.' _" .~"

.:}'

R7

'_,;EXTENSION OF LANGMUIR'S ($, 1]) TABLES FOR A'

PLANE DIODE WITH,A MAXWELLIAN. DISTRIBUTION OF' THE ELECTRONS

('SuminaryLa n gmu ir t s (E, 1]) tables, being too .rough' for many practicalapplications, have been extended by graphical interpolation andby calculation . (table I and 1I, fig. 2). The depth of the potential,minimum and the distance for a given temporature are calculated(fig. 3 and 4) ; with the aid of 'a table' given in the text the lattermay be calculated for any temperature.' For the convenience ofthe user a 'few numerical examples have been worked out.

by P. H. J. A. 'KLE'~NENI •

'S3!.S43(083.S)

List of symbols"Unless otherwisecatated all the quantities are supposed be measured in

unrationalized m.k.s. units. The final choice of the units has no influence :,upon the dimensionless quantities $ and 17.; In formulae such as (4), quan-titi,es having the dimensions of a length such as x, dm and Imay be expressedin m or in cm provided the current density la is accordingly measured inAfm2, or' Afcm2, respectively. _,,' . ',' '. _ ' ,-e ':elèctron charge; e '. 1.602 X 10-19 'abs' coulk Boltzmann's constant; k .=~1.3805 X 10-23 abs jouler Kla anode current density in Ajm2 (or Ajcm2) , • '

Is saturation current density' (idem) ,T absolute température of the cathode

. d distance anode-cathode ID m (or cm)a", 'distance potential minimum to cathode (idcm)Va potential difference anode-cathode iri V,

,-v'n value of the potential minimumV potentlal. at a distance x from cathodeV T defined by eV T "'7 kT or, V T =' Tj1l605 abs volt

1. I ntroductioliAs i~'well known"}, the (la,' Va) 'cha~actetistic of a diode always ~on.sists,

.of three parts:., ' .

1) Saturated region

T,dV " "-d' > Q; Ia = Is ,x .

,2) Space-charge-limued region .'av '• ':'-d < 0, 0 < x -< dm ;.' X

dY " .:cJx>O, dm~x<d;

'.a potential minimum V = ~ Vm exists for x -: dm ;. _... , ,eVm ':,Ia = Is e- kT, (1)

, ,. 'I

.) _ A Iull exposition of. thc- tb~ory may be found in a paper by -A. van der Ziel, «Extension and applfenflnnof Lan~ri:Î.uirts eulculatloua. on a plane diode with maxwellten velocity dlatriburlon. of tbc eleétrons'", Philips.Res, Rep. 1, 97, 1946.' ", . ,

":-:. (g=;+ between anode and potential 'minimum, ,-'-g - g- between,potentlal minimum and cathode, so that g- and ~+ are always positive),La n gm u i.r solved Poisson's differentlal equations and compiled (g,1})tables. ',' . - ", .

For carryingout practical calculations, Langmuir's tables proved to befar too succinct; hence we have extended them by calculation and bygraphical interpolation, For small values of g and 'YJ the following

. formulae were used: " , .

., ..

/

82 ' , P. H. J. A. KLEYNEN

'l~I

I-'d§

I12 r--

I

: . ' /i 10 2I

,', I, I\

' ,/i .

I1,5 ~,

I ,, . V,6

/, ,.1,0

\ 1

/\

1-\ V

/~ ,

\

\\q5 V I

/ ..\ '!'-

LV.......... ..-

- -, V-.;. --1--- -- --r-- --r-,

4 o 2 4 6

Fie:. 1. The function '1](-). representing the potential distribution in the space-oharge-limited region, and its second derivative (''')7 rèpresentiug the spaoe-charge in that region.

.3) Anode-voltage-limited. region or exponentiel r~gion

dV ',+eVa-Z-<O,.V,.<O,Ia=I.e kT.lX (2)

. ,

In order to ca1culate the Ia,;'Va ~haractetistic of a diode it is sufficient toconsider the space-charge limited region only.

, After introducing the two di~ensionless quantiti,eè '

'fj = e Cv+ 1(,,,) = V + v,,, ' p! +- V~)' Ü605 ,kT VT ~ T. .x - dm .' ~ - t ,~,'.' .,..g = l' = 9.186 X 10 T, L ~x - dm) ,

(3)

(4)'

,.;----, -

EXTENSION OF LANGMUIR'S (~. 7}) TABLES 83

r= 2'YJ'/.-0.376126'YJ....:..,..0.0251956'YJ·,.~0~OOÖ691 Ó31j2+ ,,'+ 0.0~099409'YJ·J,+: .. · (Sa)

'YJ= 1/4 (~T)2_0.047 016 (~+)3+ 0.012627 (~+)4-0.003 820 (~+)5+:'. (Sb): ,>, ,I'''' T •

,obtained by developing and integrating La ngm u ir ïs formula:. .'I d

~~= [[e]'-1 ± e.u erf (;afl) =f 2 (ujnf/.]'ï; '. (6)

where. . 1'- '.2 u •

erf (u'/.) = 1/-,( e-=' dz.n'o

, I

The higher values, obtained by' graphical interpolation may show aninaccuracy of une 'or two units in ,the' fourth figure. The' results .are

