The Bell System Technical Journal : devoted to the Scientific and Engineering aspects of Electrical Communication, Vol. 22, No. 2

P - 2 S / V 3

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED T O THE SCIENTIFIC A N D ENGINEERING ASPECTS OF ELECTRICAL CO M M U NIC ATIO N

A M ineral Survey for Piezo-Electric Materials

— W. L. Bond 145

The Fundamental Equations of Electron M otion (Dynam­

ics of High Speed Particles) . . . . L. A. M acColl 153

Quartz Crystal A p p lic a t io n s W. P. M ason 178

M ethods for Specifying Quartz Crystal Orientation and Their Determination by Optical M eans . W. L. Bond 224

A N ote on the Transmission Line Equation in Term s of I m p e d a n c e ... J. R. P ierce 263

Abstracts of Technical Articles by Bell System Authors . 266

Contributors to this I s s u e ...268

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EDITORS

R. W. King J. O. Perrine

F. B. Jewett O. E. Buckley S. Bracken

EDITORIAL BOARD

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Copyright, 1943

American Telephone and Telegraph Company

P R I N T E D I N U . S A .

F or the purposes of record an d assistance to librarians, a n d for th e inform ation of subscribers, it is to be noted th a t th ere was no A pril 1943 issue of th e Bell S ystem T echnical Journal.

I t is also to be noted th a t th ere was only one issue of V olum e 21 of th e Bell System T echnical Jo u rn a l. I t was th e issue of Ju n e 1942.

T h e Bell System T echnic

Vol. X X I I July, 1943

A M ineral Survey for Piezo-E lectric M aterials

B y W . L. B O N D

T J E C A U S E of th e increasing in te rest in piezoelectric m a teria ls in m an y branches of science an exhaustive stu d y of th e m inerals was u n d e r­

ta k e n w ith th e ob ject of finding all th e m aterials th a t could possibly be of use for piezo-electric elem ents. M uch help was derived from existing d a ta.1

C onsiderations of sym m etry show us th a t for a cry sta l to be piezo-elec- tric ally activ e it m u st belong to a cry sta l class th a t h as no center of sym ­ m e try (th e P en tag o n alico sitetred ral class of th e cubic system , how ever, alth o u g h i t 'h a s no center of sy m m etry ca n n o t be piezo ac tiv e) . 2 T his m akes tw e n ty classes of possible piezo a c tiv ity an d tw elve classes th a t could n o t possibly be active. A b o u t 9 0% of th e crystals found in n a tu re fall in those classes h av in g centers of sym m etry.

A lthough th e m ineralogical d a ta are incom plete in th e ir assignm ent of m inerals to definite classes in th e seven system s, th e existing d a ta give a s ta r t in th e choosing of m inerals likely to h av e useful piezo-electric properties.

All available d a ta were gone th ro u g h to o b ta in th e following list of m in ­ erals classified b y cry sta l stru ctu res. As m any of th e non-centric ones as were o b tain ab le in th e U n ite d S tate s were te ste d by th e m eth o d of Geibe a n d Scheibe3 (resonance in a therm ionic oscillator circuit). W henever the au th o ritie s differed on th e classification of a m ineral it w as so exam ined if obtainable.

In th e m ineral list, each m ineral is num bered according to th e nu m b er of th e class in G ro th ’s Physikalische K ristallographie, as follows: (*) in d ic at­

ing classes of possible a c tiv ity :

1 D ana— A System of M ineralogy, Ford— D a n a ’s T extbook of M ineralogy; Groth—

Chemische Kristalographie; L andölt Börnstein— Tabellen; International Critical Tables;

Zeitschrift fur Kristalographie.

2 W. V oigt, K ristal physik.

3 Z e its/ P hysik 33, pg. 761 (1925).

145

*3 Sphenoidal]

4 D om atic \ M onoclinic system 5 Prism atic J

*6 Bisphenoidal'l

*7 Pyram idal j- Orthorhombic system 8 Bipyram idal j

*9 Bisphenoidal

*10 Pyram idal

*11 Scalenohedral

*12 Trapezohedral 13 Bipyram idal

*14 D itetragonal Pyramidal 15 D itetragonal Bipyramidal

T etragonal system

*16 Pyram idal 17 Rhom bohedral

*18 Trapezohedral

*19 Bipyram idal

*20 D itrigonal pyramidal 21 D itrigonal Scalenohedral

*22 D itrigonal Bipyram idal

*23 Pyram idal ]

