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Introduction to Subatomic Physics "Revolution in the science can be done only by excellent, perfectly simple theory and on the base of very carefully performed and unambiguously interpreted experimental measurement." Vladimír Wagner Nuclear Physics Institute of ASCR, Řež E_mail:[email protected] 1) Three levels of submicroscopic domain 2) Tools for microscopic domain description 3) Relativistic properties 4) Quantum properties 5) Measurement in submicroscopic domain 6) Structure of matter 7) Collision kinematics 8) Cross section quantity WWW: http://hp.ujf.cas.cz/~wagner 9) Basic properties of nucleus and nuclear forces 10) Models of atomic nuclei 11) Radioactivity 12) Basic properties of nuclear reactions Recommended textbooks: 1) W.S.C. Williams : Nuclear and Particle Physics, Oxford Science Publications, 2001 2) Ashok Das, Thomas Ferbel: Introduction to Nuclear and Particle Physics, John Wiley & Sons, 1994 3) B. Povh, K. Rith, Ch. Scholz, F. Zetsche: Particles and Nuclei. An Introduction to the Physical Concepts, Springer 2004 4) A. Beiser: Concepts of Modern Physics, McGraw-Hill Companies Date Published, 1995 5) Glen F. Knoll: Radiation Detection and Measurement, John Wiley & Sons, Inc., 2000 Three size levels of submicroscopic domain studies Variety of our ordinary surroundings (macroscopic world) is consisted of atoms and molecules originating in chemical bonding of atoms Scale in Description of the submicroscopic domain: Scale in Atom Atomic physics – physics of an electron cloud of an atom, chemical bonding of atoms to molecules, only electromagnetic interaction Nucleus Nuclear physics – physics of atomic nuclei and interactions inside, interaction of a nucleus and electron shells, interaction of nucleus and elementary particles, physics of nuclear matter, strong, weak and electromagnetic interactions Proton Quark Electron Subnuclear physics (elementary particle physics or also high energy physics) – physics of elementary particles, and interactions - strong, weak and electromagnetic Scale Size [m] Atomic ~10-10 Nuclear ~10-14 Subnuclear ~10-15 1) 2) Energy 1) [MeV] ~ 0,00001 ~ 8 ~ 200 Momentum 2) [MeV/c] elmg (molecul.) 0,002 strong (nuclear) 20 strong 200 Interaction Bonding energy of an electron in a atom or energy of molecular bonding, nucleon bonding energy, energy needed for elementary particles creation (comparable with their rest energies – masses) Calculated using characteristics size and Heisenberg uncertainty principle Δp·Δx ~ ħ Characteristic rest masses: matom mnucleus = 938 260 000 MeV/c2 (mp = 1836 me ; mp = 938.27 MeV/c2 = 1,67262·10-27kg) mparticles = 0,511 MeV/c2 (electron) 91 187 MeV/c2 (Z0 boson) Characteristic times: 1/c=3,3·10-9 s /m, transit through nucleus ~ 4·10-23 s; interactions – strong ~ 10-23 s; weak ~ 10-10 - 10-6 s and electromagnetic. ~ 10-16 - 10-6 s . Science is searching for a objective description of our world 19th and 20th centuries - new tools for description (applied within the range of extreme values of physical quantities): Microscopic domain - special theory of relativity - high velocities, transferred energies - quantum physics – very small values of masses, particle distances, transferred action Megaworld - special theory of relativity – high velocities, transferred energies - general theory of relativity – very big intensities of the gravitation field Uniformity of Universe description on whole time and space scale enables the possibility of extrapolation on the base of present state → past or future state are predicted Tools for microscopic domain studies Special theory of relativity: Differences between classical Newtonian mechanics and Einstein´s special theory of relativity (Galileo and Lorentz transformations) are significant only for velocities v of the body against reference frame near to the velocity of the light c (3·108 m/s). Motion of relativistic particles in the accelerator. Dependence of special theory of relativity manifestation on velocity. Quantum physics: Show itself during processes with action transfer in the range: h = 6.626·10-34 Js = 4.14·10-21 MeV s Influence of measurement alone on the measured object. Fundamental uncertainty of measurement: px ħ Et ħ Stochastic character. Comparison of de Broglie wave length λ for objects with different mass m (me – electron mass, mj nucleus mass, rj – nucleus radius) Relativistic properties Relation between whole energy and mass: E = mc2 For rest energy of system in the rest: E0 = m0c2 For kinetic energy then: EKIN = E-E0 = mc2 - m0c2 For relativistic systems possibility to determine of energy changes by measurement of mass change and vice versa Nonrelativistic objects (EKIN m0c2) changes of mass are not measurable. (further p and v are magnitudes of momentum and particle velocity.) Relation between energy E and momentum p and kinetic energy EKIN=f(p): E2 = p2c2 + (m0c2)2 EKIN = (p2c2 + (m0c2)2) - m0c2 Nonrelativistic aproximation (p m0c) – correspondence principle: EKIN = m0c2 (p2/(m0c)2 +1) - m0c2 p2/2m0 = (m0v2)/2 (for square root were take first members of binomial development – it is valid (1x)n 1nx pro x1) Ultrarelativistic approximation (p m0c): EKIN E pc Invariant quantity: m0c2 = (E2 - p2c2) It is valid for velocity v: v = pc2/E pc2/EKIN for p m0c or m0 = 0: v = c Quantum properties The smaller masses and distances of particles – the more intensive manifestation of quantum properties. Quantitative limits for transformation of classical mechanics to the quantum one is given by Heisenberg uncertainty principle ΔpΔx ≥ћ. Correspondence principle is valid - for ΔpΔx >> ћ occurs transformation of quantum mechanics to the classical mechanics. Exhibition of both wave and particle properties. Discrete character of energy spectrum and other quantities for quantum objects (spectra of atoms, nuclei, particles, their spins …). Quantum physics is on principle statistic theory. This is difference from the classical statistic theory, which assumes principal possibility to describe trajectory of every particle (the big number of particles is problem in reality) It is done only probability distribution of times of unstable nuclei or particles decay Objects act as particles as well as waves Quantum mechanics (as well as classical) must include: 1) Description of state of the physical system in the given time. 2) The equations of motion describing changes of this state with the time. 3) Relations between quantities describing system state and measured physical quantities. Classical theory Quantum theory Particle state is described by six numbers x, y, z, px, py, pz in given time Particle state is fully described by complex function (x,y,z) given in whole space. Time development of state is described by Hamilton equations: dr/dt = ∂H/∂p dp/dt = - ∂H/∂r where H is Hamilton function Time development of state is described by Schrödinger equation: iħ∂Ψ/∂t = ĤΨ where Ĥ is Hamilton operator. Quantities x and p describing state are directly measurable. Function is not directly measurable quantity. Classical mechanics is dynamical theory. Quantum mechanics has stochastic character. Value of |(r)|2 gives probability of particle presence in the point r. Physical quantities are their mean values. Measurements in the microscopic domain Man is macroscopic object – all information about microscopic domain are mediated. The smaller studied object → the smaller wave length of studying radiation → the higher E and p of quanta (particles) of this radiation. High values of E and p → affection even destruction of studied object – decay and creation of new particles. Main method of study – bombardment of different targets by different particles and detection of produced particles at macroscopic distances from collision place by macroscopic detection systems. Important are quantities describing collision (cross sections, transferred momentums, angular distributions …). Stochastic character of quantum