F o = a e E.VE . (2.4) 2 We estimate the polarisability, a , by a = (ea ) /Ae, where a is the Bohr radius of

hydrogen, and Ae is the energy required to excite the lowest opposite parity state.

Taking AE as 0.1 eV, we find

F a /F = 2.74x 10" 1 3 !E.VEI , (2.5)

where E is in volts/metre, and VE in volts/metre^. Even with the very large values

of E = 10 V/m, and VE = 10 3 V/m 2 we see that F /F = 3 x 10"9 « 1, and hence electric ** g

field gradient forces are negligible.

4

( d ) Magnetic Field: Charged particles are constrained to move in helices

around the magnetic field lines. The radius of such a helix is given by r, where

r = mV t/eB (2.6)

= 5.7 x 10" 1 2 (Vt/B) metre (electrons), and

= 1.04 x 10 ' 8 (Vt/B) metre (protons),

where Vt, the velocity component transverse to the magnetic field, is in

metres/sec, and B is in Tesla.

With B = 10 Tesla and Vt = 10 m/sec, r = 10 nm (electrons) and 10 urn

(protons). Thus the charged particles follow the magnetic field lines very closely

by comparison with the scale of typical apparatus. The longitudinal magnetic

6

field may be made large enough to render the effects of transverse forces

negligible.

( e ) Magnetic field gradients: Let us assume that B is almost parallel to g

which defines the z-axis of a cylindrical coordinate s>stem. We assume that the

deviations from field uniformity are small and are only in the radial direction, i.e. A A

B = B z z + Bpp with IBZI » IBpl. Using Maxwell's and Newton's equations for the tield

and panicle, respectively, we readily deduce the following:

Paramagnet ism: The particle velocity V transverse to the symmetry axis

(z-direction) of the magnetic field gives rise to a force F ?'ong that axis due to the

magnetic field component B . Classically

v

F z = 2 m V t B i n B z / 3 z l ( 2 7 )

If the magnetic gradient forces on the spin are included, wt must treat the whole

problem quantum mechanically. In this case we find

Fz. = - h «>c (n + \ ± \ti \Mn B?/dz\ , n = 0.1,2,3 (2.8)

where the cyclotron frequency co = leB /ml , (2.9)

7

and y = 1.001 159 (electrons), and

= 2.792 844 (protons).

For electrons

F z /F = 2xl0 6 (metreATesla) l|n + j ± ^ y |3B z/dz! . (2.10)

The condition F /F « 1 requires an unachievably uniform B .unless z g z

n = 0 and the lower (-) sign holds. In this case F / F 0 = lxlO 3 (metre/Tesla) I3B /3zl , (2.11)

z g z

which is achievable with care.

For protons and negative hydrogen ions, the classical formula suffices. With

kinetic temperature T,

IF Z/F gl = (kT/mg)l3 In B z/3zl

= 0.84x103 (m/K) TO In Bz/ozl « 1 . ' * (2.12)

i : 13 In Bz/3zl = 1/00 4 m), this would require T « 12 K. D i a m a g n e t i s m : As the negative hydrogen ion is expected to behave

diamagnetically it would be repelled from regions of high magnetic energy

density. This diamagnetism is mainly due to its charge radius. In terms of the diamagnctic polarisability, a , the diamagnetic force is

F z = + a m B z < 3 V ^ o . (2.13)

The polarisability is given as follows am = ' T a o a ( r / V • ( 2 1 4 )

The root-mean-square radius, r, is of order a , the Bohr radius, and a is the

fine structure constant. We find

IFZ/F I = 1.6 x 10 ' 3 (r /a Q ) 2 (B z aB z/3z)/(Tesla 2 m" 1) . (2.15)

This force is quite negligible.

The temporal stability of the magnetic field is not critical either, as a

fractional increment in the magnetic field, AB /B , at most generates a similar

fractional change in thr kinetic energy of the particle. Even without

superconducting solenoids such stability is easily achieved.

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( f ) Radiation Pressure: If S is the radiant power per unit area travelling in

a fixed direction, the force it exerts on a particle of radiation cross-section o is

F r = (S/c)a r . (2.16)

For the scattering of photons of energy too « mc 2 from a free point particle of

charge e and mass m, a is given by the Thomson scattering formula (Sakurai.

1967. §2-5).

° r = (8TI/3) r* s a Q , (2.17)

where r c =e 2 /47 te 0 mc 2 (2.18)

is the classical radius of the particle. We see that while ilti electron cross-section

is much larger than that of the proton, neither is significant.