*24 Trapezohedral

25 Bipyram idal \

*26 Dihexagonal Pyramidal 1 27 Dihexagonal BipyramidalJ

*28 Tetrahedral-Pentagonal-Dodecahedral 29 Pentagonal Icositetrahedral

30 D yakis-D odecahedral

*31 Hexakis-tetrahedral 32 H exakis Octahedral

Rhom bohedral system

H exagonal system

Cubic system

In a d d itio n to th e above classification, th e following list of m in erals is a n n o ta te d w ith th e following sym bols:

A = active by test I = in active by test R = unavailable or rare

M = mineral occurs only m assive, am orphous or in other unsuitable form

S = crystal alw ays very small

H = mineral is alw ays non-hom ogeneous U = unstable

C = electrically conducting

? = class not absolutely certain

A ctinolite A delite Aegirite A enigm atite Aeschynite A labandite A lam osite Albite

lgondonite ctite

C L A SS IF IE D L IS T OF M IN E R A L S 5 Allanite

5 A llem ontite 5?I Allophane

2 Alm andite 8 A ltaite

*311 Alum inite A lunite Alunogen Am blygonite 5

2 H

5?SI A m esite

5 A m osite M

21 A m pangabeite 8?U I

M A m phibole 5?H I

32 A nalcim e 32

32 A ncylite 8

M A ndalusite 8

21 A ndesine 2

M A ndorite 8

2 Andradite 32

5 A nem ousite 2

A nglesite 8

A nhydrite 8

A nkerite 17

A nnabergite 5

A nnerodite 8

A nom ite S

A northite 2

A northoclase 2

A nthophyllite 8

A ntigorite 5?H

A ntlerite M

A p atite 251

A phrosiderite ?I

A p h th italite 21

A pophyllite 15

A ragonite 8

Ardennite 8

Ardunite M

A rfvedsonite 5

Argen tite 32

Argentojarosite I

A rgyrodite 32

Arrhenite H

Arseniosiderite 8

Arsonolite 32

A rsenophyrite 8

Ascharite M

A strakanite 5

A strophyllite 8?I

A tacam ite 8

A uerlite 15

A ugite 51

Aurichalcite M

A utom olite 32

A venturine 2

A xinite 2

B abingtonite 2

B addeleyite 5

Baldaufite PR

B arkevikite 5

Barite 8

B arytocalcite 5

B astn äsite I

Baum hauerite 5

B auxite M

Beaverite PS

B echilite M

Beckelite 32PS

B em en tite 8PI

B enitoite *221

Beraunite I

Bertrandite *71

Beryl 27

Beryllonite 8

Berzelianite M R

Berzelite 32PI

B etafite 32

Bindheim ite M

Binnite 32 PI

B iotite 5

Bischofite 5

B ism ite 21PI

B ism uthinite 8

B ism utite M

B löd ite 5

Blom strandine 8PMI

B oleite 15 PI

B oracite *7A

Borax 5

Borickite M

Bornite *111

B oulangerite 8

Bournonite 8

Braunite 15

B reithauptite *201

B ritholite 27PS

B rochantite 8

Brom yrite 32

B rookite 8

B rucite 21

Brushite 5

Bunsenite 32

B ytow n ite 2

Cabrerite 5

C acoxnite M

Calam ine *7A

C alaverite 5

Calciothorite M

Calcite 21

Caledonite 8

Calomel 15

Cam pylite 25

Cancrinite 27

Canfieldite 32

Cannizzarite PI

Carnallite 8

Carnotite I

Carpholite 5

Caryocerite 21

Cassiterite 15

Castorite 5

Casw ellite I

C atapleite 5

C elestite 8

Celsian 5

Cenosite 8PI

Cerargyrite 32

Cerite 8

Cerrusite 8

C ervantite 8?

C habazite 21 PI

C halcanthite 2

C halcedony 8PM

Chalcocite 8

Chalcolam prite 32

C halcophyllite 21 PI

Chalcopyrite *11C

C halcosiderite 2

C halcostibite 8

Cham osite M

C hiastolite 8

Childrenite 8PI

Chillagite 10?

