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On the theory of electron scattering in gases
O N T H E T H E O R Y OF E L E C T R O N S C A T T E R I N G I N GASES. BY
ALLAN C. G. MITCHELL, Ph.D., Fallow, Bartol Research Foundation of The Franklin InstRute.
A GENERAl. wave-mechanical theory of FOUNDATION collisional processes has been developed by Com~,,~c,tionNo.3S. Born 1 and applied by \Ventzel 2 to the scattering of alpha-particles by heavy atoms, and by Elsasser 3 to collisional processes in atomic hydrogen. Recently Sommerfeld 4 has applied the m e t h o d to the scattering of electrons in argon and has found agreement with the experimental results. T o carry out the calculation it is necessary to know the potential of the a t o m v(r) as a function of the distance from the nucleus. Wentzel first assumed the potential of an atom of atomic n u m b e r Z to be given by the Coulomb law v(r) = Ze/r, neglecting the effect of the electrons around the nucleus but found t h a t this led to integrals which would not converge. On assuming the potential function to be v(r) = Ze/re -T/R, where R is of the order of m a g n i t u d e of an atomic radius, he was able to obtain a scattering formula which, for alpha-particles, agreed with the Rutherford scattering law. On the other hand Elsasser found the potential of the hydrogen a t o m in terms of its eigenfunctions, the potential for the hydrogen a t o m in its normal state being similar to t h a t used by Wentzel, thus ]ustifying his assumption, and showed t h a t a scattering law for atomic hydrogen could be obtained. Finally, Sommerfeld considered scattering from other atoms by treating all the electrons of the a t o m as united in the K-shell without m u t u a l l y influencing each other. " T h e action of these electrons toward the outside would then
BARTOL RESEARCH
1 M. Born, ZS.f. Physik, 37, 863; 38, 803 (1926). 2 G. Wentzel, ibid., 40, 590 (1927). 3 W. Elsasser, ibid., 45, 522 (1927). * A. Sommerfeld, " A t o m b a u und SpektraUinien," Wellenmech. band, pp. 226 et seq.
Ergiinzungs753
754
ALLAN C. G. MITCHELL.
[J. F. I.
be represented wave-mechanically by the Z-fold electron cloud of hydrogen in its normal state, whose charge density is P
eZ 4 ~_ - - e-2r(Z/a), rra a
where a is the radius of the first Bohr circle." He then showed that the potential energy of an electron in the field of the atom is V(r)=
-
e2Z ( I- -4- Z )
(')
By using this potential energy in the Born collision equations he arrived at a scattering formula, which for high velocity alpha-particles and for n o t too small scattering angles, agrees with the Rutherford scattering law. T h e formula also agrees with the experiments of Kirchner 5 on the scattering of electrons i n argon (elastic collisions). It will be noted t h a t expression (I) for the potential energy takes into account shielding by the electrons of the atom. It is the purpose of this paper to carry through the calculations using a somewhat different m e t h o d of obtaining the potential of the a t o m at a distance r from the nucleus. T h o m a s 6 and Fermi 7 have recently shown t h a t if the electrons of an atom of atomic n u m b e r Z be considered to be a perfectly degenerate gas in the sense of the Fermi statistics, s the potential at a distance r from the nucleus is given by v =
--
/,
•
,
(2)
where 32/3h 2
# = 21~/37r4/ame2Z1/z
(m = mass of electron). This function ¢(rlg) is u n i t y when r is zero and decreases to zero for increasing r, such t h a t for distances a little greater t h a n the atomic radius the potential will be zero. No 5 F. s L. r E. s E.
Kirchner, Ann. der Phys., 83,969 (1927). H. Thomas, Proc. Cambridge Phil. Soc., 23, 542 (I926). Fermi, ZS. f. Phys., 48, 73 (I928). Fermi, ibid., 36, 902 (1926).
June, 1929.11
ELECTRON
SCATTERING IN GASES.
755
analytical expression is known for the function ¢ over its whole range, but a table of values is given by Fermi? He has used this function to calculate the Rydberg correction terms of the alkali atoms, 1° generally thought of as due to the shielding of the nonradiating electrons. In this paper we shall consider this potential function as giving the true potential of an atom at a distance r from the nucleus, inclusive of shielding, and use the potential energy derived therefrom in conjunction with the Born collision theory t() calculate the scattering from different atoms. To calculate the scattering of an electron by an atom situated at O (Fig. I), we consider a plane wave for an electron ,4
Q
,
0
X
~
p Fro. I.
approaching the atom from the negative X-direction. The wave equation for the total system, electron plus atom, is 8 7r2m
v2~ +-V-(E
-
9 See ref. 6, p. 75. 10 E. Fermi, ZS. f. Phys., 49, 550 (I928).
V ) ~ = o.