processes in the microscopic domain → statistical character of measurements. Relation between wave length and kinetic energy EKIN for different particles. (rA – size of atom, rj – radius of nucleus, rL – present size limit) Relation between accuracy of mean value determination of stochastic quantity on number of measurement N (assumption of independent measurement – Gaussian distribution). Structure of matter: Standard model of matter and interactions Hadrons – baryons (proton, neutron …) - mesons (π, η, ρ …) Composed from quarks, strong interaction Bosons - integral spin Fermions - half integral spin Interactions and their character Interaction – term describing possibility of energy and momentum exchange or possibility of creation and anihilation of particles The known interactions: 1) Gravitation 2) Electromagnetic 3) Strong 4) Weak Description by field – scalar or vector variable, which is function of space-time coordinates, it describes behavior and properties of particles and forces acting between them. Quantum character of interaction – energy and momentum transfer through v discrete quanta Exchange character of interactions – caused by particle exchange Real particle – particle for which it is valid E m02c4 p 2c2 Virtual particle – temporarily existed particle, it is not valid relation (they exist thanks Heisenberg uncertainty principle): E m02c4 p 2c2 Four known interactions: Interaction intermediate boson interaction constant range Gravitation graviton 2·10-39 Weak W+ W- Z0 7·10-14 *) 10-18 m Electromagnetic γ 7·10-3 infinite Strong 8 gluons 1 10-15 m infinite *) Effective value given by large masses of W+, W- a Z0 bosons Interaction intensity α Interaction intensity given by interaction coupling constant – its magnitude changes with increasing of transfer momentum (energy). Variously for different interactions → equalizing of coupling constant for high transferred momenta (energies) Exchange character of interactions: Mediate particle – intermedial bosons strong SU(3) unified SU(5) weak SU(2) electromagnetic U(1) Range of interaction depends on mediate particle mass Magnitude of coupling constant on their properties (also mass) Energy E [GeV] Leveling of coupling constants for high transferred momentum (high energies) Example of graphical representation of exchange interaction nature during inelastic electron scattering on proton with charm creation using Feynman diagram Introduction to collision kinematic Study of collisions and decays of nuclei and elementary particles – main method of microscopic properties investigation. Elastic scattering – intrinsic state of motion of participated particles is not changed during scattering particles are not excited or deexcited and their rest masses are not changed. Inelastic scattering – intrinsic state of motion of particles changes (are excited), but particle transmutation is missing. Deep inelastic scattering – very strong particle excitation happens big transformation of the kinetic energy to excitation one. Nuclear reactions (reactions of elementary particles) – nuclear transmutation induced by external action. Change of structure of participated nuclei (particles) and also change of state of motion. Nuclear reactions are also scatterings. Nuclear reactions are possible to divide according to different criteria: According to history ( fission nuclear reactions, fusion reactions, nuclear transfer reactions …) According to collision participants (photonuclear reactions, heavy ion reactions, proton induced reactions, neutron production reactions …) According to reaction energy (exothermic, endothermic reactions) According to energy of impinging particles (low energy, high energy, relativistic collision, ultrarelativistic …) Nuclear decay (radioactivity) – spontaneous (not always – induced decay) nuclear transmutation connected with particle production. Elementary particle decay - the same for elementary particles Set of masses, energies and moments of objects participating in the reaction or decay is named as process kinematics . Not all kinematics quantities are independent. Relations are determined by conservation laws. Energy conservation law and momentum conservation law are the most important for kinematics. Transformation between different coordinate systems and quantities, which are conserved during transformation (invariant variables) are important for kinematics quantities determination. Rutherford scattering Target: thin foil from heavy nuclei (for example gold) Beam: collimated low energy α particles with velocity v = v0 << c, after scattering v = vα << c The interaction character and object structure are not introduced Momentum conservation law: m v0 m v mt v t .. (1.1) and so: square: m v 0 v t v t m m m v02 v2 2 t v v t t m m Energy conservation law: ….. (1.1a) 2 2 v t ….. (1.1b) mt 2 2 2 1 1 1 2 2 2 v v vt (1.2a) and so: 0 m v 0 m v m t v t m 2 2 2 m Using comparison of equations (1.1b) and (1.2b) we obtain: v 2t 1 t m For scalar product of two vectors it holds: .. (1.2b) 2v v t ………… (1.3) a b a b cos so that we obtain: Reminder of equation (1.3) If mt<<mα: m v 2t 1 t m 2v v t 2 v v t cos Left side of equation (1.3) is positive → from right side results, that target and α particle are moving to the original direction after scattering → only small deviation of α particle If mt>>mα: Left side of equation (1.3) is negative → large angle between α particle and reflected target nucleus results from right side → large scattering angle of α particle Concrete example of scattering on gold atom: mα 3.7·103 MeV/c2 , me 0.51 MeV/c2 a mAu 1.8·105 MeV/c2 1) If mt =me , then mt/mα 1.4·10-4: We obtain from equation (1.3): ve = vt = 2vαcos ≤ 2vα We obtain from equation (1.2b): vα v0 Reminder of equation (1.2b): v 02 v2 mt 2 vt m Then for magnitude of momentum it holds: meve = m(me/m) ve ≤ m·1.4·10-4·2vα 2.8·10-4mv0 Maximal momentum transferred to the electron is ≤ 2.8·10-4 of original momentum and momentum of α particle decreases only for adequate (so negligible) part of momentum . Maximal angular deflection α of α particle arise, if whole change of electron and α momenta are to the vertical direction. Then (α 0): α rad tan α = meve/mv0 ≤ 2.8·10-4 α ≤ 0.016o Reminder of equation (1.3) m v 2t 1 t m 2v v t 2 v v t cos Reminder of equation (1.2b): v 02 v2 2) If mt =mAu , then mAu/mα 49 mt 2 vt m We obtain from equation (1.3): vAu = vt = 2(mα/mt)vα cos 2(mαvα)/mt We introduce this maximal possible velocity vt in (1.2b) and we obtain: vα v0 2 mt 2 m t 2m α v α 4m 2 2 2 2 v2 v v v v v v2 because: 0 t m m m t mt Then for momentum is valid: mAuvAu ≤ 2mvα 2mv0 Maximal momentum transferred on Au nucleus is double of original momentum and α particle can be backscattered with original magnitude of momentum (velocity). Maximal angular deflection α of α particle will be up to 180o. Full agreement with Rutherford experiment and atomic model: 1) weakly scattered - scattering on electrons 2) scattered to large angles – scattering on massive nucleus Attention remember!!: we assumed that objects are point like and we didn't involve force character. Inclusion of force character – central repulsive electric field: Thomson atomic model Electrons Thomson model – positive charged cloud with radius of atom RA : Electric field intensity outside: Electric field intensity inside: Positive charged cloud Rutherford atomic model Electrons Q 40 r 2 E(r R A ) 1 1 Qr 40 R 3A The strongest field is on cloud surface and 2eQ force acting on particle (Q = 2e) is: FMAX 2eE(r R A ) 40 R 2A This force decreases quickly with distance and it acts along trajectory L 2RA t = L/ v0 2RA/ v0 . Resulting change of particle momentum = given transversal impulse: 4eQ p FMAX t 40 R A v 0 Maximal angle is: Positive charged nucleus E(r R A ) tan p /p 4eQ 40 R A m v 02 Substituting RA 10-10m, v0 107 m/s, Q = 79e (Thomson model): rad tan 2.7·10-4 → 0.015o only very small angles. Estimation for Rutherford model: Substituting RA = RJ 10-14m (only quantitative estimation): tan 2.7 → 70o also very large scattering angles. Possibility of achievement of large deflections by multiple scattering Foil at experiment has 104 atomic layers. Let assume: 1) Thomson model (scattering on electrons or on positive charged cloud) 2) One scattering on every atomic layer 3) Mean value of one deflection magnitude 0.01o. Either on electron or on positive charged nucleus Mean value of whole magnitude of deflection after N scatterings is (deflections are to all directions, therefore we must use squares): 2 N N i i2 N 2 i 1 i1 2 N …..