For the case of radiation scattering from a bound electron in an atomic

syuem, such as the negative hydrogen ion. the cross-section is given by the

Rayleigh scattering formula modified to include radiation damping,

(Sakurai. 1967, § 2-6). Specifically,

°r = °0 2 fl + , ,2 • ( 2 - 1 9 )

(coQ - (o ) + (yu)

where cr is the Thomson scattering cross-section and too is the energy of the

lowest excited atomic state coupled to the ground state by a dipole transition. The

cross-section rises as co4 to a peak at to = to with width-at-half-hcight (due to

radiation damping) of y = (2/3)co (r /c).

For the case of thermal radiation at temperature T, with a worst case solid

angle of 2K. wc expect a radiation force

9

ftco^dco K f a r ( ( , ) ) , „ w/kT < 2 - 2 0 >

J r ( 2 7 t c ) 3 ( e ' l < 0 / k T - l )

Wc illustrate the integrand of this expression in Figure 1 for the case of the H" ion with too = 0.747 eV (the first electron detachment energy, C.R.C. Handbook,

1980. p.E-67) and T = 300K, whence hw0 » kT. This exhibits a low energy term F r '

near the peak in the black body spectrum where the cross-section behaves as

(<o/o) )"*, and another term F r " near the peak in the cross-section where the black

body spectrum is exponentially decreasing:

F r ' = (<TT 4/C) 16aQ (jtkT/too) 4

= (a o/c)(2jtkT 2//»(0O) 4 a , (2.20a)

where a is the Stefan-Boltzman constant, and F r " - ( 2 o 0 » r c / c 2 ) c - ' , ( , ) o / k T . (2.20b)

Under these conditions we then Find

F r /F g = 2 x 10-8 « 1

and wc see that even for thermal radiation pressure at room temperature, this

force is negligible.

3. Residual Gas Scattering

We consider the interaction of the residual helium gas with the beam

particles. Some examples of particle trajectories are shown in Fig. 3. A single

upward impulse imparted to a particle (Fig. 3-b) would decrease its TOF if it would

have reached H anyway, and it might increase its TOF beyond t if it had already

reached its turning point below H (Fig. 3-c). Downward impulses cannot increase

the TOF beyond t unless the particle is subsequently scattered upward to H by a

second interaction (Fig. 3-d).

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A point particle of charge q will predominantly interact with a neutral

background gas atom by the electrostatic attraction caused by the induced electric

dipole moment. The force may be shown to be

2a c q" F = 7 7 . (3.1)

( 4 J I £ ( ) ) V

where a is the molecular polarizability of helium. e

We note that even at 4.2K, the mean speed, Vp. , of the helium atoms will be

about 140 m/s. much faster than the speed, V , of the beam panicles, which in a

typical drift tube experiment are expected to be a few m/s.

We therefore assume, afte.- Witteborn, Lockhart and Fairbank (1988), an

impulse approximation for the interaction and estimate the velocity imparted to

the virtually stationary particle q, AVq, as a helium atom passes with closest

approach distance r 0 :

iw i 2a„q 2

*\Jf——'-2 • < 3 - 2 > M q V H e r 0 ( 4 r t £ 0 >

4

We may therefore define a, the cross-section for the particle to be scattered with a

velocity change of at least AVq:

2 3* 1 2 a

C 1 2 !/2 c(AV ) . * r 0 * * < T . . — — j ) (3.3)

q He q (4JI£Q)

The cross-section, together with knowledge of the velocity spectrum,

temperature and pressure of the background gas in a particular experimcnial

configuration, may be used to calculate the mean time x between scatters. Recall

that the guide magnetic field should ensure that only the vertical component of

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the imparted velocity changes atfects the TOF. although the probability or further

scatters may be affected by the transverse component.

Under the conditions of the WF and LWF experiments, AE > 10 eV was

considered unacceptable. Assuming a Maxwell distribution for the helium atom -13 velocities with temperature 4.2K and pressure 10 torr, we find t = 5 s. This is

much larger than t c = 0.428 s for a = -g = -9.8 m/s2 and H = 0.91 m.

If the pressure in the tube varied, say with temperature then the mean time

between collisions mignt become less than the flight time and significantly affect the TOF distribution, especially near t Wc illustrate in Fig. 4 the results of a

Monte Carlo computer simulation. Note the increased cut-off timrs at higher

pressures, and the similarity between t^; results for a = -g and a = 0 at p = 10"

torr. This may have been significant in the LWF experiment results, where a

reduction in pressure due to cryopumping near 4.2K may have contributed to a

strong temperature dependence.

Under the conditions for the proposed antipro'on experiment, we Find x = 230 s, which is very much greater than the expected t . This experiment is

therefore much less sensitive to pressure deterioration and is unlikely to be

affected by residual gas scattering.