(3)
756
ALLAN C. G. )e{ITCHELL.
[J. F. I.
We consider V, the potential energy of the system, as a perturbation term and write ¢ = ~0 + ~1 + . . . .
8 7r2mE h2 we have
Substituting in (3) and writing k 2 V~(~o+~l+...)+k
~- ~ - ~
(4)
(~o+~1+...)=o,
(5)
where V26o + k26o = o,
(6)
since V~I is to be neglected in comparison with V2¢0 and k2~0. F r o m (3) and (4) we have V v2~1 + k2¢,~ = k 2 ~ ¢0, V V2~2 -~- k2~2 --- k 2 -~ 1~1,
(7)
2V
V2@n "4- k2ff/n = k --~ if~n--1 = Fn-1.
It will be noted t h a t (6) is the equation for the oncoming electron since the interaction term has been neglected. The solution of (6) for the plane wave of the electron is ~o = e ikx.
We m u s t now solve the inhomogeneous equation (7), which m a y be accomplished by Green's theorem in the following manner. If ~ and . are functions satisfying the usual conditions, Green's theorem gives
fff
(iv:. -
nV~)dT =
~~
-- " 0 s ]
We shall let eikr
-
r
,
(9)
a function satisfying the homogeneous equation v ~ + k2~ = o,
(io)
June, 192o,]
ELECTRON
757
SCATT1-RING IN (~.\SV.S.
and r/ = ¢~.
(II)
We take the origin of r as the point P, at which we shall seek the value of ~,. We shall take for our boundary of integration two surfaces, one a sphere of radius R about P, which sphere shall eventually be made infinite, and for the other boundary a small sphere of radius r0, which shall eventually 1)e made zero. Substituting (9) and (IO) in (8) we obtain
(ffe, r
•~
j'f(
_ _ (V25 n + k2¢n)d r = r
)
~ &k,~ _ ,t, Oe" d,~ -~r
r n Or
(i2)
where the negative sign in the integral over the sphere of radius r0 is incorporated to permit of the m)rmal being drawn outwards from P. Since ~,, by the fundamental conditions of the Schroedinger theory must vanish to the order of ! , where e > o (a rE conclusion which will subsequently be seen to be true) the I
integral over the surface R will vanish to the o r d e r - - • re Consider now the integral over the surface r0. Let M be the maximum value of O~b,/Or then
j'~eik~ J~o 7
eikro O~Tr~da < 1"0 M4rrr°2
and is zero in the limit.
Now
¢ ~ Or
r
r'-' 4 ~rr2dr = -- 4~-¢~
in the limit. Hence (I2) becomes eikr
¢.=
I
4~r.
VOL. 207, No. 1242--52
; f f
r (v2¢~ ÷
k~¢~)dr'
758
ALLAN C. G. ~ITCHELL.
[J. F. I.
and r e m e m b e r i n g (7) we obtain
¢,,
x
--
Fn-a(rQ) rpQ
_ .f.(/
dTQ
03)
T a k i n g the first a p p r o x i m a t i o n we have ¢1(v, -
4~-E a .
a
V(rQ) --rpQ e'er°dr'"
(I4)
F r o m Fig. I we have
rQI
t e e = [r -
= [r 2 + 2r.rQ + rQ231/2.
If r >> re then rpQ
(
=
7"
=
r --
I
--
r- 2- r • r Q
) (I5)
e.rq,
where e is a unit v e c t o r in the direction r.
W e also write
XQ ~- eo.r~
06)
where e0 is a unit v e c t o r in the direction X. becomes
erEk~e;rJ'ff
@I(P) =
E q u a t i o n (I4)
V(rQ)e'*(e-°°)'wdr'"
(17)
W e n o w change to polar co6rdinates taking O A , parallel to ]e0 - e l , as the polar axis, ~ as the longitude and 0 as the co-latitude. T h e n
p = rQ (e0 -
(i8)
e).rQ = °[e0 - e l p cos 0.
