The thickness of the nuclear surface

TEIE THICKNESS

OF THE

NUCLEAR

SURFACE

ERICH VOCT Energy of CanadaLixuited, Cl&k Rfver, O&&o, Canads

Atomic

Reccji\red 4 April 1962 The surke thickness, a (cf. equation (2) below), of a target nuc;Eeusaffects the reflection of bomrding pkticfes. In low energy nuclear reactions resonance cross sections or average cross sectior?s described by the optical model) we show e prir(cQ3.I effect is to multiply the convennuclear penetrability by a factor which deon cx and which has a value of between two and tkiree for B-r! usual estimates of cI. The WignerTefcknann sum-rule limits I) for nucleon reduced ‘c~idtbsmost be azititiplied by the same reflection &t0r. An acillrultie estimate of the nuclear surface thickness is o#ained from the data for the lowenergy @-wave meutron strength function. &irify the role of Q in nuclear reactions we by comparing the scattering and absorption t’s neutrons by a square complex wtential,

q7r1 -: - y*

+ iW*)/El + e (a-R,VQj

l

parts of (1) and (2) exhibit the principal effects of ct. The two real potentials are shown on fig. 1 along with the wave function for each potential at a l.ow-energy resonance. The scattering cross section is related to the phase shift, 6, by 4n Q= ---&n26

(3)

l

In the case of the real square well an expression for 6 can be immediately written down but with the Saxon-Woods potential this is not possible. Thus to compare the cross sect%.s for the two ( 8889 we use the language of resonmce theory a). For each potential a matching radius, R, is chosen and a set of standing waves or resonant states, Xl, is con-

(21

at 10~ energy typica parameters i .25 fermis (where A is the target

r), Vo--52 fernais. The scy\lase such a tin-Woods we3.l is chosen to e 01 ~‘@lz such that the lowest unbound ~%~?mances of the two wells coincide. Fur this bomb resonance a reduce8 width. is dethe Sa~n-Woods well at a matching radius R (R rd necessarily equal to Ro)~ The square W?ll radius. RI , is chosen to be equal to R (RI = R)

L

v--w

---

1

Fig. 1. Ths m equivalentred well&Iu[1c1 tkeir vmve fumtlma at, 8 low energy resonanca At low anergiee a remnmx3 oocum when tha wave funotion has mm slope outaide the potentialwell.

Vohme 1, mu&et 8

PIiYSfCS LETTERS

t&tctird for the regionr sR by meansof the wave equatton and,a bomdary oorrdition v d@dv

=-trX,

at the matchingracbus. The phase shift then takes the form a=t@n-‘[PCIZI(14%?+cp, where the %-functie~l, 92, contaW3all the properties of the i&rior region, r ;5R, and where the pene+aMlityt P, the shift functkm, S, arid the hard-spherb shfft ‘p are defimd in terms of the values at R of the regular and ffregulisur solutions, F and G, of the ra&al wave equ&on in the exterior region (Y r R). (F(W) irs equal sin kr and G(k) to co8 k~ for r so large that the potentSaX (1) or (2) can be neglected in the wave equation.) We have P s kR/EF2(kR) + d(kR)] s = - b + P f%dW’dp q~ = tad

[F(kR)/G(kR)]

,

W

+ C(ddG/dp

1,&R

9

.

(7) (8)

The Q-function may be written in the form familiar from resonance theory 2 CRc c&(B, - E) ? i9) wher the Eh are the +kracteristic energies and the Qs the reduced widths of the resonant states. If we retain only one level, x =’0, in (9) and use the resulting approximate %!-functionin (5) and (3) we obtain the well-known Breit-Wigner single level formula, GZ-

;2

I

2

r0

2hkReikR

- {Jy” + & + E) - ire/2

(lo)

where To (s 2.I$) is the total width of the level and A. (I Sy$) is the level shift. For the square well, we choose the matching radius R equal to the square well radius RI: then P and (o both hve the value kR1 and S + b vanishes identically. If we further choose b = 0 then S and ,&-,vanishidentically, each yt has the value R2/mR& where m is the nucleon mass, and the resonant states lie above the bottom of the well by an amount Ei :=(p12J2m) (4 - 4)2 9,/R?