…. (1) We deduce equation (1). Scattering takes place in space, but for simplicity we will show problem using two dimensional case: Deflections i are distributed both in positive and negative directions statistically around Gaussian normal distribution for studied case. So that mean value of particle deflection from original direction is equal zero: N N i 1 i 1 i i 0 Multiple particle scattering i the same type of scattering on each atomic layer: i2 2 Then we can derive given relation (1): 2 N 1 N N 1 N N N 2 N N 2 i i 2 i j i 2 i j i2 N 2 i 1 ji 1 i 1 j i 1 i 1 i1 i1 i 1 Because it is valid for two inter-independent random quantities a and b with Gaussian distribution: 1 N 1 M 1 N M 1 NM abk ab a b ai b j ai b j N M N i 1 M j1 N M i 1 j1 k 1 And already showed relation is valid: N We substitute N by mentioned 104 and mean value of one deflection = 0.01o. Mean value of deflection magnitude after multiple scattering in Geiger and Marsden experiment is around 1o. This value is near to the real measured experimental value. Certain very small ratio of particles was deflected more then 90o during experiment (one particle from every 8000 particles). We determine probability P(), that deflection larger then originates from multiple scattering. If all deflections will be in the same direction and will have mean value, final angle will be ~100 o (we accent assumption each scattering has deflection value equal to the mean value). Probability of this is P = (1/2)N =(1/2)10000 = 10-3010. Proper calculation will give: P e 2 We substitute: P 90o e 90 o 1o 2 e 8100 10 3500 Clear contradiction with experiment – Thomson model must be rejected Derivation of Rutherford equation for scattering: Assumptions: 1) particle and atomic nucleus are point like masses and charges. 2) Particle and nucleus experience only electric repulsion force – dynamics is included. 3) Nucleus is very massive comparable to the particle and it is not moving. Acting force: Charged particle with the charge Ze produces a Coulomb potential: Ur 4 r 0 Two charged particles with the charges Ze and Z‘e and the distance r r experience a Coulomb force giving rice to a potential energy : Vr 1 Ze ZZe 2 40 r 1 Coulomb force is: 1) Conservative force – force is gradient of potential energy: Fr Vr 2) Central force: Vr V r Vr Magnitude of Coulomb force is ZZe 2 Fr and force acts in the direction of particle join. 40 r 2 1 Electrostatic force is thus proportional to 1/r2 trajectory of particle is a hyperbola with nucleus in its external focus. We define: Impact parameter b – minimal distance on which particle comes near to the nucleus in the case without force acting. Scattering angle - angle between asymptotic directions of particle arrival and departure. First we find relation between b and : Nucleus gives to the particle impulse Fdt particle momentum changes from original value p0 to final value p: p p p 0 Fdt …………. (1) Using assumption about target fixation we obtain that kinetic energy and magnitude of particle momentum before, during and after scattering are the same: p0 = p = mv0=mv Geometry of Rutheford scattering. We see from figure: 1 p m v 0 sin p 2m v 0 sin 2 2 2 ……….. (2) Because impulse is in the same direction as the change of momentum, it is valid: …………… (3) F cos dt Momenta in Rutheford scattering: where is running angle between F and p along particle trajectory. We substitute (2) and (3) to (1): 2m v0 sin F cos dt 2 0 2m v 0 sin 2 We change integration variable from t to : ………….....................……(4) 1 2 -1 2 F cos dt d d …. (5) where ddt is angular velocity of particle motion around nucleus. Electrostatic action of nucleus on particle is in direction of the join vector r F 0 force momentum do not act angular momentum is not changing (its original value is mv0b) and it is connected with angular velocity = d/dt mr2 = const = mr2 (d/dt) = mv0b 2 then: dt r d v 0 b 1 2 2m v b sin Fr 2 cos d 2 1 2 2 0 we substitute dt/d at (5): We substitute electrostatic force F (Z=2): We obtain: 1 2 2 2Ze 2 Fr cos d 40 1 2 1 2 because it is valid: 1 2 1 2 1 2 2Ze 2 F 4 0 r 2 1 Ze 2 d cos d sin cos 1 2 1 2 ................................… (6) 0 cos 2 2 sin 2 cos 2 2 2 2 Ze cos We substitute to the relation (6): 2m v b sin 2 0 2 2 Scattering angle is connected with 20 m v0 b 40 E KIN cotg b collision parameter b by relation: Ze 2 Ze 2 2 2 0 The smaller impact parameter b the larger scattering angle . … (7) Energy and momentum conservation law Just these conservation laws are very important. They determine relations between kinematic quantities. It is valid for isolated system: nf nf E E Conservation law of whole energy: k 1 m c ni k j j1 k 1 m c E m c E nf k 1 nf 2 0 k k 1 nj ni KIN k j1 j j1 E KIN m c nj k j1 0 2 E KIN M f0 c 2 E fKIN M i0 c 2 E iKIN 2 0 0 2 KIN j Nonrelativistic approximation (m0c2 >> EKIN): EKIN = p2/(2m0) M f0 M i0 M f0 c 2 M i0 c 2 Together it is valid for elastic scattering: E f KIN E ni p2 p2 k 1 2m 0 k j1 2m 0 j nf i KIN Ultrarelativistic approximation (m0c2 << EKIN): E ≈ EKIN ≈ pc E E f i E f KIN E i KIN nf ni k 1 j1 p k c p jc nf Conservation law of whole momentum: ni p k pj k 1 j1 nf ni k 1 j1 pk p j j We obtain for elastic scattering: Using momentum conservation law: 0 p1 sin p2 sin and p1 p1 cos p2 cos We obtain using cosine theorem: p22 p12 p12 2p1p1 cos Nonrelativistic approximation: Using energy conservation law: p12 p12 p22 2m1 2m1 2m 2 We can eliminated two variables using these equations. The energy of reflected target particle E‘KIN 2 and reflection angle ψ are usually not measured. We obtain relation between remaining kinematic variables using given equations: m m m p12 1 1 p12 1 1 2 1 p1p1 cos 0 m2 m2 m2 m m m EKIN1 1 1 E KIN1 1 1 2 1 E KIN1EKIN1 cos 0 m2 m2 m2 Ultrarelativistic approximation: Using energy conservation law: p1 p1 p2 p2 p1 p1 2p1p1 2 We obtain using this relation and momentum conservation law: 2 2 cos 1 and therefore: 0 Conception of cross section 1) Base conceptions – differential, integral cross section, total cross section, geometric interpretation of cross section 2) Macroscopic cross section, mean free path. 3) Typical values of cross sections for different processes Chamber for measurement of cross sections of different astrophysical reactions at NPI of ASCR Ultrarelativistic heavy ion collision on RHIC accelerator at Brookhaven Introduction of cross section. Formerly we obtained relation between Rutherford scattering angle and impact parameter ( particles are scattered): 40 E KIN cotg b Ze 2 2 ………………… (1) The smaller impact parameter b, the bigger scattering angle . Impact parameter is not directly measurable and new directly measurable quantity must be define. We introduce scattering cross section for quantitative description of scattering processes: Derivation of Rutherford relation for scattering: Relation between impact parameter b and scattering angle particle with impact parameter smaller or the same as b (aiming to the area b2) is scattered to a angle larger than value b given by relation (1) for appropriate value of b. Then applies: (b) = b2 ……………….……....