3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in...
-
Upload
stephanie-clarke -
Category
Documents
-
view
212 -
download
0
Transcript of 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in...
![Page 1: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/1.jpg)
3D scattering of electrons from nuclei
Finding the distribution of charge (protons) and matter in the nucleus
[Sec. 3.3 & 3.4 Dunlap]
![Page 2: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/2.jpg)
The Standford Linear Accelerator, SLAC
![Page 3: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/3.jpg)
Electron scattering at Stanford 1954 - 57
Professor Hofstadter’s group worked here at SLAC during the 1960s and were the first to find out about the charge distribution of protons in the nucleus – using high energy electron scattering.
1961 Nobel Prize winner
cA linear accelerator LINAC was used to accelerate the electrons
![Page 4: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/4.jpg)
Electron scattering experiments at SLAC 1954 - 57
e-
![Page 5: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/5.jpg)
Why use electrons?• Why not alpha’s or protons or neutrons?• Why not photons?
Alphas, protons or neutrons have two disadvantages
(1) They are STRONGLY INTERACTING – and the strong force between nucleons is so mathematically complex (not simple 1/r2) that interpreting the scattering data would be close to impossible.
(2) They are SIZEABLE particles (being made out of quarks). They have spatial extent – over ~1F. For this reason any diffraction integral would have to include an integration over the “probe” particle too.
Photons have a practical disadvantage: They could only be produced at this very high energy at much greater expense. First you would have to produce high energy electrons, then convert these into high energy positrons – which then you have to annihilate. And even then your photon flux would be very low. Energy analysis of photons after scattering would be also very difficult.
![Page 6: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/6.jpg)
Why use electrons?• Why not alpha’s or protons or neutrons?• Why not photons?
Electrons are very nice for probing the nucleus because:
(1) They are ELECTRO-MAGNETICALLY INTERACTING – and the electric force takes a nice precise mathematical form (1/r2)
(2) They are POINT particles (<10-3 F – probably much smaller). [Like quarks they are considered to be “fundamental” particles (not composites)]
(3) They are most easily produced and accelerated to high energies
![Page 7: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/7.jpg)
Concept of Cross-section
Case for a single nucleus where particle projectile is deterministic
Case for multiple nuclei where projectile path is not known.
The effective area is the all important thing – this is the Cross-Section.
Nuclear unit = 1 b = 1 barn = 10-24cm-2 = 10-28m-2 = 100 F2
![Page 8: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/8.jpg)
Rutherford scattering of negatively charged particles
Alpha scattering
Electron scattering
2csc.
)4(22csc.
16
1 4
2
200
242
0
m
Zzes
d
d
2csc
)4(24
2
200
2
m
Ze
d
d
![Page 9: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/9.jpg)
Rutherford scattering of negatively charged relativistic particles
2sin1.
2csc
2
. 22
204
2
20
cm
cZ
d
d
137
1
)4( 0
2
c
e
Fine structure constant
2sin1.
2csc
2
. 22
204
2
0
cp
cZ
d
d
Which for extreme relativistic electrons becomes:
2cos.
2csc
2
. 24
2
pc
cZ
d
d c0
2sin1.
2csc
)4(22
2
204
2
200
2
cm
Ze
d
d
Extra relativistic kinematic factor
Z<<1
22242
222 )(
2cos
2csc
4
)(.
fZT
cZ
d
d
Mott
TEpc
More forward directed distribution
Known as Mott scattering
![Page 10: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/10.jpg)
Mott Scattering
2cos
2csc.
4
1 242
22
T
cZ
d
d
Mott
![Page 11: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/11.jpg)
Mott differential scatteringTake the nucleus to have point charge Ze - e being the charge on the proton.
If that charge is spread out then an element of charge d(Ze) at a point r will give rise to a contribution to the amplitude of
im efdrd ).(.)()(
Where is the extra “optical” phase introduced by wave scattering by the element of charge at the point r compared to zero phase for scattering at r=0
r
dΨ
chargeunit per angleat amplitude scatteringMott theis )( where
)(.)(2
cos2
csc4
)(.
