I propose, therefore, to generalize my previous assumption so as to admit secondary beams to emerge from the scattering atom in all directions. This amounts to an averaging of the matrix elements defined in my former note over all possible directions. According to this new assumption, only beams emerging with zero velocity should give rise to completely polarized radiation, i.e., only radiation corresponding to the short wave-length limit will be completely polarized. This is in agreement with experiments of P. A. Ross. Furthermore, only radiation near this limit would give zero intensity in the direction of the incident beam; other parts of the continuous spectrum will, of course, furnish intensity in this direction because of the emergent beams forming an angle with the incident wave. This difference in the radiation from the limit and from other parts of the spectrum was already mentioned by Kulenkampff

Sommerfeld, A.

Caltech Authors - Main

PHYSICS: A. SOMMERFELD
ABOUT THE PRODUCTION OF THE CONTINUOUS X-RAY
SPECTRUM
BY A. SOMMURPULD
CALITORNLA INSTITUTS oF TECHNOLOGY, PASADENA
Communicated March 18, 1929
Twenty years ago, I gave an account' of the angular intensity distribu-
tion of the general x-radiation, "Brems-Strahlung," of an anticathode,
on classical basis, with the result that the maximum intensity should be
shifted from the position 0 = 7r/2 to smaller angles, 0 being counted
from the direction of propagation of the cathode ray to that of the x-ray
observed. The assumption was made that the stopping process of the
incident cathode particles can be considered as rectilinear and coinci-
dent with the direction of the impinging beam. During the stopping
process, all velocities had to be considered between the initial velocity v,,
and the final velocity v2 = 0, and the radiations corresponding to these
different velocities had to be summed up. Because, according to the
classical theory, or what is the same, according to the special relativity
theory, electromagnetic field and radiation intensity in any state (vv) of
the stopping process are given by:
ev sin ] e v2 sin (
c2r (1- cos 3)3 47rc3r2 (1- A COS 0)6'
the resultant radiation during the whole process of stopping (v = const.,
v decreasing from v to 0) would be2
= e2v sin2 1 - l. (2)
167rc2r2 cos0 (1 - #cos 0)4
The shift of the maximum according to (2) of course is smaller than the
shift according to (1), if iB = v/c corresponds in both cases to the initial
velocity v. Also, the polarization first observed by Barkla of the general
x-radiation is accounted for by the hypothesis of the rectilinear stopping.
At that time, only observations with solid anticathodes were available,
the results of which seemed to agree in a general way with our formula (2).
The observed polarization could be, of course, not complete, because of the
multiple scattering and bending in the anticathode material. But, now-
adays, by the work of Duane3 on mercury vapor, by Webster4 and Kulen-
kampff5 on very thin foils (thickness, e.g., 0.5k), we are informed about
the stopping process in the single atom. We will show in this note that
especially with the observations of Kulenkampff formulas (1) hold if the
velocity ratio is occurring in (1) is interpreted in a convenient way; we
VOL. 15, 1929 393
PHYSICS: A. SOMMERFELD
will point, also, to the method, by which the total problem of production
of general x-radiation is to be treated in wave mechanics.
1. General Assumptions.-In the older quantum-theory, one would
start with an hyperbolic orbit of the incident cathode electron and would
consider the transition to another hyperbolic orbit of smaller energy, the
difference of energies being radiated as an x-ray wave. But this view
does not correspond to the spirit of wave mechanics. One must treat
the incident cathode beam as an electronic wave, modified by scattering
on the nucleus of the atom, and one must treat as well, the emergent electron
as an electronic wave of smaller energy, modified also by scattering on the
nucleus. Both waves are present in the whole space, the incident wave as
well behind the nucleus as before, the emergent as well before the nucleus
as behind. Multiplying both one has to compute the combined electric
density p = #4,'14 of the initial and the final state (#* meaning the conjugate
complex to At) and one has to compute the combined electric moment
M of the transition, which is responsible for the energy emitted. This
treatment is exactly equivalent to the usual treatment of the line spectra
where one computes, e.g., M. by the "matrix-element"
M= fx01dT, (3)
the initial eigenfunction 4,1, and the final #,2, being defined for all space
and time and working together in the determination ofM according to (3).