,

(11)

where the x’s are integers beginning with x = 1. The %?-function of the real square well can be then summed up simply as

where K is the wave vector inside the square well. The choice b = 0 may be regarded as a “natural” boundary condition for the real square well, in as much as it makes the standingwaves in the well

1wlyleCn

rWemble the actual physical statds at resonance as closely as possible. Fig. 1 shows that the wave function at resonance has the logarithmic derivative b = 0. Equivalentlythe shift functionvanishes for this choice of b 80 that the energy, El, of the standingwave coincides w&tithe peak cross section energy4 Fkrthermore, oofthany other choice of b the one-level approximation as in (10) is no longer valid: the resonance then occurs in the wrong place and has too narrow a width *. For the real Saxon-Woods potentialthe best choice of the matchingradius is not obvious. Efwe choose R larger than h by manytimes the surface thiekn&ssa then P, v and S + b have the same values as for a square well of that radius. As R ts brought toward Ro, cpremains almost constant (and small at low energy) but P and S + b change by large amounts. For R near R. the %akuraP choice of zi is net b = 0 but rather the one which makes S vanish, or equivalently, the one equal to the logarithmic derivative of the physical state at resonance as on fig. 1. With this %aturaP choice of b we can make the whole ~-function of the real Saxon-Woodswell equal to the %!-functionof the corresponding real square well. To show this we take a case (A w 155) where Table 1 The level energies and reduced widths of a real SaxonW&s potential and of th& equivalent square well, with an appropriate %aturaP boundary bond&ion, b, applied to aaoh potential at the matching radius, R . The parameters of the Saxon-Woods well are V. = 51 MeV, a = 0.5 fermia mdR, = 6.72 fermis in order to produce the 4s level (x = 4) near zero energy. A matching r-118 of R = 6.845 fermfa is choeen. The square well parameters are R1 = R and VI = 53.88 MeV. For the square well the boundary condition is bz 0 and for the Saxon-Woods well b r 6.12. The reduced widths of the square well are all equal to h2hR 2 md the reduced widths of the SaxonWoods well are given fn unitts of A2/mR 2. Results are given for only the lowest nine levels. Because of the non-zero value of the natural boundary condition for the Saxon-Woods well, the very lowest level of this potential lies below the bottom of the well and has an ano.lmalous reduced width. Level Energy, MeV h

Square well

Saxon-Woods well

1

-52.79 -44.06 -26.61 - n_ *A9 -z 34.48 78.12 130.49 191.58 261&O --_II_

-53.41 -43.69 -26.53 - n AR

2 3 41 5 6

Saxon-Woods well Reduced width% i ( p12,fmR2) -7 -I 4.260

1.043

1.030 I-._-nnn 0.989 0.987

0.987 0.989 0.989 -.. .-- -

* A calculation of the effect of the boundary condition on

the one-level approximation to the square well scatterfng croaa section ia described on fig. 1 of the review artfcle by E. Vogt 3, . 85

radius

R such that

tie

re-

- this radius is just

t3 make the 4s level lie of the %cm-Woods

lity by a reflection facon the surface thickness .67 for Q = 0.5 fermis. S-?rlrrA\‘E NEUTRON SCATTERiNG BY REAL POTENTiALS AT !5QkeV

crams sections

on because the equUty of the %-functions extends beyond the one-level appro&.nation. Whenwe add an imaginary part to the potential well the principal effect of the surface thickness a is still to modify only the penetrabilityP. The transition from the seal square well to the complex square well can be made by @a&g E, in (S), by B + iW. By the sane means a sqt&rreima@nary well may be added to the results for t&ereal SaxonWoods potentialand the results obtained in table 1 show that the c)2-functioru3of the two equivaient wells are then still almost equal. However, because W is small compared to the real potential at the matching;radius, such a square imaginary well does not cause any appreciable &mountof reflection. Hencewe obtain approximatelythe same phase shifts by simply replacing the square imaginary well by a Saxon-Woods imaginarywell, thus proving thatthe two wells (1) and (2) differ principally %ntheir penetrability, or equivalentlyin the rh of their resonant states. For the complex wells the scattering cross section near a resonance is given by (10) if we re-

of the two

of at0

I

L

I60

160

-A Fig.

I

200

220

rrtrength fun&ion data of ref. 6), aa 8 function 02 atomic weight A, and optical model computationa of the strength fun&ion for varloua surface thi&ne~eea, 81; in. fermie. Eaoh computed cqlrve corresponds to a S -Woo& potential ST, R. = 1.25 A 3 fermie sIl1d with ightly to In& tha 43 giant reaooccur at the BBme place for each curve.

