Distorted wave expansion for scattering problems

2.1L

A urlear Physics 27 (1961) 620---631 ; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

AVE EXPANSION FOR SCATTERING PROBLEMS P. SWAN Department of Physics, University of Melbourne, Australia Received 17 April 196''. Abstract : The properties of a distorted wave method for the scattering wave function Vj(rj and phase-shift 61 are examined . For 16,1 _ 16, > in, whereas the Born series are not. ß It is concluded that VI M (r) and tan(') 6 1 are approximate summations of the corresponding Born series for 1611 < in, and therefore include contributions from the Born expansions of all orders . The summations for pa(") (r) and tan 6j (") are also valid for weak potentials for n > 1611 > In, having the character of Born expansions in inverse powers . The method is illustrated on zero energy neutron-proton scattering .

Introduction

distorted wave method, giving simple analytical expressions for the elastic scattering wave function ip,(r) and phase-shift 8a, has been applied to a number of iruclear collision problems 1, 2, 3) . It takes, as basis the trial wave function at-ga(r), (1) VI M (r) = fa(kr)cos 4-1' a(kr)sin 8a , where fl (kr) and Xl (kr) are the regular and irregular plane wave solutions for the wave equation with zero potential. They are related to the spherical Bessel and Neumann functions by the relations

Ir a(x) = xnl (x) = (2nx)INL+I(x),

where the definitions of j a (x) and n,(x) are those given by Schiff 4) . In eq. (7), ga(r) is a form factor, which must satisfy the boundary conditions 9, (r -> 0) ~

r21+1~

1-gt(r -> oo) ^' U (r) -> 0,

(3)

where U(r) = (2y/h2) V (r) is the interaction potential. Results are practically independent of the detailed form ofg, (r), provided conditions (3) are satisf ied i, 2, . s) 62G

8201

DISTORTED WAVE EXPANSION

Comparison of (1) with the asymptotic wave function ya(r) . fi (kr)cos ôa -X,(kr)sin

8a

(~)

shows that V, (°) (r) itself gives no information about the phase-shift 6, . However, it may in effect be iterated to give a value by substitution in the integral : equation for the phase-shifts) sin b a

= - kf

00

U(r)f a(kr)ipa(r)dr,

where the parameter A is a measure of the strength of the interaction AU (r) . This leads to tan 6a(1) = AB1j1( 1-AB2a)) (6) where B = -

k

U (r)

o

a2 (kr) dr,

X a(kr)dr . k o g (r)tJ(r)~a(kr)

B2a =

(7) (8)

Here, Bia is the first Born approximation integral, and B2a an integral measuring the distortion of the wave function due to U(r), and introducing a strong interference effect in 6a ; B2a(k) is a function which initially decreases from a maximum value B2,(0) and oscillates with diminishing amplitude thereafter . That (6) corresponds to an order of approximation one higher than (1) may be seen by the following argument. The wave equation and its associated boundary conditions on VY, (r) maybe expressed in the equivalent integral equation form 6 ) ya(r)

=

fa(kr)cos ô,-,t

Jo dr'G I (r, r')U(r')tpa(r'),

~9)

the Green's function Ga (r, y') being given by

=-

Gi(r, ;"')

k

fa(kr<)X&(kr>),

(1o1

where r< and r> are the smaller and larger of r, r' respectively. We next define the integrals B (r)

_ - 1k

B2, (r) B3 , (r)

=

B., (r) = -

o

dr'U (r' ) o~a2 (kr' )

T

1

ƒ dr'{a(r~)U(r')~a(kr')`4a(kr'), k o I k 1 k

( 11 )

F dr'U(r') ƒa(kr')X,(kr'), r

dr'ga(r')U(r')Xa2(kr') .

