Transcript Slide 1
16.451 Lecture 12: The neutron Particle Data Group entry: 14/10/2003 1 ???
• slightly heavier than the proton by 1.29 MeV
(otherwise very similar)
• electrically neutral (q/e < 10 -21 !!!) • spin = ½ • magnetic moment = - 1.91 N
(should be zero if pointlike: Dirac)
• unstable, with a lifetime of about 15 minutes:
n
p
e
• accounts for a little more than half of all nuclear matter
v e
Recall the nuclear “landscape” from lecture 1:
100 Heavy nuclei, N > Z
light nuclei have N Z
0 0 neutron number, N 140
http://www.nscl.msu.edu/future/ria/science/toi.html
2
Neutron electric form factor: G e n
(from elastic electron scattering, etc.)
3 • difficult to measure! no free neutron target ... • recall the form factor expansion from lecture 8: (compare 1 H and (magnetic contribution dominates) 2 H, etc...) • very small contribution to total cross section, since net charge = 0
F
(
q
2 ) 1
i q
r
(
q
r
) 2 / 2 ...
(
r
)
d
3
r
q
2
r
2 6 ...
for
(
r
)
d
3
r
0 !
All the world’s data (2003): Positive slope implies negative
G e n (0) = 0 Various quark model theories
What does negative
r
2
r
2 (
r
)
d
3
r
r
2 4
r
2 (
r
)
dr
• charge density must have both –ve and +ve regions, since net charge = 0 • integral is weighted with r 2 more negative charge at large radius 4
Neutron magnetization distribution: about the same as the proton Neutron 5 Both plots show ratios to “dipole” fit:
G
D
1
Q
2 1 / 0 .
71 GeV 2 2 Proton Recall: G M (0) = , i.e. the magnetic moment is the “magnetic charge” ...
Isospin and the nucleon:
(Krane, 11.3)
6 • the neutron and proton are very similar apart from a small mass difference (0.1%) and of course the difference in electric charge • both play an equally important role in determining the properties of nuclei • postulate that n,p are two “substates” of a “nucleon”, with “Isospin ½”, by analogy with ordinary spin s
(Heisenberg, 1932)
for spin, S:
s
1 2 ,
s
2
s
(
s
1 ),
s z
m s
1 2 e.g. electron: spin “up” and spin “down” states have different values of m s , but this is a trivial difference – both are electrons! for Isospin, T:
T
1 2 ,
T
2
T
(
T
1 ),
T z
m t
1 2 by convention, the proton has m
t
= + ½ , and the neutron has m
t
= - ½ ; these are two “substates” of the nucleon (N) with isospin T = ½ ! (PDG table uses I)
Nucleon states: 7
E
( MeV ) 939 .
6 938 .
3 Nucleon, N
m
t
1 / 2 ,
neutron m
t
1 / 2 ,
proton
• both neutrons and protons have spin S = ½ • S and T are independent quantum numbers • S is “real” in that it has classical analogs in mechanics (intrinsic angular momentum) and electrodynamics (magnetic moment) = g s S N • T has no classical analog; it is a quantum mechanical vector, literally “like spin” (iso = ‘like’), so it follows the same addition rules as S, L, J, etc...
• in this language, (n,p) are isospin-substates of the nucleon, N • as far as the strong interaction is concerned,
m
t
is all that distinguishes a
Why isospin?
8 • It turns out to be rather a lucky guess that isospin is a symmetry of the strong interaction: both m
t
and T are conserved in strong scattering and decay processes. • The electromagnetic interaction breaks isospin symmetry; i.e. it can distinguish between different values of m t There is a simple relation between m t and electric charge for all hadrons, (particles made up of quarks, exhibiting strong interactions...)
Nucleon: N = (n,p)
T = ½ isospin doublet, m
t
= ½ electric charge (q/e) = m
t
+ ½ (mass ~ 940 MeV)
Delta:
(1232) = ( ++ , + , °, ) T = 3 / 2 isospin quartet, m
t
= (
3
/
2
, ½, - ½ , -
3
/
2
)
electric charge (q/e) = m
t
+ ½ (mass ~ 1232 MeV)
Pion or
-meson = ( + , °, ) T = 1 isospin triplet, m
t
= (1, 0, -1) electric charge (q/e) = m
t
(mass ~ 140 MeV )
Conservation Laws: A conserved quantity is the same before and after an interaction takes place, e.g.: total energy linear momentum angular momentum (quantum vector) electric charge parity isospin (exception: weak interaction) (strong interaction only) from classical mechanics quantum mechanics 9 Example: resonance decay, + p + ° in the rest frame:
M
“before”
p
o
“after” Total energy and momentum conservation: M( ) = m(p) + m( ) + K(p) + K(
p p
p
0
what about the other quantities?
Adding angular momentum
(review: lecture 3, hydrogen atom...)
Whether we are adding “spin” or “orbital” or “total” angular momentum (s, same rules apply, so we will use “j” in the formalism here: Consider:
j
1
j
2
J
l
, j), the • the total angular momentum has quantum number
J
and z-projection
m J
• the z-projections add linearly:
m j
1
m j
2
m J
• the solutions for
J
must be consistent with a complete set of configurations which can be found by writing down all possibilities,
as in lecture 3, slide 11 m J ,
• this leads to the general rule:
J
(
j
1
j
2 ), (
j
1
j
2 1 ) ...
| (
j
1
j
2 ) | • an exact prescription is beyond the scope of this course, but it involves writing the quantum state |J,m J > as a linear superposition of configurations |j 1 ,m 1 ,j 2 ,m 2 >:
J
,
m J
m
1 ,
m
2
a
(
j
1 ,
m
1 ,
j
2 ,
m
2 ,
J
,
m J
)
j
1 ,
m
1 ,
j
2 ,
m
2
(The coefficients a(j 1 ,m 1 ...) are just numbers; they are called “Clebsch-Gordon” coefficients in advanced books on quantum mechanics.)
10
Application: + p + ° (the quantum numbers have to add up!) 11 “before” Angular momentum: J = 3/2 Parity: + Isospin: T = 3/2, m
t
= ½
p
o
“after” Angular momentum: proton: s = ½ pion: s = 0 orbital:
L
1 2
L
J
L
1 Parity: proton: + pion: orbital: (-1)
L
( Isospin: proton: T = ½, m t pion: T = 1, m t = ½ = 0 )( )( 1 )
L
T m
t
( 3 1 / / 2 2 , 1 / 2 ) All the conservation laws are observed. Reaction proceeds in the “T=3/2 channel”
Isospin and quarks: There are a total of 6 quarks in the Standard Model (u,d,s,c,t,b – more later!) only two play a significant role in nuclear physics: u and d.
but Not surprisingly, isospin carries over into the quark description: the “up” quark has isospin T = ½ “up” and similarly for the “down” quark: Quark “flavor” u (“up”) Spin, s 1/2 Charge, q/e + 2/3 Isospin projection, 1/2
m
t
12 d (“down”) 1/2 - 1/3 -1/2 Isospin addition for the proton: p = (uud), m
t
neutron: n = (udd), m
t
= ½ + ½ - ½ = ½ = ½ - ½ - ½ = - ½ What about the delta? Addition of 3 x isospin- ½ vectors: T = 1/2 or 3/2; T = 3/2 is the : ++ = (uuu), + = (uud), ° = (udd), = (ddd) What about antiquarks? same isospin but opposite m
t
e.g. pion: ( + , °, )
u d
,
m t
1 2 1 2 1 ,
etc
...