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Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW) Citation: Everdingen Jr, E. van, The HALL-effect and the increase of resistance of bismuth in the magnetic-fieldat very low temperatures. II, in:KNAW, Proceedings, 3, 1900-1901, Amsterdam, 1901, pp. 177-196 This PDF was made on 24 September 2010, from the 'Digital Library' of the Dutch History of Science Web Center (www.dwc.knaw.nl)

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By adding the two last conditions, we find:

I •

a form to WhlCh we mayalso rcduC'e the form given befol'c. From the value for the first diffel ential coefficients we deduce, that for substances, which are perfect]y miscible, the partial pressure of one component dccreases, when the second component is substituted for a part ,of it. From this follows that the total pressure must be smaller than the sum of the tensions of the separate components. If 1+:1'1 (I-xl) f.lzJ should be negative, the partial pressure of a component increases on the other hand by substItution by the second component. Then it wIll be the question whether the partial pressure cannot rise so high, that it exceeds the initial value.

This question, howeyer, cannot be solved without the knowledge of the proper ties of the function p.

Physics. - Dr. E. VAN EVERDINGEN JR.: "The HALL-effect and tJee increase of 1'esistance ot bismuth in the magnetic field at very low tempel'atures." Il. ~ Communication N°. 58 from the Physical Laboratory at Leiden, by Prof. H. KAMERLINGH ONNES).

1. From the measurementb of the HALL-effect in bismuth at the boiling-point of liquid nitrous oxide anel hquid oxygcn, described in the Proceedings of 29 October 1899, p. 221 and 30 December 1899, p. 380, it appeared that the HALL-coefficient increased cOllsiderably with falling temperatUl'es; it hence seemed desirabie to determine thIS increase with greater accuracy. The measuremellts in liquid nitruus oxide had shown that the strength of the magnetic field had

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a considerable infl.uence on the temperature-coefficient. Rence a]so the measurements in liquid oxygen ought to be taken with different strengths of the neld. For the theory of the phenomenon -it is necpssary to know the resistance of the bismuth in the magnetic field at the same temperatures, in order to, be able to calculate the angle through which the equipotential lines are turned. 80 I at first decided to measure, in say fi ve different fields, HALL-effect, resistance and increase of resistallce at the temperature of the room and at the boiling points of liquio nitrons oxide and oxygen. Af tel'­wal'ds a series of measurements at the boiling point of methyl chloride was added, and fillally I completed the research by repeating the same rneasurements at the boiling point of water. As earlier researches 1) have shown, that at still higher temperatures both H.A.LL­effect and increase of resistance become very smalI, it might be cOllsidered superfl.uous to further l'aise the limit of temperatnres.

Rence the temperatures range from - 1820 C. te + 1000 C. or from 910 to i:n3° on the absolute scale.

2. Experimental arrangements. In all the experiments except those at 1000 C. the experimental plate of bisrnuth was mounted in the apparatu~, described in § 6 of the communication of 30 December 1899 2). The only change made in this since th en is that the streng­thening-rim at the lower end of 83 (see fig. 2 of that communication) has been omitted, and repldced by two glass tubes, fixed at both àides of the vessel b alld over which the thin pa.per of 83 is stretched.

In the modified section of the apparatus, fig. la, these tubes are indicated hy the letters i. T!lis enabled me to adjust the vessel between the polc-pieces and to take it out again without altering tbe distance of the pole·pieces. In tbis manner 1 st the apparatus remained quite closed at t,j" 2nd during the whole research the distance between the poles, and hence the strength of field for a given magnetising current, remained unaltered, and 3ld the repair of smaH faults in the val'ious leads during the expel'iments was facilitated.

The apparatus continued to p:ive satisfactory results and wad at the end of the whole research still in good condition.

For tbe experiments at 1000 C. the plate was placed in a copper

1) See for instance LEunnT, Vers!. der Verg. Kon. Ak. 'Jf. Wetensch. vnn 28 Sept. 18!15, p. 108, Comm. Pl1ys Lub. Leiden NO. 19, p. 26 j HENDEJtSON, Wied. Anu. 53, p. 912, 1894.

~) Versl. der \'erg. vun 30 December 1899, p. 380, Comm. N°. 53, p. 10.

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~. \.