(13) (14)

622

P. S'VWAN

Substitution of (1) for V,(0) (r) in the right hand side of (9) gives

va(') (r) =

,(kr)[ { 1+AB3,(r)}cos ól+AB,&a(r)sin ój -,X,(kr)[ÄB  (r)cos da +AB2,(r)sin bj . (15)

From the definitions (11)--(14) and (6), (7), we have B,,(oo) = Bia,

B21( 00 ) =

= 0, Bsa(oo) = B4,(oo)

B2,)

so that the asymptotic form of (15) for large r is -> oo) =

,(kr)cos 6,--.V,(kr)[AB ia cos 6,+AB 2asin áa] .

(16)

Onequating the coefficient of --XI (kr)tosinb a (c.f. (4)),wegeteq . (6) fortanó,(') . Equation (6) for tan a 1(1) has been extensively tested for a number of potential well shapes and t values in the range 0 6, s n, giving an overall accuracy of about 2-3 %, dependent on the care with which the form factor g,(r) is chosen 1. 2,3) . Potentials singular at r = 0 may give somewhat lower accuracy 3) . It is preferable, although not necessary, to (,hoose an arbitrary parameter inside g, (r) so as to give the correct zero-energy scattering length a. The latter may itself be estimated easily and accurately for the potential in question by using a recent variational procedure 7 ) . At low energies, k2 Z 1, eq. (6) reduces to an approximate form of the shape-independent formula 2) k cot 60(1) = A +Bk2+0 (k4),

(17)

where A -, --1/a, B sw 2r® , ro being the effective range of the interaction. A choice of the parameter inside g, (r) to make A = -1 /a thus normalizes the method at zero energy.

its

da (e) Approximation

Validity for the ta

Provided IAB2J < 1, the expression (6) may be expanded in the series tan 6a(i)

-__ íîB1,

«0

Âl` Bea,

n®0

18

which for sufficiently high energy, k2 » JAU(r)l, reduces to the first Born approximation for tan b, :

tan S, % bIB,, (19) (k2 _> 00 ) .

oweeer, interactions singular at the origin, such as the Yukawa potential, do not satisfy the necessary condition k2 » (AU(r)i for r sufficiently small, which accounts for the poor accuracy obtained for the Born approximation both for tan 6, and for the scattering amplitude /(0) with this type of potential. The accuracy of (6) is also affected to a lesser degree 3) .

DISTORTED WAVE EXPANSION

62 3

Eq. (13), although much simpler, bears a sufficient resemblance to the Born expansion of tan 6, to suggest that (6) is a good approximation to the Born series, illustrated strikingly by the example in sect . 4. The known accuracy of (6) supports this hypothesis. In sect. 3 we show that an alternative expansion for vpj (r) using; (1) as initial function to iterate the integral equation (9), reduces in the limit n ->. oo to the Born series, provided 1 6 ,1 < 1 2a, the latter being also convergence of the Born the condition for expansion. Convergence conditions for Born expansioi_r have been dealt with by a number of authors 8-12), both for partial wave and full wave expansions. The condition IAB2z1 < 1 necessary for the validity of the series (16) for tan 6,(1) corresponds to {6,U,1 < -1,n, provided we assume 6t(1) is in the range 0 :-&- 1a1MI x (all k), so that B2a(k) has one zero only. That (1), (6) and (1 .5) are invalid for 16,U) I ® x (k 0), follows from the following arguments : 1) The trial wave function (1) for tV,(o) (r) is non-degenerate only within the region -®x < a =(1) < x. Thus if 1611 > a(k 0), there is no reason why iteration of (1) in (9) should lead to a convergent expansion, since the initial function is a very bad choice. Physically this corresponds to the 'inside' wave function being a distorted `outside' wave function, whereas for 16,1 > 7r, the potential is so strong that the correct 'inside' wave function has a quite different character, coupled with a large wave number. The function 1Y,(®) (r) cannot describe the case of strong distortion even approximately, as the 'inside' wave function is dependent largely on the potential and very little on 6,, except at sufficiently high energies 2) The method breaks down for 16,1 = x (k 0), as tan 6 í (1) = 0 contra dicts B1 , 0 in (6) . However, 16,1 = a (k = 0) is acceptable as in this case ia ® 0. Two-body interactions may be conveniently divided into two main classes : 1) The weak potentials, for which 16, (k)1 < ;r (k 0 0), 61(0) = 0 or n 13) . Examples are the n-p interaction and the equivalent two-body interactions between the very light nuclei, such as e n--ac system. Our formula (6) for tan 6,(1) should always be valid in these cases. 2) The strong potentials, for which 16, (k)1 > ;r (k lt < k < k2t), where k 1 , = 0 for attractive potentials which do not change sign, and k,, =A 0 for repulsive potentials, as 6,(O)= 0 13) . Eq. (6) is clearly invalid for kla < k < k2g in both cases, and also in the latter case for 0 < k < k1a, although 61 may be small, as the 'inside' wave function cannot be represented by the weakly distorted function (1) . Examples include the optical model potentials for interactions between all but the lightest nuclei, where we have by Levinson's theorem 6 a (0)-bt (oo) = na,

