m

13 1 W’S iM i l f t KvOnderzoekingen over het Paramaguetisme

bij Lage Temperaturen.

L. C. JACKSON.

Universiteit Leiden

1 481 141 3

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RESEARCHES ON PARAMAGNETISM AT LOW TEMPERATURES.

Druk van Eduard IJdo. Leiden.

Onderzoekingen over het Paramagnetismebij Lage Temperaturen.PROEFSCHRIFT

TER VERKRIJGING VAN DEN GRAAD VAN

Doctor in de Wis- en NatuurkundeAAN DE RIJKSUNIVERSITEIT TE LEIDEN,

O r GEZAG VAN DEN RECTOR-MAGNIHCUS

D r . l v a n i t a l l i e ,

HOOGLEERAAR IN DE FACULTEIT DER WIS- EN NATUURKUNDE,

VOOR DE FACULTEIT TE VERDEDIGEN

M aandag 2 6 Maart 1923, des nam iddags te 3 uur,

DOOR

LEONARD CECIL JACKSON,GEBOREN TE NOTTINGHAM, ENGELAND.

LEIDEN. — EDUARD IJDO. — 1923.

f

AAN MIJNE OUDERS.

i

Bij het voltooien van mijn proefschrift grijp ik gaarne degelegenheid mijn warmen dank te betuigen aan allen, die totmijn wetenschappelijke vorming hebben medegewerkt.

In het bijzonder dank ik U, Hooggeleerde B arton, voor alleswat ik onder Uw leiding heb mogen leeren.

Hooggeleerde K amerlingh Onnes, Hooggeachte Promotor,V bovenal ben ik ten zeerste dankbaar voor de voortdurendebelangstelling en welwillendheid, die ik steeds van U mochtondervinden. He tijd, welke ik in Uw laboratorium hebmogen werkzaam zijn, behoort tot de aangenaamste van mijnstudietijd.

Hooggeleerde L orentz en E hreneeot, ik beschouw het alseen groot voorrecht Uw Colleges te hebben mogen volgen.

Aan U, Zeergeleerde Crommelin en W oltjer , ben ik veelverplicht voor Uwe hulp en belangstelling.

Het is mij een aangename plicht de “Royal Commissionersfor the Exhibition of 1851” dank te zeggen voor den finan-cieelen steun, dien zij mij tijdens de bewerking van ditproefschrift hebben verleend.

B H H H M B B W H

C O N T E N T S .

Page.In tro d u c tio n .................................................................................... 1

P A R T I.

M easurem ents on Pow dered M a te r ia ls ................................. 9

C H A P T E R I.

P revious W o r k .............................................................................. 11

C H A P T E R I I .

M ethod of M easurem ent and M a te r ia ls .................................. 17§ 1. M easurem ent of S u s c e p tib il i ty ......... 17§ 2. M easurem ent of F ie ld S tren g th ......... 18§ 3. M easurem ent and Control of the T em perature 19§ 4. M a te ria ls ..................................................... 20

C H A P T E R I I I .

Observations and R e s u l ts .................................................................... 21§ 1. Specim en O b se rv a tio n s ............................. 21§ 2. F in a l R e s u l t s .......................................... 22

I. Anhydrous Cobalt Sulphate ......................................... 22II. Cobalt Sulphate Heptahydrate.................................. 24

III. Cobalt Ammonium Sulphate.........................................26IV. Anhydrous Nickel Sulphate............................ ; 27V. Nickel Sulphate Heptahydrate.................................. 28

VI. Nickel Ammonium Sulphate . ............................ 29VII. Anhydrous Ferrous S u lp h a te .................................. 30

VIII. Ferrous Sulphate H ep tah y d ra te ............................ 31IX. Ferrous Ammonium Sulphate . . . . . . 33

C H A P T E R I V

Discussion of R e s u l t s ................................................................... 35§ 1. E xam ination fo r the L aw A m {T %) = C . 35§ 2. D eviations from the W eiss L a w ....... 36§ 3. Com parisons of R esults w ith E x is tin g Theories 38

Page.P A R T I I .

M easurem ents of th e P rin c ip a l S usceptib ilities ofC r y s t a l s ................................................................................................ 45

C H A P T E R I.

P rev ious W o r k ........................... 47

C H A P T E R I I .

M e t h o d ....................................... . . . 52§ 1. Theory of the M ethod of O bservation . . . 52§ 2. A p p ara tu s and M a te r ia ls ................................................. 58§ 3. M ethod of O b se rv a tio n .................................................... 64

C H A P T E R I I I .

O bservations an d R e s u lts ............................................................. 70§ 1. O bservations................................................................... 70

Calibration of Torsion W i r e s .............................................. 70Preliminary Test of A pparatus.............................................. 71Specimen Set of Observations.............................................. 72Final Observations.................................................................... 76Cobalt Ammonium Sulphate............................................. 77Nickel Sulphate H eptahydrate...................................... 78

§ 2. P in a l Results .......................................79Cobalt Ammonium Sulphate ....................................................79Nickel Sulphate Heptahydrate.............................................. 82

C H A P T E R IV .

Discussion of R e s u l t s ..........................................................................85§ 1. E xam ination for the Law % (T -j- A) = C . ■ 85§ 2. D eviations from the W eiss L a w ................................87§ 3. Connection of A w ith C rystal S truc tu re . . . 88§ 4. Com parison w ith E x isting Theories . . . . 91

S u m m a r y ................................................................................................95S t e l l i n g e n ...................................... 99

X

I N T R O D U C T I O N .

The investigation of paramagnetism, in particular as afunction of the temperature, is of great interest on accountof the valuable information which it can contribute to thestudy of two problems of outstanding importance in modernphysics viz: — the internal constitution of the atom and theenergy relationships of the atoms of a solid as functionsof the temperature. In both these problems we are broughtinto immediate contact with the present day developmentsof the quantum theory and it is of the greatest interest andimportance to determine in how far this theory as it existsat present can furnish an explanation of the phenomena ofparamagnetism and especially whether the quantum theorycan, firstly, explain the phenomena previously explained interms of the classical theories and secondly those for whichthe classical theory failed to give an explanation.

In connection with the first of these problems to whichthe study of paramagnetism can bring data of value, wemay note that, according to any of the theories of para­magnetism, the experimental results enable us to evaluatetwo quantities of especial interest: — the magnetic momentof the atom and the effect of the interaction of the atomicforces which may be summarised in the term “ molecularfield” introduced by W eiss. The evaluation of the atomicmagnetic moment leads directly to the conception of themagneton, the ultimate unit of magnetic moment, so largelydeveloped by W eiss. In the light of the later ideas of thestructure of the atom, the question must be raised as to theactual reality of the experimentally determined magnetonof Weiss. I s it necessary or desireable to replace the latter

1

2

by the magneton derived from the quantum theory accordingto B ohr’s conception of the constitution of the atom andwhich has a value almost exactly equal to five times thatof the W eiss magneton and, in doing so, accept the conceptionof the “ effective magneton number” introduced by P a u l i1)with such fruitful results in connection with the para­magnetism of gases ? Any ultimate theory of the constitutionof the atom must be capable of explaining the existence ofthe magnetic moment of paramagnetic atoms and of theapparent rationality in terms of the W eiss magneton and ofpredicting their actual values. A considerable step in thesolution of the first of these problems has been achievedby B ohr 2) in his more recent conception of the atom. Itwill thus be seen from the few examples mentioned abovethat the further study of paramagnetism may provide thetheoretical physicist with a powerful weapon for the attackon the problems of atomic structure.

The determination of the effect of the interaction of theatomic forces on the phenomenon of paramagnetism involvesthe consideration of several factors such as the structureof the atoms concerned, the their movements and hence thethermal properties of the substance and, in the case of cryst­alline materials, the arrangement of the atoms in the crystal.We have at present no precise knowledge of the nature ofthese interatomic forces. In introducing the conception ofthe molecular field into the theory of paramagnetism W eiss

specially pointed out that this field need not necessarily bemagnetic in nature and, indeed, the exceedingly greatmagnitudes demanded for it on the supposition that it isof magnetic character (of the order 107 gauss) suggest thatit is of some other nature. W eiss himself in a later paperbrings forward evidence that the molecular field is not ofmagnetic origin and also that it cannot be of an electrostaticcharacter. As regards the connection of the molecular field

1) Phys. Zeits. 21, 615, 1920.2) Zeits. f. Physik. i», 1 and 2, 1, 1922.

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as expressed by the A in the W eiss law % {T -{- A) = C l)with the arrangement of the atoms in crystals, it will beshown in Part II that the evaluation of A may probablyfurnish valuable evidence as to the actual structure of acrystal.

In all the theories of paramagnetism yet put forward themagnetisation can be expressed as a function of the quantityH/T (in which H is the strength of the applied magneticfield and T the absolute temperature). In the case ofLangevin’s theory which is founded on the classical electro­dynamics, the magnetisation is directly proportional to thisquantity, the susceptibility thus varying inversely as theabsolute temperature as given by the experimentally dis­covered law of Curie % T = constant. Since the expressionwas obtained by assuming the law of the equipartition ofenergy as required by the classical theory and on the assump­tion of the perfect freedom of rotation of the elementarymagnets2), it would not be expected that the theory wouldhold in the case of solids at the low temperatures at whichwe know that the equipartition law ceases to be valid. Thisis generally the case with the paramagnetic solids and allthe later theories have endeavoured to obtain a better pictureof the probable behaviour of these substances at low tempe­ratures. The deviations from the Curie law cannot, however,be explained simply by allowing for the fact that the thermalenergy of the substance is no longer directly proportionalto the temperature at the low temperatures as was done inthe early attempts of Oosterhuis 3 4) and Keesom *) to explainthe phenomena of paramagnetism at low temperatures. Thecurious fact remains however, that the susceptibility of

1) X — susceptibility, T = absolute temperature, A = constant of themolecular field, C = Curie constant.

2) Ehrenfest has shown (Leiden Comm. Rupp. 44t) that the Curielaw can be derived without the assumption of the rotational freedomof the elementary magnets.

3) Leiden Comm. Supp. 31.4) Leiden Comm. Supp. 32.

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gadolinium sulphate was found to be inversely proportionalto the absolute temperature down to the lowest temperatureat which investigations were made (2° K. in a bath of liquidhelium boiling under reduced pressure) although the thermalenergy of the atoms is far from being proportional to thetemperature at these extremely low temperatures. Similarlythe susceptibility of ferric alum was found to follow the lawof Curie down to the lowest temperature investigated (in thiscase about 14° K.). It would therefore appear that thereis some mechanism acting in the case of these two substancesby means of this strict inverse proprotionality to the tempe­rature is maintained in the regions in which we should expecta priori that the law would fail. It wTill be seen that thephenomena of paramagnetism raise many points of interestin the region of the temperature dependence of the thermalenergies of substances.

All the above considerations apply equally well to thecases of solid substances whether in the form of crystalsor in the amorphous state either naturally or artificiallyisotropic in the form of powders. The very great majorityof investigations on paramagnetism in general have beencarried out on substances in the form of powders in whichit is the mean of the principal susceptibilities of thecorresponding crystals that is measured. The investigationof the actual principal susceptibilities of crystals offershowever a field of work from which results of considerableadditional interest may be expected as bringing to lightphenomena which are hidden in the mean values.

In the study of the phenomena of paramagnetism thedomain of low temperatures immediately suggests itself asone in which investigation promises results of particularvalue. At these low temperatures obtainable with liquidhydrogen the magnitudes of the various phenomena are manytimes greater (15—20 times) than at ordinary temperatures,thus making it possible to investigate the phenomena withgreater precision without however increasing to any greatextent the difficulty of the measurements. Again the dis-

5

turbing action of traces of ferromagnetic impurities, thepresence of which it is very difficult to avoid completely,becomes proportionally less as the temperature is lowered.

Again, since the magnetisation is a function H/T, we mayregard this quantity as the “ effective magnetic field”. Ininvestigating the connection between the susceptibility andthe field strength (the susceptibilities of paramagneticsubstances are in the great majority of circumstances in­dependent of the field strength) we may reach higher valuesof the “ effective field strength” either by increasing H orby decreasing T. There is however a very definite limitbeyond which the magnitude of H cannot be increased byany means at present at our disposal, fields as great as50.000 gauss being obtainable only with difficulty and byspecial means. Yet with the greatest value of H obtainablein the laboratory the quantity a = MH/RT (M = magneticmoment per molecule, R = gas constant per molecule) ofLAngevin’s theory can, at atmospheric temperatures, betreated as a small quantity and saturation phenomena whichare predicted by the theory as ocurring for large valuesof a cannot be observed. However by lowering the tempe­rature the “ effective field strength” may be increasedconsiderably without correspondingly increasing the experi­mental difficulties. Thus an “ effective field” of 30.000 gaussat atmospheric temperatures may without further demandson the magnet be increased to 450.000 gauss by lowering thetemperature to about 20° K. (the boiling point of liquidhydrogen) and to 4.500.000 gauss by employing a bath ofliquid helium with which a temperature of 2° K. can beobtained.

Further, since at the low temperatures the heat motionsof the atoms are very much smaller than at atmospherictemperature, decreasing in reality much more rapidly thanthe temperature itself, the influence of the heat motions ofthe atoms on the phenomena of paramagnetism may beexpected to have decreased correspondingly, thus resultingin a simplification of the conditions. The phenomena shown

6

by paramagnetic substances at low temperatures are thus ofconsiderable interest in connection with the thermal energyof the atoms. In reality the phenomena occurring at theselow temperatures appear to be more complicated than thoseobtaining at higher temperatures as will be shown later.

The present work is intended as a contribution to thestudy of the paramagnetism of solids at low temperaturesin continuation of the previous studies in this domaignalready carried out at Leiden1).

The work naturally falls into two parts, the first dealingwith measurements of the susceptibilities of powdered sub­stances and the second in which measurements on crystalsare described. I t was decided to investigate the followingseries of compounds in the powder form v iz : — the anhydrousand heptahydrated sulphates and the ammonium doublesulphates of ferrous iron, nickel and cobalt. Of these compounds,data for the anhydrous and heptahydrated ferrous sulphateshave already been published2). They are considered herefor the sake of completion in the series. The choice of thesubstances was regulated by the following considerations.The substances form three series of compounds of preciselysimilar constitution for the three strongly ferromagneticelements which themselves form a special group in theperiodic system. I t would thus be expected that results ofinterest might be obtained in thus being able to determinethe effect on the general behaviour of these paramagneticsubstances as the paramagnetic atom is changed from ironthrough nickel to cobalt. Again the substances form threeseries of compounds of increasing “ magnetic dilution” (thesignificance of the term will be explained in the nextchapter). As a result of the previous work at Leiden it wasconsidered that the deviation from the law of Curie, at leastas concerns the relatively higher temperatures, was smallerthe greater the “ magnetic dilution” of the substance under

1) Leiden Comm. Nos. 116, 122, 124, 128, 182, 138, 140.2) Leiden Comm. Nos. 122, 124, 129, 139.

7

consideration. The above series of compounds was chosenwith a view to further testing the idea.

In Part II will be found an account of the measurementof the principal susceptibilities of two crystals, cobalt am­monium sulphate and nickel sulphate heptahydrate, downto the lowest temperature obtainable with liquid hydrogen.These measurements constitute the first data availableconcerning the principal susceptibilities of crystals at thesevery low temperatures. The substances were chosen asrepresentatives of different classes of crystals. Thus cobaltammonium sulphate belongs to the monoclinic system andhas a considerable “ magnetic dilution”. Nickel sulphateheptahydrate is a rhombic crystal; its “ magnetic dilution”is rather less than that of cobalt ammonium sulphate. Itwas chosen also partly on account of the interesting devia­tions from the W eiss law X (T + A) — C which were shownby its mean susceptibility as determined from measurementson the powdered material.

I desire to mention that some of the work here describedhad been previously undertaken at Leiden by Dr. K. Hof in1915—’16 but owing to most of the notes of the observationshaving been lost during his return to Switzerland and tothe fact that the measurements could not at that time bemade over any very extensive range of temperature, thework was entirely repeated by me. Where the notes are stillavailable, I have added the results obtained by Dr. Hof

to those obtained later so that a comparison may be madebetween the two. The apparatus for the measurement of thecrystals was made for Dr. Hof at Leiden and he carriedout a series of measurements with it on various crystalsbut unfortunately none of the notes containing the finalobservations and results are now available. The work wastherefore begun once more quite independently of theprevious work. I wish however to make my acknowledge­ments for the considerable assistence which such of the notesas still remained afforded during the early stages of the work.

PART I.

MEASUREMENTS ON POW DERED MATERIALS.

:__

C H A P T E R I.

PREVIOUS WORK.