_I, OOIl.ted with UbeiitOl! Knd packed in .... 001, .... hieh was cloeed at tbe upper end by mCllDI of cork and Ihrongb whieb liteam

WI8 100. Whilst tbeufore tlle baths of eonstll.nl \emperatn~ pl"ll8eDloed no

lIplICi&1 difficullies, we had to bestew greIIt care 10 BOOure good .:.lubel et the H UL- lIud rwiefaoce elootrodes. In earl ier expen­meuta, told~,ing wa. doomod u n suil~ble, on acooupt of the daoger of Bpoiling tbe purity of Ihe bismufb So here we bRd 10 u&e damp.eleccrodeA. Ic 600U appMroo tbllt, tor the contact8 10 {Mist Ule intense eooliog, they ought 10 he made elastic. Steel sprin8' could nol bil uaed, beC3uae of the d;8lurbanoo they ,,"ould hAre CIIUse<:l in tbe magnetic field; bras! eprings, ,.hieh were tried 6rd, appeared to llU'k 8ul6cient elalticity, w t\tat (he contact! welt! oovettlwJ8$8 spoiled on cooling. These difliculties were overoome, fot the mOlt part al least, by UaiDg 'pcings made ora.n IIIloy of Platinum with 90 pOL. of Iridinm.

With reIen:lnce to fig. 1 .... e will DOlT deI\Cribe how Ihe plate of bismutb was H:roo in tbe carrier in itl! Hna.!. form.

For thls purpose we fit$!; direct OUr atteotion 10 tbe perspectirc drawing on thc righl 'fhere we se<) the plate of bismuth P ,tu~k Ihrough tbe verticaI sJit ... f the fr&rne R to whieh tbe Pla.lioum­Iridium ~pringl! VI and 17, are filel! by mea.na of JlCrews SJ and Ss. 'fe the eltremitîes of the!!l! 6pringrs little Pbtinum pens are atlaclted, willeh go through eylinoiriea! holes in Rand end on the horiwnW lioe in the middle of Ihe plBIe P. These oonstitute !he rosislaneo clectrodJ:fl, and are ftoout 10 mmo ap!lrt. The ~pringa VI Md V. are plaoed in s1ita of the frame in order Dot to iOQrta&6 tbc toUil

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brcadth and are thus at the same time protected against damagc. To make sure th at the liquefied gases reach the plate the central portion of thc vertical slit is widened and moreover large holes arc made in the sides of the frame.

In thc middle of the upper and lower planes there is a small hole O. 'fhrough this enter the secondary electrodes of the plate­carrier l, drawIl in section in the figure. The primary-electrode" El has been screwed through the brass strip Al and thl'ough the frame l aDd is therefore not elastic. E2 on the contrary has been screwed only through the brass plate A 2 and goes freely through l, whilst .(12 is elastically attached to l. The secondary electrodes E3 and E4 are also elastically attached; they go freely through the brass plates As and A4, but are screwed through the nuts MI alld M2' which are pressed iuwards by spirals of Platinum-Iridium, whilst a pin prevents them from tU1'lling together with the screws. The plate Al is connected by a thin insulated wire to the copper wire D4' To the wireB Dl'" D4 the thin copper wires mI' m2' nl' 112 (SPe.'

fig. 1 Oomm. N°. 53) are fixed by means of screw-connexions. FroID the screws SI and S2 copper wires of 0,1 mmo diameter go out of the apparatus through the same glass tubes as mI and m2 for the measurement of the resistance.

In the experirnents at 100° C. the plate-carrier l was made of wood and the frame R of ivory; in the othol' experiments both were made of ebonite.

3. Measuremenfs of the HALL-effect. The plate of bismuth wbich served in all experiments was not tbe same as tbat used in tbc preliminary experiments of Communication N°. 53, as the latter was broken, when further observations had already been made at some temperatures. The new plate was ho wever obtained in the same mannel' by electrolysis; the CUlTent for this was chosen somewhat smaller than on the former occasion. The resistivity of this plate appeared to be a little smaller than that of tbe other, while the HAIJL-effect and the illcrease of resistance were somewhat larger; this indicates that this bismuth is a little purer. That ho wever com­plete purity is not yet a,ttained follows from the most sensitive cri­terion : the resistance out of the magnetic field at low temperatures, to which we will drawattention once more fUl,ther on.

rrhe dimensions of the plate were: length 21 mm., breadth 9,1 mm., thickness 0,795 mmo

Of tho method of observation nothing new need be mentioned 1).

1) Verslag d. Vergadering van 30 Mel 1896, p. 47. Comm. NO 26, p. 3.

