Scattering Partial-Wave Equations and Resonance Equations

Transcription

1 Scattering Partial-Wave Equations and Resonance Equations UCRL-14193, 1 May 1965(Revised Aug 010) L. David Roper Web address: Preface to August 010 Revision 1. I have made a few corrections and added a few equations.. I have put the figures in a separate file, UCRL14193_RoperLD_Figures.pdf, so that it can be viewed beside this document. The captions for the figures are included here as well as in the figures file. 3. In some cases the figures did not scan very clearly from the original document. Abstract This report contains equations that relate the phase shifts and partial-wave amplitudes for elastic scattering. The effect of inelasticity on elastic scattering is included. Also, resonance equations are given, and many curves show the behavior of a resonance as a function of the various resonance parameters for the case of p scattering. In particular, the unusual behavior of highly absorptive resonances is emphasized. (For the latest results for p scattering see Contents I. Introduction II. Partial-Wave Amplitude Equations III. Resonance Formulas IV. Resonance Width Energy Dependence A. Elastic Width B. Inelastic Width C. Important Observation V. Numerical Examples of Resonance Phase Shifts, Absorption Parameters, and Partial-Wave Amplitudes A. Comparison of the Four Forms for the Width Energy Dependence B. Comparison for Various Values of x using Form (d) for the Width Energy Dependence C. Comparison for Various Values of r 0 D. Comparison for Various Values of k 0 E. Comparison for Various Values of E r F. Comparison for 0, 1,, and 3 G. Important Observations More Kinematics References

2 Distribution in 1965 I. Introduction Herein are listed many familiar equations regarding partial-wave scattering amplitudes and phase shifts. The unusual behavior of inelastic resonances is elucidated algebraically and graphically. We shall always consider the case where absorption occurs, as the elastic case can be obtained by setting 1. We define: S e i S matrix element for elastic scattering (unitarity requires that S S 1) A S1 i 1 i ei 1 Partial-wave amplitude for elastic scattering Phase shift for elastic scattering e Absorption parameter (a measure of the incident particles removed from the beam due to inelastic scattering) ( is the imaginary part of the phase shift. We shall always use rather than.) Subscripts denoting the angular momentum, parity, and isotopic spin have been suppressed on the quantities above. k c.m. momentum (in units of incident particle mass) q 0 Incident particle c.m. total energy (in units of incident particle mass) E Incident particle lab. kinetic energy (MeV) The equations relating the last three quantities are k M T M i EE M i M T M i M T E and q 0 k 1, where M T and M i are the target particle and incident particle masses (MeV), respectively. The total c.m. energy is W p 0 q 0, where the target particle c.m. total energy is p 0 k both in units of the incident particle mass. (The speed of light c is set to 1.) More kinematics equations are at the end. II. Partial-Wave Amplitude Equations The partial-wave amplitude is M T M i A 1 ei 1 e i sin i e i sin 1 1, i 0 (See Fig. 1 for loci of constant and. See Ref. 1 for relationship between partial-wave amplitudes and observables.) 1

3 Therefore: Re A 1 sincos sin Re A 1, Im A 1 sin 1 cos 1 0 Im A # tan Re A 1 Im A, Re A 1 Im A 3 tan Im A 1 Re A 1 Im A Re A Im A 1 Re A Im A 1 Re A Fig. 1. Loci of partial-wave amplitudes in the complex plane for constant [circles of radii / centered at (0,1/)] and for constant [radial lines emanating from (0,1/) to a distance of 1/]. An arbitrary amplitude must lie inside or on the outer circle. Other ways to write the relationships listed above: S e i 1 i tan 1 i tan. 5 therefore, A S 1 i III. Resonance Formulas Define: q 0r E r el in el in i1 1 tan 1 i tan 1 tan 1 i tan 1 cot i 4 ; 6 1 cot i ; 7 A 1 Im A A A 8 1 Resonance position incident particle total c.m. energy (in units of incident particle mass) Resonance energy in terms of lab. kinetic energy of incident particle (MeV) Elastic full width at half maximum (in units of incident particle mass) Inelastic full width at half maximum (in units of incident particle mass) Total full width at half maximum x el Fractional elastic width q 0 r q 0 Distance from the resonance position in units of the half width at half maximum The Breit-Wigner resonance formula is,3