4. Surface Patch Potentials and Cor'amination.

Surfaces of metals may exhibit patches of different electric potential due to

the presence of crystal facets of different work functions, or due to regions with

adsorbed surface contaminants. We thus expect a spatially random electric

potential along the axis of a drift tube. To investigate whether the large number

of fluctuations along the tube integrate to cause a significant effect, we consider

the energy integral for the time-of-flight t of a particle of charge q and total

energy E moving vertically in a tube of height H, where at each point the total

potential q<t>(z) < F:

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r H

t (mx 1/2 x\- l /2 t = ( T ) J dz (E-q<D(z)) (4.1)

expanding the intergrand in (q<t>(z)/e) we find

, . <m/2e>"2 / " , , ( , + l « M i l + | ( « « i y ^ ( ^ 3 + . . . . ) , 4 . 2 )

0

The potential may be written as the sum of a smooth potential <&.(z), including that

due to gravity, and the random potential q<6 (z),

<6(z) = O s(z) + <Dr(z) . (4.3)

The change of the time-of-flight due to * is then

M r = (m/2e) 1 / 2 J d z ( | — T ~ + 8 ~ 2 ~ ^ ~ + 8 i " - + " ) • ( 4 4 )

r* C t

The first term in the expansion (4.4) may average to zero over the distance

H, however the even terms will always contribute to the time-of-flight. In

particular, the third term is related to the mean square potential or. axis, < * 2 > .

Comparing its contribution. At' , with t , for the case when gravity is the only

force, we require:

A t ' r 3 e 2«t> 2> c (mgH)

where e = mgH has been used. Thus, for H = 0.9 m, we require

<E)RMS = V<ct 2> < < 0.1 nV (electrons), and

« 0.2 uV (protons). (4.6)

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To calculate the expected value of « R M j , we consider a cylinder of infinite

length and radius R which has a potential distribution <t>(R,0,z) on its interior

surface. We shall later take this distribution to be zero outside the range 0 < z < H.

The potential on the axis of the cylinder due to the surface potential may be

shown, [Opat, Moorhead, Rossi, 1990] to be given by

^ J ^ " g(z-z') <D(R,e\z') . (4.7)

where

T dk e

i k < z - z ' > «<«'>•* J ^ T T ^ R T • ( 4 8 )

and I (kr) is the Modified Bessel Function of zeroth order (Abramowicz and

Stegun, 1965).

Equation (4.7) shows, that the axial, potential is formed by averaging the

surface potential around the tube, and convoluting the resulting average with the

smoothing function g(z-z'). This smoothing function vanishes with exponential

rapidity as lz-z'l becomes large. Thus, surface potentials at points far from the

axial observation point do not contribute to the potential at that point

significantly. In fact

exp(- j f t . Iz-z'l/R) g(z-z') T-T—T (4.9)

V J or

for lz-z'l » JIR, where

L . = 2.405 is the first zero of the Besscl function of zeroth order, JQ , and

1 4

J j ( j Q . ) = 0.51915 is the value of the Bessel Function of first order at that zero.

If we introduce the azimuthally averaged potential on the surface

In f d8 <D(R.z) = J T- <D(R,9,z) , (4.10)

ZJt o

t hen

oo

*(z) = J Y g(z-z')0(R,z') . (4.11)

The mean square potential on the axis is given by

<s£> = 1 J <t>(z)2 dz ( 4 . 1 2 )

_ R 7 4*1 7 aV H ' R ' R G ( Z " " Z , ) *( R .z")*(R.z') . <4-13)

-oo -oo

where

G(z"-z') = J ~ g(z-z') g(z-z") (4.14)

_ J RjMc e i k < z " - z ' >

-oo 2 K i J ( k R ) (4.15)

To proceed further a model of the surface potential is required. We introduce the

patch potential model as follows:

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<i>(R.e.z) = Z V.A.(8,z) . (4.16) i '

V- is the potential of the i l h patch, which covers a region R- and

Aj(8,z) = 1 (8,z) e R-j

= 0 (G,z) e Rj . (4.17)

We assume that the patches are small in linear dimension compared with H, that the i t h patch has area A-, that the potential V- is uncorelated with the size and

shape of the patch R- , and that the average surface potential is zero. It follows

that

< v i v j > = 8ij v o 2 • < 4 - 1 8 >

VQ being the root-mean-square surface potential.

The mean patch area is defined by

A = I A*/1 A- . , (4.19) i ' i ' f

Assuming that averages over the height of the tube :»re equivalent ot ensemble

averages we find:

V A °° 0 f

<<I>2> = : J dzG(z)C(z) , (4.20) 2KR1 . M

where C(z) is a patch-patch correlation function defined as follows:

az^z ' ) = K < SjU")^/') > , (4.21)

1 6 -

where

(7.) ^ J 5j(7.) ^ J ^ Aj(9.z) (4 .22)

ith and 2:rR 8J(Z) is the width of the i t n patch at height z.