· given in tables I and II, table. I gives ~-. as a function of 1], table TIgives 'fi as a function ofc;+. Eig.L, in which 'YJhas been plotted as/a functionof s. gives a impression of the general trend of the functions.The dotted line in fig.' 1 represents the function:

d2r, I,} ',,-' ld~l , '2 e7} ,11± erf (y'YJ) ç (7)

obtained from (10) by differentiation. According to (3) and (4) we· have: ",

/

d2'fi' . d2Vdé = const "-dX2 '

which cxpression is proportional to the space charge: As is' evident'from fig. 1 the space charge is practically concentratëd near- the cathodeand decreases rapidly with increasing distance from the éathode., ,Denoting the value of17and =f ~ at the cathode and at the anode by (IJe, e-;)· and (1]0' ~:) respectively, we have according to (1),'" ". '

andVa+ Vm Va'

'YJa= -V~= VT+'YJr, (9)

whereas

1:- =~'!!.": l'

(10).'J!_ ,4 tdm - l~; = ~-;T /9.186 X 105 Ia, (lOa)

and, \

I:a+ d - dm d ' _, . ",l!. t ' -'" ==" 1 = i - ~e = 9.186 105 ~ T. 4 Ia d ~ ~c • (11)

2. The calculation of the' characteristic""' . . ",,' ,',' - I',' '. T -'. ' , ••••

The calculation of the (IQ, Vu) characteristic riow proceeds in the follo-wing way, which is illustrated in fig. 2. .. . . . "

r=

84 P. H. J, A. KLÈYNEN

.Assuminga certain valûe of Ja, '1]e follows'from' (5), ~-;from Langmuir's+ .'tables, ~a from eq. (8), 1]a from Langmuir's tables and Va from eq. (6a).

For small- values of la and small distances d the value for ~: calculatedwith the aid of (8) may benegative. This means that no potential minimumexists between cathode and anode, so that the value of la belongs to the'exponential region of the charactêristic.

r , ' _ • 4SBH ..

Fig. 2. [Calculation'[of the anode voltage in. the space-chargeglimited rçgion::when the.. a_nodecurrent is given:, . " ,

!l.3. Examples' , .

Same conditions but ar{odecurrent 1.5 mA ..la = 1 mA/C!ll2; according to eq. (5) 1]e = In 100/1 ..:..4.605 -t:= 2.411 and.dfl = .9.734, hence,t: < I; The' a~ode voltage 'Playthen ~e calculated with,the help .of eq. (2) and (5). " .

kT . I. -T -1160 .. v..=-:-'--;-InT:= - VT 1]e = 11605 'Y}~= 11605 X 4.605= .,-~.46volt.. ,

4. Position and value of the potentiel minimum .:

In order'to obtain some idea about thé manner in which the fieldbetweenth,~cathode and the anode is influenced bythe initial velocity distribution 'of'the electrons we calculate the depth Vm of the potential minimum andits distance dm from the cathode. T~e depth Vno is given byeq, (Sa); in,

.....EXTENSION OF LANGMUIR'S (i, 11) TABLES 85

Oe oK r!t700 973 174.2800 1073 187.5

........ 900 ·1173 200.4.·1000 1273 213.12400 .2673 371.8

i'" Ts=fOA/cm'j'). ,.Is!!,f

r-

15=0,1

3,0

0,

.~ ._ ..~- .i-'- -

~._-,

-L!2OOI (- 1-- <,

" --

r-, - .I"--.. ...... 100 "I"-..... ...... __tlX.

19.r-; ....... D5 7V1

r--:.... <,~~ I()°C<; R ...... C&. I{j_~c 11"'-

r-.......? , <, -,- z-- ::--.' ":::----:, K5 r- ss ::::-~ ~:s- . 20C t

-5 -4 '-3 -2

2,0

t,5

1,0

0.1

_

.0.01

_

,OO1!:,-.l-L-l-I-L..l.':!!:;--.L......L..J....U.l.Uf---'--"-:L..1..1.~---;.--:--'';;L-'-J~0.01 0,1 1 fa I. (mA) 100Fig.4. Distánce 'between the potential mini~~and the cathode as a function of the anode current.. at a cathode temperature of 700 .cC, for differentvalues of the saturation current. '

fig: 3 Vm is plotted againstlul I. 'for different valuesof the cathode tempe-rature. The distance dm isgiven by eq. (lOa); infig. 4dm is plotted against the. anode current density la ' .for different values of thesaturation current densityI. at a càthode : tempe- ,ratnre of 700° C (973° K).As dm is proportional to

n' .1'4 similar curves can becalculated for differentvalues of the' cathodetemperature T,' with the"aid of the table on -theright side of fig. 4.',From these figures itmay .

be seenthat Vm may takeon values comparable withthe anode voltage Va, anddm values ,comparable withthe distance d between theanode and the cathode .

Fig. 3. Value of the potentialminimum as a function of theanode current at differentcathode temperatures ..

Eindhooen, May 1945 .

"..'

86 P.H.I.A.KLEYNRN

TABLE Ie- e-. . e""" ~-" ,rJ rJ .. rJ rJ

0·0040.