4 A R el q 0r q 0 i x i 1 1 x1 0 x0. 9 (See Fig. ) (x 1 represents pure elastic scattering. x 0 represents no elastic scattering at all). S R 1 ia R 1ix i i i x1 Fig.. Loci of resonance partial-wave amplitudes 1 in the complex plane for constant x (circles of radii x/ centered at 0, x/ and for constant (radial lines emanating from 0. 0 to the outer circle). In general x is a slowly varying function of, so that the actual loci approximate circles. Therefore: and or and Re A R tan R x 1 x 1 x x1 1 0 x0 1 x0, Im A R x 1. 1 x1 1 0 x0 (i.e., x1 R /) (i.e., R 0 for all values of ) x0 R x x1 x1/ 1 1 x0 tan R x 1 1 x 1 x tan R 0 x 1 x 1 x x1 (i.e., R /) x1/ 0 (i.e., R 0 ) x1/ 11, 1 ; 13 q 0r q 0 Equations (11) through (14) exhibit the following behavior (see Fig. 5): Considering x as constant with respect to (more about this later): (a) for x 1/ the phase shift ( R ) passes through 90 at 0 and asymptotically approches 180 as ; the absorption parameter ( R ) symmetrically dips (with respect to ) to a minimum ( Rmin x 1) at 0 and asymptotically approaches 1 as ; (b) for x 1/ the phase shift ( R ) passes through 0 at 0 after having reached a 14

5 maximum [ Rmax 1 tan1 x/ 1 x 45 ] at 1 x. It reaches a minimum [ Rmin 1 tan1 x/ 1 x 45 ] at 1 x and asymptotically approaches 0 at ; the absorption parameter symmetrically dips to a minimum ( Rmin 1 x) at 0 and asymptotically approaches 1 as. (c) for for x 1/ the phase shift ( R ) at 0 as a function of x is discontinuous at x 1/ [ R 0, x 1/ 90, R 0, x 1/ 0 where is a small positive number, as shown below]; the absorption parameter symmetrically dips to zero at 0 and asymptotically approaches 1 as. and tan R x 1/ (i.e., R 90 ) 0 (i.e., R 45 ) tan R x 1/ (i.e., R 0 ) 0 (i.e., R 45 ) However, the partial-wave amplitudes at 0 are, of course, not discontinuous for x 1/. That is, Re A and Im A are the physical quantities. and are parameters in expressions for the physical quantities. The same general behavior occurs for non-constant widths as shown in Section V. We make the following definition: Other useful relations are: Elastic resonance: xe E r 1 Inelastic resonance: Absorptive resonance: 1/ xe E r 1 Highly absorptive resonance: 0 xe E r 1/. ImS R 1ReS R ReA R ImA R x ReA R ImA R ImA R x ReA R ImA R ReA R el q 0r q 0 Resonance Width Energy Dependence All widths are in units of the incident particle mass. We use pion-nucleon scattering in all of the numerical examples. A. Elastic Width The simplest assumption would be to set el constant with. However, the width should have a threshold behavior k 1, so we could use el k 1. Resonance theory as given by Breit and Weisskopf gives el kr 0 V kr 0, where 15

