Atomic Lattices and Atomic Dimensions

Voi. 13, 1927

227

PHYSICS: J. K. MORSE

ELC(1 1-02LC \

KwLC \2 -@2c

approximately, where co, K, L are constant and C variable. Hence, even if we neglect the term in K and associate Y with the fringe displacement, s, an equation in this form is not serviceable in identifying As c AC along linear elements, unless w2LC is small compared with 1. This would not be the case with the fundamental or any harmonics of the cylindrical pipe. Even if X refers to the frequency of the spring break taken at pitch a, the equation remains inapplicable. This suggests a simpler approach through the capacity equation Q = CV whence As cc Ai = (d V/dt) AC; or the slopes of the linear elements of the graphs are to be associated with the effective time rate at which the potential of the condenser changes. The value of d V/dt depends on the form of residual wave on which the new impulse is superimposed. Moreover a reason for the broken linear relations of s and C is now apparent; for the fringe displacement s measures the difference of level of the surfaces of mercury in the U-gage. It, therefore, also measures the potential energy localized in the stationary wave at the point of the pinhole probe, though it does this with a coefficient which may be either positive or negative. The stream lines run from the outside to the inside of the pinhole embouchure. * Advance note from a Report to the Carnegie Inst. of Washington, D. C. 'These PROCZZDINGS, 13, pp. 52-56, 1927.

ATOMIC LATTICES AND ATOMIC DIMENSIONS By JARBD KIRTLAND MORSB'

DZPARTMUNT

OF PHYSICS,

UNIVzRSITY Or

CHICAGO

Communicated March 4, 1927

By means of the concept of a spherical atom W. L. Bragg2 and W. P. Davey3 have independently computed the radii of atomic spheres of influence from the distances of closest approach obtained from X-ray measurements on crystals. In this paper these spherical atoms are specialized and the cubic atom proposed by Lewis and developed by Langmuir is extended to simple polyhedrons inscribable in spheres. It is shown in detail how these simple models can be built up into various types of observed cubic and hexagonal lattices and how the lattice constants are geometrically related to the atomic radii. From these relations possible atomic radii are computed and tabulated.

2,28

P'ROC. N. A. S.

PHYSICS: J. K. MORSE

Using cubic models piled together in such a manner that the edges of adjacent cubes are common, a face-centered cubic lattice can be constructed in the following manner. The coordinates of the cube centers and corners are expressed as fractions of the lattice constant ao as is customary in space group theory. FAC -CONThRED CuBIc LATTIcs-TYPs I Cube Centers: 0; 0; 0; 1/2, 0, 1/2; 0, 1/2, 1/2; 1/2, 1/2, 0; Cube Corners:

1/4, 1/4, 1/4; 1/4, 3/4, 3/4; 3/4, 1/4, 1/4; 3/4, 3/4, 1/4;

1/4, 1/4, 3/4; 1/4, 3/4, 3/4; 3/4, 1/4, 3/4; 3/4, 3/4, 3/4; If R is the radius of a sphere circumscribed about any one of these cubes the relation between R and the lattice constant is

ao-/3.(1)

R

4

If the cubes are piled so that adjacent corners are common, we can form a second type of face-centered cubic lattice thus: Cube Centers: Cube Corners:

FAc -CENThRED CuBIc LArTicI -TYP II O, 0, 0; 1/2, 0, 1/2; O, 1/2, 1/2; 1/4, 0, 1/4; 3/4, 0, 3/4; 1/4, O, 3/4; O, 1/4, 1/4; 1/2, 1/4, 1/4; O, 1/4, 3/4; 1/4, 1/2, 1/4; 3/4, 1/2, 1/4; 1/4, 1/2, 3/4; 0, 3/4, 1/4; 1/2, 3/4, 1/4; O, 3/4, 3/4;

R Cube Centers: Cube Corners:

0, 1/4, 1/8, 1/8, 1/8, 3/8, 3/8, 3/8, 5/8, 5/8, 5/8, 7/8,

DIAMoND 0, 0; 1/4, 1/4; 1/8, -1/8; 3/8, 3/8; 5/8, 7/8; 1/8, 1/8;

3/8, 5/8, 1/8, 3/8, 5/8, 1/8, 7/8, 3/8, 7/8, 5/8,

3/8; 7/8; 3/8; 5/8; 7/8; 1/8; 5/8; 5/8;