(IV)
If n o w we introduce the scattering angle O, we see from the figure 0 [e0 - e I = 2 s i n - . (20) 2
M a k i n g the a b o v e s u b s t i t u t i o n s (I7) becomes ¢1(P) =
4
e
; (oVoTo r
V(p) e ~2~~ o/2ooo, Op2dp sin OdOd,~. (2 I)
W e now use the potential energy derived from the Fermi
June, J9291
759
ELECTRON SCATTERING IN GASES.
statistics, which is
V(B) -
Ze~dP(~
for an electron at a distance 0 from the nucleus. (2 I) then becomes /~2 r z
l]/l(p ) --
2
.*2rr
twzr ,,1,0o
/
(22) Equation
\
~ AeErreikrtdo .lot Jot o~-)tP*e*2ko~l.o,'2cO.OpdpsinOdOd¢.(23)
We first carry out the integration with respect to 0 keeping p constant. Writing ~ = 2kp sin 0/2 we have ~0 T e~c°~° sin
2 . 6, OdO= ~sln
(24)
and (23) becomes after integrating with respect to ¢, ff1~e~ - 2E sinO r J0 ¢
sin 2kp sin-2 dp.
(259)
2
The analytical expression of ¢(p/#) is not known, hence the integration of (25) must be carried out graphically. We form ff~ I~ , and remembering that 16012 = I we obtain the quotient ~0 P=
',~ = [kZeZ]~ (O) ( {:)) do]~. (26/ *o 2E~-r_!s-in~I [~oo . ~°q)./~.sinx2kpsin--~ 2
Equation (26) may be interpreted as the number of particles per unit solid angle which are scattered through an angle 0 in relation to the number which fall on the atom. We shall now compare this with the expression obtained by Sommerfeld using the Born method. He obtains
P =
[e2Z]Z[ cd" ) 212 ' 0I + ( 4E~A sine2 + " 2 sin202 + a2
Z where a = ~ ;
a = radius of first Bohr circle.
(27)
It will be
760
ALLAN C. G. MITCIIELL.
[J. f,'. 1.
noted that the above formula gives exactly the Rutherford scattering law for alpha-particles if we neglect a2 in comparison to sin 2 0/2 and if we introduce the proper factor into the constant to take care of the double charge of the alphaparticle. It has been shown by Wentzel that for alphaparticles and heavy nuclei a" m a y be neglected in comparison to sin" 0/2 for not too small angles of scattering. On the other hand equation (27) gives for 0 = o
p=[dZ]24 1_4Er -~,
(28)
i.e., a finite number of particles scattered at 0 = o instead of an infinite number as given by the Rutherford law. Kirchner has measured the number of electrons of IO to 4o kv. energy scattered in argon by the Wilson cloud expansion chamber method and the results are in good agreement with expression (27). The number scattered at 0 = o is finite and can be calculated from the value of a2 given above. 11 We have now to investigate whether our expression (26), derived on the assumption t h a t the potential in the atom is t h a t given by the Fermi statistics, leads to a finite number of particles scattered at 0 = o. It will be readily seen that on putting 0 = o (26) results in an indeterminate form. Differentiation of both numerator and denominator yields a second 11 W h a t Kirchner actually measured was the number of electrons scattered through angles greater than O, which is given by N sins 2 q- aS On using the correct value of a s (calculated for an atom of atomic n u m b e r Z and for electrons of a given velocity), the experimental data were found to fit the theoretical curve (see curves in Sommerfeld's book p. 225). Unfortunately, (26) does not readily admit of the integration performed above, owing to incomplete knowledge of the function ,~(x), hence we cannot expect to get numerical agreement with experiment. Although it is very difficult to perform the integration indicated in Note II, one can at least get an indirect comparison with experiment by comparing the results obtained from (27) with those obtained from (26), after correct substitution of the constants involved. A comparison has been made, making use of a graphical integration of (26), and the results are to appear shortly in the Proceedings of the National Academy.
June, ~929.]
76I
I~LECTRON SCATTERING ~IX GASES.
indeterminate form P.=o
_ __~oq~(;)sin P 2kC[foo (2kpsin ¢o)dpl sin ~0 cos
(2kp sin w)dp]
where
e2Zk ]2
0
C=k~-~rj'!
co=-.2
Finally on differentiating b o t h n u m e r a t o r and d e n o m i n a t o r a second time and placing ~0 = o we o b t a i n
W e n o w m a k e the s u b s t i t u t i o n x = P~=0 =
p/~, then
[;0 xa~(x)dx
4k2~4C
.
