Tài liệu Early Atomic Models – From Mechanical to Quantum (1904-1913) pptx

1
Early Atomic Models – From Mechanical to Quantum (1904-1913)

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2
1 Introduction

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1
Thomson, W., p. 15. An English translation by Tait had recently appeared [Phil. Mag. 33: 485-512] of
Helmholtz, H. 1858. Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen
entsprechen. Journal für die reine und angewandte Mathematik 55: 25-55. William Thomson also makes
mention of papers from Rankine (1849-50) on "Molecular Vortices".
2
Kragh, H. 2012. Niels Bohr and the Quantum Atom, Oxford University Press, Oxford.
3
Heilbron, J. L. 1981. Historical Studies in the Theory of Atomic Structure, Arno Press, New York.

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4
Davis, E. A. and Falconer, I. J. 1997. J. J. Thomson and the Discovery of the Electron, Taylor & Francis,
London.

6
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5
There would be an insufficient restoring force if the centers of mass for the uniform ring and the planet
ever deviated from equilibrium, eventually leading the ring and planet to crash into each other. The only
exception was most unlikely: an otherwise uniform ring would require an additional point mass located at
its outer edge, equal to 0.82 of the mass of the total ring. [Maxwell 1859, p. 55]

7
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6
Maxwell 1861, p. 165.

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7
Thomson, W. 1878. Floating Magnets. Nature 17: 13-14.

9
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Figure 1. Static arrangements for n = 2 – 20 floating magnetic needles (all with their

south poles oriented vertical), under the attractive influence of a central bar magnet. If
multiple static configurations were possible for a given n, only those denoted by an "a"
were stable against perturbations. [From Mayer, p. 248-9](
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8
His first conceptions were in analogy with electrolysis. [Davis and Falconer, p. 77-8; see also p. 77-138
for details on the evolution in Thomson's thinking about gaseous discharge leading up to his discovery of
the electron.]

11
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/1.*10( 12( 41$-8'4%&'( *0( #( -1'*.*6&%+( 4"#$E&7( $&E*10( 12( '-#4&9( #07( 410.#*0&7( 01(
6*'418'( 7#/-*0E( .&$/'9( /&#0*0E( ."&( &%&4.$10'( B&$&( 2$&&( .1( /16&( #C18.( "*'( #.1/(
B*."18.( $&'*'.#04&K((L"1/'10R'( 7+0#/*4( /17&%( 418%7( ."&$&21$&( C&( 6*&B&7( #'( #(

9
31'*.*6&%+( 4"#$E&7( '8C#.1/*4( -#$.*4%&'( "#7( C&&0( -$1-1'&7( C8.( 0&6&$( 1C'&$6&7K Goldstein had
discovered positively charged rays during cathode experiments reported in 1886, but they were only known

at the time to have a charge-to-mass ratio similar to ionized hydrogen. [Davis and Falconer, p. 199] It was
not until after the concept of isotopy was developed that physicists could identify these "positive rays" as
singly-ionized tritium (an isotope of hydrogen, with one proton and two neutrons), and not triatomic
hydrogen (as proposed by Thomson in 1913). [Kragh, p. 96-100]
10
Letter to Sir Oliver Lodge from 11 April 1904. [Quoted in Davis and Falconer, p. 195-96]

12
E&0&$#%*Z#.*10(12(."&(#.1/(&Q-1807&7(C+(i&%6*0(*0(#0(#$.*4%&(2$1/(G><W9(B"&$&("&(
$&6*'*.&7( ."&( `10&( 2%8*7( ."&1$+( 12( &%&4.$*4*.+a( 12( X$#0Z( A&-*08'( \GTWH^G;<W_K((A'(
7&'4$*C&7(C+(i&%6*09(A&-*08'(-$1-1'&7(."&(&Q*'.&04&(12(#(0&E#.*6&%+(4"#$E&7(&%&4. $*4(
2%8*7(."#.(-&$/&#.&7(#%% (12('-#4&9(#07(2%1B&7(2$&&%+(`#/10E(."&(#.1/'(12( - 107&$#C%&(
/# &$Ka( O&E#.*6&( \#07( -1'*.*6&_( 4"#$E&( 7*'.$*C8.*10'( 418%7( ."8'( C&( 6*&B&7( #'(
$&E*10'(12('-#4&(B*."(#0(&Q4&''(\1$(7&2*4*&04+_(12(."*'(*7&#%(2%8*7K((i&%6*0q'(-$1-1'#%(
B#'(."#.('84"(#(2%8*7(418%7(#4.8#%%+(C&(/#7&(8-(2$1/(#(/8%.*.87&(12(.*0+9(0&E#.*6&%+(
4"#$E&7( `#.1/'( 12( &%&4.$*4*.+a
GG
( ."#.( /16&7( B*."18.( $&'*'.#04&( *0'*7&( 12( -1'*.*6&%+(
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4"#$E&'
GW
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."&(#.1/na(kL"1/'10(G><=9(-K([;>l(
(

11
Hence the title of his paper, "Aepinus Atomized."
12
Kelvin asserted that these negatively charged electrions “no doubt occupy finite spaces, although at
present we are dealing with them as if they were mere mathematical points…” [Kelvin, p. 258]
13
“As a tentative hypothesis, I assume for simplicity that…” [Kelvin, p. 258; emphasis added.]

