Transcript Measurement of nuclear radius
Measurement of nuclear radius
• Four methods outlined for charge matter radius: – Diffraction scattering – Atomic x-rays – Muonic x-rays – Mirror Nuclides
Measurement of nuclear radius
• Three methods outlined for nuclear matter radius: – Rutherford scattering – Alpha particle decay – -mesic x-rays
Diffraction scattering
ki
kf q -kf
ki
k i
k f
k
q
2
k
sin( /2) • q = momentum transfer
k i
Diffraction scattering
k f
k
q
2
k
sin( /2) • Measure the scattering intensity as a function of the
distribution of charge in the nucleus
, *
f
i dv e iq
r
dv
equation 3.4
4
q
sin
e r d r
to infer
Diffraction scattering
4 sin
e r d r
q
2 to infer the
distribution of charge in the nucleus
• •
e
is the
inverse Fourier transform
of is known as the
form factor
for the scattering.
• c.f. Figure 3.4
; what is learned from this?
Diffraction scattering
•
Density of electric charge
e
constant in the nucleus is ≈ constant
e
A
4
R
3 4
R
3
A R
R o A
1 / 3
Diffraction scattering
•
The charge
distribution does not have a sharp boundary – Edge of nucleus is diffuse - “
skin
” – Depth of the skin ≈ 2.3 f –
RMS radius
is calculated from the charge distribution and, neglecting the skin, it is easy to show
r
2 3 5
R
2
Atomic X-rays
•
Assume the nucleus is uniform charged sphere
.
• Potential
V
– is obtained in two regions:
Inside
the sphere
Ze
2 4
o R
3 2 1 2
r R
2
r
R
–
Outside
the sphere
Ze
2 4
o r r
R
Atomic X-rays
• • For an electron in a given state,
its energy
depends on -
V
*
n V
n dv
•
Assume does not change appreciably if V pt
V sphere
V
r
R
*
n V
• Then,
E = E sphere
n - E pt dv
r
R
*
n V
n dv
n can be
giving
(3.12)
Atomic X-rays
•
E
between sphere and point nucleus for 1,1(1
s
)
E
1
s
2 5
Z
4
e
2 4
o R
2
a o
3
E
1
s
(
pt
) •
Problem!
• Compare this
E
to measurement and we have
E
1
s E
1
s
(
sphere
) • We will need
two measurements
to get R - • Consider a 2p 1s transition for (
Z,A
) and (
Z,A’
) where
A’
= (
A
-1) or (
A+
1) ;
what x-ray does this give?
E K
E K
E
2
p
E
1
s
E
2
p
E
1
s
Atomic X-rays
E K
E K
E
2
p
E
1
s
E
2
p
E
2
p
E
2
p
E
1
s
E
1
s E
1
s
•
Assume that the first term will be ≈ 0. Why?
• Then, use
E 1s E K
from (3.13) for each
E 1s E K
E
1
s
E
1
s
term.
Why?
2 5
Z
4
e
2 4
o
1
a o
3
R o
2
A
2 / 3 2 / 3
Atomic X-rays
E K
E K
• This
x-ray energy difference
is called the
“isotope shift”
•
We assumed that R = R o A 1/3 .
Is there any authentication?
• How good does your spectrometer have to be to see the effect?
• We assumed we could use hydrogen-line 1s wavefunctions Are these good enough to get good results?
• Can you use optical transitions instead of x-ray transitions?
Muonic X-rays
• • • Compare – – – – this process with atomic (electronic) x-rays:
Similarities Differences Advantages Disadvantages
n
,
l
,
m
2
Z a o
3 / 2
e
Zr a o n
1,
l
0,
m
0
a o
4
o m e
2 2
E n
32
mZ
2
e
4 2
o
2 2 What is
a o
? Pauli Exclusion principle for muons, electrons?
n
2
Coulomb Energy Differences
• Calaulate the
Coulomb energy
of the charge distribution directly
E C
E C
5 3 5 3
Q
2 4
o R e
2 4
o R
Z
2 Consider
mirror nuclides
:
Z
1 2
Z Z
A
1 ;
N A
2 1 ;
N
2
A
2 1 2
A
1
E C
3 5
e
2 4
o R
2
Z
1
Z
A
1 2
A
2
Z
1
E C
3 5
e
2 4
o R o A
2 / 3
Measure Assume R
E C; How?
is same for both nuclides. Why?
Measurement of nuclear radius
• Three methods outlined for nuclear matter radius: – Rutherford scattering – Alpha particle decay – -mesic x-rays