1
Lecture 4 : Beta stability, the LD Mass Formula, and Accelerators
Simplest form of LD Mass Formula
TBE =
C1A  C2A2/3  C3Z2/A1/3  C4(N-Z)2/A2 + C6/A1/2
<BE> =
C1 
C2A1/3  C3Z2/A4/3  C4(N-Z)2/A3 + C6/A3/2
E.
Line of Beta Stability – Isobars
1. Beta Decay – Form of Radioactive Decay
n ↔ p
∴
conversion inside nucleus
A doesn't change; just N/Z ratio  ISO BARS
Most probable N/Z ratio ≡ Line of Beta Stability
2.
Example:
A = 75 chain
N/Z= 1.5
N/Z= 1.08
75 Zn  75Ga  75Ge  75As  75Se  75Br  75Kr
30
31
32
33
34
35
36
n→p
3. Isobaric Mass Formula
BetaStable
Nucleus
n←p
Since in  decay the mass number remains constant, it is useful to develop
an isobaric mass formula (“iso” means same in Greek).
 A is constant in Binding Energy Equation ( & <BE> form)
  AZ X  = Z H + N n  TBE
plug in LD Mass Equation

  AZ X  = d1 Z2 + d2Z d3 + d + d4 , where di = f (Ci, A)
 This is equation for a parabola; minimum defines most stable nuclide
for a given A. These values define the "valley of stability".
Atomic number of this nucleus is ZA.
2
Two Cases
a. Case I: odd-A nuclei ( = 0)
 Single parabola:   A X  = d1 Z2 + d2Z + d3 ; one Parabola
Z
X)
  (A
RESULT: ONE
Z
= 2d1 ZA + d2
 Most probable charge:
STABLE ISOTOPE
Z
PER MASS NUMBER
M(Z)-M(ZA)
in MeV
Z
A=125 mass parabola
3
Consider the two mass parabolas of A=75 and A=157. What do you notice?
4
( =  1)
(  A X ) = d1 Z2 + d2Z + d3   d4
2
RESULT: Two parabolae ; even-Z always lower
CAN HAVE 1, 2 OR 3 STABLE NUCLEI PER A
b. Case II: even A nuclei
 A=128
 Upper parabola is odd-odd; Lower parabola is even-even.
5
Digrression : How
H do we know
k
that th
he mass of thhe neutron iss
m(n) = 1.008 664
4 916 37(82)) u
alue of the Neutron
N
Ma
ass. The absoolute waveleength of the gamma-ray
Moree Precise Va
produ
uced in the reaction
r
n+p d+ (2.2
2 MeV) was measured w
with a relative uncertaintyy
of 2 × 10-7 using the NIST IL
LL GAMS4 crystal
c
diffraaction facilitty at the Insttitut LaueLang
gevin in Gren
noble, France. This waveelength meassurement, exxpressed in eenergy units
and corrected
c
forr recoil, is the binding en
nergy of the nneutron in ddeuterium. A previous
crystaal diffraction
n measuremeent of the deeuteron bindiing energy hhas an uncerttainty 5 timees
largerr than this neew result. Th
he neutron mass
m followss directly froom the reactiion expresseed
in ato
omic mass un
nits: m(n) = m(2H) - m(1H) + S(d) wh
where S(d) is the separatioon energy off
the neeutron in deu
uterium. Thee uncertaintiies of the atoomic mass diifference, m(2H) - m(1H)),
and th
he new deterrmination off S(d) are 0.7
71 × 10-9 u aand 0.42 × 100-9 u, respectively, wherre
u is unified
u
atomiic mass unit. The new, more
m
precisee value for thhe neutron m
mass,
m(n) = 1.008 664
4 916 37(82)) u, has an un
ncertainty w
which is 2.5 times smaaller than thee
ue. [E. Kessler and M.S. Dewey (Diiv 846)]
previious best valu
n from http://physics.niist.gov/Tech
hAct.98/Divv842/div842h
h.html
Taken
6
Accelerators
INTRODUCTION: Uses of Accelerators
World wide inventory of accelerators, in total 15,000.
The data have been collected by W. Scarf and W.
Wiesczycka (See U. Amaldi Europhysics News, June 31,
2000)
Category
Number
Ion implanters and surface modifications
7,000
Accelerators in industry
1,500
Accelerators in non-nuclear research
1,000
Radiotherapy
5,000
Medical isotopes production
200
Hadron therapy
20
Synchrotron radiation sources
70
Nuclear and particle physics research
110
Good Overview of accelerators (no equations) :
http://nobelprize.org/physics/articles/kullander/
I.
Electrostatic Devices (constant E field)
Van de Graaf/Cockroft-Walton Accelerators
High Voltage Devices
A.
Principle of Operation: One or Two Big Kicks
1.
E = qeV where
q = atomic charge state (ion charge)
e = electric charge in units of eV
V = potential difference in Volts
7
2.
Liimitations on V
Ellectric discharge: V
V  25  1006 Volts (O
Oak Ridge))
E =  (qe) 25 M
MeV; SF6 aas insulatoor
8
Scheematic of a Van de Graaf.
G
Typiccally a volltage of 200 kV cab bbe reached.
Prob
blems: belt moves at ~ 60 km/hrr; Belt dusst  sparkking; Need for an
insullating gas (SF
( 6);
3.
Taandem Van
n de Graaff: Two-steep Acceleraation
neegative
io
on source
Xqq
V
V
X+Z

