Size effects in superfluid field emission

A detailed experimental investigation has been made of the influence of chamber dimensions on field emission characteristics in liquid He. 4 The current was found to be a strong function of the emitter-collector separation, in satisfactory quantitative agreement with the predictions of a simple theory

Size effects in superfluid field emission P. V. E. McClintock and A. G. Webster

The chief importance of field emission in the liquid heliums arises from the relatively large current which may be injected by this means, so a field emitter may be used as a high flux ion source for a variety of experiments aimed at probing the nature of either liquid. The phenomenon is, however, also of considerable interest in its own right and has therefore been investigated in detail both for liquid He3 1 and liquid He4. 2,3,4,5 Careful measurements of the diode characteristics have enabled the ionic mobility to be determined down to 0.25K for He3 6 and to about 1.6 K for He4 2,S,Tand in the case of He4 it has also proved possible to deduce values of the probability of vortex nucleation by an ion, s,9 and of the probability that the ion will escape once more from the vortex on which it is trapped, 5 by fitting the data to a simple model. 1° None of these experiments has examined systematically the influence of chamber dimensions on the emission characteristics. For the temperature ranges above 1.6 K, where the effect of superfluid vortices may be ignored, the currentvoltage characteristics are well described by Halpern and Gomer's spacecharge equation 2

V = Vo + 3.097 x 10s ( e ~ p ) '/2

(1)

where Vis the emitter potential in volts, Vo a constant, R the radius of the collecting electrode in mm, I the current in amps, art the emission cone solid angle in sr, e the dielectric constant, and # the ionic mobility in m2V-1 ~1. The main purposes of the present work were, firstly, to find out whether the size dependence of the current, as implied by the presence of R in the second term o f ( l ) occurred in practice, and, secondly, to ascertain whether the temperature at which the maximum in the current occurred depended on R in the manner which had been predicted s earlier on the basis of a simple model.

Apparatus The ideal cell for this investigation would consist of the emitter and, surrounding it, a spherical collecting electrode whose radius could be continuously varied; but, in the face

The authors are with the Department of Physics, University of Lancaster, Lancaster, UK. Work supported in part by the Science Research Council under contract B/RG/60948. Received 16 August 1974.

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of the formidable technical difficulties which would be involved in designing and constructing such a device, we have chosen to depart altogether from spherical geometry. Instead, we have used a point-to-plane configuration. Provided that the separation D of emitter from plane collector is very much larger than the collector radius, we can regard the plane as infinite. Thus, although we do not know a priori what will be the effective average value of R corresponding to any given value of D, we may safely assume that the geometry of the situation remains unchanged except by a scaling factor while D is varied, and that the value of R for insertion in (1) will therefore be proportional to D. Little information is lost through this change of geometry since, on the basis of earlier work in a segmented spherical chamber, 11 it is known that the emission is in any case anisotropic, so that an is some average emission angle which must in practice be determined by experiment for each emitter and shape of chamber. The lower portion of our cryostat insert is illustrated diagrammatically in Fig.1. The tungsten emitter, a, prepared by electrochemical etching as described previously, 1 was spot-welded to a length of 0.5 mm diameter nickel wire, b, which was itself held in the nylon mounting, i, by means of a grub screw. The nylon mounting was supported by a brass disk, h, which was held in position on the two brass rods, g, by means of a pair of grub screws. The collecting electrode, c, which consisted of a brass disk of diameter 50 mm, was attached to, but insulated from, the movable component, e, which was free to slide up and down the rods, g, in a direction parallel to the axis of the emitter. This motion was effected through rotation of the captive threaded rod, f, which then acted on the threaded portion of e. Movement of f was accomplished by rotation of the thin-walled stainless steel operating tube, n, acting through a key and socket arrangement, m. All the components were ultimately supported by the 6 mm thick brass base plate, p, which was itself connected to the cryostat top-plate by the two 10 mm od thin-walled stainless steel supporting tubes, k and 1, the latter also serving as a siphon guide. The eht lead, q, to the emitter was insulated by a 4 mm od, 1 mm id polythene tube, j, which passed through the other supporting tube, k; and the current collected by c was taken via the screened lead r to a Keithley electrometer, type 610 B. The whole arrangement, shown in Fig.l, was immersed in helium inside a glass cryostat of conventional design, which could be pumped to about 1.3 K. The temperature was measured by means of a 470 ~ Speer carbon thermometer, calibrated against the He4 vapour pressure.