………. (2) (then dimension of is m2, barn = 10-28 m2) We assume thin foil (cross sections of neighboring nuclei are not overlapping and multiple scattering does not take place) with thickness L with nj atoms in volume unit. Beam with number NS of particles are going to the area SS. (Number of beam particles per time and area units – luminosity – is for present accelerators up to 1038 m-2s-1). The number of target nuclei on which particles are impinging is: Nj = njLSS. Sum of cross sections for scattering to angle b and more is: (b) = njLSS. Fraction f(b) of incident particles scattered to angle larger then b is: Reminder of equation (2) (b) = b2 Reminder of equation (1) N ( b ) n j LS S f( b ) n j L b 2 NS SS SS 40 E KIN cotg b 2 2 Ze 2 We substitute of b from equation (1): Sketch of the Rutherford experiment Ze 2 cot g 2 ………………… (3) f ( b ) n jL 2 40 E KIN Angular distribution of scattered particles Reminder of pictures Reminder of equation (3) 2 Ze 2 cot g 2 f ( b ) n jL 2 40 E KIN During real experiment detector measures particles scattered to angles from up to +d. Fraction of incident particles scattered to such angular range is: 2 Ze 2 cotg sin 2 d df n j L 2 2 40 E KIN We can write for detector area in distance r from the target: dS (2 r sin )( rd ) 2 r 2 sin d 4 r 2 sin cos d 2 2 Number N() of particles going to the detector per area unit is: 2 Ze 2 cotg sin 2 d N S n j L dN N S df 2 2 40 E KIN dS dS 4 r 2 sin cos d 2 2 Such relation is known as Rutherford equation for scattering. dN dS N S n jLZ 2 e 4 80 2 r 2 E 2KIN sin 4 2 … (4) Different types of differential cross sections: angular d ( , ) d d ( ) d d (E) dE spectral spectral angular d ( E, , ) dEd double or triple differential cross section Integral cross sections: through energy, angle Values of cross section: Very strong dependence of cross sections on energy of beam particles and interaction character. Values are within very broad range: 10-47 m2 ÷ 10-24 m2 → 10-19 barn ÷ 104 barn Strong interaction (interaction of nucleons and other hadrons): 10-30 m2 ÷ 10-24 m2 → 0.01 barn ÷ 104 barn Electromagnetic interaction (reaction of charged leptons or photons): 10-35 m2 ÷ 10-30 m2 → 0.1 μbarn ÷ 10 mbarn Weak interaction (neutrino reactions): 10-47 m2 = 10-19 barn Cross section of different neutron reactions with gold nucleus Macroscopic quantities: Particle passage through matter: interacted particles disappear from beam (N0 – number of incident particles): dN n j dx N N x dN N N n j 0 dx 0 ln N – ln N0 = – njσx N N0 e n j x Number of touched particles N decrease exponential with thickness x: Number of interacting particles: N0 N N0 (1 e For x→0 : e n j x n j x ) 1 n j x N0 – N N0 – N0(1-njx) N0njx and then: dN N 0 N n j x N N0 Absorption coefficient = nj Mean free path l = is mean distance which particle travels in a matter before interaction. l 1 n j Quantum physics all measured macroscopic quantities , l are mean values (l is statistical quantity also in classical physics). Phenomenological properties of nuclei 1) Introduction - nucleon structure of nucleus 6) Magnetic and electric moments 2) Sizes of nuclei 7) Stability and instability of nuclei 3) Masses and bounding energies of nuclei 8) Nature of nuclear forces 4) Energy states of nuclei 5) Spins Introduction – nucleon structure of nuclei. Atomic nucleus consists of nucleons (protons and neutrons). Number of protons (atomic number) – Z. Total number nucleons (nucleon number) – A. Number of neutrons – N = A-Z. A Z Pr N Different nuclei with the same number of protons – isotopes. Different nuclei – nuclides. Different nuclei with the same number of neutrons – isotones. Nuclei with N1 = Z2 and N2 = Z1 – mirror nuclei Different nuclei with the same number of nucleons – isobars. Neutral atoms have the same number of electrons inside atomic electron shell as protons inside nucleus. Proton number gives also charge of nucleus: Qj = Z·e (Direct confirmation of charge value in scattering experiments – from Rutherford equation for scattering (dσ/dΩ) = f(Z2)) Atomic nucleus can be relatively stable in ground state or in excited state to higher energy – isomers (τ > 10-9s). Stable nuclei have A and Z which fulfill approximately empirical equation: Z A 1.98 0.0155A 2/3 Reliably known nuclei are up to Z=112 in present time (discovery of nuclei with Z=114, 116 (Dubna) needs confirmation). Nuclei up to Z=83 (Bi) have at least one stable isotope. Po (Z=84) has not stable isotope. Th , U a Pu have T1/2 comparable with age of Earth. Maximal number of stable isotopes is for Sn (Z=50) - 10 (A =112, 114, 115, 116, 117, 118, 119, 120, 122, 124). Total number of known isotopes of one element is till 38. Number of known nuclides: > 2800. Sizes of nuclei Distribution of mass or charge in nucleus are determined. We use mainly scattering of charged or neutral particles on nuclei. Density ρ of matter and charge is constant inside nucleus and we observe fast density decrease on the nucleus boundary. The density distribution can be described very well for spherical nuclei by 0 relation (Woods-Saxon): (r ) ( r R ) 1 e where α is diffusion coefficient. Nucleus radius R is distance from the center, where density is half of maximal value. Approximate relation R = f(A) can be derived from measurements: R = r0A1/3 where we obtained from measurement r0 = 1,2(1) 10-15 m = 1,2(2) fm (α = 1,8 fm-1). This shows on permanency of nuclear density. Using Avogardo constant or using proton mass: Am p 4 R3 3 mp 4 r03 3 1.67 1027 kg 4 1.2 10 3 15 m 3 High energy electron scattering (charge distribution) smaller r0. Neutron scattering (mass distribution) larger r0. Larger volume of neutron matter is done by larger number of neutrons at nuclei (in the other case the volume of protons should be larger because Coulomb repulsion). Distribution of mass density connected with charge ρ = f(r) measured by electron scattering with energy 1 GeV we obtain 1017 kg/m3. Deformed nuclei – all nuclei are not spherical, together with smaller values of deformation of some nuclei in ground state the superdeformation (2:1 3:1) was observed for highly excited states. They are determined by measurements of electric quadruple moments and electromagnetic transitions between excited states of nuclei. Neutron and proton halo – light nuclei with relatively large excess of neutrons or protons → weakly bounded neutrons and protons form halo around central part of nucleus. Experimental determination of nuclei sizes: 1) Scattering of different particles on nuclei: Sufficient energy of incident particles is necessary for studies of sizes r = 10-15m. De Broglie wave length λ = h/p < r: Neutrons: mnc2 >> EKIN → h/ 2mE KIN → EKIN > 20 MeV Electrons: mec2 << EKIN → λ = hc/EKIN → EKIN > 200 MeV 2) Measurement of roentgen spectra of mion atoms: They have mions in place of electrons (mμ = 207 me): μ,e – interact with nucleus only by electromagnetic interaction. Mions are ~200 nearer to nucleus → „feel“ size of nucleus (K-shell radius is for mion at Pb 3 fm ~ size of nucleus) 3) Isotopic shift of spectral lines: The splitting of spectral lines is observed in hyperfine structure of spectra of atoms with different isotopes – depends on charge distribution – nuclear radius. 