3. The neutron

“p4

I

Voltnne1, numer 2

PHYSICS

place E by IS+ iWo. Since Wo is genera3ly much bigger th&n ro/2 the wfdths of the giant resonances of the complex potentials are controlled by W. and are nearly independent of surface thickness. HQWever, &, occurs in the numerator of (IO) and hence the s&ace thickness u,determines the height of the orotis section= More general@ it can be shown that the absorption cross seetim of a diffuse edge potential lies everywhere above that of the eguivalent square well by the reflection factor discussed above. Th2 factor is about I+ 6.7 -2, for u less than a Fermi, and appears to be almost independent of the shape or value of the imaginary part of the potential. Fig. 3 shows the data 6) for the s-wave neutron strength function together with computed curves for a square well and for equivalent Saxon-Woods wells of varying surface thickness. The constant value of the ‘black nucleus” strength function is also shown. The “black nucleus*’ strength function is actually a “square black nucleus” because it corresponds to taking the square complex well and averaging out the structure of the giant resonances. As cz is increased from the value zero the average strength function (averaged from KR = 3n to KB = 4rron the figure) increases from the “black nucleus” value of 0.867 x W4 by an amount equal to the reflection factor of 1 + 6.7 dr2. This increase of the average strength function over the black nucleus value is very nearly independent of the magnitude of the imaginary part of the potential, W, md of the spatial distribution of W or of splittings 5~6) in the giant resonance produced by shape deformations in the target nucleus, so that the estimate of a made from the increase is essentially model independen:. The data of fig. 3 taken for the whole 4s giant resonance yields an average value of 1.42 f 0.06 x log4 for the strength function. The consequent value of a is 0.31 f 0.02 fermis. The “black nucleus” strength function changes slightly with the choice of radius - for Ho = 1.4 A’j-we wopld have found a = 0.25 f 0.02 and for I& = 1.07 As, a = 0.37 f 0.02, so that the estimate a = 0.31 + 0.06 encompasses all reasonable variations of the well parameters. The surface thickness obtained from experiments with high energy electrons 7) is a = 0.55 f 0.06 and differs significantly from our result, Nucleon elas*****

LETTERS

1 Il&cly1222

tic scattering at moderately high energies appears to yield values of ct in agreement with the electron scattering value, although u tends to decrease with nucleon energy 6) and may be reasonably extrapolated to our low value at zero energy. Such an energy dependence of a for nucleons may have its explanation in the polarization of the target nu cleans by the incoming particle and the tnability of a slow incoming particle to shake off the nucleons it attracts. The effect of the surface thickness in nuclear reaction theory and resonance theory will be described in another paper.. We point out here that Teichmann-Wigner sum-rule limits for reduced widths are based on the reduced widths and penetrabilities of the square edge. For a finite surface thickness, Q, a true single particle level uxnAd have a total width larger than that of the square edge nucleus by the reflection factor, F, discussed above. If the conventional penetrability is used in analyzing such a level then the reflect? ‘%&or must multiply the reduced width. The Mxlg sum rule limit for nucleons is then F A2/mR6. Many parts of tfifs work have benefitted from discussions with R. Chrien, A. Jain and H. Palevsky and with the author’s colleagues at Chalk River. The computations of figs. I, 2 and 3 and of table 1 were carried out by means of computer r;~ograms prepared under the direction of Dr. 3. M. Kennedy whose essential help in this problem is gratefully . acknowledged.

References 1) -T. Teichmann and E. P. Wigner, Phys. Rev. S? (1952) 123. 2) A, M. Lane and R. G. Torn-, Rev. Mod. Phys-. 30 (1958) 257. 9) E. Vogt in Nuclear Reactions, edited by P. M. Endt and M. Demeur (North-Holland Publishing Company, 1958). 4) K.Kikuchi, Progr. Theor. Phys. 1’7 (1957) 643. R.L.ZimmermannandR.E.Chrien, 5) D.J.Hughes, Phys. Rev. Letters 1 (1958) 461. 6) B. MargoPis and E. S. Trou3xtskoy. Phys. Rev. 106 (1957) 105. Rev. !Ivlod. Phys. 30 (1958) 430. 7) D.G.Ravenhall, C.B.Duk@ and M.A.Melkanoff, Phys. 8) J.S.Nodvik, Rev. 125 (1962) 975.