LETT]~I~:E AL NUOVO CIM~NTO
VOL. 34, N. 11
10 Luglio 1982
The Spin Kinetic Energy and Intrinsic Mass of Elementary Particles. I. Massive Particles of Spin One-Half. L. S. LEVITT
College o] Science, University o/ Texas - E1 Paso, Tex., 79968 ( r i c e v u t o il 5 Aprile 1982)
F o r e • [z~, p• a n d n ~ s = 1 a n d t h e spin a n g u l a r m o m e n t u m is
po(~) = V s ( s + 1)~, = V 3 ~ / 2 .
(1)
T h e spin r o t a t i o n of a p a r t i c l e m a y also b e e x p r e s s e d 0) in t e r m s of its m o m e n t of i n e r t i a I a n d its spirt a n g u l a r v e l o c i t y ~o:
po(~)- Io).
(2)
T h e a s s o c i a t e d k i n e t i c e n e r g y of spin EKc~) i~, t h e r e f o r e , (3)
El<(,)- 10)2/2 ~ p2o~)/2I.
w h i c h , w i t h eq. (1) b e c o m e s (4)
E~(,~ : 3h~/8I.
B u t , since I = rn~r~, w h e r e ~t~ is t h e (( i n t r i n s i c ~>m a s s of t h e s p i n n i n g p a r t i c l e a n d r 0 i t s (( r a d i u s )), one o b t a i n s (5)
E~,,) = 3ti~-/8~,~r~,.
W e c o n c e i v e t h a t t i m o b s e r v e d ((rest m a s s )> of a n e l e m e n t a r y p a r t i c l e m o is in f a c t c o m p o s e d of t w o t e r m s , its i n t r i n s i c r e s t m a s s m i a n d t h e m a s s e q u i v a l e n t of its spin k i n e t i c e n e r g y m~. T h u s , (6)
(1)
m o = ml + m~.
A. J . SONNESSA: I ntroductio~ to Molecular Spectroscopy ( N e w Y o r k , N. Y., 1966), D. 3 t .
333
334
L . s . LnV~TT
T h e r e f o r e , t h e t o t a l <(r e s t m a s s >> e n e r g y of a p a r t i c l e is (7)
E T ~ El)
~- EK(s) =
mo c2 ,
w h e r e Ex(~) is t h e kinetic e n e r g y due t o its spin a n g u l a r m o m e n t u m , po (except for p a r t i c l e s like p i o n s w i t h s = 0). T h e t o t a l (( r e s t ~>e n e r g y is t h e r e f o r e , as usual, mo c2, b u t its <,intrinsic )> r e s t e n e r g y is m~c 2, w h e r e m~ < m0. I t is seen, t h e r e f o r e , t h a t
(s)
ET
moce
=
2
2
talc 2 + po/2mlro
Or
(9)
mo c~ = m~c 2 + 3h~lSm~r~.
E q u a t i o n (9), w h e n m u l t i p l i e d t h r o u g h b y mi gives a q u a d r a t i c in m~:
(lO)
m ~ c 2 - - momic~ + 3h2/8r~ = O,
w h i c h , w h e n s o l v e d for m~ r e s u l t s in
(11)
mo c2 -1- ~/ (moc2) 2 - - 4c2(31i2/8r~)
mi =
2C 2
or
(~2) S e t t i n g t h e p a r t i c l e or (>r a d i u s e q u a l to its C o m p t o n w a v e - l e n g t h (2) (h/m~c), one o b t a i n s a f t e r s o m e m a n i p u l a t i o n
(13)
m,= ~
( 3)1
+
.3~-~.z] = m o 1 + 3 ~
'
or
(14)
m i = m0/(1.00951 ) = 0.9906m 0 .
I t is seen t h a t m s = m o - m
(15)
i, a n d
m e = (1 - - 0.9906)m o = 9 . 4 2 . 1 0 - 3 m e .
T h e <~i n t r i n s i c >>r e s t m a s s t h e r e f o r e c o m p r i s e s 99.06% of t h e o b s e r v e d r e s t m a s s , a n d t h e m a s s e q u i v a l e n t of t h e spin k i n e t i c e n e r g y is 0 . 9 4 % of m 0. W h e n t h e p a r t i c l e is a n n i h i l a t e d , t h e n , t h e o b s e r v e d t o t a l e n e r g y release is still E T = c~(mL + m,) = moc 2 . I t m a y b e of s o m e i n t e r e s t t o c a l c u l a t e n u m e r i c a l v a l u e s for s o m e of t h e q u a n t i t i e s d i s c u s s e d above.
L. S. LEVI'rT: B u t t . A m . P h y s . See., 2, 211 (1956); Cimento, 12, 537 (1975).
(2)
Experienfia, 14, 223 (1958);
Lett. N u o v o
TIIE
SPIN
KINETIC
335
]~N~IR.GY ~TC.