CRYOGENICS. DECEMBER 1974

Experi mental results In making the measurements we have determined changes in the distance D separating a and c by counting the number of turns N of the knurled knob attached to the upper end of n, and have then used the fact that the pitch of the thread is 1.05 ram. To ensure that emitter and collector did not inadvertently come into contact, and to provide a definite base position from which the number of turns o f n could be counted, a pair of adjustable stops (not shown in Fig. 1) could be attached to the slide rods, g, by means of grub screws: in practice, the stops were usually set such that the minimum distance Do between c and a was about 2 ram. To reduce the effects o f backlash, the position o f the collector plate was always set by turning the adjusting knob clockwise, that is, with the plate rising. The collected current was measured for a range of emitter potentials, tipcollector separations and temperatures T in the range: 2.0 ~< V~<3.5 k V ; 2 ~
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4

5

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N Fig.2 The c u r r e n t / reaching the c o l l e c t o r as a f u n c t i o n o f the e m i t t e r - - c o l l e c t o r separation f o r various e m i t t e r p o t e n t i a l s V a t 4.2 K: N is the n u m b e r o f turns o f the operating rod, w i t h N = O d e f i n e d as being w h e n D = O o, w i t h the c o l l e c t o r in c o n t a c t w i t h the stops; the full curves are visual guides.

precision corresponding to about N = 0.2. Measurements of the I (T) characteristics for constant V and a number of fixed values of D (Fig.3) showed that the temperature Tm, where the maximum occurred in the currerLt, was reduced slightly as D decreased.

Discussion Before results such as those of Fig. 1 can be compared quantitatively with (1) it is necessary to relate R to N. We assume that R = OD where 0 is a numerical constant of order unity, in which case we can write R = 0 (Do + NO)

(2)

where 0 is the pitch of the threads on e and f. Combining (2) and (1), and re-arranging slightly, we find t ~ = AO N + ADo J

Fig.1 Diagram o f the l o w e r part o f the c r y o s t a t (slightly m o d i f i e d so as t o appear in a single vertical section) a -- tungsten f i e l d e m i t t e r ; b -- nickel m o u n t i n g w i r e ; c - c i r c u l a r plane brass c o l l e c t i n g electrode; d -- n y l o n insulating disk; e -- movable brass m o u n t f o r c o l l e c t o r ; f -- captive brass screw; g -- brass slide rods; h -- brass disk; i -- n y l o n e m i t t e r m o u n t ; j -- t u b u l a r p o l y t h e n e insulator; k -- stainless steel s u p p o r t i n g tube; I -- stainless steel s u p p o r t i n g t u b e and s y p h o n guide; m -- elliptical cross-section brass key and socket; n -- stainless steel t ~ b u l a r o p e r a t i n g rod; p -- c i r c u l a r brass base-plate; q -- lead t o eht s u p p l y ; r - screened lead to e l e c t r o m e t e r .

CRYOGENICS. DECEMBER 1974

(3)

where A = 9.6 x 10 l° 0/[o~e/~ (Vs-Vo) a ] is constant for fixed /l and V. Plotting 1-1 as ordinate against N as abscissa should therefore, for constant T and V, yield a straight line whose intercept on the abscissa is is - Do/4). The result of applying this procedure to the experimental results of Fig.2 is shown in Fig.4. Within experimental error a series of straight lines, with a common intercept at N = - 1 . 6 , may be drawn through the data points. Similar results were obtained for a number of different temperatures above Tin. The data for T > Tm are thus in excellent agreement with ( I ) provided that, for this particular experiment, - Do/4) = 1.6, or Do = 1.7 ram. Direct measurement of D o is not straightforward. Because of thermal contraction, an optical measurement at room temperature is unsatisfactory; and a similar measurement in situ at liquid helium temperature was not feasible owing to the poor optical

651

where it is assumed that the escape probability and ionic mobility may to good approximations be written as

8

P = 1' exp

-

and

1.7

"3

where 7,/3, U, and A are constants. The chamber dimensions enter the equation in two ways: first, explicitly through R 2 in the denominator, and secondly, for given Vs, by affecting the average electric field in the chamber and so modifying U, which is the potential barrier the ion must surmount in order to escape. Because the electric field at any radius r in the chamber is proportional to (rR) -~2, s and D = R, we conclude that the average electric field, in the chamber, which determines the effective value of U, F cc D-I. Using ,Schwarz's 12 measurements of/a for negative ions and Padmore's empirical relation 13

