Comparative study of scattering by hard core and absorptive potential

Transcription

1 6 Comparative study of scattering by hard core and absorptive potential Quantum scattering in three dimension by a hard sphere and complex potential are important in collision theory to study the nuclear reactions [64, 87]. The former is used for modeling the repulsive core potential in nucleon-nucleon scattering and the later is used to describe the situations when the incident flux is absorbed into nonelastic and reaction channels. This is analogous to the use of complex refractive index for the passage of an electromagnetic wave in an 127

2 absorptive medium. In 3D quantal scattering, the total scattering cross section for a complex potential is a sum of the total elastic cross section σ el and the total absorption cross section σ a. These quantities can be determined by the solution of appropriate Schrödinger equation satisfying the required boundary conditions. In complex potential scattering, the imaginary part of the potential causes the absorption of the incident flux. In our study on 1D quantum transmission across an absorptive rectangular barrier [23] we found that absorption A b reaches a maximum for a finite value of the absorption strength W 0 for fixed energy and then steadily decreases with an increase in W 0. As W 0, A b 0 and reflection R 1, the absorptive domain becomes almost fully reflective, showing some similarity with an infinite barrier, since for a real barrier of infinite height at a fixed energy the transmission T = 0 and R = 1. Similarly in this chapter we show that in three dimension the absorption cross section σ a for a rectangular absorptive well potential as a function of W 0 for a given energy has at least one maximum and σ a monotonically decreases to zero [89] as W 0. We further explore whether an absorptive potential with infinitely large strength W 0 is similar to a hard core potential as far as the cross sections are concerned. It is well known that the classical differential cross section of a hard sphere of radius a is isotropic and the classical total cross section is given by σ cl,t = πa 2. The quantum mechanical scattering cross section is 2πa 2 in the high-energy limit; the additional πa 2 is due to the dominance of diffraction, particularly in the forward angle region, showing that quantum scattering is a wave phenomena. 128

3 Quantum mechanical scattering by a hard sphere of radius a by projectiles having incident energy E = 2 k 2 /2m and corresponding momentum k is dominated by partial waves with l ka. That is, the partial wave S -matrix S l is almost unity for l reasonably larger than ka and generates significant phase shifts that contribute to the cross section for l ka. Whereas there is no absorption in hard-core scattering, for scattering by an infinitely absorptive sphere we expect that S l 0 for l ka and S l = 1 for l reasonably larger than ka. Such S -matrix elements contribute to both the elastic and absorption cross sections. That is, the behavior of S l is very different for an absorptive potential compared to a hard sphere. We will determine to what extent the differential scattering cross section σ(θ) and the total cross section σ T by a strongly absorptive sphere are similar to those generated by a hard core. In addition the absorptive potential scattering provides good example of the threshold behavior of phase shifts for small energy for a given partial wave and for the asymptotic behavior of S l as a function of l for a given energy. The former is well studied in textbooks and in this chapter [24] we examine the latter in detail. The asymptotic behavior of S l with l will illustrate how classical scattering is affected by quantum mechanical behavior for partial waves around l ka. This chapter is organized as follows: In Section 6.1 we describe briefly the three dimensional potential scattering theory and in section 6.2 we give the comparative study of quantum hard sphere scattering with the classical hard sphere scattering. In Section 6.3 we study the scattering by an absorptive well and hard sphere and in section 6.4 we describe the mathematical and numerical results of our study about the variation of absorption cross section as a function 129

4 of W 0 and energy E. In section 6.5 we explain the threshold behavior of σ a and summarize the mains results of our study in section Scattering by absorptive and hard core potentials In this section we summarize several well known results of 3D potential scattering theory (see for example [87]) relevant to our study. When the mutual interaction between the projectile and the target depends on their relative separation, the two body problem of scattering can be reduced to equivalent one body problem of scattering by a potential. For well behaved spherically symmetric potential the quantal scattering can be studied by using time independent Schrödinger equation: 2 2µ 2 Ψ( r) + U(r)Ψ( r) = EΨ( r), (6.1) where µ is the reduced mass of the interacting particles, is the reduced planck s constant, r is the relative distance between the interacting particles, U(r) is the potential and E is the energy of the system. The asymptotic behavior of the scattering wave function Ψ(r, θ, φ) is a combination of incident plane wave assumed to be moving along z-axis and the scattered spherical wave having amplitude f (θ, φ): Ψ(r, θ, φ) r A c [e ikz + f (θ, φ) eikr r ]. (6.2) 130