4) Study of α decay: The nuclear radius can be determined using relation between probability of α particle production and its kinetic energy. Masses of nuclei Nucleus has Z protons and N=A-Z neutrons. Naive conception of nuclear masses: M(A,Z) = Zmp+(A-Z)mn where mp is proton mass (mp 938.27 MeV/c2) and mn is neutron mass (mn 939.56 MeV/c2) where MeV/c2 = 1.78210-30 kg, we use also mass unit: mu = u = 931.49 MeV/c2 = 1.66010-27 kg. Then mass of nucleus is given by relative atomic mass Ar=M(A,Z)/mu. Real masses are smaller – nucleus is stable against decay because of energy conservation law. Mass defect ΔM: ΔM(A,Z) = M(A,Z) - Zm + (A-Z)m p n It is equivalent to energy released by connection of single nucleons to nucleus - binding energy B(A,Z) = - ΔM(A,Z) c2 Binding energy per one nucleon B/A: Maximal is for nucleus 56Fe (Z=26, B/A=8.79 MeV). Possible energy sources: 1) Fusion of light nuclei 2) Fission of heavy nuclei 8.79 MeV/nucleon 1.4·10-13 J/1.66·10-27 kg = 8.7·1013 J/kg (gasoline burning: 4.7·107 J/kg) Binding energy per one nucleon for stable nuclei Measurement of masses and binding energies: Mass spectroscopy: Mass spectrographs and spectrometers use particle motion in electric and magnetic fields: Mass m=p2/2EKIN can be determined by comparison of momentum and kinetic energy. We use passage of ions with charge Q through “energy filter” and “momentum filter”, which are realized using electric and magnetic fields: FE QE FB Qv B for Bv is FB = QvB and then F = QE The study of Audi and Wapstra from 1993 (systematic review of nuclear masses) names 2650 different isotopes. Mass is determined for 1825 of them. Frequency of revolution in magnetic field of ion storage ring is used. Momenta are equilibrated by electron cooling → for different masses → different velocity and frequency. Comparison of frequencies (masses) of ground and isomer states of 52Mn. Measured at GSI Darmstadt Electron cooling of storage ring ESR at GSI Darmstadt Excited energy states Nucleus can be both in ground state and in state with higher energy – excited state Every excited state – corresponding energy→ energy level Quantum physics → discrete values of possible energies Scheme of energy levels: Deexcitation of excited nucleus from higher level to lower one by photon irradiation (gamma ray) or direct transfer of energy to electron from electron cloud of atom – irradiation of conversion electron. Nucleus is not changed. Or by decay (particle emission). Nucleus is changed. Three types of nuclear excited states: 1) Particle – nucleons at excited state EPART 2) Vibrational – vibration of nuclei EVIB 3) Rotational – rotation of deformed nuclei EROT (quantum spherical object can not have rotational energy) it is valid: EPART >> EVIB >> EROT Energy level structure of 66Cu nucleus (measured at GANIL – France, experiment E243) Spins of nuclei Protons and neutrons have spin 1/2. Vector sum of spins and orbital angular momenta is total angular momentum of nucleus I which is named as spin of nucleus Orbital angular momenta of nucleons have integral values → nuclei with even A – integral spin nuclei with odd A – half-integral spin l r p . At quantum physic by appropriate operator, Classically angular momentum is define ˆ ˆas ˆ which fulfill commutating relations: l l i l There are valid such rules: ̂ 1) Eigenvalues I 2 are ˆI 2 I(I 1) 2 , where number I = 0, 1/2, 1, 3/2, 2, 5/2 … angular momentum magnitude is |I| = ħ [I(I-1)]1/2 2) From commutation ̂ 2 relations it results, that vector components can not be observed individually. Simultaneously I and only one component – for example Iz can be observed . 3) Components (spin projections) can take values Iz = Iħ, (I-1)ħ, (I-2)ħ, … -(I-1)ħ, -Iħ together 2I+1 values. 4) Angular momentum is given by number I = max(Iz). Spin corresponding to orbital angular momentum of nucleons is only integral: I ≡ l = 0, 1, 2, 3, 4, 5, … (s, p, d, f, g, h, …), intrinsic spin of nucleon is I ≡ s = 1/2. ˆ ˆ 5) Superposition for single nucleon j l ŝ leads to j = l 1/2. Superposition for system of more particles is diverse. Extreme cases: LS-coupling, where ˆI Lˆ Sˆ , Lˆ ˆli , Sˆ ŝi i i jj-coupling, where ˆ ˆ I ji i Magnetic and electric momenta Magnetic dipole moment μ is connected to existence of spin I and charge Ze. It is given by relation: g jI g j I where g is gyromagnetic ratio and μj is nuclear magneton: Bohr magneton: j B e 3.15 1014 MeVT 1 2mp c e 5.79 10 11 MeVT 1 2m e c For point like particle g = 2 (for electron agreement μe = 1.0011596 μB). For nucleons μp = 2.79 μj and μn = -1.91 μj – anomalous magnetic moments show complicated structure of these particles. Magnetic moments of nuclei are only μ = -3 μj 10 μj, even-even nuclei μ = I = 0 → confirmation of small spins, strong pairing and absence of electrons at nuclei. Electric momenta: Electric dipole momentum: is connected with charge polarization of system. Assumption: nuclear charge in the ground state is distributed uniformly → electric dipole momentum is zero. Agree with experiment. Electric quadruple moment Q: gives difference of charge distribution from spherical. Assumption: Nucleus is rotational ellipsoid with uniformly distributed charge Ze: 2 Q Z(c 2 a 2 ) (c,a are main split axles of ellipsoid) deformation δ = (c-a)/R = ΔR/R 5 Results of measurements: 1) Most of nuclei have Q = 10-29 10-30 m2 → δ ≤ 0.1 2) In the region A ~ 150 180 and A ≥ 250 large values are measured: Q ~ 10-27 m2. They are larger than nucleus area. → δ ~ 0.2 0.3 → deformed nuclei. Stability and instability of nuclei Stable nuclei: for small A (<40) is valid Z = N, for heavier nuclei N 1,7 Z. This dependence can be express more accurately by empirical relation: A Z 1.98 0.0155A 2/3 For stable heavy nuclei excess of neutrons → charge density and destabilizing influence of Coulomb repulsion is smaller for larger number of neutrons. N Z number of stable nuclei Even-even nuclei are more stable → existence of pairing even even 156 even odd 48 odd even 50 odd odd 5 Magic numbers – observed values of N and Z with increased stability. At 1896 H. Becquerel observed first sign of instability of nuclei – radioactivity. Instable nuclei irradiate: Alpha decay → nucleus transformation by 4He irradiation Beta decay → nucleus transformation by e-, e+ irradiation or capture of electron from atomic cloud Gamma decay → nucleus is not changed, only deexcitation by photon or converse electron irradiation Spontaneous fission → fission of very heavy nuclei to two nuclei Proton emission → nucleus transformation by proton emission Nuclei with livetime in the ns region are studied in present time. They are bordered by: proton stability border during leave from stability curve to proton excess (separative energy of proton decreases to 0) and neutron stability border – the same for neutrons. Energy level width Γ of excited nuclear state and its decay time τ are connected together by relation τΓ ≈ h. Boundery for decay time Γ < ΔE (ΔE – distance of levels) ΔE~ 1 MeV→ τ >> 6·10-22s. Nature of nuclear forces The forces inside nuclei are electromagnetic interaction (Coulomb repulsion), weak (beta decay) but mainly strong nuclear interaction (it bonds nucleus together). For Coulomb interaction binding energy is B Z (Z-1) B/Z Z for large Z non saturated forces with long range. For nuclear force binding energy is B/A const – done by short range and saturation of nuclear forces. Maximal range ~1.7 fm Nuclear forces are attractive (bond nucleus together), for very short distances (~0.4 fm) they are repulsive (nucleus does not collapse). More accurate form of nuclear force potential can be obtained by scattering of nucleons on nucleons or nuclei. Charge independency – cross sections of nucleon scattering are not dependent on their electric charge. → For nuclear forces neutron and proton are two different states of single particle - nucleon. New quantity isospin T is define for their description. Nucleon has than isospin T = 1/2 with two possible orientation TZ = +1/2 (proton) and TZ = -1/2 (neutron). Formally we work with isospin as with spin. Spin dependence – explains existence of stable deuteron (it exists only at triplet state – s = 1 and no at singlet - s = 0) and absence of di-neutron. This property is studied by scattering experiments using oriented beams and targets. Tensor character – interaction between two nucleons depends on angle between spin directions and direction of join of particles. Expect strong interaction electric force influences also. Nucleus has positive charge and for positive charged particle this force produces Coulomb barrier (range of electric force is larger then this of strong force). Appropriate potential has form V(r) ~ Q/r. In the case of scattering centrifugal barier given by angular momentum of incident particle acts in addition. Models of atomic nuclei 1) Introduction 2) Drop model of nucleus 3) Shell model of nucleus Octupole vibrations of nucleus. (taken from H-J. Wolesheima, GSI Darmstadt) Extreme superdeformed states were predicted on the base of models Introduction Nucleus is quantum system of many nucleons interacting mainly by strong nuclear interaction. Theory of atomic nuclei must describe: 1) Structure of nucleus (distribution and properties of nuclear levels) 2) Mechanism of nuclear reactions (dynamical properties of nuclei) Development of theory of nucleus needs overcome of three main problems: 1) We do not know accurate form of forces acting between nucleons at nucleus. 