C hloanthite 30

C hlorastrolite H

Chlorite 5

Chloritoid 5

C hlorm anganokalite 21

Chloropal M

C hloraphoenicite I

Chlorospinel 32

Chondrodite 5

Chrom ite 32

Chrysoberyll 8

Chrysolite 8

Cinnabar *181

C laudetite 5

C lausthalite M

C leveite 32

Clinochlor 5

C linoclasite 5

Clinohedrite *4A

C linohum ite 5

C linozoisite 5

C obaltite *28C

C ohenite M

C olem anite 5

Collinsite I

Collophanite M

Coloradoite M

Colum bite 8

C onnellite 25

C ookeite M

Cordylite 21

C ornetite PI

Corundum 21

Corynite 281

C otunnite 8

C ovellite *18?I

C restm oreite M

Cristobalite M

Crocidolite M

Crocoite 5

C ronstedtite *16A

Crookesite M

Cryolite 5

C ryolithionite 32

Cuprite 32

Cuproscheelite I

C yanite 2

Cyrtolite I

D ahllite M

D anburite 8

D atolite 5

D aw sonite M

D echenite 8

D elessite PSI

D ellafosite I

Delorenzite 8

D elvauxite M

D em antoid 32

D eschloizite 8

Desm ine 5

D ew eylite M

D iam ond 31 ?I

D iaphorite 8

Diasporę 8

D iopside 5

D iop tase 17

D ixenite ?SI

D olem ite 17

D om eykite 8

D ouglasite 5

D ufrenite 8

D ufreneysite 5

D um ortierite 8

D y sa n a ly te 32

D yscrasite 8

E dingtonite *6A

Eleonorite ?R

Ellsw orthite M R

Elpidite 8

E m bolitę 32

Em erald 27

E m m onsite ?SI

Em plectite 8

Enargite 8

E n sta tite 8

Eosphorite 8?H I

Epidesm ine 8?SI

Epididym ite 8

Epidote 5

E pistilbite *4?A

E p istolite 5

Epsom ite *6A

Erikite 8

E rythrite 5

Erythrosiderite 8

Euclase 5

Euchroite 8?I

Eucolite 21

Eucairite M

E u d ialyte 21

E udidylite 5

E u ly tite *311

Euxenite 8

Fairfieldite 2

F assaite 5

F aujasite 32

F ayalite 8

Ferberite 5

Fergusonite *101

Ferrierite I

F lorencite 21

Fluocerite 27

Fluorite 32

Forsterite 8

Forshagite M

Fouquerite I

Fow lerite 2

Francolite 25

Franklinite 32

Freibergite *31C

F reyalite M

Frieseite 8

Fritzscheite 15

Fuchsite I

Gadolinite 5

G ageite I

G ahnite 32

Galena 32

G anom alite I

Garnet 32

G astaldite 5

G ay-L ussite 5

G edrite 8

G ehlenite 15

G erm antite 32

Gersdorffite 30

G eyserite M

G ilsonite M

G ism ondite 5

Glaserite 21

G lauberite 5

G laucodot 8

G lauconite M

G laucophane 5

Gm elinite 17

G oethite 8

Goslarite *61

Graphite 21

Greenockite *20IS

Griffithite M

Grossularite 32

G uanajuatite 8?

G um m ite M

G ym nite M

G ypsum 5

H ackm anite I

H aidingerite ?S

H alite 32

H alloysite M

H am bergite 8

H ancockite 5?S

H anksite 27

H ardystonite M

H arm otone 5

H atch ettolite 32?I

H auerite *281

H ausm annite *111

H au yn ite *311

H edenbergite 5

H edyphane M

H ein tzite 5

H ellandite 5

H eloite *28?R

H elv ite *311

H em atite 21

H ercynite 32

Herderite 8

Herrengrundite 5

H essite 32

H etaerolite M

H eu lan d ite 5

H ielm ite 8?I

H iera tite 32

H illebrandite M

H iortdah lite 2

H isin gerite M

H od gk inson ite 5?I

H oeferite M

H ok u tolite H

H olm q u istite 5?H I

H opeite 8

H ow lite M

H uebnerite 5

H um ite 8

H ussakite *13

H u tchin sonite 8

H yalophane 5

H ydroboracite 5

H ydrom agnesite 5

H ydrozincite M

H ypersthene 8

Ilm enite 17

Ilm inerutile 15

Ilsem annite M

llv a ite 8

Inesite 2

Iodem bolite 32 ?I

Iodobrom ite 32

Iodyrite *26?I

Iolite 8

Jadeite 5

Jam esonite 5?SI

Jarosite 21

Jefiersonite 5?I

Jenkinsite M

Jezekite 5?

