- Email: [email protected]
Product representation in effective range theory
Nuc~leur Pht~s/cs A303 (1978) 412-424 ; © Nort/~-Holland Publishing Co., Arnaterdatx
Not to be reproduced by photoprint or microfilm without written permiuion from the publisher
PRODUCT REPRFSENTATION IN EFFECTIVE RANGE THEORY A. DELOFF lnstitute,/br Nuclear Reseurch, Warswr, Poland
Received 13 July 1977 (Revised l5 February 1978) Abstract : The scattering length and the effective range regarded as functions of the potential strength have been represented in ¢n infinite product form . Each factor in the product contributes a pole and a zero, the sequence of poles and zeros being sufTicient to determine the effective range parameters . For six commonly used potentials the nearest singularities have been computed and tabulated . It is shown that with only a few factors excellent accuracy is obtained .
1 . Introduction The scattering length and effective range have been known to provide a convenient means to investigate two-body interactions close to threshold and have been inferred from the experimental data for a variety of processes. Although these parameters bear directly on the strength and range of the underlying interaction, it is in general not easy to establish such a relation. On a phenomenological level one introduces a simple local or non-local potential and by solving the appropriate wave equation evaluates the effective range parameters in terms of the depth and the range of the potential . The extensive calculations made have revealed that for a fixed range the scattering length is a rapidly varying function of the potential depth, going to infinity whenever the strength becomes sufficient to support a zero energy bound state. The effective range exhibits also a singular behaviour. It is due to the presence of poles that most of the attempts to devise approximate procedures to calculate these parameters more directly have failed, except perhaps for very weak forces . For some local potentials frequently used in various calculations tables ofnumerical results are available. The early computations by Blatt and Jackson t) performed because ofthe application in the n-p system have been extended by Levee and Pexton a) to cover a wider range of values of the strength parameter and might be therefore applied directly also in other systems, viz. A-p, E-p, K-p, etc. The numerical results show that the scattering length regarded as a function of the potential depth behaves qualitatively in a very similar manner, irrespective of the shape of the potential. This observation indicates that these two quantities, viz. the scattering length and the potential strength, are tied up by a relationship of a fairly universal character. Since ïn most problems the shape of the potential is not known, the derivation of such a general relation would be of considerable theoretical interest, for different potentials ate
PRODUCT REPRESENTATION
41 3
could be directly compared . In many situations the scattering length remains the only piece of evidence on the low energy interaction and a general formula connecting the scattering length with the interaction strength would be a good starting point for various approximations, e.g. for a given scattering length, one could estimate the strength of the interaction for a wide class of potential shapes . In a search for an analytic expression, de Swart and Dullamond ') suggested for the scattering length a simple one pole term formula which was theft extended to the form pole times a polynomial whose coefficients were fitted to the tabulated values in ref. Z). Such a formula, obtained at the expense of a considerable numerical effort, was quite accurate (to about 1 ~) around the strength values sufficient to bind the 1 s state. For larger values of the strength parameter the accuracy deteriorates and finally the formula necessarily fails in the vicinity of the strength giving the 2s bound state. In principle, the latter limitation may be lifted if the scattering length is sought as an infinite sum of poles, and, in fact, such a formula is well grounded theoretically, e.g. by the Weinberg expansion 4). In practice, however, one has to truncate the series retaining just a few pole terms and it is difficult then to estimate the error introduced by neglecting the "background" . Alternatively, one may try to approximate the "background" by a polynomial at the price of some extra parameters to be adjusted . This procedure does not seem therefore to be practical . In this paper we consider an infinite product formula for the scattering length as a function of the potential depth. This form generalizes the old expression of ref. a) and takes into account an infinite number of poles corresponding to 1 s, 2s, 3s, . . . zero-energy bound states . Since between two neighbouring poles the function necessarily goes through zero, we will also have an infinite number of zeros and the functional dependence is determined by the positions of the interlacing poles and zeros. In most applications the contribution from distant singularities is negligible and with only a few factors an excellent approximation is obtained. A similar product representation is proposed for the effective range. It turns out that the effective range exhibits a second order pole at the zero of the scattering length, except for the zero at the origin where the effective range has a simple pole. The zeros are sought for the difference, effective range minus the intrinsic range. The latter function has a pair of zeros between two neighbouring poles. We illustrate the method by considering six commonly used potentials for which the nearest singularities have been found either analytically, or computed from the wave equation and tabulated. 2. The product representation Our considerations will be based on the quasiparticle method developed by Weinberg a) which is essentially the Hilbert-Schmidt theory applied to the LippmannSchwinger equation. In this method the T-matrix is expanded in terms of the eigenvectors of the kernel of the Lippmann-Schwinger equation. Thus, one has to solve
414
A . DELOFF
the eigenvalue problem for the kernel (W-Ho)-lU~~~i = q~(W)I~ g i,
(1)
where W is a parameter which may take complex values, Ho is the kinetic energy operator, U is the interaction potential, and rl (i~ is the eigenvalue . Assuming a local and central potential U(r), eq . (1) reduces to the quasi-Schrödinger equation - d2 + 1(1+ 1) + ~~dW ; r) = 2pW~~a(W ; r), dr2 rZ 2~U(r)J q ,(W) where ~ is the reduced mass of the system . Consider now scattering from a potential V(r) = sU(r),
(2)
(3)
where s is a dimensionless strength parameter. Since the depth of the potential U(r) may be scaled arbitrarily we have some freedom in the definition of s. We choose to define s as the integral s = =2p ~~ V(rhdr, 0