In our case, we deal with asymptotically plane waves and may put,
following a paper by Temple6 on the scattering of a particles,
411 = ei"kxL(ul), #2 eik*xL(u2), (4)
{ 27rw 2w 2wr 2wrIk1=-_=-mvl, k2 =-=-_mv2,
X1 h X2 h (5)
l= iklq, u2 = ik2q,
v meaning the parabolic coordinate, which is symmetrical around the
axis of incidence (x-axis), vl and v2 meaning the initial and final velocity
of the cathode particle; L(u) is a certain solution of the differential equa-
tion of Laguerre (not a Laguerre's Polynomial, but a transcendental func-
tion, so to speak, a polynomial of an imaginary degree). The frequency v
of the radiation emitted is determined by the time factor of 4,14i#, which
gives according to de Broglie's fundamental assumption:
V 1262= m (k2-k2). (6)
h 2 h 8w2m
As to the origin of the formula (4), we may mention that "scattering,"
or, what is the same, "diffraction" of an electron by a nucleus is a one-
394 PRoc. N. A. S.
PHYSICS: A. SOMMERFELD
electron problem. So, we understand that we need for this problem the
eigenfunctions of the hydrogen atom, the above-used function L(u)
being, indeed, the eigenfunction of hydrogen in parabolic coordinates
characteristic for the continuous hydrogen spectrum. If we had to treat,
on the other hand, the diffraction of an electronic wave by an hydrogen
atom, we would need the eigenfunctions of the helium atom, because we
deal in this case, with a two-electron problem, though of course the "helium"
atom would have the nuclear charge one instead of two; and so forth.
While those problems could be handled until now only in an approximate
way, by the method of perturbations, it can be shown that an exact and
complete solution can be written down valid for any atom, the difficulty
consisting only in the imperfect knowledge of those eigenfunctions. For
the production of x-rays, it is evident that we need not consider com-
plicated atomic structures, but may restrict ourselves to the consideration
of the bare nucleus.
We are not concerned here with the detailed computation of the moment
M; but will draw only some general conclusions from the form of the
expression (3). In the first place, M has only the one component Mx,
My and Mz being equal to zero because of the symmetry7 of the incident
and the emergent beam with respect to the axis of x. This statement is
equivalent to our old assumption as to the rectilinear way of stopping and
provides complete polarization, the plane of polarization being given by
the incident cathode beam and the emitted x-ray. We mention that
Ross8 observed complete polarization with a thick anticathode for the short
wave-length limit, and that we have to expect from our present (or former)
theory the -same result with a sufficiently thin anticathode for any part
of the continuous spectrum.7
In the second place, we remark that our density distribution i,64 is
not, as in the optical case, restrained to a portion of the space small com-
pared with the wave-length of radiation, but this distribution is in our
present case extending over the whole space. So we have to refine the
expression (3) by a "retardation factor" and have, also, to compute it not
for a single frequency v (cf. Eq. 6), but for an element (v,Av) of the
continuous spectrum. But for what follows the general form of the expres-
sion (3) is sufficient without any detailed knowledge of its numerical value.
We now come to the main point, viz., the angular distribution of the
radiation emitted by the oscillating electric moment (3). In the optical
case, we deal with the radiation from an atom at rest, so that we are justified
in putting if = 0 in (1). We have to replace in (1), of course, meantime,
ev by the general expression M. In this way, we get from (1) the well-
known radiation law of a dipole.
VOL. 15, 1929 395
J = C sin2 0 (7)
PHYSICS: A. SOMMERFELD
C not depending -on 0. But this would be wrong in our case of x-ray
production. The radiation is emitted really by a moving charge distribu-
tion, the initial velocity being vl, the final v2. It is obvious to take as the
mean velocity of motion during the emission process
Vl+ V2 v V 4+< 2 (8
V_=
__
=
2e- vV (8)
2 c ~mc2 2
V1 meaning the voltage of the incident cathode beam, V2 the voltage of that
particular range of the continuous spectrum to be considered. Doing so,
we get from- (1) instead of (7)
sin2 0(1-3 COS 0)6
0 as a 10
We see, at once, from this expression, as mentioned already in connection
with equation (2), that the shift of the maximum intensity toward small
angles 0 will now be larger than in our former theory. We see, moreover,
that the shift will be larger for the longer wave-lengths of the continuous spec-
trum than for the short wave-length limit. This conclusion seems at the
first view somewhat paradoxical, but it is checked, as we will see, by
Kulenkampff's experiments.
If we consider that the denominator in (1) comes from the relativity
transformation, say the motion of the radiating source relative to the
observer, the interpretation of ,3 given in (8) seems to be quite sensible.