6 reduced width, r 0 interaction range (in units of incident particle Compton wavelength), and V r 0 k barrier penetration factor given by V r 0 k 1 r 0 k j r 0 kn r 0 k k0 r k k Reference for the spherical Bessel functions j kr 0 and n kr 0 y kr 0. For example: V 4 r 0 k V 0 r 0 k 1, V 1 r 0 k r 0k 1r 0 k, V r 0 k V 3 r 0 k r 0 k 4 93r 0 k r 0 k 4, and r 0 k 6 545r 0 k 6r 0 k 4 r 0 k 6. r 0 k r 0 k 135r 0 k 4 10r 0 k 6 r 0 k 8 Fig. 3a contains plots of V kr 0 for values from 0 to 5 and r ( 1 fermi for pion-nucleon scattering). Fig. 3b contains plots for V r 0 k for and various values of r 0. Layson 4 has derived el by means of the Klein-Gordon equation rather than the Schrödinger equation. His result is el 4M q 0 q kr 0 V r 0 k, 0r where M is the target-particle mass (in units of the incident-particle mass.). Thus, our possible form for el are: (a) el C (b) el C k 1 (c) el kr 0 V r 0 k (d) el 4M q 0 q 0r kr 0 V r 0 k 17 In Fig. 4 we compare the four forms by setting C, C, and such that el E E r is the same for all four forms. There are two examples. In both examples, r and E r 600 MeV. (1) el E E r (This corresponds to C 0. 5, C , , ) (Dashed curves) () el E E r 1. 0.(This corresponds to C 1. 0, C , , ) (Solid curves) It is obvious that one cannot use a constant width or el C k 1 and expect better than 10% accuracy beyond 50 MeV from the resonance position. In Ref. 1, where the P 11 partial wave has a zero at about 150 MeV, well below the resonance position at about 560 MeV, a modification of the Layson elastic-width formula is

7 used to allow the zero: el q 0 q z q 0 4M q 0 q 0r r 0 kv r 0 k. 17a See where this formula is used. In Fig. 5a we plot form (d) for el with different values of r 0. We use, E r 600 MeV, and el E E r r fermi r fermi r fermi r fermi r fermi r fermi Fig. 5b shows form (d) for and different values of r 0. (This corresponds to el E E r 1. 0 for r ) We use and E r 600 MeV. In Fig. 6a we use form (d) and compare el [normalized such that el E E r 1. 0 ] for values of from 0 to 3. We use r and E r 600 MeV. The values of for each value are: 0 : : : : Fig. 6b shows form (d) for values of from 0 to 3 for (This corresponds to el E E r 1. 0 for 0.) We use r and E r 600 MeV. B. Inelastic Width Below threshold k k 0 for inelastic scattering, in 0. (For pion-nucleon scattering k ) So the simplest assumption would be in C in k k 0, where we use the step function k k 0 1 for k k 0 0 for k k 0. Or, we could give it the threshold behavior in C k k 0 k k The problem here is that we do not know what value of to use. Assume that the inelastic final state is a two-body state. This final state may have a different orbital angular momentum than the initial state; the final state value is the one that should be used. For example, the process p p, where is a 0 low-mass meson, may be the dominant mechanism for inelastic scattering in the P 11 pion-nucleon state. If so, the initial 1 and the final 0. Thus the threshold behavior should be k k 0 rather than k k 0 3. In the numerical examples given below we assume that the final and initial s are identical. Analogous to the elastic case, we could use the following forms for in :

8 (a) in C in k k 0 (b) in C in k k 0 1 k k 0 (c) in in k k 0 r 0 Vk k 0 r 0 k k 0 4M (d) in q 0 q 0r in k k 0 r 0 Vk k 0 r 0 k k 0 18 Actually, for a two-body inelastic final state one should use k, the final-state c.m. momentum, rather than k k 0.Its relation to the total c.m. energy W is k 1 W W M 1 M 1 M M 1 4 M 1 M 1 W (in units of incident-particle mass), where the final state particle masses are M 1 and M (in units of the incident-particle mass). In terms of these final-state masses, the inelastic threshold c.m. momentum is k M EE M 1 ME, where the threshold incident particle lab. kinetic energy (in units of the incident-particle mass) is E M 1 M M 1 M k 1 M 1 M M 1 M M 1 M 1 M M 1 4M. When there are several possible two-body inelastic final states, in in M 1i, M i i where the sum goes over all possible final states. Of course, there may be three- or many-body inelastic final states possible, also. For purposes of illustration we use k k 0 rather than k and only one inelastic width. In Fig. 7 we compare the four forms for the energy dependence of in. We set in E E r 0. 5 for all four forms, which dictates that C in 0. 5, C in , in , and in We use, r k and E r 600 MeV. In Fig. 8a we plot form (d) for in with different values of r 0. We use, E r 600 MeV and in E E r 0. 5.