,0 ' , Ó 0·1250 0'Q080' 0·1759 0·0120 0·21450·0001' 0·0200 0·0041 0·1265 0·0081 0·1769 0·0121 0·21540·0002 0·0282 0·0042 0·1280 0·0082 0·1780 0·0122 0·21630·0003 0·0345 0·0043 0·1295 0·0083 0·1791 0·0123 0·21720·0004 0·0398 0·0044 0·1310 0·0084. 0·1801 0·0124 0·2180

0;0005" -. 0·0445 0·0045 ' 0·1325 0·0085. 0'.1812 0·0125 0·21890·0006 0·0488 0:0046 0·1339 0·0086 0·1822 0·0126 0·21980·0007 0·0527 0·0047 0·1354 0·0087 0·1833 0·0127 0·22060·0008 0·0563 0·0048 0·1368 0·6088 0·1843 0·0128 0·2215 .Q·0009 ,0-0597 ' 0·0049 0·1382 0·0089 0,1853, ,0,0129 0·2223·

-, 0·0010 0·0629 0:0050 0·1395 0·0090 0·1863 ' 0·0130 0;2231

,0,0011 0·0659 0·0051 0·1409 0.0091 0·1874 ' 0·01~1 0·2240·0·0012 0·0688 0.0052 0·1423 ' 0·0092 ' 0·1884 ' 0·0132 ' 0·22480·0013 0·0716 ' ,0,0053 0·1436 0·0093 0·1894 0·0133 0·22560·0014 0·0743 0·0054 0·1449 0·0094 0·1904 0·0134 0·2265 '

, .'\ 0:0015 0·0769 .0·0055 0·1462 ,0-0095 0·1914 0·0135 0·2273 ;:, 0·0016 0·0794. 0·0056 0·1476 ' 0·0096 0·1923 0·0136 0·~281 ,;0·0017 0·0818 .0·0057 ' 0·1489· ~.0.0097 0·1933 0,0137 0·2289 '"; 0·0018 0-0842 0·0058 0·1501 ,'0·0098 0·1943 0·0138 0·2298,";0·0019 0·0865 0·0059 0·1514 0·0099 0·1953 0·014 Ó·2313 .'

,'0·0020 0·0887 0·0060 0·1527 ,0,0100 0·1962 . ' 0·015 0,·2393'0·0021 ' 0·0909 0·0061 . 0·1539 0·0101 0·1972 . 0·016 0·2470 ~{0·0022 .0·0930 0·0062 0·1551 ' 0:0102 0·1981 ' 0·017 0·2544'0·0023 0·0951 0·0063 0·1564 0·0103 0·1991 O'01S 0·26160·0024 0·0971 0·0064 0·1576 0·0104 0·2000 0·019 0·2685

,;

0·0025 0·0991 ·0·0065 0·1588 0·0105 0·2010 0·020 0·27530·0026 0·1010 0·0066 0·1600 0·0106 0·2019 '0·021 0·28190·0027 0·1029 0·0067 ,0,1612 0·0107 0·2029 0·022 0·2883,0'0028 0·1048 0·0068 0·1624' 0·0108 0·2038 ,0,023 0·29460'0029 0·1066 0·0069 0·1635 0·0109 0·2047 0·024 0·3007

0·0030 0·1084 0·0070 0·1647 0·0110 0·2056 0·025 0·3067 '0·0031 0·1102 0·0071 0·1658 0·0111 0·2065 0·026 0·31260,0032, 0·1l19 ·0·0072 0·1670 0·0112 0·2074 0·027 0,318,40·0033 0·1136 ' 0·0073· 0·1681 0·0113 '0·2083 0·028 0·32410·0034 0·1153 0·0074 ' 0·1693 0·0114 0·2092 0·029 0·3296

0.0035 0·1l70· 0·0075 0·1704 0·0115 0·2101 0·030 ' 0·33500·0036' 0:1186 0·0076 0·1715 0·0116 0·2110 0·031 ·0·3403

. 0·0037 0·1203 0·0077 0,1726, 0·0117 0·2119 '0·032 0·3455,0,0038 0·1219 0:0078 0·1737 O·Oll8 0·2128 0·033 0,3506,'0·0039 0·1234 0·0079 0·1748 0'OiI9 0·2137 0·034 0·3557 .

I -EXTENSION.OF LANÇMUIR'S (~, '1) TABLE!': 87

'YJ ~- 'YJ ~- 'YJ ~--:- 'YJ ~-0·035 0·3608 0·075 0·5190 ,0,25 0·903 0·65 1·355;0·03li ' 0·3658 0·076 0·5222 0·26 0,919 0·66 1·3630·037 0·3707 . 0·077 0·5254 0·27 ' 0·934 ; 0·67 ...,.1·3120·038 ' 0·3755 0·078 0:5286 0·28 0,949 0·68 1·3800·039· 0·3802 0·079 0·5318 0·29 0·964 0·69 -1·388

.'"0·040 . 0·3848 0·080· 0·5350 0·30 0·979 0·70 1~3960·041 0·3894 0·081 0·5381 P'31 0'993 0·71 1-404,0,042 0·3939 0·082 0·5412 0·32 1·007 0·72 1-412 ...-0·043 0·3984 0·083 ,0,5443 0·33 1·020. 0·73 1·4200·044' 0·4028 ' 0·084 0·5474 0·34 1·033 0·74 1-421

0·045 0·4071 . 0·085 0·550~ '0.35 1·046 0·75 1·4340·046 0·4114 0·086 0·5535 . 0·36 -1·059 0'76 ]·442 .0:047 . 0-4156 0·087 0·5565 0·37 ]·072 0·77 1·4490·048 0·4198 0·088 ' 0·5595 0·38 1·084 0·78 1·4560·049 0-4240 0·089 0·5625 0·39 1·096 0·79 ' ' 1·463

0·050 0-4281 0·090 0·5655 0·4.0 1·]08 0·80 1-4100·051 0·4322 0·091 0·5684 0·41 i-iao ' 0·81 1·4770·052 0'43q2 . 0·092 0·5713 0-42 1·132 0·82 1·4840·053 0·4402 0·093 0·5742 0-43 1·143 '0·83 1-4910·054 0·4441 0·094 0·5771 0·44 1:154 , 0·84 1·498