(4)
assuming that this integral exists . It should be emphasized that this parameter is different from the strength s introduced by Blatt and Jackson 1). Their parameter s has been defined by the requirement that the attractive potential U(r) yields a zeroenergy bound state (ls state) . It is known from potential theory s) that the Jost function in the Ith partial wave F,(-k, s) is identical with the Fredholm determinant (Jost-Pais theorem), i.e. we have where W = k Z/2p. The right-hand side of (5) may now be expressed in terms of the eigenvalues n~(W) restricted to the Ith partial wave b~ evaluating the determinant in the representation that diagonalizes the kernel . One obtains a very important formula which expresses the Jost function in terms of the eigenvalues
The Jost function, which is a function of two variables, has been written in such a form that the dependence on k enters solely via rl , whereas s has been separated out. Since the low energy scattering parameters, viz. the scattering length a and the effective range ro are readily obtained from the Jost function, eq. (6) provides direct means to study the s-dependence of these parameters . The properties of the eigenvalues n ,(Wj have been examined by Weinberg 4). For real negative values of W tha eigenvalues are real and ~l ,(W) are analytic functions
PRODUCT REPRESENTATION
41 5
in the complex W-plane except for a cut along the positive real axis . The discontinuity along this cut is of the W} type. For attractive (repulsive) potentials U(r) and W real and negative, the eigenvalues are always real and positive (negative) . Let us assume that the potential U(r) is attractive and for simplicity confine our attention to s-waves discarding everywhere the index 1. For real negative W close to zero, we can write nn(W) = An+Bn( -2~W)}+Cn(-2~W)+Dn(-2~W) }+ . . .,
(7)
where the coefficients A , B , Cn and Dn have to be real and positive if the nn(W) are to stay positive over the whole range of W from - oo to 0. Additional properties of the eigenvalues may be deduced if the potential V(r) is less singular at zero than r - Z and has a sufficient fall-off at infinity. Specifically, we shall be interested here in the so-called short range potentials for which both the scattering length and the effective range exist. Short range is here taken to mean that for some ~. > 0,
and the "range" is the greatest lower bound of ~. satisfying (8). Weinberg shows a) that for such potentials nn(W) obey the sum rule n= 1
sgn(W) _
-(2~c/k) ~~ V(r)ei~ sin krdr. 0
(9)
This relation gives for the coefficients in (7) the following sum rules : s ~ A n = -2~c ~~ V(r)rdr, n o
(l0a)
s ~ Bn = -2~ ~~ V(r)rZdr, n o
(lOb)
s ~ Cn = - 3~ ~~ V(r)r3dr,
(lOc)
0
s ~ Dn = -~~ ~ V(rh4dr.
(lOd) n 0 Having assembled all the necessary tools, the effective range parameters can be readily obtained by expanding the Jost function in powers of k k cot S =
F(k, s)+F(-k, s) , 1 tk = - - +~rokZ + . . .. F(k, s)-F(-k, s) a
(11)
416
A . DELOFF
Using (6) and (7), one finds a=
r0
-S~ "
B" , 1-sA"
(12)
= 3~a 3 + Y (S)~~aZ+
( 1 3)
where the function Y(s) is defined as z Y(s) - ~ 1 sA" C3D" +3C"B" 1 sA" +Bp \1 sA"~ ~ ~
(14)
As seen from (12), the scattering length regarded as a function of s has an infinite number of poles at s = s" where s" = 1/A" . Thus, all poles are real and positive. Physically, the presence of poles is connected with bound states and at s = s" (n = 1, 2, 3, . . .) the strength of the potential becomes sufficiently large to support a zeroenergy 1 s, 2s, 3s, . . . bound state, respectively . The scattering length a will also have a number of zeros and all of them are real and positive. To prove this suppose that a = 0 for s = a + iß, where a and ß are real numbers. For this to be true, due to (12), following equality should hold : B"
Brr`z "
_
(15)
but since A" and B" are positive, we conclude that ß = 0, which proves our assertion. For s -~ 0, we have a
where C can be obtained from (12) and (lOb) C-~
B"
-
Jo
(16)
-Cs,
V(r)rzdr
I
Jo
V(r)rdr.
(17)
(The above result is of course identical with the first order Born approximation.) Between two neighbouring poles the scattering length decreases monotonically. Indeed, differentiating (12) with respect to s, one finds
Since a is positive at s = s"+e and negative at s = s" + , -e and monotonically decreases, we conclude that there must be always exactly one zero z" between every two neighbouring poles s" and s" + ,. These properties of a lead to the following
PRODUCT REPRESENTATION
417
representation which explicitly exhibits the presence of poles (s,~ and zeros (z,~ a = -sC ~
1 - (s/z")
(19)
As seen from (12) and (19), for repulsive potentials (s < 0) a will be always positive. The convergence of the infinite products in (19) can be easily established. The product in the denominator of (19) is convergent if the series ~ (sls,~ converges. Recalling that s" = 1/A", we have in view of (4) and (l0a) the following sum rule for the inverse of s" (20)
"
which ensures the convergence of the denominator in (19) . The infinite product in the numerator also converges because the series ~, (s/z") does, which immediately follows from the inequality s" < z", setting an upper bound ~,z - R < ~, .s - ,i,. This complets the convergence proof of (19) . Actually, the sum ~ z  may be easily evaluated. To this end it is sufficient to evaluate the contribution to (19) of the order of sZ. But the latter is identical with the second Born approximation and may be directly evaluated. The comparison of the coefficients multiplying sZ yields the following sum rule for the inverse zeros : 1 " z"
2 ~~ V(r)rZdr ~~ V(r')r'dr' ~ V(r)rZdr ~ V(r')rdr' 0
0
In practical applications the infinite product in (19) will be approximated by the first few factors, which in most cases should provide sufficient accuracy. The proposed method works rather well because of the fast convergence of the infinite product. Indeed, it can be proved e) that for large n the s" increase like nZ. This may be also seen by using the WKB approximation to determine the strength s" necessary to support a zero-energy nth excited level. The strength may be obtained from the equation J
rl [- 2pV(r) -(l+Z) Z/rZ]~dr = ~(n-i)-rzl,
(22)
where rl , a are the usual turning points . For l = 0 and a sufficiently large value of the strength, the centrifugal term may be neglected and the resulting integral will be proportional to s~. Thus, eq . (22) leads to the large-n behaviour s" = const x n2. Let us turn now to the effective range. Note that even though both Y(s) and a are singular at s = s", the effective range will be finite at these points . Taking the limit
418
A . DELOFF
s -" s" from (13) one obtains '~~."