A short summary will be given here of the principalresults of the measurements of the susceptibilities of para­magnetic substances at low temperatures dealing especiallywith the researches which have been carried out in the cryo­genic laboratory at Leiden. *)

From the theory of paramagnetism developed by Langevin * 2 3)we should expect that paramagnetic substances would obeythe law previously found experimentally by Cu rie*) viz: —that the product of the susceptibility and the absolute tempe­rature is constant or % T = C ; the introduction of the ideasof Weiss concerning the presence of a molecular field in solidsubstances would somewhat modify this result, leading insteadto the law % (T -\- A) — C in which A is a constant charact­eristic of the substance. These two laws have been much usedin guiding many investigators along the various lines possiblein the study of paramagnetism and the results obtained herewill be discussed in the light of them.

As a result of the investigations of P lessner4), Koenigs-berger5 6), Meslin*), H onda 7), W eiss and Foëx8), Ow en9),

•) Eor a collected account of the magnetic researches carried outin the Leiden laboratory see Leiden Comm. Sapp. No. 44.

2) L angevin , Ann. de chim. et de phys. 70, 1905.3) Curie, Oenvres, 276, 277.4) P lessner, Wied. Ann., 39, 386, 1889.5) K oknigsbrrger, Wied. Ann., 66, 698, 1898.6) Meslin, Ann. de chim. et de phys., (8), 7, 145, 1906.7) H onda, Ann. der Phys., 82, 1027, 1910.8) W eiss and F oëx, Tables Int. des Const, et Données, No. 1911.9) Owen, Ann. der Phys., 37, 657, 1912.

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H onda and Sonó1), I shtwara2), T heodorides 8) and Foëx4)among others, it has been shown that in general theCurie law does not hold for the case of solid substances hutthat the W eiss law is applicable over a range of some 100°below and 200° above 0° C. The graphs therefore of l/%against T consist of straight lines which do not in generalpass through the zero. In some cases3), *) it was found thatthe graphs consisted of straight lines but that slope of thelines changed suddenly. These results have been interpretedby the W eiss school as being due to a sudden change in thenumber of magnetons present in the substance. The actualresults obtained by the various investigators in so far as theyconcern the substances studied in the present work will befound in a later chapter together with the values obtainedas a result of the present measurements.

The work of the above mentioned investigators has gene­rally been carried out at temperatures higher than atmos­pheric and in no case have the temperatures been lower thanthose obtainable by the use of liquid air. I t was therefore ofthe greatest interest to determine whether the phenomenashown by paramagnetic substances at the higher temperaturesalso occur at the lowest temperatures obtainable. With thisend in view a number of solid substances in the form ofpowder and gaseous, liquid and solid oxygen have beenmeasured in Leiden5). As has been mentioned in the intro­duction it was found that two of the substances measured,gadolinium sulphate and ferric alum, followed the Curie lawdown to the lowest temperatures investigated. In no other casewas it found that the Curie law was rigorously obeyed overany extended range of temperatures. As examples of sub­stances showing the various types of deviation from the law

1) H onda and Soné, Sci. Rep. 8, 139, 1914.2) I shi war a, Sci. Rep., 8, 303, 1914.3) T heodoridks, These Zurich 1921, Journ. de Phys., 8, 1, 1922.4) FoRx, These Strasbourg 1921, Ann. de physique, 16, 174, 1921.5) Leiden Comm. Nos. 116, 122, 124, 129, 132, 134, 139, 140.

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of Curie we may mention liquid oxygen, solid oxygen, anhy­drous and hydrated manganese sulphates and anhydrous andhydrated ferrous sulphates.

The study of liquid oxygen, both when pure and whenmixed with known amounts of liquid nitrogen, brought tolight the fact that the results could be explained by thepresence of a negative molecular field proportional to theconcentration of the paramagnetic substance. Thus the obser­vations could all be represented by the law % ( T -f- A ) = Cin which the constant A has different values dependent onthe concentration and hence on the mean distance apart ofthe paramagnetic molecules.

Similarly solid oxygen also followed the Weiss law butshowed a very definite discontinuity in the value of thesusceptibility at a temperature of about 32° K.

The cases of the anhydrous and hydrated manganese sul­phates are instructive. The experiments showed that thehydrated salt, MnS04. 4H20, follows the Curie law verynearly down to about 65° K. but at lower temperaturesdeviates therefrom in such a manner that the susceptibilityincreases less rapidly with fall in temperature than is givenby this law. On the other hand anhydrous manganese sulphateobeys the Weiss law with a A equal to 24 down to nitrogentemperatures but at lower temperatures deviates from thelaw in the same manner as the hydrated salt but to a muchgreater degree. Thus the susceptibility becomes practicallyindependent of the temperature for temperatures in theneighbourhood of 15° K.

In the case of hydrated ferrous sulphate, Fe S04. 7H20,we have an example of a substance which follows the Weisslaw with a small A (= 1 ) dwon to the lowest temperatureinvestigated (14° K.). Anhydrous ferrous sulphate on theother hand obeys the Weiss law with a A equal to 31 downto nitrogen temperatures, while at lower temperatures adistinct maximum of susceptibility was found. At the timeit could not be determined whether the change in susceptibilityin this region was gradual or sudden since the region of

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the exact maximum can only be reached with the aidof a hydrogen vapour cryostat, the use of which in themagnetic measurements presents great difficulties. In thelight of the results announced in the present work it becomesprobable that the change is a gradual one, perhaps as shownin the graph fig. 3 on page 32.

The work so far published from the Leiden laboratorywould thus indicate the possibility of the existence of sub­stances

(1) which obey the Curee law down to the lowest tempe­ratures obtainable,

(2) which obey the Weiss law with A varying over a widerange,

(3) which show deviations from the Weiss law at lowesttemperatures such that the susceptibility increases lessrapidly with the fall in temperature than is given bythis law 1 2),

(4) which show a maximum of susceptibility at low tempe­ratures.

Another point which arises from these investigations is thedependence of A on the various factors which may conceivablyinfluence it. A is, of course, independent of the temperaturebut, as the measurements showed is, at least in the case ofliquid oxygen-nitrogen mixtures, very nearly proportional tothe concentration of the paramagnetic substance present.Some such dependence was to be expected, since on Weiss’stheory A expresses the influence of the molecular field onthe behaviour of the paramagnetic substance and indeed onthis theory A should be equal to CN pn 2) in which n isprobably equal to 1. In this connection it was assumed (andthe measurements seemed to justify the assumption) that theliquid nitrogen (which is weakly diamagnetic) acts only as

1) The term cryomaguetic anomaly has been given to this pheno­menon by Kamerlingh Onnes. See Rap. Conseil Solvay held at Brus-sells Apr. 1921 and Leiden Comm. Sapp. 44.

2) c = Curie constant, N = coefficient of molecular field, p= density .

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a diluent and exerts no direct action on the behaviour ofliquid oxygen.

This idea was extended to the case of solid substances byK amerlingh Onnes and Oooterhuis ') who suggested a con­nection between the A and the distance apart of the para­magnetic atoms in substance. If thus we can consider thatin a substance, such as MnS04. 4H20 for example, the partof the molecule other than the paramagnetic atom Mn actsonly as a diluent and is without direct influence on theparamagnetic part of the molecule (the contribution of theanion and the water of crystallisation to the actually measuredsusceptibility can be allowed for and is, of course, not meanthere). Thus the larger the rest of the molecule comparedwith the paramagnetic part, the greater the average distanceapart of the paramagnetic atoms will be. We may thusintroduce the conception of the “ magnetic dilution” of asubstance, a substance being the more “ magnetically dilute”the larger the “ inert” portion of the molecule. This conceptionholds of course only for those substances in which themolecule contains a weakly diamagnetic part such as waterof crystallisation or the anion SOit Cl etc. and not for suchsubstances as oxides. It was suggested that the greater' the‘ ‘ magnetic dilution ” of a substance the smaller the A will be.This was supported by examining the cases of anhydrous andhydrated manganese sulphate, in which A = 24 and 0 respect­ively and-anhydrous and hydrated ferrous sulphate, in whichA — 31 and 1 respectively. In each case the substances whichhave the smaller A are those in which the “ magnetic dilution”is the greater, the hydrated compounds being considered moredilute than the corresponding anhydrous substances on accountof the presence of the water of crystallisation. Again ferricalum Fe2(S04)3. (NH4)2S04. 24H20, which may be consideredas very dilute magnetically, has a value of A too small formeasurement.

1) Leiden Comm. No. 139.

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It will be seen from the present work that whilst thisrelation holds in the case of the cobalt sulphates, it requiressome modification in the cases of the nickel sulphates andthe double sulphates.

All the substances for which investigations have yet beenpublished for very low temperatures have been found toposses a positive x) value of A if they obey the W eiss law. Thework of other investigators at higher temperatures has broughtto light the existence of substances for which the A ’s havenegative values 1 2). Should these substances continue to followthe W eiss law down to the lowest temperatures, then weshould expect that the substances would become ferromagneticbelow the temperature corresponding to the value of A. Inthe present work an example of a substance with a negativevalue of A is to be found in nickel sulphate heptahydrate.This substance does not however become ferromagnetic belowtemperatures below that corresponding to A (-59) since itshowed very considerable deviations from the W eiss law attemperatures below about 160° K., the susceptibility remainingbelow the value given by the above law.

1) An investigation of the anhydrous chlorides of nickel and cobaltby K amerlinöh Onnes and W oltjer (and yet unpublished) has shownthat the A’s for these substances are negative but that the substancesshow the usual cryomagnetic anomaly and hence do not definitelybecome ferro magnetic at the lowest temperatures. See Leiden Comm.Supp. 44, footnote 3, p. 25.

2) W eiss and FoBx, Arch. Sc. Phys. et Nat., (4), 31, 4 and 89, 1911.

C H A P T E R I I .

METHOD OF MEASUREMENT AND MATERIALS.

§ 1. Measurement of Susceptibility.

The method adopted for the measurements of the suscepti­bilities of the powdered substances was that which may betermed the rod. method. In this method a rod of the substanceto be investigated is placed in the field of an electromagnetin such a manner that one end of the rod is situated in astrong magnetic field (that on the axis of the interferrumof the magnet) while the other end is outside the field orin only a very weak field. A force then acts on the rodgiven by the expression

in which F = Force on rod,% = Susceptibility per unit mass,m = Mass of substance,I — Length of rod,

Hx, H% = Fields at ends of rod.

The actual apparatus used in the measurements has beendescribed in detail in Leiden Comm. Nos. i39a an(j 139^ tjjeformer giving the description of the apparatus and methodof working and the latter the modification of the apparatusintroduced by Oooterhuis. A very brief description will there­fore suffice here. The powdered material is contained in a

2

18

cylindrical glass tube of uniform cross section, the lower halfof which is filled with the substance, while the upper halfis evacuated or contains helium at a low pressure, the latterbeing introduced to assist in the unification of the tempera­ture over the specimen. To the upper end of the tube isattached a glass rod, this in turn being fastened to a systemfloating in mercury in such a way that the upper end ofthe cylinder of powder is situated on the axis of the poles ofthe magnet. The floating system carries a scale-pan on whichweights can be put. The values of F, H 1 and H 2 in theexpression given above are then determined as follows. Aknown weight is placed on the scale-pan and the currentthrough the magnet coils is then switched on and increaseduntil the floating system, owing to the upward force actingon it due to the magnetic field, is brought back into theposition it occupied in the absence of the field before theweight had been placed on the scale-pan. The values of Fand H 1 (and H 2) are then given immediately by the weightadded and the current through the magnet coils respectively.When ™aking the measurements at low temperatures the tubecontaining the powder is of course surrounded by a Dewarvessel containing a suitable liquified gas.

§ 2. Measurement of Field Strength.

The electromagnet used in the present work was a smallsize W eiss magnet. It was mounted on a turn-table so thatit could be turned into any position about a vertical axis aswas required in making the measurements on the crystalsdescribed later. The positions of the magnet could be read ona circular scale divided into degrees and attached to themagnet proper. The field strengths on the axis of the polepieces, i.e. the H1 of the above formula, had previously beendetermined with great care for a large number of currentstrengths by both the ballistic method and that of the magneticbalance of Cotton. A table had then been prepared from theobservations giving the field strength on the axis for the parti-

19

cular pole distance used (16.8 mms.) as a function of thecurrent in the magnet coils. Further the ratio of the strengthof the field at any point in an equatorial plane to that onthe axis of the magnet had also been determined thus givingthe value of H2. In general, with the lengths of tube usedin the present work the value of H* was small comparedwith H| and could be neglected or introduced as a smallcorrection. In the present work the values of H1 were deter­mined by reading the value of the current in the magnet coilsand referring to the table mentioned above.

§ 3: Measurement and Control of the Temperature.

The measurements of the atmospheric temperatures weremade with a mercury thermometer which had been comparedwith a standard mercury thermometer calibrated at the Physi-kalisch Technische Reichsanstalt. The lower temperatures werecontrolled and measured as follows. The temperatures wereobtained by means of baths of the various liquified gasesboiling under constant known pressures. For temperatureslower than the boiling point of any of the liquified gases,the pressure under which the gas was boiling was reducedby means of a pump and could be kept constant by meansof a valve. The pressure was determined by the aid of anopen mercury manometer, an auxiliary oil manometer beingemployed to show any small changes in the pressures. Thisoil manometer could be maintained at a definite pressure towithin a few millimeters, thus ensuring that the liquified gas,which was always kept well stirred by moving the floatingsystem up and down, was boiling at a temperature which didnot vary by so much as one tenth of a degree absolute. Thetemperatures were obtained by measuring the pressure underwhich the liquified gases were boiling and then referring tothe vapour pressure curves previously obtained by measure­ments in the laboratory. The temperatures were thus knownto within one tenth of a degree, which was sufficientlyaccurate for the present purpose.

§ 4. Materials.

The substances used were, with the exception of cobaltammonium sulphate1), Merck’s guaranteed pure reagents. Theheptahydrated sulphates and the double sulphates wererecrystallised before use, dried on filter paper and powdered.The anhydrous sulphates were prepared from Merck’s puresamples of the corresponding heptahydrated sulphates bycareful heating until all the water of crystallisation had beendriven out as shown by the constancy of the weight of thematerial and the agreement of the loss in weight with thatcorresponding to the loss of seven molecules of water.

20

1) Supplied, by H. T rommsdohff, Chemische Eabrik, Erfurt.

C H A P T E R I I I .

OBSERVATIONS AND RESULTS.

§ 1. Specimen Observations.

The following set of observations may serve to show thenature of the results obtained in the present investigation.

Cobalt Sulphate Heptahydrate.Mass of substance used = 5.565 grms.Length of column of substance = 8.85 cms.

Observations with bath of liquid nitrogen at 764.5 mms.pressure.

Hence temperature = 77°.3 K.

I. Mass in Scale-Pan = 3.1624 grmsCurrent in magnet coils to balance above mass =

9.61 amps.Field Strength = 8846 gauss. ■Susceptibility = 126.1 X 10—6.

II. Mass in Scale-Pan = 4.4877 grms.Current in magnet coils to balance above mass =

11.41 amps.Field Strength = 10490 gauss.Susceptibility = 127.2 X 10—6.

III. Mass in Scale-Pan = 4.4460 grms.Current in magnet coils to balance above mass =

11.38 amps.Field Strength = 10463 gauss.Susceptibility = 126.7 X 10—6.

22

IV. Mass in Scale-Pan = 6.5762 grms.Current in magnet coils to balance above mass -

14.16 amps.Field Strength = 12700 gauss.Susceptibility = 127.2 X lO "6.

V. Mass in Scale-Pan = 7.6501 grms.Current in magnet coils to balance above mass =

15.81 amps.Field Strength = 13701 gauss.Susceptibility = 127.1 X 10—6.

Mean Susceptibility = 126.9 X 10- 6 .Range of Field Strengths = 8846 — 13701 gauss.

§ 2. Final Results.

I. Anhydrous Cobalt Sulphate.

The results of the measurements on anhydrous cobaltsulphate are given in the table below.

T % X 106 Xm( T + 44.9)

289?5 62.2, 0.00964 103., 3.224

286.6 62.78 0.00978 102., 3.224

169.66 96.90 0.0150, 66.5] 3.22,

77.27 169., 0.0263, 38.0, 3.21,

64.95 194., 0.0300, 33.2, 3.30,

20.45 378 .8 0.0687, 17.0, 3.84,

17.44 420 ., 0.0652, 15.3, 4.06,

14.77 461.„ 0 07160 13.9, 4.27,

In the above table % denotes the specific susceptibility,%m the molecular susceptibility, T the absolute temperature.The values of the inverses of the molecular susceptibilities

23

given in column 4 are plotted against the absolute temperaturein graph 1 fig. 1. The numbers in column 5 are intendedas a test for the Weiss law Xm (T + A) = C 1) and will bediscussed in the next section.

