Transcript ppt - RHIP
III. Effective Interaction Theory Topics to be covered include: Intuitive ideas Brief review of operator formalism Watson scattering formalism Kerman, McManus, Thaler scattering formalism Feshbach scattering formalism Brueckner nuclear matter formalism Relativistic multiple scattering General References: 1) Ray, Hoffmann, Coker, Physics Reports 212, 223 (1992). 2) Rodberg and Thaler, “Introduction to the quantum theory of scattering,” (Academic Press, 1967). 1 The inituitive ideas that led to effective interaction theory The basic ideas for dealing with the many-body, strong (non-perturbative) nuclear interaction problem began with scattering – so that’s where I will start. The seminal idea was due to Leslie L. Foldy who as a recent Physics B.S. graduate (in 1941) was working on sonar during WWII in New York City. Building on his experiences with acoustical waves Foldy in 1945, just before entering graduate school to work with J. R. Oppenheimer at U.C. Berkeley, and after his work was declassified, published the landmark paper (L. L. Foldy, Phys. Rev. 67, 107 (1945)). Foldy described projectile scattering from a nucleus as a wave propagating through many, dense scattering sources with a complex (absorptive) index of refraction. His essential idea was to express the total scattered wave in terms of individual N+N scattered waves, rather than in terms of the very strong N+N interaction which cannot be expanded in a perturbation series, and may even diverge in the case of hard-core N-N interactions. A nice tribute to Foldy is: http://ising.phys.cwru.edu/plt/PapersInPdf/206Foldy.pdf “His 1945 theory of the multiple scattering of waves laid out the fundamentals that most modern theories have followed (and sometimes rediscovered),…” Not bad for a pre-graduate student! 2 The inituitive ideas that led to effective interaction theory In 1950 Geoffrey Chew introduced the “impulse approximation” as a suitable way to simplify the intractable A+1 – body problem (e.g. p + A, n + A) to an effective two-body scattering problem. Three papers established the basic ideas for what would become known as Multiple Scattering Theory: Chew, Phys. Rev. 80, 196 (1950). Chew and Wick, Phys. Rev. 85, 636 (1952). Chew and Goldberger, Phys. Rev. 87, 778 (1952). The basics are: 1) the full A+1 scattering can be accurately represented as a coherent sum of individual hadron + nucleon scatterings and re-scatterings from nucleons in the target nucleus 2) at high energies the free-space hadron + nucleon scattering amplitude is unaffected by the nuclear medium 3) the hadron + nucleus scattering amplitude should be expanded in terms of the two-body scattering amplitudes, rather than directly in terms of the N+N potential. In 1951 Melvin Lax (Rev. Mod. Phys. 23, 287 (1951), Phys. Rev. 85, 621 (1952)) extended these approaches to obtain an effective interaction potential, later called the optical potential to represent the effective p + A interaction. This was the first representation of such an effective potential and introduced the so-called “tr” form, where r is the nuclear density and t is an effective N+N interaction. Hans Bethe learned about this and in a seminar passed this new idea on to Roy Thaler (my 2nd mentor in the late 70’s and 80’s) who devoted his career to multiple scattering formalism. 3 The inituitive ideas that led to effective interaction theory In 1953 Kenneth M. Watson gathered up all these emerging ideas and published the first, formal scattering solution for the p + A problem in K. M. Watson, Phys. Rev. 89, 575 (1953). His theory will be presented in later slides. K. M. Watson In 1959 Arthur Kerman, Hugh McManus and Roy Thaler corrected a double counting problem in the Watson theory by re-organizing the expansions (Ann. Phys. 8, 551 (1959)) which paved the way for accurate application of Chew’s impulse approximation and led to many applications for scattering experiments. 