" #~ A" -A~ B~ The value of ro at s = s, is called the intrinsic range of the potential and has been customarily denoted by b. Thus, the intrinsic range b is given by the formula
The zeros ofa at s = z" in general lead to a second order pole in r o , as seen from (13) . An exception is the zero at s = 0 which gives a simple pole in ro. In the limit s -~ 0, using (12}{ 14), one obtains where the constants D and E are both positive : D
2 ~ D" "
=
"
(~B") z
=
2 3
~~ V(r)rdr ~~ V(~~4dr' 0
0
~~~ V(r)rzdr /z 00
2 ~ (D"A" + C"B") "
~
(26)
(12)
The first term in (25) is identical with the first order Born approximation. In contrast with the properties ofajust discussed, the effective range for s > 0 may have complex zeros (actually, for most potentials of practical interest ro stays positive for s > 0 and ro will have no real zeros in this region.) This behaviour leads to the following product representation D
°° 1+a s+ß sz
(28)
where a" and ß" are real coefficients . For negative s close to zero the effective range is negative and eq. (25) indicates that there will be a real zero in ro for small negative values ofs. The other zeros may be real or complex depending on the detailed shape of the potential. The series ~a" and ~ß" both converge and may be easily summed by expanding (28) in powers of s and then comparing term by term with the appropriate perturbative expansion of ro. With this result the convergence of the infinite product in formula (28) immediately follows. The complex zeros make formula (28) somewhat unwieldly and this has led us to investigate the difference ro -bin the hope that the latter expression no longer has complex roots. In the examples considered in sect. 3 this guess was indeed found to
PRODUCT REPRESENTATION
41 9
be correct but we have not been able to prove that ro - bin a general case will have only real zeros. Thus, guided solely by model considerations we suggest in place of (28) the following formula : (29) where x and y are real and x <_ z <_ y . The root x o is negative and its presence follows from the small-s behaviour. As mentioned above, some years ago de Swart and Dullamond 3) suggested on phenomenological grounds two simple formulae for a and r o which, with the notation and conventions used in this paper, are a b-
s/sl q(s), 1 - s/sl
h = 1- Q(s) ~ ,
(30) (31)
where q(s) and Q(s) were assumed to be slowly varying functions of s. Comparing (19) and (29) with (30) and (31), respectively, we obtain for q(s) and Q(s) the following expressions
Q(s) = bz C1-
Cs1 ~
s ~ °° 1- s/z~
xolCl
s i l ~ (1 - s/z xl -s/s~)
(32)
(33)
Thus, as long as s is away from the nearest singularity, i.e. for s < z,, q(s) and Q(s) will be slowly varying and may be approximated by polynomials z). To conclude this section a few comments are in order. The scattering length has a very simple structure as a function of s and the behaviour of a is qualitatively the same for all potentials which do not change sign . This is a consequence of the fact that a is determined completely by the first two coefficients in the expansion (~. The structure of r o is considerably more complicated and qualitatively the behaviour of r o as a function of s may vary a great deal for different potentials. This is not surprising since the number of parameters is doubled and ro is specified by the first four parameters in the expansion (7), which leads to a much richer family of curves . Therefore we could make much less definite predictions concerning the behaviour of ro without specifying the potential. The results of thi$ section may be immediately extended to higher l-values . The generalized effective range expansion has the form kz~+ 1cot8, _ - a-1 + ~ roks + . . .
420
A. DELOFF
and expressions (19) and (29) remain valid. Of course, for l > 0 the parameters a and ro will no longer be of the dimension of length, which is directly seen from the generalized expressions for C and D, __
1
J~
V(r)rz' + zdr 0
~~ V(r)rz~ +
0
J0
°° V(r)rzi+zdr~ z CJo
(35)
3. Examples In this section we shall illustrate the method suggested in the preceding section by investigating six commonly used potentials characterised by two parameters, viz. the depth and the range. Since the range can be used to scale the unit of length, the strength s remains the only variable and the poles and zeros becomejust pure numbers characteristic of a given shape . Consequently, the task of finding a and ro as functions of s boils down to calculating a few numbers. Of course, the method should work equally well when the potential contains more than two parameters (e.g. SaxonWoods shape), only in the latter case the s , z , etc. become functions of the various parameters specifying the shape of the potential . In particular, the hard core potentials belong in general to this category, i.e. the poles and zeros may depend upon the core radius . In the following we choose to consider potentials of the following shapes : (i) square well (SW), (ü) cut-off Coulomb (CC), (iii) exponential (EX), (iv) Hulthén (HU), (v) Yukawa (YU), and (vi) Gaussian (GA). For the first two (SW and CC) analytic expressions are available for a, ro, s and z for all l. For the second pair (EX and HU) only the ! = 0 analytic solution exists from which a, ro and s may be obtained . Finally, for the third pair (YU and GA) no analytic result is available and everything has to becomputed numerically. Denoting the range and depth parameters by R and Vo, respectively, the above potentials can be written as V(r) _ - Vof (x~
(36)
where x = r/R and ~(x) is a dimensionless function specifying the shape of the potential. In table 1 we have listed the various forms of f(x) for the potentials considered together with their s-values . We shall now present, whenever possible, the analytic expressions for a and ro obtained from the appropr}ate solution of the wave equation.
PRODUCT' REPRESENTATION
42 1
TABLE I Values of the various parameters defined in the text _
!( .r) .. s C/R s `n
D/R
SW -I .) . . .
~(x p Vo R z i énz (2n z - ~) 2~312,n
s
CC aC =~~/c 2N Vo R z ]
~~ô~. .
EX
HU
YU
GA
exp(-x) 2~e Vo R z 2
[exp( .r)-I ]-' ~n z p Vo R z 12s(3)/n z bn z z
exp(-x)/x 2~ Vo R z 1
exp(- .r z ) Fi Vo R z # rz irz
4
~zt(5)/{(3)z
4
2/n i~z
âJô.~
4lz,q
i
Here 0(x) is the step function, {(.r) is Riemann zeta function, j,, . is the nth root of the Bessel function J,(r) .