Johnstrupite 5

Jordanite 5

Joseite M

K ainite 5

K alinite 30

K aolinite 5

K asolite I

K elih au ite 5

K entrolite 8

K erm esite 5?SI

K ieserite 5

K lap roth olite 8

K lebelsbergite ?S

K n op ite 32PI

K ob altm anganerz M

K oen en ite 21

K op p ite 32

K ornerupine 8

K rennerite 8

K roeh n k ite 5

K u n zite 2

Labradorite 2

L angbanite 17

Langbeinite *28A

Langite 8

Lanthanite 8

Lapis-lazuli H

L aum ontite 5

Laurionite 8

Laurite *28

Lautarite 5

L avenite 5

Law sonite 8

Lazulite 5

Lazurite 5

L eadhillite 5

Lehnerite I

Lehrbachite M

Leonite 5

Lepidolite 5

Lepidom elane H

Leucite *31 ?I

Leucophanite *6A

Leucopboenicite 5?I

Libethenite 8

Lim onite M

Linarite 5

Linnaeite 32?

Licroconite 5?I

Liskeardite M

Lithiophilite 8

Loew eite 15

Loellingite 8

Loparite I

Lorandite 5

Loranskite 8?

Ludlam ite 5?I

Ludw igite M

M agnesite 21

M agn etite 32

M agnetoplum bite I

M alachite 5

M alacon I

M allardite M I

M anganhedenbergite 5?I

M anganite 8

M anganophyllite I

M anganosite 32

M anganotantalite 8?

M arcasite 8

M argarite 5?R I

M argarosanite 2

M argasite 5

M arialite 13

M arignacite 32

M arm olite M

M arshite *311

M artite 32?I

M ascagnite 8

M atlock ite 15?I

M aucherite i5?r

M eion ite 151

M elanite 32

M elanocerite 21

M elanophlogite ?SI

M elanterite 5

M elilite 15

M eliphanite *9?A

M ellite 15

M endozite 30

M enilite M

M erw inite I

M esolite 5

M etacinnabarite *311

M eta Torbernite I

M etavoltin e ?SI

M iargyrite 5

M icrocline 2

M icrolite 32

M icroperthite ?HS

M icrosom m ite ?SI

M iersite *31R

M ilarite 27*

M illerite *201

M im etene 25

M im etite 25, 231

M inium ?S

M irabilite 5

M izzonite 13

M olyb d en ite 27

M olyb d ite 8

M onazite 5

M onticellite 8

M ontm orillonite M

M ontroydit 8

M orensonite 6

M organite 27

M osandrite 5

M ossite 15

M ottram ite M

M uellerite M

M u scovite 5

M uthm annite *7R

N adorite 8?I

N agyagite 8

N atrolite 8

N atron 5

N aum annite 32

N em alite M

N eotan talite 32

N eo to cite M

N ep h elite *231

N ephrite M

N ep tu n ite 5

N esquehonite 8?I

N iccolite *201

N ickolsonite 8

N ick elb lu ete 5

N ickeleisen 32

N iter 8

N ocerite 21PSI

N orthrupite 32

N oselite *311

N ow m eite M

Ochrolite I

Octahedrite 15

Okenite M

O ligoclase 2

Olivenit'e 8

O livine 8

O m phacite M

Onofrite 31

Opal M

Orpim ent 8

Orthoclase 2

Osmiridium 21

O tavite 21

O ttrelite 2?I

Pachnolite 5

Panderm ite 5

Paragonite 5

Parahoepite 2

Paralaurionite 5

Paratakam ite 21?

P aravavxite I

Pargasite 5

Parisite 21

P atronite M

Pearceite 5

P ectolite 5

Penninite 5

P entlandite 32

P ercylite 32?I

Periclase 32

Peristerite 2

Perovskite 8?

P erthite ?H, S

P etalite 5

P etzite 32?

