SHOT EFFECT OF SECONDARY EMISSION. I

b y M. Z I E G L E R

Natuurkund ig Laborator ium der N.V. Philips' Gloeilampenfabrieken Eirtdhoven-Holland

R 6 s u m 6

Le choc d ' u n 61ectron de certaine vitesse contre une 61ectrode m6talli- que cause 1'6mission d'61ectrons secondaires qui peuvent gtre recueillis par une autre 61ectrode. Le courant secondaire peut 6tre repr6sent6 par

la s6rie~-~j ~ 6n T~m oh 6. ~ m est la fraction de courant primaire dont

0 ( x ) les ~lectrons l ib~rent chacun n ~lectrons secondaires 6n = 1 .

0 Dans l 'art icle su ivan t on suppose que le choc primaire et y6mission

secondaire sont p ra t iquement simultan~s de sorte que les ~ ~lectrons secondaires lib~r~s par le m~me ~lectron primaire peuvent ~tre consid6rgs comme un seul ,,~lectron mult iple" quan t k l ' impulsion de courant qu'i ls produisent.

Selon la formule bien connue du Shot effect (ou Schrot-effekt), applicable

k des courants satur6s, les f luctuat ions ( I - ~ de chaque fraction ~1. 6n Iprlm du couran t secondaire sont proportionelles au courant m~me et k la valeur des impulsions de courant 616mentaires qui const i tuent ce courant , donc proportionelles k n~ 6. ~ . Les f luctuat ions de chaque fraction de couran t 6 tant ind6pendantes les unes des autres, les carr6s

moyens (I - T) 2 s ' a jou ten t les u~s aux autres de sorte que les f luctuations

totales du c0urant secondaire sont proportionelles k ~ . - n 2 6-~,~m. A l 'aide

0 de cette th6orie il est ais6 de d6montrer que dans une triode oh la grille et la plaque sont positives par rapport k la cathode et peuvent 6mettre cha- cune des 61ectrons secondaires et oh, en outre, les densit6s de charge entre les 61ectrodes sont n6gligeables, la diff6rence ( I g - ~ ) ~ - (I t - ~)~ des f luc tua t ions des courants de grille et de plaque dolt fitre proportionelle

la difference ( I g - Ip) de ces courants. • Cette pr6diction se trouve confirm6e exactement par l 'exp6rience ; il y a

donc lieu de conclure que l 'hypoth6se formul6e est exacte.

Physiea I I I I

2 M. ZIEGI.ER

§ 1. In a vacuum tube containing an emitter of particles, all with the same electric charge q, and another electrode brought to such a potential that all the emit ted particles flow from the emitter to that electrode, the instantaneous intensity I off the electric current in the tube fluctuates about its mean value I as a result of the random distribution in t ime of the current impulses originating from the individual particles (shot effect).

In these f luctuations all frequencies are represented equally between 0 and the very high frequencies where transit times Come into play. The classical formula for the shot effect may be written in the form:

d, (1__~2 = 2qY (1) dv

where v is the frequency. When we have to measure unknown fluctuations, it is a great help

to compare them with the well defined fluctuations caused by this pure shot effect as represented by (1). It can easily be arranged so (see experimental part) that the current fluctuations of a diode D caused by pure shot effect are just equal to the unknown current fluctuations of a tube under investigation T. We then have:

= (zD--z.) = 2 7-.

(e = charge of the electron). The diode current I-o maybe used as a measure for the fluctuations.

We define, therefore,

d d-~ ( I t - - I t ) 2 (2)

iT=ID = 2e

For a current Iq consisting of particles with charge q, (2) takes the simple form

i, = 2 q I , = q_ ~ (3) 2e" e "

§ 2. Now, electrons which strike an electrode with a certain velocity cause the emission of a number of secondary electrons, which depends on the constitution of the electrode and the velocity

S H O T E F F E C T O F S E C O N D A R Y E M I S S I O N . I " 3

and direction of the primaries. The number of secondary electrons can be several times the number .of primaries: thus one primary electron may release several secondaries.

As the intensity of the secondary emission caused by the impact of a certain large number of primaries is a continuous function of the primary voltage, all pr imary electrons cannot have the same influ- ence: a part 90 has no effect, a part ~1 releases each 1 secondary *), a fraction releases each 2 secondaries, etc., so that

O O

~n = 1 (4) o

O 0 • -

while the current of secondaries I,., is equal to n ~n times the 0

primary current Ip,~m O 0

o

The values of the ~'s depend on the constitution of the electrode, primary velocity, etc.