5 The function f (θ, φ) is known as scattering amplitude. For spherically symmetric potential it is independent of φ and hence from now onwards we use the symbol f (θ). The constant A c signifies the overall normalization. In the case of spherically symmetric potential, the differential scattering cross section σ(θ) is: σ(θ) = f (θ) 2. (6.3) Partial wave analysis of the scattering amplitude is very useful in the study of scattering by well behaved central potentials, resonance phenomena etc particularly at lower energies when the number of significant partial waves are not too large. The partial wave expansion of the scattering amplitude f (θ) is given by: f (θ) = 1 2ik (2l + 1)[e 2iδ l 1]P l (cosθ), (6.4) l=0 where δ l is the partial wave phase shift and is related to the partial wave S - matrix S l by the relation: S l = e 2iδ l. (6.5) In the case of absorptive potential δ l is complex with positive imaginary part such that S l 1 and for real potential S l = 1. For short range potential, the asymptotic behavior of the radial Schrödinger equation solution R l (r) is given by: lπ sin(kr R l (r) + δ r eiδ l 2 l). (6.6) kr It is convenient to use the modified radial wave function u l (r) = rr l (r) which 131

6 satisfies the differential equation: ( ) d 2 u l (r) + k 2 l(l + 1) V(r) u dr 2 r 2 l (r) = 0, (6.7) where V(r) = (2µ/ 2 )U(r) and k 2 = 2µE. One can use this equation to evaluate 2 phase shift δ l of the l th partial caused by the potential V(r). Consider the absorptive potential: V 0 iw 0, r < a, W 0 > 0, V(r) = 0, r > a. (6.8) V(r) approaches hard sphere potential when V 0 and W 0 = 0. Similarly when W 0, V(r) approaches infinitely absorptive sphere. For r < a the acceptable solution of Eq.(6.7) for the potential V(r) is of the form: u l (r) = A l (αr) j l (αr), r < a, (6.9) where α 2 = (k 2 V 0 + iw 0 ) and A l is the normalization constant. In the absence of the potential, the resulting free modified radial equation has two independent solutions, namely (kr) j l (kr) and (kr)η l (kr), where j l (kr) and η l (kr) denotes spherical Bessel and Neumann functions [90]. Let us designate these as F l and G l respectively. For large r, u l (r) can be written as a linear combination of F l and G l. Asymptotically they have the following behavior as kr : F l sin(kr lπ/2), kr G l kr cos(kr lπ/2). Beyond the range a of the potential, the solution of modified radial schrödinger 132

7 equation u l (r) is: u l (r) = C l [cosδ l F l sinδ l G l ], r a. (6.10) Phase shift δ l can be calculated by matching the logarithmic derivative of u l (r) at r = a: ( ) 1 du l u l dr r<a = r=a ( ) 1 du l u l dr r>a, (6.11) r=a and one notices that equations Eq.(6.11) is independent of A l and C l. Denoting ) = γ l, it is straight forward to obtain: r=a ( 1 ul du l dr r<a tanδ l = kf l γ lf l kg l γ lg l, (6.12) where superscript prime indicates the derivative with respect to r and S l = (1 + itan δ l) (1 itan δ l ). (6.13) In the case of hard core potential we get: tanδ l = j l(ka) η l (ka), (6.14) and the corresponding S l for hard sphere potential scattering can be expressed as: S l = h(2) l (ka) h (1) l (ka). (6.15) In the case of complex square well potential when W 0 becomes very large, i.e. 133