2) Equations describing motion of nucleons at nucleus are very complicated – problem of mathematical description. 3) Nucleus has at the same time so many nucleons (description of motion of every its particle is not possible) and so little (statistical description as macroscopic continuous matter is not possible). Real theory of atomic nuclei is missing only models exist. Models replace nucleus by model reduced physical system. reliable and sufficiently simple description of some properties of nuclei. Models of atomic nuclei can be divided: A) According to magnitude of nucleon interaction: Collective models (models with strong coupling) – description of properties of nucleus given by collective motion of nucleons Singleparticle models (models of independent particles) – describe properties of nucleus given by individual nucleon motion in potential field created by all nucleons at nucleus. Unified (collective) models – collective and singleparticle properties of nuclei together are reflected. B) According to, how they describe interaction between nucleons: Phenomenological models – mean potential of nucleus is used, its parameters are determined from experiment. Microscopic models – proceed from nucleon potential (phenomenological or microscopic) and calculate interaction between nucleons at nucleus. Semimicroscopic models – interaction between nucleons is separated to two parts: mean potential of nucleus and residual nucleon interaction. Liquid drop model of atomic nuclei Let us analyze of properties similar to liquid. Think of nucleus as drop of incompressible liquid bonded together by saturated short range forces Description of binding energy: B = B(A,Z) We sum different contributions: B = B1 + B2 + B3 + B4 + B5 1) Volume (condensing) energy: released by fixed and saturated nucleon at nuclei: B1 = aVA 2) Surface energy: nucleons on surface → smaller number of partners → addition of negative member proportional to surface S = 4πR2 = 4πA2/3: B2 = -aSA2/3 3) Coulomb energy: repulsive force between protons decreases binding energy. Coulomb energy for uniformly charged sphere is E Q2/R. For nucleus Q2 = Z2e2 a R = r0A1/3: B3 = -aCZ2A-1/3 4) Energy of asymmetry: neutron excess decreases binding energy 5) Pair energy: binding energy for paired nucleons increases: + for even-even nuclei B5 = 0 for nuclei with odd A where aPA-1/2 - for odd-odd nuclei We sum single contributions and substitute to relation for mass: TZ2 (Z A 2) 2 B 4 a A a A A A Volume energy Surface energy Coulomb energy Asymmetry energy M(A,Z) = Zmp+(A-Z)mn – M(A,Z) = Zmp+(A-Z)mn–aVA+aSA2/3 + aCZ2A-1/3 + aA(Z-A/2)2A-1±δ B(A,Z)/c2 Binding energy Weizsäcker semiempirical mass formula. Parameters are fitted using measured masses of nuclei. (aV = 15.85, aA = 92.9, aS = 18.34, aP = 11.5, aC = 0.71 all in MeV/c2) Shell model of nucleus Assumption: primary interaction of single nucleon with force field created by all nucleons. Nucleons are fermions one in every state (filled gradually from the lowest energy). Experimental evidence: 1) Nuclei with value Z or N equal to 2, 8, 20, 28, 50, 82, 126 (magic numbers) are more stable (isotope occurrence, number of stable isotopes, behavior of separation energy magnitude). 2) Nuclei with magic Z and N have zero quadrupol electric moments zero deformation. 3) The biggest number of stable isotopes is for even-even combination (156), the smallest for odd-odd (5). 4) Shell model explains spin of nuclei. Even-even nucleus protons and neutrons are paired. Spin and orbital angular momenta for pair are zeroed. Either proton or neutron is left over in odd nuclei. Half-integral spin of this nucleon is summed with integral angular momentum of rest of nucleus half-integral spin of nucleus. Proton and neutron are left over at odd-odd nuclei integral spin of nucleus. Shell model: 1) All nucleons create potential, which we describe by 50 MeV deep square potential well with rounded edges, potential of harmonic oscillator or even more realistic Woods-Saxon potential. 2) We solve Schrödinger equation for nucleon in this potential well. We obtain stationary states characterized by quantum numbers n, l, ml. Group of states with near energies creates shell. Transition between shells high energy. Transition inside shell law energy. 3) Coulomb interaction must be included difference between proton and neutron states. 4) Spin-orbital interaction must be included. Weak LS coupling for the lightest nuclei. Strong jj coupling for heavy nuclei. Spin is oriented same as orbital angular momentum → nucleon is attracted to nucleus stronger. Strong split of level with orbital angular momentum l. without spin-orbital coupling with spin-orbital coupling Number Number per shell Energy per level Sequence of energy levels of nucleons given by shell model (not real scale) – source A. Beiser Total number Radioactivity 1) Introduction 2) Decay law 3) Alpha decay 4) Beta decay 5) Gamma decay 6) Fission 7) Decay series Detection system GAMASPHERE for study rays from gamma decay Introduction Transmutation of nuclei accompanied by radiation emissions was observed - radioactivity. Discovery of radioactivity was made by H. Becquerel (1896). Three basic types of radioactivity and nuclear decay: 1) Alpha decay 2) Beta decay 3) Gamma decay and nuclear fission (spontaneous or induced) and further, more exotic types of decay Transmutation of one nucleus to another proceed during decay (nucleus is not changed during gamma decay – only energy of excited nucleus is decreased). Mother nucleus – decaying nucleus Daughter nucleus – nucleus incurred by decay Sequence of follow up decays – decay series. Decay of nucleus is not dependent on chemical and physical properties of nucleus neighboring (exception is for example gamma decay through conversion electrons which is influenced by chemical binding). Electrostatic apparatus of P. Curie for radioactivity measurement (left) and present complex for measurement of conversion electrons (right) Radioactivity decay law Activity (radioactivity) A: A dN dt where N is number of nuclei at given time in sample [Bq = s-1, Ci =3.7·1010Bq]. Constant probability λ of decay of each nucleus per time unit is assumed. Number dN of nuclei decayed per time dt: dN dt dN = -Nλdt N N t dN Both sides are integrated: N dt N0 0 N N 0 e t ln N – ln N0 = -λt Then for radioactivity we obtain: A dN N 0 e t A 0 e t dt where A0 ≡ -λN0 Probability of decay λ is named decay constant. Time of decreasing from N to N/2 is decay ln 2 N0 T half-life T1/2. We introduce N = N0/2: T Mean lifetime τ: 1 2 For t = τ activity decreases to 1/e = 0,36788. Heisenberg uncertainty principle: ΔE·Δt ≈ ħ → Γ · τ ≈ ħ where Γ is decay width of unstable state: Γ = ħ /τ = ħ λ N 0e 12 12 Total probability λ for more different alternative possibilities with decay constants λ1,λ2,λ3 … λM: M M k k k 1 k 1 Sequence of decay we have for decay series λ1N1 → λ2N2 → λ3N3 → … → λiNi → … → λMNM Time change of Ni for isotope i in series: dN /dt = λ N - λ N i We solve system of differential equations and assume: i-1 i-1 i i N1 C11e 1t N 2 C21e 1t C22e 2 t … N M CM1e 1t ... CMM e M 2 t i1 i j Coefficients with i = j can be obtained from boundary conditions in time t = 0: Ni(0) = Ci1 + Ci2 + Ci3 + … + Cii For coefficients Cij it is valid: i ≠ j Cij Ci 1, j Special case for τ1 >> τ2,τ3 … τM : each following member has the same number of decays per time unit as first. Number of existed nuclei is inversely dependent on its λ. → decay series is in radioactive