(i) The effective range parameters for the SW potential are
a
.lt(~)
R = -1-t( ro _ R 1
1R 2 3 aZ
2s),
1 R
2s a '
(37) (38)
where j,(x) is the spherical Bessel function . Using the familar infinite product representation for the Bessel functions formula (37) takes the form (19) and one obtains immediately the s and z expressed in terms of the zeros of the Bessel functions. They are listed in table 1 where ja, denotes the nth zero of the Ja(x) Bessel function . (ü) For the CC potential, one Rnds a _
R
JZ(2s~) }) , Jo(~
ro _4 R 3
2 3s
2R 3a'
and similarly as before, the poles and zeros obtained from the product form of the Bessel functions are given in table 1 . Note, that ro will have a first order pole at s = z and a term RZ/a2 does not occur. (iii) For the EX potential the exact formulae for a and ro are
I~
J o(2s~)
a LA - 2B a~ ,
(42)
A. DELOFF
422
A=
n
1 (it .)
~
(11 .)
n=t n
S~ _ ~ m-~,
j =
n.= t
1, 2, 3, . . .,
where y is Euler's constant and Yo(x) is the Bessel function of the second kind . The poles sn are immediately found as zeros of the denominator Jo (2s#), but the zeros zn have to be computed numerically. (iv) Finally, for the HU potential, one has a/R = 2y + ~(1 + 6s/n)+ ~(1- 6s/a), o__ R 3 R
_
2
3 (a)
~
~
C(3)-
~
n=t
s > 0,
zl CnZ- 6s/n /
(43)
3
where ~i(x) is the logarithmic derivative of the gamma function . The poles of a are identical with those of the last term in (43) and the values of s n are given in table 1 . The zeros have to be computed numerically. Taa~
2
The values of the nearest poles and zeros in the scattering length formula (19) for various potentials
s,
z,
sZ z~
s3 zj
s,
SW
CC
EX
HU
YU
GA
1 .233701 10.09536 11 .10331 29 .83976 30.84252 59 .44934 60.45132
1.445796 6.593648 7.617815 17 .71125 18 .72175 33 .75521 34 .76007
1 .445796 5.502608 7.617815 15 .91830 18 .72175 31 .31950 34.76007
1 .644934 4.703272 6.579736 12 .44660 14 .80441 23 .50597 26 .31189
1.6799 4.4671 6.4504 11 .852 14 .364 22 .441 25 .457
1 .3420 7.0542 8.8979 20 .399 22 .787 40 .093 42 .982
TABLE 3
The valubs of the nearest zeros in the effective range formula (29) for ditïerent potentials SW b/R
1 .0
xa x,
- oo - oo
x2
s~
y,
p2
x3 y3
10 .983
CC
EX
HU
YU
GA
0.872226
3.54079 -86.63 4 .6579 9.7179 14 .336 23 .505 28 .979 42 .727
3.0 -69.27 4.0148 8.1002 11 .220 I8 .150 21 .719 31 .807
2.12016 -62.27 3.7107 7.9248 10.485 17 .611 20 .409 30 .913
1 .43523 -439 .3 6.4960 11 .020 19 .328 27 .397 38 .541 50 .451
- oo z,
8.8154
z2
30 .796
20.364
60 .427
36 .625
s,
z,
PRODUCT REPRESENTATION
423
The above expressions (37}{44) may be readily continued to negative values of s (repulsive forces) by means of the substitution s} -~ i(-s)~ . In order to examine the rate of convergence of the expressions (19) and (29) and also to facilitate the computation of the scattering length and the effective range, we present in tables 2 and 3 the values of the nearest poles and zeros. In most problems the values of the strength parameter s lie around s1 or below and in this case formulae (19) and (29) with the parameters listed in our tables ensure an accuracy of about 0.3 ~, or better, and, correspondingly about 3 ~ for s-values around s3 . Comparing the different entries in table 2 we can see that, in accordance with our previous assertion, the s values grow roughly as nZ, or even faster. The effective ranges for the potentials considered do not show a unique structure, unlike the scattering length . As mentioned above, for the CC potential, owing to cancellations (x = z,~, the effective range will have only first order poles at s = z . For the remaining potentials ro has a second order pole at s = z . The SW potential, however, goes to - oo for s -~ z , whereas for EX, YU, HU and GA, the effective range tends to + oo for s -" z and never becomes negative for s > 0. When one retains only the first term in the infinite product in (19) (which still gives about 3 ~ accuracy for s around s l), the relation (19) may be readily inverted to give s as a function of a/R. Hence, for a given shape one can immediately express the potential depth in terms of the scattering length which may be important in problems where the latter is the only datum and one has to construct a potential. 4. Conclusions We have derived product formulae for the scattering length and the effective range. This representation has several advantageous features. (i) The formulae obtained provide a simple means to envisage the behaviour of a and ro as functions of the potential strength without specifying the shape of the potential. The effective range parameters are determined by the location of the corresponding poles and zeros. (ü) Owing to a fast convergence the task of finding a and ro in practice reduces to the evaluation of a few constants. The proposed representation may be also looked at as an interpolation formula. (iii) With expressions (19) and (29) and tables 1-3 for six widely used potentials we have generalized the previous results tabulated by Levee and Pexton Z). Furthermore, we provide justification for the old formulae of de Swart and Dullamond 3) . We have also extended the accuracy and the range of applicability. (iv) The product formulae (19) and (29) may be readily inverted and then used to evaluate the strength of the forces for a given value of the low energy scattering parameters.