8 k << α, as α, u l behaves as: u l α ( sin αr lπ ). (6.16) 2 Hence γ l = u l u l for a fixed energy and l, but large complex α can be approximated as: ( γ l αcot αr lπ ), (6.17) 2 and γ l α, (6.18) and hence one finds that for large α, S l will gradually approach the hard core S l. For completeness we give below the partial wave expansions for total elastic scattering σ el, total absorption cross section σ a and total cross section σ T generated by an absorptive potential: σ el = π k 2 (2l + 1) ( 1 S l 2), (6.19) l=0 σ a = π k 2 (2l + 1) ( 1 S l 2), (6.20) l=0 σ T = σ a + σ el = 2π k 2 When the potential is real, σ a = 0. (2l + 1) [1 Re(S l )]. (6.21) l=0 134

9 6.2 Hard sphere scattering classical and quantal Hard sphere of radius a is represented by the potential:, r < a, U(r) = 0, r > a. (6.22) For s-wave scattering l = 0, so j 0 (ka) = (ka) 1 sin(ka) and η 0 (ka) = (ka) 1 cos(ka). Hence using the Eq.(6.14) and Eq.(6.15), one gets the s-wave phase shift δ 0 as: tanδ 0 = tan(ka), δ 0 = ka, (6.23) and a the hard core radius is the scattering length. S -matrix for s-wave is: S 0 = e 2ika. (6.24) Significant contribution to scattering cross-section [10, 87] will be from s-wave when ka << 1. Hence, in this case the scattering cross-section can be expressed as: σ T = 4π k 2 sin2 (ka), (6.25) and lim k 0 σ T = 4πa 2. In the high energy limit the total cross section is given by [87]: σ T = 2πa 2 [ (ka) 2/3 +...]. (6.26) 135

10 Eq.(6.26) implies that the quantum mechanical total scattering cross section does not approach the classical total scattering cross section σ cl,t = πa 2 in the high-energy limit but approaches 2σ cl,t. This behavior is due to the domination of diffraction in the forward direction. Comparison of classical and quantum differential cross section by a hard sphere A comparison of the quantum mechanical differential cross section σ(θ) with σ cl (θ) is not discussed in detail in textbooks including the extent to which σ(θ) is isotropic. It is interesting to compare the quantum mechanical scattering differential cross section with the corresponding classical result in the entire angular range. To do so, we show the differential scattering cross section σ(θ) for energies E = k 2 = 4 and E = k 2 = 36 generated by a hard core interaction in Figure 6.1. The classical differential scattering cross section σ cl (θ) is a 2 /4 = 9/4 = 2.25 is isotropic. It is clear from Figure 6.1 that outside the forward angle region, the cross section approaches σ(θ) = 2.34 fm 2 from below. This result shows that in the large angle region, hard-core and soft-core scattering is almost independent of angle and agrees closely with the classical result. The additional nonclassical terms in σ T are mostly due to the forward angle diffraction peak. The first diffraction minima in Figure 6.1 occur at angles that are in the vicinity of the corresponding results for diffraction of light by a circular aperture. In quantum scattering the coherent addition of partial waves in three dimension leading to the forward diffraction peak is due to the fact that P l (cos θ) in the partial wave expansion of the scattering amplitude is close to unity for θ sufficiently close to the forward angle for all values of l. At larger angles the lack of such phase correlation leads to large scale cancellations among partial wave amplitudes resulting in more or less 136

11 Figure 6.1: Variation of the differential cross section σ(θ) as a function of θ for energies k 2 = 4 and k 2 = 36 for a hard core potential of radius a. isotropic behavior of σ(θ) at larger angles. For low energies, the diffraction peak becomes broader and, hence, the quantum mechanical cross section is significantly higher than the corresponding classical cross section, which helps to explain the result σ T = 4πa 2 in the low energy limit. We find that as energy increases, except for forward angle region, σ(θ) is practically same as the corresponding classical differential cross section and quantal effects dominant only at forward angles. This is primarily due to the interference of incident and scattered quantal wave function in the forward region. 6.3 Scattering by an absorptive well and hard sphere In this section we make a comparative study of scattering by a hard sphere and the strongly absorptive potential. We had generally discussed the nature of S l for these two cases in section 6.1. In this context it is of interest to study the variation of S l 1 with l which helps to understand the subtle features of 137