13
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#$1807(#($*0E(\C8.(*0(."&(#C'&04&(12(#0+(-1'*.*6&(4"#$E&_K((L"*'(B#'(&Q-%*4*.%+(710&(*0(
#0.*4*-#.*10(12( 28.8$&(B1$F9( #07("*'($&'8%.'(B18%7( "#6&(*/-1$.#0.( */-%*4#.*10'(21$(
."&($#7*#.*6&('.#C*%*.+(12("*'(#.1/'K(
L"&(7+0#/*4#%(#.1/(*0*.*#%%+(-$1-1'&7(C+(L"1/'10(*0(b#$4"(G><H(410'*'.&7(
12( #( 80*21$/( '-"&$&( 12( -1'*.*6&( 4"#$E&( #07( #( '*0E%&( $*0E( 12( 0&E#.*6&%+( 4"#$E&7(
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GH
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1."&$(0&E#.*6&(4"#$E&'K(

Figure 2. Diagram of Thomson’s 1904 atomic model.
17

A uniform sphere of positive charge (shaded region, of
radius b) contains n negative point charges arranged at
equal intervals around a circle (of radius a). The ratio
a/b = 0.6726 for a static ring of n = 6 charges (see
below).
(
(

14
Thomson continued to refer to negative atomic charges as “corpuscles” throughout the article, and for
many years after. George Stoney claimed in 1894 to have coined the term electron; Kelvin used the term
electrions for his 'atoms of electricity'. Rutherford’s 1911 nuclear model was agnostic on the actual form

and distribution of the negative charge. Nagaoka (1904), Haas (1910), Nicholson (1912) and Bohr (1913)
all identified their negative atomic charges as electrons.
15
L"&('-%* *0E(12('-&4.$#%(%*0&'(&/* &7( C+(&Q4*.&7(#.1/'(*0(#(/#E0&.*4(2*&%7K((V.(B#'( #''8/&7( ."#.(
."&$&(B#'(#(10&? .1 ? 10 &(41$$&'-107&04&(C&.B&&0(&#4"(r&&/#0(%*0&(#07(#0(1$C*.*0E(&%&4.$10K(
16
If the positive charge distribution had no inertial mass, there would be no constraint on the amount of
positive charge available to neutralize an atom filled with thousands of electrons.
17
He did not reproduce any illustrations of his model, perhaps because it was so simple that a picture would
be superfluous. Except for the positive charge distribution, it is clearly described in the title alone: “On the
Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of
Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the results
to the Theory of Atomic Structure.”

14
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# $#4.*6&( 21$4&( 10( &#4"( 4"#$E&( \&D8#%( .1( -'
8
"s5
9
( 21$( #0( &%&4.$*4#%%+( 0&8.$#%( #.1/_(
/8'.( C#%#04&( ."&( 0&.( $&-8%'*6&( 21$4&( 12( ."&( 1."&$( -( ^( G( 4"#$E&'K( ( L"&( #Z*/8."#%(
41/-10&0.( 12( ."*'( $&-8 %'*6&( 21$4&( B18%7( C&( Z&$1(78&( .1('+//&.$+9( #07( ."&( $#7*# %(
41/-10&0.(:
)
(E*6&0(C+I
G;

(
(
((
( (
, , , , ,
.,, , , , , ,,,,,,,(1)(
(
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(
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(

18
Thomson did not explicitly derive this formula in his paper, but the derivation is presented here as an
example of the numerous trigonometric quantities that appeared throughout.
19
The indexed quantities L, M & N appearing in (2) represent various sums of trigonometric functions,
similar to the quantity S defined in (1).

F
r
=

F
k
sin k
π
n
( )
k =1
n −1
∑
=
e
2
sin k
π
n
( )
a − a cos 2k
π
n
( )
( )
2
+ a
2
sin
2

2k
π
n
( )
k =1
n −1
∑
=
e
2
2a
2
sin k
π
n
( )
1 − cos 2
π
k n
( )
=
e
2
2a
2
sin k
π
n
( )
2 sin

2
k
π
n
( )
k =1
n −1
∑
k =1
n −1
∑
=
e
2
4a
2
csc k
π
n
( )
k =1
n −1
∑
≡
e
2
4a
2
S
n

3e
2
4a
3
S
k
+ L
k
− L
0
− mq
2
⎛
⎝
⎜
⎞
⎠
⎟
N
0
− N
k
− mq
2
( )
= M
k
− 2m
ω
q

( )
2
k = 0,1, 2 n − 1
( )

