Chapter 5

Intrinsic Properties of a Nucleus

◎ Total Angular momentum and Nuclear spin

● Parity

◎ The Electric field outside an arbitrary charge distribution

● Nuclear Electromagnetic moments

5-1 Total Angular momentum and Nuclear spin

For nuclei:

The nucleus is an isolated system and so often acts like a single entity with has a well defined total angular momentum.

It is common practice to represent this total angular momentum of a nucleus by the symbol I and to call it nuclear spin.

[Associated with each nuclear spin is a nuclear magnetic moment which produces magnetic interactions with its environment.]

For electrons in atoms:

For electrons in atoms we make a clear distinction between electron spin and electron orbital angular momentum and then combine them to give the total angular momentum.

Angular momentum

For nuclei:

total angular momentum of a nucleus = I ― "nuclear spin"

This is the VECTOR sum of:

the intrinsic spins of the individual nucleons (S)

HALF-INTEGER

the intrinsic orbital angular momentum of the individual nucleons (L)

INTEGER

Angular momentum

For nucleons:Every nucleon has quantum numbers: L, S, J

L - from orbital angular momentum quantum number ― INTEGERS - from spin quantum number ― HALF INTEGERJ - total angular momentum quantum number of single nucleon ― HALF INTEGER

Because the value of the J quantum number is always half-integral, if there is an even number of component angular moments J, then I will be integral

If there is an odd number, then I will be half-integral. In nuclear systems it is J that is good quantum number.

for odd A nuclei I is half integralfor even A nuclei I is integral.

All nuclei with even Z and even N have zero total nuclear angular momentum, I = 0.

Use I for total nuclear spin; use J for nucleon spin

(1) A single valence nucleon may determine the spin so: I = J.

(2) Two valence nucleons may determine the spin so: I = J1 + J 2 there can be several values of I.

(3) A valence odd particle and the remaining core contribute: I = Jparticle + Jcore

Total angular momentum of the nucleus - nuclear spin - is the vector sum of each nucleon = Total angular momentum quantum number of NUCLEUS = I

)1(22 III

),......,( IImmIZ

(1) The nucleus behaves like a single entity with angular momentum I(2) In ordinary magnetic fields I splits into 2I +1 Ze

eman effect(3) In strong fields individual nucleon states split in

to 2J +1

Notation and concepts

“sharp” s L = 0“principal” p L = 1“diffuse” d L = 2“fundamental” f L = 3

for example: for a nucleon, n = 2, L= 1 the state is designated 2p

The vector coupling of L and S suggests J = L +1/2 or L - 1/2Thus for L = 1 (p) we have p3/2 and p1/2

we can count higher states e.g. 2p3/2 .. 3p3/2

If we want to describe a nuclear state (for a nucleus) completely we need to identify its parity. Strong nuclear interactions will not alter the “parity”. The parity is conserved under strong nuclear interactions. Every nuclear eigenstate has its own “parity”.

Real mathematical functions can be categorized into three types:

1. If f(-x) = f(x) then f(x) is called an “even function”, or a function of even parity.2. If g(-x) = -g(x) then g(x) is called an “odd function”,

or a function of even parity.3. There are functions of mixed parities. Ex. h(x) = x2 + x

parity cover

5-2 Parity

An even function An odd function

It is quite often that functions we are dealing with are of mixed parities.

In terms of quantum mechanical wave functions we may get from the Schrödinger’s equation solutions of even and odd parity separately.

)()( xx

)()( xx

A wave function of even parity

A wave function of odd parity1-dimensional case

In the 3-dimensional case

)()( rr

)()( rr

),,(),,( zyxzyx

),,(),,( zyxzyx

even parity

odd parity

rr parity transformation

(1)

(2)

z

x

y

r

r

θ

φ

(π+ φ)(π-θ)In fact the parity transformation is to change r into – r.

In the spherical coordinates it is to change

and leave the radial coordinate r unchanged.

),,(),,( rrthat is

r

x

y

r-

θ

φ

(π+ φ)therefore if l = even then

Example:

In describing the hydrogen atom we solve the Schrödinger equation and come up the solution which is written as

),()(=),,(=)( φθYrRφθrψrψ lmnlnlmnlm

where n, l, and m are quantum numbers. Rnl(r) is the radial part of the wave function and Ylm

(θ,φ) is the angular part of the wave function. Ylm(θ,φ) is generally referred as the spherical harmonics

)(rψnlm

In this case

),()(),,()( lmnlnlmnlm YrRrr

),()1(),( lml

lm YY

is a function of even parity.

if l = odd then

)(rψnlm is a function of odd parity.