12 quantal scattering. To study the asymptotic behavior of S l for hard core, we express the phase shifts for hard core scattering in terms of Bessel s functions. Using Eq.(6.15) and the relations between the cylindrical and spherical Bessel functions [90], we get: S l 1 = 2J λ(ka). (6.27) (ka) H (1) λ Here λ = l + 1/2 and the Hankel function is given by: H (1) λ = J λ (ka) + iy λ (ka). (6.28) For a given ka as λ, the asymptotic behaviour [90] of J λ (ka) and Y λ (ka) is: J λ (ka) = ( ) λ 1 eka, (6.29) 2πλ 2λ Y λ (ka) = Using Eq.(6.27)-(6.30), we get: 2 πλ ( ) λ eka. (6.30) 2λ S l 1 λ e 2λ(ln eka ln 2λ) 0. (6.31) Eq.(6.31) shows that the partial wave amplitude falls off exponentially for large λ. In classical scattering the angular momenta that have an impact parameter greater than the hard-core radius a do not contribute to the scattering cross section. In the corresponding quantum mechanical case the transition is less abrupt and a significant number of partial waves beyond λ = ka contribute to the scattering amplitude. This contribution is illustrated by the variation of S l 1 with l = 0, 1, 2,... for a = 4 and k 2 = 9 as shown by the points 138

13 Figure 6.2: Variation of the partial wave amplitude 1 S l as a function of l for hard core (HC) scattering and complex well (CW) scattering. The curve is drawn as an aid to the eye. on the curve in Figure 6.2. Also note that the rapid decrease of S l 1 as l is a general property of well-behaved short-range potentials including the rectangular absorptive potentials. In Figure 6.2 we also show the behavior of S l 1 with l for a rectangular absorptive well with parameters V 0 = 5, W 0 = 3, a = 4, and k 2 = 9. Both potentials produce similar asymptotic behavior; however, for small l, S l 1 for the complex potential, signifying the dominance of absorption. For an absorptive potential the variation of S l 1 as a function of l demonstrates the decrease of absorption with increasing l due to the dominance of the centrifugal term in Eq.(6.7). Its asymptotic behavior is close to the corresponding hard core result. 139

14 A comparative study of differential scattering cross sections of hard sphere and Absorptive potential S l (k) for an absorptive potential with large W 0 approach the hard core S l matrix. As a consequence of this, we expect that the scattering by a repulsive core with large V 0 will be similar to the corresponding result for scattering by an absorptive sphere with large W 0. The small difference between the differential cross sections σ(θ) for a repulsive core and an absorptive well shown in Figure 6.3 demonstrates this similarity. In both cases, σ(θ) is isotropic for larger angles and the forward angle region diffraction peaks are similar. Because of the presence of absorption, σ(θ) for the absorptive well is smaller than that for the repulsive potential. This similarity is due to the fact that a highly absorptive potential makes the wave function negligibly small in the absorptive region, which is akin to the impenetrability of the wave function into the hard core. To obtain a similar angular distribution pattern we have to use a substantially larger value of W 0 than the corresponding V 0 because most of the attenuation inside the repulsive potential approximating the hard core is governed by the term Im(k 2 V 0 ) 1/2, whereas for the absorptive well attenuation is governed by Im(k 2 V 0 + iw 0 ) 1/2. 140

