16.451 Lecture 12: The neutron 14/10/2003 1

Particle Data Group entry:

• 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: • accounts for a little more than half of all nuclear matter

evepn

???

Recall the nuclear “landscape” from lecture 1:

140

neutron number, N

pro

ton

num

ber,

Z

0

0

100

http://www.nscl.msu.edu/future/ria/science/toi.html

light nucleihave N Z

Heavy nuclei,N > Z

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Neutron electric form factor: Gen (from elastic electron scattering, etc.)

• difficult to measure! no free neutron target ... (compare 1H and 2H, etc...)• very small contribution to total cross section, since net charge = 0 (magnetic contribution dominates)• recall the form factor expansion from lecture 8:

!0)(...6

)(...2/)(1)( 3

22

322 rdrforrq

rdrrqrqiqF

All the world’sdata (2003):

Gen(0) =

0

Positive slope impliesnegative <r2> !

Various quarkmodel theories

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What does negative <r2> mean?

drrrrrdrrr )(4)( 22322

• charge density must have both –ve and +ve regions, since net charge = 0• integral is weighted with r2 more negative charge at large radius

4

Neutron magnetization distribution: about the same as the proton

Neutron

Proton

Both plots show ratios to “dipole” fit:

222 GeV71.0/1

1

QGD

Recall: GM(0) = , i.e. the magneticmoment is the “magnetic charge” ...

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Isospin and the nucleon: (Krane, 11.3)

• 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:

e.g. electron: spin “up” and spin “down” states have different values of ms, but this is a trivial difference – both are electrons!

2

122

1),1(, smzsssss

6

for Isospin, T:

by convention, the proton has mt = + ½ , and the neutron has mt = - ½ ;

these are two “substates” of the nucleon (N) with isospin T = ½ ! (PDG table uses I)

2

1

2

1),1(2, tmzTTTTT

Nucleon states:

E6.939

3.938)MeV(

neutronmt ,2/1

protonmt ,2/1

Nucleon, N

• 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) = gsS 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, <Tz> = mt is all that distinguishes a

neutron from a proton

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Why isospin?

• It turns out to be rather a lucky guess that isospin is a symmetry of the strong interaction: both mt 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 mt

There is a simple relation between mt and electric charge for all hadrons, (particles made up of quarks, exhibiting strong interactions...)Nucleon: N = (n,p) T = ½ isospin doublet, mt = ½

electric charge (q/e) = mt + ½ (mass ~ 940 MeV)

Delta: (1232) = (++, +, °, -) T = 3/2 isospin quartet, mt = (3/2, ½, - ½ , -3/2)

electric charge (q/e) = mt + ½ (mass ~ 1232 MeV)

Pion or -meson = (+, °, -) T = 1 isospin triplet, mt = (1, 0, -1)

electric charge (q/e) = mt (mass ~ 140 MeV)

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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 (exception: weak interaction) isospin (strong interaction only)

from classical mechanics

quantum mechanics

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Example: resonance decay, + p + ° in the rest frame:

“before”

M

p o

“after”

Total energy and momentum conservation:

M() = m(p) + m() + K(p) + K(), 0 pp p

what about the other quantities?

Adding angular momentum (review: lecture 3, hydrogen atom...) 10

Whether we are adding “spin” or “orbital” or “total” angular momentum (s, l, j), thesame rules apply, so we will use “j” in the formalism here:

Consider:

• the total angular momentum has quantum number J and z-projection mJ

• the z-projections add linearly:

• the solutions for J must be consistent with a complete set of configurations mJ , which can be found by writing down all possibilities, as in lecture 3, slide 11

• this leads to the general rule:

• an exact prescription is beyond the scope of this course, but it involves writing the quantum state |J,mJ> as a linear superposition of configurations |j1,m1,j2,m2>:

(The coefficients a(j1,m1 ...) are just numbers; they are called “Clebsch-Gordon” coefficients in advanced books on quantum mechanics.)

Jjj

21

Jjj mmm 21

|)(|...)1(),( 212121 jjjjjjJ

2211,

2211 ,,,),,,,,(,21

mjmjmJmjmjamJmm

JJ

Application: + p + ° (the quantum numbers have to add up!)

p o

“after” “before”

Angular momentum: J = 3/2

Parity: +

Isospin: T = 3/2, mt = ½

Angular momentum:

proton: s = ½pion: s = 0

orbital: L

Parity: proton: + pion: - orbital: (-1)L

Isospin: proton: T = ½, mt = ½ pion: T = 1, mt = 0

JL

2

1

L)1)()((

1L

2/1)2/1,2/3(

tmT

All the conservation laws are observed. Reaction proceeds in the “T=3/2 channel”

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Isospin and quarks:

There are a total of 6 quarks in the Standard Model (u,d,s,c,t,b – more later!) butonly two play a significant role in nuclear physics: u and d.

Not surprisingly, isospin carries over into the quark description: the “up” quark hasisospin T = ½ “up” and similarly for the “down” quark:

Quark “flavor” Spin, s Charge, q/e Isospin projection, mt

u (“up”) 1/2 + 2/3 1/2

d (“down”) 1/2 - 1/3 -1/2

Isospin addition for the proton: p = (uud), mt = ½ + ½ - ½ = ½

neutron: n = (udd), mt = 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 mt

e.g. pion: (+, °, - )

...,1,2

1

2

1 etcmdu t

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