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As with the last measurements of Oomm. N° 53 the resistance in the secondary circuit was measured for both direntions of the mag­netic field immediately aftel' the determination of the resistance required in the compellsative-circuit. In order to be a,ble to quickly perform this measurement I proceeded in the following way. As a commencemcnt I observed the deflection of the galvanometer cam;ed in the secondary circuit by a WEsToN-element when aresistanee of 50.000 Ohms was inserted and the poles of the element were connected to two of the mercury-cups of the commutator of this circuit 1), the other two mercury-cups being connected by a short copper wil'e. As this deflection remained constant during a series of observations, it need be observed only once. For the mea­surement of tbe resistance the deflection was observed again aftel' a shunt had been made between the two first mentioned mercul'y­cups witb a known resistance [.bout equal to that of the secondary circuit, so that the deflection was reduced to ab out one half of its former value. Jf we can a the total deflection, b the reduced deflec­tion and w the resistance of the shunt, then the resistance of the

d '" a-b secon ary C'll'cmt IS 10-­

b :First we give the results for the HALL-coefficient R in vanous

magnetic fields M (in O. G. S. units).

HALL-coefficient R.

'1' e m per a t ure in d e g ree s een tig rad e.

+ 100° \ + 11° 5 \ - 23° \ - 90° \ - 1820

M/R \M/R /M/R /M/R /M/R

1000 7 23 1050 13.24 1060 16 00 1020 27.88 1050 61.8

2200 7 16 2100 12.69 2120 15.83 2140 24 80* 2100 54 5

3920 6.99 3100 12.06 3110 L4.98 3070 22 87 3830 46.2*

4800 6.87 4440 11.42 3770 14.55 3730 21 85 6050 39.8

4870 6.82 6010 10.61 4320 14 13 4370 21.10

5970 6 75 44040 13 94 5180 19 97

5260 13.39 6050 18.68·

6010 12.90

1) See my thesis, plnte lIL

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These results are represented graphically in fig. 2, where the little crosses indicate observed points.

JU -.111 f"'-60

"'-.. ,---'-.... ~

I

- = '-.... ............. V ""Rl

........

> ~ fII) /: WI1!fJ

L

JO L -$0 ............. /1 ;--....

v :---i"'-- H-- I----"" -90' / t-- -V ....---vl--

-23 _?C

,/ ./' F= .u'S

/ --- ~ -JO

~ ::;:::;- --- - --- ~ +.m' i>" ~ ':'--f- -

Z, ~ f:::::: t:::-- :lIto I

0 J(; .. qo

Fig. 2.

When a series of ohservations was finished usually one or more of the determinations we re repeated; on the one hand this gave a means of te&ting tbe accuracy of the measurements, on tbe other hand of testing the constancy of the temperatures. With + 100° a. the two measurements in a field of about 4800 may serve for a test; with - 23° a. thoee in a field of about 4400; with the tempe­l'atures - 90° aDd - 182° an asterisk in the table indicates that the measurement was l'epeated and gave the same 1'esult; tbe value 46.2 in the field 3830 at - 18~0 a. is the reault of t!tree meas­urements in complete agreement 1).

Usually befare or aftAr a series of measurements a determination at ordinary temperature was made; these toa agreed always weIl, so that, at least during tbe three months occupied hy the research, no traces of a variation with time are to be detected.

With the experiments in methylchloride and nitrous-oxide no trouhle was experionced in keeping the vessel filled with liquid for about 5 hours, 80 that there was abundant time for observations.

1) The results, obbLÏned witb the plate afterwards broken, usually a]so agree with those above mentioned. /

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With the experiments in liquid oxygen however the vessel was usually nearly empty before a new guantity of liquid could be admitted, hence in this case the number of measurements was somewhat reduced.

During the experiments at 11 °,5 C. air appal'atus in order to ensure equilibrium surroundings.

, was sucked through the of tempel'ature with the

The results wholly confirm the rule formulated before l ): 'that the variation of the HALL-coefficient with the magnetic field is larger the lowel' the temperature, or: that the infiuence of temper­ature on the HALL-coefficient is ]argest in weak fields.

The value 81.,8 of the H.ALL-coefficient in the weakest field at - 1820 C., is again considerably larger tban the highest value obtained hefore 2) (in the magnetic field 4400).

By means of the curves drawn tbrough or between the ohserved points the HALL·coefficients in the fields ,1000, 2000 ... 6000 were interpolated, and multiplied by the corresponding magnetic fields. Tbe values of the thus obtained product RM, which may be consi­dered as a measure of the total transverae difference of potential, are represented in the same figure. 1'11e scale value of these ordinates is indicated on the right band side.

Finally In fig. 3 the variation with temperatul'e of the HALT,-

" 8l.

""~ coop

" 00"\ l'l I'~ t\ \ ~..,~ h\\ 1\

" !m

~ l\\ 0 "\ ~ \\

~ ~ ~

" ~ ~ ~ \~ ~ ~

I

::-...