Pharm acolite 5

Pharm acosiderite *311

Phenacite 17

Phillipsite 5

Phlogopite 5

Phosgenite 15

Phosphoferrite M

Phosphophyllite 5

Phosphosiderite I

Phosphuranylite M

Pickeringite M

P icotite 32

Picromerite 5

Piedm ontite 5

Pinakiolite I

Pinguite M

P in ite M

P innoite *101

Pirrsonite *7

P isolite M

Pitchblende 32

Plagionite 5?I

Plattnerite 15

P leonast Plum bojarosite P olian ite P ollu cite P olyb asite P olycrase P o ly d y m ite P olyh alite P olym ign ite P ow ellite P rehnite Priorité Prism atine Probertite Prochlorite P rou stite Pseudobrookite P seudom alichite P silom elane P sitta cin ite P tilolite P ucherite P um pellyite Pyroargyrite P yrite Pyroaurite Pyrochlore Pyrochroite P yrolusite Pyrom orphite Pyrope P hyrophanite P yrop hyllite Pyropissite Pyrosm allite P yrostilpn ite P yroxene Pyroxm angite P yrrhotite Quartz Q uenselite Q uercyite Q uisqweite R alston ite R am m elsbergite R aspite R ealgar R habdophanite R hodochrosite R hodolite R h od on ite R hom ite R ichterite R ichardite R iebeckite R in k ite R in n eit R ip id olite R isorite R iversideite

32 R om eite 21 R oscoelite 15 R osenbushite

?I R ow landite

5 R uby

8 R um pfite 32? Rutherfordine 5?I R u tile

8

13C Safflorite

*71 Sal-am m oniac 8?I Salite

8 Samarskite

M Sanidine

5?I Sapphirine

*20C Sarcolite 8?S Sartorite M Sassolite M Scheelite M Schefferite

?S Schirmerite 8 Schizolite I Schorlom ite

*201 Schreibersite 30 Schrockingerite 21 Schrotterite 32 Schw artzengergite

*20?I Schw etzite 8?H I Scolecite

25 Scorodite 32? S em seyite

17 Senarm ontite 8 Sepiolite M Serpentine

I Serpiente 5 Shortite 5 Siderite 2 Sillim anite

*20?C Sipylite Skem m atite

*18A Skutterudite I S m altite M Sm ithsonite M Sodalite

Sodaniter 32 Spencerite

8 Spessartite 5 Sphalerite 5 Spinel

M Spodum ene

21 Spurrite I Staffelite 2 S tannite 2 Staurolite I Steenstrupine M Stephanite

5 Sternbergite 5 Stibiconite 21 Stibiotantalite

5 Stibnite 32?I Stichtite

M Stilbite

32?I Stilpnosiderite

?S S tolzite 5 Strengite M Strom eyerite 21 S trontianite M . Struvite

5 Sulfoborite 15 Sulfur

S u lvanite 8 Sussexite

*28 Svanbergite 5 Sych n od ym ite 8 S ylvan ite 5 S y lv ite 5 Sym plesite

*10?I Syndalphite 5?S

9

S yn gan ite

Z

131 T achyaphaltite 5 T achyhydrite

M T alc

2 T an talite 32 T apiolite M T arbu ttite

8 Tasm anite M T eallite

?S Tengerite

*31 Tennantite

*4A Tenorite 8 Tephroite 5?I Tetradym ite

32 Tetrahedrite

M Thalenite

5 Thaum asite 8?S Thenardite

*7A Thermonatrite 21 Thom senolite

8 Thom sonite

*101 Thorianite

H Thorite

30 T hortveitite

*381 Thuringite 21 Tiemannite

*31?I Tiger-eye 21 Tilasite

5 Titanite 32 Titanm agneteise

*31A Topaz

32 Topazolite 5 Torbernite 5?I Tourm aline

M Trechm annite

*111 Trem olite 8 T rid ym ite 21 Trim erite

*7RI T rip h ylite 8 T riplite M Triploidite

*7A T ritom ite 8?I Troegerite

I T roih te

5? Trona

T roostite 17

Tschefikinite M , H

Tscherm igite 30

T ungstenite M

T u n gstite 8

Turgite I

Turquois 2

T ych ite 32

T yrolite M

T yson ite 27

U lexite M

U llm annite 30

U ralite PHI

U raninite 32

U ranocricite 8?

Uranophone M

Uranopilite M

U ranosphaerite M

Uranospinite 8?