The normalisation constant is so chosen that

J C(z)dz = 1 . (4.23)

v C(z) vanishes when Izl exceeds a typical patch dimension, which is assumed small

compared to R.

As G(z) changes on a scale commensurate with R, we find from equations

(4.20), (4.23).

< <D2 > = VQ A G(0) /2JIR 2

= V 0 / N e f f ' < 4 - 2 4 )

where

N ,, = effective number of patches

= 2rtR2/AG(0) . (4.25)

We expect something like a normal distribution of potential on the axis with

*RMS = a s ^ e s t a n d a r d deviation. For R = 2.5 cm, \ A = 0.1 um and V = 100 mV,

and given G(0) = 0.4353, we have N c f f = 9.0 r. 10 1 1 and * R M S = 0.1 uV. This may be

acceptable for protons, but is certainly not for electrons. If the patch dimensions

17

and voltage are reduced to V A = 10A and V Q = 10 mV, ^ j ^ * ; =0 .1 nV, which is

adequate for protons, but barely adequate for electrons.

It is essential to keep <I>r(x) as small as possible in these experiments. This

may be accomplished by choosing materials or preparation techniques resulting

in sufficiently small patches, or relying on shielding mechanisms associated with

material condensed on the surface.

5. Gravity Induced Electric Field.

(a) Theory

The gravity induced electric field both internal and external to a metallic

conductor has been investigated by a number of authors (Schiff and Barnhill,

1966; Dessler et al, 1968; Schiff, 1970; Herring, 1968; Stmad, 1971; Rieger. 1970;

Leung et al, 1971; Leung, 1972; Peshkin, 1968 and 1969; Harrison, 1969; Held, 1967;

Davis, 1987; Davis and Opat, 1988). In essence, a metallic object supported against

gravity will undergo a redistribution of its electrons and nuclei, thereby

producing an electric field.

The equilibrium state of the conductor, at constant temperature and volume,

can be determined by the. condition that • the Helmholtz free energy, suitably

generalized to include gravitational potential energy, is minimized. The system is

then characterized by a complete chemical potential of the electrons which

is constant throughout the conductor. Here, the complete chemical potential, ii, is

a generalization of the electrochemical potential and includes the gravitational

potential energy of the electrons, my = mgz, where z is the vertical height. Thus,

u is given by

\i = u - e4>iiu + my , (5.1)

where u is the ordinary chemical potential (containing kinetic, exchange and

correlation contributions to the electron energy), and • e < t , ; n t ' s t n e internal

18

electrostatic potential energy, averaged over distance scales larger than the

screening length but much smaller than the size of the conductor. We take the

energy of an isolated electron at rest at infinity to be zero.

The internal electric field can be calculated by taking the gradient of (5.1)

and noting that Vu. = 0, viz..

= i m s - v * i n t = + ! r 1 - c V ^ ( 5 2 )

Of particular relevance to the drift tube experiments is the external electrostatic potential <t>„_.. The calculation of the electrostatic potential is

C X I

complicated at the surface region of the metal, which extends of order a Fermi

wavelength (4.6 A for copper) on either side of the last iop at the surface (Lang

and Kohn, 1971; Wigner and Bardeen, 1935; Bardeen, 1936; Herring and Nichols, 1949). We will show that A . is related to the work function, W, of the surface.

—> A Consider two points, x ±n8 , symmetrically placed about the surface charge

- » A

at x , where n is a unit vector normal to the surface. The distance 6 is chosen to

be larger than the characteristic depth of the surface region, in particular.

sufficiently large that the image potential for an electron at the exterior point is negligible. At the interior point, <>• , is given by (5.1). We know from classical

electrostatics that the only reason 4> ., at the exterior point, differs from <t>:nt, at

the interior point, is that there is some effective surface dipole moment density in

between which displaces the electrostatic potential on either side, viz.

-> A _> A D U S )

W* s

+ n8> - W , - " * ) - ~^~ ( 5 3 )

where D is the normal component of the surface dipole moment density and e is

the permittivity of free space. Substituting for 0 j n ( from (5.1) we have

19

-» —» A —> A ' * S ~> A

e<> e x t ( x s + n8) = - my ( x g -n8) + (-e — -\i( x g - n5) )+ p. (5.4)

For the distance scales chosen above, the bracketed term in (S.4) may be identified

as the work function of the metal [23-26], viz.