The values of 1 / %m are plotted against T in graph. 1, fig. 1.From the above data we may calculate the number of

magnetons present in the substance from the expressionsgiven by Weiss viz:

n = <rm l 1123.5 <r2m = 3%m (T + A ) . B = S CR

in which n = magneton number, am = magnetic moment pergrm. mol., R = gas constant per grm. mol. We then findfor n the value 25.2.

We may now compare the above given values with thedata already to be found in the literature. This substancehas been measured by Ishiwara (loc. cit.) from atmospherictemperatures down to about 95° K. and by Theodorides(loc. cit.) from atmospheric temperatures up to about500° C. The value for the susceptibility at 25° C. found byTheodorides is 62.25 X 10—6 while the data given byIshiwara are to be found below.

t) Before comparing the experimental results with, any theory ofparamagnetism, a correction must be applied te them on account ofthe diamagnetic properties of the anion, the water of crystallisationetc. The correction is however negligibly small in the present caseand has been left out of account. I t is applied to all the resultsgiven later for the cases in which it has an appreciable value. Thedata previously published from Leiden have not been so corrected whencalculating %m(T + A) and Cabrera (Ann. Fis. Quim. 13, 2B6, 1915)has drawn attention to this fact in the passage, „En éstos, como entodos los demas cueros estudiados por los fisicos holandeses, laconstante de Curie se ha calculado para la molécula completa >proceder que no esta de acuerdo con la teoria, principalmente en lassales hidratadas, para las cuales la corrección por diamagnetismo delanión y el agua llegan a exceder del 1 %". This is quite true but itmay he pointed out that in calculating the magneton numbers fromthe Leiden observations, the correction for the diamagnetism of theanion etc. has been applied.

24

T X X 106 T X x 106 T X X 106

22.4 C. 59.6 - 69.2 85.1 — 145.8 127.3— 1.1 65.1 — 86.6 91.6 — 161.9 145.1- 22.0 70.8 — 106.8 100.6 - 179.3 170.3— 45.4 77.4 — 127.2 113.2

The value given by Theodorides for atmospheric tempera­tures is somewhat higher than that found in the presentwork but the present values are in good agreement withthe data of Ishiwara except for his lowest temperatures.

II. Cobalt Sulphate I I eptahydrate.The values of the susceptibility of this substance found

in the present work are given in the table below.

T X x 1 0 6 Xm, Xm 1/Xm Aim ( 7 * + 1 3 .7 )----------------------- 1

292.0 37.0 0.01040 O.OlO50 95.2, 3.210 \291.4 36.9 0.0103, 0.0104, 95.4, 3.196 [ 3.193288.2 37.0 0.0104, 0.0105, 95.1, 3.172 )169.5 61.6 0.0173, 0.0174, 57.8, 3.193 3.19377.48 126.5 0.0355 0.0356 28.0 3-252 177.3 126.9 0.0356, 0.0357, 27.9, 3.256 ] 3,26463.85 146.2 0.0411„ 0.0412o 24.2,

3195 ̂ 317863.8 144.6 0.0406, 0.0407, 24.6, 3.160 i 317820.4 403.0 0.113, 0.113„ 8.81, 3'779 I 378120.32 404.4 0.113, 0.113, 8.78, 3.783 ) 8-7817.47 464.4 0.130, 0.130, 7.65, 4.084 4.08414.76

'635.9 0.150, 0.160, 6.63, 4.292 4.292

In the above table denotes the molecular susceptibilitycorrected for the diamagnetism of the anion and the waterof crystallisation. The values of the corrections used werethose derived principally from the work of Pascal. Thevalues required for the present case will be found collected

25

together in a paper by W eiss and Bruins (Proc. Acad.Amst., 18, 246, 1915). It must be noted however that thevalues of the corrections mentioned above were determinedat atmospheric temperatures but that they are here appliedto all the readings at whatever temperature. This is probablyquite justifiable as the susceptibilities of diamagnetic sub­stances are in general independent of the temperaturealthough exceptions to this rule are known. It cannot thusbe stated with certainty that the values of the correctionsdetermined at atmospheric temperatures will be valid at thelower temperatures but, in the present state of our know­ledge, the best thing we can do is to assume that they arevalid.

300 T 350 °KPig. 1.

The values of 1/%̂ , are plotted against T in graph. 2, fig. I.

26

The magneton number calculated from the above data isfound to be 25.04.

In the following table are given the observations madeon cobalt sulphate heptahydrate by Dr. Hof.

T X X 10® X m

290°.0 37.0, 0.0104,20°.33 400., 0.112,1B°.5 478., 0.134„14°.8 540.0 0.151,

The above data may be compared with the results foundby previous investigators.

Meslin (1906) == 34.5 X Id- 6 (atmospheric temperature— value not stated).

Weiss and Foëx (1911) = 31 .9 X ld ” 6 (2d° C.).

III. Cobalt Ammowium Sulphate.

The observations on cobalt ammonium sulphate are givenin the table below.

T X x 106 X m Xm 1 / X m X m { T + 22)

290°.8 24.61 0.0097, 0.0099, 100., 3.10077.28 78.4 0.0309, 0.0311, 32.0, 3.09520.3 241 0.0953, 0.0956, 10.4, 4.04016.71 287 0.113, 0.113, 8.77, 4.40914.78 321 0.127, 0127, 7.85, 4.682

The values of 1 /%£ are plotted against T in graph 3, fig. 1.The value of the magneton number calculated from the

above data is 24.75.

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IV. Anhydrous Nickel Sulphate.

The measurements on anhydrous nickel sulphate gave theresults to be found in the following table.

T X X 1 0 ® 1 l%m + 79.4)

289.0 26.7 0.00413 242.0 1.625169.2 39.9 0 .0061, 161., 1.53677.4 62.8 0.0097, 102. , 1.52420.33 81.7 0 .01264 79.1 1.26120.33 82.1 0.01270 78.7 1.26716.77 79.7 0 .0123, 81.1 1.18616.63 79.6 0 .0123, 81.2 1.18314.7 80.6 0 .0124, 80.2 1.17314.6 80.9 0 .0125, 79.9 1.175

The values of V Xm are plotted against T in graph 1, fig. 2.The value of the magneton number calculated from the

above data is 16.9. To the above results may be added thoseobtained by Dr. Hof as below.

T X 'Wïi

290.0 25.8 0.0039f,20.33 82.1 0.0127»14.8 82.1 0.01270

Anhydrous nickel sulphate has been measured by Ishiwarawith the following results.

T X X 1 0 ® T X X 1 0 ® T X x 10®

24.3 C 27.1 — 52.5 34.8 — 122.6 46.36.1 28.5 — 72.7 38.3 — 144.1 52.1

— 11.7 30.5 — 86.7 39.9 — 162.7 58.0- 27.5 32.0 — 105.8 42.7 — 182.8 68.4

28

The values of Ishtwara are throughout somewhat higherthan the present observations but the dependence on tempe­rature is exactly the same in the two cases in the range oftemperature common to both.

I t may also be mentioned that the following values havebeen found by various investigators as a result of measure­ments on solutions of nickel sulphate.

Quincke1) 29.9 X 1 9 - 6 (19° C.).J aeger and Me y er2) 27.9X19—6 (17°.5 C.).Koenigsberger 3) 30.5X19—6 (22° C.).L iebknecht and W il e s4) 27.0 X 10—'0 (18° C.).

V. Nickel Sulphate Heptahydrate.

The results of the measurements on nickel sulphate hepta­hydrate are given in the following table.

T X x 1 0 6 %m %m 1 Ix'm Xm{T— 59 )

292?2 16.0 0.0045 0.0046 218 1.073169.6 34.8 0.0097, 0.0098, 102., 1.09277.29 77.3 0.0217 0.0218 45.9 0.89964.65 91.6 0.0257 0.0268 38.8 0.14620.33 222 0.0624 0.0625 16.0 — 2.41716.65 270 0.0757, 0.0758, 13.2 — 3.21214.6 294 0.0826 0.0827 12.1 ^ 3.672

The values of 1/%m are plotted against T in graph 2, fig. 2.The value of the magneton number calculated from the

data is 14.62.The following table gives the values found by Dr. H of.

1) Wied. Ann., 24, 347, 1885.2) Wien. Ber., 106, 594 and 623, 1897; 107, 5, 1898.3) Wied. Ann., 66, 698, 1898.4) Ann. der Phys., (4), 1, 178, 1900.

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T X X 10® %m

290° 0 15.8» 0.0044,20°.33 218 0.0612o15°.17 286., 0.0804,14°.8 294., 0.0825,

Nickel sulphate heptahydrate has been measured by Meslin(loc. cit.) who found the following value: — 17.0X10—6 atatmospheric temperature (precise value of temperature notstated).

300 1 " 350 °KVI. Nickel Ammonium Sulphate.The results of the measurements on nickel ammonium

sulphate are given the following table.

30

T X x .10® X m Xm 1 / X m

10.6, 0.0042, 0.00440 227., 1.273

169.5 18.0, 0.0071, 0.0073, 137.o 1.270

77.16 39.5, 0.0156, 0.0158o 63.2, 1.282

64.6 48.0, 0.0189, 0.0191, 52.1, 1.314

20.3 150.* O 0594, 0.05960 16.7, 1.448 1

16.8 179., 0.0707, 0.0709, 14.0, 1.476

j 14.5 207., 0.0819, 0.0821* 12.1, 1.520,

m e values OI / A>m arc l«uiw;u ■» ----1 °The value of the magneton number calculated from the

above data is 15.9.

VII. Anhydrous Ferrous Sulphate.

The results of the measurements on this substance havebeen already published (see Leiden Comm. N°. 1295) hut aregiven for the sake of completion of the series.

Xm{ T + Z 1)x x

290.2169.6

77.870.464.820.117.814.4

67.6107.,200.*215.,227.s402379335

Xm 1 / X j t

0.0102,0.0163,0.0804,0.0326,0.0345*0.0610,9.0575,0.0509o

97.3,61.4032.8*30.6028.9,16.3,17.3,19.6,

3.2993.2678.2983.3473.3093.1212.8102.311

The values of 1/ X m are plotted against T in graph 1, fig. 3.The value of the magneton number calculated from the

above is 25.5,.The results given above may again be compared with the

values found by other investigators. Thus we find thefollowing values given for the susceptibility at atmospherictemperatures as derived from measurements on solutions offerrous sulphate.

31

Quincke (loc. cit.) 75.8 X 10"~6 (19° C.).J aeger and Meyer (loc. cit.) 91.0 X 6 (17°. 5 C.).Koenigsberger (loc. cit.) 69 .9X 10—6 (22° C.).Liebknecht and W ills (loc.cit.) 80.4X 10—6(18°. C.).Townsend1) 70 .1X 10—6 (10° C.).Cabrera, Moles and Marquina 2) 78.2 X 10—6 (22°.4 C.).

Ishiwara has measured anhydrous ferrous sulphate in thepowder form with the following results.

T X X 106 T X X 106 T X x 106

33?3 C. 71.0 — 563 97.5 — 146.1 153.98.5 76.9 — 78.4 107.8 -1 6 2 .8 177.0

— 17.0 83.8 — 102.2 120.1— 41.0 92.5 —128.8 139.5

VIII. Ferrous Sulphate Heptahydrate.

Data for this substance have also been published fromLeiden (see Leiden Comm. N°. 122a). They are to be foundin the table below.

T X X 106 £V ƒAm l/* m {T + 1)

289.6 41.4, 0.0115, 0.0116, 85.9, 3.38077.45 154., 0.0429, 0.0430, 23.2, 3.37864.24 186.! 0.0517, 0.0518, 19.2, 3.38220.33 655., 0.164, 0.154, 6.47, 3.29518.73 589., 0.163, 0.164o 6.09, 3.23617.01 641., 0.178, 0.178, 5.60, 3.21415.47 695., 0.193, 0.193, 5.16, 3.18618.93 757., 0.210, 0.210, 4.74, 3.144

1The values of 1/Xm are plotted against T in graph 2, fig. 3.The value of the magneton number calculated from the

above data is 25.8*.

■*) Proc. Roy. Soc. 60, 186, 1896.‘0 Ann. Pis. Qnim., 18, 256, 1915.

32

Ferrous sulphate heptahydrate has also been investigatedby K oenigsberger, Meslin and I shiwara. The values foundare given below.

K oenigsbergek 34.5 X 10—6 (22° C.).Meslin 46.2X10—6 (temperature not stated).

The data given by Ishiwara are to be found in the tablebelow.

T X x 106 T X X 106 T X X 106

23°0 C. 41.9 — 42.0 63.6 — 120.4 81.812.6 43.0 - 63.6 66.8 — 143.4 96.12.3 44.5 -1 66.1 61.2 — 159.6 108.6

- 7.6 46.3 — 97.4 | 72.0 — 178.6 122.7— 31.7 60.4 — 109.7 76.6

nr.FeS04 (n h 4] 2s c ,

-II.reS 0 4 7 H20------

I.Fe 5 0 4.

Fig. 3.

33

IX. Ferrous Ammonium Sulphate.The results of the measurements on ferrous ammonium

sulphate are given in the table below.

T X X 106 Kn 1/%'m %m( T + 3)

290?3 32.6, 0.0127, 0.0129, 77.2, 3.79877.1 121 6 0.04768 0.0478, 20.8, 3.83320.3 413., 0.1620 0.162, 6.16, 3.77916.74 488., 0.191, 0.191, 5.22o 3.78214.5 547.o 0.214, 0.214, 4.66, &757

The value of the magneton number calculated from theabove data is 27.5.

To the above measurements may be added unpublishedobservations on ferrous ammonium sulphate made byDr. Oosterhuis at Leiden. They are given in the followingtable.

T X X 106 %'m 1/%'m Xi'm{T ■+• 3)

290?0 31.9 00126, 0.0126, 78.8o 3.718169.0 55.5 0.0217, ■ 0.0219, 45.5, 3.774

77.2 121., 0.0476, 0.0478, 20.9, 3.83664.6 144.„ 0.0564, 0.0566, 17.6, 3.82920.2 419., 0.164, 0.0164, 6.08o 3.81614.7 549.0 0.215, 0.215, 4.64, 3.814

Both the above sets of observations are plotted in graph 3,fig. 3.

I t will be seen that the present values (circles) and thoseof Oosterhuis (triangles) are in very good agreement withthe exception of the observations at atmospheric temperature,for which, as seen from the graph, the value in the first tablewould seem to be the more nearly correct.

Ferrous ammonium sulphate has also been investigated byF inke1), who determined the principal susceptibilities of a

i) Ann. der Phys., 81, 149, 1910.3

34

crystal of the substance and by Fóëx *), who carried outmeasurements on the powdered substance.

For the principal susceptibilities (per unit volume) atatmospheric temperature, F inke gives the following valuesx, = 84-32 x 10-6 f „2 = 78-74 x Id"6 , *3 = 76’20 x 10"6.

Calculating the specific susceptibilities (density = 1.899)and taking the mean we find % = 42.0 X 10- 6 .

Foëx gives the following two series of observations ofthe molecular susceptibility corrected for the diamagneticproperties of the molecule. .

T %m{T- 22)

288?9 0.01278 3.411275.9 0.01346 3.417223.4 0.01695 3.414200.5 0.01903 3.397

88.0 0.0529 4.6686 0.0693 5.96

283.1 0.01283 3.414

290.7 0.01271 —89 0.0404 -88 0.0420 —87 0.0432 —86 >0.0480 —

! Although the values for atmospheric temperature are ingood agreement with the present observations, the values atlower temperatures are very different in the first series.Again, in the present measurements no trace was found ofgreat deviation from the Weiss law at the lower temperaturesas found by Foëx, neither was the A negative. It may beobserved that of the two series of measurements by Foëx,which differ from each other by a very considerable amount,the present observations agree better with the second thanwith the first.

1) Thèse, S trasbourg 1921, Ann. de P h y s , (9), 16, 174, 1921.

C H A P T E R IV.

DISCUSSION OF RESULTS.

§ 1. Examination for the law %m (T -+- A) = C.

In the tables of the results given in the previous chapterwill be found columns giving the values of Xm (T -1-A), inwhich A is a contant characteristic of the substance. I t willbe seen that in all cases but that of nickel sulphate hypta-hydrate the values in these columns are constant to withinthe experimental error down to the temperatures obtainablewith liquid nitrogen (i. e. about 70° K.). Hence it maybe stated that these substances follow the W eiss law Zm(T A) = C over this range of temperatures. In the case offerrous ammonium sulphate it will be seen that this law holdswith the same accuracy down to still lower temperatures andindeed to the lowest temperature investigated. This is broughtout by an examination of the graph given in fig. 3. Thisgraph shows that the observations for this substance fall ona straight line when */ Xm is plotted against T.