9 r fermi in r fermi in r fermi in r fermi in r fermi in r fermi in Fig. 8b shows form (d) for different values of r 0 and in (This corresponds to in E E r 0. 5 for r ) We use and E r 600 MeV. In Fig. 9a we use form (d) and compare in, [normalized such that in E E r 0. 5] for values of from 0 to 3. We use r , k and E r 600 MeV. The values of in for each value are: 0 : in : in : in : in Fig. 9b shows form (d) for values of from 0 to 3. for in (This corresponds to in E E r 0. 5 for 0.) We use r , k and E r 600 MeV. C. Important Observation Figs. 4 and 7 make it evident that the choice of the resonance-width energy dependence may be important, even in the vicinity of the resonance position. V. Numerical Examples of Resonance Phase Shifts, Absorption Parameters, and Partial-Wave Amplitudes A. Comparison of the Four Forms for the Width Energy Dependence Four numerical examples are shown in Fig. 10 in each of which we compare the four forms for the energy dependence of the elastic and inelastic widths. In all four examples,e E r 1. 0, r k and E r 600 MeV. The first two examples are in Fig. 10a and the last two are in Fig. 10b. (1) el E E r 1. 0, in E E r 0.(This corresponds to C 1. 0, C , , and C in C in in in 0.) Of course, 1 everywhere for this example and Re A and Im A reach their unitary limits. (Solid curves) () el E E r 0. 5, in E E r 0. 5.(This corresponds to C 0. 5, C 0. 3, , and C in 0. 5, C in , in , in ) The discontinuity in at 0, x 1/ is plainly visible. Note that the peak of Im A is shifted to the low-energy side in the case of non-constant widths. (Dashed curves) (3) el E E r 0. 5, in E E r (This corresponds to C 0. 5, C , , and C in 0. 75, C in , in. 0343, in ) Form (a) for this example and the

10 next example give identical s. Note that the dip in is shifted to the low-energy side in the cases of non-constant widths. (Dashed curves) (4) el E E r 0. 75, in E E r 0. 5.(This corresponds to C 0. 75, C , , and C in 0. 5, C in , in , in ) Form (a) for this example and the previous example give identical s. Note that the dip in is shifted to the low-energy side in the cases of non-constant widths. (Solid curves) (B) Comparison for Various Values of x Using Form (d) for the Width Energy Dependence From here on all examples will use form (d) for the elastic and inelastic widths. Nine numerical examples are shown in Fig. 11a for nine different values of x el / at E E r. In all nine examples,e E r 1. 0, r , k and E r 600 MeV. (1) xe E r (This corresponds to 0 and in arbitrary) This example is no elastic scattering at all (A 0, 0, 1). () xe E r (This corresponds to and in ). (3) xe E r (This corresponds to and in ). (4) xe E r (This corresponds to and in ). (5) xe E r (This corresponds to and in ). The discontinuity at 0, x 1/ is plainly visible. (6) xe E r (This corresponds to and in ). (7) xe E r (This corresponds to and in ). (8) xe E r (This corresponds to and in 0). This example is pure elastic scattering ( 1). In Fig. 11b we compare elastic resonances xe E r 1 for different values of. We use, r ,and E r 600 MeV. (1) E E r () E E r (3) E E r (4) E E r (5) E E r () E E r Comparison for Various Values of r 0 In the examples given here, k , and E r 600 MeV. We use form (d) for widths. Numerical examples for different values of r 0 at two values of xe E r are given in Fig. 1a. (1) el E E r 0. 5 and in E E r That is: and in for r fermi and in for r fermi and in for r fermi and in for r fermi