15 Figure 6.3: The differential scattering cross section σ(θ) as a function of θ for large strengths of the repulsive core (I) and absorptive (II) potentials. The parameters are shown in the figure. 6.4 Absorption cross section as a function of absorption strength W 0 In our study of 1D quantum transmission across an absorptive potential, we found that absorption A b 0 as W 0 0 and W 0, showing that between 0 < W 0 <, A b has one maximum value. In this chapter we analyze whether the absorption cross section σ a in 3D absorptive potential scattering will show the similar trend like A b in 1D. Unlike the absorption A b in one dimension, σ a has contributions from all relevant partial waves. Thus σ a as a function of W 0 for a given energy will reach at least one maximum in the range 0 < W 0 <. We analyze this for a rectangular complex potential. In order to do this we 141

16 write the wave function u l (r) in the form: u l (r) = 1 2ik [F l( k) f l (k, r) F l (k) f l ( k, r)]. (6.32) The functions f l (±k, r) are the Jost solutions given by: f l (k, r) = i l 1 (kr)h (2) l (kr) r e ikr, (6.33) The functions: f l ( k, r) = i l+1 (kr)h (1) l (kr) r e ikr. (6.34) F l (±k) = u l f l (±k, r) u l f l(±k, r), (6.35) are known as Jost functions and are independent of r. The asymptotic form of the u l (r) can also be expressed as: ( u l (r) r eiδ l sin kr lπ ) 2 + δ l. (6.36) In the absence of scatterer, the free wave function has the form: ( u l (r) r sin kr lπ ). (6.37) 2 The phase shift δ l is the only difference between the radial wave function in the presence of the scatterer and in its absence. From Eq.(6.32), one finds that the S -matrix element S l can be expressed as (see for example [10]): [ S l = e iπl F l (k) F l ( k) = f eiπl l (k, r) γ ] l f l (k, r) f l ( k, r) γ l f l ( k, r) r=a, (6.38) 142

17 where γ l = u l /u l and the prime denotes a derivative with respect to r. The S l measures the response of the scattering target by determining the phase of the resulting outward-traveling spherical wave relative to what it would be if there were no target. If S l = 1, it implies the absence of absorption or emission. We use Eq.(6.32) and Eq.(6.38) and rewrite u l as: u l = C l [ fl (k, r) e iπl S l f l ( k, r) ], (6.39) where C l = F l( k). If we use the Schrödinger radial equation satisfied by u 2ik l and u l, we find: 1 S l 2 = 1 W a 0 u C l 2 l 2 dr. (6.40) k 0 As k, the right hand side of Eq.(6.40) goes to zero, implying that σ a 0 as the energy becomes infinitely large. Also (1 S l 2 ) = 0 if W 0 = 0. The behavior of σ a for large value of W 0 can be studied by analyzing the behaviour of F l ( k) for large W 0. For this, we use the result of Eq.(6.9), u l = αr j l (αr) and the result for f l (±k, r) given by Eqs.(6.33) and (6.34) for r a. For large W 0, we set u l = sin(αr lπ 2 ) and using the properties of h(1) l (kr) and h (2) l (kr), we obtain the expression for F l ( k) for large W 0 as: F l ( k) 2 e2α ia 4 α + k 2, (6.41) 143

18 where α = α r + iα i. The leading behavior of u l in the range 0 < r < a is e2α i r 4. Hence the right-hand side of Eq.(6.40) can be written as: 1 S l 2 = 4kW a 0 F l ( k) 2 0 (αr sin lπ 2 ) 2 dr = 2kW [ 0 sinh(2αi a) ( 1) l sin(2α ] ra). F l ( k) 2 2α i 2α r (6.42) By using Eq.(6.41), we find that (1 S l 2 ) = O(W 1/2 0 ) as W 0, because α i W 1/2 0. Thus Eqs.(6.40)-(6.42), show that: 1 S l 2 W0 0 0, (6.43) 1 S l 2 W 0 W 1/2 0. (6.44) Since (1 S l 2 ) 0 in the range 0 < W 0 <, (1 S l 2 ) will have at least one maximum. Alternatively, the result for (1 S l 2 ) = O(W 1/2 0 ) as W 0 can be shown as follows: If we use Eq.(6.38) and set: [ f ] η ± l (±k, r) l = f l (±k, r) r=a, (6.45) and note that when k is real, η + l = (η l ) and [ f l (k,a) f l ( k,a)] = 1, we obtain: 1 S l 2 4kIm(γ l ) = k 2 + γ l 2 2Re(γ l η l ), (6.46) 1 S l 2 = (η+ l ) γ l + η + l γ l ((η l ) γ l + γ l η l ) η l 2 + γ l 2 2Re(γ l η l ). (6.47) 144

19 We note that η ± l is independent of W 0 and for large ka, η + l ik and η l ik. From Eq.(6.17) for a given k 2 and V 0, and for large W 0 we have: γ l α = O(W 1/2 0 ). (6.48) Eq.(6.48) gives the dominant behavior of (1 S l 2 ) as: 1 S l 2 = 4kα r k 2 + α 2 + 2kα r O ( ) W 0 W 1/2 0. (6.49) Eq.(6.49) shows that as W 0, the absorption cross section approaches zero. Here (1 S l 2 ) O(W 1/2 0 ) as W 0. On the other hand, (1 S l 2 ) 0 as W 0 0. Thus σ a = 0 when W 0 = 0 and σ a 0 as W 0. Hence, σ a must have one or more maxima between W 0 = 0 and W 0 = as illustrated in Figure 6.4. (i) Numerical Analysis of σ a as a function of absorption strength and energy To confirm our mathematical analysis of variation of σ a as function of W 0 and energy, we studied the variation of numerically computed σ a, σ el and σ T obtained for a rectangular absorptive potential well as a function of W 0 for fixed energy and as a function of energy for fixed W 0. In Figure (6.4) we show the variation of σ a, σ el and σ T as a function of W 0 for a potential with fixed V 0 = 0, a = 2 and k 2 = 9. Similarly in Figure (6.5) we show σ a, σ el and σ T as a function of E which begins slightly away from E = 0 for V 0 = 0, W 0 = 10, and a = 2 because we discuss the threshold behavior in next section. We find that σ a has a broad peak in the vicinity of k 2 = W 0 and then gradually decreases [89], tends to zero as k and as k 0 σ a diverges as k 1 for finite W 0. However, the peak in σ a is much broader than the corresponding result for A b 145

20 Figure 6.4: Variation of σ a, σ el, and σ T as a function of W 0 ; σ a is a maximum in the vicinity of W 0 = k 2. in one dimension because unlike in one dimension σ a has contributions from several partial waves and the collision cross section is shared between only two channels corresponding to absorption and elastic scattering. In one dimensions T and R play mutually opposing roles resulting in a sharper absorption peak. 146

21 Figure 6.5: Variation of σ a, σ el, and σ T as a function of energy E. Note that E begins slightly away from E=0, because we discuss the threshold behavior separately. In Figures 6.5 and 6.6 we show more examples of the variation of σ a and σ el as a function of E for different sets of V 0 and W 0. These figures show that the dominance of σ a over σ el can occur for purely absorptive imaginary potentials without any real part. In contrast, when an attractive or repulsive real part of the potential is added to an absorptive potential, σ el dominates over σ a. In particular, we note that σ a rises with a decrease in energy for potentials with an attractive real part, which is not the case when the real part of the potential is repulsive. This increase is due to the additional damping of the wave function due to the repulsive real part. This increase does not occur when the real part of the potential is attractive. 147

22 In our study of variation of σ a as a function of W 0 for a rectangular absorptive potential in 3D, σ a is maximum in the vicinity of W 0 = E, similar like A b in 1D rectangular absorptive potential. Hence it is quite interesting to examine the variation of numerically computed W 0 (E) as a function of E, where W 0 (E) specifies the value of W 0 that gives an absorption maximum at E. We demonstrate this correlation in Figure 6.7 for three absorptive potentials. We again see the expected result that the attractive real part of the potential facilitates absorption in comparison to the repulsive real part. It is clear that W 0 (E) need not be a monotonic function of E. Unlike the absorption A b in one dimension, the absorption cross section has contributions from all relevant partial waves. Thus σ a as a function of W 0 for a given energy will reach at least one maximum in the range 0 < W 0 <. 148