V
B
Beam
Strippeer foil
E
E1 = qe 
;
Ground
E2 = Ze V
9
a.
Total Energy Gain: E = E1 + E2 = ( q + Z) e V
b.
Example:
S2 ion ; terminal voltage = 25 MV
E = { 2 + 16} 25 eMV = 450 MeV
c.
Large V leads to higher charge state in second stage.
Tandem accelerator at Brookhaven National Lab. (BNL)
110
B.
Propertties
1.
Ions: most of periodic tabble
2.
V ≲ 25 MV ; high preccision, simpple operatiion
3.
I ~ 10
0A
4.
Time structure: Continuouus
5.
Uses
I
t
Largeely applicattions todayy; e.g., ion implantatiion,
charged-particlee activationn analysis ; 14C datinng.
E
namic (Tim
me varyin
ng E and B fields)
II. Electrodyn
A.
A Cyclotron (Lawrrence, 1929
9, Nobel P
Prize)
Id
dea: Confin
ne the mottion of the particle with a magnnetic field w
while you
Accelerate
A
it.
11
1.
Equations of Motion for a Charged Particle in a Magnetic Field
Particle mass: M
Charge state: qe
Magnetic field: H
H
radius:
r
M, q
a.
Trajectory is Circular path of radius r
2
Fcentripetal = Mv =
r
Fmagnetic =
H vqe
c
The two forces are balanced, so equate them!
Mv 2 c  Mc 
v
r

Hvqe  Hqe 
b.
; i.e. r = f(v) (classically)
Orbit time: v << c
t = 2 r  2 Mcv = 2 Mc
v
v Hqe
Hqe
c.
CONSTANT!
CYCLOTRON PRINCIPLE: orbit time is independent
of particle energy for classical motion
(classical research: ion-cyclotron resonance)
Frequency-
2 Hqe


t
Mc
for q/M ~ 0.5 (e.g., 4He+2, 12C+6),
 ~ 10-30 MHz for H ~ 1.5 tesla.
(lower end of FM frequency.)
Notice that for a fixed magnetic field H, the cyclotron frequency is
proportional to q/M of the particle.
2.
Acceleration
a.
Supply radiofrequency energy for each revolution
i.e., E = qe V, where V ~ 50-250 kV
b.
Result: velocity increases and particle spirals outward
12
c.
Energy is limited by magnetic field H and radius r ($)
d.
Total energy: defined by number of orbits required to reach
maximum radius, rmax = n:  E = n (qe) V
e.g. for n = 500, V = 200 kV (q = 2) ,
3.
E = 200 MeV
Classical Kinetic Energy: EK
cyclotron
EK = 1/2 Mv2 = 1/2 M
 r 2 H 2 q 2 e2 