(3 o)
F e r m i 9 gives a table for q~(x) for values of x from o.oo to 20.0. • (x) is given b y the a p p r o x i m a t e formula 0 ,I, (x) =
( i;, I44
x 3
where X ( a b o u t ~) is an a r b i t r a r y constant.
F o r v e r y large
values of x the a b o v e e q u a t i o n becomes • = I44
X3 •
gral
f0 x~(x)dx
Theinte-
will converge and hence a finite value o f
P~=0 will be o b t a i n e d from e q u a t i o n (3o), in a g r e e m e n t with t h e results o b t a i n e d using the other theory. Finally, we shall show t h a t the general form of the scattering curves is the same on the basis of the t w o theories, n a m e l y t h a t the scattering is a m a x i m u m for ~o = o and a m i n i m u m for 00 = 7r/2. Consider, first, e q u a t i o n (27). By taking the first and second d e r i v a t i v e of P with respect to w (where ~0 = 0/2), and applying t h e criteria for m a x i m a and minima, it is easily seen t h a t P has a m a x i m u m at w = o and
762
ALLAN C. G. MITCHELL.
[J. F. I.
a m i n i m u m at ~0 = ~/2. S i m i l a r l y for e q u a t i o n (26), based on t h e F e r m i p o t e n t i a l s , we o b t a i n for t h e first d e r i v a t i v e
dP
dco -
2 C c o s w [ /'~ s-in-7~ kdo ~ sin I
]
(2kp sin co)dO .Qo
sin co[ ~o ~sin(2kpsinco)dp]
"1
T h i s will o b v i o u s l y be zero for w = ~r/2. d i t i o n s for I sin s o0
dP ~ =
[£
F o r co = o the con-
o are
• sin
(2kp sin
co)dp
]
~ oo
at
= o
(3I)
and
'[20"
sin co
q
sin
(2kp sin oo)dpl
-.I
= 2k
f0
(3 2 )
p~ cos (2kp cos w)dp.
T h e first c o n d i t i o n gives an i n d e t e r m i n a t e form, w h i c h on e v a l u a t i n g gives I
sin 2
[fo
@ sin (2kp sin co)do
]
= 2k2[fo~p2c~sin(2kpsinw)dp]. = 0
for
co = o.
T h e second c o n d i t i o n likewise is fulfilled for ~ = o, since on e v a l u a t i n g t h e i n d e t e r m i n a t e f o r m on t h e left h a n d side of (32) we o b t a i n t h e r i g h t h a n d side of (33) for ~0 = o. P is obvio u s l y a m a x i m u m for co = o. T o s h o w t h a t it is a m i n i m u m for co = 7r/2 we t a k e t h e second d e r i v a t i v e w i t h respect to ~0 dco ~ -
sin co
X
• sin
{ i[;o sin ~
(2kp sin oa)dp ¢ sin (2kp sin co)dp
]
June, I929.1
763
ELECTRON SCATTERING I N GASES.
+ 2k
[;0
0¢ cos (2ko sin w)do
+ A cos ~0(- • .).
The last term indicates a number of factors which are multiplied by cos ¢0 and which vanish for ¢0 = ~/2. The condition that P should have a minimum at w = 7r/2 is sin o~ •
1 [foo
¢ sin (2kp sin ¢o)dp + 2k
1
pC cos (2kp sin ¢o)dp > o.
A numerical integration of the integrals involved shows that this condition is fulfilled. In conclusion we m a y summarize the results of this investigation as follows. By using the Fermi statistical potentials in conjunction with the Born collision theory, we have derived a formula for the scattering of electrons in gases. The scattering formula here developed gives a finite number of particles scattered at small angles, 0 = o; the maximum scattering occurs at 0 = o, and the minimum at 0 = 7r. This is in qualitative agreement with the results obtained by 50mmerfeld, by considering all the electrons to be gathered together in the K-shell and treating the result as a hydrogenlike atom. The method of Sommerfeld, although it gives results in agreement with experiment, seems somewhat illogical in the light of our knowledge of the periodic table. On the other hand, the argument presented in this paper appears to be logical in so far as the calculation of atomic fields by the method of Fermi is logical. The latter, however, seems to be in agreement with the laws of the periodic system, since Fermi has used it to calculate in which atoms the s, p, and d terms appear and has found agreement with the facts. The method of Fermi has the disadvantage of mathematical complexity which does not allow us to readily calculate numerical values of scattering which can be compared with experiment. The writer wishes to acknowledge his indebtedness to his colleagues of the Bartol Research Foundation, in particular to Prof. W. F. G. Swann and Dr. A. Bramley for m a n y helpful suggestions in connection with this work.
