Notes for Lecture

Feb 23, 2002

Notes for Lecture Notes for Lecture #17

Derivation of the hyperfine interaction

Magnetic dipole field

These notes are very similar to the notes for Lecture #12 on the electric dipole field.

The magnetic dipole moment is defined by

m =

ó
õ

d³r^¢ r^¢ ×J(r^¢),

(1)

with the corresponding potential

A(r) =

m₀

m ×

^
r

r²

(2)

and magnetostatic field

B(r) =

m₀

ì
ï
í
ï
î

^
r

(m ·

^
r

) - m

r³

m d³(r)

ü
ï
ý
ï
þ

(3)

The first terms come form evaluating Ñ×A in Eq. 2. The last term of the field expression follows from the following derivation. We note that Eq. (3) is poorly defined as r® 0, and consider the value of a small integral of B(r) about zero. (For this purpose, we are supposing that the dipole m is located at r = 0.) In this case we will approximate

B(r » 0) »

æ
è

ó
õ

sphere

B(r) d³r

ö
ø

d³(r).

(4)

First we note that

ó
õ

r £ R

B(r) d³r = R²

ó
õ

r = R

^
r

×A(r) dW.

(5)

This result follows from the divergence theorm:

ó
õ

vol

Ñ·V d³r =

ó
õ

surface

V · dA.

(6)

In our case, this theorem can be used to prove Eq. (5) for each cartesian coordinate of Ñ×A since Ñ×A = [^(x)] ( [^(x)] ·(Ñ× A) ) + [^(y)] ( [^(y)] ·(Ñ×A) ) + [^(z)] ( [^(z)] ·(Ñ×A) ). Note that [^(x)] ·(Ñ×A) = -Ñ·([^(x)] ×A) and that we can use the Divergence theorem with V º [^(x)] ×A(r) for the x- component for example:

ó
õ

vol

Ñ·(

^
x

×A) d³r =

ó
õ

surface

(

^
x

×A) ·

^
r

dA =

ó
õ

surface

(A ×

^
r

) ·

^
x

dA.

(7)

Therefore,

ó
õ

r £ R

(Ñ×A) d³r = -

ó
õ

r = R

(A ×

^
r

) ·(

^
x

^
y

^
z

) dA = R²

ó
õ

r = R

(

^
r

×A) dW

(8)

which is identical to Eq. (5). We can use the identity (as in Lecture Notes 12),

ó
õ

^
r

|r - r^¢|

4 p

r_<

r_>²

^
r^¢

(9)

Now, expressing the vector potential in terms of the current density:

A(r) =

m₀

4 p

ó
õ

d³r

J(r^¢)

|r - r^¢|

(10)

the integral over W in Eq. 5 becomes

R²

ó
õ

r = R

(

^
r

×A) dW =

4 pR²

m₀

4 p

ó
õ

d³r^¢

r_<

r_>²

^
r^¢

×J(r^¢).

(11)

If the sphere R contains the entire current distribution, then r_> = R and r_< = r^¢ so that (11) becomes

R²

ó
õ

r = R

(

^
r

×A) dW =

4 p

m₀

4 p

ó
õ

d³r^¢ r^¢ ×J(r^¢) º

8 p

m₀

4 p

(12)

which thus justifies the so-called ``Fermi contact'' term in Eq. 3.

Magnetic field due to electrons in the vicinity of a nucleus

In Lecturenotes #15, we showed that the current density associated with an electron in a bound state of an atom as described by a quantum mechanical wavefunction y_{nlm_l}(r) can be written:

J(r) =

-e (^h/_2p) m_l

^
f

m_e r sinq

|y_{nlm_l}(r)|².

(13)

In the following, it will be convenient to represent the azimuthal unit vector [^(f)] in terms of cartesian coordinates:

^
f

= - sinf

^
x

+ cosf

^
y

^
z

×r

r sinq

(14)

The vector potential for this current density can be written

A(r) = -

m₀

e (^h/_2p)

m_e

m_l

ó
õ

d³r^¢

^
z

×r^¢

|r - r^¢|

|y_{nlm_l}(r^¢)|²

r^¢² sin²q^¢

(15)

We want to evaluate the magnetic field B = Ñ×A in the vicinity of the nucleus (r ® 0). Taking the curl of the Eq. 15, we obtain

B(r) =

m₀

e (^h/_2p)

m_e

m_l

ó
õ

d³r^¢

(r-r^¢) ×(

^
z

×r^¢)

|r - r^¢|³

|y_{nlm_l}(r^¢)|²

r^¢² sin²q^¢

(16)

Evaluating this expression with (r ® 0), we obtain

B(0) = -

m₀

e (^h/_2p)

m_e

m_l

ó
õ

d³r^¢

r^¢ ×(

^
z

×r^¢)

r^¢³

|y_{nlm_l}(r^¢)|²

r^¢² sin²q^¢

(17)

Expanding the cross product and expressing the result in spherical polar coordinates, we obtain in the numerator [^(r)]^¢ ×([^(z)] ×[^(r)]^¢) = [^(z)](1-cos²q^¢)-[^(x)]cosq^¢sinq^¢cosf^¢-[^(y)]cosq^¢sinq^¢sinf^¢).

In evaluating the integration over the azimuthal variable f^¢, the [^(x)] and [^(y)] components vanish which reduces to

B(0) = -

m₀

e (^h/_2p)

m_e

m_l

ó
õ

d³r^¢

^
z

r^¢² sin²q^¢

r^¢³

|y_{nlm_l}(r^¢)|²

r^¢² sin²q^¢

(18)

and

B(0) = -

m₀ e (^h/_2p) m_l

^
z

4 pm_e

ó
õ

d³r^¢|y_{nlm_l}|²

r^¢³

º -

m₀ e

4 pm_e

L_z

^
z

r^¢³

(19)

File translated from T_EX by T_TH, version 2.20.
On 23 Feb 2002, 19:22.