0-055 0·4480 0·095 0·5800 0·45 " 1·165 0·85 1·5050,0$6 0·4519 0·096 0·5829 0·46' . 1·176 9·86 1·5120·057 -, 0·4557 0·097 0·5857 0-47 1·187 . , 0·87 1·518

, 0·058 0·4595 0·098 ' 0·5885 0-48 l·.l97 0·88 . 1·5250059 0·4633 <)·099 0·5913 0·49 1·207 Q·89 1·532

0·060 0·4670 0·100 0·5941 0·50 1·217 0·90 1·5380·061 0·4707 0·11 0·621 0·51 1·227 0·91 1·5440·062 0·4743 0·12 0·646 0·52 1·237 0·92 1·5510·063 0·4779 0·13 0·672 0:53 1·247 0·93 1·5570·064 0·4815. 0,14, 0·695 0·54 1·256 0·94 1·563

0·065 0·4850 ·0·15 0·717 0·55 ~·266 , 0·95 1·5690·066 0·488:; 0·16 .0·738 0·56 1·275 . 0·96 . 1·5750·067 0·4920 0:17 0·759 0·57 1·284 0·97 1·5820·068 0·4955 . .0;18 0·779 ' 0·58· 1·294 0·98 1·588

c.-

, 0·069 0-4990 0·19 0·798 0·59 1·303 0·99 1·594

0·070 0·5024 0·20 0·1n7 0·60 1·312 r . 1·00 . 1·6000,071 0·5058 0·21 . 0·835 . 0·61 '1·321 1·01 1·6050·072 0·5091 ·0·22 0·853 0·62 ' 1·330 1·02 1·6110·073 0·5124 .0·23 0·870 0·63 1·338 I·O~ 1·611.0·074 0·5151 0·24 0·887 0:64 ,1,347 1·04 1·623

.J

"._ r.,_ _.... . '

·88 1'. H. J.\.. l(LEYN!'N •,

.~

'IJ ~- 'IJ ~- 'IJ ~- .1'J ~--

1-05 1-628 1·45 1-822. 2·10 2-042 ' 2'90 -, 2-217,'~1·06 '1·634 1-46 1-826 2-12 2·047 ,2·92 ~·220 :,1·07' :- 1-640 1-47 . i-830 ,2-14 2-053 2-94 2·224i-08 1-645 1-48 . 1-834 _ 2-16' 2·058 . 2-96 . 2·2271-09 - 1·651 1·49 -, 1·838 2-18 2-063 2-98 ~-.2-230

, \"

1-10 _ 1-656 , I-50 ' , 1·842 2-20 2-068 3-00 2-234i-n 1-661 I-51 1-846 2-22 - 2-073, 3·02 2·2371·12 1·667 1·52 1·850 2·24 2;078 3·04 2·240 _1·13 1-672 . 1·53 1·854 - 2·26 2·083 3·06 - 2·2431·14 1·677 1·54 1·858 2·28 . 2·088 3·08 2·246

1·15 1·682 H,5 1·862 ' 2·30 2·093 3·10 2·249 ''1·16 1·688 1·56 1·866 2·32 2·098 3·12 2·2531·17 1·693 ]·57 1·870 2·34 2·103 3·14 2·2561-18 1-698 I-58 1-874 2-36 2-107 3-16 ' 2-2591-19 1-703 I-59 1-877 2-38 a-na 3·18 2-262

,1-20 ' 1-708 1-60 1-881 2·40 2·ll7 3·20 2-265' '1-21 1-713 1-62 1-888 ,2,42 2'121 3-22 2·2681-22 1-718 1-64 1-896, 2-44 2·126 3·24 2-2711-23 1-723 1-66 - 1-903 2-46 2-130 3-26 2·2741-24 1-728 1-68 1·910 2-48 ' . 2-135 3·28 2-277

1-25 1-732 1-70 ' 1-917 2-50 2-139 3·30 2-2791-26 1-737_ 1-72 1·924 2-52 2:143 3·32 ,2,2821-27 1-742 1·74 1·931 2-54 ' 2·148 3;34 /2-2851-28 ·1-747 1-76 ' 1-938 2·56 '2·152 3-36 2·2881-29· 1·752 1-78 1-945 2-58 2·156 3-38 2·290. '

1-30 1-756 1-80 1·952 2·60 ' 2-160 3·40 2-293, , 1-31 1-761 1·82 1·958 2-62 2·164 3-42 2·296

1-32 1-765 1·84 1-965 2-64 2-168 3·44 ,2,·2981-33 1-770 1-86 1-971 2·66 2-172 3-4~ 2·3011-34 .1-775 1-88 1-977 2·68 2'J76 3-48 2·303

,"

' 1-35 1-779 ,1-90 1·983 2·70 ' 2-180 3·50 2-306, 1-36 1-784 1·92 1-989 2-72 2-184 ,3·52 2·3081-37 1·788 1-94 1·995 2·74 2·188 3-54 2·3ll

-.; '1-38 1-792 1·96 2-001 2·76 2·19~ .' 3-56 2·3131·39 1-797 1·98 2-007 2·78 2-195 ,3·58' 2·316

, ~,

1-40 1·801 2·00 , 2·013 2-80' 2·199 3-60 ,2,318, 1-41 1·805 2-02', 2-019 2·82 2-203 3-62 ' 2·3211-42 1-809 2-04 ,2-025 2·84 , 2·206 3·64 2-323

, 1-43 1·814 2·06 '2·031 2·86 2·210 3·66' , 2-3251-44 1-818 2·08 " 2;03q ,2,88 2·213 3·68 2-328

v , - 7";0' -~ -_.---. _- " _--- _---.~- ' .. ' '_-,

. EXTENSION OF LANGMUIR'S (E. >I) TABLES 89

'11 ~- . 'YJ ~'-'. 'YJ ~- 'YJ . ~-:-3·69 2·329 6·Ö ' ,2'483 8·5 " 2·534 11 2,548/, .