Uranothallite 8

Uranothorite M

Uranotile 2

U tah ite ?S

U van ite 8?S

U varogite 32

V alentinite 8

V anadinite 251

Variscite 8?S

Vauxite I

Verm iculite I

V esuvianite 15

V illiaum ite 32?I

V ivianite 5

Volborthite I

V oltaite 32PI

V onsenite 8?I

Wad M

W agnerite 5

W alpurgite 2

W arwickite I

W avellite I

W ernerite *101

W hew ellite 5

W hitneyite M I

W iikite I

W ilkeite I

W illem ite 17

W ilsonite I

W itherite 8

W ittichenite 8

W oehlerite 5

W olfachite 8

W olframite 5

W ollastonite 5

W ulfenite *10

W urtzite *20A

X an thoconite 21

X an th op h yllite 5 PI

X an thoxenite 5PS

X en otim e 15

Y ttrialite M

Y ttrocerite M

Yttrofluorite 32

Y ttrokrasite 8

Y ttrotan talite 8

Zeratite M

Zaophyllite I

Zeunerite 15

Zincite *19?I

Zinkenite 8

Zinw aldite 5

Zircon 15

Zirkelite 32

Zoisite 8

Zorgite M

Zunyite *31AS

Of th e 830 m inerals listed 70 belong to classes th a t allow piezo -activ ity b u t only 17 are fo u n d to be ac tiv e b y th e Giebe a n d Scheibe te st. (O ur te s t of Io d y rite w as n egative b u t G reenw ood a n d T o m b o u lian4 found it to be activ e; on th e o th e r h an d , we found Scolecite to be ac tiv e w hile th e y re p o rt it inactive.) I t m a y b e t h a t o th e rs of th e rem aining 56 classes h av e such sm all piezo-electric c o n stan ts as to be u n d etectab le. O th ers m a y be in ­ correctly classified as to sym m etry.

Of these ac tiv e m aterials, q u a rtz is th e m ost im p o rta n t. B ecause of its excellent m echanical p ro p ertie s (stab ility , etc.) as well as for its rela tiv e cheapness it seems d estined to rem ain one of th e m ost im p o rta n t piezo m a ­ terials.

T o u rm alin e is also im p o rta n t because of th e high m a g n itu d e of its elastic m oduli in certain directions; how ever, it c a n n o t be o b ta in ed in large pieces of sa tisfac to ry hom ogeneity.

S p halerite is v ery difficult to han d le because of its m a n y cleavage planes, a n d ap p ears to give little prom ise of becom ing p rac tica lly useful. I ts a c tiv ­ ity is q u ite m arked.

H om ogeneous cry stals of calam ine a p p e ar to be v ery rare, so th a t w ork­

able crystals large enough for o rd in ary piezo-electric app licatio n are u n ­ obtain ab le. M o st of th e m a te ria l occurs m assive.

4 On Piezo E lectricity— Greenwood and Tom boulian— Zeits. f. K rist. Jan. 1932.

E p so m ite gives a m a rk e d response b u t th e cry sta ls are gen erally sm all and t ey do n o t w ea th er well. T h ere is som e p o ssibility, how ever, th a t they can be m a d e artificially.

B o rac ite gives a m a rk e d response, b u t b o rac ite a lte rs slowly. I ts imper­

m anence m a y b a r it for som e uses.

S tib io ta n ta lite occurs only in th in scales, a n d th e necessary cuts m ust be m ade in th e m o st w asteful w ay. T w inning is p re v a le n t a n d the composi­

tio n v aries w idely.

Scolecite occurs only as sm all cry sta ls a few m illim eters in diam eter and a ce n tim ete r o r so in length, uniform ly tw inned.

Io d y rite h a s been found to be ac tiv e b y o th e r investigators. I t is electrically conductive, v ery soft a n d n o t v ery com m on.

S tru v ite is soft, u n sta b le , a n d occurs only in sm all crystals.

Z u n y ite occurs only in m in u te crystals.

L an g b e in ite slowly changes its c ry sta l stru c tu re . I t m ay be m ade a r ­ tificially so m a y b e of som e use if it can b e k e p t from alteration.

L eu c o p h an ite a n d M e lip h a n ite are re la te d m inerals. N either seem s to occur in good (i.e., hom ogeneous a n d untw in n ed ) crystals of usable size.

W u rtz ite does n o t a p p e a r v ery ac tiv e b u t good crystals w ere n o t obtain ab le.

T ie m an n ite cry stals were also u n o b tain ab le, b u t fragm ents of m assiv e tie m a n n ite responded. C ry sta ls m ig h t respond more energetically if th e y w ere o b ta in ab le, b u t m inerals t h a t are too difficult to g et w ould n o t b e of p ra c tic a l use.

E p istilb ite occurs only in sm all specim ens, uniform ly tw inned.