15
K( ( ( ( (3)(
(
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W
( B*%%( C&( 0&E#.*6&( \/#F*0E( ;( */#E*0#$+_9(
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4"#$E&'( B18%7( C&( /&4"#0*4#%%+( 80'.#C% &9( 01( /# &$( *.'( #0E8%#$( 6&%14*.+9( 1-/'++( #0(
#77*.*10#%( 0&E#.*6&( 4"#$E&( B&$&( -%#4&7( #.( ."&( 4&0.&$( 12( ."&( '-"&$&K
W<
((P*."( '*Q(
4"#$E&'( *0( #( $*0E( #07( #( '&6&0."( %14#.&7( #.( ."&( 4&0.&$9( ."&( $11.'( .1( ."&( '8*.#C%+(
/17*2*&7(2$&D8&04+(&D8#.*10'(B18%7(#%%(C&(-1'*.*6&(*0(;
W
9(#07(."&$&21$&(1'4*%%#.1$+K((
Y&(28$."&$(21807(."#.(*2(."&(08/C&$(12(18.&$(4"#$E&'(*04$&#'&7(C&+107(&*E".9(/1$&(
#07(/1$&(*0.&$*1$(4"#$E&'(B18%7(C&($&D8*$&7(21$(."&(7+0#/*4('.#C*% *.+(12(."&('+'.&/K(
kL#C%&(Vl(
(


5
6
7
8

9
10
15
20
30
40

0
1
1
1
2
3
15
39
101
232

Table I. Number of charges n contained in a single atomic ring, and the minimum
number of interior charges p required for the dynamic stability of that ring. [From
Thomson 1904, p. 254.] The last entries represent the largest number of electrons in a
single atom mentioned by Thomson in this paper.
(
(
L"1/'10(."18E".( ."&'&( *0.&$*1$(4"#$E&'(B18%7(\%*F &( b#+&$R'( /#E0&.'_( #%'1(
#$$#0E&(."&/'&%6&'(*0.1(4%1'&79('.#C%&(1$C*.'9(#07(."#.(#.1/'(B*."(-(u(;(18E".(.1(C&(
/#7&(8-( C+(#('&$*&'(12(4104&0.$*4($*0E'K(( L"&(08/C&$(12( 4"#$E&'(B18%7( 6#$+(2$1/(
$*0E( .1( $*0E9( B*."( ."&( E$&#.&'.( 08/C&$( *0( ."&( 18.&$/1'.( $*0E9( #07( #( 7&4$&#'*0E(
08/C&$( 12( 4"#$E&'( 1448-+*0E( &#4"( 12( ."&( *00&$( $*0E'K( kL#C%&( VVl( L"1/'10( ."&0(
-$14&&7&7(.1(#$E8&("1B(."&(-&$*17*4*.+(12(."&'&(4102*E8$#.*10'(\#07(."&*$($&'8%.#0.(

-$1-&$.*&'9(7&.&$/*0&7(C+(."&(08/C&$(1448-#.*10(#07('.#C*%*.+(12(."&($*0E'_(4%1'&%+(
/#.4"&7( ."&( -&$*17*4*.+( 12( ."&( 0#.8$#%( &%&/&0.'K( ( O1.*4*0E9( 21$( &Q#/-%&9( ."#.( ."&(
*0.&$*1$(4102*E8$#.*10(21$(#0(#.1/(B*."(=(o([<(4"#$E&'(B18%7(C&(*7&0.*4#%(.1(."#.(12(((((((

20
A single negative charge added at the center introduced a positive quantity to the first term in parentheses
of (2) & (3), so that the offending term in (3) would be manifestly positive. [Thomson 1904, p. 251-2]
−
14 − 8 3
8
e
2
a
3
− mq
2
⎛
⎝
⎜
⎞
⎠
⎟
58
e
2
8a
3
− mq
2
⎛

⎝
⎜
⎞
⎠
⎟
= 4m
2
ω
2
q
2
n
p

16
=(o( H<9( L"1/'10(-$1-1'&7( ."#.( E$18-'(12( '84"( #.1/'(k&#4"(10&( 7&$*6&7(2$1/(."&(
-$&6*18'(C+(."&(#77*.*10(12(#('*0E%&($*0E9(#'(B*."(."&(=(o(=9(GG9(WH9(H<(c([<(#.1/'(*0(
L#C%&(VVl(B 18%7("#6&('*/*%#$(4"&/*4#%(#07('-&4.$#%(-$1-&$.*&'9(#07('"18%7(."&$&21$&(
2#%%(*0.1(."&('#/&(6&$.*4#%(41%8/0(*0(."&(3&$*17*4(L#C%&K(kL"1/'10(G><H9(-K(W]>l(
(