ALLAN C. G. MITCHELL, Ph.D., Fallow, Bartol Research Foundation of The Franklin InstRute.
A GENERAl. wave-mechanical theory of FOUNDATION collisional processes has been developed by Com~,,~c,tionNo.3S. Born 1 and applied by \Ventzel 2 to the scattering of alpha-particles by heavy atoms, and by Elsasser 3 to collisional processes in atomic hydrogen. Recently Sommerfeld 4 has applied the m e t h o d to the scattering of electrons in argon and has found agreement with the experimental results. T o carry out the calculation it is necessary to know the potential of the a t o m v(r) as a function of the distance from the nucleus. Wentzel first assumed the potential of an atom of atomic n u m b e r Z to be given by the Coulomb law v(r) = Ze/r, neglecting the effect of the electrons around the nucleus but found t h a t this led to integrals which would not converge. On assuming the potential function to be v(r) = Ze/re -T/R, where R is of the order of m a g n i t u d e of an atomic radius, he was able to obtain a scattering formula which, for alpha-particles, agreed with the Rutherford scattering law. On the other hand Elsasser found the potential of the hydrogen a t o m in terms of its eigenfunctions, the potential for the hydrogen a t o m in its normal state being similar to t h a t used by Wentzel, thus ]ustifying his assumption, and showed t h a t a scattering law for atomic hydrogen could be obtained. Finally, Sommerfeld considered scattering from other atoms by treating all the electrons of the a t o m as united in the K-shell without m u t u a l l y influencing each other. " T h e action of these electrons toward the outside would then
BARTOL RESEARCH
1 M. Born, ZS.f. Physik, 37, 863; 38, 803 (1926). 2 G. Wentzel, ibid., 40, 590 (1927). 3 W. Elsasser, ibid., 45, 522 (1927). * A. Sommerfeld, " A t o m b a u und SpektraUinien," Wellenmech. band, pp. 226 et seq.
Ergiinzungs753
754
ALLAN C. G. MITCHELL.
[J. F. I.
be represented wave-mechanically by the Z-fold electron cloud of hydrogen in its normal state, whose charge density is P
eZ 4 ~_ - - e-2r(Z/a), rra a
where a is the radius of the first Bohr circle." He then showed that the potential energy of an electron in the field of the atom is V(r)=
-
e2Z ( I- -4- Z )
(')
By using this potential energy in the Born collision equations he arrived at a scattering formula, which for high velocity alpha-particles and for n o t too small scattering angles, agrees with the Rutherford scattering law. T h e formula also agrees with the experiments of Kirchner 5 on the scattering of electrons i n argon (elastic collisions). It will be noted t h a t expression (I) for the potential energy takes into account shielding by the electrons of the atom. It is the purpose of this paper to carry through the calculations using a somewhat different m e t h o d of obtaining the potential of the a t o m at a distance r from the nucleus. T h o m a s 6 and Fermi 7 have recently shown t h a t if the electrons of an atom of atomic n u m b e r Z be considered to be a perfectly degenerate gas in the sense of the Fermi statistics, s the potential at a distance r from the nucleus is given by v =
--
/,
•
,
(2)
where 32/3h 2
# = 21~/37r4/ame2Z1/z
(m = mass of electron). This function ¢(rlg) is u n i t y when r is zero and decreases to zero for increasing r, such t h a t for distances a little greater t h a n the atomic radius the potential will be zero. No 5 F. s L. r E. s E.
Kirchner, Ann. der Phys., 83,969 (1927). H. Thomas, Proc. Cambridge Phil. Soc., 23, 542 (I926). Fermi, ZS. f. Phys., 48, 73 (I928). Fermi, ibid., 36, 902 (1926).
June, 1929.11
ELECTRON
SCATTERING IN GASES.
755
analytical expression is known for the function ¢ over its whole range, but a table of values is given by Fermi? He has used this function to calculate the Rydberg correction terms of the alkali atoms, 1° generally thought of as due to the shielding of the nonradiating electrons. In this paper we shall consider this potential function as giving the true potential of an atom at a distance r from the nucleus, inclusive of shielding, and use the potential energy derived therefrom in conjunction with the Born collision theory t() calculate the scattering from different atoms. To calculate the scattering of an electron by an atom situated at O (Fig. I), we consider a plane wave for an electron ,4
Q
,
0
X
~
p Fro. I.
approaching the atom from the negative X-direction. The wave equation for the total system, electron plus atom, is 8 7r2m
v2~ +-V-(E
-
9 See ref. 6, p. 75. 10 E. Fermi, ZS. f. Phys., 49, 550 (I928).
V ) ~ = o.