" - r-r""'= ~

ff ,0 ,.FO IJ"" Mo I 100 1$0 I JOO JSO +

Fig. 3. \

coefficient fol' given values of the field was represented fol' the fields

1) Ver~l. 30 Mei '96, p. c9. Oomm. :No 26, p. 20. 2) If 1$0 Dec. '90, p. 382. Oomm. N0 53. p. 13.

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1000 to 6000. The data for this figure are also given 10 the table below.

HALL-coefficient R.

Magnetic fielt! in O. G. S. units.

Tabs. I I I I I 1000 2000 3000 40{JO 5000 6000

I -91 62.2 55.0 49.7 45.8 42.6 40.1 -

183 28.0 25.0 22.9 21.5 20.2 18.9

250 17.0 16.0 15.1 14.3 13.6 12.9

284,6 13.3 12.7 12.1 Il.5b U.05 10.6

373 7.28 7.17 7.06 6.95 6.84 6.72

It appears that in all fields the increase of the HA.LL-coefficient with falling temperature is approximately proportional to T-a , where a is greater tban unity.

4. Measurements of 1'esistivity and inC1'ef!Se of 1·esistivity. For the method of observation reference may be made to Comm. N°. 48 1).

(The resistance in the circuit containing the resistance-electrodes was measured in the same manner as that of the sccondary circuit for tbe HA.LL effect).

In the experiments before a determination of the resistance of the bismuth in the magnetic field a determination without magnetic field was always made, and the increase of resistance èaused by the magnetic field was calculated, by a direct comparison of these resistances.

Rence we obtained for the resistance out of the magnetic field as many values as there were measurements. The agreement between thrse values was always very satisfactory, once more confirming that the tempeI'ature remained constant during the experiments. For, these observations were made at the same time with those for the HALL-effect.

For tbe calculation of the resistivity we further want the dimensions of the plate in the transverse section, which were known accurately, and the distance of the l'esistance-electrodes, which could not be obtained with the same accura,cy, were it only because the pJaries of contact were rather large as compared with their distance. In

1) Versl. 25 Maart '99, p. 486. Oomm. N°. 48, p. 6. \

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order to get nevertheless at the various temperatures a good co1'­respondance in the values of the l'esistivity, immediately before or af ter a series of lllcasurements a determination of the resistance of thc bismuth at ordinary temperatul'e was made. In this manner we got an accurate determination of the ratio between this resistance and that at the lower or higher temperature. Finally.for the resist­iv!ty at 11°.5 C. a value was accepted as righr and from this thc values at other temperatures were calculated. The difference between the value calcuJated in this mannel' and that obtained directly was in the most unfavourable case only 2 pCt.

A correction for contraction of the plate of bismuth and the plate­carrier by cooling would be too small to be worth considering.

We first communicate the results for the percentage increase of the resistivity in the magnetic field.

Percentage increase of r~sistivity b,r.

Tem per a t ure i n d e g ree s een tig rad e.

+ 1000 90°

111 I Ar \

MIM M I t.r ! M r Ar M ! Ar

! 2200 0.9 1050 0.9 10liO 1.5 1020 3.5 2050 35.9

3950 2.6 20GO 3.0 ~140 5.2 2140 12.5 3730 90.2

4830 4·.0 3060 5.8 3110 9.7 3100 22 4 4740 127.1 . 6100 (..8 HSO 10.4 37!l0 13.2 3760 29.9 6000 175.7

- G030 16 6 4410 16.6 4150 34.6

5250 21.5 5200 47.7

I I 6050 26.51 6110 59.6 ,

It appeared that the formula for the increase of resistance given in Comm. nO. 48

repl'esents very s[ttisfactol'ily the dcterminations at all the tempemtur'es of observation. For brevity I shall not mention here the calculated

1) Vers!. 25 Maort '99, p. 485. Comm. N0. ·1.8, p. 4 .•

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values and communicate only the values of Cl and C2, and tho largest deviations bet ween calculation and observation.

Tabs Cl C2

I Largest

I deviation.

91 0.312 14.027 3.7

183 o 285 4.381 0.69

250 0.219 1 6S1 0.34

2846 0.187 0.968 0.29

373 o 069 0.220 0.12

In fig. 4 the curves are drawn through the calculated points, the crosses indicating the observations. 'I'hey also show a good agreement, while the deviations are not systematically distributed.

~, o_ I -vo

v

100 ~-r--r-+--+--r-4--r~I~~1--~-r~--r-~ ~+-~-+--~.+-~-+~/~-+-,r-+-~._-r--

/

1/ -90

1/ /

Fig. 4.

By rneans of the values of the percentage increase calculated from this formula and the results for the resistivity out of the magnetic

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field (second column) the following values for the resistivity in the magnetic field have been found.