11 () el E E r 1. 0 and in E E r That is: and in for r fermi and in for r fermi and in for r fermi and in for r fermi Examples for different values of r 0 with and in are given in Fig. 1b. (This corresponds to el E E r 1. 0 and in E E r 0. 5 for r ) D. Comparison for Various Values of k 0 In the examples given here, r , and E r 600 MeV. We use form (d) for the widths. Numerical examples for different values of k 0 at two values of xe E r are given in Fig. 13a. (1) el E E r 0. 5 and in E E r That is: and in for k MeV one-pion production and in for k MeV two-pion production and in for k MeV N 33 production (1) el E E r 1. 0 and in E E r That is: and in for k MeV one-pion production and in for k MeV two-pion production and in for k MeV N 33 production Examples for different values of k 0 with and in are given in Fig. 13b. (This corresponds to el E E r 1. 0 and in E E r 0. 5 for k ) E. Comparison for Various Values of E r In the examples given here, r , and k We use form (d) for the widths. Numerical examples for different values of E r at two values of xe E r are given in Fig. 14a. (1) el E E r 0. 5 and in E E r That is: and in for E r 400 MeV and in for E r 600 MeV and in for E r 800 MeV () el E E r 1. 0 and in E E r That is: and in for E r 400 MeV and in for E r 600 MeV and in for E r 800 MeV Examples for different values of E r with and in are given in Fig. 14b. (This corresponds to el E E r 1. 0 and in E E r 0. 5 for E r 600 MeV.) F. Comparison for various values of In the examples that follow r , k and E r 600 MeV. We use form (d) for the widths. Examples for different values of at two values of xe E r are given in Fig. 15a.

12 (1) el E E r 0. 5 and in E E r That is: and in for and in for and in for 0. 9 and in for 3 (1) el E E r 1. 0 and in E E r That is: and in for and in for and in for and in for 3 Examples for different values of with and in are given in Fig. 15b. (This corresponds to el E E r 1. 0 and in E E r 0. 5 for 0.) G. Important Observations In Fig. 11 we see that, even though the phase shift may be small for an inelastic resonance, there is distinctive behavior that may enable one to identify such a resonance in a phase-shift analysis; viz., the phase shift passes downward through zero at the resonance position and the absorption parameter has a deep dip slightly to the low-energy side of the resonance position. Stated in terms of the partial-wave amplitude the same words apply for an elastic resonance; viz., the real part of the amplitude passes downward through zero at the resonance position and the imaginary part has a peak slightly to the low-energy side of the resonance position. Of course, a large background in the same state as the resonance could obliterate some of these distinguishing features. More Kinematics Here are some more useful equations: s W k m k M q0 p 0 mm ME k sm M 4m M 4s 1 W W4 M m W M m 1 W W M m W M m References q 0 1 W W4 M m W M m 1 W W M m p 0 Wq 0 1 W W M m 1 L. D. Roper, R. M. Wright, and B. T. Feld, Phys. Rev. 138, B190 (1965) 1a L. D. Roper, Phys. Rev. Letters 1, 340 (1964) J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley and Sons, New York, 195), pp.33, 361, M B. Watson, M. Ferro-Luzzi, and R. D. Tripp, Phys. Rev. 131, 48 (1964) 4 W. M Layson, Nuovo cimento 7, 75 (1963) Distribution in 1965

13 LRL Internal Distribution Information Division Paul Csonka Paul Finkler Ronald Bryan M. J. Moravcsik R. M. Wright David Bailey L. D. Roper LRL Berkeley V. Perez-Mendez R. K. Wakerling External Distribution TID-4500 (41st Ed.), UC-34, Physics Dr. R. D. Tripp, CERN, Geneve 3, Switzerland Dr. W. M Layson, c/o Charles Carroll, Pan American World Airways, Guided Missile Range Division, Patrick Air Force Base, Florida Brian DeFacio, Physics Department, Texas A&M University, College Station, Texas Professor B. T. Feld, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts