**456** **R. S. Bartlett.**

3, but is of the order of 1/10 too small. From (25) and (7) the absorption in the Lj levels is 1 /8 of that in the K-levels, and the Ln and Lin absorption is from (42) and (46) much smaller than the Lx absorption for small wave-lengths.

To sum up, we may say th at with regard to the variation of oc„ with X, and the ratio of the absorption in the K and Lj levels and in the different L levels, the present theory is reasonably in agreement with experiment. Further, by comparison with Oppenheimer’s results for the photo-electric effect, the theory indicates th at the ratio between internal and external absorption varies only slightly over a wide range of wave-lengths in agreement with experiment. On the other hand, the order of magnitude of the theoretical internal absorption coefficient is everywhere of the order of 1 /10 too small.

It is hoped to carry out the calculations with a better approximation to the characteristic functions of a heavy atom than th at given by those of a hydrogen atom.

*The Increase o f Thermionic Currents from Tungsten in Strong* *Electric Fields.*

By Bu ssell S. Ba r t l e t t, Ph.D., Yale University.

(Communicated by 0. W. Richardson, F.R.S.—Received August 10, 1928.) Schottky* has pointed out th at there should be an increase in thermionic currents above the saturation value with the application of strong electric fields. His theoretical expression is based on the assumption that an electron is prevented from escaping by the attraction of its electric image in the emitting wire, and that the external field neutralises part of this image force field. In the same paper he reports an approximate experimental verification of his theoretical law, though the details are lacking. Since that paper, reference has been made to this increase in current, and it has been applied as a correction in determining the thermionic work function. Becker and Muellerf have discussed the problem, but apparently assume the correctness of Schottky’s expression. Under the circumstances it seemed that an experimental investigation was needed, all the more because Schottky’s theoretical expression seems open to question on certain points.

* * Phys. Z.,’ vol. 15, p. 872 (1914).

t ‘ Phys. Rev.,’ vol. 31, p. 431 (1928).

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**Theoretical ***Cons***.***

**Let us suppose that a single electron is escaping from a plane electrode, and **
**that the image force alone is acting upon it.**

**d V __e ****y __ e fz d x __ e _ e****dx ****x2’ ****i ) z 0%2 ****4a?0 ** **4x***

**Suppose now an external field E is applied, in a direction to assist the escape **
**of the electron.**

**V' =** **e****4:X0**

**_e**

**4x****E x,****d T __ _e___**

**dx ****4x2**

**And, for the potential minimum,**
**V ' _ **

**V1 “** **2 V e /E**

**E ** **f e **

**2 V E** **<£o — V Ee.**

**Putting this expression in the thermionic emission equation, we obtain**

**—<fro + Vlflg ** **«/****e****« ** **4*38 V****e**

**I = AT2e*T ** **™ = I oSfcT , ****I =- I 0e ** **t** **.**

**Where E is expressed in volts per centimetre.**

**For fields between 300 and 30,000 volts per centimetre, the distance of the **
**potential minimum from the surface ranges from 10~6 to 10-6 cm. ** **With **
**concentric cylinders for electrodes, the departure from the plane electrode case **
**will be negligible if the radius of the emitting wire is greater than 10-3 cm. **

**From the dimensions of the electrodes it is possible to calculate the electric **
**field at the cathode in terms of the applied voltage, if the effect of space charge **
**is neglected. ** **As a second approximation we may take E = C (Y — V0) **
**where V0 is the voltage necessary to produce saturation. Referring now to **

**Schottky’s expression, we find that**

**log I /I 0 = 4 - 3 8 \/E /T = 4-38 ****a****/C ( V - V 0)/T.**

**So plotting log I /I 0 against ****a****/ Y — V0 should give a straight line if the image **
**force is the only force effective.**

**Experimental Procedure.**

**The experiments were performed with a tungsten wire, concentric in a **
**molybdenum cylinder, and maintained in position by a small bow spring at **
**each end, the whole mounted in a quartz tube which permitted thorough **

*** Based on Schottky’s derivation, loc. cit.**

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**458** **R S. Bartlett.**

**bathing in an electric furnace. The heating current for the tungsten cathode **
**was furnished by accumulators of sufficient capacity to maintain the tempera**

**ture sensibly constant throughout a series of observations. The voltage **
**between electrodes was furnished by banks of cells amounting to about 1000 **
**volts in all. The multiplying factor introduced by the dimensions of the **
**apparatus gave maximum fields between 30,000 and 60,000 volts per centimetre **
**for the different wires used. The thermionic currents were measured by **
**galvanometers and microammeters, suitably shunted. Filament temperatures **
**were determined from the resistance, using Worthing and Forsythe’s* data, **
**and checked by the saturation thermionic emission currents.**

**A considerable amount of manipulation was required before the following **
**satisfactory procedure was devised. After preliminary out-gassing in the **
**electric furnace, the tungsten wire was glowed out at about 3000° K. Then **
**the cylindrical anode was raised by bombardment to a bright red. After this **
**it was possible to maintain the vacuum at better than 10-6 mm, of mercury. **

**But to obtain consistent and reproducible results it was necessary to repeat **
**the glowing and bombardment before each series of observations. When that **
**had been done, measurements of the thermionic emission current were made at **
**intervals of about 40 volts in the potential between electrodes.**

**Experimental Results.**

**Satisfactory results were obtained with two wires : A, ordinary commercial **
**tungsten, not thoriated, of 0*08 mm. diameter; and B, specially pure tungsten, **
**obtained through the kindness of the General Electric Company Research **
**Laboratories, of 0*095 mm. diameter. Runs were made at five different **
**temperatures with A, and at eleven with B, between 1500° and 2400° K. **

**Figs. 1 to 4 show typical results obtained with these wires—first I plotted against **
**V, and then log I against \ / V and against V V — V0. For all runs the curves **
**obtained were of the same general form.**

**According to the image force theory, the points (log I ~ v V — V0) should **
**lie on a straight line. Though the space charge correction, substituting **
**V — V0 for V, brings the points into much better alignment, it was impossible **
**to select any value for V0, even absurdly large, which would completely **
**eliminate the curvature. A close study of all the observations forces one to **
**the conclusion that Schottky’s expression does not quite fit the facts. Certainly **
**the residual curvature is greater than can be accounted for by experimental **
**error.**

*** ‘ Phys. R ev.,’ vol. 18, p. 144 (1921).**

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**l-20r **
**log*/t **

1-15

**HO-**

**n ' ’'r ““‘i** t **r** ^{32 } ^{y/F -}

**o o °**

**/ /****f**

**105**

1-00

**0-95**

*** o o**

0-9 0 -

**°**

**i ** **i**

10 20

0 200 400 600

**F****ig****. 1.—Series A 11. T = 1900° K . I 0 = 3 -5 **x **10-6 amp./sq. cm.**

800 1 000

**V —+-**

1 000

**F****ig****. 2.— Series A 12. T = 1945° K. I 0 = 9-2 X 10-6 amp./sq. cm**
I

**However, it is still possible to obtain a good value for the average slope of **
**the log I ~ V V — V0 curves, from which we can investigate the variation of **
**this slope with temperature. The theory predicts that it should be inversely**

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**460** **R S. Bartlett.**

**proportional to T, and fig. 5 shows to what extent this is borne out by **
**observation. In interpreting these points due weight must be given to the**

**V**

**F****ig****. 3.—Series B 6. T = 2040° K . I 0 = 1*27 **X **10-3 am p./sq. cm.**

**27**

**A**

**25**

**200** **400** **600** **800**

**V -+**

1000

21 ^{-}

**19 -**

**I / y**

**'ogf/lWso**

**'ogf/lWso**

log/1

**1-4**

**0 ** **8 ** **16 ** **_ 24 ** **32**

**j v —***

**F****ig****. 4.—Series B 8. T = 2200° K. I 0 = 9 ’94 x 10~8 amp./sq. cm.**

**experimental difficulties and uncertainties. In the first place, the heat treat**

**ment caused a certain amount of wasting of the tungsten wire between runs,**

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**which in turn introduces some uncertainty into the determination of tempera**

**ture and electric field. Further, at the lowest temperatures it was necessary**

**1600** **1800 ** **2 ooo**

**T e m p era tu re**
**F****ig****. 5.**

**2 200** **2400**

**to correct for an insulation leak inside the tube, while at the highest temperature **
**an error m ay be introduced b y the effect of space charge.**

**I t is seen from th e figure th at the change of slope w ith temperature is in the **
**right direction, but much greater than th a t predicted b y theory. Even if the **
**end points are discredited to some exten t for the reasons given above, it is **
**still impossible to reconcile the results w ith the theory.**

**Quantitatively these results agree w ith the theory at about 1900° K. For **
**higher temperatures the experim ental slope is too small by amounts varying **
**up to 25 per cent., while for lower temperatures the departures are much **
**greater, and in th e opposite direction, the experimental slope being about **
**six tim es the theoretical at 1560° K.**

**In thermionic emission work questions are always asked as to the purity of **
**the tungsten, and the possibility of surface contaminations. I t was partly to **
**settle th at difficulty th at the vigorous heat treatm ent was used, and as a **
**guarantee of the purity of the tungsten, the temperatures calculated from **
**emission data were consistently lower than those calculated from the resistance, **
**showing th at the emission was, if anything, less than what is normal for pure **
**tungsten. The result of th at heat treatm ent is perhaps best illustrated by a **
**comparison of figs. 6 to 7 for runs taken with filament A without any vigorous **
**heating just before the observations, with similar runs after heat treatment**

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**462** **B. S. Bartlett.**

1050**v —***

- **1*2**

**F****ig****. 6.—A 5. T = 1750°. I 0 = 3-86 X 10~I ***** 3**
Filament not glowed out.

250 450 650 850 **V—***

1050

30 l

**/v*

**log;/'**
**1*5**

**o ° o **
**o ** **o**

V - - - . ,-- **!**_____________**1_,____u**

0 20 40 60 **o ° o ° **^{o }^{o}

**o ** **o **
**_ o _ o **

**o ** **o**

**Zv.*

**o ** **o **

o o

**O ** **o **

**o ** **o **

**o ** **o **

**o ** **o **

**o ** **o**

**l°Sf4F5obr****o log**

I I_____*J*_____ l i I_____ |_____ L

0 8 16 24 „

**F****ig****. ****7.—A 6. T = 1970°. I 0 = 6-86 x 10~3. **

Filament not glowed out.

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**(figs. 1 and 2). I t is noticeable that without the heat treatment there is **
**nothing that can be called a saturation current. Similar, though less marked, **
**results were obtained for filament B, the especially pure tungsten.**

**Despite this lack of saturation, it is still possible to bring these points into **
**fairly good agreement with theory, by choosing Y0 of the order of 100 volts.**

**The insulation and gaseous discharge leak inside the tube was measured **
**immediately following each series of observations, and corrected for where **
**necessary. This leak amounted to about 20 per cent, of the thermionic current **
**at the lowest temperature, and was, of course, entirely negligible at higher **
**temperatures.**

**Discussion of Results.**

**It was pointed out above that there is no correction necessary for the **
**cylindrical electrodes in this experiment. Further, account is taken, to a **
**good degree of approximation, of the effect of space charges in diminishing the **
**applied field. Even then Schott ky’s theoretical expression is not exactly **
**verified in observations at constant temperature, still less so, apparently, when **
**it comes to variations of the effect with temperature. A probable reason for **
**this discrepancy is not far to seek.**

**The theoretical expression was derived on the assumption that a single **
**electron was moving out from the metal surface acted upon by the attraction **
**to its electric image in the metal. Though the electrons escaping from the **
**metal are comparatively few in number, there must be a great stream, of 1020 **
**per square centimetre per second or more, passing through the surfaces and **
**being turned back by the work function force. We saw above that for the **
**range of this experiment the work function extends out from the surface a **
**distance of 10~5 to 10~*6 cm. So that for the electron just escaping there are **
**other electrons as close as is the surface charge, and these electrons themselves, **
**and their images in the metal surface, must have some effect upon the **
**particular electron under consideration. An attempt has been made upon this **
**problem, somewhat analogous to the problem of space charge, but so far it has **
**yielded nothing in as close agreement with experiment as the Schottky **
**equation.**

**Summary.**

**Experimental results for the increase of thermionic currents with applied **
**electric field at constant temperature show general agreement with theory, **
**but the departures from a predicted straight line are greater than can be **
**accounted for by experimental deficiencies. In the dependence of this rate of**

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**464** **L. H. Thomas.**

increase upon temperature the failure of experiment to agree with the theory is still more marked, even after due allowance is made for certain experimental difficulties.

It is suggested that Schottky’s equation should be modified to take account of the influence of neighbouring electrons close to the surface—electrons that emerge from the surface, but do not completely escape.

Attention is drawn to the marked effect of surface impurities in the cathode upon the experimental results.

The author wishes to express his thanks to the laboratory staff at King’s College, London, where this work was carried out, for their kindness and assistance ; to Dr. A. J. Waterman for many helpful discussions ; and in particular to Prof. Richardson for his interest and encouragement.

*On * *the Rate * *at which Particles take up Random Velocities from* *Encounters according to the Inverse Square Law.*

By L. H. Thomas, Ph.D., Trinity College, Cambridge.

(Communicated by R. H. Fowler, F.R.S.—Received August 17, 1928.)
*Summary.*

This paper deals theoretically with the changes in the velocities of particles which move through a cloud of particles interacting according to the inverse square law. The formulae obtained—for low densities—lead to “ effective mean free paths ” shorter than might have been expected. I t seems not impossible that the rapid rate at which beams of electrons moving through highly ionised gases have been observed by Langmuir* to take up Maxwellian velocity distributions may be explained in this way.

*1. Introduction.*

This problem has been discussed by Jeansf for stellar encounters. He takes into account all the encounters for which the closest distance of approach is less than the normal distance between adjacent stars and neglects the remaining

* ‘ Phys. Rev.,’ vol. 26, p. 585 (1925).

t ‘ Astronomy and Cosmogony,’ p. 309.