6·1 2·487 ·8·6'- ' .2·535. 12 2·5503·7 2·330 6:2 2·490 8·7 2;536 13 2·5523·8 ,2,341 6·3 2·493 8·8 2·537 14 2·5533·9 . 2·352 6·4 .2·496 ,8,9 2:'538 15 2·553 '

'.,- .,4·0 2·362 6·5 2·499 9·0 . ,2,538 ' 16 ' 2·5534·1 2'37,1 6·6 2·502 9·1 2·539 17 2·554

,4·~ ! 2·380 6·7 2·504 9·2 2·540 18 2·5544·3 2·388 6·8 2·507 9·3 ,2·540 19 2·554 »<

4·4 2·396 6·9' 2·509 9·4 ' 2',541 20 2·554<

.: 4·5 2·404 7·0 " 2·511 9·5 2,542, ' 25 . 2·5544·6 " " 2·411 7·1 2'513 9·6 2·542 1000 2·554

" 4·7 2·418 7·2 ,2,515 9·7 2·5434·8 2·425 7·3 ' - 2·517 9·8 2·543 .4·9 '", 2,432 7·4 ' , 2·519 9,9, 2'544 ,-,

5·0 ' 2·438 7·5 2·521 ~O·O 2:5445·1 ; 2·443 7·6 2·522 10·1' 2·5455·2 ',.

2·449 ,7,7 2·524 10·2 ,2,545 - ,5·3 2·454 7·8 2·525 10·3 2·5465·4 ,2,459 7·9 2·527 10·4 2·546

5·5 2·463 8·0 2·528 10.~5 2·5465·6 ,2,468 8·1 2·529 10·6' 2·5475·7 2·472 8·2 ' 2~531 10·7 2·5475',8 2-476 8·3 2·532 10·8 2·5475·9 2·480 8·4

. 2·533 10·9 2·548

,-

TABLE II

~+ 'YJ ~+ 'YJ ~+ 'YJ ~+ 17

0 0·000"

0·10 0·00245 0·20 0·00964 0·250 0·014940·01' 0'000025 '0·11 0·00296 0·205 0·01013 0·255 0,0155,2 ' , ,

0·02 0·00010 0·12 0·00352 0·210 0·01063 0·260 0·016110·03 0·00023 0:13 . ,0·00412 0·215 0·01113 0·265 ' 0·016n0·04 0·00040 0·14 0·00477 0·220 0·01164 0'270 0·01732

0·05 0·00062 0-15 0·00547 0·225 0·01216 0·275 0·01794', •0'06 0·00089 0:16 0·00621 0·230 0·01269 0·280 0·018580·07 0·00121 0·17 0·00700 0·235 0·01323 - 0·285 0·01924 ' ,0·08 0·00157 0·18 ,0,00783 0·240 0·01379 _ ' 0:290 0·019920·09 ; 0·00199 0·19 0·00871 0·245 0·01436 0·295 0·02061

.' ..-90 P. H. .T. A. KLE:YNEN

~+ 'YJ ~+ 'YJ ~+ '1 ·e+ 'YJ

.. 0·300 0·02132 0·380 0·033.75 0·460 0·04881 0·585 0·07740·302 0·02159 0·382 0:03409 0~462 0·04922 0:590 0·0787,0,304 0·02186 0·384 0·03444 0·464 0·04963 .,0,595 0·07990·306 0·02214 0·3R6 0·03479 0·466 0·05004 0·600 0'0813,0,308 0·02242 0·388 0·03514 0·468 ,O·~5045 0·605 0·0826

:o-sio 0·02271 0·390 0·03549 0·470 0·05087 0·610 0·08390·312. 0·02300 0·392 0·03584 0·472 0·05129 0·615 0·08510·314 0·02329 0·394 0·03620 0·474 0·05171 0·620 0·08650·316 0·02358 0·396 0·03656 0·476 0·05213 0·625 0·0878 '0·318 0·02388 0·398 0·03692 0·478 0·05255 0·630 0·0891

0·320 0·02418 0·400 0·03728 0·480 0·05298 0·635 0·09050·322 0·02447 0·402 0·03764 0·482 0·05340 0·640 0·09180·324 0·02477 0-404 0·03800 0·484 0·05383 0·645 0·09320·326 0·02507 0·406 0·03836 0·486 0·05426 0·650 0~09460·328 0·02537 0·408 0'03873 0·488 0;05469 0·655 0·0960'. .