=

ao\2. 4

Typo CuBIc LATTIcE 1/2, 1/2, 0; 1/2, 0, 1/4, 3/4, 3/4; 3/4, 1/4, 1/8, 1/8, 3/8; 1/8, 1/8, 1/8, 3/8, 5/8; 1/8, 5/8, 1/8, 7/8, 1/8; 1/8, 7/8, 3/8, 1/8, 3/8; 3/8, 1/8, 3/8, 3/8, 7/8; 3/8, 5/8, 3/8, 7/8, 3/8; 3/8, 7/8, 5/8, 1/8, 5/8; 5/8, 1/8, 5/8, 3/8, 7/8; 5/8, 5/8, 5/8, 7/8, 1/8; 5/8, 7/8, 7/8, 1/8, 5/8; 7/8, 1/8, 7/8, 3/8, 7/8; 7/8, 5/8, 7/8, 7/8, 1/8; 7/8, 7/8,

R

=

1/2, 1/2, O, 1/2, 1/4, 3/4, 1/2, 1/2, 3/4,

3/4,

aO.V3,

0;

3/4; 3/4; 3/4; 3/4;

(2) 1/2; 3/4; 7/8; 3/8; 5/8; 5/8; 1/8; 5/8; 7/8; 1/8; 3/8; 7/8; 1/8; 5/8;

O, 1/2, 1/2; 3/4, 1/4; 3/8, 1/8; 5/8, 5/8; 7/8, 7/8; 3/8, 1/8;

3/4, 1/8, 1/8, 1/8, 3/8, 3/8, 3/8, 5/8, 5/8, 5/8, 7/8, 7/8, 7/8,

5/8, 7/8, 3/8, 5/8, 7/8, 3/8, 5/8, 7/8,

5/8; 7/8; 1/8; 3/8; 5/8; 3/8; 3/8; 7/8;

(3)

229

PHYSICS: J. K. MORSE

VOL,. 13, 1927,

GRAPHITh TYPs HZXAGONAL LATTIcs This lattice can be built up of cubes and for purposes of convenience is considered as a special type of orthorhombic lattice in which aO = box/3 instead of referring it to the hexagonal axes in general use. Cube Centers: 0; 0; 1/3, 0, O, 0, 0; 5/6, 1/2, 0; 1/2, 1/2, 1/6, 1/2, 1/2; O, O, 1/2; 1/2, 1/2, 1/2; 2/3, 0, 1/2; Cube Corners: O, O, 11/18; O, O, 8/9; O, 0, 1/9; O, O, 7/18; O, 1/2, 29/54; 1/12, 1/4, 1/27; 1/12, 1/4, 25/54; 0, 1/2, 1/27;

1/12, 1/6, 1/4, 1/3, 5/12, 7/12, 2/3, 3/4, 11/12, 1/2, 5/6,

1/27; 1/12, 3/4, 25/54; 1/6, 7/18; 1/6, 1/2, 11/18; 1/4, 1/27; 1/4, 3/4, 29/54; 1/3, 1/27; 1/3, 1/2, 25/54; 5/12, 29/54; 5/12, 3/4, 26/27; 1/2, 1/27; 7/12, 1/4, 25/54; 7/12, 0, 7/18; 2/3, 0, 11/18; 2/3, 1/4, 1/27; 3/4, 1/4, 29/54; 3/4, 1/4, 29/54; 11/12, 1/4, 26/27; 11/12, 1/2, 1/9; 1/2, 1/2, 8/9; 1/2, O, 1/27; 5/6, O, 25/54; 5/6,

3/4, 1/2, 3/4, 1/2, 3/4, 1/4,

R= boV3; R4V

R = ao

4V2-'

The axial ratios are a:c

=

bo:co

=

29/54; 1/27;

0, 1/4, 0, 1/4, 0, 3/4, 1/2, 3/4, 3/4, 1/2,

29/54; 1/27; 1/27; 29/54; 1/27; 29/54; 7/18;

1/2,

1/9;

1/9;

1/6, 1/4, 1/3, 5/12; 1/2, 7/12, 2/3,

O, 1/4, O, 1/4, O, 3/4, 1/2, 3/4, 3/4, 11/12, 3/4, 1/2, 1/2, 5/6, 1/2,

2X'2-

The axial ratios

are a:c

=

bo:co

26/27; 29/54; 26/27;

11/18;

8/9;

1:2.76.