From this point of view, it would be meaningless to take, e.g., vl - v2
instead of the mean velocity ((v1 + v2)/2); indeed, this assumption would
be in contradiction with the experiments. We saw in (6), that v -2
determines the wave-length of the radiation, but v1 - v2 does not at all
determine the geometrical relation between the source and the observer,
which is responsible for the intensity distribution.
396 PRoc. N. A. S.
PHYSICS: A. SOMMERFELD
Working with the corpuscular notion of v1 and 2 in our treatment of
electronic waves, it is perhaps not out of place to point to the well-known
fact that vl and v2 mean, in wave mechanics, the group velocities for the
incident and for the emergent wave.
It is true that our treatment of the emitted intensity is not a genuine
wave mechanical one. But the same happens at the present state of the
theory with any intensity calculation. I do not think that it is in the
present case less approximate than in the optical case.
2. Comparison with Experiments.-From (9), we get immediately for
the position of Jmax:
(1-2 sin ) (cos - cos2 - 3f sin2 0), (10)
(1 - Cos 0)7
so we have the quadric:
2IBx2+x= 3i, x =cos Omax,
the solution of which is
X =1 {it + 3(4X)_1}()
For j = 0, we get x = 0, 06max = 7r/2, as is known from the classical
oscillator at rest, equation (7); for # = 1, we get x = 1, Omax = 0, which
means "needle radiation." Figure 1 shows x as function of (3. The
points X and 0 are observed values of 0max under conditions which will
be explained immediately.
Kulenkampff gives three distribution curves of intensity corresponding
to the same initial voltage, V1 = 31.0 kv., but to different wave-lengths
of the continuous spectrum, viz.:
X = 0.435, X = 0.53, X = 0.73 A,
the short wave-length limit being
'Xm = 0.39
We may rewrite equation (6), dividing it by vmax = C/Xm and noticing
that V2 = 0 for X = Xm:
Xm V1 V2 V2
_
= 1- --
X V1 VI
This gives
V2 = V1 1 - X
VOL. 15, 1929 397
PHYSICS: A. SOMMERFELD
So we have for the three values of X mentioned above:
V2 = 3.2, 8.2, 14.4 kv. (12)
From (8) now, we get
(3 = 0.230, 0.263, 0.292,
and from (11)
x 0.550 0.600 0.640 cac.
Omax. 56.60 53.10 50.20
x 0.57 0.62 0.67 obs.9
Omax. 550 520 480
The agreement between the calculated and observed values is very satis-
factory. The observed values of x are marked by X in figure 1.
Kulenkampff gives another set of observations corresponding to different
initial voltages, viz.:
V1 = 37.8, 31.0, 24.0 16.4 kv. (13)
The distribution curves for all these voltages are "taken for the short
wave-length limit." But we may remark that in the first of the former
cases (12), the observation has been made "as closely as possible to the
short wave-length limit." Therefore, we may be justified in admitting
that also in the present cases, the limit could not exactly be reached and
we may take the same voltage
V2 = 3.2 (14)
as in the former case, for the final energy of the emergent electron. Then
we get from (8) with the values (13) and (14),
f3 = 0.248, 0.230, 0.209, 0.182,
and from (11)
x 0.578 0.550 0.515 0.467 calc.
Omax. 54.70 56.60 59.00 62.20
x 0.643 0.574 0.500 0.422 obs.10
Omax. 500 550 600 650
Also, here the agreement with the observed values is satisfactory, the
shift to smaller 0 no increasing with decreasing wave-lengths (increasing
initial voltages) in opposition to the former case where the shift was
increasing with increasing wave-lengths (increasing final voltages). The
observed values of x are marked in figure 1 by 0.
We shall not enter here in the discussion of the shape of the distribution
curves as indicated by the experiments of Kulenkampff on the one side,
and by the theory on the other side, because we would need for that
398 PROC. N. A. S.
PHYSICS: A. SOMMERFELD
purpose a more detailed computation of our moment M. I must leave
that for a fuller paper to be published later in the Annalen der Physik.
In conclusion, I think that the old problem of the stoppage of cathode
particles, attacked by Stokes, J. J. Thomson and Wiechert immediately
after Roentgen's discovery, can now be solved by the mathematical methods
of wave mechanics in a satisfactory way, as is shown by the foregoing dis-
cussion of the few experimental data available for the shift of the intensity
maximum in the continuous x-ray spectrum.
Addendum.*In the above, I have treated from the standpoint of wave
mechanics, the continuous spectrum of x-rays produced in a thin target.