.... 0·330 0·02567 0·410 0·03910 0·490 0·05513 0·660 0·09740·332 0·02597 0·4]2 0·03947 . 0·492 0·05556 0·665 0·0988

. '0·334 0·02627 0·414 0·03984 0·494 0·05599 0·670 0,1002,0·336 0·02658 0·416 0·04021 0·496 0·05642 0·675 0·10160·338 0·02689 0·418 0·04058 0·498 0·05686 0·680 0·1030

0·340 0·02720 0·420 0·04096 0·500 0·05730· 0·685 0·10440·342 0·02751 0·422 0·04134 0·502 0·05774 0·690 0'~059'0·344 !)o02782 0·424 0·04172 0·504 0·05818 0·695 0·10740·346 0·02813 0-426 0·04210 0·506 0·05863 0·700 0·10890·348 0·02845 0·428 0·04248 0·508 0·05908 0·705 0·1104

0·350 0·02877 0·430 0·04286 0·510. 0·05953 0·710 0·11190·352 0·02909 0·432' . 0·04325 0·515 0'0606' 0·715 0·1134 .0·354 0·02941 0-434 . 0·04'364 0·520 0·0617 0·720 0·1149

.0·356 0·02974 0-436 ' 0·04403 ,0·525 0·0628 0·725 0·11640·358 0·03007 '0·438 0·04442 0·530 0·0640 0·730 0·1179

.. ~" 0·360 0·03040 0·440 0·04481 0·535 0·0652 0·735 0,1194,

0·362 0·03073 0·442 0·04520 0·540 0·0 65 0·740 0·1209.. 0·364 0·03106 0·444 0·04560 0·545 0·0677 0·745 0'1224:. ·0·366 0·03139 0·446 0·04600 0'5,50 0·0689 0·750 0·1239

0·368 0·03173 0·448 0·04640 0·555 0·0701 0·755 0·1255

·0·370 0·03206 . 0·450. 0·04680 0·560 0·0713 .. 0·760 0·1271 .0·372 . 0·03239 0·452 0·04720 0·565 0·0725.: 0·765 0·1287 ,0,374, 0·03273 0·454 0·04760 0·570 0·0737· ·0·770 0·1303 .0·376 0·03307 0·456 . 0·04800 0·575 . 0·0749 0'775 0·1319

·0·378 0·03341 0·458 0·04840 0·5jJO 0,0761, 0·780 0·1335~..

, '

92 P. H. J. A. KLEYNEN

" s+ 'YJ I ~+-; 'YJ ~+ 'YJ ~+ 'YJ. , ',2·95 ~.-1·4822·15 '0·8538 2,55 1·152· 3·35 1·843

2·16 0·8608 2·56 1·159 2'96 ' 1·491 3·36 1·8522·17 0·8678 2·57 1·167 2'97 1·499 . 3·37 1·8622·18' 0·8749 2·58 1·175 2·98 i·508 3·38 1·87i2·19 0·8820 2,59, 1·183 " 2·99 1·517 3·39 1·881

...,2·20 ' 0·8891 2·60 1·191 3·00 1·526

r •3·40 1·890

2·21 0·8962 2·61' '1·199 3·01 . 1·535 3·41 1·899··2·22 0·9033 2·62 1·207 ' 3·02 1'544 3·42 1'9092·23 0'9104 2·63 1·216 3·03 ~ 1·552 , 3·43 1'918'. 2,24, 0'9176 2·64 1·224 3·04 1·561 3·44 1·928

2·25 0'9248 ,2·65 1·232 3·05 1·570 3·45 1·9382·26 ' 0'932Q ,2·66' 1·240 3·06 1·579 3·46 l'9472·27 0·9393 2·67 1·248 3·07 1·588 . 3·47 1·957

, 2·28 0'9466 '. 2·68, 1·256 3·08 1·596 3·48 1:9662·29 0·9539 2·69 1·264 3·09 1·605 . ,3,49 1'976

,2,30 0'9612 2·70 1·272 3'lO 1·614 3·50 1·986, , 2·31 ' 0'9686_ 2·71 1·281 3·11 1·623 3·51 1·995,

2·32· 0'9760 2·72 1·289 3,12, 1·632 3·52 2·005.',2,33 0·9834 2·73 1·297 - 3·13 - 1·641 , 3,53, 2·014

2'34: ,0,9909 2'74 1·305 - 3·1~ 1·650 3·54 2',024

,2,35· 0·9984 2·75 1·313 3·15 1·659 . 3·55 . 2·0342·36 1·0056 2·76 1·322 3·16 1·668. 3·56 2·0442·37 . 1·0131 2'77 1·330 3·17 1·677 3·57 2·0532·38 1'021 4'78 1·338 3·18 1·686 3·58 ' 2·063

, 2·39 1·028 2"79 1·346 3·19 1·6!)5 3·59 2,073

2·40 1·036: ' 2·80 1·354 3·20 1·704 3·60 2·0832·41 1·043 2·81 1·363 3·21 1·714 3·61 2·092~,. 2·42 1·051 2·82 .1·371 3·22 1·723 3·62 2·1022·43 1·058 2,83,. 1·379 3·23 '1·732 3·63 2:112 "2·44 1·066 2·84 1·388 3·24 1·741 3·64 2·122

2·45 . 1·074 2·85 ' 1·397 '3;25.; '1·750 : 3·65 . 2;1322·46 ' 1·082 2·86 1·405 3·26 1'759 3·66 2·142

.-2·47 ' 1·089 ·2·87 - 1·414 3·27 - 1·768' -3·67 2·1522·48,' 1·097 ' 2·88 1·422 3·28 1·778 -3,68 2·1622·49 ,1,104 2·89 1·431 3·29 1·787 ' 3·69 2·172

,2·50 1·112 2·90 1·439 3·30 1·796 3·70'. 2·182

• '2'51 1·120 2·91 1·448 3·31 . 1·806 3·71 2·1922·52, . 1·128 -2·92 ' 1-456 3·32 1·815 3·72 2·2012:53 '1.136 '2·93 1·465 3·33 1·825 3·73 2·1112·54 1:144 2·94 1-473 3·34 1·834 _ 3·74-- 2·221

".