W(x s ) = - c — - u ( x s - n 8 )

= A « D ( 7 s ) - u ( ^ s - n 8 ) , (5.5)

where A«t> = -eD/e is the energy expended by an electron in going through the

dipole layer. This definition of the work function corresponds to the minimum

energy required, on average, to remove an electron from the interior to rest just

outside the surface region. v

- » A

The u.( x - n8) component of W reflects the state of the interior of the

metal, while D( x ) reflects the state of the surface and the result of the evanescent wave functions of electrons tunneling into the vacuum. We note that

—> in addition to gravitational effects, W contains the variation of D( x ) over the

crystal facets (the patch effect) and adsorbed dipoles from contamination, as

discussed in Section 4. 4

From (5.4) and (5.5), the external electrostatic potential is given by,

- C 6 c x t ( x s + n8) = -my (x % - n8) + W ( * s ) + \L (5.6)

The determination of the potential in the space external to the conductor

requires the value of the potential on the boundary. This potential boundary

value is provided by equation (5.6). The tangential component of the external

electric field, just outside the surface, is immediately given by

A - > A A m - > 1 1 E c x . - - t V ^ e x t = t < 7 g + e V W ) ( 5 " 7 )

20

A

where t is a unit vector tangent to the surface.

The first term in (5.2) reflects the fact that, in the absence of any

deformation of the crystal lattice or redistribution of the nuclei, an electric field

must be set up by a redistribution of the electrons to balance the force of gravity

on them. The effect is analogous to the acceleration induced emf in electron-

inertia experiments (Davis and Opat, 1988, and references therein). This field is

continued just outside the surface in (5.7), as is evident from the arguments

leading to equation (5.3).

Equation (5.7) implies that whatever else happens to the distribution of

electrons and nuclei under gravity, it must be reflected in a change in the work

function. Although we are still left with the difficult task of determining exactly

how gravity produces a gradient in the work function, it can nevertheless be

determined experimentally. The contact potential difference between two metal

surfaces in equilibrium is equal to the o'fference of 'heir work functions. Thus

the second term in (5.7) corresponds to a continuous contact potential Variation

along the surface; it can be investigated by a contact potential measurement

which reproduces the effect of gravity o>; the conductor.

An exact first principles quantum mechanical treatment of W would be a

formidable task even for an ideal metal surface, let alone one coated with oxide or

other contaminants such as was the case in the drift tube used in the Witteborn

and Fairbank experiments. We can, however, qualitatively describe the physical

effects involved.

Firstly, there will be a non-uniform deformation of the metallic lattice

calculable by elasticity theory. In such deformation, the mobile conduction

electrons will closely follow the distribution of positive ions, otherwise huge

electric fields would result. In turn, this will produce gradients in u in the

interior as a result of the deformation of the primitive cell, with the change in

density likely to dominate. Because the internal potentials are altered, the

21

tunneling imo the vacuum, and hence D will change. D is also changed because

the deformation changes the areal density of the dipole moments. All of these

perturbations will alter u,. The mobile conduction electrons will respond creating

a net surface charge whose electric field makes u. constant throughout the system.

Secondly, as gravity shifts the position of the heavier nucleus away from

the centre of its tightly bound cloud of core electrons, the ion will become

polarised thereby creating an additional electric field. A model including this

effect was examined by Shegelski (1982). However, we do not expect this tc

produce a net electric field, due to screening by the conduction electrons.

(b) Estimates of the Fields

Schiff and Barnhill (1966) were the first to calculate the gravity induced

external electric field concluding that the dominant effect was due to the first

term in (5.7), which places the field at -5.6 x 10" ^ V/m. The result of Witteborn

and Fairbank, that the acceleration of free electrons in their copper drift tube was

0 ± 0.09g, seemed to agree with this. In contrast, DMRT (Dessler et al, 1968)

estimated the work function contribution to be a few orders of magnitude larger.

Herring has discussed this discrepancy and- it seems generally agreed that the

DMRT calculation is essentially correct.

For a vertically standing drift tube, sufficiently far from the supports, the

deformation consists of longitudinal compression which varies linearly with

height:

u = Zujj= H (z-0 (5.8)

n = n Q ( l - u ) (5.9)

where u-- is the strain tensor, u is the dilation, K is the bulk modulus of

compressibility,p is the mass density, / is the height of the tube, n and n are the

1 e

aw dz ~~

. Y M£ ' e

Y n o

" 3 K aw au •

22

number density of atoms before and after deformation. Eqn.(5.8) contains the

effects of poisson dilation as a detailed calculation will show. Assuming that W

varies only with dilation, the second term in (5.7) is given by

(5.10)

where n „ aw

(5.11)

and M is the atomic mass. This is the DMRT field, which is proportional to the

strain derivative of the work function.