It may also be noted that with the exception of nickelsulphate heptahydrate all the substances, for which A is notzero, have a positive value of A. The value of A for nickelsulphate heptahydrate is however negative (-59). This sub­stance is therefore of considerable interest since according toW eiss’s theory it should become ferromagnetic at temperaturesbelow that corresponding to A (Curie point) provided thatthe W eiss law is obeyed down to this temperature. It was,however, found that this substance did not become ferro­magnetic at the lowest temperatures since it ceased to obeythe W eiss law, showing very large and interesting deviationstherefrom.

36

On examining the present data in the light of the suggestionof K amerlingh Onnes and Oosterhuis as to the possibleconnection between ‘ ‘ magnetic dilution and the value ofthe A in the W eiss law, we find that they are not altogetherin agreement with this idea. Taking first the case of thecobalt compounds, we see that the A’s for CoS04, CoS04. 7H20and CoS04. (NH4)2S04. 6H20 are 44.9, 13.7 and 22 respect­ively. Thus while it is true that for CoS04 and CoS04. 7H20the A is smaller in the substance with the greater magneticdilution, the value of A for CoS04. (NH4)2S04. 6H20 isgreater than that for CoS04. 7H20 although one would expectthat the former substance could be considered as the moredilute magnetically since it is derived from the latter by thesubstitution of one molecule of ammonium sulphate for onemolecule of water of crystallisation.

In the case of the nickel compounds we find that the A’shave the following values:— NiS04, 79.4, NiS04. 7H20, 59,NiS04. (NH4)2S04. 6H20,4. The heptahydrated sulphate hereappears to be exceptional.

For the ferrous compounds we find the values FeS04,31,FeS04. 7H20,1, F eS 0 4 . (NH4)2S04. 6H20,3. In the case ofthe two sulphates the A is, of course, the smaller for thesubstance of the greater magnetic dilution since this was oneof the cases considered by K amerlingh Onnes and Oosterhuis

in making the suggestion referred to above but here againthe double sulphate has a larger A than the heptahydratedsulphate. The values of A are, however, very small and hencethey are not determinable with any great accuracy from theobservations.

I t would thus seem that while the values of A are frequentlygreater the smaller the magnetic dilution of the substance,more especially in the case of the hydrated and the corres­ponding anhydrous compounds, this rule does not always hold.

§ 2. Deviations from the Weiss law.

An examination of the tables and the graphs will makeit immediately obvious that, with the exception of ferrous

37

ammonium sulphate, all the substances show deviations fromthe Weiss law at low temperatures and that these deviationsare of various types.

We will now examine in detail the deviations from the lawXm (T -|- A) = C which occur at low temperatures, consideringfirst the double sulphates, then the heptahydrated sulphatesand finally the anhydrous sulphates, this order being chosenhere as then the type of deviation becomes progressivelymore complex.

Considering first the double sulphates and referring to thetables and the graphs, we see that the deviations from theWeiss law are such that the susceptibility increases morerapidly with fall in temperature than is given by this law.The deviation is considerable in the case of cobalt ammoniumsulphate, less with nickel ammonium sulphate and too smallfor measurement with ferrous ammonium sulphate. I t willbe seen that at about 14° K. the susceptibility of cobaltammonium sulphate is 1.4 times greater than the valuecalculated for this temperature from the law Xm (T 22) = C.This type of deviation from the Weiss law is thus the exactcounterpart of the type previously found, for example, withanhydrous manganese sulphate, and now known as the cryo-magnetic anomaly.

Turning now to the heptahydrated sulphates and takingfirst the case of the nickel compound, we find here a morecomplicated type of deviation. Thus, as the temperature fallsthe susceptibility first increases less rapidly with fall intemperature than is given by the Weiss law, later the suscep­tibility begins to increase more rapidly. Hence the curve ofil%rn against T shows an inflection, the curvature changingsign at about 120° K. The susceptibility remains throughout,however, less than the value for any particular temperaturebelow about 160° K. as calculated from the Weiss law. Cobaltsulphate heptahydrate shows somewhat similar phenomenabut on a much smaller scale; also the susceptibility at thelowest temperatures is greater than the value given by thelaw Xm (T 13.7) - C which is obeyed at relatively high

38

temperatures. On the other hand, ferrous sulphate bepta-hydrate shows a small deviation from the W eiss law of theolder type i. e. the ‘‘cryomagnetic anomaly” .

Considering now the anhydrous sulphates and takingfirst the case of the nickel compound, we see that (1) the.susceptibility for temperatures below about 75° K. has avalue throughout which is less than that given by the W eiss

law and (2) there are a distinct maximum and minimum ofsusceptibility. The minimum, which is very sharp, occurs inthe region of temperature obtainable with liquid hydrogen.In the case of anhydrous ferrous sulphate the points actuallymeasured are shown on the full curve. I t is immediatelyobvious that a maximum value of the susceptibility occursbetween hydrogen and nitrogen temperatures. Nothing canbe said with certainty of the shape of the curve in the regionbetween the lowest nitrogen temperature (about 64° K.) andthe highest hydrogen temperature (about 20°.3 K.) as it isalmost impossible to make accurate magnetic measurements inthis region on account of the cryogenic difficulties but it isprobable in the light of the phenomena shown by anhydrousnickel sulphate that the curve runs something as shown inthe dotted portion. I t will be observed that in this case nominimum of susceptibility has been found. I t would be ofthe greatest interest to continue the measurements down tohelium temperatures so as to determine whether a minimumoccurs or whether the susceptibility continues to decreasesteadily.

Anhydrous cobalt sulphate also shows some indications ofa m a v i m n m and a minimum value of the susceptibility at thelowest temperatures but the phenomena are on a much smallerscale than in the case of the corresponding nickel and ironcompounds.

§ 3. Comparison of Results with Existing Theories.

The theories of paramagnetism can be divided into twoclasses, those based on the classical electrodynamics and thosebased on the quantum theory. A short account will be given

39

here of these theories in so far as they concern the phenomenashown by paramagnetic substances at low temperatures.

The theories of paramagnetism based on the classical electro­dynamics are all more or less intimately connected with thetheory of Langevin. According to this theory the susceptibilityshould be independent of the field strength except for verygreat fields and should be inversely proportional to theabsolute temperature, that is it should follows Curie’s law.It is now known that in general this is not the case withsolid substances. W eiss later showed that the introduction ofa molecular field into the theory of paramagnetism leads tothe law A ( T + %) = C in which the symbols have their usualsignification. I t has been found that this law is almostinvariably obeyed at relatively high temperatures but ingeneral it does not hold at low temperatures.

Other writers have proposed various modifications of thetheory, in particular Gans 1) allowed for the effect of themutual interactions of the molecules. He found that ifthe molecule possesses a figure axis the substance will beparamagnetic but the dependence on temperature of thesusceptibility at low temperatures given by theory is not ingood agreement with the experimental facts.

Foëx 2) has extended the theory of Weiss by consideringthe effect of the introduction of the following ideas (1) amolecular field equal to i u (<r = magnetisation) in which nhas in general different values along the principal axes ofthe crystal and (2) there exists a potential energy which isa function of the direction in thé crystal. An account of thistheory as far as it concerns the principal susceptibilities ofcrystals will be found in Part. II. Foëx also considers inconsiderable detail the possible phenomena shown by the meanof the principal susceptibilities, i. e. the quantity determinedby measurements on powdered materials. He shows that thetheory predicts the existence of the following phenomena (1)

1) Gans, Ann. der Phys., 49, 149, 1916.2) These Strasbourg, 1921, Ann. de physique (9), 16, 174, 1921.

40

At high temperatures the W eiss law holds, (2) at the lowertemperatures the susceptibility may increase either morerapidly or less rapidly with fall in temperature than is givenby the W eiss law, (3) the curve of the inverse of the suscep­tibility against the absolute temperature may posses a pointof inflection so that the curve changes from concave towardsthe temperature axis to convex and vice versa, (4) a maximumand a minimum of susceptibility may occur at the lowesttemperatures. I t will thus be seen that the theory covers allthe phenomena found as yet, including those announced inthe previous pages. Foëx’s theory is, however, purely quali­tative, so that it cannot he asserted with any certainty thatthe theory is really a true representation of the facts.

The earliest of the theories of paramagnetism based on thequantum theory is that of Oosterhuis1). The susceptibilityaccording to Langevin’s is inversely proportional to the kineticenergy of rotation of the molecules. Oosterhuis replaces theclassical expression for this energy by the quantum expressiongiven by E instein and Stern2), viz:

1 .+ 2 hv

and obtains finally a formula which degenerates into the W eisslaw at relatively high temperatures and which is capable ofreproducing the experimental results for the anhydrous andhydrated manganese sulphates with a good accuracy over thewhole temperature range investigated. The theory was laterextended by K eesom 3).

Theories which are more in accordance with the laterdevelopments of the quantum theory have been given byReiche 4) and Smekal 5) as extensions to two degrees of

1) Leiden Comm. Suppl. No. 31.2) Ann. der Phys., 40, 551, 1913.3) Leiden Comm. Suppl. No. 32.4) Ann. der Phys., 54, 401, 1917.5) Ibid, 57, 376, 1918.

41

freedom of an earlier work by W eYsbenhoff *) which dealtonly with one degree of freedom. In these theories theHamilton-Jacobian differential equation of motion is developedfor the rotations of a molecule with two degrees of freedom,a permanent magnetic moment m and a moment of inertia Jin a magnetic field H. The rotations are then quantisedaccording to the conditions laid down by Sommerfeld andW ilson. The distribution of the molecules in the “ phasespace” is then considered by R eiche from the point of viewof the so-called P lanck’s second quantum theory and bySmekal from that of the first quantum theory. R eiche thusarrives at the following formula for the susceptibility

„ N .4h* S

in which S , = | — ^ I S = 2 n e

N = Avogadro number,h = Planck’s constant,Jc — Gas constant,

h?17 9>ir2J kT

The values calculated from this formula are in good agree­ment with the experimental data for the anhydrous andhydrated manganese sulphates, anhydrous ferric sulphate andhydrated ferrous sulphate. The formula derived by Smekal

does not, however, reproduce the observed results with thesame accuracy.

R eiche’s theory can thus give a quantitative explanation ofthe phenomena shown by several substances but is open toseveral serious objections. Firstly the value for the magneticmoment required to secure the agreement of the calculatedand the observed values is altogether different from that

■*) Ann. der P h y s , 61, 285, 1916.

42

found from measurements on solutions by applying the Curie

formula although this formula is the limiting expression forT = «5 according to R eiche’s theory. Secondly the theoryexplains only the type of deviation from the W eiss law inwhich the “ cryomagnetic anomaly” occurs and not the othertypes of deviation shown to exist in the present work. Itmay also be objected that no account is taken in the theoryof the possible intramolecular actions but Reiche himselfacknowledges this and notices it as a point requiring impro­vement.

An almost identical theory was published almost at thesame time as that of Reiche hut quite independently of thelatter by R otszajn1). .

Gans2) has also published a theory of paramagnetism basedon the quantum theory. In this theory the mutual interactionsof the molecules is allowed for and the quantum theory isintroduced by equating the mean energy of the molecule to

2 hvhv ’ ;

ekT- l

the value given by the original P lanck theory for an oscillatorwith two degrees of freedom. He then arrives at a formulawhich reproduces quite well the observations on platinum,the anhydrous and hydrated manganese sulphates, hydratedferrous sulphate, anhydrous ferric sulphate, and liquidoxygen-nitrogen mixtures. It must be observed, however, thatthe method of the introduction of the quantum theory isnot very satisfactory theoretically and that the theory doesnot explain the newer types of deviation from the W eiss lawat low temperatures.

In reviewing the present position of the theory of para­magnetism, it may be said that no satisfactory quantativetheory embracing all the known facts exists at present. Thus,

1) Ann. der Phys., 67, 81, 1918.2) Ibid, 60, 163, 1916.

43

though Foëx’s theory explains the existence of the varioustypes of deviation, it is in no wise quantitative, while thequantitative theories such as that of Reiche, which may beconsidered as fairly satisfactory theoretically, give no explan­ation of the other types of deviations from the Weiss lawthan that known as the “ cryomagnetic anomaly” .

PART II.

MEASUREMENTS OF THE PRINCIPAL SUSCEP­TIBILITIES OF CRYSTALS.

C H A P T E R 1.

PREVIOUS WORK.

Very few instances of trustworthy quantitive measurementsof the magnetic properties of crystals are to be found in theliterature and still fewer data on the temperature dependenceof", the principal susceptibilities.We may pass over withoutfurther reference the earlier more or less qualitative invest­igations of P lückkk1)^ B e e k 2), Grailich and v. L a n g3),H ank el4), T yndall5), R owland and J acques6), Strenger7)and K önig8). The first reliable measurements of the principalsusceptibilities of crystals are those of V oigt and K inoshita9).The latter investigators developed a new method of measure­ment and determined the properties of the following substancesat atmospheric temperature: — regular — rock salt, alum, leadnitrate, fluor-spar, pyrites, zinc blende; magnetically uniaxial— calc-spar, dolomite, quartz, beryl, rutile, turmaline, apatite,zircon; crystals with three different principal susceptibilities —topas, coelestine, aragonite.

Their work was continued by F inke 10) who investigated

1) P lücker, Pogg. Ann., 72, 76, 77, 78, 81, 82, 86, 110, 1847—1860.2) B eer, Pogg. Ann. 81, 1860 ; 82, 1858.3) G railich and L ams, Wien. Ber., 82, p, 43, 1868; (Ila), 108

p. 561, 19004) H ankel, Ber. d. Sachs. Ges., p. 99, 1851.5) T yndall, Phil. Mag., (4), 2. p. 174, 1851,6) R owland and J acques, Sill. Jou rn ., (3), 18, p. 360, 1879.7 ) 'Strenger, Wied. Ann., 20, p. 304, 1883; 85, p. 331, 1888.8) K önig, Wied. Ann, 81, p. 273, 1887 ; 82, p. 222, 1887.9) Voigt and K inoshita, Ann. der Phys., 24, p. 492, 1907.

10) F inke, Ann. der P hys., 81, p. 149, 1910.

48

the magnetic properties of a number of paramagnetic anddiamagnetic substances, principally crystals of the monoclinicsystem at atmospheric temperatures. His results are givenin the following table.

C rystal. X 106 X 106 %3 X 106 *

Angite * .......................... 26.6» 12.8» 22.6, — 7*0'Hornblende * .................... 23.9, 16.6, 17.9, - 21*65'Adnlar ............................... — 1.09 — 0.98 - 0.81 — 13*20'Cane Sugar *. . . . . - 0.60 — 0.55 — 0.57 — 1*50'FeSo4 .(N H 4),S 04 .6 H ,0 . 85.4» 76.6, 74.9„ — 18*58'CoS04 . (NH4).2S 04. 6H ,0 55.9, 43.5, 45.60 — 27*31'N iS04 . (NH4),S 04 . 6H ,0 19.6, 15.7, 17.9, — 16*17'CoS04 . CuS04 . 6HjO * . 28.6, 23.0, 25.5, - 52*64'CoS04 . K ,S04 . 6H ,0 . . 66.5, 49.6, 77.1, — 21*20'FeSo4 . 7H ,0 . . 84.3, 78.74 76.2o + 30*10'CoS04 . 7 H ,0 ..................... 70.9 64.0 68.5 — 39*40'

N iS04 .7 H ,0 ..................... 29.0o 29.2» 29.9, —

%1> %2, %3 give the three principal susceptibilities of thecrystal, Xi and %2 being the susceptibilities in the symmetryplane of the monoclinic crystals and %3 along the symmetryaxis, is the angle between the axis of greatest susceptibilityand the crystallographic c axis. The numbers for the substancesmarked with an asterisk denote susceptibility per unit mass,the other numbers susceptibility per unit volume.

Of .the investigations of the temperature dependence of themagnetic properties of crystals, we may note the early workof F araday extended later by L utteroth 1 2). The latterinvestigator determined the temperature dependence of thedifferences of the susceptibilities of a number of crystals intwo mutually perpendicular planes orientated in known direct-

1) F araday, Expt. res., Ser., 22, 1848, 80, 1866; Pogg, Ann., 100,p. 439, 1857.

2) L utteroth, Wied. Ann., 66, p. 1080, 1898.

49

ions with regard to the crystallographic axes. The substancesinvestigated were the following: — monoclinic paramagnetic —nickel potassium sulphate, cobalt potassium sulphate, cobaltsulphate, monoclinic diamagnetic — zinc potassium sulphate,zinc ammonium sulphate, rhombic paramagnetic — nickel sul­phate, rhombic diamagnetic — zinc sulphate.

Sections of each of these crystals were suspended about avertical axis by means of a quartz fibre and the differencein susceptibility of the crystal in a plane perpendicular tothe axis of suspension determined over a range of temperatureof from 0° C. to about 50° C. In every case it was foundthat the results could be represented by the linear relation

dt — d0 (1 +- x t),in which d = difference in susceptibility at temperature t.