4 K M T The inituitive ideas that led to effective interaction theory Herman Feshbach and collaborators, at the same time, developed a powerful projection operator formalism which they used to generate a perturbation expansion of the optical potential and which could be applied to reactions other than elastic scattering [Ann. Rev. Nucl. Sci 8, 49 (1958); Ann. Phys. (NY) 5, 357 (1958); Ann. Phys. (NY) 19, 287 (1962)]. Keith Brueckner The success of multiple scattering formalism and the effective interactions was noticed by nuclear structure theorists. These ideas were incorporated into theories of infinite nuclear matter by Keith Brueckner and Levinson in 1955 (Phys. Rev. 97, 1344 (1955)). Infinite nuclear matter (no Coulomb, no surface, no symmetry energies to worry about) was a first step on the way to a theory of nuclear structure. The resulting effective interactions, called “g-matrix” in the literature, is ubiquitous in nuclear structure calculations. 5 Introduction to Operator Formalism I assume that most of this notation is familiar to you and here offer only a quick review of those expression which are most useful in nuclear scattering and which we will need to understand the effective interaction formalism. The following are copied from Rodberg and Thaler, Ch. 6. 6 Introduction to Operator Formalism 7 Introduction to Operator Formalism 8 Introduction to Operator Formalism 9 Introduction to Operator Formalism Local Potential Schrodinger equation in 10 operator form Introduction to Operator Formalism 11 Introduction to Operator Formalism 12 Introduction to Operator Formalism Principal value : f ( x) f ( x) f ( x) P dx dx lim 0 x x x State vector in operator form 13 Introduction to Operator Formalism Moller operator 14 Introduction to Operator Formalism Definition of T-matrix 15 Introduction to Operator Formalism Summary: () T V VG T T V G V () 16 Watson Scattering Formalism (Based on Watson, Phys. Rev. 89, 575 (1953) with updated notation) 17 Watson Scattering Formalism 18 Watson Scattering Formalism Intractable, non-perturbative… 19 Watson Scattering Formalism This auxiliary t-matrix is readily approximated in terms of the free N+N scattering amplitude (Chew’s impulse approximation.) Perturbative expansions can be constructed in which many-body corrections are included. (see slides 22-24 for details) 20 Watson Scattering Formalism (see slides 22-24 for details) Perturbative expansion of the optical potential in terms of nuclear correlations Solve Sch.Eq. for elastic wave function and scattering amplitude. (Add/subtract elastic channel) 21 Watson Scattering Formalism Details from preceding slides A A i 1 i 1 T vi vi G ( ) ( E )AT , just write G ( ) for brevity. t vi vi G ( )QtiW , solve for vi W i vi tiW 1 1 G ( )QtiW 1 tiW W () 1 ti G Q A A 1 1 W t tiW G ( ) AT i W () W () Q Q i 1 1 ti G i 1 1 ti G T A define T Ti , in terms of the transitio n amplitude for target nucleon i i 1 Ti 1 1 W t tiW G ( ) AT , left operate with 1 tiW G ( )Q i W () W () 1 ti G Q 1 ti G Q 1 t W i G ( )QTi tiW tiW G ( ) AT Ti tiW tiW G ( ) AT tiW G ( )QTi A A A i 1 i 1 i 1 T tiW tiW G ( ) AT tiW G ( )QTi For antisymmet ric nuclear states all Ti are equal. Hence Ti multiple scattering expansion A A 1 A 1 Ti T A i 1 A 1 A W () T t t G AT ti G QT . Project out the elastic scattering channel A i 1 i 1 i 1 W i W i () A A i 1 i 1 PTP PtiW P PtiW G ( ) P QTP A A 1 A W () Pti G QTP , and we also need QTP A i 1 1 A QTP Qt P Qt G P QTP QtiW G ( )QTP A i 1 i 1 i 1 W i W i () 22 Watson Scattering Formalism A A i 1 i 1 PTP PtiW P PtiW G ( ) P QTP A A i 1 i 1 PtiW P PtiW G ( )PTP 1 A W () Pti G QTP A i 1 Details from preceding slides A 1 A W () Pti G QTP A i 1 Solve the following for QTP : A A QTP QtiW P QtiW G ( ) P QTP i 1 i 1 A A QtiW P QtiW G ( )PTP i 1 i 