, -, ,

"

,, "

EXTENSION OF LANGMUIR'S (;, 1/) TABLES 93

E+ '1] E+ 1] E+ 1] ~+ ,1]'

3·75 2·231 4·30 -:.2·807 5·10 3·723 6'75 5,871,, 3·76 2·242 4·32 ~'829 5·12 3,7,,1,7 6·80 5·942 ',3,77 2·252 4,,34 2·851 5·14 3·771 6·85 6·0123'·78 - 2·262 4·36 2·873 5·16 3·796 6·90 ' 6·0823,,79 2·272· 4·38 ' 2·895 5·18 3·820 6'95 6·153-3·80 2·282 4·40 2·917 5-20 ' 3·844 7·00 ,6,224

3·81 2·292 4·42 2·939 5·22 3·868 7·05 6·2953·82 ,2,303 4·44 2'961 5·24 3·893 7·10 ' 6,367,3·83 2·313 4·46 2:983 5·26 ' 3·917 7·15 6·4393·84 2·323 4,48" 3·006 , 5·28 3·941 q·20 6·511

3·85 2·333 4·50 3·028 5·30 3·966 7·25 6·582,3,86 2,3·13 4·52 3·051 7'3Q 6·6543,87' 2·353 4·54 3,073 5·35 4·029. 7·35 6·727 '3·88 2·363 ' 4·56. '3'096 5-40 4·092 7·40 , 6·8003·89 2·374 ' 4·58 3·118 5-45 4·l!i4 7·45 6·874

3,90 2·384 4·60 3,141' .5-50 4·217 7·50 6:9483,91, 2·394 4·62 3·164 5·55 4·279 7·55 7·0233·92 2,4,04 4·64 3·187 L 5·60 4·342 7·60 7·097 '3·93 2·415 4·66 '3·209 5·65 4·405 7·65 7·171 .,'

3·94 2·425 4·68 3·232 ,5'70 4-467 7,70 7·245

3·95 2,4,36 4·70 3·255 5·75 4·531 ' 7·75 7·320,3,96, 2,446, .. 4,72, ';3·278 _", . 5·80 ~'4;595 ':',-'#' '7·80 7·3953·97 2·456 4:74 3·301 5·85 ' 4·660 7·85 7-47.13·98 2·467 4·76 3·324 5·90 4·725 7:90 7·5463·99 2·477 ,4,78 3·347 5,95, 4·790 7·95 ' 7·622

4·00 2·488 4·80 3·370 6·00 4·856 8·00 , 7,698,4·02 2·509 4,,82 3'393 ,6,05 '4,921 8·05 7·7"754·04 ' 2·530 4·84 3·416 6·10 4·987 8·10 ' 7·8524·06 2·551 4·86. 3'439 6·15 5·053 8·15 7,9294·08 2·572 ' 4·88 3·462 6·20 , 5·119 8·20 8·006

4·10 2·592 ' 4.90 3·486 6·25 5'186' 8·25 8·083 '4·12 ·2·614 4·92 3·509 , ·6·30 ' 5·253 8·30 8·1504·]4 2·635 ' ' 4"94'- 3·533 " :6·35 ,: 5·321 8·35 8·237I , 4·16 2·657 ~:96 ' 3·556 <,

6·40 5·389 8·40 8·315 "4,18" 2·678 4·98 3·580 ,6,45. 5·457 8·45 .8·393 ,

4·20 2·700 5·00, 3·604 . 6·50 5·526 8·50 : 8·471~·22 2·722 5·02 3·628 . 6·55 . 5·595 8·55 8·550 .4·24 2·743 5·04 '3,652 6·60 5·663 ,8·60. 8·6294·26 2·764 5·06 ' 3·676 6·65 5·732 8:65 8·708 .4·28 2·785 5·08 ' '3·699 6·70 ,5,810 ,8·7()· 8·787 ,. .'

~ 94 ." P. H .. r. A: KLEYI'1EN ., -:'

e+ . 1] e+ . '. 1] e+ iJ' e+ iJ

8·75 8·867 11·5 13·57 15·5 ·21·38 .19·5 30·178·80 8·948 11·6 . 13·75 15·6 . 21·59 19·6 . 30·40 . ')'8·85 . 9·028 11·7 13;93 15·7 21·80 19,7, . 30·638·90 9·109 ll·8 14·12 15·8 22·01 19·8 30·868·95· 9·190 .11·9 14·30 15·9 22·22 19·9 31·09

,

9·00 9·271 12·0 14·49 16·0 : 22·43 20·0 31·339·05 9·353 12·1 14·68 16·1 22·64 20·1 31·569·10 9,434, 12·2 14·86 16·2 22·85 . 20·2 31·799·15 9·515 12·3 . '15·05. 16·3 23·07 20·3 32·02

, 9·20 9·596 12·4 15·23 I 16·ij,· 23·29 20·4 32·26

9·25 9·678 . 12·5 15·42 16·5 .23·51 20·5 . 32·509·30. 9·760 12·6 15·61 i6·6 23·72 . 20·6 32·749·35 9·842 ·12·7 . 15·80 16·7 23·94 20·7 32·98.9-40 .9'924 12·8 15·99 16·8 24·16 ' 20·8 . 33·:::29·45 10·01 12·9 16·19 16·9 24·38 20·9 33·56

9·50 10·09 13·0 16·38 17·0 24·60 21·0 33·709·55 10·17 13·1 16·58 17·1 .24·81 21·1 33·9.t9·60 ]0·25 13·2 16·77 ' 17·2 25·02 21,2 34·17 .9·65 ' '10·34 13·3 16·97 \ 17·3 25·24 21·3 34-40 r

I9·70 10·43 13'4 17·16 17-4 25-46· 21·4. 34·64' .

