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Mesons and the structure of nucleons. Part IV. The nucleon-nucleon potential
ANNALS
18, 47-64
OF PHYSICS:
Mesons
and
(1962)
the Structure Nucleon-Nucleon
of Nucleons. Potential*
Part
IV. The
G. COSTAR AND B. T. FELD Physics
Department
and Laboratory Technology,
for Nuclear Science, Massachusetts Cambridge, Massachusetts
Institute
of
The atomic model, in which physical nucleons are described as compound systems of a nucleon core plus a single pion, has been used as the basis for a computation of the low-energy interaction between two physical nucleons. Advantage is taken of previous work for the specification of the interaction between the pion and the nucleon core in the different possible states of t.otal angular momentum and isotopic spin. An initial simplification, where we replace one of the “atoms” by an “effective scattering center” is only moderately successful in reproducing the gross features of the observed nucleonnucleon interaction. The full computation (approached in the spirit of the Heitler-London approximation to the H-H interaction) improves on the qualitative fit, but it turns out to be impossible to reproduce simultaneously the asymptotic behavior of both the central and tensor terms in the nucleonnucleon potential using a reasonable set of pion-core interactions. A significant result is the derivation, using the simplified model, of a nucleon-nucleon spin-orbit interaction whose range is shorter than that of the other terms in the potential. Our computation thus provides a strong indication that the spin-orbit interaction may arise from the same features of the pionnucleon interaction as give rise to the resonant pion-nucleon scattering in the state of T = J = 3/t?. INTRODUCTION
In an earlier communication (1) the “atomic” model of the physical nucleons, previously developed (9, 3)) has been applied to the problem of pion-nucleon scattering.’ The model is a semiclassical one, in which the physical (observed) nucleon wave function is approximated by the combination of a nucleon LLcore” and a single (real) pion, bound in a p-state with total angular momentum -9; ( pljz) and isotopic spin -J/z. The results of the previous investigations demonstrate the usefulness of the atomic model in providing a reasonable and intuitive picture of the main proc* This work was supported by a joint program of the Atomic Energy Office of Naval Research and the Air Force Office of Scientific Research. t Present address: Division Theorique, CERN, Geneva 23, Switzerland. 1 References (1-9) will be referred to as Parts III, I, and II, respectively. 47
Commission,
the
48
COSTA
ASD
FELD
esses bet,wcen pions and nucleons based on the generally accepted Yukawa interaction. In order to investigat’e further t,his feature, we have ext’ended the applicat,ion of the at’omic model to the problem of t,he nucleon-nucleon interaction. One would hope that the model might provide an understanding of the static interactions between two nucleons, and possibly give a simple picture of the origin of the spin-orbit roupling. The present communication is devoted t’o t’he investigat,ion of these questions. Neglect’ing any pion-pion interaction (as we have also in Part III), a straightforward application of the atomic model reduces the interaction between two nucleons to the interaction between the (two) pions and the (t’wo) nucleon “cores” in different states of the two (physical) nucleon system. The problem is analogous to the problem of the H? molecule, a problem which has been treated by an approximation of the Heitler-London type. We have maintained here the description of the P-wave interaction between a pion and the nucleon-core in terms of a set of square-well potentials, with strengths vjt depending on the angular momentum and isotopic spin of the (j, t) st.ate in question and with a common ra,nge r. (1). The resulting expression for the potential is not expected to have physical meaning at distances less than TO(assumed considerably less than the pion Compton wavelength) ; in other words we hope to obtain the long-range part of the nucleon-nucleon potential. This, however, corresponds to that part of t’he potential about which most is known, both experimentally and theoretically. In fact, both the conventional field-theoretic and the double-dispersion investigations, based on an analysis of the forces according to t’heir ranges, are limited to the long-range potential. However, we are not primarily concerned with the derivation of a two-nucleon potential which will provide a precise fit to the experimental data; instead, we limit ourselves to an attempt at understanding the qualitative features of the potential. Towards this end, we have further simplified the comput8ation by adopting an approximation in which the “atomic” description (i.e., the assumption of a compound ( 1, 1) bound state of a nucleon core and a pion) has been used for only one of the two physical nucleons; the effect of the other (nucleon “core” plus pion) is replaced by that of a simple scattering center. In this approximation, then, only one pion is explicitly present at any time; t#his pion is assumed to intera,ct with the core to which it is bound through the afore-mentioned potential vjt and with the other (physical) nucleon through a P-state potential, V, whose strength VJT depends on the total angular momentum J and isotopic spin T of the pion plus physical nucleon system. From an empirical point of view, we might define these potentials V,, in terms of the observed P-wave pion-nucleon scattering phase shifts. However, we shall not strictly adopt this approach but, rather, in the following we consider the VJT as parameters, which must still be consistent with the requirements of the observed pion-nucleon scattering.