We can express 7— in terms of the density parameter r , the Wigner-Seitz

radius,

n e = 3 r c r s , (5.12)

where n is the average concentration of conduction electrons. Then

(5.13) aw rs aw au 3 drs

and

V Y " 9K

s aw d r s

(5.H)

A naive fixed potential well model of the work function would ignore the

changes in D and take u as the free electron Fermi energy, ep:

u = Ep - constant,

. *2 9 _ 2/3 -2 where e p = 7 ^ (-^-) r s . (5.15)

23

Figure 5 shows the density dependence of Ep. From the above equations we deduce

that

/ 2 n o \ y = V ^ M p ( 5 1 6 >

For Copper (Kittel, 1976), e p = 7.00 eV, n Q = 8.45 x 1 0 2 8 m" 3, K = 1.37 x 10 1 1 Nm"2 •

M = 1.0568 x 10" 2 5 Kg, and Y= 0.1 j . This would make the DMRT field about 18,000

times larger than the Schiff-Barnhill field.

DMRT considered a more accurate expression for u from Wigner and

Bardeen (1935, 1936), and estimated D and its dependence on density from Herring

and Nicholls (1949). They concluded that this would lower y from the free electron

estimate but, barring accidental cancellation between D and %\i, would still leave a

field orders of magnitude larger than the Schiff-Bamhill field.

In fact, modern density functional theory, using the jellium model for a

metal surface predicts that y m a y be much smaller. Heine and Hodges (1972) have

discussed the physical reasons for there being a large degree of cancellation

between the components of W, A<I> and \x. in equation (5.5). If u. increases, raising

the internal electron levels /closer to the vacuum, the electrons tunnel further

out, making D more negative. Thus, A<& increases and compensates for u. This is

evident in Fig. 5, taken from the calculations of Lang and Kohn (1971) for W and

its components, including the high density results of Perdew and Wang (1988).

While A<1> and u vary considerably, W is relatively insensitive to density. These

calculations predict that 3W/dr , and hence y, is zero for certain densities

[(r s/a ) = 0.6, 1.0, 1.8, where a = 0.53 A is the Bohr radius]. For copper,

(r s/a ) = 2.67 and aW/a(r s /a Q ) = - 0.44 eV. Thus, y= -0.013, an order of magnitude

smaller and of opposite sign to the above estimate. This still leaves the DMRT field

about 1500 times larger than the Schiff-Barnhill field and so should dominate over

24

the force of gravity in drift tube experiments with electrons. For anti-proicns it

would be slightly smaller.

We must point out, however, that the Lang and Kohn calculation for the

work function of copper is about 20% lower than experimentally measured values

for polycrystalline samples. This model is more successful with simple metals.

Moreover, the effect of oxide layers, other contaminants and surface states is

uncertain. The discussion so far has been limited to the density dependence of the

work function. If we consider the full crystal lattice, we would expect W to depend

on the general deformation. There is also evidence that W varies with the

dislocation density in a real metal specimen under strain (Andreev and Paligc,

1964; Mints et al, 1975). At present, it must be admitted that the theoretical and

experimental understanding of the processes involved cannqj.. predict the value of

the gravity induced electric field in metals.

6. Thermoelectric Field

A temperature gradient in a conductor will produce an additional external

electric field. To find this additional field we generalize equation (5.6) to include

the temperature dependence of the work function and of the complete chemical 4

potential:

- e<$> c x t(t s + n5) = - m V ( t s - n6) + W("x S.T) + £(T) (6.1)

where T is the temperature. Differentiating this expression gives the external

electric field. The result for the gravity-induced field, in the previous section,

followed from the fact that Vp. = 0 in an isothermal conductor carrying no electric

current. When T varies, this is no longer true; in this case, Vji must be derived

from the appropriate phcnomcnological equations.

25

Davis and Opat (1988). have discussed heat and electric current flow in a

non-isothermal conductor subject to gravity and strain, obtaining the following

familiar pair of equations:

l = ^ - a S V T (6.2)

J = f STVn - (K + oS2T)VT , (6.3)

where I ii the electric current density, J is the heat current density, o is the

electrical conductivity, tc is the thermal conductivity, S is the Seebeck coefficient

(or absolute thermoelectric power) and ji has been generalised to the complete

chemical potential used above. If no electric current is allowed to flow in the

conductor, then only equation (6.2) is required and we have

V£ = eSVT. (6.4)

For a vertically standing drift tube with a temperature gradient along its axis and

no electric current, equations (6.1) and (6.2) give the external electric field just

outside the surface as

**•(?+;%>+*• < * 5 >

where the first two 'pirns are the gravity-induced field derived previously, and E T ,

the thermoelectric field, is given by

^ = ( c ? f + S ) d 7 ' ( 6 6 )

26

where S is a generalized thermoelectric coefficient that includes the temperature

dependence of the work function. This gives the thermoelectric field in terms of

two experimentally measurable quantities, 3W/3T and S.