The incident cathode ray was regarded as a plane de Broglie wave and the
emergent beam was treated also as a plane wave of de Broglie type, having
the same direction as the primary beam. Meanwhile I learned from the
experimental results of Professor W. Duane, which he presented at the
present meeting of The American Physical Society, that my previous as-
sumption as to the direction of the emergent beam is too specialized.
While this assumption would provide complete polarization for any part
of the continuous spectrum, Duane observes, in general, incomplete polari-
zation. While, furthermore, my assumption excludes radiation in the
direction of the incident beam, there is evidence, according to Duane, of
some radiation even in this direction.
I propose, therefore, to generalize my previous assumption so as to admit
secondary beams to emerge from the scattering atom in all directions. This
amounts to an averaging of the matrix elements defined in my former note
over all possible directions. According to this new assumption, only beams
emerging with zero velocity should give rise to completely polarized radia-
tion, i.e., only radiation corresponding to the short wave-length limit will
be completely polarized. This is in agreement with experiments of P. A.
Ross. Furthermore, only radiation near this limit would give zero in-
tensity in the direction of the incident beam; other parts of the continuous
spectrum will, of course, furnish intensity in this direction because of the
emergent beams forming an angle with the incident wave. This difference
in the radiation from the limit and from other parts of the spectrum was
already mentioned by Kulenkampff.
The main point of my previous note, viz., the account for the angular
distribution of x-rays observed by Kulenkampff, is certainly not essentially
influenced by this modified viewpoint.
I am indebted to Professor E. C. Kemble for having suggested to me this
generalization, and also to Professor Kennard for having emphasized the
analogy of this procedure with the usual treatment in the case of line spec-
tra. Indeed, when calculating intensities of line spectra, we have to sum
over all the different orientations of the atom characterized by the mag-
netic quantum number m, giving one weight to each of the magnetic levels.
VOL. 15, 1929 399
PHYSICS: FRUMKIN AND WILLIAMS
Corresponding to this is in our case the averaging over all possible directions
of the emerging beam, assuring equal weight for all of them.
* Read before the Academy, April 22, 1929.
1 Sommerfeld, A., Physikal. Zeitschr. 10, 969, 1909.
2 Illustrated in Atombau and SpektraUinien, 4th Edition, Fig. II, p. 37 and p. 755.
3Duane, W., these PROCEEDINGS, 13, 662, 1927; 14, 450, 1928.
4Webster, D. L., Clark, A., Yeatman, R. M., and Hansen, W. W., these PROCEUDINGS,
14, 679, 1928.
c Kulenkampff, H., Ann. Phys., 87, 597, 1928.
6 Temple, G., Proc. Roy. Soc., 121, 673, 1928.
7However, the symmetry is disturbed by the retardation to be introduced for the
emitted radiation as remarked below. As a consequence of this retardation the polari-
zation would be complete only for an angle of observation perpendicular to the incident
cathode beam, but in this case for every part of the continuous spectrum, so far as our
assumption about the direction of the emergent cathode beam is justified. (Note added
in proof.)
8 Ross, G. A., J. Opt. Soc. America, 16, 375, June, 1928.
9 Cf. Fig. 9 in Kulenkampff's paper, loc. cit.
10 Cf. Fig. 11, Ibid.
THE RELATION BETWEEN THE ELECTRIC MOMENT AND
THE POTENTIAL DIFFERENCE AT AN INTERFACE
By A. FRUMKIN AND JOHN WARRSN WILLIAMS
LABORATORY OF PHYSICAL CHUMISTRY, UNIVERSITY OF WISCONSIN
Communicated March 22, 1929
The relation between the electrical properties of interfaces and the
orientation of molecules was first recognized by Hardy' and discussed by
Harkins, Davies and Clark.2 It was stated in a definite way for the
case of the water-mercury interface by Gouy3 in 1917. One of US4 has
studied these potential differences at interfaces in their relation to the
polarity and orientation of different linkages in the molecule. Quantita-
tive calculations of the electric moment of the molecules from the potential
difference data have been made by Guyot' and later by Rideal.6 At an
earlier date quantitative comparisons were made by Frumkin.7 Rideal
calculates the electric moment of the butyric acid molecule to be
= 0.305 X 10-18,
assuming the orientation to be complete and the dielectric constant of the
surface layer to be unity, and compares it with the value calculated by
Smyth8 using the Gans theory
iA = ca 2.0 X 10-18.
400 PROC. N. A. S.


About the Production of the Continuous X-Ray Spectrum

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