5 The phase function. Examples

5 The phase function. Examples

In this chapter we discuss some explicit examples of phase functions, which have been evaluated by numerical integration of the phase equation. The first case considered is the attractive Yukawa potential V(r)

=

-(lO/r)e-r.

(1)

The phase function associated with this potential is plotted in Figs. 1-4, for various values of I and k. I n the first three figures the phase function corresponding to a given value of I is plotted for several values of k. From Fig. 1 and the discussion of the preceding chapter we infer that the potential of Eq. (1) possesses two S-wave bound states; it would possess only one S-wave bound state if it were amputated of its part extending beyond P , with 0.15 5 F 5 1.15, and no bound state if i 5 0.15. (For a check of these results, see Fig. 2 1 . 3 Similarly, we infer from Figs. 2 and 3 that the potential of Eq. (1) possesses one P-wave and no D-wave bound states. (Incidentally, the phase function for k = 0.1 has not been plotted in Fig. 3 because it turns out to be too small.) Note that for lower partial waves the phase function increases with k for small t and decreases with k for large Y . T h e former behavior corresponds to the dominance of the centrifugal barrier, which is better pierced by more energetic particles; the latter, to the attractive nature of the potential, as represented by Levinson’s theorem. For I = 2 the centrifugal repulsion is sufficient to forbid any bound state, and accordingly the asymptotic values of the phase function do not increase monotonically with k; they must, in fact, vanish at both small and large k. I t should of course be kept in mind that the asymptotic value of the phase function is determined by a scattering experiment at best only up to a mod(rr) ambiguity. Thus it is not 21

22

5-

CHAPTER

5

U k-1.0

-

4 -

k-2.0

3-

2k-10

-

1-

k-100

0

a5

in

1 .5

20

25

r

30

FIG. 1. S-Wave phase function for an attractive potential. Note that this potential possesses two S-wave bound states.

30

2.5

2 .o

1.5

1.0

a5

0

05

in

15

20

25

30

FIG. 2. P-Wave phase function for an attractive potential. Note that this potential possesses one F-wave bound state.

23

THE PHASE FUNCTION. EXAMPLES

k = 20

k =10 07.

06-

k-10 04.

k-SO

03-

k-100

k 0

2

1

3

4

-

5

0.5

r

6

FIG. 3. D-Wave phase function for an attractive potential. Note that this potential possesses no D-wave bound state.

4.

61 4

3.

3 2. 2

1.

1.0

1

c.1

1.2 ,---------------r -2

0 0

0

1

---

-,Me---

a

2

3

-

I

4

FIG.4. Phase function for an attractive potential.

S

r

24

CHAPTER

5

possible, in general, to infer from the values of the scattering phase shift whether a bound state does or does not exist. However, by examining the behavior of the phase shift as a function of k and of 1 and by comparing it with graphs such as those discussed here and below, one may try to build up a rough image of the potential. I n comparing the results for different values of the angular momentum 1 it should be noted that different scales have been used in Figs. 1-3; a more convenient comparison is afforded by Fig. 4,where the phase functions corresponding to different values of E are plotted on the same graph. I n conclusion, we may abstract the following qualitative rules for the energy dependence of the phase shift produced by an attractive potential. If the energy is small so that K P (E l), where P is some measure of the range of the potential, the centrifugal effect dominates the behavior of the phase shift, which therefore increases with energy. T h e more energetic the scattering particles, the more they penetrate into the potential region, and therefore the larger the scattering phase shift. This rule, however, applies only if the potential is not strong enough to possess bound states; otherwise it is Levinson’s theorem that dominates the low-energy behavior of the phase shift. This may be physically interpreted as being due to a sort of tunnel effect, by which the particles penetrate into the inner region and are kept there by the attractive interaction, when the attraction is strong enough to bind one or more bound states. On the other hand, if the energy is large, so that kP (I l), then the dominant factors in determining the phase shift are the time spent by the particle in the potential region (which is longer the slower the particle moves) and the ratio of the potential to kinetic energy; both factors collaborate in increasing the phase shift as k decreases. We consider next the repulsive Yukawa potential

< +

> +

T h e phase function associated with this potential is plotted in Figs. 5-8 for various values of 1 and k. I n the first three of these figures the phase function corresponding to a given 1 is plotted for several values of k; in the last one, the phase function is plotted on the same graph for different values of E and k. Of course the repulsive nature of the potential is reflected by the fact that the phase function is negative and vanishes for both large and small values of k. This potential is

25

THE PHASE FUNCTION. EXAMPLES

0 -D2 -0.4

k=100

k- 05 -0.6

k - 50 -0.8 k.10 -10

-12

-1 4

k-20

-1 6

80 -1.8

k - 10 0

0.5

1.o

15

20

25

r

3.0

FIG. 5. S-Wave phase function for a repulsive potential.

k - 1.0

- 0.4 -06

k- 2.0 -08

-

El

- 10 -12

4 -1.4

-

0

k- 10 0.5

10

1.5

I

I

2.0

2.5

FIG. 6. P-Wave phase function for a repulsive potential.

r

3.0

26

CHAPTER

5

k ' 1.0

k-m k

\

-

50

J

-

k - lo

- l4l0

0.5

1.0

1.5

20

;5

FIG. 7. D-Wave phase function for a repulsive potential.

FIG.8.

Phase function for a repulsive potential.

r

1

3.0

THE PHASE FUNCTION. EXAMPLES

27

similar in shape to the attractive one discussed above, except that its range is shorter and its strength greater; but of course the behavior of the phase function is very different, especially at small k, because the interaction has changed from attractive to repulsive. However, there are certain common features, notably that at short distances the phase function increases with k and that the accumulation of the phase begins at larger values of r the larger the value of I and the smaller the value of k. I n fact, both these effects are due to the centrifugal barrier and are therefore essentially independent of the nature of the potential. Finally, we consider the potential that obtains by adding the two potentials previously considered:

v(l)= (20/~)e-~" - (IO/r)e-".

(3)

This potential is repulsive at short distances and attractive at great distances; it changes sign in the neighborhood of r = 0.7. T h e phase function associated with it is, of course, decreasing and negative for r < 0.7, and increasing for r > 0.7; it is plotted in Figs. 9-12 for various values of I and k. It is apparent from the first three of these figures that this potential possesses only one S-wave bound state and no P-wave (or, of course, D-wave) bound states. For a given value of I, the asymptotic value of the phase function is positive for small K and negative for large k; this corresponds to the fact, already noted above and clearly displayed by the behavior of t l e phase function in all cases, that at large k it is the inner part of the potential that plays a dominant role in the growth of the phase function, with the converse happening at small k. I n conclusion, let us summarize the main qualitative features of the behavior of the phase function. Its growth begins at a value of r that is larger, the larger 1 is and the smaller k is; this is a centrifugal effect, with an obvious semiclassical interpretation. For large values of k, the accumulation of the phase shift occurs mostly at small values of r ; thus the values of the phase shift for large energies are probes of the inner potential region. (From the point of view of the phase equation, this is an effect of the factor k-l in the r.h.s., which makes the derivative Sl'(r) small for large k as soon as the potential is not large.) For small values of k, the accumulation of the phase shift occurs mostly for larger values of r, the more so the larger I is. (This is again an effect of the factor K-1 in the r.h.s. of the phase

28

CHAPTER

5

3.0

25

6,

v

20

(r)-(20/r)

-

iZr (io/r) 6'

1.5

1.0

0.5

0

-0.5

- 10 0

10

15

2.0

:

25

FIG. 9. S-Wave phase function for a potential that is repulsive at short distances and attractive at large distances.

-..P -

k -100

k-50 k -x)

- 0.4 -0.6

0

1

1

2

3

4

5

r

6

FIG. 10. P-Wave phase function for a potential that is repulsive at short distances and attractive at large distances.

29

THE PHASE FUNCTION. EXAMPLES

06

I

I

I

I

,

I

k * 2.0

-0.31 0

J

1

-.'I

I

1

I

2

3

4

5

r

6

FIG.11. D-Wave phase function for a potential that is repulsive at short distances and attractive at large distances.

-0.6 -0.8

I\ -

1=O

FIG.12. Phase function for a potential that is repulsive at short distances and attractive at large distances.

CHAPTER

5

equation, which makes Sz’(r) large when k is small, except near the origin, where the other factor that multiplies the potential is small, + ~ . ) remarkable effect at large k is the wiggles of order ( k ~ ) ~ ~Another in the behavior of the phase function; their width is of order rjk (in fact, somewhat less for attractive potentials, more for repulsive potentials), and their origin is clearly related to the periodic vanishing of the factor that multiplies the potential in the r.h.s. of the phase equation. (This is particularly evident in the S-wave case, but is valid in general as soon as kr 2 1 1, so that the Riccati-Bessel functions may be replaced by their asymptotic expressions.) Finally the major role played by Levinson’s theorem, for attractive interactions, in affecting the behavior of the phase function for small k, should be mentioned.

+