2
2
 M C 

1  r 2 H 2 e2 
2  c2 
ion
 q2 




 A
K
OP
EK = Kq2/A , for v < c ; limited by relativity
where K is the figure of merit for the accelerator
Inserting the values for the constants we get:
EK = 5.05  103 H2 r2 (q2/A)
MeV/tesla2-cm2
B.
Properties
1.
Ions: Most of periodic table (electron cyclotron resonance (ECR)
sources
high q)
ion sources permit up to U ions
2.
Higher energy, less precision than Van de Graafs
3.
Energy limits: H and He: K = 215 (IUCF)
Heavy ions: K = 1200 (MSU)
relativity and size
of magnets limit
energy
30 ns
I ≲ 10A
4.
Intensity:
5.
Time structure of beam: Pulses
t ~ 200 ps
I
t
13
More historical information: http://www.aip.org/history/lawrence/
Original paper on cyclotrons: http://prola.aps.org/abstract/PR/v40/i1/p19_1
Facts about the IU cyclotron (IUCF) :
http://www.iucf.indiana.edu/whatis/facts.php
III.
Synchrotron
A.
Principle of Operation
1.
Fixed “Circular” Path
Trajectory is controlled by magnets placed around rings;
Vary H with velocity to bend particles and keep orbit
constant (ramping).
Computer-controlled process
Approach overcomes both magnet size and relativity
limits.
H
UP
1-5 s
DOWN
t
IUCF Synchrotron (“Cooler”)
K=500 MeV for this machine
14
2.
Result:
B.
Properties
1.
Ions: p , p , e , e+ ; up to U at RHIC (Brookhaven)
2.
3.
Energy: FNAL = 1.6 TeV (V/c ~ 0.999)
Storage rings: inject beam and store in ring ; unless
particles collide, will circulate continually.
Light sources:
e    biochemistry and
materials science
4.
C.
Maximum energy depends on radius (real estate) and
strength of ring magnets; r & H = f ($)
FNAL ~ 2 TeV = 2  1012 eV
Uses:
Primarily nuclear and high-energy physics
(increasingly condensed matter studies)
For more on synchrotrons:
http://accelconf.web.cern.ch/AccelConf/e96/PAPERS/ORALS/FRX04A.PDF
15
III.
Linear Accelerators
A.
Principle of Operation: Multiple Kicks
E = V




q e n: , where ni is the number of stages
Inside the tubes (drift tubes) the voltage is the same i.e. no acceleration
The voltage on the tubes is varied at radiofrequency so that as a particle moves
between tubes it experiences an acceleration.
As all the drift tubes are pulsed at the same frequency, and we want the particle to
always reach the gap at the same moment, we write:
V

T
where  
and T is the period.
L V  
2
2
c
After n gaps,
1
2
MVn  n( q )U 0
2
Rearranging,
1
 2qU 0  2
Vn   n

M 

1

 2qU 0  2 T
Ln   n

M  2

Notice that the drift tubes have to get longer as the particle accelerates so that the
particle always reaches the gap at the same time.
16
B.
Properties: special purpose machines
1.
Projectiles: Light ions (H) ; injector stages of FNAL and AGS
Heavy ions (Li U): ATLAS, FSU
Electrons:
SLAC
2.
Currents: Up to ~ 1mA
pulsed machines
I
etc.
3.
Radiofrequency cavity Boosters
IUCF  CIS
Superconductor technology: ATLAS
t
17
V.
VI.
Coupled Accelerators
Most physics accelerators today couple several different parts.
A.
IUCF:
B.
RHIC:
Van de Graaf and linear accelerator + synchrotron +
synchrotron
C.
US Facilities
Summary
RfQ + Cyclotron; RFQ + Linac