SINGLE-STATE METHOD WITHIN THE HORSE (J-MATRIX) FORMALISM Andrey Shirokov Lomonosov Moscow State University East Lansing, June 2018
COLLABORATORS: v J. Vary, P. Maris (Iowa State University) v A. Mazur, I. Mazur (Pacific National University) v G. Papadimitriou (LLNL) v R. Roth, S. Alexa (Darmstadt) v I. J. Shin, Y. Kim (RISP, Daejeon, Korea)
GENERAL IDEA: v NCSM + HORSE = continuum spectrum
No-core Shell Model • NCSM is a standard tool in ab initio nuclear structure theory • NCSM: antisymmetrized function of all nucleons Y • Wave function: Ψ = A φ i ( r i ) i • Traditionally single-particle functions are φ i ( r i ) harmonic oscillator wave functions • N max truncation makes it possible to separate c.m. motion
No-core Shell Model • NCSM is a bound state technique, no continuum spectrum; not clear how to interpret states in continuum above thresholds − how to extract resonance widths or scattering phase shifts • HORSE ( J -matrix) formalism can be used for this purpose • Other possible approaches: NCSM+RGM; Gamov SM; Continuum SM; SM+Complex Scaling; … • All of them make the SM much more complicated. Our goal is to interpret directly the SM results above thresholds obtained in a usual way without additional complexities and to extract from them resonant parameters and phase shifts at low energies. • I will discuss a more general interpretation of SM results
J -matrix (Jacobi matrix) formalism in scattering theory • Two types of L 2 basises: • Laguerre basis (atomic hydrogen-like states) — atomic applications • Oscillator basis — nuclear applications • Other titles in case of oscillator basis: HORSE (harmonic oscillator representation of scattering equations), Algebraic version of RGM
J -matrix formalism • Initially suggested in atomic physics (E. Heller, H. Yamani, L. Fishman, J. Broad, W. Reinhardt) : H.A.Yamani and L.Fishman, J. Math. Phys 16 , 410 (1975). Laguerre and oscillator basis. • Rediscovered independently in nuclear physics (G. Filippov, I. Okhrimenko, Yu. Smirnov): G.F.Filippov and I.P.Okhrimenko, Sov. J. Nucl. Phys. 32 , 480 (1980). Oscillator basis.
HORSE : • Schrödinger equation: H l Ψ lm ( E, r ) = E Ψ lm ( E, r ) • Wave function is expanded in oscillator functions: • Schrödinger equation is an infinite set of algebraic equations: ∞ X ( H l nn 0 − δ nn 0 ) a nn 0 ( E ) = 0 . n 0 =0 where H=T+V , T — kinetic energy operator, V — potential energy
HORSE: • Potential energy matrix elements: • For central potentials: • Note! Potential energy tends to zero as n and/or n ’ increases: • Therefore for large n or n ’ : A reasonable approximation when n or n ’ are large
HORSE: • In other words, it is natural to truncate the potential energy: • This is equivalent to writing the potential energy operator as • For large n, the Schrödinger equation takes the form ∞ X ( T l nn 0 − δ nn 0 E ) a n 0 l ( E ) = 0 , n ≥ N + 1 n 0 =0
General idea of the HORSE formalism And this looks like a natural extension of SM where both This is an exactly potential and kinetic energies are solvable algebraic problem! truncated
Asymptotic region n ≥ N • Schrödinger equation takes the form of three-term recurrent relation: • This is a second order finite-difference equation. It has two independent solutions: where dimensionless momentum For derivation, see S.A.Zaytsev, Yu.F.Smirnov, and A.M.Shirokov, Theor. Math. Phys. 117 , 1291 (1998)
Asymptotic region n ≥ N • Schrödinger equation: • Arbitrary solution a nl ( E ) of this equation can be expressed as a superposition of the solutions S nl ( E ) and C nl ( E ), e.g.: • Note that
Asymptotic region n ≥ N • Therefore our wave function • Reminder: the ideas of quantum scattering theory. • Cross section • Wave function • δ in the HORSE approach is the phase shift!
HORSE solutions • Schrödinger equation • Inverse Hamiltonian matrix: • Phase shifts: • 𝑇 #$ (𝐹) and 𝐷 #$ 𝐹 are the functions which can be expressed analytically
� � � � � J -matrix, P -matrix, R- matrix HORSE is a discrete analogue of the P -matrix approach, 𝑄 = 𝑆 ,- . • Oscillator expansion: Ψ = ∑ 𝑏 # 𝜒 # • . # At large quanta 𝑂 , the oscillator function 𝜒 # is a high-oscillating function • at distances up to the classical turning point 𝑐 5$ = 𝑠 7 2𝑂 + 3 and rapidly decreases at 𝑠 > 𝑐 5$ ; hence only the vicinity of 𝑐 5$ contributes to the integral ∫ 𝜒 # 𝑠 𝑔 𝑠 𝑒𝑠 and 𝜒 # (𝑠) #→@ 𝐵 # 𝜀(𝑠 − 𝑠 7 2𝑂 + 3 ) . • Truncating potential matrix within HORSE at very large 𝑂 is equivalent to • P -matrix formalism with channel radius 𝑐 = 𝑠 7 2𝑂 + 7 . If 𝑂 is not extremely large, HORSE is a discrete analogue of the P -matrix formalism with a natural channel radius 𝑐 = 𝑠 7 2𝑂 + 7 ; the oscillator function 𝜒 # differs essentially from the 𝜀 -function, but the matching to free solutions is defined not in the coordinate space but in the discrete space of oscillator functions that seems to be more natural for RGM, shell model and other approaches utilizing oscillator basis (see details in Bang, Mazur, AMS, Smirnov, Zaytsev, Ann. Phys. (NY), 280 , 299 (2000))
� Natural channel radius 𝑆 F = 𝑐 5$ = 𝑠 7 2𝑂 + 7 is the optimal choice for 𝑆′ .
