Nuclear_Physics_Lecture_2.pdf
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(2) Bulk Properties To estimate the density of nuclear matter Nuclear mass Nuclear volume. M n = Zm p + Nmn ≈ Am p 4 3 4 3 V = πR = πr0 A 3 3. Am p mp Mn = = Matter density ρ = 4 3 4 3 V πr0 A πr0 3 3. Independent of A. m p ≈ 10 −27 kg , r0 ≈ 10 −15 m mp ≈ 1017 kg / m 3 ρ= 4 3 πr0 3. Density of water 103 kg/m3 !.
(3) Bulk Properties 2. Nuclear spin (I): Nucleons have intrinsic spin angular momentum S = 1/2 (in unit of ħ) In addition, nucleons posses orbital angular momenta about the CM of nucleus – quantum number L Total angular momentum of the nucleus (commonly termed as nuclear spin). I=L+S. Quantum Mechanically Spin angular momentum. pS = S ( S + 1)h. Spin angular momentum pL = L( L + 1)h Spin angular momentum. pI = I ( I + 1)h. Ground state spin of He4 is 0 & Li7 is 3/2.
(4) Bulk Properties 3. Statistics of Nuclei: Nuclear spin can be 0 or some integer or half – integer Accordingly nuclei follow Bose – Einstein or Fermi – Dirac statistics Nuclei having I = n (n = 0, 1, 2, 3, ….) follow BE statistics Nuclei having I = (n + 1/2) [n = 0, 1, 2, 3, ….] follow FD statistics He4 (I = 0) follows BE statistics & Li7 (I = 3/2) follows FD statistics.
(5) Bulk Properties 4. Parity of Nuclei: r. Quantum mechanically the nucleus is described by a wave function ψ (r ). r r The space inversion ( r → − r ) is described by the parity operator P̂ which. r r operates as Pˆ ψ ( r ) = ψ ( − r ) If the Hamiltonian of nuclei remains invariant under space inversion, the. r r r ˆ change in wave function under parity operation is Pψ ( r ) = ψ ( − r ) = ±ψ ( r ) r r The nucleus is said to have even parity for ψ ( − r ) = +ψ ( r ) & odd parity r r for ψ ( − r ) = −ψ ( r ) Ground state parity of He4 is even & Li7 is odd.
(6) Bulk Properties 5. Magnetic dipole moment of Nuclei: The nucleons, like electron, carry intrinsic magnetic moment. The intrinsic magnetic moment for proton is μp = 2.7927 μN & for neutron μn = – 1.9131 μN eh -27 J/T µN = is the nuclear magneton = 5.0571 × 10 2m p eh [Bohr magneton µB = = 9.2849 × 10-24 J/T ] 2me. Neutron, though electrically neutral, has intrinsic magnetic moment!.
(7) Bulk Properties 5. Magnetic dipole moment of Nuclei: In addition to intrinsic magnetic moment, the contribution comes from orbital motion as well, but for proton only. No contribution to the nuclear magnetic moment comes from orbital motion of neutrons Total nuclear magnetic moment is the vector sum of the intrinsic magnetic moments of protons and neutrons, and magnetic moment due to orbital motion of the proton.
(8) Bulk Properties 6. Electric moments of Nuclei: Nucleus is positively charged with azimuthally symmetric charge distribution Electrostatic potential due to this charge distribution has multipole components Most dominating component is due to monopole – equal to total charge (+Ze) The electric dipole moment of a nucleus in its ground state vanishes Next higher order term comes from quadrupole moment defined as. Q = ∫ (3z '2 − r '2 )ρ ( r ' )dτ '.
(9) Bulk Properties 6. Electric moments of Nuclei: Q = 0 for spherical charge distribution, Q < 0 for oblate and Q > 0 for prolate charge distribution Measurement of Q yields an idea on the shape of nucleus.