424
A. DELOFF
of the potential in the product formulae is represented by a number of constants (s , z , etc:). In practice, at the expense of a few constants we can examine the behaviour of a and ro for a wide class of potential shapes . (v) The shape
References I) 2) 3) 4) 5) 6)
J. M. Blatt and J . D. Jackson, Phys . Rev. 76 (1949) 18 R. D. Levee and R. L. Pexton, Nucl. Phys . 55 (1964) 34 J. l. de Swart and C. Dullamond, Ann. of Phys . 19 (1962) 458 S. Weinberg, Phys . Rev. 131 (1963) 440 R. G. Newton, Scattering theory of waves and particles (McGraw-Hill, New York, 1966) K. Chadan, Nuovo Cim. ASS (1968) 191
Not to be reproduced by photoprint or microfilm without written permiuion from the publisher
PRODUCT REPRFSENTATION IN EFFECTIVE RANGE THEORY A. DELOFF lnstitute,/br Nuclear Reseurch, Warswr, Poland
Received 13 July 1977 (Revised l5 February 1978) Abstract : The scattering length and the effective range regarded as functions of the potential strength have been represented in ¢n infinite product form . Each factor in the product contributes a pole and a zero, the sequence of poles and zeros being sufTicient to determine the effective range parameters . For six commonly used potentials the nearest singularities have been computed and tabulated . It is shown that with only a few factors excellent accuracy is obtained .
1 . Introduction The scattering length and effective range have been known to provide a convenient means to investigate two-body interactions close to threshold and have been inferred from the experimental data for a variety of processes. Although these parameters bear directly on the strength and range of the underlying interaction, it is in general not easy to establish such a relation. On a phenomenological level one introduces a simple local or non-local potential and by solving the appropriate wave equation evaluates the effective range parameters in terms of the depth and the range of the potential . The extensive calculations made have revealed that for a fixed range the scattering length is a rapidly varying function of the potential depth, going to infinity whenever the strength becomes sufficient to support a zero energy bound state. The effective range exhibits also a singular behaviour. It is due to the presence of poles that most of the attempts to devise approximate procedures to calculate these parameters more directly have failed, except perhaps for very weak forces . For some local potentials frequently used in various calculations tables ofnumerical results are available. The early computations by Blatt and Jackson t) performed because ofthe application in the n-p system have been extended by Levee and Pexton a) to cover a wider range of values of the strength parameter and might be therefore applied directly also in other systems, viz. A-p, E-p, K-p, etc. The numerical results show that the scattering length regarded as a function of the potential depth behaves qualitatively in a very similar manner, irrespective of the shape of the potential. This observation indicates that these two quantities, viz. the scattering length and the potential strength, are tied up by a relationship of a fairly universal character. Since ïn most problems the shape of the potential is not known, the derivation of such a general relation would be of considerable theoretical interest, for different potentials ate
PRODUCT REPRESENTATION
41 3
could be directly compared . In many situations the scattering length remains the only piece of evidence on the low energy interaction and a general formula connecting the scattering length with the interaction strength would be a good starting point for various approximations, e.g. for a given scattering length, one could estimate the strength of the interaction for a wide class of potential shapes . In a search for an analytic expression, de Swart and Dullamond ') suggested for the scattering length a simple one pole term formula which was theft extended to the form pole times a polynomial whose coefficients were fitted to the tabulated values in ref. Z). Such a formula, obtained at the expense of a considerable numerical effort, was quite accurate (to about 1 ~) around the strength values sufficient to bind the 1 s state. For larger values of the strength parameter the accuracy deteriorates and finally the formula necessarily fails in the vicinity of the strength giving the 2s bound state. In principle, the latter limitation may be lifted if the scattering length is sought as an infinite sum of poles, and, in fact, such a formula is well grounded theoretically, e.g. by the Weinberg expansion 4). In practice, however, one has to truncate the series retaining just a few pole terms and it is difficult then to estimate the error introduced by neglecting the "background" . Alternatively, one may try to approximate the "background" by a polynomial at the price of some extra parameters to be adjusted . This procedure does not seem therefore to be practical . In this paper we consider an infinite product formula for the scattering length as a function of the potential depth. This form generalizes the old expression of ref. a) and takes into account an infinite number of poles corresponding to 1 s, 2s, 3s, . . . zero-energy bound states . Since between two neighbouring poles the function necessarily goes through zero, we will also have an infinite number of zeros and the functional dependence is determined by the positions of the interlacing poles and zeros. In most applications the contribution from distant singularities is negligible and with only a few factors an excellent approximation is obtained. A similar product representation is proposed for the effective range. It turns out that the effective range exhibits a second order pole at the zero of the scattering length, except for the zero at the origin where the effective range has a simple pole. The zeros are sought for the difference, effective range minus the intrinsic range. The latter function has a pair of zeros between two neighbouring poles. We illustrate the method by considering six commonly used potentials for which the nearest singularities have been found either analytically, or computed from the wave equation and tabulated. 2. The product representation Our considerations will be based on the quasiparticle method developed by Weinberg a) which is essentially the Hilbert-Schmidt theory applied to the LippmannSchwinger equation. In this method the T-matrix is expanded in terms of the eigenvectors of the kernel of the Lippmann-Schwinger equation. Thus, one has to solve
414
A . DELOFF
the eigenvalue problem for the kernel (W-Ho)-lU~~~i = q~(W)I~ g i,
(1)
where W is a parameter which may take complex values, Ho is the kinetic energy operator, U is the interaction potential, and rl (i~ is the eigenvalue . Assuming a local and central potential U(r), eq . (1) reduces to the quasi-Schrödinger equation - d2 + 1(1+ 1) + ~~dW ; r) = 2pW~~a(W ; r), dr2 rZ 2~U(r)J q ,(W) where ~ is the reduced mass of the system . Consider now scattering from a potential V(r) = sU(r),