® 0 » » f j » 11

t = temperature in degrees Centigrade.The final results are given in the following table.

Substance. xx Xy xz xz *y

NiK,(S04) , . 6HsO . . . • _ - 0.0,6739 — 0.0,5512 + 1.223 _CoK,(S04), . 6H,0 . . . — -0.0,6210 — 0.0,6037 -+1.232 i —ZnK,(S04) ,, 6H,0 . . . — — 0.0,5208 - 0.0,4259 + 1.223 . —Zn(NH4),(S04) ,. 6H,0 . — -0.0,4064 — 0.0,3310 +1.227 -■ —NiS04.7H,0................ +0.0,1887 —0.0,5087 — — — 0.871ZnS04. 7H,0 . . . . + 0.0,4985 —0.0,1341 ; — — j. — 0.372CoSo4. 7H,0 . . . 4-0.0,2302 — 0.0,4000 . r* 7— — 0.575

In the above table xx, xy, xz denote the coefficients obtainedwhen the crystal was suspended about the axes X, Y and Zrespectively. X is the axis of the greatest, Y that of themiddle and Z that of the least susceptibility. I t will beobserved that xy and xz for the monoclinic double sulphatesare both negative, that is dt decreases with rise in tempe­rature. In the case of the rhombic sulphates xx is positive

4

50

and ocy is negative and similarly with the monoclinic cobaltsulphate.

Thus the difference of susceptibility is dependent on thetemperature whether the crystal is paramagnetic or dia­magnetic. Again it will he seen that the ratios of the a’sarrange themselves in classes determined by the symmetry ofthe crystal.

The first instance of the determination of the principalsusceptibilities of a crystal over an extended range of tempe­rature to be found in the literature is the measurement ofsiderose by Foëx1). This substance is a naturally occurringferrous carbonate containing appreciable quantities of thecarbonates of manganese and the alkaline earth metals,belonging to the tetragonal system. We therefore know apriori that the three principal magnetic axes will lie alongthe crystallographic axes and that two of the former will beequal. Foëx therefore made measurements with a cylinder cutfrom the crystal in such a direction that its axis was parallelto the ternary axis of the crystal. Measurements were madegiving the susceptibilities (1) parallel to the axis of thecylinder and (2) and (3) in two directions perpendicular tothe axis and to one another. The results are given below.

T°K.

X parallelto axis.

X \ perpendic.to axis.

X a perpendic.to axis X mean.

87° 581.6 180.8 180.3 314.1200° 231.3 111.6 111.2 151.1295°.6 142.3 84.2 84.2 103.6315°.8 129.8 79.4 79.45 96.2835° .1 120.9 76.1 75.9 90.9856°.7 111.9 72.6 72.2 85.6875°.5 105.8 69.5 69.3 81.4401°.5 97.0 66.1 65.9 76.8

The above figures give the susceptibilities calculated perunit mass of paramagnetic carbonate present and not per unit

t) F oëx, T hèsb, Strasbourg 1921 ; Ann. de Phys., (9), 16, 174, 1921.

51

mass of the crystal itself. They are apparently to bemultiplied by 10—6 but this does not appear to be stated. Theequality of the columns 3 and 4 furnishes a test of thehomogeneity of the crystal and of the accuracy of mounting.

The following important points may be noticed. The twocurves of the above results *) for moderately high tempe­ratures are very approximately straight lines i. e. the suscep­tibilities follow the law X (T - f A) = C and they are parallelto one another, showing that the Curie constant C is thesame for the two cases. I t will also be seen that the A’s inthe law % (T -j- A) = C rruiy he both positive and negativein the same crystal. (A1 = — 60° and A2 = -f l0 3 °) . Thirdlythe curves show deviations from the straight line law at lowertemperatures and these deviations are in opposite directionsin the two cases. These deviations occur however practicallyonly at the very lowest temperature investigated and hencesome small doubt may. be attached to their reality. Anextension of the work to lower temperatures would have beenof considerable value as enabling this point to be settleddefinitely.

An account of the previous work on the magnetic propertiesof crystals as regards the theory will be found in a latersection of the present work.

1) See page 85 of the thesis of F oëx.

C H A P T E R I I .

' METHOD.

§ 1. Theory of the Method of Observation.

Two methods present themselves as suitable for thedetermination of the principal susceptibilities of a crystal atdifferent temperatures viz: (1) the Curie method1), in whichthe attraction on a small body placed in a non-homogeneousmagnetic field at the position of the maximum of H.dH/dx(x being the coordinate at right angles to the direction ofthe field is measured

and (2) the W eiss method2), in which the maximum coupleexerted on a small cylinder of the crystal suspended in ahomogeneous field is measured. The first method has theadvantage that the observations give directly the value of thesusceptibility of the crystal in a direction parallel to the xcoordinate, whereas the second method gives only thedifference of the susceptibilities in the plane perpendicularto the direction of suspension of the crystal cylinder. TheCurie method however presents considerably greater difficultiesfrom the cryogenic point of view than the W eiss method, sothat it was decided to apply the latter method to the presentmeasurements.

For each crystal the three principal susceptibilities Xv %2,

1) P. Curie, J. de Phys. (3) 4, p. 197 and p. 263, 1895. See alsoC.R. 116, p. 136, 1893.

2) P. W eiss, J. de Phys. (4), 6, p. 655, 1907.See also P. Weiss and H. K amerlingh Onnes. Leiden Comm.

N®. 114.

53

%3 have to be determined and, for all crystals not belonging'tb one of the rectangular crystal systems (regular or cubic,tetragonal and rhombic), also the angles defining the orient­ation of the magnetic axes with respect to the crystallographicaxes. To determine these n unknowns we must have of coursen independent equations involving them. Of these n equationswe may obtain n—1 equations by measuring the differencesof susceptibilities of crystal sections orientated in n—1 differentdirections with respect to the crystallographic axes. The finalequation must involve some absolute value of the susceptibilitiesin order to make it possible to calculate the values of the nunknowns, since the other m—1 equations involve only differ­ences of susceptibility. This last requirement could not ofcourse be obtained with the Weiss method but it did notinvolve further measurements, since the value of the mean ofthe three principal susceptibilities of the crystals underconsideration had already been determined by the measure­ments on the powders described in the previous section ofthe present work and this quantity could be used in thecalculations.

The theory underlying the W eiss method as applied to themeasurements of the susceptibilities of crystals will now begiven. Let Zt, X2, X3 by the principal susceptibilities of thecrystal. Then the formule for the induction ellipsoid givesimmediately

+ X2Y 2 + X3Z 2 = 1.

Let a, b, c be the principal axes of the induction ellipsoid;then we have

%! = 1 /o2 , %2 = 1/6* , %3 = 1/c2.

The susceptibility 3 r corresponding to a definite direction r,of which the direction cosines are x, (3, y, is then given by

*2 m m io2 + + c2: = r 2"

54

or + %2(P 4- %sr 2 = - 2 = % r ..........................(1)

Let the crystal now be brought into the magnetic field andlet the direction r be parallel to the field; then the energyresulting from the introduction of the crystal into the fieldis equal to

E ^ ^ X r E \Ju

If now the crystal is suspended about a vertical axis andthe magnetic field is in a horizontal plane, then the crystalwill tend to turn into such a position that the energy is aminimum. In this connection only the susceptibilities, whichlie in that ellipse, which is the section of the induction

ellipsoid by the ho­rizontal plane, areto be taken intoconsideration.

In fig. 4 let aand b be the axes ofthis ellipse and %\and X't the corres­ponding susceptibili­ties. From consider­ations of symmetryit follows that equi­librium occurs for

r — a and r = b. For every other direction there occurs acouple, which is determined by

Fig. 4.

d e “(?**■)dp dp

Since H is constant, this gives

55

d%d<P

1 cos2 (p sin2 cpr 2 a2 62

( I , )\ r 2/ — 2 cos sin <p 2 cos cp sin <J>

o2 ó2

= sin 2<?) ( ~ — — ) = (%'2 — sin 2<p

— ' = ~ (%'2 — 3^) sin 2cp H 2 = couple = (7 . . . . (3)

We will now describe how the susceptibilities along theprincipal magnetic axes and the positions of the latter forthe two substances as yet investigated (nickel sulphate hepta-hydrate and cobalt ammonium sulphate) can be obtained byobservations of the couple, whose value is given by equation (3).This couple is measured for various section plates cut indifferent and definitely known directions, from the crystal.

We will consider first the case of nickel sulphate,NiS04. 7H20. A s the heptahydrated form crystallises in therhombic system, the positions of the magnetic axes are fixeda priori, since they must be parallel to the crystallographicaxes. We thus have to determine %it X2, %3. For this purposewe require two cylinders cut from the crystal in knowndirections. Nickel sulphate heptahy-drate crystallises in the form shownin fig. 5. The crystals are elongatedalong the direction of the crystallo­graphic .c axis and are also some-

1 times tabular parallel to the mface (110). It was therefore decidedto cut the crystal sections in thefollowing manner-viz: (1) axis ofcylinder parallel to the c axis and(2) axis of cylinder perpendicularto the m face.

Then a measurement of the couple exerted on cylinder (1)

will give the value of X a — X x> = c and the cylinder (2)will give %c — %m = m, where X a, X b, X c, X m are the suscep­tibilities parallel to the directions a , b, c, m respectively i . e.perpendicular to the planes (100), (010), (001), (110). NowX a, %6, X r are respectively equal to %v %2> %3 an<̂given by

56

X m = cos2 <p + %2 sin2 <p (see fig. 6)

in which <p is equal to tan t (fe/o) = tan 1 (1/0.9815 =45°32' *).

Rewriting the above relations and adding the value of themean susceptibility derived from the measurements on the

Fig. 7.Fig. 6.

powdered material, we have the following equations fromwhich to determine X v X 2 and X 3.

X, — X 2 = c . ................. (1)X 3 — ( X t cos2 <p X 2 sin2 <p) = m ............... (2)

-4- %2 “b Xg = 3 a .........* • • (3)

i) See Groth. Chem. K ry s t , 6. I I , p. 434.

57

Solving these equations, we find for the principal suscepti­bilities the following values

= 1/3 (3a — 2c — m — c cos2 <p). \%2 = 1/3 (3a — c — m — c cos2 <p). >.(4)

= 1/3 (3a — c — 2m — 2c cos2 (p). )

Cobalt ammonium sulphate CoS04 . (N H J4S 04 . 6H20.This substance belongs crystallographically to the monoclinic

system. In such a system we only know a priori that one ofthe magnetic axes, let us say %3, lies along the direction ofthe symmetry axis of the crystal, i. e. the b axis, and thatthe other two magnetic axes lie in the symmetry plane, i. e.the plane containing the crystallographic a and c axes. Wehave thus to determine the positions of the magnetic axes inthe symmetry plane in addition to the values of the principalsusceptibilities. For the former purpose it will be sufficientto determine the angle which %i , the axis of greatestSusceptibility, makes with the c axis (see figure).1)

Cylinders were ground from the crystal with their axesrespectively perpendicular to the plane (100), (010), 001).Then measurements of the couples exerted on these crystalcylinders will give the values of

3̂ — %c — a .............. | .................. (1)^3 %a = b • • ................................... (2)

— C ............ ..................... (3)

From these quantities and adding the value of the meansusceptibility derived from measurements with the powderedmaterial we obtain the following equations for Xv X2, %3 and p.

%3 — (%i cos2 — %2 sin2 \p) == a . . . (1)^ 3 fai cos2 (ip + 0) — %2 sin2 (\p (3)) = b . . . . (2)

1̂ 2̂ = c . . . . (3)%! + + %3 = 3m .. . (4)

This mode of defining the positions of the magnetic axes isadopted so that the results may be directly comparable with thoseof PlNKE loc. cit.

58

in which (3 is the known crystallographic angle between thea and c axes.

From (1) with (3) and (4) we obtain%8 — c cos2 \p — = a

and from (2)%3 — c cos2 (t// (3) — %2 =

Subtracting we have

c { cos2 (4> + (3) — cos2 1/ ' } — a — b.

This may he transformed into

c j (Z cos — n sin 4>)z — cos2 'P } = a —

in whichI — cos (3 , n = sin (3 , Z2 n 2 = 1.

From this we obtain

{(Z2 _ n 2 _ i) cos2 $ — 2In c o s 1/(1 — cos2^)} = (a — b) / c — n 2 = k

4 Z2ra2 cos2 ^ (1 — cos2 \p) = *2 + 4n4 cos4 ^ + 4rc2ifc cos2

Solving for cos2 ■■I we find

( I V _ w2^) ± ] / j ( I V — 7J2*)2 — (w4 — Z2n2) A:2 }006,2 * = 2(»4 + Z2n2)Treating \p now as known we obtain from equations (1),

(3) and (4)

% i= -K (3m — 2c — a — c cos2 4>) jO I

%2 = i ( 3 m — c — a — ccos2■$) \ . (5)ö I

%3 = (3m + c — 2a -+- 2c cos2 \p) ]O ƒ

§ 2. Apparatus and Materials.

The apparatus, by means of which the method theoreticallydescribed in the previous pages was applied to the measure­ment of the principal susceptibilities of crystals at low tem-

59

peratures, will now be described. As it was necessary to shutoff from the air the space in which the crystal was suspendedand which also contained the liquefied gas and its vapour,a fairly complicated cryogenic apparatus had to be employed.This is shown in section in PI. 1. I t consists principally ofthree tubular portions which, naming from the outside inwards,we may call the cover, the adjusting tube f and the holder b.The cover consists of a silvered vacuum tube a, a brasstube B, a glass tube C and a cap D which closes theapparatus.

The crystal holder b is made from a long thin brasstube bb. At the lower end of this tube a copper ring r isfastened. A small copper cylindrical cup fits into this ringwith considerable friction and holds the crystal, which isfastened into the cup with the smallest possible amount ofwax or Canada balsam. The whole holder is very light andis constructed of practically non-magnetic material in orderto make the corrections, which must be applied to the obser­vations for the inherent magnetism of the apparatus, assmall as possible. The holder is connected to the rod k bythe wire g2 and the spring gt. In order to make the equil­ibrium of the suspended system stable and to prevent thecrystal from being attracted to the poles of the magnet theholder is held fast underneath by a thin wire d, fastenedto a small projection on the lower side of the crystal holder;the other end of this wire is held in place by a small coneóf brass which fits into a hole in the bottom of the adjustingtube. By means of the spring gx a small tension can be puton the wires g2 and d without risk of breaking them. Forthis purpose the spring g1 is so constructed so that it canbe easily elongated in the direction of g2 but serves, withall the accuracy necessary, as an immoveable connection asfar as torsion is concerned.

The turning of the crystal in the magnetic field is broughtby the tube bb to the mirror b1 by means of which thedeflections of the crystal holder can be read through theopening f 10 and the window C2 on a glass scale divided into

v ■ *

Pla te 1.

61

millimeters, placed at 2.2 meters distance and illuminated bymeans of spherical mirror strips.

The tension on the wires can be regulated by the rod kwhich passes through the packing-gland Dr in the cap D2.The rod k may be moved up and down by turning thenut D3 and at the same time preventing the rotation of k.

As the holder is very light and under tension, it has agreat tendency to vibrate under the influence of smallshocks. In some preliminary experiments this tendency wasfound to be so great as to make it impossible to read thescale when machinery was in motion in the building. Adamping arrangement, similar to that used by K amerlingh

Onnes and P errier1) was therefore attached to the upperend of the tube bb. It consists of a small cylinder to whichare attached four vanes which dip into an annular oilbath.The vanes must be entirely immersed in the oil in order toprevent the occurrence of capillary actions which lead tocouples on the suspended system which are not negligible.As .Weiss and Kamerlingh Onnes2) have observed, it is alsonecessary to free the oil from all volatile substances and toprevent the occurrence of bubbles of air under the oil, sincethe apparatus has to be evacuated after it has been completelymounted. With this damping arrangement (supplemented instrong magnetic fields by the damping due to the Foucaultcurrents induced in the well conducting holder) the deflectionsof the suspended system were easily and accurately readable.

The whole holder and the wires hang in the adjustingtube ƒ, the upper end of which is screwed to the cap D2,which also carries the rod k and itself rests on the glasstube Cj. The cross section of the adjusting tube which consistsof three parts f —/ 3, decreases three times.; the narrowestlower part / 3 surrounds the crystal holder as closely aspossible. Against the bottom f 4 rests the small cone c, towhich the lower wire d is soldered and which serves to

Leiden Communication Nn. 1162) Ibid N". 114, p 20.