3
11
15
20
24
30
35
40
45

50
55
60

3
3
8
5
10
1
7
12
3
8
13
5
10
15
1
6
12
16
3
8
13
16
4
10
14
17

1
5
11
15
18
1
7
12
16
19
3
8
13
16
20

Table II. Sampling of charge arrangements for a number of different atoms, from the
inner rings (upper rows) to the outer rings (lower rows), where N represents the total
number of charges in each atom. The N = 3, 11, 24, 40 & 60 atoms would form a single
vertical column in the Periodic Table, each member of the group derived from the
previous member by the addition of a single ring. [Adapted from Thomson 1904, p. 257.]
(
(
V0(#('*/*%#$(6&*09(#.1/'(*0(."&( E$18-(=(o(]>(^([T(&#4"("#7( W<(4"#$E&'(*0(."&*$(
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'.#C%&(."&(18.&$($*0E'(B18%7(C &9(/#F*0E(."&(18.&$($*0E(21$(=(o(]>(."&(%&#'.('.#C%&(12(
."&(E$18-K((L"1/'10(4"#$#4.&$*Z&7(."*'($*0E(#'(`6&$+(0&#$(."&(&7E&(12('.#C*%*.+9a('1(
."#.( *.( /*E".( &#'*%+( /%+'( 10&( 12( *.'( 4"#$E&'9( -&$"#-'( 78&( .1( '1/&( &Q.&$0#%( 21$4&K((
Y#6*0E( 710&( '19( *.( B18%7( . "&0( C&"#6&( %*F&( #( '.$10E%+( &%&4.$1-1'*.*6&( #.1/( #07(
*//&7*#.&%+(# $#4.(#0+(0&E#.*6&(4"#$E&'(*0( *.'(6*4*0*.+9('1(."#.(*.(B18%7(C&(80#C%&(.1(

/#*0.#*0( *.'( 4"#$E&7( '.#.&9( #07( #4.( *0( ."&( %10E( .&$/( #'( ."18E"( *.( B&$&( 4"&/*4#%%+(
*0&$.K(kL"1/'10(G><H9(-K(W[Gl(
(
(
(
(
(
N

17

Table III. Distribution of charges from inner rings (top rows) to outer rings (bottom
rows) for the group of atoms containing N = 59 – 67 charges. Each has 20 charges in its
outermost ring, placing them all in a single horizontal row of the Periodic Table. [From
Thomson 1904, p. 258]
(
(
P*."( #77*.*10#%( *0.&$*1$( 4"#$E&'9( '844&''*6&( #. 1/'( *0( ."*'( E$18-( B18%7( C&(
%&''(#07(%&''(&%&4.$1-1'*.*6&9
WG
(&6&0.8#%%+(.1(."&(-1*0.(B"&$&(."&+(418%7("!!'2#(10&(
1$( .B1( &Q.$#( 4"#$E&'( B"*%&( '.*%%( /#*0.#*0*0E( '.#C*%*.+j( ."&'&( # .1/'( B18%7( "#6&( #0(
*04$&#'*0E%+(&%&4.$10&E#.*6&(4"#$#4.&$K((b#Q*/8/('.#C*%*.+(B18%7(1448$(21$(=(o([T9(
B"*4"( B18%70R.( C&( #C%&( .1( $&.#*0( #0+( &Q.$#( 4"#$E&'( 21$( %10E( C&4#8'&( =( o( [;(
41$$&'-107&7(.1(#(D8#'*?'.#C%&(18.&$($*0E(12(WG(4"#$E&'K((@*/*%#$(.1(=(o(]>9(."*'(#.1/(
B18%7(D8*4F%+(%1'&(#0+(#77*.*10#%(4"#$E&(*.("#7(E#*0&79(#07(#%'1(#4.(#'(."18E"(*.("#7(
01( 6#%&04+K( ( L"1/'10( #$E8&7( ."#.( ."&( #.1/*4( -$1-&$.*&'( -$&7*4.&7( C+( "*'( /17&%(
B&$&($&2%&4.&7(*0(#('&$*&'(12(F01B0(&%&/&0.'9(B"&0(#$$#0E&7(#'I(

He

Li
Be
B
C
N
O
F
Ne
Ne
Na
Mg
Al
Si
P
S
Cl
Arg

L"&(2*$'.(#07(%#'.(&%&/&0.'(*0(&#4"($1B("#6&(01(6#%&04+9(."&('&4107(#$&(/1016#%&0.(
&%&4.$1-1'*.*6&9( ."&( '&4107?.1?%#'.( /1016#%&0.( &%&4.$10&E#.*6&9( #07( '1( 10K(((
kL"1/'10(G><H9(-K(W[<?[Wl(
Y&(#%'1(7$&B(# &0.*10(.1(."&$&(C&*0E(#7%('&.'(12(6*C$#.*10'(#''14*#.&7(B*."(
"*'(/17&%I(10&(41$$&'-107*0E(.1(."&(#0E8%#$($1.#.*10(12(."&($*0E'9(."&(1."&$(#$*'*0E(
2$1/(."&(1'4*%%#.*10(12( ."&(4"#$E&'(#C18.(."&*$(&D8*%*C$*8/( -1'*.*10'K((L"18E"("&R7(

21
The outer rings would become more stable, so that they would be less likely to lose a charge and behave
like an electropositive atom.