MESONS
AS-D
NJCLEONS.
49
IV
The following sections summarize the results obtained for the static potential and for the spin-orbit term. As will be clear from the next section, a number of features of the observed static potential are inadequately reproduced by this calculation. In order to decide whether t,his situation is due to the approximations used, or whether it is connect’ed with t,he intrinsic failings of the atomic model itself, we have attempted a more accurate evaluation of the static potent’ial, taking into account explicitfly the “atomic” structure of both physical nucleons. We discuss the details of this calculation in a separate section. THE
STATIC
NUCLEON-KUCLEON
POTENTIAL
We start from a static Heitler-London approximation, Hamiltonian of the system of two physical nucleons is H = -VI’
in which the total
- vse + VA(l) + UB(2) + U4(2) + V,(l),
(1)
where vx(;) represents t’he interact’ion between the nucleon core X and the pion i, as given in Part III. However, instead of solving the scattering problem with the Hamiltonian (1) , we adopt a simplification which ignores the structure of one of the two nucleons, as not.ed in the preceding. Thus, the int,eraction between the two nucleons is described in terms of a potential V acting directly between t,he pion associated with an “atomic” nucleon and a simple scattering center describing the other pion-nucleon combination. In analogy with the expression used in Part III for t,he potential 8, we assume li = 681fqrr,)
c liJTOJT , J,T
(2)
in which B(r-ro) represents the radial dependence of the pion-nucleon core interaction and the OJT stand for “projection operators” which single out the appropriate states of a pion and physical nucleon with the total angular momentum J and isotopic spin T. Equation (2) can be rewritten in terms of explicit expressionsfor the project.ion operators OJT : v = [a + b( d . 1) (< . t,) + c( d * 1) + d(7 . t,)]s,, B(r-0)
(3)
where a = $gv,,
+ 4V33 + 2V13 + 2VS’,,l
b = gs’[vl, + v,, -
VI3 - v311
(3a) (3b) (3c) (34
The symbols n, 7 represent the spin and isotopic spin of the physical nucleon; t, is the isotopic spin of the pion; 1 is the relative angular momentum between t.he pion and the scattering (physical) nucleon center.
50
COSTA
The Hamiltonian
AND
FELD
(1) then reduces to the form H = -v12 + 2)8(l) + V,(l)
(4) or to the alternate one obtained by interchanging either A and B or 1 and 2, or both. The choice of the indices is, in this approximation, immaterial. We have now to evaluate the expectation values of the Hamiltonian (4) for given states of a, ab, , a, llcS, where 0 and PCare the total spin and isotopic spin of the two physical nucleons. This is performed by writing down the corresponding eigenfunctions for the two physical nucleon states, and replacing the eigenfunction of one of the nucleons by the one appropriate to t,he atomic model (see ref. 1). Taking into account the fact that the Hamiltonian exhibits only cylindrical symmetry, one obtains different interaction energies in the substates a5, = 0 and $3, = fl of the triplet spin state. We shall denot’e by 3Vl , 3Vo the expectation values of the potential corresponding to .&52= fl and 0, = 0, respectively. This dependence of the potential on the relative orientation of the total spin of the two physical nucleons implies a tensor force. If we apply to the triplet potential the standard decomposition into a central and a tensor term (4)) 3v = “v, + s,, “VT XAB = f (da.r)(dB.r)
(5)
- (dA.de)
(5’)
we can deduce the two terms 3Vc and 3VT from the expressions for 3V1 and “VO “v,
= $$(2”V, + 3vo) ;
3v, = ;,fg”v,
- 3vo).
(f-3) The even and odd character of the potential is connected with the eigenvalues of the isotopic spin IIc = 0, 1 through the Pauli principle. Hence, by evaluating separately the matrix elements for the two values of the isotopic spin, one obtains the six functions denoted in the literature by ‘Vc+, IV,-, 3Vc+, 3Vc-, 3vT+, 3VT- which describe completely the static interaction between two nucleans. The six functions can be summarized by the following expressions” ’ [V,, F”‘(z)
* The potentials
+
V,,G(‘)(x)]
are in units of r/137 = I Mev (P is the pion mass).