2.7

P = 2.6 x 1011 exp -

[(Ps/P) (61.5 - 3 . 8 4 F + O.11FZ)/T]

I

O

1.4

1.5

1.6

1.7

1.8

1.9

T,K Fig.3 The collector current / as a function of temperature 7-, showing that the temperature T m of the maximum in I(T) increases slightly with emitter--collector separation D; the full curves are visual guides

quality of our dewars. At the conclusion of all the current measurements for any particular emitter, therefore, we moved the stops (without touching any of the other components), refilled the cryostat with helium, and then counted the number of turns of the operating rod which were necessary to establish direct electrical contact between a and c, a procedure which was, of course, final because it resulted in the destruction of the emitter.

where F is in kV cm "l and Ps/P is the superfluid fraction, For Y and U, we have fitted (4) to our Tm(D) data at one point, choosing the average field F to be such that Tm = 1.76 K at D = 5.9 mm, in agreement with our measurements. We have assumed that, to a good approximation, D = R under the In. We have then calculated the variation

o 2.0 ~7 2.5 [] 3.0 A

3.5

6-

In the event, 2 1/8 turns were required to establish contact and, in view of the overall backlash of about % turn in the mechanism, we conclude that the direct measurement of D O is indeed consistent with the value derived from the emission characteristics on the basis of (1). The existence of maxima in I (T), such as those of Fig.3, has been attributed 3,5,1o to the influence of quantized vortices in the superfluid. At low temperatures the ions become trapped on slowly moving vortices, a process which tends to increase the spacecharge density and therefore reduces the current. As the temperature is raised the influence of the vortices decreases, with a corresponding rise in the current, because the increasing escape probability P reduces the average length of time each ion spends trapped on a vortex until, above Tin, the vortices have a negligible effect and the current falls with/a as described by (1). It has been shown on the basis of a simple model that the maximum in I should occur at a temperature s U+A Tm =

652

k In [4"},R2 A/3U~Vs-Vo)]

(4)

-2

-I

O

I

2

3

4

5

6

7

N Fig.4 Plot of (current i)-1 against N for emission at 4.2 K with various emitter potentials V. The straight lines have been drawn by eye through the data points to demonstrate consistency with (3)

CRYOGENICS,

D E C E M B E R 1974

1.9

Conclusion

1.8

We have found that field emission characteristics in liquid He4 are strongly influenced by the physical dimensions of the chamber, and we have been able to account for our experimental results on the basis of existing theories of the phenomenon.

1.7-

We are much indebted to N. Bewley and I. Miller who constructed the cryostat insert, to C. N. Barber who rendered assistance in etching the emission tips and to S. S. Swarbrick for his help in running the cryostat.

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References

mm

1 2 3

Fig.5 C o m p a r i s o n o f the observed dependence o f T m u p o n D (points) and the b e h a v i o u r p r e d i c t e d b y (4) (full curve)

4 5 6 7 8 9

of Tm with D on the basis of(4), using an iterative procedure to allow for the relatively weak temperature dependence of U arising from the Ps/P factor, and have obtained the solid line of Fig.5. The curve has not been extended below D = 3 mm where Padmore's formula would become a poor approximation. Within experimental error, the size dependence of Tm for D > 3 mm appears to be in excellent agreement with (4).

10 11 12 13

McClintock, P. V. E. J L o w Temp Phys" 11 (1973) 15 Halpern, B., Gomer, R. J Chem Phys 51 (1969) 1031 Hickson, A., McOintock, P. V. E. Proc LT12 Kanda, E. (ed) (1970) 95 Phillips,A., McClintock, P. V. E. Phys Lett 46A (1973) 109 Phillips,A., McClintock, P. V. E. (forthcoming) McClintock,P. V. E. J L o w T e r n p P h y s l l (1973) 277 Sitton, D. M., Moss, F. Phys Lett 34A (1971) 159 Gavin, P. J., McClintock, P. V. E. Phys Lett 43A (1973) 257 Phillips,A., McClintock, P. V. E. J Phys C: Solid State Phys 7 (1974) Ll18 McClintock,P. V. E. J P h y s C: Solid State Phys 6 (1973) L186 McClintock,P. V. E. Read-Forrest, H. Cryogenics 13 (1973) 371 Schwarz,K. W. Phys Rev A6 (1972) 837 Padmore,T. C. Phys Rev A5 (1972) 356

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