Resistivity r (multiplied by 10-5).

Magnetic field in C. G. S. units.

Tabs.

I I I I I I 0 1000 2000 3000 4000 5000 6000

91 1.711 1.894 2.316 2.826 3.418 4.054 4.718

188 1.526 1.578 1.701 1.853 2.023 2.219 2.423

250 1.600 1 623 3.683 1.744 1.828 1.920 2.020

2846 1.690 1.703 1.738 1.783 1.839 1. 904. 1.967

373 2.094 2.098 - 2.111 2.129 2.]52 2.180 2.212

These results are represented graphically in fig. 5. If we compare them with those of FLEMING and DEWAR 1), it

40 ~

1:: 600c

\

" D .m;o \

\ \ s 1\ \

13 } 1\

\\ I~ \ 1\ I

\ \ I

:\ .. ~ \ \ '\ \-

" '\ "- ""- ~ ~ r--.... k:::; ~ • JOD~ ""- " i'-. r--. --::::: ..... " v .... ........... ...........

J,3

0,.

Is:'" S() JI» ~() %OD S'D ._ JS. '"

Fig. 5.

1) Proc. Roy. Soc. 50, p. 425, 1896.

13 Proaeedings RO;Y'!II Aalld. Amsterdllw. Vol. lIl.

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appeal's that the general chal'actcl' of the curves is the same. The resistivity out of the magnetic field howevcr does not continuously decrcase here, as with thc electrolytic bismuth from HARTMANN alld BRAUN, but reaches a minimum and then rises again to about thc va,lue it showed at ordinary temperature. FLEMING and DEWAR 1) found a similar behaviour with some samples of bismuth carcfully prepared by chemical means; my curve happens to eoincide prae· tically with tbe one they found with "M.A.TTHEY'S Bismuth (E)". Hence very likely tbe bismuth of my plate also contains only a very slight impurity; for the present research this impurity is of no consequence.

5. .An.qle of rotation of the equipotential lines. A combination of the results of § 3 and 4 enables us to calculate the angle, through

- whieh the equipotential lines are turned in the maguetie field. Thus if the product RM is devided by the resistivity r, the quotient is equal to the tangent of that angle. Again the quotien~ of Rand r is equal to that same tangent for a magnetie field 1, a quantity first introduced by LEDUC 2) and which we shall eaU D as he did. From what follows it will appeal' that this quantity has a simpier theoretical meaning tban the HALL-coefficient R.

The results of the ealculation of D are found in the following table and in fig. 6.

Rotatiomtl cOE'fficient D (multiplied lIy 105).

Maglltltic field in C. G. S. units.

Tabs.

I I I I I ]000 2000 3000 4.000 5000 6000

{)1 32.!! 23 8 17.6 13.~ 10.5 8.5

183 17.75 14.7 12.3" 10.6 IJ 1 7 8

250 10.4.8 10.48 !!.50 8 66 7 82 6.39

284" 7.81 7.31 G.7!! 6.28 5.80 5.39

373 3.47 3 4.0 3.32 3 23 3.14 3.04

In order to facilitate a survey of the intluence of temperature

1) Proc. Roy. lust. Juna 5, 1896, p. t6. 2) La Lumière Electrique 2!!, p. 230, 1888.

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r-I

V -1--r--= "" V / -~

\ v V I--./'

1'\ / ./ V L.--~

'\

/ V v ~ /"-" / v v h( v .,/

1/ / I'>< 17 V V

'" / ~ ~ ~L /' l::>< ~ l.--~

/ '1 I/V -----k ~ f<:: 1-4 ----~

/ 1/ ---

w - t.c:::

/ '// ""JS ---.;;

-------.... ,

~ ---- i :m, - .. \0 ''''

Fig. 6.

and of magnetic field on D, I tl'ied to represent thc results fol' each of the five temperatures by a formula of the same form, and succeeded very weU with the formula

Do D = --___ -,.~-___ __:_:-::-1 + Dl V.'1ls + Ds M2

As in the case with the increase of l'esistance I shall give here only the constants, and the largest deviations from the calculated values. M was expressed here \ in the unit 1000 O.G.S., D in the unit 10-5 O.G.S.

Largest Do Tabs. Do Dl Ds

deviation Dl

91 47.18 0.3708 0.06603 0.14 128

183 21.13 0.1850 o 01714 0.18 114

250 11.50 , 0.0885 0.00736 0.03 130

2845 8.40 o 0663 0.0045l 0.03 126

373 3 53 0.0155 0.00188 0.002 (227)

Fig. 7; contains the curves . drawn to represent these formulae. 13*

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( 190 )

0

f 0\

1\ I 0

\ -t- --+-- --\

!

''\ 1\

0-- 1\ \ -\ \ I

l\ 1\ ~

U"'~ \ \ t--

--. 1"'- i\ :\ L i\- \ I

JIX'O

i'-- !"'- \ ,\ ~'W. "- '\ .\ \. !~ ------ "' ~ ~ ~ ...... ó~ ---- .::::: ~~ ~ ~ ~ I

;--;--- t """ ~ ~ tr

SO I- o.I. I ,." ·f· I JJO JJO ... :I!'ig. 7.

Do is evidently the limit to which D approaches in very weak fields; this quantity is best adapted to form a judgment of the influence of temperature alone, as the influence of the magnetic field on the l'esistivity of bjsmuth may be neglected in sueh weak fields.

The coefficients ])0 and Dl show a somewhat parallel course, as

Do 0 0 appears from tbe column headed Dl. nIy the ratio at T = 373

differs considerably; but here tbe line is almost straight and the whole variation with M flmall, 80 ihat the interpolation and with this the deduction of the coefficients becomes a little arbitrary. If we put Dg = 0 fol' T = 373 0

, then the results fol' that ternperature are l'eprcsented to within differenees of at most 0,03 by a formula

D with Do = 3,56 and Ih = 0,0206, aDd hence with -E. = 134.

Dl All coefficients are approximately proportional to T- a, where a is

greatel' than 1, especially fol' Dg. Whilst, as we saw hefore, even at tbe lowest temperature tbe

product RM increases throughout with M, we see here in fig. 6 that

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the product DM (drawn on a Ecalo 1000 time. smaller than D) shows a maximum at -182° O. 1). This peruliarity also is borne out by the formula, in which above a definite value of M the term D2 M2 dominates; one might even deduce from it thut in strong magnetic fields (17.000 to 20.000) the product would have its highcst value at the highest temperature, which does not seem very probable. But it is possible that with the very high values which thc trans­vërse differenre of potentifkl reuches in this plato a disturbance is cam~ed hy the electrodes for the primary cUl'rent, which are 3,5 m~f. thick and hence ean certainly not be eonsidered a~ mere points, so that we do not measure the full HALL-effect. VON ET'l'INGSHAUSEN

and NERNST 2) found that the fuIl HALTJ-effect was almost reached when the ratio of breadth to length was ai! 2 to 3, and the primary electrodes fu11y covered the sides. In their r€search howevor they did not obtain nearly sueh high values of R. If therefore my pre­sumptioll is justified, one might suppose that the true effect is repre­sented by the same farm uIa with D2 ' O. DM approaches th en for

all temperatures to a common limit Do. Dl

6. Remadcs on the theot'y of tlze pltenomenon. These results may contribute to the determination of the temper­

ature-functions 1Ie' and el in VOIGT'S thermo-dynamical theory 3). The theory of the HALL-phenomenon, based upon the recent theo­

ries of the conduction of electricity in metals, such as that of LORENTZ 4) or as a p~rt of the "Electron-theory of metals" ,. al'ter DRUDE 5), is at this moment still in the nascent state. Yet I think it possible even now to draw from the foregoing some eOllclusions with respect to that theory and to indicate how far Fbis theory is able to give an explanation of the infl.uence of temperature and of rnagnetic field on the constants which represent the HALL-effect.

As yet the 0111y completely elaborated theory on this basis is that of RmCKE 6). This gives the following formula 7) for the HAT,L ·coefficient:

I) This maximum 0,536 is smnller thnn tlle vnlue 0,7<10, giv8n in Commllnicntion N0. &3. There however a preliminary value for t11e resistnnce wus nssumed w11ich now appenrs to have been too low.

2) Wien. Sitz. Ber. 94, p. 568, 1887. 3) Wied. Ann. 67, p. 717, 1899. 4) Versuch einer Theorie der electrischen und optisch en Erscl1eiulIngen in bewegtcn

Körpern. Leiden, 1895. Ii) Ann. der Physik. 1, p. 566, 1900. 6) Wied. Ann. 66, p. 345 en 545, 1898. i) ib. p. I) 6S.

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p = _~ ~t2 Un-V'), gp

r lt .qn+V gp

(r = conductivity, 1& and v velocities of the charged partic1es caused by a potcntial gradient 1, Up and gil velorities caused by a temperature gradient 1).

Hence for D we find:

According to Rmclm's theory u and v, and likewise gp and gn, are to be multiplied by the eame factor for val'iations of temperature. From tbis it fo11ows immediately that with change of temperature D is mIlltiplied by the same factor as 1& or v.