F o r all p a r t i c l e s of s p i n 89 h a v i n g a r e s t m a s s , it is o b v i o u s t h a t m~/m o = 0 , 9 9 0 6 ~ n d m ~ / m 0 = 9.4- 10 -3. T h e s p i n m o m e n ~ o f i n e r t i ~ f o r e l e c t r o n s a n d p o s i t r o n s is I = - - m~r0z ~ ( 9 . 1 0 . 1 0 -2s g ) ( 2 . 4 3 , l 0 -a~ c m ) 2 ~ 5 . 3 7 . 1 0 47 g c m 2 a n d [ e ~ n also b e w r i t t e n a s
(16)
7
=
h2/~.~ i e
2 .
S i n c e I v a r i e s i n v e r s e l y w i t h m~, t h e v ~ l u c f o r p r o t o n s , , ~ n t i p r o t o n s a n d n e u t r o n s is m u c h s m ~ l l e r a n d e q u a l t o 2 . 9 2 - 1 0 - 5 ~ c m 2. T h e s p i n a n g u l a r v e l o c i t y o~ f o r elect r o n s ~ n d p o s i t r o n s is po/l a n d is e q u a l t o 1.71.1019 s - L F o r n u c l e o n s a n d a n t i n u c l e o n s , w = 3 . 1 4 . 1 0 2 e s - L C o m b i n i n g e q s . (1), (2), (4) ~ n d (16) g i v e s eo = V~g~m~c~/h. T h e r o t a t i o n a l f r e q u e n c y , /, is t h e n s e e n t o b e :
(17)
T h e r o t a t i o n a l f r e q u e n c y f o r clc(~trollS a n d p o s i t r o n s is 2 . 7 2 . l 0 is s -1 a n d i s 5.00.1021 s -1 for nucleons and antinucleons. A m o s t i n t e r e s t i n g r e s u l t ia o b t a i n e d on c a l c u l a t i n g t h e p e r i p h e r a l (linear) s p i n velocity (18)
v :
roe) =
/- .
C o m b i n i n g c q . (18) w i t h (16), w(' t h e n h a v e
\talc ] h2/mic 2
(19)
It
4~ '
TABLE I. -- Calculated quantities /or particles having a rest mass a~td spin 1. Quantity
Particle eL
~•
p•
no
Ao
po(~) (erg s)
0 . 9 1 3 . 1 0 -27
0 . 9 1 3 . 1 0 -0-7
0 . 9 1 3 . 1 0 -27
0 . 9 1 3 . 1 0 -27
E~:(s) (MeV)
4.8
1.01
8.84
8.85
r o (cm)
2.43
l 0 -l~
1.17 . 1 0 -12
1.32 . 1 0 is
1.32 9 10 -12
1.11 . 1 0 -13
I
5.3
10 -47
2.59
9 10 -49
2.92
9
10 -5o
2.92
9 10 -50
2.46
9 1020
3.14
9
1020-
3.14
9
( g c m 2)
9 10 -3
0 . 9 1 3 . 1 0 -27 10.7
9 10 -50
~o ( s - 1 )
1.71
l019
3.53
1022
3.71
9
J (s -1)
2.72
10-is
5.62 9 1019
5.00 9 1021
5.00 9 1021
5.91
9 102t
v (cm/s)
4.14
109
4.14
4.14
4.14
4.14 9 109
m~ (MeV)
4.8
10 -3
t.01
m~ (MeV)
0.506
104.7
929.5
930.8
1105.5
m o (MeV) (3)
0.511
]05.7
938.3
939.6
1116.0
(3) E.
- 109
8.84
9 109
8.85
9 109
1022
10.5
•. FRAUENFELDER and E. ~,I. HENLEY: Subato~nic Physics (Englcwood Cliffs, N . J . , SEORI~: Nuclei and Particles (Reading, Mass., 1977).
1974);
336
L.s.
L~VlTT
w h i c h is seen t o b e t h e sanle for all p a r t i c l e s of spin 89 since Po is t h e s a m e (0.913.10 -27 erg s). T h e v a l u e for v is t h u s 4.14.109 cin/s a n d fi = 0.138. T a b l e I gives a s u m m a r y of all c a l c u l a t e d q u a n t i t i e s . I n v i e w of t h e foregoing, s h o u l d n o t t h e a n n i h i l a t i o n e n e r g y of o r t h o p o s i t r o n i u m (4), Psi' ~, be g r e a t e r t h a n t h a t of p a r a - P s ~ b y an a m o u n t e q u a l to 2msc: = 9.8 K e V ? W e leave t h e r e a d e r to p o n d e r t h i s question.
(')
R . A . FERREL: Phys. Rev., 110, 1355 (1958); A. BISI, G. CONSOLATI,G. GAMBARINIand L. ZAPPA: Nuovo Cimento B, 65, 442 (1981); L. S. LEVITT: Left. Nuovo Uimento, 15, 314 (1976).
~) by Societk Italiana di Fisica Proprietk letteraria riservata Direttore responsabile: RENATO ANGELO RICCI Stampato in Bologna dalla Tipografla Compositori r tipi della Tipografia Monograf Questo faseieolo i~ stato lieenziato dai torehi il 6-VII-1982