R

=

8/9;

26/27; 29/54; 25/54;

(4)

R

CLOSE PACKmD HEXAGONAL LATTIcc This lattice can be built up of cubes and the following coordinates to orthorhombic axes for convenience. Cube Centers: O, 0, 0; 1/2, 1/2, 0; 1/6, 1/2, 1/2; O, 0, 3/8; 1/6, 1/2, 1/8; 1/3, 0, 1/8; Cube Corners: 5/6, 1/2, 1/8; 2/3, O, 1/8; O, 0, 5/8; 1/3, 0, 5/8; 1/2, 1/2, 5/8; 5/6, 1/2, 5/8;

R=bo-3_.

26/27; 29/54;

=

2+;ao

R

=

_.o

are

also referred

2/3, 0, 1/2; 1/2, 1/2, 3/8; 1/6, 1/2, 7/8; 2/3, 0, 7/8;

3c0

(5)

1:1.633.

THr BODY-CENTERED This lattice can also be built of cubes. Cube Centers: 0; 1/2, 0, 0, Cube Corners: 1/4, 1/4, 1/4; 1/4, 1/4, 1/4, 3/4; 1/4,

CUBIc LATTIcE-TYPE I

1/2, 1/2; 3/4, 1/4; 3/4, 1/4, 1/4; 3/4, 3/4, 1/4; 3/4, 3/4; 3/4, 1/4, 3/4; 3/4, 3/4, 3/4;

R/aOV 4

(6)

PHYSICS: J. K. MORSE

230

PROC. N. A. S.

Tn9 BODY-CSNTGRED CUBIC LATTICS-TYPE II Another type of body-centered cubic lattice can be formed from rectangular parallelopipeds derived from cubes and circumscribable in spheres. O, 0, 0; 1/2, 1/2, 1/2; Centers: Corners: 0, 1/2, 1/4; 1/2, 0, 1/4; 1/2, 0, 3/4; 0, 1/2, 3/4;

R = O5 4

(7)

If the center positions are identified with atomic centers and the corners are considered as possible electron positions the radius R of the circumscribed sphere becomes the radius of the atom and can be defined as the maximum distance between the nucleus and the electron or electrons on the periphery of the atom. Then equations- (1)-(7), inclusive, can be used to compute possible atomic radii for such atoms as unite with each other to form lattices by the sharing of electrons. It is considered that the lattices of the elements with the possible exception of argon are formed in this way. It is also possible that these lattices may in some cases be formed by the packing together of spherical atoms in which case the radius of these spheres can be found by halving the distance of closest approach as was pointed out by W. L. Bragg2 and Wheeler P. Davey.3'4 The following tables show the values of the atomic radii computed by equations (1)-(7) from the lattice constants together with the radii for spheres. The values of the lattice constants are taken from the International Critical Tables, 1926. For graphite the lattice used is taken from Bernal's5 investigations. FAC9-CUNTUROD CUBIC AT. NO.

EXMNT Aluminum

13 18 Argon 20 Calcium 26 27 28 29 45 46

Iron y

Cobalt Nickel Copper Rhodium Palladium Silver Cerium Iridium Platinum

47 58 77 78 79 Gold 82 Lead 90 Thorium

ae X 108 cm.

R-aoN13/4

4.043 5.43 5.56 3.63 3.554 3.499 3.603 3.820 3.859

1.751 2.351 2.408 1.572 1.539 1.515 1.560 1.654 1.671 1.766 2.217 1.655 1.694 1.758 2.130 2.18

4.079 5.12 3.823 3.913 4.06 4.92 5.04

R

a44 1.429 1.915 1.966 1.284 1.257 1.237 1.274 1.351 1.364 1.442 1.810 1.352 1.384 1.436 1.736 1.78

-

SPENNEIS

R X 1O cm.

1.429 1.915 1.966 1.284 1.257 1.237 1.274 1.351 1.364 1.442 1.810 1.352 1.384 1.436

1.736 1.78

231

PHYSICS: J. K. MORSE

VOL. 13, 1927

DIAMOND TYPE CUBIC LATTICE ao X 108 CM.

6 14 32 50

R =

aoV/3/8

0.77 1.173 1.217 1.399

3.56

Carbon Silicon Germaiium Tin (Grey)

5.42 5.62 6.46

BODY-CENTERED CUBIC LATTICE AT. NO.

3 11 19 23 24 26 42 73 74

8L.MENT

Ao X 108 CM.