(20)

n being the number of composlt~" bound states with non-zero binding energy .

It is usual to define b t (oo) = 0, giving 6, (0) = nn. These potentials may be very strong compared to the n-p interaction, the optical potential between a neutron and the nucleus A -= 125 giving n = 4 for l = 0. Simple arguments based on a square well of depth Ua and width b show that the minimum interaction strength UQ b2 required, for n -= 4 is 49 times that needed for n = 1 (l = 0) . The same arguments show that eq. (6) must fail for strong potentials for bt(i) small contradicts Bit large. ®n the other hand, 1611 = a--Ej, £t < 1, as tan eq. (6) must be valid under Born approximation conditions, k2 » JU (r) l, so that b a is small. An intermediate region exists for which eq. (6) but not the Born appproximation is valid, defined by bi,(k) decreasing monotonically from either 2-a or a maximum value bjkm) towards zero, whichever is the smaller. In the former case, this requires 0,8 2,1 < 1 . However, ABU(k) is a function of energy oscillating with decreasing amplitude, so that AB2t(k) --1 may have a number of zeros say at k = ki, k2 , . . ., kn. The modified condition becomes 1AB21(k > kn) I

< 1.

In the intermediate region eq. (6) is valid in the sense that it is an approximation to the Born series (cf . sect. 3) . However, for l sufficiently large, 6,(k) may be small for all k, although neither the first Born approximation nor eq. (6) are valid except at high energies . The Born and distorted wave series of sect. 3 are valid for all k, but may be very slowly convergent for small k. Kohn 9) has noted that the convergence of Born series for V,(Y) and tan 6,(k) improves with increasing l, so that if eq. (6) is valid for l = 0 for k2 > k' 2, the same condition should suffice to make (6) valid for all t > 1 . Many-body interactions may be classified similarly, although the size of bt is often dependent on symmetry considerations rather than the strength of the potential. In these cases, b t satisfies the theorem ba(0 ) -- bt0o) = (n+m)x,

(21)

where n is the number of composite bound states of non-zero binding energy and m is the number of states excluded by the Bauli principle 14) . Taking bt (oo) = 0, we see that bt (0) and hence b t (k) can have large values even although the potential is weak in the sense that n = 0. Thus electron scattering by the rare gases He, hle, A, Kr and Xe gives bo(0) = a, 2n, 3r, Oar, 5;r respectively, although the inert character of the gases shows that n = 0 in each case, corresponding to the non-existence of negative ions. Instead we have m --- 1, 2, 3, 4 and 5 corresponding to the number of full electron shells (excluded states) . Examples for which 1 b a (k) l > a are treated exactly as if they were streng potentials in category 2) above. The tan b,(i) formula (6) may be used safely only i J B21 (k > kn 1 < 1, where Bet has the modified form given in sect . 4