62

maintain the tension on the suspended system. A small slotin the bottom allows the small cone to be placed in position.In order to mount the adjusting tube, already containing thecrystal holder, in position in the cover, the cap D2 is fittedinto the bronze ring, which is fastened to the glass tube C\of the cover and the joint is made air tight by means ofthe rubber ring D4. This latter is smeared with rubbersolution and bound tight with wire. The lower end of theglass tube is fastened to a second tube B t of the cover,the middle part of which carries a ring n, in which therods B3 holding the vacuum glass terminate.

The Dewar glass itself consists of a lower narrow part,which is completely silvered and an upper wider part, whichis silvered as far as a; the upper part is left transparent sothat it may be possible to ascertain that too much of theliquefied gas is not allowed to enter the vacuum-glass. Thelatter fits into the brass tube B1 and is protected by awooden ring. The rods B3 hold the glass in its place and atsuch a height that it is just free of the wooden ring. Theconnection between the vacuum-glass and the cover is madeairtight by means of a rubber ring, smeared with rubbersolution and fastened with copper wires. The upper part ofthe vacuum-glass has an outer diameter of 4.9 cms. and aninner diameter of 3.8 cms.; the lower part has an outerdiameter of 1.85 cms. and an inner diameter of 1.45 cms.

Into the tube B1 is soldered the steel capillary of a heliumthermometer with a german-silver reservoir and a glassstem S2. The quantity of helium is so chosen that at theboiling point of nitrogen the mercury stands at a mark inthe lower portion of the stem and at the melting point ofhydrogen at one in the upper portion. The thermometer isnot used to measure the temperature of the liquefied gasbut to indicate the level of the latter in the vacuum^glass.As soon as the level sinks below the upper end of thereservoir of the thermometer, the mercury sinks in thestem S1.

The liquefied gas is introduced into the apparatus through

63

the german silver tube B7. The gas formed by evaporationescapes through B s and through valves to a gasholder or toa vacuum pump. By means of the valves the vapour pressureis regulated and its value is read off on a manometer. Beforeintroducing the liquefied gas through the tube B1t which isclosed by a rubber tube with a glass stopper, the air ispumped out of the apparatus through Bs.

To prevent the cooling of the upper portion of the appa­ratus containing the torsion wire by the escaping vapours, anumber of large openings are made in the tube f 3 arrangedin such a way that its resistance to torsion is not impared.In addition to this, copper baffle-plates surrouding f 2 andsoldered to B3 are arranged so that the tube can move withslight friction in them. The refilling of the apparatus withliquefied gas takes place in the same manner as the firstfilling. Generally several series of observations can be carriedout with a single filling.

The Electromagnet,

This is the same as was used in the measurements of thesusceptibilities of the powders described in the previous sectionof this work. I t was mounted on a turntable so that it couldbe rotated into any position about a vertical axis. The azimuthof the magnet and hence the direction of the magnetic fieldcan be read on a circular scale, divided into degrees andattached to the turntable. The pole distance used in thepresent measurements was 30 mms.

The Crystals.

The crystals, from which the cylindrical sections used inthe measurements were ground, were prepared by the slowevaporation of a saturated solution at atmospheric tempera­ture, according to the method described by Tutton (Crystallo­graphy and Practical Crystal Measurement Part. I, p. 14).By this means large crystals of excellent quality can easilybe obtained. The best crystals in a number of crops werechosen and employed in preparing the sections. Small cylinders

64

had then to be ground from the crystals in such a way thatthe axes of the cylinders were orientated in known directionswith regard to the crystallographic axes. As the accuracy,with which the measurements could be made with the cryo-magnetic apparatus, was of the order one per cent., it wasnot thought necessary employ an accurate grinding goniometerfor the preparation of the crystal sections. A crystal wasmounted on the end of a brass rod in such a manner thatthe direction in the crystal, in which the cylinder axis wasto lie, was coaxial with the rod. This adjustment could bemade with all the accuracy that was necessary in the presentcase by taking advantage of the well developed faces of thecrystal. The rod was then mounted in a lathe and the crystalcarefully ground to a circular shape with fine emery. Thecrystal was then removed from the rod and the ends groundflat perpendicular to the length of the cylinder. No attemptwas. made to polish the cylinder as this was quite unnecessaryfor the present purpose and as the use of the usual polishingmaterials, since they themselves possess strongly magneticproperties, would render the measurements doubtful or evenuseless.

§ 3. Method of Observation.The course of the observations is very simple when one

the apparatus has been set up in order. We have to determinethe maximum value of the couple exerted on the crystal bythe magnetic field. This couple is, by theory, connected withthe azimuth of the magnetic field by the relation

( 7 = 1 / 2 . {Xi —-£ 2) • rn . H* . sin 2 0in which 0 is the angle between one of the magnetic axesor %2 and the direction of the field. The couple should thenhave a maximum value corresponding to the value 45° pf 0and hence as the azimuth of the magnet is turned through360° we should obtain four positions of maximum value ofthe couple distant 90° front the another. A test of theaccurate working of the apparatus and of the magnitude ofthe corrections to be applied to the actual readings may

65

thus be made measuring the couple exerted on the crystal fora number of azimuths of the maignet and comparing theresultant curve connecting the observations with a true sinecurve. Such an example is given in the next section.

As however it is the maximum value of the couple thatis required, this maximum may he determined directly andnot by interpolation on the experimental curve obtainedconnecting the magnitude of the couple and the azimuth ofthe magnetic field. Those azimuths of the electromagnet arefirst determined tentatively for which the couple is a maximumon either side of the zero. I t is sufficient to perform thesetrials two or three times with a suitably chosen field strength,since the azimuth of the maximum couple changes but littlewith the field and for other values of the field we can,without risk of error, have recourse to interpolation. Thesepositions of the magnet can be determined quite easily towithin 0°.5 or 1°, which is sufficiently accurate as the valueof the couple changes but little with the azimuth in theimmediate neighbourhood of the m n.xim nm

After this has been effected the observations are taken inthe following manner. The magnet is set in one of thepositions of maximum couple, the current through the coilsis switched on and adjusted to a definite value. Before anyreadings are taken however, the current is reversed severaltimes to ensure a well defined field, for, as a coupleproportional to the square of the field has to be measured,any inaccuracies in the value of the field strength, whichmight be obtained notwithstanding the fact that the ironof the electromagnet core was extremely soft, would haveconsiderable influence on the results. Having adjusted thecurrent finally to a definite amount, the value of the couplewas estimated as given below; the current was then reversedand the couple determined again. The same observations werethen repeated with the magnet in one of the other positionsof maximum couple. The whole process was carried out everytime for a whole series of different values of the fieldstrength.

5

66

Measurement of couple.'When small values of the field strength were employed at

atmospheric temperatures, the deflection of the suspendedsystem was generally of such a magnitude as to be accuratelyreadable on the scale. In this case then the couple wasestimated by measuring the deflection, both torsion wiresthus contributing to the opposing torsional couple. For greatervalues of the field however and for the lower temperaturesat which the susceptibilities of the crystals are much greater,the deflection of the suspended system was too great to beread on the scale. The constant of the suspension wires hadbeen purposely chosen so that this might occur and hencemake possible the application of the method of observationnow to be described.

When working at the low temperatures obtainable with thevarious liquefied gases, the upper part of the apparatus, onaccount of the presence of the baffle plates at / s, remainsthroughout at atmospheric temperature; hence of the twowires g2 and d, g2 is always at atmospheric temperatureand d takes up the temperature of the bath. If thereforeboth wires were used in the observations at low temperaturesby allowing the suspended system to be deflected, we shouldrequire a knowledge of the elastic properties of the wire atthe low temperatures so that a correction might be appliedto the value of the constant of the wire for its change withtemperature. This, however, may be eliminated by the followingproceedure.

After the current in the magnet has been switched on andthe suspended system has suffered a deflection, the graduatedhead S attached to the top of the rod k is turned until thesuspended system is nearly in its original position in the absenceof the magnetic field as shown by the scale and telescope.The angle through which the torsion head has been turnedis then read by means of the small reading microscope M.No attempt is made to bring the crystal holder exactly intoits original position but the torsion head is turned throughan exact number of divisions of the scale engraved on S

67

until the deflection has been reduced to not more than oneor two centimeters on the glass scale. This was done becausethé accuracy with which the deflections could be read wasthereby much increased, for, as the scale on the torsion headwas not very finely divided, readings obtained by estimatingparts of a division might contain considerable errors. Thenthe value of the couple could be obtained as follows. Let2 a, be the angle through which the torsion head had to beturned, x1—x2 the difference in scale readings respectively inthe presence and in the absence of the magenetic field, d thedistance of the scale from the mirror on the suspendedsystem, and C1 and C2 the constants of the upper wire andthe combined upper and lower wires respectively (both beingthe values corresponding to ordinary temperature). Then thecouple exerted on the crystal is given by

+ c2 tan-1 ) .................. (1)

which was used for the calculations in the form

C ,|.+(A ±iO to“" ( i5 ^ i) t ........ (la,

in which \ and l2 are the lengths of the upper and lowersuspension wires respectively.

In employing (1) (actually in the form (la) ) to calculatethe couples obtained at low temperatures, it will be noticedthat the value C2 of the constant of the suspension wiresused is actually the value holding at atmospheric tempera­tures, although the lower wire, which contributes a part ofthe couple, is at the low temperature. This is permissible asthe constant of the lower wire will not change greatly withtemperature in the region of low temperatures and as theintroduction of the effect of temperature on the value of C2will only bring with it a fairly small correction to a termwhich is itself already a small correction to the value of thecouple.

68

Measurement of Temperature.For the low temperatures this was carried out as described

in Part I p. 19 by determining the pressure under whichthe particular liquefied gas was boiling at the time of theexperiment.

Measurement of Magnetic Field.The field was, as before, characterised by the strength of

the current in the coils of the electromagnet, reference thenbeing made to a table of field strengths compiled as a resultof previous calibrations.

Sources of Error and Corrections.We will now examine the sources of error incidental to the

method of experiment and any corrections 'which must beapplied to the actual readings. As posible sources ofuncertaining we may note

(1) Inhomogeneity of the field.(2) Influence of the crystal holder.(3) Asymmetries in the apparatus and its mounting.We will deal with these in turn.(1) An inhomogeneity of the magnetic field will bring

with it an additional couple dependent on the lack of homo­geneity of the field in a plane perpendicular to the axis ofsuspension of the crystal. As the inhomogeneity in the presentease was such that the change of the field strength over avolume of one cubic centimeter did not amount to more thanone part in a thousand, the correction to be applied to theobservations fell within the limits of accuracy of the experi­ments and hence might be neglected.

(2) Although the crystal holder was made of materialwhich was free from ferromagnetic substances and wasbut weakly magnetic itself, the apparatus showed certaindeflections when mounted precisely as when the actualobservations were taken but without any crystal in the holder.At atmospheric temperatures these deflections were systematicbut quite small, so that a small correction could be applied

to the observations to allow for the effect of the holder itself.At low temperatures however, the deflections due to theholder itself were quite unsystematic, being on opposite sidesof the zero for opposite directions of the current in themagnetic coils for the same position of the magnet and werenot proportional to the square of the field. The effect wasthus to bring in certain (small) asymmetries into the readingsat low temperatures. It was however found that the effecton the results of these asymmetries could be eliminated byemploying the method of observation previously described.

(3) Asymmetries in the apparatus and in its mountingalso cause asymmetries in the deflections which are alsoeliminated by the method of observation.

69

C H A P T E R I I I .

OBSERVATIONS AND RESULTS.

§ 1. Observations.

Calibration of Torsion Wires.The torsion wires were calibrated in the following manner.

A small metal body of suitably large moment of inertia wasattached to an accurately measured length of wire cut fromthe same reel as that used for the torsion wires. The systemwas then suspended from the free end of the wire andcaused to perform torsional oscillations, the period of whichwas then determined. A body of known moment of inertiain the shape of a ring was then added to the system andthe period of the torsional oscillations again determined. Thenthe constant of the wire per unit length could be calculatedas follows.

Let t1 be the time of oscillation with ring.f2 »• » „ » » without ring.

K 1 „ „ moment of inertia of ring.K 2 „ „ „ „ „ „ system without ring.

Then for K 2 we have the relation

at2 =K,t\

Further we have

71

Then D, the constant of the wire, is equal to4?rz K

'

or constant per unit length = * —.

The mean of a number of determinations gave the valuenoted, below for the constant of the wire per unit lengthper radian angular displacement.

D = 414.7.

Preliminary Test of Apparatus.In order to test the good working of the apparatus when

set up in the usual manner, a crystal was mounted and thedeflections determined as a function of the azimuth of themagnet for a number of field strengths. The theory indicatesthat the observations should lie on a sine curve. Anexamination of the results showed that the theory andobservation were in satisfactory agreement and hence thatthe apparatus was working well. An example of such aseries of observations is given below.

Cobalt Ammonium Sulphate. Crystal c.mass of crystal = 0.1250 grms.

temperature = 16°.0 C. length of upper wire = 8.5 cms.pole distance = 21.4 mms. „ „ lower „ = 5 .52 cms.

current in magnet coils = 4 amps,distance of mirror from scale = 227 cms.

MAGNET.SCALE READINGS.

Current off on direct off on reserve.

90 38.65 37.62 38.68 37.7080 38.66 38.70 38.80 38.7070 38.80 39.50 38.75 39.6160 38.80 40.35 38.80 40.3050 38.80 40.90 38.80 40.85

72

MAGNET.SCALE READINGS.

Current off on direct off on reserve.

40 38.80 41.15 38.80 41.1030 38.70 40.89 38.70 40.7020 88.78 40.45 38.70 40.5010 38.40 40.22 38.53 40.350 38.63 39.60 38.53 39.43

350 38.80 38.76 38.30 38.68340 38.20 87.80 38.10 37.70830 38.16 36.86 37.95 36.95320 37.90 36.90 38.00 35.91310 38.00 35.70 37.95 35.75800 38.00 35.70 38.00 85.35290 38.10 35.50 38.10 36.62280 38.20 36.10 38.10 36.00270 38.16 86.93 88.04 36.87260 38.10 37.92 38.10 37.80250 38.10 38.70 38.20 38.80240 38.20 39.70 38.30 39.75230 88.20 40.35 38.20 40.40220 38.20 40.80 38.20 40.78210 38.25 40.70 38.20 40.80200 38.40 40.80 38.30 40.50190 38.40 40.38 38.40 40.30180 38.40 39.80 38.40 39.70

The numbers under the heading “ magnet” denote thepositions of the magnet in degrees as shown on the scaleattached to the turn table. The figures in the other columnsdenote the readings of the glass scale expressed in centimeters.The observations are exhibited graphically in fig. 8.

Specimen Set of Observations.

The set of observations given below may serve as aspecimen of the results obtained in the measurements.

73

=C.s in.2 0 .

Fig. 8.

Cobalt Ammonium Sulphate.

Crystal c2 (crystal cut with axis of cylinder perpendicularto c face)’.

Mass of crystal = 0.087 grms. Distance of mirror fromscale = 220.7 cms.

Length of upper torsion wire = 8.37 m a» » lower „ „ = 5.90 cms.

Pole distance = 30 mms.

74

Measurements in liquid nitrogen at 756.2 mms. pressure.Hence temperature = 77°.3 K.

Zero at 125 & 43.5 cms.Magnet current = 8 amps.

M A G N ET. R E A D IN G S .

18° ( 0 116 & 46.68 cms.(A) 116 „ 42.80 ,

283° (A) 134 „ 35.80 „

(0 134 „ 40.50 „M agnet current = 10 am ps

283° (1) 141 & 41.10 cms.(A) 141 „ 54.60 „

13° (A) 109 „ 35.5 „(0 109 n 35.7 „

Zero after experim ents = 125 & 43.3 cms.Further observations — data as before.

Zero a t 125 & 43.1 cms.M agnet current = 14 amps.

13° (0 96.1 & 40.8 cms.(A) 96.1 „ 37.4 „(A) 96 „ 36.5 „

(repeated a t end o f expt.)283° (A) 154 & 48.2 cms.

(I) 154 n 53.6 „M agnet current = 12 amps.

283° (9 146 & 41.3 cms.(A) 146 „ 45.2 „

13° (A) 102 „ 38.1 „(0 102 . 29.4- „(last observations repeated)

13° (A) 102 & 34.3 cms.(0 102 „ 29.8 „

M agnet current = 10 amps.13° (0 n ° & 45.2 cms.

(A) 110 , 39.9 „283° (A) 140 n 47.5 1)

(I) 140 » 4 6 6 „

75

M AGNET. R E A D IN G S.

M agnet current = 8 amps.283° (I) 135 & 46.5 cm s.

(A) 135 „ 51.0 „13° (A) 114 „ 36.7 „

(I) 114 „ 37.0 „

The figures in the first column under the heading“ Readings” denote the readings on the scale attached to thetorsion head; the numbers followed by “ cms.” denote thereadings on the glass scale expressed in centimeters. Theletters I and h in brackets indicate the positions low andhigh of the reversing switch and hence the directions of thecurrent in the magnet coils.