Our considerations on the fluctuations of the secondary current are based upon the following supposition:

The impact of every primary electron is followed by the emission l)f n. secondary electrons (n may be 0, 1, 2, 3, 4 . . . . . etc.). The ,,emission-time", i.e. the lapse of time between primary impact and corresponding secondary emissions is so small, that the transit of these n secondaries may be considered as a single current impulse, coinciding in time with the impulse originating from the primary electron,

Thus n secondaries of charge e, released by the same primary electron will have equal influence on the fluctuation of the secondary current as a single particle with charge n e.

We may thus consider each part n ~ Ipam of the secondary current as being composed of particles with charge n e and may write, for each part, according to (3)

• *) P r i m a r y e l e c t r o n s w h i c h a re go ing to b e r e f l ec ted a re cons ide r ed to b e l o n g to t he

s a m e g r o u p .

4 M. ZIEGLER

Now we remark that the impulses due to the charges n e, which form each part n ~. Ip.,. of the current, are quite independent in phase of all the other impulses so that we may add the values of i~ just as might be done with the shotfluctuation energies of a number of seFarate tubes.

Thus we obtain for the fluctuation of the total secondary current1) •

o o

o

E n= (5. is generally greater than E n ~. ; this means physically o o

that a secondary current shows a greater fluctuation than a pr imary current of the same magnitude.

§ 3. A direct experimental check of (6) is not possible as the o o

individual terms of the series Z ~, are unknown. The preceding 0

theory, however, makes it possible to predict a relation which must be satisfied in a vacuumtube containing a cathode (emitter of primary electrons) and two other electrodes e.g. grid and plate, when both grid and plate are positive with respect to the cathode, while care has been taken to have a good vacuum and practically no space charge_

One part Ig p,~m of the electrons emit ted by the cathode reaches the grid, the other part Ip p,~ strikes the plate, and both grid and plate emit secondary electrons.

We shall consider first the simple case in which all secondaries of the plate reach the grid and no secondaries of the grid can reach the plate, which may be realized practically by means of a suitable potential difference. The plate behaves, therefore, chiefly as an emitter of secondaries, the grid as a collector. The functions of grid and plate can be inverted.

According to (5) the grid current is then equal to

CX3

0

The fluctuations of Ig are obtained by adding the completely

SHOT E F F E C T OF SECONDARY EMISSION. I 5

independent fluctuations of Ig ~,~,. and Ip s~. According to (3) and (6) we then get:

0

For one primary electron hitting the plate, n secondaries are emitted. According to our supposition the emissions of the s'econd- aries are correlated with the primary impacts in such a way that the impulse -he caused by the transit of the n secondaries may be superposed on the corresponding primary impulse e. The impact on the plate of one primary electron gives, therefore, together with the escape of its secondaries an impulse (1 -n)e , so that the total plate current can be represented by

O 0

L = ( 1 - . ) (9) o

where the impulses of each part have the value ( 1 - n)e and are quite independent.

Similarly to eq. (6) we may write, therefore :

O O

ip = ~ (I - n) 2 ~. I~ p,,~ (I0) o

As we remarked in a previous paper u), we obtain from the 4 last equations

i s - ip = I~ - I p (1 I)

This relation is also valid when only a part X o/ the secondaries emitted by the plate are collected by the grid, the other par t ( I - k) falliug back on the plate. In that case we have only to consider those secondaries, that are collected (the other having no influence). We may then speak of an apparent secondary emission I~ (equal to X times the total secondary emission) with a different distribution expressed by means of the coefficients ~; (7) and (9) are then transformed into:

oo

I-~ 0

6 M. Z I E G L E R

O O

0

While (8) and (10) take the form O O

o O O

i # = ~ (1 - n)2~, I-~,am 0

so that (11) holds again. We assumed at the beginning that no secondaries of the grid could

reach the plate: the validity o] (11), however, is not limited by this restriction. When secondaries of the grid reach the plate and in- versely the whole phenomenon may be split into two independent parts f9r which (11) is sure to hold.