equilibrium. Creation of radioactive nuclei with constant velocity – irradiation using reactor or accelerator. Velocity of radioactive nuclei creation is P: dN/dt = - λN + P Solution of equation (N0 = 0): λN(t) = A(t) = P(1 – e-λt) It is efficient to irradiate number of half-lives but not so long time – saturation starts. Development of activity during homogenous irradiation Alpha decay High value of alpha particle binding energy → EKIN sufficient for escape from nucleus → Relation between decay energy and kinetic energy of alpha particles: A Z X AZ42Y 42 He Decay energy: Q = (mi – mf –mα)c2 Kinetic energies of nuclei after decay (nonrelativistic approximation): EKIN f = (1/2)mfvf2 EKIN α = (1/2) mαvα2 From momentum conservation law: mfvf = mαvα → From energy conservation law: EKIN f + EKIN α = Q We modify equation and we introduce: vf m v mf ( mf >> mα → vf << vα) (1/2) mαvα2 + (1/2)mfvf2 = Q 2 1 m m mf 1 m 1 m f v m v2 m v2 1 E KIN Q 2 mf 2 mf 2 mf Kinetic energy of alpha particle: E KIN mf A4 Q Q m m f A Typical value of kinetic energy is 5 MeV. For example for 222Rn: Q = 5.587 MeV and EKIN α= 5.486 MeV. Barrier penetration: Particle (Z,A) impacts on nucleus (Z,A) – necessity of potential barrier overcoming. For Coulomb barier is the highest point in the place, where nuclear forces start to act: Z Ze 2 1 Z Ze 2 VC 40 r0 (A1α 3 A1 3 ) 40 R 1 Barrier height is VCB ≈ 25 MeV for nuclei with A=200. Problem of penetration of α particle from nucleus through potential barrier → it is possible only because of quantum physics. Assumptions of theory of α particle penetration: 1) Alpha particle can exist at nuclei separately 2) Particle is constantly moving and it is bonded at nucleus by potential barrier 3) It exists some (relatively very small) probability of barrier penetration. Probability of decay λ per time unit: bound state quasistationary state λ = νP where ν is number of impacts on barrier per time unit and P probability of barrier penetration. We assumed, that α particle is oscillating along diameter of nucleus: E KIN E KIN c 2 v 1021 2 2 2R 2m R 2E 0 R Probability P = f(EKINα/VCB). Quantum physics is needed for its derivation. Beta decay Nuclei emits electrons: 1) Continuous distribution of electron energy (discrete was assumed – discrete values of energy (masses) difference between mother and daughter nuclei). Maximal EEKIN = (Mi – Mf – me)c2. → postulation of new particle existence – neutrino. mn > mp + mν → spontaneous process n p e neutron decay τ ≈ 900 s (strong ≈ 10-23 s, elmg ≈ 10-16 s) → decay is made by weak interaction Relative electron intensity 2) Angular momentum – spins of mother and daughter nuclei differ mostly by 0 or by 1. Spin of electron is but 1/2 → half-integral change inverse process proceeds spontaneously only inside nucleus Process of beta decay – creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus. Z is changed by one. A is not changed. Electron energy Schematic dependence Ne = f(Ee) at beta decay According to mass of atom with charge Z we obtain three cases: 1) Mass is larger than mass of atom with charge Z+1 → electron decay – decay energy A A is split between electron a antineutrino, neutron is transformed to proton: ZYZ1Y e 2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 – 2mec2 → electron capture – energy is split between neutrino energy and electron binding energy. Proton is transformed to neutron: A Y e- A Y Z1 Z 3) Mass is smaller than mZ+1 – 2mec2 → positron decay – part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron. Proton changes to neutron: Discrete lines on continuous spectrum : YAZY e A Z1 1) Auger electrons – vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Röntgen photon. Its energy is only a few keV → it is very easy absorbed → complicated detection 2) Conversion electrons – direct transfer of energy of excited nucleus to electron from atom Beta decay can goes on different levels of daughter nucleus, not only on ground but also on excited. Excited daughter nucleus then realized energy by gamma decay. Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei. During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done. Gamma decay Excited nucleus unloads energy by photon irradiation After alpha or beta decay → daughter nuclei at excited state → emission of gamma quantum → gamma decay Multipole expansion and simple selective rules: Different transition multipolarities: Electric Magnetic EJ → spin I = min J, parity π = (-1)I MJ → spin I = min J, parity π = (-1)I+1 Transition between levels with spins Ii and If and parities πi and πf : I = |Ii – If| pro Ii ≠ If I = 1 for Ii = If > 0 π = (-1)I+K = πi·πf K=0 for E and K=1 pro M Electromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist Energy of emitted gamma quantum: Eγ = hν = Ei - Ef More accurate (inclusion of recoil energy): Momenta conservation law → hν/c = Mjv 1 1 h 2 Energy conservation law → E i E f h 2 M j v h 2M c j 2 2 h E h E i E f E i E f E R 2M jc 2 where ΔER is recoil energy. 2 Mean lifetimes of levels are mostly very short ( < 10-7s – electromagnetic interaction is much stronger than weak) → life time of previous beta or alpha decays are longer → time behavior of gamma decay reproduces time behavior of previous decay. They exist longer and even very long life times of excited levels - isomer states. Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states. Usually transitions, for which change of spin is smaller, are more intensive. System of excited states, transitions between them and their characteristics are shown by decay schema. Example of part of gamma ray spectra from source 169Yb → 169Tm: Decay schema of 169Yb → 169Tm: Internal conversion Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons): Ee = E γ – Be Energy of emitted electron: where Eγ is excitation energy of nucleus, Be is binding energy of electron Alternative process to gamma emission. Total transition probability λ is: λ = λγ + λe The conversion coefficients α are introduced: It is valid: dNe/dt = λeN and dNγ/dt = λγN and then: Ne/Nγ = λe/λγ and λ = λγ (1 + α) where α = Ne/Nγ We label αK, αL, αM, αN, … conversion coefficients of corresponding electron shell K, L, M, N, …: α = αK + αL + αM + αN + … The conversion coefficients decrease with Eγ and increase with Z of nucleus. Transitions Ii = 0 → If = 0: only internal conversion not gamma transition The place freed by electron emitted during internal conversion is filled by other electron and Röntgen ray is emitted with energy: Eγ = Bef - Bei characteristic Röntgen rays of correspondent shell. Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Röntgen photon. Nuclear fission The dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus. A Z XAZ11Y1 AZ22Y2 Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1, Z2 > Zα = 2) → the lightest nucleus with spontaneous fission is 232Th. Example of fission of 236U: Energy released by fission Ef ≥ VC → spontaneous fission After supplial of energy – induced fission – energy supplied by photon (photofission), by neutron, … Energy Ea needed for overcoming of potential barrier – activation energy – for heavy nuclei is small ( ~ MeV) → energy released by neutron capture is enough (high for nuclei with odd N). Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) → further fissions are induced → chain reaction. 