From the above data we may calculate first the anglescorresponding to the differences in scale readings and secondlythe values of the deflections i. e. the quantities which, whenmultiplied by Cx (see (la) p. 66), gives the values of thecouple acting on the crystal.

Angles from scale readings.

CURRENT. ANGLE.

8 amps. (1) 0 .825 , (2) 0 .182 , (3) 2 .154 , (4) 0.77910 , (1) 0 .6 2 9 , (2) 2 .860 , (3) 2 .077 , (4) 2.02614 „ (1) 0 .696 , (2) 1 .480, (3) 1 .323, (4) 2.710

(3) 1 .713 ,%12 , (1) 0 .4 6 7 , (2) 0 .535 , (3) 1.298, (4) 1.932

(3a) 2 .284 , (4a) 3.68010 „ (1) 0 .635 , (2) 0 .8 3 0 , (3) 1 .141, (4) 0.908

8 „ (1) 0 .895 , (2) 2 .048 , (3) 1 .905, (4) 1.584

76

CURRENT DEFLECTION.

8 amps (1) 19.996, (2) 17.660, (3) 21.211, (4) 17.88110 „ (1) 83.523, (2) 25.106, (3) 26.977, (4) 27.09814 „ (1) 56.325, (2) 54.106,

(2a) 53 856(3) 64.800, (4) 61.446

12 „ (1) 43.130, (2) 40.705, (3) 42.860, (4) 41.326(3a) 40.476, (4a) 37.340

10 „ (1) 31.294, (2) 27.993, (3) 27.238, (4) 27.8038 „ (1) 17.835, (2) 15.158, (3) 17.393, (4) 18.169

From the above data we can calculate the values of thedifferences of susceptibility as given by the various series ofobservations. The results of the calculation are given below.

Magnet current. Difference o f susceptibility.

8 amps. 11 .34X 10-6 \10 „ 10.67X 10-6 114 „ 10 .54X 10-6 f o v x m «> mean = 10.73 X 10-612 „ 10 .85X 10-6 (10 „ 10.84 X 10-6 ]8 * 10.16 X 10-6 '

Final Observations.

In the present section are given the final observations ofthe differences of susceptibility of the crystal sections ofcobalt ammonium sulphate and nickel sulphate at the differenttemperatures investigated. The figures are throughout meanvalues derived from observations with various field strengths,no systematic dependence óf the differences of susceptibilityon the field strength being observed.

77

Cobalt Ammonium Sulphate.

I. Crystal c — axis perpendicular to (001) plane.

Temperature. Difference of susceptibility.

290° 1.11 X 10-6i7 . 10.7 X lO -620.3 46.8 X 10-616.7 66.8 X 1 0 -614.5 65.7 X 1 0 -6

II. Crystal a — axis perpendicular to (100) plane.

Temperature. Difference of susceptibility.

290°5 0.403 X 10-677.18 1.88 X 1 0 -620.33 9.69 X 1 0 -616.75 12.10 X 10-614.8 14.07 X 10-6

III. Crystal b — axis perpendicular to (010) plane.

Temperature. Difference of susceptibility.

2897 3.88 X 10-677.4 32.6 X 10-620.3 121.0 X 10-617.0 163.8 X 1 0 -614.6 170.9 X 10-6

78

Nickel Sulphate Heptohydrate.The final results for one of the crystal sections of nickel

sulphate are given below in full as an example of the measure­ments on this crystal.

Crystal c.

Difference of susceptibility.

1.36 X 1 0 - 71.29 X 1 0 -71.28 X 1 0 -7

6.79 X 1 0 - 75.77 X 1 0 -7

1.65 X >0-61.70 X 10-61.72 X 10-6

2.28 X 10-62.20 X 10-62.22 X 10-6

mean1.29 X 10 -7

mean5.78 X 1 0 -7

mean1.69 X 10-6

mean2.23 X 10-6

Temperature. Magnet current.

10 amps.12 „

14 „

14 „16 „

14 „16 „12 „14 „16 „12 „10 »12 „

14 „

10 „12 „14 „

10 „12 „

12.04X10-® )12.26 X 10-6 |12 .26X 10-6 )

16 .47X 10-6 )17.02 X 10-6 (16.96 X I » -6 )

21.29 X 10-6 121 .71X 10-6 )

mean12.18 X 10-6

mean16.91 X 10-6

mean21.60 X 10-6

287°.5 <

169°.5 I

77°.2 |

65°.6 |

20°.43 |

17°.07 |

14°.65 I

With the other crystal section of nickel sulphate hepta­hydrate the noteworthy result was obtained that the differenceof susceptibility decreased with fall in temperature instead ofincreasing as is the case with the other crystal sections. Thisresult is in agreement with the observations of Lutteroth1).The final values for this crystal section are given below.

i) Loc. cit.

79

Crystal m.

Tem perature Difference o f susceptib ility .

•289.0 8.1 X 1 0 - 8169.B 2.6 X 1 0 - 8

77.29 2.2 X 1 0 - 864.6 2.1 X 1 0 - 8

20.83 2.0 X 1 0 - 816.66 2.0 X 10 ~814.6 1.96 X 1 0 -8

§ 2. Final Results.

Cohalt Ammonium Sulphate.

From the observations given in the previous section wemay draw graphs and with their aid read off a series ofvalues of a, h and c (see page 77) in the immediate neigh­bourhood of the observations. The following are the valuesof a, h and c finally adopted in the calculation of theprincipal susceptibilities.

T a x 1 0 6 b X 106 c x 1 0 ®

290° 0.40 1.11 3.8877°.2 1.88 10.7 82.620°.3 9.60 46.8 12116°.7 12.1, 66.8 15414°.6 14.2 65 7» 175

From the above data we may next calculate the values of ■■p.p is however to be determined from the roots of a quadraticequation involving cos2 \p. We thus obtain the following twoseries of alternative values for cos2 \p.

80

T Cos1 2 * p.

290° 0.7340 or 0.449077°.2 0.7756 „ 0.494920°.3 0.7920 „ 0.515616°.7 0.7843 „ 0.505214°.6 0.7845 „ 0.5090

A calculation enables us- to distinguish the applicable andinapplicable values (positive or negative) of the square rootsto be used to determine cos p and hence p. We thus arriveat two alternative values for the angle viz:— about 27°40'and 135°31'. As the first value is in good agreement withthe value 27°31' given by F inke, this value has been used infurther calculation of the results.

Then calculating the principal susceptibilities*) from theexpressions (5) on page 58 we obtain the following finalvalues for these quantities.

T Xt x 1 0 6 %2 X 106 %3 X 1 0 6 Xm X 1 0 6 P

290° 26.1 21.9 25.5 24.6 31°18'77° .2 91.2 58.6 85.6 78.5 28°17'20°.3 286.5 165.. 277.4 241 27°8'16°.7 346 192 325 288 27°41'14°.5 388 213 365 322 27°40'

It will be seen that, with the exception of the value atatmospheric temperature, for which the precision of thedetermination is necessarily small, the values of P are fairlyconstant, showing that the magnetic axes preserve their orient­ation with regard to the crystallographic axes down to thelowest temperature investigated.

1) These values are calculated per unit mass. Hence the term“principal susceptibilities” should perhaps be more correctly “principalspecific susceptibilities.”

81

In the following table are given the values of the molecularsusceptibilities.

T ^ im ^ 2 m %Sm

290° 0.0103, 0.0086, 0.0100,77°.2 0.0360u 0.0231, 0.0338,20°.3 0.113, 0.0654, 0.109,16°.7 0.136, 0.0769, 4.128,14°.7 0.1534 0.08420 0.144,

For purposes of comparison with any theory of para­magnetism the above values must be corrected for the dia­magnetic proporties of the anion, ammonium sulphate vandthe water of crystallisation. The corrected values are givenbelow.

CoS04 NH4 2 S 04 6H20

T 3 5 0 °K

Eig. 9.

6

82

T <y'•im 2m % '3 m ] I K m m

290° 0.0105b 0.008S4 0.0102, 95.24 113.2 97.4577°. 2 0.03618 0.0233, 0.0340, 27.64 42.83 29.3620°.3 0.113, 0.0656o 0.109, 8.812 15.24 9.11716°.7 0.1870 0.0761, 0.128, 7.300 13.13 7.77114°.5 0.153, 0.0843, 0.144, 6.511 11.85 6 9 2 1

In the above table the values of the inverses of thecorrected molecular susceptibilities are added for convenience.The latter values are plotted as functions of the absolutetemperature in fig. 9.

In testing the results for a W eiss law % (T A ) = Cwe obtain the results given below.

T %[m ( T + 9.8) ^ ( 2 ’+52.4) %;m( r + i4.6)

290° 3.129 3.026 3.125• 77°.2 3.148 3.026 3.127

20°.3 3.416 4.769 3.82716°.7 3.629 5.263 4.02314°.5 3.732 6.645 4.205

Hence we have At = 9.8, A2 = 52.4, A3 = 14.6.

Nickel Sulphate Heptahydrate.The results of the calculations of the principal suscepti­

bilities of nickel sulphate heptahydrate will now be given.The final values for a, c and m use in the calculations willbe found in the table below.

T c 3 a m

0000cq 1.29 X 1 0 - 7 4 9 .8 3 X 1 0 - 6 3.10 X 'O -8169°.6 5.78 X 1 0 - 7 104.3 X 1 0 -6 2.6 X 1 0 - 877°. 29 1.69 X 1 0 - 6 231.9 X 1 0 -6 2.2 X l O - 864°.5 2.30 X 1 0 -6 274:5 X 1 0 -6 2.1 X lO -820°.33 12.32 X 1 0 -6 666.0 X 1 0 -6 2.0 X 1 0 -816°.65 17.41 X 1 0 -0 810 X 1 0 -6 2.0 X 1 0 - 8I4°.6 21.57 X 10-6 882 X 1 0 - 6 1.95 X 1 0 -8

83

From the above data the following results were obtainedin calculating %i , %2 and %3 from the expressions (4). onpage 57.

T X 106 X2 X 106 %3 X 106

288°169°.577°.2964° .52Q°.3316°.6514°.6

16.6635.178.1

.92.6,228.,27&v304. „

16.5334.576.4,90.3,

216.,255.,283.,

16.6334.877.39L 6

219,261.,290.4

Calculating the molecular susceptibilities we obtain thefollowing values.

T %i m %2 m

288°169?.577°.2964°.520°.3316°.6514°.6

0.0046,0.0098,0.02190.02600.06410.07830.0856

0.0046,0.0096,0.0215002540.06060.07180.0796

0.0046, '0.0097,0.02170.02570.06180.07340.0816

These values must now be corrected for the diamagneticproperties of the anion and the water of crystallisation. Thecorrected figures are. given below together with the inversesof the corrected molecular susceptibilities.

T %1 m % 2m % 3m 1 / % 2m 1/ % 3m

288° 0.0047, 0.0047, 0.0047, 209.2 211.0 209.7169°.5 0.0099, 0.0097, 0.0098, 100.5 102.2 101.377°.29 0.0220 0.0216 0.0218 45.45 46.30 45.8764°.5 0.0261 0.0255 0.0258 38.31 39.22 38.8020°. 33 0.0642 0.0607 0.0619 16.68 16.48 16.1616°.65 0.0784 0.0719 0.0735 12.76 13.91 13.6114°.6 0.0857 0.0797 0.0816 11.67 12.65 12.26

84

Testing now for a Weiss law we obtain the followingresults with A, = — 59.9, A2 = — 58.0, A3 = — 59.7.

T X \ m { T — 59.9) 58.0) % 3 m ( T — 59.71

288° 1.090, 1.090, 1.090169°.B 1.090, 1.090, 1.08,

77°.29 0.382, 0.416, 0.383,64°.5 0.120, 0.165, 0.123,20°. 33 — 2.540 - 2.287 - 2.37,16°.65 — 3.391 — 2.973 — 3.16,14°.6 - 3.882 -3 .4 5 9 — 3.68,

—y

C H A P T E R IV.

DISCUSSION OF RESULTS.

§ 1. Examination for the law X (T + A) = C.

A glance at the graph (fig. 9) which gives the inverseof the principal susceptibilities of cobalt ammonium sulphateas functions of the absolute temperature will show immediatelythat down to about 80° K. the graphs consist of threeapproximately parallel straight lines, none of which passesthrough the zero. Prom this fact we may draw two importantconclusions viz: —

(1) that each of the principal susceptibilities follows theW eiss latv X (T A ) —-C (and• not the simpler Curie law% T = C ) down to about nitrogen temperatures and

(2) the Curie constant C is approximately the same in thethree cases since C is given by the slope of the curve.

The similar result of Foëx*) thus receives confirmationfrom the present work.

In columns 2, 3 and 4 of the second table on page 82 aregiven the values of X\m(T + A1), %2„, (T A2) and X'zm{T + As)derived from the observations. The values of A1( A2, A3 are9.8, 52.4 and 14.6 respectively, the greatest value belongingof course to the axis of smallest susceptibility. The valuesof C derived from the calculations will be seen to be equalwithin the limits of accuracy of the experiment.

An examination shows that similar phenomenon are shown

1) Loc. cit.

86

by nickel sulphate heptahydrate 1). In this case also each ofthe principal susceptibilities follows the law % (T -\- A) = Cbut only down to temperatures not lower than about 160° E .The Curie constant obtained for the three principal directionsis again the same in the three cases. In columns 2, 3 and 4of the last table on page 84 are given the results obtainedin testing for the W eiss law. It'w ill he seen that the valuesof the A’s are all negative, At , A2, A3 being respectively equalto — 59.9, — 58.0 and — 59.7.

From the above results it is immediately obvious that theCurie constant is a quantity of great significance in connectionwith the magnetic properties of crystals since it is a quantitywhich is equal along the principal magnetic axes. The Curieconstant is of special significance in the theory of W eisssince from it the number of magnetons 2) present in a sub­stance is to be calculated from the following expressions.

n = / 1123.5 , <r2 = 3 % (T + A) R = 3 CR.

in which n — number of magnetons,5-= magnetic moment per gram-molecule,

R — gas constant per gram-molecule.

Performing such calculations for the three principal suscep­tibilities of a crystal we obtain, of course, the same numberfor the three cases, this number being, within the limits ofaccuracy of the knowledge of the A’s, equal to the numberobtained from similar calculations on the mean susceptibilitiesas determined from measurements with the powdered sub­stance. This was to be expected as the latter number is themean of the already equal numbers relating to the crystal.

1) No graph is given of the principal susceptibilities of nickel sul­phate as the differences of the susceptibilities are so small as not toshow clearly on a graph on any reasonable scale. The shape of thecurves may however be visualised by comparison with the “mean”curve, the individual curves being all similar to the latter.

2) The experimental magneton of W kiss is of course meant hereand not the theoretical magneton of Bohk.

87

It would therefore seem that the magnetic moments of theparamagnetic atoms in crystals are symmetrical with regardto the principal magnetic axes.

§ 2. Deviations from the Weiss law.

Considering first the case of cobalt ammonium sulphate, wewill now examine in how far the law % (T -J-A) = C holdsfor the lowest temperatures. It will be seen on examinationof columns 2, 3 and 4 of the second table on page 82 andof the graphs in fig. 9 that this law ceases to be valid attemperatures lower than about 75° K. Thus the values of% (21 a ) instead of remaining constant become graduallygreater, showing that the principal susceptibilities increasemore rapidly with fall in temperature than is given by theW eiss law. These deviations from the Weiss law appear tobe the exact counterpart of the deviation shown by anhydrousmanganese sulphate in which the susceptibility increased lessrapidly with fall in temperature than is given by theW eiss law.

Again the magnitudes of the deviations from the straightline law are in the inverse order of the magnitudes of thesusceptibilities,. the axis of smallest susceptibility showing thegreatest deviation and vice versa.

Reference to the last table on page 84 will show thatthe principal susceptibilities of nickel sulphate heptahydratedeviate in a particularly. interesting manner from the law ofW eiss at low temperatures. Thus this substance commencesto deviate from this law at a temperature of about 160° K.,the susceptibilities remaining throughout legs than would begiven by the W eiss law. The curves for 1J%m as functions ofthe absolute temperature are however somewhat complicatedas they posses a point of inflection such that for temperaturesabove 70° K. and below 170° K. the curve is convex to thetemperature axis but at temperature below 70° K. it is concavetowards the temperature axis.

In each of the cases here considered the three principalsusceptibilities when plotted as in the figures given here

88

follow the same type of curve as the mean susceptibilityderived from measurements on the powdered material.

Thus the principal susceptibilities do not show phenomenonwhich are hidden when the mean is taken. This is, however,probably not a general phenomenon for Foëx from his measure­ments on siderose concludes that the principal susceptibilitiesof this show different types of deviation from the "Weiss lawfor the various axes (see page 51).