HORSE applicability • HORSE was successfully used within RGM • HORSE was successfully used in various cluster models, e.g., 11 Li disintegration • Coulomb interaction can be accounted for within HORSE • Inverse scattering HORSE theory has been developed and used, e.g., for constructing JISP16 NN interaction • However there are problems with a direct HORSE extension of modern shell model calculations
Problems with direct HORSE application to NCSM • A lot of E λ eigenstates needed while SM codes usually calculate few lowest states only • One needs highly excited states and to get rid from CM excited states. • are normalized for all states including the CM excited � n 0 | λ ⇥ ones, hence renormalization is needed. • We need for the relative n -nucleus coordinate r nA but � n 0 | λ ⇥ NCSM provides for the n coordinate r n relative to the � n 0 | λ ⇥ nucleus CM. Hence we need to perform Talmi-Moshinsky transformations for all states to obtain in relative � n 0 | λ ⇥ n -nucleus coordinates. • Concluding, the direct application of the HORSE formalism in n -nucleus scattering is unpractical.
Example: n α scattering
Single-state HORSE (SS-HORSE) Suppose E = E λ : tan δ ( E λ ) = S N +1 ,l ( E λ ) C N +1 ,l ( E λ ) Calculating a set of E λ eigenstates with different ħ Ω and N max within SM, we obtain a set of values which we can approximate by δ ( E λ ) a smooth curve at low energies.
Single-state HORSE (SS-HORSE) Suppose E = E λ : Note, information about wave tan δ ( E λ ) = S N +1 ,l ( E λ ) function disappeared in this formula, any channel can be treated C N +1 ,l ( E λ ) Calculating a set of E λ eigenstates with different ħ Ω and N max within SM, we obtain a set of values which we can approximate by δ ( E λ ) a smooth curve at low energies.
Convergence: model problem
Universal function " % f nl ( E ) = arctan − S nl ( E ) $ ' C nl ( E ) # &
⎛ ⎞ f nl ( E ) = arctan − S nl ( E ) scaling property ⎜ ⎟ ⎝ ⎠ C nl ( E ) r 2 E n � Limit n → ∞ : ~ Ω S nl ( q ) ≈ q √ r 0 ( n + l/ 2 + 3 / 4) 1 4 j l (2 q p n + l/ 2 + 3 / 4) ≈√ r 0 ( n + l/ 2 + 3 / 4) − 1 4 sin[2 q p n + l/ 2 + 3 / 4 − π l/ 2] C nl ( q ) ≈ − q √ r 0 ( n + l/ 2 + 3 / 4) 1 4 n l (2 q p n + l/ 2 + 3 / 4) ≈√ r 0 ( n + l/ 2 + 3 / 4) − 1 4 cos[2 q p n + l/ 2 + 3 / 4 − π l/ 2] 2 E q = Ω P QRS,T (U) is a function of 𝜁 = U XY So 𝑔 HI-,$ 𝐹 = arctan Z , V QRS,T (U) ℏ] ℏ] where scaling parameter 𝑡 = ⁄ = ^HI$I _ ` #I _ ` ⁄
Universal function scaling E cm (MeV) ⇒ ε = E cm [2( N + 1) + l + 3 2] Ω f N+1,l = - arctan(S N+1,l /C N+1,l ) h � = 20 l=2 l =2 N+1 = 5 N+1 = 10 150 N+1 = 15 N+1 = 20 f nl (degrees) 100 50 0 0 5 10 � = E [2(N+1)+l+3/2] /h �
Eigenstate behavior in the presence of resonance ℏΩ ℏΩ tan δ ( E λ ) = S N +1 ,l ( E λ ) 𝑡 = = 2𝑜 + 𝑚 + 7 2 𝑂 + 7 2 e e C N +1 ,l ( E λ )
S -matrix at low energies 1 Symmetry property: S ( − k ) = S ( k ) S ( k ) = exp 2 i δ p Hence δ ( � k ) = � δ ( k ) , k ⇠ E, p p p E ) 3 + F ( E ) 5 + ... E + D ( δ ' C √ As δ ` ∼ k 2 ` +1 ∼ ( E ) 2 ` +1 k → 0 : b ( k ) = k + ik ( i ) S ( i ) b , Bound state: k − ik ( i ) b s E ⇤ ⇤ ⇤ E ) 3 + f ( E ) 5 ... δ 0 ⇥ π � arctan | E b | + c E + d ( r ( k ) = ( k + κ ( i ) r )( k − κ ( i ) ∗ ) r S ( i ) Resonance: ( k − κ ( i ) r )( k + κ ( i ) ∗ ) r p δ 1 ' � arctan a E p p c = � a E ) 3 + ..., E � b 2 + c E + d ( b 2 .
How it works
n α scattering: NCSM, JISP16 E λ ( ~ Ω , N max ) = E A =5 ( ~ Ω , N max ) − E A =4 ( ~ Ω , N max ) λ λ ~ Ω s = ( N max + 2 + � + 3 / 2) . tan δ ( E λ ) = S N +1 ,l ( E λ ) C N +1 ,l ( E λ )
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