MESONS
ASD
NUCLEONS.
51
IV
where x = LYTand LYis the “range” parameter which characterizes the length scale of the pion bound state eigenfunction as used in Part III. The subscripts C and T, which go, respectively, with the signs + and -, refer to the triplet central and tensor interactions, respectively. Expressions for the functions F(xj and G(x) are given in the Appendix as polynomials in increasing powers of (l/x), In the asymptotic region (x >> 1), FT and GT decrease like (l/x) while the others approach constants, with the following relationships among them: Fc3’(x) C
‘v G”‘(x).
F”‘(x)
> G”‘(x)
F$3’(x) =
9 ;
Gg’(x)
‘v F”‘(x)
(84
FA3’(x)
< Gd3’(x)
(8b)
(8c)
G?‘(x)
The asymptotic form of the space dependence of the potentials is given, for this model, by (e-“/x)‘, which is the square of the usual Yukawa form characteristic of the predictions of meson theories in the asymptotic region. This feature is an inevit’able failing of the atomic model, in which the pions are treated as real particles rather than as quanta of a field; it arises, we believe, from the difference in normalization of the pion wave functions in the two approaches. Since it is our purpose, however, to investigate the qualitative features (i.e., attractive vs. repulsive, strong vs. weak) of the potential, we propose to ignore this deficiency of the atomic model. Specifically, we compare the product (e-“lx)-“V predicted by our model with (e-“lx)-‘V for any of the models based on meson field theory. In this qualitative comparison, we lean on the collection of experimental and TABLE COMPARISON Term
1vc+ vc+ VT+ ‘VcycClVT-
OF
“THEORETICAL"
TM0 (5) Attractive strong* Repulsive medium Attractive strong Repulsive strong Uncertain weak Repulsive weak
Q See also, Feshbach, *Strong, medium, (!io-0).
AND
H (7) and B (8) Attractive strong Uncertain weak Attractive strong Repulsive strong Uncertain w-eak Repulsive medium Lomon and xveak
I
“EMPIRICAL" G (9) and GT (IO) Attractive strong Attractive medium Attractive strong Repulsive medium Uncertain weak Repulsive weak
NUCLEON-NUCLEON
POTENTIALS~
BW (11) Attractive strong Attractive weak Attractive medium Repulsive strong Attractive weak Repulsive weak
so (I?) Attractive strong Repulsive weak Attractive strong Repulsive strong Repulsive weak Repulsive weak
and Tubis (13) and Cirelli and Stabilini (16). are intended to stand, roughly, in the ratio
l:($$-Sg):
52
COSTA
AND
FELD
theoretical information on the nucleon-nucleon interaction contained in the review by Cammel and Thaler (5) and on some unpublished notes by H. Feshbath and A. de-Shalit.3 Table I presents a summary of the properties of the static nucleon-nucleon potentials in the asymptotic region (X >> 1) as derived by a number of investigators from meson field theoretic computations and from empirical fitting of the available data. The entries in column 2 (TMO) represent the results of a meson-theoretic computation by Taketani, Machida, and Onumu (6) taking account of both one-meson and two-meson exchange terms (the so-called OMEP plus TMEP potentials). (These ent’ries are identical with t,hose of column 2 of Table II, in which we give the results of our computation of the asymptotic nucleon-nucleon potentials.) The rest of Table I exhibits the results of a number of attempts t’o fit the observed nucleon-nucleon interactions with potentials which, in the asymptotic limit, are assumed to join smoothly onto the meson-theoretic potentials (although the different attempts do not always use the same version of the asymptotic meson-theoretic potentials). In particular, these fits make different assumptions concerning the size and nature of the “hard core” in the region of small 12 5 0.5; they also involve different assumptions concerning the spinorbit interaction, not shown in Table I. Table I indicates common agreement on the singlet potentials: strongly attractive and repulsive for ‘Vo+ and ‘lice, respectively; and on the tensor potentials: strongly attractive for the 3VT’ and weakly repulsive for “V,-. Both of the triplet central potentials appear to be relatively weak, but there is no general agreement on their signs. For the comparison of our results with Table I, it is necessary to make specific assumptions about the parameters VJ, . We assume first that Vz3is the dominant term of the potential V, in accordance with the observations on low-energy pion-nucleon scattering, and disregard completely the interactions in the other (J, T) states. The nonvanishing terms of (7) then reduce to lvc+ ‘v (3
(; v34 G’Yx)
vc- s (;>’ (; v33)G%?$ 3v*- ‘v We note that the dominant rise to a static nucleon-nucleon
43
(attractive) interaction
3 We are indebted to these authors, especially material available and for clarifying discussions.
(9)
(; Va3) G&3’(2).
V33 pion-nucleon interaction gives in only three states; the properties to Professor
Feshbach,
for
making
this
-
I.
Attractive strong” Repulsive medium Attractive strong Repulsive strong Uncertain weak Repulsive weak
Meson theory (TMO)O
a See Table
Term
-
-
-
TABLE
II
Attractive medium Repulsive weak
Attractive strong
V33 only
-
-
_
I
Uncertain weak Repulsive weak Repulsive strong
-
-
Repulsive strong Attractive weak Repulsive strong
-
-
-
I
Repulsive medium At.tractive weak Repulsive strong
-
1
Attractive strong Repulsive weak Repulsive weak Repulsive medium Attractive weak Attractive weak
X=1
-
model term only)
Attractive strong Repulsive weak Repulsive weak Repulsive medium Attractive weak Attractive weak
x=2
Improved (exchange
COMPARISON OF RESULTS FOR THE ASYMPTOTIC STATIC POTENTIAL
Attractive strong Repulsive weak Attractive weak Repulsive medium Attractive medium Repulsive weak
.x >> 1
+q
z
3 Ei
m + i3
2
E
54
C’OSTA
AND
FELD
of these potent’ials are summarized in column 3 of Table II. Although these potentials show some of the main features of the corresponding meson-theoretic potentials, as given in Table I, they leave much to be desired. In particular, we observe that the dominant I/a, interaction is not at all involved in determining the properties of the potentials in three of the nucleon-nucleon configurations, of which especially the 3VTf and ‘V,- are both strong. This may be a special feature of the simplification which we have introduced into this computationi.e., that of replacing one of the physical (pion plus core) nucleons by a simple scattering center; this question will be examined in a section to follow. On the other hand, the deficiencies of the above approximation may indicate the importance of the other, small pion-nucleon potentials for the nucleonnucleon interaction. In this case, the low-energy pion-nucleon scattering provides little guidance in the choice of the other Ir,, , since the small P-wave phase shifts are only very poorly known (1). At the same time, the atomic model, as developed in Part III, is quite ambiguous concerning the properties of the V.,, other than VZ3 . Nevertheless, we shall examine the dependence of the nucleon-nucleon potentials, Eqs. (7)) on the small V.,r , to determine whether these may be brought into closer accord with the observations by the choice of an appropriate set of VJT . Let us consider, then, those terms in the nucleon-nucleon potential (Eq. 7) which do not contain V33 . Of these, we would like to obtain a strongly repulsive 3VT+. Taking into account the ‘Vc-, a weak 3VC+, and a strongly attractive asymptotic relations for x >> 1 (Eqs. (8) ) , it is easy to demonstrate that these requirements cannot be met simultaneously.4 Thus, attainment of the required sign for ‘V,- and weak 3Vcf 1s * possible by choosing VII and V3, of the same order of magnitude but with opposite signs, i.e., VI, > 0, Vsl < 0, (column 4, Table II). However, this choice leads to a repulsive 3VT+. Other choices, for instance, the assumption of the dominance of either VII or Val , can lead to the same sign for ‘V,- and 3Vc+, although, if the dominating term were V3131 , we could thereby obtain the opposite sign for 3VT+. These conclusions are summarized in columns 4 and 5 of Table II. In any event, the reasonable fitting of the IV,+, 3Vc-, and 3VT- potentials need not be disturbed if we maintain the predominance of the V33 term; indeed, the agreement shown in column 3 of Table II can even be improved by an appropriate choice of VI3 . But there does not appear to be any choice of the pa4 However, in until rather ships apply With Table
these II.
the asymptotic large values
values,
relations among of .z. For example,
the functions F and in the range z = l-2,
G, Eqs. (S), do not set the following relation-
F(” ‘v GU’ y $‘fi’cW N 2Gc’V k 30 (FTC31 = &(a)). the choice of 1’ 31 _V - 2Vll > 0 leads to the results
shown
in column
(8’) 6,
MESONS
AND
NUCLEONS.