For metals, S is of order luV/K at room temperature; semiconductors exhibit

much larger values of order lmV/K; superconductors in contrast have S = 0. At

low temperatures, thermodynamics predicts that S -» 0 as T -» 0. Gold et al. (1960)

have measured the thermoelectric power of Cu below 20K finding values as small

as 0.05 uV/K, although impure samples exhibited anomalous values as high as 16

uV/K.

The temperature dependence of the work function is less well known.

Gartland et al. (see the review by HOlz and Schutle, 1979) find iC 20 U.V/K for Cu

monocrystals at 300K. In contrast, Lee et al. (1969), find a value of about 1 mV/K

for cesiated tungsten surfaces at 300K. As in the case of S, it is expected that

3W/3T -> 0 as T -» 0 (Herring and Nicholls, 1949). Data for semiconductors seems

scarce but probably ~ ~-fF °* S. Shott and Walton (1977), have measured the

temperature dependence of the work function of superconducting tin below the

critical temperature finding the surprisingly large value of 2mV/K.

The force on a charged particle of mass m and charge e compared to gravity

is given by:

^ = ^- f . (6.7) m g m g dz

Assuming S ~ luV/K, this ratio is:

1.79 x 104 T (electrons), and (6.8a) dz

9.78 — (protons) . (6.8b)

27

Thus, we require

— « 6 x 10"5 K/m (electrons), and (6.9a)

0.1 K/m (protons) . (6.9b)

It might be thought that a semiconducting oxide surface layer on a drift

tube (i.e. CuO and/or Cu^O on Cu) would exhibit larger thermoelectric fields since S

- 1 mV/K instead of 1 uV/K for most metals. However, in this case, there will be

thermoelectric currents flowing along the oxide layer and back into the metal

substate. In a simplified model with a thin oxide layer on an infinite metal

substate and a fixed temperature gradient along the junction, equation (6.1) gives

^oxide / e \ / „ s dT - l ^ " = V c — ; ) l z , o x i d e + ( c S o x i d e ) £ • < 6 1 0 >

where

dT ^.oxide = CToxide ( S metal " S oxide ) dZ • (6.11)

The resulting expression for E j is

1 a W o x i d e dT % = (e <TT + S metal> dZ ' ( 6 1 2 )

This calculation would have to be modified at 4.2 K where most semiconductors

behave essentially as insulators, since the carriers are frozen out. In any case, if

the work function of the oxide layer had an anomalously high temperature

dependence, say of order 1 mV/K, then a severe limit would be placed on

permissible temperature gradients in experiments with electrons or positrons.

28

The work function of a surface may be strongly temperature dependent due

to adsorption of dipoles from a background gas. A temperature gradient along the

surface will produce a gradient in the amount of adsorbed species and also in the

work function. De Waele et al. (1973) found the temperature dependence for Cu to

be around -lOuV/K at 4.2K. However, the presence of sufficient He gas to cover

the sample with a monolayer produces a very strong temperature dependence

near 4.2K of about 3mV/K. This levels off, to the value it had before admission of

He, at about 8K (see Figure 6). They interpret this as strong desorption of He above

4.2K.

We would expect a drift tube with adsorbed gases and a temperature

gradient to show a strongly temperature dependent electric field due to such

desorption phenomena as were observed by de Waele et al. v This would provide a

natural explanation for the temperature dependent force observed by LWF if

there was more adsorbed gas on their drift tube than was originally supposed.

7. Image Charges

( a ) Image Charge Force

A charged particle outside a conductor create? and is attracted to its image charge.

For a panicle enclosed by and on the axis of a perfect, conducting cylinder this

force is zero due to symmetry. The particle is, however, unstable against radial

attraction and must be maintained on or near the axis by a longitudinal magnetic

field (Section l.d). For this reason also, the internal surface should oc

manufactured to be as exactly cylindrical as possible so that geometric

imperfections cause insignificant perturbations to their trajectories.

2 9

( b ) Image Current Dissipation.

A group of particles of charge Q outside a conductor creates an image charge -Q on

its surface. When this group of particles moves in any direction (normal or

parallel to the surface), the image charge follows in an appropriate way. This

redistribution of surface charge -Q is accomplished by the flow of electric

currents in the bulk (not surface) of the conductor. Such currents dissipate

energy ohmically, and as a consequence, the particle group experiences an

effective frictional force. Under certain circumstances such a frictional force

can heavily damp the motion of the group. We investigate this problem.