T h e m ineral clinohedrite is stro n g ly active b u t crystals are v e ry ra re . C ro n ste d tite a n d E d in g to n ite are v ery weakly active. C ry sta ls of th e se are v ery rare.

(D ynam ics of H igh Speed Particles)

By L. A. MacColl

I . In t r o d u c t i o n

In w ork relatin g to th e m otion of electrons a n d o th e r p article s it is fairly com m on to assum e th a t th e p article s obey th e laws of N ew to n ian dynam ics.

T h a t is, briefly, it is assum ed t h a t th e rec tan g u la r coordinates (x, y, z) of th e p article u n d er consideration satisfy th e differential equ atio n s

m x = X , m y = Y , m'z — Z ,

w here m is th e m ass of th e p article (assum ed c o n sta n t), X , Y , a n d Z are th e com ponents of th e applied force, an d th e d o ts in dicate d ifferentiation w ith respect to th e tim e t.

H ow ever, it is well recognized now t h a t th e above equ atio n s are n o t stric tly correct, a n d t h a t th e y m erely rep resen t an appro x im atio n w hich is ad e q u ate w hen th e speed of th e p a rtic le is sufficiently sm all com pared w ith the speed of light. T h e system of dynam ics based upon th e correct eq u a­

tio n s1 (which will be exhibited presently) is com m only called relativistic dynamics, n o t because an y know ledge of th e th e o ry of re la tiv ity is essential to its u n d ersta n d in g a n d use2, b u t b ecause it is in ag reem ent w ith th e th e o ry of re la tiv ity (which N ew tonian dynam ics is n o t), because it w as first de­

veloped in connection w ith w ork on th e th e o ry of rela tiv ity , a n d because even y e t v irtu a lly all of th e expositions of th e su b ject are to be found in books a n d p ap e rs dealing p rim a rily w ith th e th e o ry of rela tiv ity .

J u s t w here th e dividing line should be set betw een cases in w hich N ew ­ to n ia n dynam ics is an ad e q u a te ap p ro x im atio n a n d cases in w hich it is necessary to use rela tiv istic dynam ics is, of course, a ra th e r vague question which ca n n o t be answ ered sim ply a n d definitely. W e m a y note, how ever,

1 I t is not the purpose of this article to discuss questions of fundam ental physics, or the physical valid ity of any particular equations. For purposes of discussion, we assum e outright that relativistic dynam ics is at least more nearly correct than is N ew tonian dynam ics.

2 T he theory of relativity can be described briefly as a theory of the relations betw een the descriptions of phenom ena in term s of different system s of reference. W e shall not be concerned w ith this theory, because we shall be em ploying the sam e reference system throughout m ost of our discussion. In the final section of the paper we shall consider purely geom etrical transformations of the coordinate system . These transformations, how ever, involve nothing th at is really characteristic of the theory of relativity in the usual sense.

153

th a t according to rela tiv istic d ynam ics th e m ass of a five th o u sa n d volt electron is a b o u t one p e r cen t g re a te r th a n th e m ass of a n electron a t rest.

F ro m th is we can infer th a t, while N ew to n ia n dyn am ics m a y be adequate for m a n y p urposes in o u r stu d ies of electron m otion, we do n o t have any g re a t a m o u n t of m argin, a n d t h a t it will b e necessary to use relativistic d ynam ics w henever we wish to o b ta in really good resu lts concerning the m o tio n of even m o d e ra tely high speed electrons.

T h is article is p u rely expository. I ts p u rp o se is to set forth the funda­

m e n ta l eq u a tio n s a n d theorem s of re la tiv istic p a rtic le dynam ics in a clear a n d concise form , u n encum bered w ith a n y m a te ria l rela tin g to the theory of re la tiv ity pro p er. A lm ost all of th e m a te ria l is to be regarded as already know n, b u t a p p a re n tly it is only to be fo und in an inconvenient and scattered form . T h e incom plete b ib lio g ra p h y a t th e end of th e p a p e r gives references to some of th e m ore accessible sources of th is a n d o th e r related m aterial.

I I . Th e El e m e n t a r y Di f f e r e n t i a l Eq u a t i o n s o f Mo t io n

O ur discussion m ig h t be beg u n in a n y one of a num ber of ways, a n d no d o u b t th e different app ro ach es w ould appeal unequally to different read ers.