235 U + n → 236U → fission → Y1 + Y2 + ν∙n Average number η of neutrons emitted during one act of fission is important - 236U (ν = 2.47) or per one neutron capture for 235U (η = 2.08) (only 85% of 236U makes fission, 15% makes gamma decay). How many of created neutrons produced further fission depends on arrangement of setup with fission material Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor k: Its magnitude is split to three cases: k < 1 – subcritical – without external neutron source reactions stop → accelerator driven transmutors – external neutron source k = 1 – critical – can take place controlled chain reaction → nuclear reactors k > 1 – supercritical – uncontrolled (runaway) chain reaction → nuclear bombs Fission products of uranium 235U. Dependency of their production on mass number A: (taken from A. Beiser: Concepts of modern physics) Decay series Different radioactive isotope were produced during elements synthesis (before 5 – 7 miliards years). Some survive: 40K, 87Rb, 144Nd, 147Hf Beta decay: A is not changed Summary of decay series: A The heaviest from them: 232Th, 235U a 238U Alpha decay: A → A - 4 Series Mother nucleus T 1/2 [years] 4n Thorium 232Th 1.39·1010 4n + 1 Neptunium 237Np 2.14·106 4n + 2 Uranium 238U 4.51·109 4n + 3 Actinium 235U 7.1·108 Decay life time of mother nucleus of neptunium series is shorter than time of Earth existence. Also all furthers → must be produced artificially → with lower A by neutron bombarding, with higher A by heavy ion bombarding. Some isotopes in decay series must decay by alpha as well as beta decays → branching Possibilities of radioactive element usage: 1) Dating (archeology, geology) 2) Medicine application (diagnostic – radioisotope, cancer irradiation) 3) Measurement of tracer element contents (activation analysis) 4) Defectology, Röntgen devices Nuclear reactions 1) Introduction 2) Nuclear reaction yield 3) Conservation laws 4) Nuclear reaction mechanism 5) Compound nucleus reactions 6) Direct reactions Fission of 252Cf nucleus (taken from WWW pages of group studying fission at LBL) Introduction Incident particle a collides with a target nucleus A → different processes: 1) Elastic scattering – (n,n), (p,p), … 2) Inelastic scattering – (n,n‘), (p,p‘), … 3) Nuclear reactions: a) creation of new nucleus and particle - A(a,b)B b) creation of new nucleus and more particles - A(a,b1b2b3…)B c) nuclear fission – (n,f) d) nuclear spallation from point of view of used projectile: e) photonuclear reactions - (γ,n), (γ,α), … f) radiative capture – (n, γ), (p, γ), … g) reactions with neutrons – (n,p), (n, α) … h) reactions with protons – (p,α), … i) reactions with deuterons – (d,t), (d,p), (d,n) … j) reactions with alpha particles – (α,n), (α,p), … k) heavy ion reactions Reaction can be described in the form A(a,b)B, for example: 27Al(n,α)24Na or 27Al + n → 24Na + α input channel - particles (nuclei) enter into reaction and their characteristics (energies, momenta, spins, …) output channel – particles (nuclei) get off reaction and their characteristics Cross section σ depends on energies, momenta, spins, charges … of involved particles Dependency of cross section on energy σ (E) – excitation function. Threshold reactions – occur only for energy higher than some value. Reaction yield – number of reactions divided by number of incident particles. Thin target – does not changed intensity and energy of beam particles Thick target – intensity and energy of beam particles are changed Nuclear reaction yield Reaction yield – number of reactions ΔN divided by number of incident particles N0: w = ΔN /N0 Depends on specific target Thin target – does not changed intensity and energy of beam particles → reaction yield: w = ΔN /N0 = σnx where n – number of target nuclei in volume unit, x is target thickness → nx is surface target density. Thick target – intensity and energy of beam particles are changed. Process depends on type of particles: 1) Reactions with charged particles – energy losses by ionization and excitation of target atoms. Reactions occur for different energies of incident particles. Number of particle is changed by nuclear reactions (can be neglected for some cases). Thick target (thickness d > range R): dN = N(x)nσ(x)dx ≈ N0nσ(x)dx (reaction with nuclei are neglected N(x) ≈ N0) R Reaction yield is (d > R): ΔN w n (x)dx n N0 0 E KINa 0 (E KIN ) dE KIN dx dE KIN Higher energies of incident particle and smaller ionization losses → higher range and yield w=w(EKIN) – excitation function R Mean cross section: 1 (x)dx R0 → w n R 2) Neutron reactions – no interaction with atomic shell, only scattering and absorption on nuclei. Number of neutrons is decreasing but their energy is not changed significantly. Beam of monoenergy neutrons with yield intensity N0. Number of reactions dN in a target layer dx for dN = -N(x)nσdx deepness x is: where N(x) is intensity of neutron yield in place x and σ is total cross section σ = σpr + σnepr + σabs + … We integrate equation: N(x) = N0e-nσx for 0≤x≤d Number of interacting neutrons from N0 in target with thickness d is: ΔN = N0(1 – e-nσd) Reaction yield is: w N R R (1 e n d ) N0 σ – total cross section σR – cross section of given reaction 3) Photon reactions – photons interact with nuclei and electrons → scattering and absorption → decreasing of photon yield intensity: I(x) = I0e-μx where μ is linear attenuation coefficient (μ = μan, where μa is atomic attenuation coefficient and n is number of target atoms in volume unit). For thin target (attenuation can be neglected) reaction yield is: where ΔI is total number of reactions and from this reactions. We obtain for thick target with thickness d: w I a w I n d I0 a is number of studied photonuclear I (1 e a nd ) I0 a a Conservation laws Energy conservation law and momenta conservation law: Described in the part about kinematics. Directions of fly out and possible energies of reaction products can be determined by these laws. Type of interaction must be known for determination of angular distribution. Angular momentum conservation law – orbital angular momentum given by relative motion of two particles can have only discrete values l = 0, 1, 2, 3, … [ħ]. → For low energies and short range of forces → reaction possible only for limited small number l. Semiclasical (orbital angular momentum is product of momentum and impact parameter): pb = lħ → l ≤ pbmax/ħ = 2πR/ λ where λ is de Broglie wave length of particle and R is interaction range. Accurate quantum mechanic analysis → reaction is possible also for higher orbital momentum l, but cross section rapidly decreases. Total cross section can be split: l l Charge conservation law – sum of electric charges before reaction and after it are conserved. Baryon number conservation law – for low energy (E < mnc2) → nucleon number conservation law Mechanisms of nuclear reactions Different reaction mechanism: 1) Direct reactions (also elastic and inelastic scattering) - reactions insistent very briefly τ ≈ 10-22s → wide levels, slow changes of σ with projectile energy 2) Reactions through compound nucleus – nucleus with lifetime τ ≈ 10-16s is created → narrow levels → sharp changes of σ with projectile energy (resonance character), decay to different channels Reaction through compound nucleus Reactions during which projectile energy is distributed to more nucleons of target nucleus → excited compound nucleus is created → energy cumulating → single or more nucleons fly out. Compound nucleus decay 10-16s. Two independent processes: Compound nucleus creation Compound nucleus decay Cross section σab reaction from incident channel a and final b through compound nucleus C: σab = σaCPb where σaC is cross section for compound nucleus creation and Pb is probability of compound nucleus decay to channel b. Direct reactions Direct reactions (also elastic and inelastic scattering) - reactions continuing very short 10-22s Stripping reactions – target nucleus takes away one or more nucleons from projectile, rest of projectile flies further without significant change of momentum - (d,p) reactions. Pickup reactions – extracting of nucleons from nucleus by projectile Transfer reactions – generally transfer of nucleons between target and projectile. Diferences in comparison with reactions through compound nucleus: a) Angular distribution is asymmetric – strong increasing of intensity in impact direction b) Excitation function has not resonance character c) Larger ratio of flying out particles with higher energy d) Relative ratios of cross sections of different processes do not agree with compound nucleus model