Precisely the same result is dcduced from the formula for the HALL-phenomenoii in rlectroJytes, given in my thesis for the doctorate aud in Comm. N°. 41 1), whieh wholly ag'l'ces with a formula deduced by WIND 2) for the HALL-effect in metals. This reads:

R = r (u-v) hence D = (u-v).

Jf 1& and v undergo proportional variations with temperature D here also is multiplied with thc same factor. '1'he ~ame would appIy if 1& were many times greater than v, or the reverse. Therefore we shall assumc for simplicity that the temperature-variation of D is controlled by that of u. According to tlw dedudions in my thesis and to the theories of RmcKE and DRUDE

I u=J(~ 1)

V

",here K is independent of the tempentture, l is tbe mean free path and v the mean velocity of the ehargcd pal'ticles.

The suppositions made regardillg the variations of I and v with tcmpemture will hence determine the temperaturd factor of u.

HmcKE assumes: 1= 10(1- (JI)

v = q/ T (1 + (j t)

I) Thesis p. 107. In Comm. nO. 41 (Yersl. d. Verg. 28 Mei 1898, p. 53, Comm. nO. 41 p. 9.) tl/e formulll for D contllÎns anotller numerica1 factor.

~) Verh. Kon. Ak. v. Wet. Deel v- No.. 3, § l7. 8) Thesis p. 104, RIECKE 1. C. p. 377, DRUDE 1. C. p. 575.

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-( 193 )

henoe tho tempertl.ture factor of D boeomes

where t means the temperature in centigrade degrees, and T the absolute temperature. Tbe shape of this formula remains the samp even if W'(' should ilssume that v is strictly pl'oportional to VT, as then only (Y becomes zero.

If this formula is applied to the vaJues of Do, these appoilT to be reconcilable with it if (,8+0) (or fJ in the limit) is posiûve. In tbe Jatter case this means that tho mean free path decreascs with rîsil1g temperaturcs, which is according to RJECI(E'S assumptions.

When the range of temperaturo from 373 to 91 (abs.) is sub­divided into throe parts, we find as mean values for (~+J) or ('1

373 -2845

183 -91

0,00898

0,01481

RIECKE himself calculates 1) fl and ° from tho variation of thc conductivities for heat and electricity in bismnth between 00 en 1000 0., however using arelation botween ° anel another tempprature-coef­ficient which is perhaps not unobjectionable; he finds I~ to be 0,00205, J - 0,0000103, hence

P + J = 0,00204

a result of the same order of magnitude. AIso considered ap,ut the results deduced above necd not be ealled improbable.

Calculating however the valucs for the same temperature CO(lffi.­

cient in the magnetic field 6000, we find

373 -2845

2845-183

183 -91

0,Q0382

0,00159

-0,00207

It 8eems impossible that the value of iJ could be so different in tbe magnetic field.

1) l.c. p. 573.

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( 194 ,

An explanation of this appm'ent conlradiction (Jan ~e obtained hy means of tlle hypothesis, that in the ma,qnetic field the number of free cltarged particles is diminished, the same hypothesis, which Ip,ads to an explanatian of the increase of the resistance in tbe magnetic field, and of the proportionality between longitudinal-effect and increase of resistance 1).

IndE'ed in my thesis 1 ventured the suppasition S), that .this decrease is raused in the following way, that the particles with velocities smaller tban a certain amount (say smaller than a critical velocity $) are caused to move in closed orbits in the magnetic field and cease to partake in the transferénce af the current. It is evident that the mean velo city of the remaining, free particles will be greater than tbe mean velacity af all particles. Hence in Dur farm uIa we aught to insert far v the mean velacity af all particles, multiplied by a facto?' q. In a magnetic field of definite strength the critical velocity $ has a definite value which in my thesis I assumed pra­partional to Jf. The lawer the temperature, the largel' the number of partieles with veloeities below the critical. If now far a moment we assume MAXWELL'S law for the distribution of the veloc­ities of the free particles, then it appears that the rato of increase of q is greater, the largel' tbe ratio of the critical velocity ta the meELll velocity of all partieles, or, that the rate of increase of q itself increases with falling tempemtures.

For a constant value of the maglletic field and hence of 3', this result may be intraduced easily into the formuia by giving to ~ a rather large negative value far a mngnetic field of 6000; in this mannel' the negative sign of (fI +~) would be explained.

We have not yet thc data to enquire whether our hypothesis gives a quantitative e~planation of the phenomena. But we may notice that the hypo thesis is sufficient ta alsa explain other partic­ularities in the variation of the quantity D, as may be se en by reference to fig. 6 and 7.