Lithium Sodium Potassium Vanadium Chromium

3.50 4.30 5.20 3.04 2.875 2.855 3.143 3.272 3.155

Iron a Molybdenum

Tantalum Tungsten

aoV3/4 1.516 1.862 2.252 1.316 1.245

R =

R

aoV5/4 1.957

=

2.404 2.907 1.699 1.607 1.596 1.757 1.829 1.764

1.236 1.361 1.417 1.365

SPHNRMS

R X

108 CM.

1.516 1.862 2.252 1.316 1.245 1.236

1.361 1.417 1.365

HEXAGONAL CLOSE PACKED LATTICE bo,V

R =

AT. NO.

SEIMIMNT

4 12 22 27 40 44 58 76

Beryllium Magnesium Titanium Cobalt Zirconium Ruthenium Cerium Osmium

bo X 108 CM.

Co

3.607

2.283 3.22 2.92 2.514 3.23 2.686 3.65 2.714

R

3Co 8

X 108 CM.

5.23 4.67 4.105 5.14 4.272 5.96 4.32

1.398 1.972 1.788 1.539

1.35 1.961

1.751 1.539 1.928 1.602 2.235 1.620

1.978 1.645 2.235 1.662

SPHSRUS

R X 10' CM.

1.14 1.61 1.46 1.257

1.615 1.343 1.825 1.357

GRAPHITE LATTICE ao= BoN3

AT. NO.

6

MLEMENT

bo X 108 CM.

Graphite

2.46

co

X 108 CM.

6.79

4

1

0.753

R

=

Co

0.754

SPHERES

R X 108 CM.

0.710

One point is worthy of special emphasis. It will be noted that the atomic radius of the carbon atom in diamond is 0.77 A while that computed from the graphite is 0.75 A, a difference of only two hundredths of an angstrom. Both of these lattices are built up of cubes having one corner in common. Either this is an astonishing coincidence or it means that a carbon atom in

combination can be treated as a cube, that its radius is approximately constant and equal to 0.76 A over an extremely wide range of conditions, and that a carbon atom can unite with other carbon atoms by sharing one electron. The equality of the radii of the cubic hexagonal forms of the iron, cobalt and cerium is not so important as this result would also be obtained if the atoms were spherical. A consideration of the lattice structures herewith presented indicates

22PHYSICS: P.. S. EPSTEIN

232

PROC. N. A. S.

that there are more possible electron positions available than there are valency electrons to fill them, and the question also arises as to which of the radii are to be selected in those cases when two types of lattice are possible. If we knew definitely just how a single electron scatters X-rays these two questions could be easily settled by a quantitative consideration of the intensities of the observed X-ray diffraction patterns. In the present state, however, of our knowledge of scattering, the carrying through of such computations does not seem worthwhile. In conclusion I wish to express my thanks to Professor Henry G. Gale for his interest in this work. 1 This investigation was commenced by the author under a NATIONAL RESEARCH

FSLLOWSHIP.

2 W. L. Bragg, Phil. Mag., 42, 169-189, 1920. 3 W. P. Davey, Physic. Rev., 22, 211-220, 1923. 4 W. P. Davey, Ibid., 23, 218-231, 1924. 6T. D. Bernal, Proc. Roy. Soc., 106A, 749-773, 1924.

THE MAGNETIC DIPOLE IN UND ULA TORY MECHANICS By PAUL S. EPSTEIN CALIFORNIA INSTITUTY Op TECHNOLOGY Communicated March 7, 1927

1. In the following lines we present a method for the computation of the characteristic values of the parameter contained in linear differential equations. This method is applicable in certain cases when the equation cannot be reduced to the hypergeometric type. As the special example with which to illustrate our procedure we select the motion of an electron in the combined fields of a nucleus and of a magnetie dipole attached to this nucleus. This problem has an interesting bearing on the theory of the spinning electron, as will be fully discussed in section 6. The wave equation for this case has been set up by Fock;' we prefer, however, to generalize the problem by taking in the effect of relativity neglected by Fock, and so to arrive at an equation which differs from equation (5) of our last paper2 (to which we shall refer as loc. cit.) only by a term proportional to x/r3:

dX

2 dX + A

+

2B

+ p- k(k

f X

0.

(1)

The notations are the same as loc. cit., only the constant f is introduced here for the first time and is connected with the moment ,u of the dipole by the relation3