625

DISTORTED WAVE EXPANSION

of an earlier paper 2) . However, the weak potential cases where 16, (k) (!!!9 a (all k) obey the conditions in category 1) above, so that the tan 6,U) .formula (6) is valid over the whole range (6,(k)1 ;5 n (all k) . . Series

ansions for ip#) an

tan

a

The usual Born expansion of yea (r) and tan 6,, as exemplified in the work of Kohn 2), is based on axe asymptotic wave function normalised to 1/cos 6, at r --> oo, rather than the unit amplitude of (4) . Thus sin 61 changes to tan 6, in the left hand side of the integral equation (5) for 6, and cos 6, is deleted in the integral equation (9) for Vjr) . Iteration of the initial trial function a (kr) leads to the Born series for ?p a (r) and tan 61 , convergent for 16,1 < 12n. The necessary initial assumptions are that V (r) is sectionally continuous, lim r_.0 rV (r) is finite and lim,.~ r2+g V(r) ö 0, s > 0. An alternative but equivalent procedure is to retain the unit asymptotic amplitude for ipa(r) as in (4), with integral equations (5) and (9) unchanged . The initial trial function is then f~ (kr) cos 6,, leading to the Born expansions as before. In particular, the first Born approximation is tan ô,(') = Bli . It is worth noting that with this normalization, if we take 6, as small and use an initial trial function f~(kr), we obtain for the first Born approximation b) sin 6,(') ® BI, . There is no contradiction between the two cases for 6, small, but in the latter case, iteration of the initial function leads to an inconvenient transcendental equation for the nth order approximation of the type sin ô,(") ®

n-1 m-l

AmS,(m)

COS S,(n) +ÂixS,(n)

where the S,(m) are series :summations. the coefficient of cos 6,(n) is in fact the Born series for tan 8án_1), so that in the limit n -> oo and for (6,1 < -12.-a (condition for convergence), we have AnS,(n) -} 0, laving the Born series for tan 6, (n) . However, from (5), ft (kr) cos 6= is clearly a better initial function to use than jf$(kr) for 6, not small, so it can be surmised. that results for qn) are appreciably better for the former choice for n small . It is evident that a series expansion for ipa(r) can be constructed in the same way as the Born series, but using the distorted wave (1) as initial function in iterating (6) . As (1) is a much better trial function than 'a (kr) cos 61, we may expect greatly improved results, at least for low n values. We define the

626

S . SNVAN

following multiple integrals for m z 1 : .,_. la,aa... ., (r) =

1 k

® dr'B a,a~...aa (r')U(r')

24,as. ..a 1(r) =

Sa,aq. ..a 1(r)

B4a,a,...«mg (r)

a,a, . . .a~,z(r')U(r')

=

a2 (kr') ,

(22 )

a(kr')~1(kr')~

(23 )

a,a s . . .a ,(r')ZT (r')

= -- 1 ƒ00 dr'Ba a,...a

I(kr')X1(kr'),

,(r')U(r')Xa2(kr'),

(24)

(25)

where a,,(v = 1, 2, . . ., m) may have any of the v; dues 1, 2, 3 or 4. Eqs. (11)-(14) and (22)-(25) completely specify the expansion for V,(n) (r), while that for tan 6 1(n) is covered by defining Bla,a s. . .a,,i( oo ) Bsa,ag . . .am

1(oo) =

Bla,a s . . .a 8~

B2a,as...a,»z(oo)

= ®.

`

B2ae,aa .. .a~~~t~

It is worth noting that (22)-(25) are functions of the form factor ga (r) only if the last index number a, has the value 2 or 4. Expressions for the first few 6 1 01) are listed below: V1(n) (r) and tan 1(°) (r) ys (l '(r)

= 1,(kr)cos at --gl(i')X1(kr)sin d1, = [ { 1 +ÂB3x(r)}cos ô1+AB41(r)sin ôj

(Zß) z(kr)-[AB,,(r)cos al -} ®AB 2z (r)sin S 1 ]X 1 (kr),

v1 M (r)

= [ {1+AB31(r)+Ä2(B33E(r)+B411(r » }cos

(27)

ó,+Ä2{B421(r) +B34a(r)}sin 8,]