Tbe decrease af D with increasing magnetic forces at constant temperature (fig. 6) is eXplained immerliately by thc increase of IV; for the mean velacity of all particles remains constant, and q con­sequently increases. This decrease of D is most rapid at the lowest temperature i this also is explainable, as then the critica) velocity

I) See Vers!. Kon. Aknd. v. Wetellsch. 25 Maart 1899, p.496. Comm. No. 48 p. 23-2) See p_ W~.

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( 195 )

is grèatest as éompared with thc mean velocity and the rate of increasfl of q is larger. Jf the critical velocity happened to become mllch larger than the mean velocity, v wouW become approximately independent of temperaturc; henrc this might be the explanation of the smaIl influence of the temperature in strong fields, and ren der it probablo that in a very strong field D would bccomc independent of thc temperature . .A maximum of the quaDtity IJM at - 1820 C. would not be eXplained, but one might expect an approach to a constant value, as the critical YeloClty and hence thc mean value of v for thc free particles increases proportionatply to M, so that D would decrease Dendy proportionately to the inverse of ... V, which is completely in agreement with our formula, jf we put in it D2 = O.

Our hypothesis throws also some light upon the reason why the increase of the rcsistance in the magnetic field for small strengths is proportional to a power of M higher tban first. For ifwe assumc MAXWELL'S luw, then tbe probability that a particle has a velocity

'" smaller than :IJ is proportional to J a-2e - x2 d.c; for very smaIl val ues

o

of :c we may take e x2 equal te unity, and find then that thc nllmher of particles with velocities smaller than :zo would be proportional to the third power of :zo, which meaDS to the third power of M. The large inerease of the resistanee at low temperatures can be explained by the decrease of the mean velocity as compared with the cri ti ral véloeity. Finally WA TE.'mark, that, as at - 1820 C. the resistance is increased nearly in the ratio of 1 to 3 in a magnetic field of 6000, it hence seems that about 2fs of the free particles 108e their freedom in tha.t case. At 1000 C. on the contrary this number is very smaH, so that between these temperatures q should undergo a considerable change. This is in agreement witb tbe 1arge variation of tbe vallIe of ((J + 0).

'[bis survey is of course only superficial and leaves several questions. undiseussed. I think bowever that it affords sufficient reason to aSGume, that with the introduction of this hypothesis in the electron­theory of met als ft step has been made in the right direction.

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Chemistry. - "On the system: [Bt20s-N205-H20]". By Prof. J. M. VAN BEMMELEN.

Dr. G. M. RUTTEN has occupied himself in the Inorganic chemical Laboratory of the University of Leiden with the investigaiion ofilie system

according to the phasp rule. He also has, when studying the solid phases, subjected the observations of former investigators (HEINTZ,

GLADSTONE, BEcKER, JANSSEN, RUGE, YVON, LUDDECKE, D.rTTE and others) on the basIC nitrates and the so-called "Magisterium Bis­muthi" to a critical investigation.

His results wore as follows:

A. The solid phases.

1. The neutl'al salt Bt2 Os. 3 N2 05, 10 B 20 (in futura called briefly ZIO 1)). ThiS 1 formula accepted of late years has been found correct. The salt does not possess a truc melting point as formerly stated (72°), but it decomposes at 75°,5 into a liquid alld the basic salt B/2 Os. N2 0 5, B20 (BI-I-I)'

The prismatic, triclinic crystals exhibit an angle of oxtinction of 26°. Two further hydrates of thc neutral salt were discovered: Z4

and Zg. n. The neutml salt Za (with 3 Mols. of H20). It was obtained

at the ordinary temperature from ZJO, or from Bis Os by aildition of anhydrous nitrit' acid, in regular crystals as beautifully formed rhombic dodecahedrons. It should be mentioned that its composition eould not be determined directly, bccause it was not possible to separate the crystals completely from thc syrupy mother liquor. The composition was deduced by means of SCHREJNElIlAKERS' method of calculating, from the graphical cOllstruction in an equilateral tri~ angle of the compositions: 1 st of two different mother liquors whic,h were in equilibrium with crystals of Za, and 2nd of the crystals thcmselves with some of the mother liquor still adherillg. The same applies to the salts presently to be descl.'ibed Z4 and BI- 2- I , wbich alsa could not be separated from the adhering mother liquor.

lIl. The neutral salt Z4 (with 4 mols. of H20). A definite mode

1) In future the snlts which ('onmin 1 mol. of B/20a. 8 Mol. N200 nnd 10 or 4 or 3 Mol. H20 will be briefly called ZJo, Z4. Za; similnr1y the bnsic salts will be written Bna-ua-na if they contnin nl Mol. of B,2 0a, D2 Mol. N2 O, nnd Ua Mol. H:O