§ 3. Connection of A with Crystal Structure.

As has been mentioned in Part I, K amerlingh Onnes andOosTÉRHTJisJ) have suggested that the constant A is relatedin a simple manner to the distances between the paramagneticatoms in a solid, A being greater the smaller the distancebetween the atoms in question. They also suggested that thevalues of A in the different directions in a crystal mightbe simply related to the corresponding linear concentrationsof the paramagnetic atoms in these directions. In making thissuggestion it was tacitly assumed that the diamagnetic atomsof thé crystal exert no immediate magnetic action on theparamagnetic atoms but act merely as diluents.

This latter suggestion has now been tested and extendedas will now he described. If the structure of either of thecrystals investigated in the present work had been known, asa result of X-ray analysis, it would have been a simplematter to test the above-mentioned suggestion and determine*the actual relation, if one really exists, connecting the A’swith the corresponding spacings. Unfortunatelly we have nosuch data available for either of the crystals at present underconsideration. The idea was therefore tested in the case ofnickel sulphate heptahydrate in the following manner, whichis avowedly of a provisional nature only. I t was tried whether,in the light of the available crystallographic data, the nickelatoms could he arranged on a space lattice in agreement withthe latter in such a manner that the A ’s could be calculated

*) Leiden Comm. N°. 139d.

89

immediately from a single simple law in terms of the spacings.If then the hypothetical space lattice could be tested by X-raymethods and found to be highly probable, some evidencewould have been obtained for the correctness of the viewthat a simple relation exists between the A’s and the corres­ponding spacings in a crystal. Should the hypothesis beverified by future m e a su re m e n t s fo r the cases of othercrystals we should obtain a means of determining thestructure of paramagnetic crystals at least as far as theparamagnetic atoms are concerned. The method would beparticularly useful in the case of the more complicated crystalsystems for which the X-ray analysis becomes very difficultif not impossible.

We posses the following data concerning nickel sulphateheptahydrate of use for our present purpose: —

Ai = ■ - 59.9 , A2 = — 58.0 , A3 = "— 59.7 ,a : b : c — 0.9815 : 1 : 5650,

Density = 1.95 (1.953, 1.949, 1.955).Now it was found that the values of A could be calculated

accurately by means of the following expressionsA1 = a — bdiA 2 = a — bd2A 3 = a — bd3

provided that the nickel atoms were arranged on a spacelattice as shown in fig. 102) du d2, d3 are the distancesbetween the successive planes of nickel atoms along the axes a,b and c respectively. An examination of the figure will showthat the following relations hold between the d ’s.

di : d2:d3 = 0.4907:1; 0.5656.

Unfortunately the case of cobalt ammonium sulphate could notbe tested in a similar manner on account of the difficulty of inter­preting the results of X-ray analysis in the case of a monocliniccrystal.

?) will be noticed that the suggested structure is in every wayanalogous to that found by Ogg and Hopwood (Phil. Mag. 32, 518,1916) for the case of the anhydrous alkali sulphates, which are alsorhombic.

90

Then with a = — 61.79 and b — — 3.77 the calculated valuesof the A’s are’as given below..

a4 = — 59 94 , A2 = — 58.02 , A3 = — 59.64.

I t was not found possible to arrange the nickel atoms tofit the. data assuming any other equally simple law, particul­arly that of the inverse proportionality of A and spacing,which was more particularly in mind when the suggestion ofK amerlingh Onnes and OosterhuIs mentioned above was made.

A specimen of the substancewas then sent to Prof. K eesom

of Utrecht who had the goodnessto test the correctness of thehypothetical spacelattice shownin the above figure by theD ebye-Scherrer method of X-ray analysis. He now reportsthat the tèsts have succeededwell, a good photograph hayingbeen obtained on which weretwenty-four measureable lines.In making the test 326 linesemanating from all the possibleplanes of the suggested struc­ture were calculated on the

assumption that the remaining atoms of the crystal werearranged in the same manner as the nickel atoms and withouttaking into account the influence of the remaining atoms onthe intensities of the lines. The observed and calculated lineswere then compared as regards position and intensity. As aresult of the comparison Prof. K eesom announces that thehypothetical structure suggested for the crystal is in agreementwith the photograph obtained 1). I t is, however, not possible

Fig 10.

i) To give the actual words of the report. “ De conclusie moet luiden,dat de llöntgeiiphoto in overeenstemming is met de aangegeven structuur,maar wegens de moeilijkheid van dit onderzoek bij een rhombisch

91

to state definitely that the photograph leads necessarily tothe suggested structure on account of the difficulty of inter­preting the experimental results in the case of a rhombiccrystal. The test, however, furnishes considerable evidence ofthe correctness of the suggested structure.

It would seem therefore that the conception of a simplerelation existing between the A ’s and the correspondingspacings in a crystal receives support from the present work,thus opening up a possible new field in the investigation ofcrystal structure. It remains for future work to test thepoint more fully.

§ 4. Comparison with existing theories.The great majority of the theories put forward to explain

the phenomena of paramagnetism deal only with the meanvalue of the susceptibility and not with the principal suscep­tibilities of crystals. Thus the only theories actually relatingto crystals at present in existence are those of W e is s * 1) andFoëx2). W eiss has considered independently the effect ofintroducing into the L angevin theory of paramagnetism (1)a molecular field different in different directions in thecrystal and (2) a potential energy which is a function of thedirection in the crystal and is intended to represent theinteraction of the atoms of the crystal lattice. Foëx - hascombined these two ideas and has developed them further. Wewill therefore confine ourselves here to the theory of Foëx.

This theory is based on the conceptions of the “ classical”electrodynamics and contains no reference to the quantumtheory. It rests on the following three assumptions.

(1) The “ classical” kinetic theory holds and the heatenergy of the atoms is proportional to the absolute tempe­rature.

kristal (zeer speciale assen ver hou din gen bui ter} gesloten) kannen wijniet zeggen, dat de Röntgenphoto ondubbelzinnig moet leiden tot diestructuur.”

1) C.R., 156, 1674 and 1836, 1913.2) These Strasbourg 1921; Ann. de physique, 16, 174, 1921.

92

(2) -There exists a molecular field equal to n r (r = mag­netisation) in which n is equal to nlt n2, n3 for the threeprincipal directions in the crystal.

(3) There are also forces due crystalline nature of thertiaterial such that there is a potential energy which is afunction of the direction.

The assumption is also made and pains are taken to justifyit that the two last named types of action are independentof one another.

I t is then supposed that for each orientation of the atomicmagnets there is a potential energy W ; then if ot, is the anglebetween the direction of the magnetic moment of the atomand the field H, dw the solid angle and R the gas constantas in the theory of Langevin, we find the following valuefor (t the magnetisation at the temperature T :

f w3 CH Je RTcos2xdu~r— t z e — ................................................ ( 1 )J e RT dco

in which C is the Curie constant.Generalising now for the three principal directions in the

crystal and denoting by yt, y2, y3 the quantities along theprincipal axes corresponding to the ratio of the integrals

r _ wJe cos2 xdu

y ~ T w ’J e ' ^ d x

we find for the magnetisations along the principal axes

3 CHrp • Vi

3 CH?2 ---- rp • V i

3 CH• Vs

( 2)

<Jg = T

93

Introducing now a molecular field with the three coeffi-ciente nlt n2, na we obtain the three following equations inplace of (2).

* l

°2

«■3

From these expressions we can determine the principalsusceptibilities as follows:

3 Cyx \1 T — 3«j Cyx I

x*= r 3S/V„ /........... (4)T «jWj Cy% 1r 3 Cy 3 1

8 T — 3na Cy3 JIn order to proceed any further with the theory some

definite assumption must be made as to the dependence ofthe y ’s, which represent actually the mean value of cos2 »taken over all the atoms, on the temperature. The assumptionis made in the form of two alternative graphs. Thus it isknown a priori that at high temperatures y must approximateto V, and at low temperatures must have a value betweenzero and 1. The shape of the curve of the dependence ofthe y ’s on the temperature between these points is thenassumed to be given by one of two possible graphs, iri thefirst of which the rate of change of the value of y iseverywhere of the same sign hut in the second ‘there is apoint of inflection. On this basis the shape of the 1/%.T curves may be determined. It is shown that these curvesare straight lines at the higher temperatures but at the lowertemperatures they may show any of the following types ofdeviation from the straight line law.

3 CHT

3 CHT

3 CH

( H n ^ ) Vl

(■B + na<rs) y 2

(H -4- » 3<r8) y 3

(3)

94

(1) The curves may show a gradual curvature eitherconcave or convex towards the temperature axis,

(2) they may possess a point of inflection such that thecurvature changes from concave towards the tempera­ture axis to convex or vice versa and

(3) a maximum and ^a minimum of susceptibility mayoccur at the lowest temperatures.

The theory can be more precise on certain points withoutthe necessity of introducing the above assumptions. Thus fora cubic crystal we have

n l = « 2 = W3 ^ = 2/a = 2/s = I / 3

Zi = %2 = Z3 = T — nU

Thus a regular crystal will obey the W eiss law or theCurie law depending on whether n is not or is equal to zero.This is confirmed in the case of ferric alum, a cubic crystalwhich follows the Curie law.

It may also be noticed that the Curie constant C is thesame for each of the principal axes of the crystal in agreementwith the results of Foëx for siderose and the present workon cobalt ammonium sulphate and nickel sulphate hepta-hydrate.

I t will be seen thus that the theory can give a qualitativeexplanation of all the phenomena of the temperature depen­dence of the susceptibilities of paramagnetic Substancesincluding those announced in the present work. The theorymust, however, be regarded as empirical and is from its verynature impossible of quantitative verification in its presentform.

SUMMARY.

The work described in the previous pages is divided intotwo parts. In the first part is given an account of theinvestigation of the susceptibilities of the following para­magnetic substances as functions of the temperature down tothe lowest temperature obtainable with liquid hydrogen (about14° K .) : the anhydrous sulphates, the heptahydrated sulphatesand the ammonium double sulphates of cobalt, nickel andferrous iron, all in the powder form.

Whilst all these substances follow the W eiss law— C a t relatively high temperatures, they show the

following types of deviation from this law at the lowertemperatures not found in previous investigations at the lowesttemperatures.

(1) The susceptibility increases more rapidly with fall intemperature than is given by the W eiss law. (Cobaltammonium sulphate, nickel ammonium sulphate).

(2) The curve of 1/% against T possess a point of inflection.(Nickel sulphate heptahydrate).

(3) A maximum and a minimum value of the susceptibilityoccur in the region of the lowest temperatures. (Anhy­drous nickel sulphate, anhydrous ferrous sulphate).

I t has been found that the idea of the intimate connectionbetween the “ magnetic dilution” of a substance and the Aof the W eiss law suggested by K amerlingh Onnes andOosterhuis is not of quite general application, exceptions tothe rule having been found.

In the second part of the work an account is given ofthe determination of the principal susceptibilities of twocrystals over a range of temperature from about 14° K. to290° K. These are the only data yet available as regards theprincipal susceptibilities of crystals at the lowest temperatures.The substances chosen were cobalt ammonium sulphate andnickel sulphate heptahydrate. In both cases each of theprincipal susceptibilities follows the W eiss law at the highertemperatures but deviates therefrom at the lower temperatures.

96

It was found that in each case the Curie constant is the samefor each of the three principal susceptibilities.

The curves of 1/ x against T for cobalt ammonium sulphatewhich consist of three parallel straight lines at the highertemperatures, become concave towards the temperature a$isat the lower temperatures, the magnitudes of the deviationsfrom the straight line law are in the inverse order of themagnitudes of the susceptibilities as are also the A ’s of theW eiss laws Xi (T A4) = C etc. Hence each of the principalsusceptibilities of this crystal increases more rapidly withfall in temperature than is given by the W eiss law which isobeyed at the higher temperature.

In the case of nickel sulphate heptahydrate each of thecurves of the inverses of the principal susceptibilities againstthe temperature possesses a point of inflection, the curvaturechanging from convex towards the temperature axis to concaveas the temperature falls below 70° K.

Thus in each case the principal susceptibilities show thesame type of dependence on temperature as the meansusceptibility though this in probably not a general pheno­menon as Foëx found opposite types of deviation in two ofthe principal susceptibilities of siderose.

An attempt has been made to test the suggestion ofK amerlingh Onnes and Oosterhuis that the A ’s along theprincipal axes should be intimately connected with corres­ponding spacings of the paramagnetic atoms, the A beinggreater the smaller the spacing. As a result a hypthetical spacelattice has been suggested for the positions of the nickel atomsin nickel sulphate heptahydrate by using the magnetic datawhich has now been obtained. This hypothetical space latticehas been tested by the Debye-Scherrer method of X-rayanalysis and it has been found that the X-ray photographobtained is in agreement with the lattice suggested. Shouldthe idea be confirmed by further measurement a new methodwill have been obtained for the investigation of the structureof paramagnetic crystals.

S T E L L I N G E N .

STELLI NGEN.

i .

De bewering van Foëx, dat de quanten-theorie niet noodigis voor de verklaring van het paramagnetism, is weinig over-tu igend .

Foëx, Thèse, Strasbourg, 1921;Ann. de physique, (9), 1®, 174, 1921.

I I .

Dé eenvoudige betrekking tusschen de magnetische ver­dunning van een stof en de waarde van zjjn A in de wetvan W eiss % (T -+- A) = C, zooals die opgesteld is doorK amerlingh Onnes en Oosterhuis, kan niet geheel algemeentoegepast w orden.

Leiden, Comm. N°. 139e.

Hl.

Het is waarschijnlijk, dat uit magnetische metingen be­langrijke gegevens over den bouw van paramagnetischekristallen kunnen verkregen worden.

D it proefschrift, p. 88.

100

rv.

De proeven van R ankine over de viscositeit van gassenpleiten niet meer voor het atoommodel van L angmuir danvoor dat van B ohr.

Proc. Roy. Soc, A, 9S, 360, 1921; 99, 331, 1921.Phil. Mag., 42, 601 and 615, 1921.

V.

De audion-methode voor het bepalen van dielectriciteits-constanten, beschreven door P ungs en P reuner verdient devoorkeur boven die van J ona.

J ona, Phys. Zeit., 20, 14, 1919.P ongs and P rroner, Ibid. 20, 543, 1919.W eigt, Ibid. 22, 643, 1921.

VI.

Het zuur-radikaal in esters schijnt slechts weinig by tedragen tot de waarde van hunne dielectriciteitsconstanten.

J ackson, Phil. Mag., 43, 481, 1922

VII.

Systemen van gekoppelde slingers kunnen op doeltreffendewijze dienen als mechanische analogieën bij de verklaringvan de eigenschappen van oscilleerende electrische stroomenin gesloten ketens.

B arton and B rowing, Phil. Mag., 34, 246, 1917;35, 62, 1918; 86, 36, 1918.

J ackson, Ibid, 89, 294, 1920 ; 40, 329, 1920.

101

VIII.

De aanwezigheid van „reversed halo’s” in ytterby-micawijst er op, dat de temperatuur, van de mica eens aanzienlijkhooger moet geweest zijn, dan die waarbij het gevormd is.

J olt, Nature, 109, 517, 1922.

IX.

De theorie van R ayleigh van de proef van Melde is on­volledig.

R ayleigh, Theory of Sound, vol. I, p. 81.K risnaiyar, Phil. Mag, 43, 503, 1922.

X.

De abnormale getijden in het kanaal van Bristol kunnenop eenvoudige wijze verklaard worden door eene beschouwingvan de getijden in een kanaal van rechthoekige doorsnede,waarvan de diepte en de breedte evenredig aan de lengteafnemen.

T aylor, Proc. Camb. Phil. Soc., 20, 320, 1921.

XI.

De bewering van Mc. Lennan en L owe, dat de theorie vanSommerfeld over de fijne structuur van de lijnen vanBalmer onjuist zou zjjn, is gebaseerd op foutieve waar­nemingen.

Mc. L ennan and L owe, Proc. Roy. Soc., A, 100, 217, 1921.0.LDENBBR8, Ann. der P h ys, 67, 69, 1922.

XII.

De theorie van Gans, volgens welke de susceptibiliteit vaneen diamagnetische stof tot op zekere hoogte afhankelijkzou zijn van de veldsterkte, is niet in overeenstemming metde experimenteele feiten, aangezien de waarde der uit­komsten, die door sommige onderzoekers zijn verkregen endie oogenschjjnlijk de theorie bevestigen, verminderd is doorde aanwezigheid van sporen van ferromagnetisch materiaalin de door hem onderzochte stoffen.

Gans, Ann. der Phys., 49, 149, 1916.H onda, Ibid., 82, 1027, 1910.F rivold, Ibid, 67, 471, 1918; 68, 356, 1922.

102