23 Figure 6.6: Variation of σ a and σ el as a function of E for three sets of potential parameters as shown in plots (a)-(c). The lowest value of the energy is E=

24 Figure 6.7: Variation of the absorption strength W 0 (E), as a function of energy E for different absorptive potentials. 6.5 Threshold behavior of scattering cross sections Threshold behavior of σ a as k 0 is a quite interesting special feature in complex potential scattering. The phase shift for the potential scattering to study the threshold behavior of scattering and absorption cross sections can be deduced from the following expression for tanδ l [87]: tanδ l = k j l (ka) β l j l (ka) kη l (ka) β lη l (ka), (6.50) where β l is the logarithmic derivative of the radial wave function R l = u l /r and δ l is the phase shift of the lth partial wave. We use the behavior of j l, η l, j l, and η l as k 0, and note that as k 0, β l is independent of k and find that δ l = O(k 2l+1 ). This behavior implies Re(δ 0 )=O(k) and Im(δ 0 )=O(k) as k

25 Hence, as k 0 we have: σ el = π k 2 1 S 0 2 k 0 4π k 2 δ 0 2 = 4π a 0 2, (6.51) where a 0 = lim k 0 δ 0 k. On the other hand, the corresponding σ a is given by: σ a = π k 2 ( 1 S 0 2) = π k 2 ( 1 e 4Im(δ 0 ) ) k 0 4π k 2 Im(δ 0) k 0 O ( k 1). (6.52) Eq.(6.52) shows that σ a diverges at threshold. We can see this divergence explicitly for the rectangular absorptive potential well. In this case S 0 (k) can be expressed as S 0 (k) = αcos(αa) + iksin(αa) αcos(αa) iksin(αa). (6.53) If W 0 = 0 and V 0 > k 2, then α r = 0; if W 0 = 0 and V 0 < 0, then α i = 0; in all other cases α is complex with Im(α) > 0 and Re(α) > 0. After some algebra, we obtain 1 S 0 2 = 4k2 sin(αa) 2 D m, (6.54) 1 S 0 2 = [4k Im(α cos(α a)sin(αa))] D m, (6.55) and 2(1 Re(S 0 )) = 4k2 sin(αa) 2 + 4k Im (α cos(α a)sin(α a)) D m (6.56) with D m = αcos(αa) 2 + k 2 sin(αa) 2 + 2kIm (α cos(α a)sin(αa)). (6.57) 151

26 The total scattering cross section σ T = 2π k 2 (1 Re(S 0 )). Thus, from Eqs.(6.54)- (6.56), we have σ T = σ el +σ a. The term D m approaches a constant as k 0. If we set α = α cos(θ) + i α sin(θ) and take 0 < θ < π/2, we can write Eq.(6.55) as 1 S 0 2 = 2k α N D m, where b = 2 α a and N = cos(θ)sinh(bsin(θ)) sin(θ)sin(bcos(θ)). (6.58) For a given θ satisfying 0 θ π/2, and b 0, N 0, and dn db = cos(θ)sin(θ)cosh(bsin(θ)) cos(θ)sin(θ)cos(bsin(θ)) 0, (6.59) we find N approaches a positive constant as k 0. Then (1 S 0 2 ) = 2k α N D m = O(k). Hence, σ a = π k 2 ( 1 S 0 2) k 0 O(k 1 ). (6.60) Eq.(6.60), which has been noted in the neutron-nucleus optical model [91], is of much significance. It shows that the absorption cross section σ a can become infinitely large near the threshold. (ii)numerical Analysis of threshold behavior of absorption cross section In Figure 6.8 we show the nature of σ a, σ T, and σ el near the threshold for a typical absorptive potential. We also give the results for the scattering cross section σ hc,el from a hard sphere, demonstrating the similarity between hard sphere scattering and scattering from an absorptive well. The unusually high cross section for the absorption of thermal neutrons by nuclei such as Cd and B can be meaningful from the behavior of σ a at very low energies. Such a divergence leading to an infinite value for σ a as k 0 needs clarification. We 152