59

60
61
62
63
64
65
66
67

2
8
13
16
20
3
8
13
16
20
3
9
13
16
20
3
9
13
17
20
3

10
13
17
20
4
10
13
17
20
4
10
14
17
20
5
10
14
17
20
5
10
15
17
20
N

18
.#F&0( E$&#.( -#*0'( .1( 7&$*6&( 21$/8%#'( 21$( &#4"( 12( ."&'&( #%%1B&7( 2$&D8&04*&'( \21$((((((((((
-(v([_9("&(/#7&(01(/&0.*10(12(#0+(# &/ (#.(D8#0.*.#.*6&(41/-#$*'10'(B*."(#.1/*4(
'-&4.$#K(( A2.&$( #%%9("*'(/17&%( -%#4&7(01(410'.$#*0.'( 10(."&(#0E8%#$( 6&%14*.*&'( 12(."&(

$*0E'( \1."&$( ."#0( &Q4&&7*0E( '1/&( /*0*/8/( 6#%8&_9( ."&( 4"#$E&( #07( /#''( 12( #0(
&%&4.$10( B&$&( 01.( '&-#$#.&%+( F01B0( B*."( -$&4*'*109( #07( "&( "#7( 01( B#+( 12(
7&.&$/*0*0E(."&(#C'1%8.&(08/C&$(12(4"#$E&'(410.#*0&7(B*."*0(#0(#.1/K
WW
(
L"1/'10( B#*.&7( 80.*%( ."&( 6&$+( %#'.( 12( "*'( -#-&$( .1( 2*0#%%+( #77$&''( ."&(
-$1C%&/(12($#7*#.*6&(*0'.#C*%*.+K((A'(/&0.*10&7(-$&6*18'%+9( 4%#''*4#%(&%&4.$17+0#/*4'(
-$&7*4.'( ."#.( #0+( #44&%&$#.&7( 4"#$E&( B*%%( $#7*#.&( &0&$E+( *0( ."&( 21$/( 12(
&%&4.$1/#E0&.*4(B#6&'K((L"&(F*0&.*4(&0&$E+(12(#('+'.&/(12(-1*0.(4"#$E&'(*0(-&$*17*4(
/1.*10( B18%7( ."&$&21$&( C&( 7*''*-#.&7( C+( ."&( $&'8%.*0E( $#7*#.*10( 80.*%( &Q"#8'.&7K((
L"1/'10(*/#E*0&7(#($*0E(12(4"#$E&'(*0*.*#%%+($1.#.*0E(B*."('822*4*&0.(#0E8%#$('-&&79(
C8.(."#.(
(
`n( *0( 410'&D8&04&( 12( ."&( $#7*#.*10( 2$1/( ."&( /16*0E( 41$-8'4%&'9( ."&*$(
6&%14*.*&'(B*%%('%1B%+(^(6&$+('%1B%+(^(7*/*0*'"j(B"&09(#2.&$(#(%10E(*0.&$6#%9(."&(
6&%14*.+( $&#4"&'( ."&( 4$*.*4#%( 6&%14*.+9( ."&$&( B*%%( C&( B"#.( *'( &D8*6#%&0.( .1( #0(
&Q-%1'*10(12(."&(41$-8'4%&'(knl((L"&(F*0&.*4(&0&$E+(E#*0&7(*0(."*'(B#+(/*E".(
C&('822*4*&0.(.1(4#$$+(."&('+'.&/(18.(12(."&(#.1/9(#07(B&('"18%7("#6&9(#'(*0(
."&(4#'&( 12($#7*8/9( #(-#$.( 12(."&( #.1/('"1.( 122K((V0(410'&D8&04&( 12(."&( 6&$+(
'%1B( 7*''*-#.*10( 12( &0&$E+( C+( $#7*#.*10( ."&( %*2&( 12( ."&( #.1/( B18% 7( C&( 6&$+(
%10EKa(kL"1/'10(G><H9(-K(W[]l(
(
N#7*#.*6&( *0'.#C*%*.+( /*E".(."&$&21$&( C&( #( /&4"#0*'/( 21$( ."&( -$1784.*10( 12(
C&.#($#7*#.*109(C8.(L"1/'10(-$16*7&7(01(28$."&$($#.*10#%&("&$&(21$(B"+(."*'(B18%7(
C&( #( `'%1Ba( -$14&''K( ( J8'.*2*4#.*10( !"-( C&( 218079( "1B&6&$9( *0( "*'( -#-&$( 2$1/( ."&(
-$&6*18'( +&#$9( 7&.#*%*0E( ."&( 2*&%7'( -$1784&7( C+( '+//&.$*4#%%+( #$$#0E&7( -1*0.(
4"#$E&'( 410'.$#*0&7( .1( $1.#.&( *0( #( 4*$4%&K( ( Y*'( 4104%8'*10( "#7( C&&0( ."#.( *2( ."&(
6&%14*.*&'( 12( ."&( -#$.*4%&'( B&$&( '/#%%( 41/-#$&7( .1( ."&( '-&&7( 12( %*E".9( ."&( $#.&( 12(