1 1 A QtiW G ( )QTP A i 1 A 1 A QtiW G ( )QTP A i 1 A A 1 W () W () 1 A Qti QG QTP Qti P 1 G PTP i 1 i 1 A A 1 A QTP 1 Qt kW QG ( ) A k 1 1 A Qt P1 G i 1 W i PTP , substitute into the above eqn. () A 1 A W A 1 A () PTP Pt P 1 G PTP Pti QG 1 Qt kW QG ( ) A i 1 A k 1 i 1 A W i () 1 A Qt P1 G j 1 W j PTP () 1 A A W A 1 A W A 1 A () W () Pti P Pti QG 1 Qt k QG Qt Wj P 1 G ( )PTP PU W P 1 G ( )PTP A i 1 A k 1 i 1 j 1 A 1 A W ( ) 1 A 1 A PU P Pt P Pti QG Qt kW Q A i 1 A k 1 i 1 A W W i where the operator identity * C * Starting with 1 BC 1 A Qt j 1 W j P , optical potential; 1 1 1 was used for the propagator . 1 BC C B 1 1 1 1 1, then (C 1 B)C 1 and we get C 1 1 BC 1 BC 1 BC C B 23 Watson Scattering Formalism Details continued, Add & subtract t he elastic channel A PU P PtiW P W i 1 A 1 1 1 1 W W W W W W P t Q Q t P P t P P t P P t P P t P i , j i j i j A i, j ~W ~W ~W i, j i, j A 1 A ( ) 1 ~ where W G Qt kW Q A k 1 A PU P PtiW P W i 1 A 1 1 W A W W W Pti ~ t j P Pti P ~ Pt j P A i, j W W i, j 1 PtiW P A A 1 2 i 1 A A Pt W i i, j A W 1 t P j ~W A2 Pt i, j W i 1 P ~ Pt Wj P W 1 A 1 PU P PtiW P A A 1 PtiW ~ t Wj P 2 W A i 1 A A 1 i j A W Optical potential 1 W W P t P P t P (see FGH Appendix) i j ~ i, j W Therefore the non-perturbative expansion T vi vi G ( ) ( E )AT i i has been reorganized in terms of two, perturbative expansions. The first involves many-body corrections to the quasi-free two-nucleon t-matrix; the second involves corrections to the optical potential involving correlations in the nuclear wave function. 24 KMT Scattering Formalism (Based on KMT, Ann. Phys. 8, 551 (1959) with updated notation) multiple scattering expansion in KMT 25 KMT Scattering Formalism KMT optical potential 26 KMT Scattering Formalism Free-space N+N scattering t-matrix 27 KMT Scattering Formalism Expansion of the effective, two-body interaction in terms of the free-space N+N scattering amplitude 28 Feshbach, Gal and Hufner Scattering Formalism [copied from FGH, Ann. Phys. (NY) 66, 20 (1971)] 29 Feshbach, Gal and Hufner Scattering Formalism For example : t v vgt v vgv vgvgv v (v vgv ) gv v tgv 30 Feshbach, Gal and Hufner Scattering Formalism KMTT - matrix (many-body propagator) i.e. average i if evaluated for anti - symmetric states Solve (A.6) for vi , sub. into (A.5) A is redundant with definition of in (A.3) 31 Feshbach, Gal and Hufner Scattering Formalism typo : i Leading - order approx. to KMT ~K Using operator identity : 1 1 1 1 ( B A) A B A B 00 PP 1 1 1 1 where A ~ ~ because of the anti - symmetry imposed on i ti Ati 1 N t j j 32 Feshbach, Gal and Hufner Scattering Formalism KMT 2nd-order optical potential 33 Feshbach, Gal and Hufner Scattering Formalism ( N 1) 00 2nd-order potential i j term cancels Form of the 2nd-order optical potential in above slides 34 Feshbach, Gal and Hufner Scattering Formalism Average over particle pairs in nuclear ground-state; includes correlations Product of single particle, ground state densities; uncorrelated Proportional to true, two-particle correlations in the nuclear ground-state, weighted with the effective projectile + nucleon interaction squared. 35 Brueckner effective interaction for nuclear matter and nuclear structure Brueckner applied the multiple scattering theory of Watson to infinite nuclear matter (INM), but with a few changes. The goal of the INM problem is to calculate from the bare N+N interaction the binding energy and saturation density of INM as estimated by the semiempirical mass formula (about 16 MeV/A) and the densities in the interiors of large nuclei (about 0.17 nucleons/fm3). The main difference for INM from the scattering solution involves the structure of the effective two-body operators. For scattering the idea was to formulate the many-body problem in terms of effective two-body operators which can be well approximated with the free-space scattering amplitude. For INM there is no “free space” and we should build the following into the lowest-order effective interaction operator: 1) Pauli exclusion (interacting pairs of particles may only jump in to, and out of, unoccupied states above the Fermi surface), and 2) the average nucleon potential energy in INM in the propagator. Also, in Brueckner’s theory two-nucleon correlations, discussed above, were neglected. 