(2)
(3)
where s is a dimensionless strength parameter. Since the depth of the potential U(r) may be scaled arbitrarily we have some freedom in the definition of s. We choose to define s as the integral s = =2p ~~ V(rhdr, 0
(4)
assuming that this integral exists . It should be emphasized that this parameter is different from the strength s introduced by Blatt and Jackson 1). Their parameter s has been defined by the requirement that the attractive potential U(r) yields a zeroenergy bound state (ls state) . It is known from potential theory s) that the Jost function in the Ith partial wave F,(-k, s) is identical with the Fredholm determinant (Jost-Pais theorem), i.e. we have where W = k Z/2p. The right-hand side of (5) may now be expressed in terms of the eigenvalues n~(W) restricted to the Ith partial wave b~ evaluating the determinant in the representation that diagonalizes the kernel . One obtains a very important formula which expresses the Jost function in terms of the eigenvalues
The Jost function, which is a function of two variables, has been written in such a form that the dependence on k enters solely via rl , whereas s has been separated out. Since the low energy scattering parameters, viz. the scattering length a and the effective range ro are readily obtained from the Jost function, eq. (6) provides direct means to study the s-dependence of these parameters . The properties of the eigenvalues n ,(Wj have been examined by Weinberg 4). For real negative values of W tha eigenvalues are real and ~l ,(W) are analytic functions
PRODUCT REPRESENTATION
41 5
in the complex W-plane except for a cut along the positive real axis . The discontinuity along this cut is of the W} type. For attractive (repulsive) potentials U(r) and W real and negative, the eigenvalues are always real and positive (negative) . Let us assume that the potential U(r) is attractive and for simplicity confine our attention to s-waves discarding everywhere the index 1. For real negative W close to zero, we can write nn(W) = An+Bn( -2~W)}+Cn(-2~W)+Dn(-2~W) }+ . . .,
(7)
where the coefficients A , B , Cn and Dn have to be real and positive if the nn(W) are to stay positive over the whole range of W from - oo to 0. Additional properties of the eigenvalues may be deduced if the potential V(r) is less singular at zero than r - Z and has a sufficient fall-off at infinity. Specifically, we shall be interested here in the so-called short range potentials for which both the scattering length and the effective range exist. Short range is here taken to mean that for some ~. > 0,
and the "range" is the greatest lower bound of ~. satisfying (8). Weinberg shows a) that for such potentials nn(W) obey the sum rule n= 1
sgn(W) _
-(2~c/k) ~~ V(r)ei~ sin krdr. 0
(9)
This relation gives for the coefficients in (7) the following sum rules : s ~ A n = -2~c ~~ V(r)rdr, n o
(l0a)
s ~ Bn = -2~ ~~ V(r)rZdr, n o
(lOb)
s ~ Cn = - 3~ ~~ V(r)r3dr,
(lOc)
0
s ~ Dn = -~~ ~ V(rh4dr.
(lOd) n 0 Having assembled all the necessary tools, the effective range parameters can be readily obtained by expanding the Jost function in powers of k k cot S =
F(k, s)+F(-k, s) , 1 tk = - - +~rokZ + . . .. F(k, s)-F(-k, s) a
(11)
416
A . DELOFF
Using (6) and (7), one finds a=
r0
-S~ "
B" , 1-sA"
(12)
= 3~a 3 + Y (S)~~aZ+
( 1 3)
where the function Y(s) is defined as z Y(s) - ~ 1 sA" C3D" +3C"B" 1 sA" +Bp \1 sA"~ ~ ~
(14)
As seen from (12), the scattering length regarded as a function of s has an infinite number of poles at s = s" where s" = 1/A" . Thus, all poles are real and positive. Physically, the presence of poles is connected with bound states and at s = s" (n = 1, 2, 3, . . .) the strength of the potential becomes sufficiently large to support a zeroenergy 1 s, 2s, 3s, . . . bound state, respectively . The scattering length a will also have a number of zeros and all of them are real and positive. To prove this suppose that a = 0 for s = a + iß, where a and ß are real numbers. For this to be true, due to (12), following equality should hold : B"
Brr`z "
_
(15)
but since A" and B" are positive, we conclude that ß = 0, which proves our assertion. For s -~ 0, we have a
where C can be obtained from (12) and (lOb) C-~
B"
-
Jo
(16)
-Cs,
V(r)rzdr
I
Jo
V(r)rdr.
(17)
(The above result is of course identical with the first order Born approximation.) Between two neighbouring poles the scattering length decreases monotonically. Indeed, differentiating (12) with respect to s, one finds
Since a is positive at s = s"+e and negative at s = s" + , -e and monotonically decreases, we conclude that there must be always exactly one zero z" between every two neighbouring poles s" and s" + ,. These properties of a lead to the following
PRODUCT REPRESENTATION
417
representation which explicitly exhibits the presence of poles (s,~ and zeros (z,~ a = -sC ~
1 - (s/z")
(19)
As seen from (12) and (19), for repulsive potentials (s < 0) a will be always positive. The convergence of the infinite products in (19) can be easily established. The product in the denominator of (19) is convergent if the series ~ (sls,~ converges. Recalling that s" = 1/A", we have in view of (4) and (l0a) the following sum rule for the inverse of s" (20)
"
which ensures the convergence of the denominator in (19) . The infinite product in the numerator also converges because the series ~, (s/z") does, which immediately follows from the inequality s" < z", setting an upper bound ~,z - R < ~, .s - ,i,. This complets the convergence proof of (19) . Actually, the sum ~ z  may be easily evaluated. To this end it is sufficient to evaluate the contribution to (19) of the order of sZ. But the latter is identical with the second Born approximation and may be directly evaluated. The comparison of the coefficients multiplying sZ yields the following sum rule for the inverse zeros : 1 " z"
2 ~~ V(r)rZdr ~~ V(r')r'dr' ~ V(r)rZdr ~ V(r')rdr' 0
0
In practical applications the infinite product in (19) will be approximated by the first few factors, which in most cases should provide sufficient accuracy. The proposed method works rather well because of the fast convergence of the infinite product. Indeed, it can be proved e) that for large n the s" increase like nZ. This may be also seen by using the WKB approximation to determine the strength s" necessary to support a zero-energy nth excited level. The strength may be obtained from the equation J
rl [- 2pV(r) -(l+Z) Z/rZ]~dr = ~(n-i)-rzl,
(22)
where rl , a are the usual turning points . For l = 0 and a sufficiently large value of the strength, the centrifugal term may be neglected and the resulting integral will be proportional to s~. Thus, eq . (22) leads to the large-n behaviour s" = const x n2. Let us turn now to the effective range. Note that even though both Y(s) and a are singular at s = s", the effective range will be finite at these points . Taking the limit
418
A . DELOFF
s -" s" from (13) one obtains '~~."