(3)
756
ALLAN C. G. )e{ITCHELL.
[J. F. I.
We consider V, the potential energy of the system, as a perturbation term and write ¢ = ~0 + ~1 + . . . .
8 7r2mE h2 we have
Substituting in (3) and writing k 2 V~(~o+~l+...)+k
~- ~ - ~
(4)
(~o+~1+...)=o,
(5)
where V26o + k26o = o,
(6)
since V~I is to be neglected in comparison with V2¢0 and k2~0. F r o m (3) and (4) we have V v2~1 + k2¢,~ = k 2 ~ ¢0, V V2~2 -~- k2~2 --- k 2 -~ 1~1,
(7)
2V
V2@n "4- k2ff/n = k --~ if~n--1 = Fn-1.
It will be noted t h a t (6) is the equation for the oncoming electron since the interaction term has been neglected. The solution of (6) for the plane wave of the electron is ~o = e ikx.
We m u s t now solve the inhomogeneous equation (7), which m a y be accomplished by Green's theorem in the following manner. If ~ and . are functions satisfying the usual conditions, Green's theorem gives
fff
(iv:. -
nV~)dT =
~~
-- " 0 s ]
We shall let eikr
-
r
,
(9)
a function satisfying the homogeneous equation v ~ + k2~ = o,
(io)
June, 192o,]
ELECTRON
757
SCATT1-RING IN (~.\SV.S.
and r/ = ¢~.
(II)
We take the origin of r as the point P, at which we shall seek the value of ~,. We shall take for our boundary of integration two surfaces, one a sphere of radius R about P, which sphere shall eventually be made infinite, and for the other boundary a small sphere of radius r0, which shall eventually 1)e made zero. Substituting (9) and (IO) in (8) we obtain
(ffe, r
•~
j'f(
_ _ (V25 n + k2¢n)d r = r
)
~ &k,~ _ ,t, Oe" d,~ -~r
r n Or
(i2)
where the negative sign in the integral over the sphere of radius r0 is incorporated to permit of the m)rmal being drawn outwards from P. Since ~,, by the fundamental conditions of the Schroedinger theory must vanish to the order of ! , where e > o (a rE conclusion which will subsequently be seen to be true) the I
integral over the surface R will vanish to the o r d e r - - • re Consider now the integral over the surface r0. Let M be the maximum value of O~b,/Or then
j'~eik~ J~o 7
eikro O~Tr~da < 1"0 M4rrr°2
and is zero in the limit.
Now
¢ ~ Or
r
r'-' 4 ~rr2dr = -- 4~-¢~
in the limit. Hence (I2) becomes eikr
¢.=
I
4~r.
VOL. 207, No. 1242--52
; f f
r (v2¢~ ÷
k~¢~)dr'
758
ALLAN C. G. ~ITCHELL.
[J. F. I.
and r e m e m b e r i n g (7) we obtain
¢,,
x
--
Fn-a(rQ) rpQ
_ .f.(/
dTQ
03)
T a k i n g the first a p p r o x i m a t i o n we have ¢1(v, -
4~-E a .
a
V(rQ) --rpQ e'er°dr'"
(I4)
F r o m Fig. I we have
rQI
t e e = [r -
= [r 2 + 2r.rQ + rQ231/2.
If r >> re then rpQ
(
=
7"
=
r --
I
--
r- 2- r • r Q
) (I5)
e.rq,
where e is a unit v e c t o r in the direction r.
W e also write
XQ ~- eo.r~
06)
where e0 is a unit v e c t o r in the direction X. becomes
erEk~e;rJ'ff
@I(P) =
E q u a t i o n (I4)
V(rQ)e'*(e-°°)'wdr'"
(17)
W e n o w change to polar co6rdinates taking O A , parallel to ]e0 - e l , as the polar axis, ~ as the longitude and 0 as the co-latitude. T h e n
p = rQ (e0 -
(i8)
e).rQ = °[e0 - e l p cos 0.