We consider first all the electrons constituting !p pare and the secondaries they release as if there were no more primary electrons; in that case I s pare = 0, hence I , ,~ = 0 and therefore (1 I) may be written down for the considered currents. The same thing may be done for I~ #,ira and the secondaries released by it. (In both cases the influence of ,,tertiary" emission caused by impact of secondary electrons is neglected). It is then easy to show that the addition of the partial effects leads again to the expression (I 1) for the total cu.rrents.

It may be remarked that, if there were no correlation between primary impacts and secondary escapes we would, for the most general case, obtain instead of (11) :

i . - i# = (i s #.m + i~,,o + i# ,~) - (i# #am + i~ ,~ + i , ,~)

= i8 #am - i# #am = Ig #am - I # #am

which is quite a different relation.

§ 4. Measurement o/fluctuations. The experimental arrangement is indicated schematically in fig. 1.

T is the tube under investigation, which in the figure is arranged for the measurement of the fluctuations of the plate current I~. The plate of T is connected, via a condenser of great capacity to the plate of D, a diode with tungsten filament. The total space currents of T

SHOT EFFECT OF SECONDARY EMISSION I " 7

and D are controlled by means of the filament resistances RT and RD resp. ; plate and grid voltages are high so that the tubes are free of space charge.

Fluctuations of the space currents of the tubes give rise to voltage fluctuations ill the L C circuit C, which, in their turn, are amplified by a high frequency amplifier. The circuit C and the amplifier are tuned to a frequency of about 150.000 cycles/sec, so that the results refer to the spectrum of fluctuations near 150.000 cycles/sec. The choice of this frequency is determined by practical considerations only.

z / ~ x G, I I Krnam~ i mt~t~-,~ ~r~'~7~.r7 _ ¢-4

~ ~ , ~ ' - W , . I_ • / / ! i e / - : ~3-1 I~11--t3 ~.o~,tr.//1 1 ~-~e'~.l

Fig. 1. A r r a n g e m e n t for m e a s u r i n g f l u c t u a t i o n s .

The amplified fluctuations are superposed on a carrier signal of suitable amplitude, so that they can be detected linearly and amplified still further by a low frequency amplifier. The low frequency output current flows through a thermo couple connected to a galvanometer.

To carry out a measurement the space current of D is first made zero by a sufficient increase of RD. The fluctuations of Ip cause a deviation of the output galvanometer, which is adjusted at a given value At, controlled by the amplification. From a calibration the deviation of G, say, A2 is known when the fluctuation energy.is just twice the fluctuation energy corresponding to the deviation A t. Now the space current of D is adjusted to such a value, ID, that the reading of the galvanometer attains A2. Under the conditions of this experiment the internal differential resistance of D, is high compared with the external impedance so that it makes no difference as regards the amplification of the fluctuations of T whether the diode is cold or not. (Amplification and detection are linear). The fluctu-

8 M. ZI]/GLER

ations of Ip and of ID are completely independent; when, therefore, the energy of the superposed fluctuations is twice the energy of the fluctuations of one tube alone, the fluctuation energies of the tubes are equal.

According to § 1 the value i for the unknown fluctuations is just equal to the diode current ID. All fluctuations may therefore be expressed in terms of the values of the diode currents determined in the way described. The thermal voltage fluctuations of C and the ,,noise" of the first amplifier tube could be neglected in these experi- ments.By careful shielding all external disturbances were eliminated.

In order to eliminate errors of measurement due to slow fluctu- ations of the galvanometer deviation it is advisable to repeat a measurement a few times.The numerical results given in the following § are the mean values of at least 10 independent measurements.

§ 5. Experimental check o/the theory. Our most convenient experi- mental object was a triode with the following characteristics:

Cylindrical tungsten filament of 80 1 z diameter and 20 mm length, cyl. grid of 18 mm diameter and ca. 25 mm length (80 turns of molybdenum wire of 150 ~ diameter with pitch of 300 ~). Cylindrical plate of 21 mm diameter and 20 mm length covered with a layer containing Barium and Strontium, giving a large secondary emission. From the geometry of the grid it may be deduced that when the ratio of grid and plate voltages is not too high or not too low, the primary grid and plate currents are equal.