" #~ A" -A~ B~ The value of ro at s = s, is called the intrinsic range of the potential and has been customarily denoted by b. Thus, the intrinsic range b is given by the formula
The zeros ofa at s = z" in general lead to a second order pole in r o , as seen from (13) . An exception is the zero at s = 0 which gives a simple pole in ro. In the limit s -~ 0, using (12}{ 14), one obtains where the constants D and E are both positive : D
2 ~ D" "
=
"
(~B") z
=
2 3
~~ V(r)rdr ~~ V(~~4dr' 0
0
~~~ V(r)rzdr /z 00
2 ~ (D"A" + C"B") "
~
(26)
(12)
The first term in (25) is identical with the first order Born approximation. In contrast with the properties ofajust discussed, the effective range for s > 0 may have complex zeros (actually, for most potentials of practical interest ro stays positive for s > 0 and ro will have no real zeros in this region.) This behaviour leads to the following product representation D
°° 1+a s+ß sz
(28)
where a" and ß" are real coefficients . For negative s close to zero the effective range is negative and eq. (25) indicates that there will be a real zero in ro for small negative values ofs. The other zeros may be real or complex depending on the detailed shape of the potential. The series ~a" and ~ß" both converge and may be easily summed by expanding (28) in powers of s and then comparing term by term with the appropriate perturbative expansion of ro. With this result the convergence of the infinite product in formula (28) immediately follows. The complex zeros make formula (28) somewhat unwieldly and this has led us to investigate the difference ro -bin the hope that the latter expression no longer has complex roots. In the examples considered in sect. 3 this guess was indeed found to
PRODUCT REPRESENTATION
41 9
be correct but we have not been able to prove that ro - bin a general case will have only real zeros. Thus, guided solely by model considerations we suggest in place of (28) the following formula : (29) where x and y are real and x <_ z <_ y . The root x o is negative and its presence follows from the small-s behaviour. As mentioned above, some years ago de Swart and Dullamond 3) suggested on phenomenological grounds two simple formulae for a and r o which, with the notation and conventions used in this paper, are a b-
s/sl q(s), 1 - s/sl
h = 1- Q(s) ~ ,
(30) (31)
where q(s) and Q(s) were assumed to be slowly varying functions of s. Comparing (19) and (29) with (30) and (31), respectively, we obtain for q(s) and Q(s) the following expressions
Q(s) = bz C1-
Cs1 ~
s ~ °° 1- s/z~
xolCl
s i l ~ (1 - s/z xl -s/s~)
(32)
(33)
Thus, as long as s is away from the nearest singularity, i.e. for s < z,, q(s) and Q(s) will be slowly varying and may be approximated by polynomials z). To conclude this section a few comments are in order. The scattering length has a very simple structure as a function of s and the behaviour of a is qualitatively the same for all potentials which do not change sign . This is a consequence of the fact that a is determined completely by the first two coefficients in the expansion (~. The structure of r o is considerably more complicated and qualitatively the behaviour of r o as a function of s may vary a great deal for different potentials. This is not surprising since the number of parameters is doubled and ro is specified by the first four parameters in the expansion (7), which leads to a much richer family of curves . Therefore we could make much less definite predictions concerning the behaviour of ro without specifying the potential. The results of thi$ section may be immediately extended to higher l-values . The generalized effective range expansion has the form kz~+ 1cot8, _ - a-1 + ~ roks + . . .
420
A. DELOFF
and expressions (19) and (29) remain valid. Of course, for l > 0 the parameters a and ro will no longer be of the dimension of length, which is directly seen from the generalized expressions for C and D, __
1
J~
V(r)rz' + zdr 0
~~ V(r)rz~ +
0
J0
°° V(r)rzi+zdr~ z CJo
(35)
3. Examples In this section we shall illustrate the method suggested in the preceding section by investigating six commonly used potentials characterised by two parameters, viz. the depth and the range. Since the range can be used to scale the unit of length, the strength s remains the only variable and the poles and zeros becomejust pure numbers characteristic of a given shape . Consequently, the task of finding a and ro as functions of s boils down to calculating a few numbers. Of course, the method should work equally well when the potential contains more than two parameters (e.g. SaxonWoods shape), only in the latter case the s , z , etc. become functions of the various parameters specifying the shape of the potential . In particular, the hard core potentials belong in general to this category, i.e. the poles and zeros may depend upon the core radius . In the following we choose to consider potentials of the following shapes : (i) square well (SW), (ü) cut-off Coulomb (CC), (iii) exponential (EX), (iv) Hulthén (HU), (v) Yukawa (YU), and (vi) Gaussian (GA). For the first two (SW and CC) analytic expressions are available for a, ro, s and z for all l. For the second pair (EX and HU) only the ! = 0 analytic solution exists from which a, ro and s may be obtained . Finally, for the third pair (YU and GA) no analytic result is available and everything has to becomputed numerically. Denoting the range and depth parameters by R and Vo, respectively, the above potentials can be written as V(r) _ - Vof (x~
(36)
where x = r/R and ~(x) is a dimensionless function specifying the shape of the potential. In table 1 we have listed the various forms of f(x) for the potentials considered together with their s-values . We shall now present, whenever possible, the analytic expressions for a and ro obtained from the appropr}ate solution of the wave equation.
PRODUCT' REPRESENTATION
42 1
TABLE I Values of the various parameters defined in the text _
!( .r) .. s C/R s `n
D/R
SW -I .) . . .
~(x p Vo R z i énz (2n z - ~) 2~312,n
s
CC aC =~~/c 2N Vo R z ]
~~ô~. .
EX
HU
YU
GA
exp(-x) 2~e Vo R z 2
[exp( .r)-I ]-' ~n z p Vo R z 12s(3)/n z bn z z
exp(-x)/x 2~ Vo R z 1
exp(- .r z ) Fi Vo R z # rz irz
4
~zt(5)/{(3)z
4
2/n i~z
âJô.~
4lz,q
i
Here 0(x) is the step function, {(.r) is Riemann zeta function, j,, . is the nth root of the Bessel function J,(r) .
(i) The effective range parameters for the SW potential are
a
.lt(~)
R = -1-t( ro _ R 1
1R 2 3 aZ
2s),
1 R
2s a '
(37) (38)
where j,(x) is the spherical Bessel function . Using the familar infinite product representation for the Bessel functions formula (37) takes the form (19) and one obtains immediately the s and z expressed in terms of the zeros of the Bessel functions. They are listed in table 1 where ja, denotes the nth zero of the Ja(x) Bessel function . (ü) For the CC potential, one Rnds a _
R
JZ(2s~) }) , Jo(~
ro _4 R 3
2 3s
2R 3a'
and similarly as before, the poles and zeros obtained from the product form of the Bessel functions are given in table 1 . Note, that ro will have a first order pole at s = z and a term RZ/a2 does not occur. (iii) For the EX potential the exact formulae for a and ro are
I~
J o(2s~)
a LA - 2B a~ ,
(42)
A. DELOFF
422
A=
n
1 (it .)
~
(11 .)
n=t n
S~ _ ~ m-~,
j =
n.= t
1, 2, 3, . . .,
where y is Euler's constant and Yo(x) is the Bessel function of the second kind . The poles sn are immediately found as zeros of the denominator Jo (2s#), but the zeros zn have to be computed numerically. (iv) Finally, for the HU potential, one has a/R = 2y + ~(1 + 6s/n)+ ~(1- 6s/a), o__ R 3 R
_
2
3 (a)
~
~
C(3)-
~
n=t
s > 0,
zl CnZ- 6s/n /
(43)
3
where ~i(x) is the logarithmic derivative of the gamma function . The poles of a are identical with those of the last term in (43) and the values of s n are given in table 1 . The zeros have to be computed numerically. Taa~
2
The values of the nearest poles and zeros in the scattering length formula (19) for various potentials
s,
z,
sZ z~
s3 zj
s,
SW
CC
EX
HU
YU
GA
1 .233701 10.09536 11 .10331 29 .83976 30.84252 59 .44934 60.45132
1.445796 6.593648 7.617815 17 .71125 18 .72175 33 .75521 34 .76007
1 .445796 5.502608 7.617815 15 .91830 18 .72175 31 .31950 34.76007
1 .644934 4.703272 6.579736 12 .44660 14 .80441 23 .50597 26 .31189
1.6799 4.4671 6.4504 11 .852 14 .364 22 .441 25 .457
1 .3420 7.0542 8.8979 20 .399 22 .787 40 .093 42 .982
TABLE 3
The valubs of the nearest zeros in the effective range formula (29) for ditïerent potentials SW b/R
1 .0
xa x,
- oo - oo
x2
s~
y,
p2
x3 y3
10 .983
CC
EX
HU
YU
GA
0.872226
3.54079 -86.63 4 .6579 9.7179 14 .336 23 .505 28 .979 42 .727
3.0 -69.27 4.0148 8.1002 11 .220 I8 .150 21 .719 31 .807
2.12016 -62.27 3.7107 7.9248 10.485 17 .611 20 .409 30 .913
1 .43523 -439 .3 6.4960 11 .020 19 .328 27 .397 38 .541 50 .451
- oo z,
8.8154
z2
30 .796
20.364
60 .427
36 .625
s,
z,
PRODUCT REPRESENTATION
423
The above expressions (37}{44) may be readily continued to negative values of s (repulsive forces) by means of the substitution s} -~ i(-s)~ . In order to examine the rate of convergence of the expressions (19) and (29) and also to facilitate the computation of the scattering length and the effective range, we present in tables 2 and 3 the values of the nearest poles and zeros. In most problems the values of the strength parameter s lie around s1 or below and in this case formulae (19) and (29) with the parameters listed in our tables ensure an accuracy of about 0.3 ~, or better, and, correspondingly about 3 ~ for s-values around s3 . Comparing the different entries in table 2 we can see that, in accordance with our previous assertion, the s values grow roughly as nZ, or even faster. The effective ranges for the potentials considered do not show a unique structure, unlike the scattering length . As mentioned above, for the CC potential, owing to cancellations (x = z,~, the effective range will have only first order poles at s = z . For the remaining potentials ro has a second order pole at s = z . The SW potential, however, goes to - oo for s -~ z , whereas for EX, YU, HU and GA, the effective range tends to + oo for s -" z and never becomes negative for s > 0. When one retains only the first term in the infinite product in (19) (which still gives about 3 ~ accuracy for s around s l), the relation (19) may be readily inverted to give s as a function of a/R. Hence, for a given shape one can immediately express the potential depth in terms of the scattering length which may be important in problems where the latter is the only datum and one has to construct a potential. 4. Conclusions We have derived product formulae for the scattering length and the effective range. This representation has several advantageous features. (i) The formulae obtained provide a simple means to envisage the behaviour of a and ro as functions of the potential strength without specifying the shape of the potential. The effective range parameters are determined by the location of the corresponding poles and zeros. (ü) Owing to a fast convergence the task of finding a and ro in practice reduces to the evaluation of a few constants. The proposed representation may be also looked at as an interpolation formula. (iii) With expressions (19) and (29) and tables 1-3 for six widely used potentials we have generalized the previous results tabulated by Levee and Pexton Z). Furthermore, we provide justification for the old formulae of de Swart and Dullamond 3) . We have also extended the accuracy and the range of applicability. (iv) The product formulae (19) and (29) may be readily inverted and then used to evaluate the strength of the forces for a given value of the low energy scattering parameters.
424
A. DELOFF
of the potential in the product formulae is represented by a number of constants (s , z , etc:). In practice, at the expense of a few constants we can examine the behaviour of a and ro for a wide class of potential shapes . (v) The shape
References I) 2) 3) 4) 5) 6)
J. M. Blatt and J . D. Jackson, Phys . Rev. 76 (1949) 18 R. D. Levee and R. L. Pexton, Nucl. Phys . 55 (1964) 34 J. l. de Swart and C. Dullamond, Ann. of Phys . 19 (1962) 458 S. Weinberg, Phys . Rev. 131 (1963) 440 R. G. Newton, Scattering theory of waves and particles (McGraw-Hill, New York, 1966) K. Chadan, Nuovo Cim. ASS (1968) 191