=(kr)

- [ {ÄBli(r)+A2 (B13i(r)+B219(r))}COS ót+a 2 {B22t(r) +Bi41(r)}Sin ó 1 ]X,(kr), V& W(r)

=

(28)

[{1+AB3i(r)+A2(B33E(r)+B41i(r) ) +As(B133ä(r)+B341Z(r)+B413E(r) +B4211(r » }COs al + Ä3 {B4221(r) +Bs42i(r)+B414i(r)+B,3 (r )}Sin ó=] ,OF 8(kr) -[ {AB, ,(r) +A2 (B131(r) +B21, (r» +Ä3 (B1sss (r) +B1411(r) +B213Z(r) 3 2211(r))}COS Sa+A {B2221(r)+B1421(r)+BV41(r)

tan ó 1 ( 1 D tan&,( 2 )

= =

[A

1A1- A ., , +A2(

1341(r)}sin d 1 ] X 1 (kr),

(29) (30)

131+B211)]/[ 1-12 (B221+BI,A1)] ,

(31)

DISTORTED WAVE EXPANSION

tan á1(3)

=

(A

11+A2 (

131+

219)+A

3(

1331+

1411+B2131

2211)]/[ 1- A3 (

tan á1(4D

=

[A

2 11+A (

627

2221+B1421+B2141+

131'+B311)+As(13,33,+B1411+

2131+

2211)+ 14 ( 13411+

21331-1`

21411+

1341)],

(3 2 )

13331

14131+

22131+ 22211)]/ [ 1®A4 (B22221+B14221+B21421 +B13421+ B21341+ B22141+B13341+B14141) ] .

14211

(33)

It is evident that ßp11"D (r) reduces to the corresponding Born series if the coefficients of sin 6, in eels . (20)--(29) are equated to zero, and that the terms additional to the Bon- expansion are of order A" only. Similarly tan á1(n) as exemplified by eqs. (30)-(33) reduces to the Born expansion for tan 6,0) if the A" terms in the denominator are equated to zero. By induction we can write for the general case (n 1), 29-1

2~_â

Vi

(n)

(r)

= C{1+ i E(AM 1 P")

~ â...s 1(y)}cos á1+{A" E(qp, gv)BQoQâ... goal (r)}sin áj 1(ky) 1 2a-â 2M-â E(a ,« , ) A mB« o ,. . . ,n1(r)}cos á 1 + {A» po0â. ..0-1(y)}sinójX1(kr), (34) 2 13 1

1

tn

M®1

1

1

where i E(P,., v) denotes a summation over 1V terms, each differing from its neighbour by an unspecified number of exchanges of the suffices From eq. (34), the asymptotic wave function is 2115-1 1 (kr)cos á1 ®. [{ E(ap, A 1}cos á1 (35) +(A" ao,8,.. .,8.1}sin b1]X1(kr) . 1 ®n equating the coefficient of ®X 1(kr) to sin 61, we obtain

tan á11"D = [ E(«,., oc,,) A nB .Aaâ" " % J/1 1-Ail E (ßo, Pj Bco,0,...ß.j " (36) 1 'n®1 Conditions for the coi.-ergence of the o series as given by the cos bl terms in (34), have been studied by Kohn 11), leading to radii of convergence A.(k). However, for our purpose, it is sufficient that 16,1 <-213. Then the extra terms in (34), coefficient A" sin á1, must approach zero as n --* oo, giving us the Born series 2,15-1 ") A'nBI>o9,... (r)} 1(kr) E 1( (r) = [{1 +

E

a6Œ,.. . ,,(r)}

1(kr)]cos b c,

(37)

IM-1

lim

n-.co

v&(") (r --> 00 ) = 1

a(kr) °

av)

2M-1

lim tan 6E(") =

n-+vo

n

% (ah~ 1

Am Ba0al...a,aE ~E(kr)~COS

m~1

E (ore, oc,)

n

, AmBaoal.. .a,a'

m ®1

aa, (38) (39)