C onsidering th e n a tu re a n d p urposes of th is article, the a u th o r h as deem ed it b e s t to beg in b y w ritin g dow n a t once th e differential equations of m o tio n of a p a rtic le (according to re la tiv istic dynam ics) in th e ir m ost ele m e n ta ry form . T h en , for th e p urposes of th is discussion, these eq u a tio n s will h a v e th e s ta tu s of a fu n d a m e n ta l assum ption. I t need h ard ly be said t h a t th e eq u a tio n s a re n o t w ritte n dow n a rb itra rily . On th e co n trary , th e y re p re se n t th e consensus of m odern opinion as to th e laws u nder w hich p a rtic le s re a lly do m o v e. 3 T h e grounds, experim ental and theoretical, for this opin ion are se t fo rth in v ario u s of th e w orks cited in th e bibliography.

F o r th e tim e being, u n til th e c o n tra ry is s ta te d in th e final sectio n , we em ploy a fixed re c ta n g u la r coordinate system . In s te a d of d e n o tin g th e co ordinates of th e p article b y x, y, a n d z, as we h av e done p ro v isio n a lly in th e In tro d u c tio n , we shall denote them b y Xi, x2, a n d x 3. T h e n x 1; x 2, a n d x3 denote th e com ponents of th e velocity of th e particle. ’ T h e co m p o n ents of th e force ac tin g on th e p article will be d enoted b y X h X 2, a n d X s. F o r th e tim e being we need only n o te th a t th e force m a y d ep e n d u p o n th e coordinates, th e velocity, a n d th e tim e; la te r on we shall in tro d u c e som e m ore explicit assum ptions a b o u t th e force. T h e sy m b o l c w ill b e u se d to denote th e speed of lig h t in vacuo.

3 T he validity of these law s is not unrestricted. I t is lim ited on th e one han d b y the quantum phenomena which become appreciable on the atom ic scale, and on the other hand b y certain phenomena revealed b y the general theory of r ela tiv ity w h ich becom e appreciable on the cosm ic scale.

W e assum e t h a t th e p a rtic le m oves, u n d er th e influence of th e force (X h X 2, X 3), so th a t its coordinates satisfy th e system of differential eq u atio n s

d m3 x n /1

= X n, ( 1 1 = 1, 2, 3), (1) dt \ — (v2/ c2)

w here m3 is a p o sitiv e c o n s ta n t ch a racteristic of th e p article , an d v2 is an ab b re v iatio n for th e expression x/ 1 + x22 + x 32.* T h e positive v alue of th e square root is th e significant one; a n d w herever square ro o ts a p p e a r in th e subsequent w ork it will be understood, unless th e c o n tra ry is sta te d , th a t th e p ositive values are intended.

A few rem arks m ay help b ring o u t th e significance of th e foregoing assu m p ­ tion an d its relatio n s to th e corresponding fu n d a m e n ta l assu m p tio n of N ew tonian dynam ics.

W e call th e c o n sta n t mo th e rest-mass of th e p article , a n d we assum e (in accordance w ith th e ex perim ental evidence) t h a t m0 is id en tical w ith th e m ass of th e p article w hich is used in N ew to n ian dynam ics. In re la tiv istic dynam ics th e q u a n tity m defined b y th e eq u atio n

m =

a / 1 — (v2/ c 2)

is called th e m ass of th e p article. W e no te th a t as v /c approaches zero th e m ass approaches th e rest-m ass (whence th e ap p ro p riaten e ss of th e la tte r term ), an d th a t as v/c approaches u n ity th e m ass increases w ith o u t lim it.

C onsider th e v ec to r h av in g th e com ponents pi, p 2, p3 defined b y th e form ulae

* - m o ± n (2₎

V I - (z>2/c 2)

W e call th is v ecto r th e m om entum of th e p article . T h e m o m en tu m is equal to th e velocity of th e p artic le m ultip lied b y th e m ass.

N ow eq u a tio n s (1) assert th a t th e tim e-ra te of change of th e m o m en tu m of th e p a rtic le is equal to th e applied force.

W e h av e alread y observed th a t as v/c approaches zero th e rela tiv istic m ass of a p article approaches th e N ew tonian m ass. W e now n o te th a t as v /c approaches zero th e com ponents of th e rela tiv istic m o m en tu m ap proach th e values

pn - m o X n , (2')

* W e m ight merely say that v is the speed of the particle. H ow ever, for our im m ediate purposes, it is im portant n ot to lose sight of the fact that v is a certain particular function of the com ponents of velocity.