22

Elements were identified at the time by their atomic weights; there wasn't sufficiently convincing
evidence until 1913 that hydrogen contained only a single electron. [Pais, p. 128; also Bohr 1913a, p. 30;
see below]

19
$#7*#.*6&(&0&$E+(%1''(B18%7(7$#/#.*4#%%+(&'!)'"+'(#'(."&(08/C&$(12(7*'4$&.&(4"#$E&'(
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'&&/( 4180.&$*0.8*.*6&9( C8.( $&4#%%( ."#.( #( '.&#7+( %11-( 12( 48$$&0.( -$1784&'( .*/&?
,-&'2'-&'-#( 2*&%7'9( #07( 71&'( 01.( $#7*#.&( &%&4.$1/#E0&.*4( B#6&'K( ( L"1/'10( 21807(
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12(1$C*.*0E(4"#$E&'(*04$&#'&7K(
L"1/'10(/&0.*10&7(#.1/*4(/#''(10%+(104&(*0( "*'(G><H(#$.*4%&9( *0(4100&4.*10(
B*."(."&(08/C&$(12(4"#$E&'(*0(#0(#.1/I(`P&( '8 1'&(."#.(."&(/#''(12(#0(#.1/(*'(."&(
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."&( D8&'.*10( $&/#*0&7( 1-&0( 12( B"&."&$( 1$( 01.( #0( #.1/( #4.8#%%+( 410.#*0&7( /#0+(
."18'#07'(12(&%&4.$10'(^(-&$"#-'(#%%(C8.(10&(1$(.B1(12(."&/(B&$&(.11(.*E".%+(C1807(
.1(C&('.$* &7(*0(#('.#07#$7(7*'4"#$E&(%#/-K(
(
(
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(
(
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(
(
(
(
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(
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(
Table IV. Average radiation per particle for -(v([ charges rotating at constant angular
speed >, at a distance a from the center, relative to the case of a single orbiting charge
(taken as unity). The middle column represents charges moving at 1/10 the speed of
light, the other at 1/100 the speed of light. [From Thomson 1903, p. 681.]
(
(
Number of Particles
Radiation from each particle.
a
ω
= c / 10
a
ω
= c / 100
1
1

1
2
9.6 × 10
−2
9.6 × 10
−4
3
4.6 × 10
−3
4.6 × 10
−7
4
1.7 × 10
−4
1.7 × 10
−10
5
5.6 × 10
−5
5.6 × 10
−13
6
1.6 × 10
−7
1.6 × 10
−17

20
2.3, The,number,of,corpuscles,in,the,atom(
(

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12( 0&E#.*6&( 4"#$E&'( *0( #0( #.1/9( C#'&7( 10( &Q-&$*/&0.#%( 7#.#( 21$( ."$&&( 7*22&$&0.(
-"&01/&0#I(."&(7*'-&$'*10(12(%*E".(#07(."&('4# &$*0E(12(p ?$#+'(C+(7*%8.&(E#'&'j(#07(
."&( #C'1$ *10( 12( d?$#+'( B"&0( .$#6&$'*0E( /# &$K( ( V0( &#4"( 4#'&( "&( B#'( %&7( .1( ."&(
'#/&(4104%8'*10I(."&(08/C&$(12(4"#$E&'(*0(#0(#.1/('"18%7(C&(10(."&('#/&(1$7&$(12(
/#E0*.87&(#'(*.'(#.1/*4(B&*E".K(
(
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$&'8%.'Ka(kL"1/'10(G><[9(-K(T[>l(
(
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#.1/'(#07(*04$&#'*0E(."&($&%#.*6&(-&$/* *6*.+(12(."&(E#'9(#'(/&#'8$&7(C+(."&(*07&Q(
12($&2$#4.*10(?K((Y&(*//&7*#.&%+(4104&7&7(."#.(."&(-1'*.*6&(4"#$E&(/8'.(410.$*C8.&(
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B18%7( "#6&( 01( 2$&D8&04+( 7&-&07&04&9( /#F*0E( 7*'-&$'*10( */-1''*C%&K
W=
((Y*'( 2*0#%(
&Q-$&''*10(21$(."&(*07&Q(12($&2$#4.*10(B#'(B$* &0(*0(.&$/'(12(."&(D8#0.*.*&'(A
B
(\."&(
0&.(61%8/&(1448-*&7(C+(#.1/'(-&$(48C*4(4&0.*/&.&$(12(E#'_9(=(\."&(08/C&$(12(#.1/'(
-&$(80*.(61%8/&_(#07(."&(*04*7&0.(B#6&%&0E."(CI(

(

23
Thomson considered only wavelengths that were long compared to atomic distances, so that the external
field could be taken as uniform over the length of an atom. If the inertial mass of either the positive or
negative charges were zero, there would be no frequency dependence in the response of the atom to the
time-varying electric field, and the maximum polarization would depend only on the amplitude of the
incident wave.