(3)

(4)

The parity transformation changes a right-handed coordinate system into a left-handed one or vice versa.

Two applications of the parity transformation restores the coordinate system to its original state.

It is a reasonable presupposition that nature should not care whether its coordinate system is right-handed or left-handed, but surprisingly, that turns out not to be so. In a famous experiment by C. S. Wu, the non-conservation of parity in beta decay was demonstrated.

Parity

(1) Central potentials only depend upon the magnitude of |r| and so areinvariant with respect to the parity operation, ie V(r) = V(-r).(2) Measurable effects of such potentials should also be invariant with respect to the parity operation.

(3) Observable quantities depend upon the square of the modulus of thewavefunction and so individual nuclear wavefunctions will be either even or odd but not a mixture of the two. (symmetric, antisymmetric)(4) That is to say, all nuclear states have a definite parity and conventionally this is used together with the total angular momentum to label the states.

(5) The definite parity of states means that the distribution of electric charge in the nucleus is even.

e.g. 0+ (I = 0, even parity) ; 2- (I = 2, odd parity)

Parity π

Every nuclear state also has a parity that is the product of theparity of each nucleon π = π1π2π3π4…πA = “+” or “-”Every nuclear state also has a parity: πThis is denoted: Iπ i.e. 0+, 2-, (1/2)-, (5/2)+.......

• not so far possible to know individual nucleon parity

• we have an overall parity that is a measured quantity

• no theoretical relationship between I and π

)()( rr

)()( rr

),,(),,( zyxzyx

),,(),,( zyxzyx

even parity

odd parity(5)

Much of what we know about nuclear structure comes from studying not the strong nuclear interaction of nuclei with their surroundings, but instead the much weaker electromagnetic interaction.The strong nuclear interaction establishes the distribution and motion of nucleons in the nucleus, and we probe that distribution with the electromagnetic interaction.

In doing so, we can use electromagnetic fields that have less effect on the motion of nucleons than strong force of the nuclear environment; thus our measurements do not seriously distort the object we are trying to measure.

In the absence of a magnetic field,

spins are randomly oriented.

Exposed to an external magnetic field, each spin or magnetic moment can assume two different orientations, denoted “parallel” (spin up) and “anti-parallel” (spin down) respectively.

5-3 The Electric Filed outside an arbitrary charge distribution

Let us consider an arbitrary charge distribution of density ρ(x’, y’, z’) occupying a volume τ’ and extending to a maximum distance r’max from the origin of coordinates O. We select O either within the volume or close to it. Such a distribution is illustrated in the figure.

The electric potential V at some point P(x,y,z) such that r > r’max is

∫='

0 "4

')',','(τ rπε

τdzyxρV

where r” is the distance between the point of observation P and the position P’(x’,y’,z’) of the element of charge ρ(x’, y’, z’)dτ’ :

2/1222 ])'()'()'[(" zzyyxxr

(6)

(7)

rr’

r”

Since r” is a function of x’, y’, z’ we may expand (1/r”) as a Taylor series near the origin:

where the subscript 0 indicate that the derivatives are evaluated at the origin.

where l = x/r is the cosine of the angle between the vector r and the x-axis.

,"

1

''

''

''

!2

1

"

1

''

''

''

1

"

1

2

0

0

rzz

yy

xx

rzz

yy

xx

rr(8)

32 "

'

'

"

"

1

"

1

' r

xx

x

r

rrx

230

"

1

' r

l

r

x

rx

and

We treat the terms concerning y’ and z’ accordingly and define m = y/r and n = z/r

Here l, m, and n are called “directional cosines” and satisfy the following requirement.

1222 =++ nml

(9)

(10)

The result of the calculation is that at the point P

321

0

22222

' 3

'0

20

'

4

')]13(

2

1')13(

2

1')13(

2

1

''3''3''3[1

4

')'''(

1

4

'1

VVV

dnymxl

ylmxxnlzzmnyr

dnzmylx

r

d

rV

(11)

'

01

'

4

1

r

dV

V1 is called the monopole term and is zero only if the total net charge is zero. The electrical field derived from V1 decreases as 1/r2.

' 2

02

')'''(

4

1

r

dnzmylxV

V2 is called the dipole term which varies as 1/r2. The electrical field derived from V2 decreases as 1/r3.

If we define an electrical dipole moment p

')ˆ'ˆ'ˆ'(''''

dkzjyixdrp

then the dipole term V2 can be written as0

12 4

ˆ

rp

V

r1 is the unit vector along r in the direction of P. knjmilr ˆˆˆ1̂

V3 is called the quadrupole term which varies as 1/r3. The electrical field derived from V3 decreases as 1/r4.