36 Brueckner effective interaction For INM there are no outgoing spherical waves; the uniform, isotropic symmetry requires a nucleon' s asymptotic form to be a plane wave k eik r 2k 2 and the energy to be E ( k ) U ( k ), where m is the nucleon mass and U ( k ) is the potential energy. 2m Also there is no " special" nucleon like the projectile in scattering . Operators therefore refer to any arbitrary pair of nucleons with particle labels i, j. The two - body operator equation analogous to that for scattering is : GB, v v GINM QGB, (Bethe - Goldstone equation) where v is the N N interactio n, GINM is the propagator in INM, and Q projects only unoccupied intermedia te states above the Fermi surface. For scattering the energy denominato r was E H 0 H P H A i . For INM the energy denominato r is 2 2 2 ki2 k j E E H 0 VC where H 0 sums all kinetic energy operators, U U T v i j k k k 2 m 2 m k k k VC is the potential energy of INM, and k , k sum over all nucleons other than i, j. Intermedia te states only permit excitation s of the i, j nucleon pair; the INM remains in its ground state, i.e. we only consider t he ground state matrix element gs VC gs . 37 Brueckner effective interaction From the algebra leading to the first - order optical potential this INM potential energy is gs VC gs ij GB,ij ij A i j , the sum over all diagonal elements where subscript A imposes anti - symmetric states. The above g.s. matrix element of GB is analogous to the first - order, elastic - channel matrix element of the optical potential. For INM there is no absorption and this term is real, and gives just the potential energy. The propagator for INM is therefore given by 1 GINM . E H 0 ij GB,ij ij A i j For an aribtrary N N pair (i, j) ij GB ij A ij v ij A m ,n Q mn A mn A ij v mn A Q mn GB ij 2 2 m A 2 2 n k k Ei E j ij GB,ij ij 2m 2m i j , and A if both states m and n are above the Fermi surface, and zero otherwise and the potential energy in the denominato r is normalized per nucleon pair. The average, pair - wise potential energy is given by ij G B i j i j ij GB ij A ij v ij i j A This must be solved iterativel y. i j ,m , n ij A where ij v mn A Q mn GB ij Ei E j 2 2 m 2 2 n A k k ij GB,ij ij 2m 2m i j A 38 Relativistic multiple scattering & effective interactions Ref. J. D. Lumpe, (LR) Phys. Rev. C 35, 1040 (1987). A semi - relativist ic scattering model for p A in Hamiltoni an form : A p m v pi H A E i 1 where , are the usual Dirac matrices for the projectile proton. The semi - relativist ic propagator is G p m 0 H A i , and the semi - relativist ic many - body T - matrix is 1 A A i 1 i 1 T v pi v piGAT Projecting the elastic channel and following the above steps for the Watson optical potential gives PTP PU DP PU DPGPTP , where PU DP is the Dirac p A optical potential PU DP gs t i i 1 gs A( A 1) gs tiG t j gs A ( A 1 ) i j 1 gs A2 G p m 0 E A i 1 ti gs G gs i t j j gs where E A is an average, intermedia te nuclear excitation energy, and ti v pi A v piGQti impulse approx v pi v pi gti where ti is given in S, P, V, A, T form (see Chpt. I). The nuclear ground state here is constructe d from anti - symmetric combinatio ns (Slater determinan t) of relativist ic single - particle states nj ( r )Yj ( rˆ) where Yj ( rˆ) is the spin - angle function (see Chpt. I), and unj ( r ) inj ( r )Y~ ( rˆ) j ~ are the positive energy upper and lower components , and 1 for j 1 2 . 39 This concludes Chapter 3: Effective Interaction Theory 40