The equations controlling the flow of charge in an isothermal conductor

are:

I = o E Ohm's Law v (7.1)

V.E = 0 Electrical neutrality of conductor (7.2)

The conductivity is denoted by a = 1/p, where p is the resistivity.

The results of solving the boundary value problem implied by equations 7.1

and 7.2 using Fourier techniques yield for the power of dissipation

P = Q2p(vJ + 2v5/d67tZ3) , (7.3a) 4

where Z is the distance of the charge group from the surface (sec also Boyer. 1974). V and V are the velocities of the group, parallel and normal to the

surface respectively.

The power dissipated for the case of a surface which has a resistive slab of

thickness a on a resistanccless substrate is

P = 3aQ2p(Vp + 2V^)/(327tZ4) (7.3b)

30

We may define a dissipation time x for motion parallel to the surface as

i = -V p / (dV p /d t )

For the cases above we find

x = 16itZ3M/Q2p (7.4a)

x = 32rcZ4M/3aQ2p (7.4b)

From Witteborn and Fairbank (1977), we estimate the resistivity of the OFHC Cu

drift tube to be p = 3.4 x 10 Qm at 4.2 K. For the case ofvN particles of charge e

and Z = 2.5 cm, equation (7.4a) yields

x= 8 x l 0 1 3 / N (electrons), and

= 2 x 10 1 7/N (protons).

(7.5)

4 The actual number of slow electrons in the WF and LWF experiments was of order

10-100, while the proposed antiproton experiment expects to launch groups of

about 100 antiprotons (Beverini, 1986). Given an experimental transit time of t = 0.45 sec for a one meter drift tube, this damping is expected to be negligible.

Equation (7.4b) can be used to evaluate the effect of a copper oxide layer on

a copper substrate. Hermann and Wagoner (1951, p.128) give the resistivity of

Cu->0 as about 10 iim at 293 K and, for an oxide thickness of 100 A, we estimate

x = 500/N (electrons), and

(7.6)

= 9 x 103/N (protons).

31

We see that with an oxide layer, surface charge dissipation could be a significant

effect for electrons. At cryogenic temperatures, the resistivity of the copper oxide

increases greatly but it behaves more like an insulator than a semiconductor, in

which case (7.4b) must be modified. We have not considered this case, but the

damping times may become significant even for protons.

The interior surface of a drift tube may be charged by a process of electric

current "rectification" occurring at the junction between a semiconducting

surface oxide layer and the metal substrate. Fluctuations of charge within the

drift tube may cause surface image charges to be induced, which are then trapped

by the diode nature of the surface oxide layer, forming a charged capacitor. Such

effects are difficult to calculate; they may be quite negligible, but should be

considered.

C o n c l u s i o n s

We have exhaustively listed and studied the various influences on the

motion of a charged particle moving under gravity. Many of the effects are

insignificant, however several are very important, and may require additional

theoretical and experimental study. We conclude this review by listing those that

are in this category.

Residual gas scattering;

Patch Potentials and their temperature dependence;

Strain induced electric fields.

32

A c k n o w l e d g e m e n t s

The authors are indebted to Alberto Cimmino, Joseph Hajnai, and Tony Klein

for many useful discussions. We gratefully acknowledge the support of the

Australian Research Grants Committee.

•

33

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36

Figure Captions:

Fig. 1: \ graph of the time-of-flight against the initial velocity in an ideal drift

tube experiment, for H = 0.91 m. t is the classical cut off time and v the

critical velocity.

Fig, 2 A plot log of the integrand in equation (2.20) against photon energy, in

eV.

Fig. 3 Possible paths of a particle projected upwards in a gravitational field,

where the time-of-flight to reach height H will be measured. (a) no

scatters. The TOF will always be < the classical cutoff time,

t c = v 2H/a . (b), (c), (d) scatters occur during the v flight. Only (c) and

(d) can increase the TOF beyond t c-

Fig. 4 (a) Monte Carlo simulation of the flight of particles through a vertical

height of 0.9m under gravity. The sharp cutoff at low times is due to an

upper bound on the initial velocities generated. The dotted line

indicates the classical cutoff time. The number of scatters increases 4

with pressure, (b) Similar plots of particles moving through a field free

region. The top plot with a = -g = -9.8 m/s^ is for reference. (The same

parameters as the bottom plot of (b)).

Fig. 5: Density dependence of the work function of jellium and its components

from Lang and Koh;. (1971) (solid lines) and Perdew and Wang (1988)

(dashed lines). W = AO - u, where AO = -cD/e0 and a is the Bohr radius.

Fig. 6: Change in the work function of copper with temperature, in the

presence of helium, from the data published by de Waele et al (1973).

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