222242
222
m
mmMott
f
fZfZT
cZ
d
d
![Page 12: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/12.jpg)
But the Nucleus is an Extended Object
Wavefront of incident electron
Wavefront of electron scattered at angle
)(
NOTE: All points on plane AA’ have the same phase when seen by observer at
Can you see why?
![Page 13: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/13.jpg)
Wavefront of incident electron
Wavefront of electron scattered at angle
)(
The extra path length for P2P2’ 2
sin..2
OX
2sin)(2
2sin2
kpp
p
The phase difference for P2P2’
rq
rprp
OXkOX
.
.cos
2sin..22
sin..22
/.rp
r
FINDING THE PHASE
rcos
![Page 14: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/14.jpg)
Wavefront of incident electron
Wavefront of electron scattered at angle
)(dp
r
THE DIFFRACTION INTEGRAL
Charge in this volume element is: ddrrdrdq .sin).().( 2
The wave amplitude d at is given by: )(..sin)( /.2 fedddrrrd rpi
Amount of wave
Phase factor Mott scattering
![Page 15: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/15.jpg)
THE DIFFRACTION INTEGRAL The wave amplitude d at is given by: )(..sin)( /.2 fedddrrrd rpi
Amount of wave Phase factor Mott scattering
The no of particles scattered at angle is then proportional to:
2322])([)()( rFTf
From which we find: 2)(f
Form Factor F(q)
Eq (3.14)
The total amplitude of wave going at angle is then:
2
0 0 0
3/. )()()()()(r
rpi rFTfdVerf
Eq (3.15)
Mott
Mott
d
dpF
d
d
d
drFT
d
d
2
23
)]/([
)(
![Page 16: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/16.jpg)
The effect of diffractive interference
p
dd
2)/( pF Mott
From nucleus due to wave interference
![Page 17: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/17.jpg)
Fig 3.6 450 MeV e- on 58Ni
128.2
.197
450
F
FMeV
MeV
c
Ek
c
Ekp
![Page 18: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/18.jpg)
Additional Maths for a hard edge nucleusWe can get a fairly good look at the form factor for a nucleus by approximating the nucleus to a sharp edge sphere:
dedrrr
ZdVer
ZpF
r
rpirpi
2
0 0 0 0 0
/cos.2/. .sin).(2
)(1
)/(
p
q )(cos).(2)(0 0
cos2
dedrrrqF iqr
2
20
0
0
2
0
)(
cossin3
cossin14
.sin4
.sin
)(4
qR
qRqRqR
qR
qRqRqRqZq
drqrrZq
drrqr
qrr
ZR
0
r=R0
30 4
.3
R
Z
![Page 19: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/19.jpg)
2sin22
sin2
tan
)(
cossin3)(
2
kpp
q
qRqR
qR
qRqRqR
qRqF
Condition of zeros
4.5 7.7 11 14
Spherical Bessel Function of order 3/2
qR
Wavenumber mom transfer
![Page 20: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/20.jpg)
Fig 3.6 450 MeV e- on 58Ni
1.1xR=4.5 R=4.1F
1.8xR=7.7 R=4.3F
2.6xR=11 R=4.2F
![Page 21: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/21.jpg)
Proton distributions
![Page 22: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/22.jpg)
Mass distributions
Z
Nr
rrr
P
NP
1)(
)()()(
![Page 23: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/23.jpg)
The Woods-Saxon Formula
0
0
( )1 exp ( ) /
rr R a
R0=1.2 x A1/3 (F)
0.52 0.01a F
t is width of the surface region of a nucleus; that is, the distance over which the density drops from 90% of its central value to 10% of its central value
![Page 24: 3D scattering of electrons from nuclei Finding the distribution of charge (protons) and matter in the nucleus [Sec. 3.3 & 3.4 Dunlap]](https://reader030.fdocument.pub/reader030/viewer/2022032722/56649f425503460f94c6111d/html5/thumbnails/24.jpg)
)()(
)()(
3
3
qFFTr
rFT
dddd
qF
Mott
nucleus
Charge distributions can also be obtained by Inverse Fourier Transformation of the Form Factor F(q)