In order to check our equation (11), we performed the following measurements. The plate of the triode was maintained at a voltage of 200 V, and the cathode emission Ig + Ip was maintained constant ad 100 ~A (a low value in order to reduce the space charge). For different values of the grid voltage V v the total plate- and grid current Tp and 7g and the corresponding fluctuations ip and ig were measured. The results are shown by fig. 2; the numerical values are given in the following table:

vg ] so [i00

2 -22 ~# 98 122 ig 43 83 /# 135 224

]150 170[ 190[ 198

-34 -36 -28 5 134 136 128 95 130 137 128 145 305 310 281 231

I oo 42 58

211 229

202

88 12

312 227

I I I 13oo I I vol, 118 -18 354 217

156 178 -56 -78 465 562 245 310

182 187 -82 -87 610 617 342 346

[zA ~zA ~zA ~A

SHOT EFFECT OF SECONDARY EMMISSION. I 9

pA 700

600

500

400

+ Tg-rp o~g-ip

/

i / 7/

-100 ~-

~"~'~ ~. ~ ,~ r. _ _ ~ , p ~

-2oo

/ "-------" -/P

Tg+Tp= lOO H A

~vg

Vp = 20~ P V

Fig. 2. Expe r imen t a l check of equa t ion (! 1).

I 0 M. ZIEGLER

The shapes of the 7~- and Ig curves can easily be explained by the fact that both plate and grid have an important secondary emission.

The shapes of the ¢p and i 8 curves, however, are very remarkable: if the fluctuations were due to pure shot effect of primaries the i curves would be the same as the I curves ;__actually, however, the difference is enormous (at the points where I ---- 0 e.g., the ,,noise" is not at all zero). On the basis of the above theory the whole shape may be understood quite well. The great values of ig and i~ are due to ,,multiple emissions" of secondaries, ig attains a much higher

ip; this was to be expected as Ig p,~m + ~ n 2 ~, Ip p~m v a l u e than is O 0

larger than ~ (n - I) 2 ~. Ip p,,~" (see § 3). The peculiar minimum o f

0 ip for Vg = 202 Volt is not surprising if we bear in mind that for this voltage a great fraction of the secondaries falls back on their emitter so that the influence of multiple emissions is considerably lowered. The fluctuations ig increase rapidly for V 8 > 200 Volt, as with increasing grid potential more and more secondaries from the plate are collected.

For each value of V v i 8 - i p is practically equal to I c - I ~ . The agreement between theory and experiment is very satisfactory and is an indication of the validity of the hypothesis on which our calculation is based.

Received November l l, 1935. Eindhoven, 9 September 1935.

1) Compare the calculation of P e n n i n g and K r u i t h o f , Physica 2, 793, 1935, which leads to the same result.

2) M. Z i e g l e r , Physica 2, 415, 1935. Note added in the prool. Recently two papers about shot effect of secondary emission

have been published: L u c y J. H a y n e r, Physics 6, 323, 1935, and W. A. A l d o u s and N. R. C a m p b e 11, Proc. roy. Soc. 151,694, 1935.

In both papers the fluctuations of secondary current are calculated by means of a formula previously indicated by N. R. C a m p b e 11. The form in which the formula has been obtained seems less intelligible than the form given in the present paper. I t may be remarked that it follows directly from the well known shifting theorem with regard to moments of second degree (see e.g. v o n M i s e s, Wahrseheinlichkeitsrechnung, Leipzig

CX~

193 I, p. 35) applied to our equ. 6): is~IZprim = ~0 n ' ~n = ~0 (n - - ~)' ~n +~" = ~T"F ~'.

SHOT EFFECT OF SECONDARY EMISSION. I | |

The principal conclusions of the first paper cited above, which is based on experiments carried out already in 1928 confirm our results. In discussing her results, the author however assumes that, ,,as a result of the large random factors involved in secondary emission,

OO

the mean square deviation ~]lz from wz (in our notation wz = ~ n ~n = ~--~1) should be 2

equal to wz itself" ; it can be shown that this assumption is incompatible with the author 's own experimental results. The proof hereof will be given in the second part of this publication.

In a note of the second paper mentioned, the authors suggest our equ. (11) to be ,,true c o

only when a = I" (in our notation, when ~0 ( l - - n ) ~n and hence 60= 1). This suggestion

must be based upon a misunderstanding, as the general validity of (11) can easily be proved (see section 3 of this paper).