The extra terms, coefficient A" sin áa, are of the same order of magnitude as the A" terms in the Born series, which must vanish for n --> oo. Their magnitude, and that of the Born terms in (36), may fee estimated if we use the known fact that (6) is a good approximation for tan a s for 16E < , at least for weak potentials. The case of strong potentials requires the additional proviso of sufficiently high energy . We mayrewrite (6) in the alternative form for I B211 < i, tan ô,% tan 8 E(i) = ABIE

n-1

(ABsa) ml[ 1- (AB2E)"j .

m-o

(4U)

As tan 6,01) is alre.dy a good approximation, with error of order 2 %, we have tan ô&) - tan ôE(") % AB1a

(AB2E) m~C1-AnBsa~

MM0

(41 )

Comparison of (41) with (36) gives us pen-1 1

E(ao av)Baoal... a,i

P:b

B1E .B2&

(m = 1, 2, . . ., n),

(42)

These relations are verified for n-p scattering in sect. 4. The case of a ? 16 1 1 L 1= is of particular interest, as it is known that the Born series for V,(r) and tan 6, are divergent in this region . Nevertheless we shall show that expansions (36) and (36) for VEM (r --> oo) and tan 8E(") are convergent for the weak potentials of sect. 2 (1611 S a for all k) . The former case involves the finite difference between two divergent series and the latter the finite ratio of two divergent series. The result does not apply to strong potentials, where 16E) can exceed 7:. A rigorous proof is not attempted here. For weak potentials, a > 16,1 =" corresponds to (AB2E( ~ 1 . It is interesting to note that the scattering system has one bound compcsite state in this case. Eq. (6) for tan ba(l) is then known to be a good approximation, which we proceed to expand in the form "-1 , - ABjE (AB2a)m -1 1 tan 8E s% tan 8a (') = % tan 8ê"), (44) ~ (A21)"

DISTORTED WAVE EXPANSION

equivalent to the series in inverse powers -1 ABIS tan &M (AB2a)-m/cl (AB2a) -~~ AB21

62 9

tan ô,("),

45)

which is convergent for n --* oo, provided VE2al > 1 . If IAB211 = 1, then we get tan = oo/0 = oo as required, the one point at which a pole in the taxi 6, series is obtained (but necessarily so). Equating terms in (36) and (44), we again find the approximate relations (42) and (43) hold, these leading in turn to the convergence of lpa(n) (r -> ) in (35). It is worth noting that the convergence of (36) in fact follows, independent of the above, provided only we assume 2A-1 R&(k), E(ap, av)B aoa' . . .a a}/{ E (po, l (46)

~{,1

w)=

1

where Ra is same number independent of n. The terms in the denominator of (46) involve a first integration over the four. factor g,(r), but apart from this, only differ from the numerator terms i-°1 the order in which the successive integrations are carried out, and these are summed, over 2"-1 permutations. From (42) and (43), we see RI(k) o~e B1c/B2a'

(47)

4. Application to Neutron-Proton Scattering Relations (42) and (43) hold with surprising accuracy, illustrated in the following example of neutral particle scattering at zero energy (t = 0) by :t square well potential (r < b), (r ? b) .

U(r) .= - Uo 0 We take as form factor 2)

(3r/2b-r3/2b3) go(r) = 1

(r < b), (r > b),

obtaining the values Blo = 3AUo bsk,

Beo = sAU0 b2,

so that ó0M (0) is 2m for AUo b 2 = 2.5. The correct value for zero binding energy, corresponding to á0(l) (0) = 12a by Levinson's theorem 13), is actually Uo b2 = 42 = 2.4674 . . ., indicating an error of order 2 % . Defining the zero-energy

P. SWAN

630

scattering length as a - 1im

k-o0 (

--

tan 60(k) 1

)

,

we evaluate eqs. (30)--(32) to find a(') = -A-AU® b81 [1-0 .4AUo b 2],

(48)

a(2) _ --~AUobs[1+0 .4AUob2]/ [1--0.1619(AUo bs)2 ],

(49)

a(s)