9·75 10·51 13·5 17·36 17·5 25·67 21·5 34·889·80,' 10·59 13·6· 17·55 ]7·6 ·25·89 21·6 35·129·85 10·68 13·7 17·75 17·7 26·10 21·7 35·36

.. 9·90 10·76 13·8 17·94 17·8 26·32 21·8 35'609·95 10·84 '13,9 18·13 17·9 '26·54· 21·9 35·84

10',00 10·93 14·0 18·32 18:0. 26·76 . 22·0 36·08. 10·1 • 11·10 14·1 18·52 ]8·1 26·99 22·1 . 36·a3

'. 10·2 11·27 14·2 18·72 18·2 27·22 22·2 36·5810·3 . 11·44 . 14·3 18·92 18·3 27-45 22·3 36·82

. 10·4 11·62 14·4,

19·13 18·4 27·68 22-4 37·06

10·5 ll·79 14·5 19·33 18·5 27·90 22·5 37·30- - 10·6 .11·96 14.·6 ]9·53 . ]8·6 28·12 22·6 37·54

10·7 12·14 14·7 19·74· ,:: 18·7 28·35 22·7 37·7810·8 12·32 - 14·8 19·95 18·8 '28,57 22·8 38·02

.. 10'9 : . 12·49 14·9 2Q'15 18·9 28·80 22·9 38·27

n·o 12·67 15·0- 20·35 19·0 29·02 23·0 38·52n·i 12·85 15·1 20·55. . lQ·1 29·25 . 23·1 ',38,76n-a 13·0.~ 15·2 20·76 ' 19,2, '29·48 . 23·2 39·01n·3 13·21 ]5·3 20·9'> 19~3 29·71 23·3 ,39·~611·4 13·39 . 15-4 . 21·17 . 19·4 29,94 . 23·4 39·51

."

:-,' ..

96 P. H. J. !'. KLl?YNEN ", .

~+ 'IJ ~~+ 'IJ; ~+ "'IJ ~+ 'IJ- ,70 . 190·4 . llO 357·9 150 549·8 190 760·471 194·2 III 362·4 151 ' .5'54'9 191 765·9 '72 198·1 ll2 . 366·9 ' 152 559·9 192 . 7,71'373 ' 202·0 ll3 371·5 153 565·0' 193. 776·974 205·9 n«. 376·1 154 . ·570·1 ' ' 194 ,782,5

75 209·8 ll5 380·7 155 575·1 195 788·0 .76 213·8 ll6 385·3 156 580·2 196 793·677 217·8 ll7 ' 389·9 157 . 585·3 . 197 799·278 - 221·8 ll8 394·5 158 590·4 . 198 804·8

-'. , 79 225·8- . ll9 399·2 159 595·5· 199 810·4

80 229·8 120 . 403·8 160 600.6 200 816·081. 233·8 121' 408·4 161· 605·7 201 821·682 237·9 122 413·0 162 610·8 202 827·383 242·0 , _123 417·7 163 616·0 203 832·984 246·1 124 422·4 164 621·2 ,204 . 838·5

.. 85 : 250·2 125 427·1 165 626-4 205 844·186 254·3' 126 ,4,31·8 166 . 631·6 206 .849,8."87' 258·4 127 436·5 167 636·8 207 855·4.88 262'5' 128 441·2 168 642·.0 208 861·089 . 266·7; 129 446·0 169 647·2 209 866,7"

90 270·9 130 450·8 170 652·5 210 872.4191 , 275·1 131 . 455·6 171 : 657·8 2ll 878·192 ' I 279·3 132 460,4, 172 663·1 212' 883·893 283·5 133 465·2 173 668·4 213 889·5

"

94 " 287·7,

134 470·1 174 673·8 214 895·2 ;

95 291·9 135 . 475·0 175 679·2 215 900,9,96

,

296·2 136 479·9 i76 684·6 216 906·697 300·5 137 484·8 177 , 690,0, 217 912·49'8'- 304·8 138 489·7 178 695·4 218 918·199 309,1, 139· 494·7. 179 700·8 219 . 923·8

f· "

100' , 313·5 140 499·7 180 706·1 ' , 220 929,6101 317·9" . 141 ' 504:7 181 711,5 221 935·4102

. 322·3 ' 142 509·7 182 716,9, ' ·222 941·2103 326·7 ' 143' 514·7 ·183, 722·3 . 223 947·0104 331·1· 144 ' 519·7 18~ '727·7 224 952·8

'105 335·5 145 524·7 lB5 733·1 225 958·6 '106 ' 339·9 14.6 .529·7 . 186 738·6 226 - 964·5107 ' , 344·4 147 534·7 187 ,144,0 227 970·3108 ' 348·9 148 ' 539·7 _i88 749·5 '"228 976·1109 35~-4 149 544·8 189 754,9, 230 987·9

;

. - 1 •.