21
9( ( ( ( ( (4),
(
B"&$&( 3D( #07( 'D( #$&( $&'-&4.*6&%+( ."&( .1.#%( 4"#$E&( 12( ."&( -1'*.*6&( '-"&$&( #07( #(
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( L"1/'10(B#'(80#B#$&(12(#0+(&Q-&$*/&0.#%(7#.#(10(."&(7*'-&$'*6&(-1B&$(12 (
/10#.1/*4( E#'&'9( C8.( "&( 8'&7( #( 2*07*0E( C+( h1$7( N#+%&*E"( \."#.( ."&( 7*'-&$'*10( 12(
"&%*8/(*'(12(."&(+"0'(%)&')(#'(#(7*#.1/*4(E#'_(.1(g8'.*2+(41/-#$*0E("*'(."&1$+(B*."(
7#.#( -8C%*'"&7( C+( MK( i& &%&$( 10( ."&( $&2$#4. *6&( *07&Q( 12( /1%&48%#$( "+7$1E&0K( ( A.(
#./1'-"&$*4(-$&''8$&9(i& &%&$q'(/&#'8$&/&0.'(+*&%7&7(."&(&Q-$&''*10I(
(
K( ( ( ( ( (5),
(
!1/-#$*0E( ."*'( B*."( \H_9(#07( 8'*0E('Ds0( o( GKTwG<
T
( #07( ='D( o(<K;9( L"1/'10( 21807(
."#.9(# $1Q*/#.&%+I(
(
K( ( ( ( ( ( ( ( (6),
L"&$&21$&9(."&(08/C&$(12(0&E#.*6&(4"#$E&'(-(-&$("+7$1E&0(#.1/('"18%7(01.(C&(/84"(
7*22&$&0.(2$1/(10&9(#07(."&(/#''(12(."&(-1'*.*6&(4"#$E&(@(/8'.(C&(%#$E&(41/-#$&7(.1(

-09(."&(.1.#%(/#''(12(#%%(."&(0&E#.*6&(4"#$E&'K(kL"1/'10(G><[9(-K(TTGl(
( L"&( '&4107( &Q-&$*/&0.#%( /&."179( 4104&$0*0E( ."&( '4# &$*0E( 12( p?$#+'9( B#'(
E*6&0(410'*7&$#C%+(%&''(# &0.*10(C+(L"1/'10(*0(."*'(-#-&$9(C&4#8'&(."&(."&1$+("#7(
#%$&#7+( C&&0( 7&6&%1-&7( *0( "*'( $&4&0.( C11F9( `!10784.*10( 12( M%&4.$*4*.+( ."$18E"(
f#'&'Ka
WH
((Y&(41/-#$&7(#(-$&6*18'%+(7&$*6&7(&Q-$&''*10(21$(."&($&%#.*6&(#/180.(12((
p?$#+'('4# &$&7(C+(#( E#'( 12( =( &%&4.$10'(-&$( 48C*4( 4&0.*/&.&$( B*."( 7#.#(2$1/( !K( fK(
)#$F%#9( B"1( "#7( /&#'8$&7( ."&( $#.*1( 12( '4# &$&7( .1( *04*7&0.( &0&$E+( 21$( p?$#+'(
-#''*0E(."$18E"(#*$(.1(C&(WKHwG<
?H
K((@& *0E(."&'&(&D8#%I(
(
9(( ( ( ( ( ( ( (7),

24
Published by Cambridge University Press in 1903, second edition appearing in 1906.
µ
2
− 1
µ
2
+ 2
= P
0
+ P
2
0
M
′

E
m
′
e
1
N M + nm
( )
3
π
λ
2
µ
2
− 1
µ
2
+ 2
=
1
3
2.8014 ×10
−4
+
2 × 10
−14
λ
2
⎧
⎨
⎩

⎫
⎬
⎭

M
M + nm
1
n
 1
8
π
3
Ne
4
m
2
= 2.4 × 10
−4

22
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/#''(12(0*.$1E&0(#.(# $1Q*/#.&%+(W;K(kL"1/'10(G><[9(-K(TTW?=l(
( L"&( ."*$7( /&."17( *061%6&7( ."&( #C'1$ *10( 12( β?$#+'9( #07( *'( 12( -#$.*48%#$(
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B
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$18E"%+(410'.#0.K(kL"1/'10(G><[9(-K(TT=l(

25
There was considerable debate at the time, and for long after, whether the absorption law for
monoenergetic electrons should be linear or exponential. For details, see Chapter 1 of Franklin, 2001.