-8AUo b3[1+0.4AU bs-E-0 .1619(AUo b2 ) 2]Î[1--i ~.06561(AUo b2)3]. (50)

The agreement between eqs. (36), (42), (43) and (48)--(50) is excellent, considered as an approximation. A numerical application of eqs. (48)--(50) is 1S n-p scattering at zero energy. Using the parpmeters 2) AUo = 0 .3390 X 1026,

b = 2.5860 fm,

we find results for a(") and the corresponding Born values aB("), given in table 1 . Note that as }3 20 =0.4 Uu b 2 --. 0.90681, the convergence of the Born series should be slow, as is indeed found . TABLE 1

Zero-energy scattering lengths for 1S n-p scattering (in fm). Square well, AUO = 0.8390 X 10 20 , b = 2.5860 fm .

,lil

e

Series (44)-(46) Born series Correct value

I

a(2)

aft)

-20.97 - 1 .954

-22.19 - 8.728

-22.72 - 5.852

-23 .69

-23 .69

-23.69

The initial value a( 1 ) is already quite good, while a(2) and a( 3) show rapid convergence to the correct value. On the other hand, aB(i) is very poor and aB(2) and aB(3) therefore converge slowly . Asymptotically, of course, a("0 and a,(") v-ill. converge at the same rate (n --> ao), but for n small there is no comparison between the two methods. TABLE 2 Zero- energy scattering lengths for 0S n-p scattering (in fnn) . Square well, AUO = 0.87704 X 1080, b .== 2.0048 fm .

Serves (48)--(50) Bom series Correct value

_7

a(1)

5.745 -2 .355 5.38

7-

00

5.611 -5 .676 5.38

afar , 5.669 --10.416 5.38

DISTORTED WAVE EXPANSION

631

3S

n-p scattering at zero energy involves the zero-energy phase ó®(0) = X, corresponding to the singlet bound state of the deuteron and 0.4 Uo V = 1 .4100. Table 2 lists values of a(") obtained via eqs. (44)-(46) and the Born series. The series expansions (48)---(50) again show rapid convergence of a("), but the Born values as(") have the wrong sign and diverge rapidly with n, as expected. e ere ces 1) 2) 3) 4) 5) 6) 7)

8) 9) 10) 11) 12) 13) 14)

P. Swan, Nature 187 (1960) 585 P. Swan, Nuclear Physics 18 (1960) 245 P. Swan, Nuclear Physics 21 (1960) 233 L. I. Schiff, Quantum L -%anics (McGraw-Hill, New York, 1949) N. F. Mott and H. S. W. Massey, The theory of atomic collisions (Oxford University Press, 1949) 2nd ed . J. M. Blatt and J. D. Jackson, Phys. Rev. 76 (1949) 18 L. Spruch and L. Rosenberg, Phys. Rev. 116 (1959) 1034 ; L. Spruch and L. Rosenberg, Phys . Rev. 117 (1960) 1095 ; L. Spruch and L. Rosenberg, Nuclear Physics 17 (1960) 30 ; L. Rosenberg, L. Spruch and T. F. O'Malley, Phys. Rev. 119 (1960) 164; L. Rosenberg and L. Spruch, Phys. Rev. 120 (1960) 474; R. Jost and A. Pais, Phys. Rev. 82 (1951) 840 W. Kohn, Rev. Mod . Phys . 26 (1954) 292 T. Kikuta, y :og. Theor. Phys. 12 (1954) 225, 234 Ch. Zemac i .end A. Klein, Nuovo Cim. 10 (1958) 1078 H. Davies, :+uclear Physics 14 (1959--60) 465 N. Levinson, Mat. Fys. Medd ., Dan. Vid. Selsk. 25, No. 9 (1949) P. Swan, Proc . Roy. Soc. 228 (1955) 10 ; K. R. Mather and P. Swan, Nuclear scattering (Cambridge University Press, 1958) Appendix B