(IV)
If n o w we introduce the scattering angle O, we see from the figure 0 [e0 - e I = 2 s i n - . (20) 2
M a k i n g the a b o v e s u b s t i t u t i o n s (I7) becomes ¢1(P) =
4
e
; (oVoTo r
V(p) e ~2~~ o/2ooo, Op2dp sin OdOd,~. (2 I)
W e now use the potential energy derived from the Fermi
June, J9291
759
ELECTRON SCATTERING IN GASES.
statistics, which is
V(B) -
Ze~dP(~
for an electron at a distance 0 from the nucleus. (2 I) then becomes /~2 r z
l]/l(p ) --
2
.*2rr
twzr ,,1,0o
/
(22) Equation
\
~ AeErreikrtdo .lot Jot o~-)tP*e*2ko~l.o,'2cO.OpdpsinOdOd¢.(23)
We first carry out the integration with respect to 0 keeping p constant. Writing ~ = 2kp sin 0/2 we have ~0 T e~c°~° sin
2 . 6, OdO= ~sln
(24)
and (23) becomes after integrating with respect to ¢, ff1~e~ - 2E sinO r J0 ¢
sin 2kp sin-2 dp.
(259)
2
The analytical expression of ¢(p/#) is not known, hence the integration of (25) must be carried out graphically. We form ff~ I~ , and remembering that 16012 = I we obtain the quotient ~0 P=
',~ = [kZeZ]~ (O) ( {:)) do]~. (26/ *o 2E~-r_!s-in~I [~oo . ~°q)./~.sinx2kpsin--~ 2
Equation (26) may be interpreted as the number of particles per unit solid angle which are scattered through an angle 0 in relation to the number which fall on the atom. We shall now compare this with the expression obtained by Sommerfeld using the Born method. He obtains
P =
[e2Z]Z[ cd" ) 212 ' 0I + ( 4E~A sine2 + " 2 sin202 + a2
Z where a = ~ ;
a = radius of first Bohr circle.
(27)
It will be
760
ALLAN C. G. MITCIIELL.
[J. f,'. 1.
noted that the above formula gives exactly the Rutherford scattering law for alpha-particles if we neglect a2 in comparison to sin 2 0/2 and if we introduce the proper factor into the constant to take care of the double charge of the alphaparticle. It has been shown by Wentzel that for alphaparticles and heavy nuclei a" m a y be neglected in comparison to sin" 0/2 for not too small angles of scattering. On the other hand equation (27) gives for 0 = o
p=[dZ]24 1_4Er -~,
(28)
i.e., a finite number of particles scattered at 0 = o instead of an infinite number as given by the Rutherford law. Kirchner has measured the number of electrons of IO to 4o kv. energy scattered in argon by the Wilson cloud expansion chamber method and the results are in good agreement with expression (27). The number scattered at 0 = o is finite and can be calculated from the value of a2 given above. 11 We have now to investigate whether our expression (26), derived on the assumption t h a t the potential in the atom is t h a t given by the Fermi statistics, leads to a finite number of particles scattered at 0 = o. It will be readily seen that on putting 0 = o (26) results in an indeterminate form. Differentiation of both numerator and denominator yields a second 11 W h a t Kirchner actually measured was the number of electrons scattered through angles greater than O, which is given by N sins 2 q- aS On using the correct value of a s (calculated for an atom of atomic n u m b e r Z and for electrons of a given velocity), the experimental data were found to fit the theoretical curve (see curves in Sommerfeld's book p. 225). Unfortunately, (26) does not readily admit of the integration performed above, owing to incomplete knowledge of the function ,~(x), hence we cannot expect to get numerical agreement with experiment. Although it is very difficult to perform the integration indicated in Note II, one can at least get an indirect comparison with experiment by comparing the results obtained from (27) with those obtained from (26), after correct substitution of the constants involved. A comparison has been made, making use of a graphical integration of (26), and the results are to appear shortly in the Proceedings of the National Academy.
June, ~929.]
76I
I~LECTRON SCATTERING ~IX GASES.
indeterminate form P.=o
_ __~oq~(;)sin P 2kC[foo (2kpsin ¢o)dpl sin ~0 cos
(2kp sin w)dp]
where
e2Zk ]2
0
C=k~-~rj'!
co=-.2
Finally on differentiating b o t h n u m e r a t o r and d e n o m i n a t o r a second time and placing ~0 = o we o b t a i n
W e n o w m a k e the s u b s t i t u t i o n x = P~=0 =
p/~, then
[;0 xa~(x)dx
4k2~4C
.
(3 o)
F e r m i 9 gives a table for q~(x) for values of x from o.oo to 20.0. • (x) is given b y the a p p r o x i m a t e formula 0 ,I, (x) =
( i;, I44
x 3
where X ( a b o u t ~) is an a r b i t r a r y constant.