In general there are totally six components for a quadrupole moment. If the charge distribution is of cylindrical symmetry, as it is in all the nuclides, then three components become vanished and it is convenient to define a single quantity q, often called the quadrupole moment of the charge distribution:

')''3( 2

'

2

drzq

(12)

(13)

5-4 Nuclear Electromagnetic moments

1. The electrical monopole moment of a nucleus is simply the total charge Ze.

2. Because of the nearly perfect cylindrical symmetry of all nuclei, spherical or non-spherical, the electric dipole moments are too small to be detected by the existing techniques.

3. However, nuclei with prolate- or oblate-like shape, non-spherical, can have measurable electric quadrupole moments. A nucleus of large measured quadrupole moment q is geometrically non-spherical.

Some measured electrical quadrupole moment q.

O178

K3919

Bi20983

In11549

Ho16567

Lu17571

Ta18173

Au19779

-0.026 barns

0.11 barns

-0.35 barns

1.16 barns

2.82 barns

5.68 barns

4.20 barns

0.59 barns

A circular loop carrying current I and enclosing area A has a magnetic dipole moment of magnitude ∣μ = ∣ iA. Here we want to consider a simple classical model of an orbiting point charge circling around a center O chosen to be the origin of the coordinate system. The orbiting point charge with charge e and mass m is circling around the origin O with a radius r and speed v.

ev

l μ

O rIn this case the magnitude of the magnetic dipole moment is

lm

eevrr

vr

e

22)/2(2

(14)

where l is the classical angular momentum mvr.

In quantum mechanics, we operationally define the observable magnetic moment to correspond to the direction of greatest component of l; thus we can take equation (14) directly into the quantum regime by replacing l with the expectation value relative to the axis where it has maximum projection, which is which is ml (h/2π) with ml = + l. Thus

lm

e

2

(15)

where now l is the angular momentum quantum number of the orbit.

Nuclear Magnetic Dipole Moment

lm

e

2

(15)

m

e

2

is called a magneton.

For atomic motion we use the electron mass and obtain the Bohr magneton

eV/T107884.5 5B

In nuclear system it is the proton mass that we should use to calculate the magneton and we have the nuclear magneton:

eV/T101525.3 8N

Note that μN << μB owing to the difference in the masses; thus under most circumstances atomic magnetism has much larger effects than nuclear magnetism.

For a proton of orbital angular momentum l its magnetic dipole moment due to the orbital motion is

Nllg (16)

where gl is the g factor associated with the orbital angular momentum l.

Magnetic dipole moment of a nucleon due to its orbital motion

ev

l μ

O r

Nllg (16)

where gl is the g factor associated with the orbital angular momentum l.For protons gl = 1; because neutrons have no electric charge, we can use equation (16) to describe the orbital motion of neutrons if we put gl = 0.

Magnetic dipole moment of a nucleon due to its spin

Due to proton spin its magnetic dipole moment can be described in a similar way:

Nssg (17)

where s = 1/2 for proton, neutrons and electrons and the quantity gs is known as the spin g factor and is calculated by solving a relativis

tic quantum mechanical equation.

For a spin-1/2 point particle such as the electron, the Dirac equation gives gs =2, and the measured value from the electron is gs = 2.002319304386 which is quite consistent with the theoretical predicted value. More accurate value can be obtai

ned by calculating higher order corrections from quantum electrodynamics.

Nssg (17)

For free nucleons the experimental values are far from the expected value for point particles:

0000022.05856912.5 :proton sg

0000018.08260837.3 :neutron sg(18)

It is evident that :

(1) The proton value is far from the expected value of 2 for a point particle.

(2) The uncharged neutron has a nonzero magnetic moment!

This is the evidence that the nucleons are not elementary point particles like the electron, but have an internal structure; the internal structure of the nucleons must be due to charged particles in motion, whose resulting

currents give the observed spin magnetic moments.

0000022.05856912.5 :proton sg

0000018.08260837.3 :neutron sg(18)

It is interesting to note that gs for the proton is greater than its expected value by about 3.6, while gs for the neutron is less than its e

xpected value (0) by roughly the same amount.Magnetic dipole moments of proton and neutron are vector

sums of quarks which constitute two different kinds of nucleons.

In nuclei, the pairing force favors the coupling of nucleons so that their orbital angular momentum and spin angular momentum each add to zero. Thus the paired nucleons do not contribute to the magnetic moment, and we need only consider a few valence nucleons. No nucleus has been observed with a magnetic dipole mo

ment larger than about 6 μN.

~The End ~