27 note that the complex potential scattering is essentially a two-body problem approximation to a multichannel many body problem involving a large number of inelastic channels. For example, in typical neutron-nucleus scattering the threshold for elastic scattering is k 2 = 0, but the threshold for the next nearest inelastic channel may be larger, say k 2 = ϵ r > 0 implying that in the range 0 < k 2 < ϵ r only the elastic channel is open and, hence, in this range the complex potential approximation is not applicable; therefore, σ a = 0 if k 2 < ϵ r. That is, the value of Q that represents the energy balance of the reaction associated with this channel is negative. For the reaction: n Cd 113 Cd + γ, (6.61) Q has a positive value of about 24 MeV and therefore the threshold energy for this reaction is zero. Hence, the complex potential optical model for n Cd scattering even for thermal neutrons is meaningful and the large value of σ a ( 20000b) can be understood from the complex potential scattering model. In charged particle scattering such as proton-nucleus scattering the presence of the Coulomb barrier makes it practically impossible to explore the nuclear reaction domain as k 0. However, in antiproton nucleus scattering a very high absorption cross section as k 0 is likely because of the attractive Coulomb potential and open annihilation channels. 153

28 Figure 6.8: Variation of σ a, σ el, σ T, and σ hc,el as a function of E as E Summary The results described in this chapter are in the context of our study of transmission with absorption in one dimension (chapter 5) and deal with the comparative study of quantal scattering in three dimension by absorptive rectangular potential and repulsive core potential. The main findings are: Just as in 1D transmission, the absorption cross section in 3D scattering by an absorptive potential at a given energy reaches a maximum at a critical strength W 0 of the imaginary part and further increase of W 0 gradually decreases the absorption. In the limit W 0 absorption cross section vanishes. That is, as W 0 absorptive potential becomes reflective like the hard core. Similarly, at a given W 0, if energy E is varied, the maximum absorption occurs at a critical energy. As one expects, in the limit E, for a fixed W 0 absorption cross section vanishes. 154

29 There are interesting similarities in the differential cross section generated by a highly absorptive potential and a repulsive core potential. Both show very similar and isotropic behavior at larger angles and compare closely with the corresponding results for classical hard sphere scattering; but both manifest diffraction peaks which are quantal effects in the forward angle region. As is well known, classical differential cross section by a hard sphere is isotropic. The quantal scattering by repulsive core potential and absorptive potential also help us to project additional differences as compared with the classical scattering. Whereas in classical scattering by a hard sphere, impact parameters larger than the core radius will not contribute to scattering, in the quantal scattering the transition is gradual and the magnitude of the partial wave amplitude as a function of angular momentum show an asymptotic gradual exponential decrease. This feature is common between repulsive core scattering and strongly absorptive potential scattering even though for lower partial waves they show significant differences. The important difference between repulsive core scattering and corresponding absorptive potential scattering is in the threshold behaviour. In the former case the cross section tends to a definite finite value as E 0, and has a well defined scattering length. On the other hand, the absorption cross section diverges as E 1/2 as E 0. In the corresponding one dimensional case absorption tends to zero as E 0. In repulsive core quantal scattering, the radial wave function does not 155

30 penetrate the hard core region. On the other hand the attenuation of the wave function in the absorptive region of the potential is more gradual. However, with large absorption strength W 0 attenuation becomes more rapid. This feature provides another way to understand the similarity between hard core scattering and absorptive potential scattering. Both hard core and absorptive potential scattering have important applications in nuclear collisions. In particular, nucleus-nucleus collision is governed by optical potential with strong absorption which attenuates the wave function in the interior region. This makes the theoretical study and experimental verification of the nature of nucleus-nucleus optical potential in the interior region a non trivial task. 156