F o r v e r y large
values of x the a b o v e e q u a t i o n becomes • = I44
X3 •
gral
f0 x~(x)dx
Theinte-
will converge and hence a finite value o f
P~=0 will be o b t a i n e d from e q u a t i o n (3o), in a g r e e m e n t with t h e results o b t a i n e d using the other theory. Finally, we shall show t h a t the general form of the scattering curves is the same on the basis of the t w o theories, n a m e l y t h a t the scattering is a m a x i m u m for ~o = o and a m i n i m u m for 00 = 7r/2. Consider, first, e q u a t i o n (27). By taking the first and second d e r i v a t i v e of P with respect to w (where ~0 = 0/2), and applying t h e criteria for m a x i m a and minima, it is easily seen t h a t P has a m a x i m u m at w = o and
762
ALLAN C. G. MITCHELL.
[J. F. I.
a m i n i m u m at ~0 = ~/2. S i m i l a r l y for e q u a t i o n (26), based on t h e F e r m i p o t e n t i a l s , we o b t a i n for t h e first d e r i v a t i v e
dP
dco -
2 C c o s w [ /'~ s-in-7~ kdo ~ sin I
]
(2kp sin co)dO .Qo
sin co[ ~o ~sin(2kpsinco)dp]
"1
T h i s will o b v i o u s l y be zero for w = ~r/2. d i t i o n s for I sin s o0
dP ~ =
[£
F o r co = o the con-
o are
• sin
(2kp sin
co)dp
]
~ oo
at
= o
(3I)
and
'[20"
sin co
q
sin
(2kp sin oo)dpl
-.I
= 2k
f0
(3 2 )
p~ cos (2kp cos w)dp.
T h e first c o n d i t i o n gives an i n d e t e r m i n a t e form, w h i c h on e v a l u a t i n g gives I
sin 2
[fo
@ sin (2kp sin co)do
]
= 2k2[fo~p2c~sin(2kpsinw)dp]. = 0
for
co = o.
T h e second c o n d i t i o n likewise is fulfilled for ~ = o, since on e v a l u a t i n g t h e i n d e t e r m i n a t e f o r m on t h e left h a n d side of (32) we o b t a i n t h e r i g h t h a n d side of (33) for ~0 = o. P is obvio u s l y a m a x i m u m for co = o. T o s h o w t h a t it is a m i n i m u m for co = 7r/2 we t a k e t h e second d e r i v a t i v e w i t h respect to ~0 dco ~ -
sin co
X
• sin
{ i[;o sin ~
(2kp sin oa)dp ¢ sin (2kp sin co)dp
]
June, I929.1
763
ELECTRON SCATTERING I N GASES.
+ 2k
[;0
0¢ cos (2ko sin w)do
+ A cos ~0(- • .).
The last term indicates a number of factors which are multiplied by cos ¢0 and which vanish for ¢0 = ~/2. The condition that P should have a minimum at w = 7r/2 is sin o~ •
1 [foo
¢ sin (2kp sin ¢o)dp + 2k
1
pC cos (2kp sin ¢o)dp > o.
A numerical integration of the integrals involved shows that this condition is fulfilled. In conclusion we m a y summarize the results of this investigation as follows. By using the Fermi statistical potentials in conjunction with the Born collision theory, we have derived a formula for the scattering of electrons in gases. The scattering formula here developed gives a finite number of particles scattered at small angles, 0 = o; the maximum scattering occurs at 0 = o, and the minimum at 0 = 7r. This is in qualitative agreement with the results obtained by 50mmerfeld, by considering all the electrons to be gathered together in the K-shell and treating the result as a hydrogenlike atom. The method of Sommerfeld, although it gives results in agreement with experiment, seems somewhat illogical in the light of our knowledge of the periodic table. On the other hand, the argument presented in this paper appears to be logical in so far as the calculation of atomic fields by the method of Fermi is logical. The latter, however, seems to be in agreement with the laws of the periodic system, since Fermi has used it to calculate in which atoms the s, p, and d terms appear and has found agreement with the facts. The method of Fermi has the disadvantage of mathematical complexity which does not allow us to readily calculate numerical values of scattering which can be compared with experiment. The writer wishes to acknowledge his indebtedness to his colleagues of the Bartol Research Foundation, in particular to Prof. W. F. G. Swann and Dr. A. Bramley for m a n y helpful suggestions in connection with this work.