(10) Nuclear Force Nuclear force binds the protons & neutrons inside a tiny volume (1) Nuclear force is the strongest force in nature The nuclear force is stronger than the electromagnetic & far stronger than the gravitational force The attractive (negative) force has a maximum at a distance of about 1 fm with a force of about 25,000 N Particles much closer than a distance of 0.8 fm experience a large repulsive (positive) force Particles separated by a distance greater than 1 fm are still attracted (Yukawa potential), but the force falls as an exponential function of distance Wikipedia.
(11) Nuclear Force (2) Nuclear force is short – ranged Acts in fm range Powerfully. attractive. between. nucleons at distances of about 1 fm Rapidly decreases to insignificance at distances beyond about 2.5 fm Becomes repulsive at distances less than 0.7 fm Nuclear potential. Wikipedia.
(12) Nuclear Force (3) Nuclear force is charge independent Independent of the charge of the interacting particles The force between two protons is same as the force between two neutrons or between a proton and a neutron within the nuclear distances. Symbolically. (n − n ) = ( p − p )nuc = ( p − n ). Coulomb repulsion between protons becomes important for r > 3 fm.
(13) Nuclear Force (4) Nuclear force is charge symmetric The strength of the nuclear force is same for the protons and neutrons, i.e. if all the neutrons in a nucleus were replaced by protons (or the vice-versa), the strength of the nuclear force remains unchanged. Symbolically,. (n − n ) = ( p − p )nuc = ( p − n ) (5) Nuclear force is spin dependent Experimental evidences show that nuclear force acting between the nucleons depends on mutual orientation of the spin of the nucleons In deuteron (1H2) the spins of the proton and neutron are parallel.
(14) Nuclear Force (6) Nuclear force shows saturation property One nucleon in the nucleus interacts with limited number of nucleons nearest to it (since the force is short – ranged) In heavy nuclei, nuclear size is larger than the range of nuclear force A nucleon senses approximately a constant number of neighbourhood nucleons It results in a constant binding fraction (binding energy per nucleon). Nuclear force is a fundamental interaction – strong interaction. It acts between quarks and mediated by gluons (detailed discussion to be followed – Elementary Particles).
(15) Bainbridge Mass Spectrometer A device for measurement of isotopic mass of nuclei Atoms with one or two electrons removed, become positive ions A beam of positive ions produced in a discharge tube is collimated into a fine beam by two narrow slits (S1) The fine beam enters into a velocity selector region.
(16) Bainbridge Mass Spectrometer The velocity selector consists of two plane parallel plates (A, B) which produces a uniform electric field (E), and an electromagnet which produces a uniform magnetic field (B) These two fields (E & B) are mutually perpendicular and perpendicular to the beam direction The ions with their velocity v = E/B do not experience any force within the velocity selector and pass through the slit (S2).
(17) Bainbridge Mass Spectrometer Only those ions with their velocity v = E/B enter the mass spectrograph from the velocity selector through the slit (S2) The positive ions with same velocity are acted upon by a magnetic field B’ perpendicular to v Ions are deflected in a circular path of radius r & strike the photographic plate. mv 2 B' qv = r B ' qr BB' qr m= = v E.
(18) Bainbridge Mass Spectrometer Ions with different masses trace different semicircular paths of different radii and produce dark spot on the photographic plate The distance between the opening of the chamber and the dark spot on the plate yields the diameter 2r from which r can be measured. Since B, B’, E, q are known, m can be precisely determined.
(19) Bainbridge Mass Spectrometer. https://www.youtube.com/watch?v=CxNnOf3POoA.
(20) Problems 1. In a mass spectrometer, a singly charged positive ion is accelerated through a potential difference of 1000 volt. It then travels through a uniform magnetic field of 1000 Gauss and deflected through a circular path of radius 18.2 cm. Calculate the (i) speed of the ion, (ii) mass of the ion and (iii) mass number . [CU – 2015].
(21) Problems 2. Singly ionized Argon ions are mass analyzed by a Bainbridge mass spectrograph. The electric and magnetic fields in the velocity filter are 1.5 × 104 V/m and 0.4 T respectively. After coming out of the velocity filter, the ions enter a magnetic field of 0.9 T. Find the distances between the ion focus lines on the photographic plate for three isotopes: Ar36, Ar38 and Ar40. [CU – 2016].
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