1 Lecture 4 : Beta stability, the LD Mass Formula, and Accelerators Simplest form of LD Mass Formula TBE = C 1 A C 2 A 2/3 C 3 Z 2 /A 1/3 C 4 (N-Z) 2 /A 2 + C 6 /A 1/2 <BE> = C 1 C 2 A 1/3 C 3 Z 2 /A 4/3 C 4 (N-Z) 2 /A 3 + C 6 /A 3/2 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 30 75 Zn 75 Ga 31 32 75 Ge 75 As 33 34 75 Se 75 Br 35 36 75 Kr n p 3. Isobaric Mass Formula 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) Beta- Stable Nucleus A Z X = Z H + N n TBE n p plug in LD Mass Equation A Z X = d 1 Z 2 + d 2 Z d 3 + d + d 4, where d i = f (Ci, A) N/Z= 1.08 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 Z A.
2 Two Cases a. Case I: odd-a nuclei ( = 0) Single parabola: A X Z = d 1 Z 2 + d 2 Z + d 3 ; one Parabola Most probable charge: ( A X Z ) RESULT: ONE = 2d1 Z A + d 2 Z STABLE ISOTOPE PER MASS NUMBER M(Z)-M(Z A ) in MeV Z A=125 mass parabola
Consider the two mass parabolas of A=75 and A=157. What do you notice? 3
4 b. Case II: even A nuclei ( = 1) ( A 2 X ) = d1 Z 2 + d 2 Z + d 3 d 4 RESULT: Two parabolae ; even-z always lower CAN HAVE 1, 2 OR 3 STABLE NUCLEI PER A A=128 Upper parabola is odd-odd; Lower parabola is even-even.
5 Digression : H How do we know that the mass of the neutron iss m(n) = 1.008 664 916 37(82) u More Precise Value of the Neutron Mass. The absolute wavelength of the gamma-ray produced in the reaction n+p d+ (2.22 MeV) was measured with a relative uncertainty of 2 10-7 using the NIST ILL GAMS4 crystal diffraction facility at the Institut Laueenergy units Langevin in Grenoble, France. This wavelength measurement, expressed in and corrected for recoil, is the binding energy of the neutron in deuterium. A previous crystal diffraction measurement of the deuteron binding energy has an uncertainty 5 times largerr than this new result. The neutron mass followss directly from the reaction expressed in atomic mass units: m(n) = m( 2 H) - m( 1 H) + S(d) where S(d) is the separation energy of the neutron in deuterium. The uncertainties of the atomic mass difference, m( 2 H) - m( 1 H) ), and the new determination of S(d) are 0.71 10-9 u and 0.42 10-9 u, respectively, where u is unified atomic mass unit. The new, more precisee value for the neutron mass, m(n) = 1.008 664 916 37(82) u, has an uncertainty which is 2. 5 times smaller than the previous best value. [E. Kessler and M.S. Dewey (Div 846)] Taken from http://physics.nist.gov/techact.98/div842/div842h.html
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 = qev where q = atomic charge state (ion charge) e = electric charge in units of ev V = potential difference in Volts
7 2. Limitations on V Electric discharge: VV 25 100 6 Volts (Oak Ridge) E = (qe) 25 MeV; SF 6 as insulator
8 Schematic of a Van de Graaf. Typically a voltage of 200 kv cab be reached. Problems: belt moves at ~ 60 km/hr; Belt dust sparking; Need for an insulating gas (SF 6 ); 3. Tandem Van de Graaf: Two-step Acceleration negative ion source X q q VV X V X +Z Beam Stripper foil Ground EE 1 = qe ; E 2 = Ze V
9 a. Total Energy Gain: E = E 1 + E 2 = ( 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)
10 B. Properties 1. Ions: most of periodic table 2. 3. 4. 5. V 25 MV ; high precision, simple operation I ~ 10A Time structure: Continuous Uses Largely applications today; e.g., ion implantation, charged-particlee activationn analysis ; 14 C dating. I t II. Electrodyn namic (Time varying E and B fields) A. Cyclotron (Lawrence, 1929, Nobel Prize) Idea: Confine the motion of the particle with a magnetic field while you Accelerate 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 F centripetal = 2 Mv r = F magnetic = H vqe c The two forces are balanced, so equate them! r 2 Mv c Mc v Hvqe Hqe ; i.e. r = f(v) (classically) b. Orbit time: v << c t = 2 r 2 Mcv v v Hqe = 2Mc CONSTANT! Hqe CYCLOTRON PRINCIPLE: orbit time is independent of particle energy for classical motion (classical research: ion-cyclotron resonance) c. Frequency- 2 t Hqe Mc Notice that for a fixed magnetic field H, the cyclotron frequency is proportional to q/m of the particle. 2. Acceleration for q/m ~ 0.5 (e.g., 4 He +2, 12 C +6 ), ~ 10-30 MHz for H ~ 1.5 tesla. (lower end of FM frequency.) 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 ($) d. Total energy: defined by number of orbits required to reach maximum radius, r max = n: E = n (qe) V e.g. for n = 500, V = 200 kv (q = 2), E = 200 MeV 3. Classical Kinetic Energy: E K cyclotron ion E K = 1/2 Mv 2 = 1/2 M rhqe 2 2 MC 2 2 2 2 1 2 rhe 2 c 2 2 2 2 q A OP K E K = Kq 2 /A, for v < c ; limited by relativity where K is the figure of merit for the accelerator B. Properties Inserting the values for the constants we get: E K = 5.05 10 3 H 2 r 2 (q 2 /A) MeV/tesla 2 -cm 2 1. Ions: Most of periodic table (electron cyclotron resonance (ECR) sources high q) ion sources permit up to U ions 2. Higher energy, less precision than Van de Graafs 3. Energy limits: H and He: K = 215 (IUCF) Heavy ions: K = 1200 (MSU) 4. Intensity: I 10A 5. Time structure of beam: Pulses I relativity and size of magnets limit energy 30 ns t ~ 200 ps t
13 More historical information: http://www.aip.org/history/lawrence/ Original paper on cyclotrons: http://prola.aps.org/abstract/pr/v40/i1/p19_1 Facts about the IU cyclotron (IUCF) : http://www.iucf.indiana.edu/whatis/facts.php III. Synchrotron A. Principle of Operation 1. Fixed Circular Path Trajectory is controlled by magnets placed around rings; Vary H with velocity to bend particles and keep orbit constant (ramping). Computer-controlled process Approach overcomes both magnet size and relativity limits. H UP 1-5 s DOWN t IUCF Synchrotron ( Cooler ) K=500 MeV for this machine
14 2. Result: Maximum energy depends on radius (real estate) and strength of ring magnets; r & H = f ($) FNAL ~ 2 TeV = 2 10 12 ev B. Properties 1. Ions: p, p, e, e + ; up to U at RHIC (Brookhaven) 2. Energy: FNAL = 1.6 TeV (V/c ~ 0.999) 3. Storage rings: inject beam and store in ring ; unless particles collide, will circulate continually. 4. Light sources: e biochemistry and materials science C. 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 n i 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: T V L V where and T is the period. 2 2 c After n gaps, 1 2 MV n n( q) U 0 2 Rearranging, V n 2qU n M 0 1 2 1 2 2 qu 0 T L n 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 3. Radiofrequency cavity Boosters IUCF CIS Superconductor technology: ATLAS I t etc.
17 V. Coupled Accelerators Most physics accelerators today couple several different parts. A. IUCF: RfQ + Cyclotron; RFQ + Linac B. RHIC: Van de Graaf and linear accelerator + synchrotron + synchrotron C. US Facilities VI. Summary