λ
δ
= 4
π
e
4
m
2
n
M
V
4
0
V
4
log
1
2
aV
2
V
0
2
m
e
2
−1
⎛
⎝
⎜

⎞
⎠
⎟

23
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H
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0
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26
He cited Rayleigh, Theory of Sound, Second Edition, Vol. I, p. 35 (1894). This is a “random walk”
argument for the average resultant of n displacements of arbitrary phase and constant amplitude
θ
.
n =
M
′
M
1.4
log
mV
2
V
2
0

a
e
2
−1
⎛
⎝
⎜
⎞
⎠
⎟

24
n ⋅
θ
9(B"*4"(.$#0'%#.&'(*0.1(
φ
m
= N
π
R
2
t ⋅
θ
(21$('4# &$*0E(*0(#(-%#.&(12(."*4F0&''(#(
"#6*0E(=(#.1/'(12($#7*8'(I(-&$(80*.(61%8/&K((L"&($&/#*0*0E(.#'F(B18% 7(C&(.1(2*07(#(
."&1$&.*4#%( 6#%8&( 21$ ( H( 8'*0E( ."&( #''8/ *10'( 12( "*'( %#.&'.( '4# &$*0E( /17&%'K(
kL"1/'10(G>G<9(-K(H[]l(
( Y&(2*$'.(7&$*6&7(#0(&Q-$&''*10(21$(. "&(/&#0(7&2%&4.*10(H
G
(78&(.1(*0.$#?#.1/*4(

&04180.&$'( B*."( ."&( =
B
( 0&E#.*6&%+( 4"#$E&7( -1*0.( -#$.*4%&'( -&$( #.1/( \#E#*09(
7*'.$*C8.&7(80*21$/%+_K((L"*'(%&7(.1(."&(&Q-$&''*10(
(
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8
K(( A'( 21$(."&(-1'*.*6&( 4"#$E&(
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I9(."&(/&#0(7&2%&4.*10(2$1/(#('*0E%&(&04180.&$(B18%7(C&(
(
K( ( ( ( ( ( ( ( (10)(
(
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7&2%&4.*10(B18%7(C&41/&(
(
9( ( ( ( ( ( ( (11)((
(
B"&$&(K(*'(."&($#.*1(12(."&(0&.(61%8/&(1448-*&7(C+(-1'*.*6&(4"#$E&'(.1(."&(61%8/&(12(
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( 1$( 9( 7&-&07*0E( 10( "1B( ."&( -1'*.*6&( 4"#$E&( B#'(
7*'.$*C8.&7K(kL"1/'10(G>G<9(-K(H[[?Tl(
L"&$&( #$&( .B1( -1*0.'( B1$."( 01.*4*0E( #C18.( ."&'&( &Q-$&''*10'K( ( X*$'.9( *2( K(
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W
( B18%7( C&( &Q#4.%+( &D8#%( .1( H
G
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θ
1
=
16
5
e
2
mv
2
1
R
3N
0
2
φ
1
=
π
4
e
2
mv
2
N

0
R
φ
2
=
θ
1
⋅ 1 − 1 −
π
8
⎛
⎝
⎜
⎞
⎠
⎟
σ
1 3
φ
m
=
θ
1
2
+
φ
2
1
( )
1 2

φ
m
=
θ
1
2
+
φ
2
2
( )
1 2

25
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G
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WT
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/
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B
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/#.&$*#%'(12(#C18.(."$&&(&%&4.$10'(-&$(80*.(12(#.1/*4(B&*E".K((L"&(1."&$(#''8/ *109(
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#( '-"&$*4#%( 61%8/&( \L"&1$+( )_9( E#6&( #( $#.*1( 12( =
B
( .1( #.1/*4( B&*E".( ."#.(*04$&#'&7(

7$#/#.*4#%%+(21$("&#6*&$('4# &$&$'9(410.$#7*4.*0E(L"1/'10q'(&#$%*&$(4104%8'*10(."#.(
."*'( $#.*1( '"18%7( C&( # $1Q*/#.&%+( 410'.#0.9( #07( 10( ."&( 1$7&$( 12( 10&K(()#'&7( 10(
."&'&($&'8%.'9(!$1B."&$(7&4*7&7(."#.(

27
The uniformity of the beam was essential for confirming this aspect of Thomson’s theory. Crowther went
into explicit detail regarding his methods for achieving a homogeneous beam, saying that certain subtleties
had been overlooked by past experimenters. He first passed the particles through a magnetic field, then
selected for a small range of velocities with a tiny aperture. Crowther pointed out that an overly sized
aperture would admit a greater range than what would naively be expected. [Crowther, p.229-33]
N
0