55
IV
rameters which will reproduce all of the observed (and meson-theoretic) erties of the asymptotic nucleon-nucleon potentials.5 THE
SPIN-ORBIT
prop-
POTENTIAL
In this section we consider the effects of the relative motion of the two nucleans. We maintain the simplified picture used in the previous section for the evaluation of the static potential, and use Eqs. (3) and (4) to describe the potential between a pion and a physical nucleon. To avoid ambiguity, we denote by A the physical nucleon whose “atomic” structure is explicitly taken into account, and by B the other. It is clear that the relative motion will affect only those terms containing the operator (dB . l,,). We adopt the following approximation: the frame of reference will be fixed on the nucleon B, with respect to which A is moving with the velocity v. For the sake of simplicity, we shall neglect the motion of the core of the nucleon A with respect to the center of mass of A itself. The momenta of the core and of the entire physical nucleon A have then the same direction, and their ratio is equal to MC/M N l(M, = mass of the nucleon core, M = mass of the physical nucleon). It is easily shown that the static value of the expectation value of cdB . lBK)St is modified to become
(dB.lBs)drIl =
(&-1Bn)st
+
5
ds-(L
+rAa
x
p>
(10)
where p is the pion mass, p is the relative linear momentum and L = R X p is the relative angular momentum of the two physical nucleons; rA,, is the corepion distance. The added dynamic term is proportional to the ratio P/M of the pion mass to the nucleon mass. The neglected corrections are of second order with respect to this ratio. We have now to re-evaluate the expectation value of (4)) taking into account that the terms (d . 1) are to be replaced in V by the dynamic expression (10). The two-nucleon eigenfunctions will be written here as the product of two parts: one describing the static di-nucleon system A-B, and the relative orientation of the spin of A and B; the other depending on the orientation of the dinucleon system with respect to a fixed direction, (the distance between A and B is kept fixed). This last part depends on the total angular momentum J = P + 85 of the two nucleons. We have to consider only the triplet state (0 and X are not necessarily constants of the motion, but owing to the fact that the symmetry of the eigenfunction is definite, sb is a good quantum number). 5 But ‘6. Table
we may II.
do better
if we look
at the intermediately-large
range
of z N 2; see column
Fixing our attention on a given lralue of 3, az = X, and parity, the angular eigenfunction of the two uucleon systems cm be written, for t.he case x =a.
In the case of opposite parity, we have to take into account the two possible values of P = JJ f 1. In (11) there appear the spherical harmonics YfL corresponding to the orientation of the di-nucleon syst#em, and TJ~ which stand for the eigenfunct’ions of the triplet state as used in the static case. We have performed the calculation for the QC = 1 state, corresponding to odd angular momentum X, and for Q = 0 corresponding to even X. The result’s are that the expect,ation value of t’he “dynamic” part of (4) cm be writ’ten as: (wdYn
= S(x) (XL . 6)
(12)
where (P . &) represents t’he expectation value of i . 6 corresponding t’o the particular choice of J and X. This proves that the resulting potentia,l has a spin-orbit character. The complete expression for this potential can then be written as:
The superscripts ( -) , ( +) stand for odd and even parit’y and correspond to the pl; = 1 and QC = 0 states, respectively. The explicit form of the function FLS(z,~) is given in the Appendix. Let us consider Eq. (13) : A phenomenological spin-orbit potential was added by Signell and Marsha.k (&V) (14) to t#he Gartenhaus potential in order to fit the 310 Mev p-p date. The spin-orbit term suggested by them (in the QC = 1 state) is intrinsically attractive; it behaves asymptotically like (l/s) (e-“/z). Okubo and Marshak (15) showed t,hat meson field theory could provide a spin orbit term of this kind, proportional to the ratio p/M, but much smaller than what was needed to fit the experiments. Spin-orbit terms were introduced also by Gammel and Thaler (5, 10)) and by other authors (7, 8) who considered also second order terms in X. The expression (13) for KS reproduces rather well some of the main features of the SM spin-orbit potential. In fact, if one considers, as we did in the case of the static potential, that the term V,, dominates, VZs is attractive; moreover, it falls off faster tha.n the static (central) part by a factor l/z. In agreement with the meson field-theoretic result, it is proportional to the mass ratio p/M;
MESONS
AND
NUCLEON&
57
IV
its magnitude is correspondingly smaller than the main terms in the static potential by this factor. However, even though the range of the spin-orbit pot’ential is predicted by our model to be less than that of the static potential (by the aforementioned l/x factor), this does still not seem to be sufficiently short-range to fit the requirements of the nucleon-nucleon scatt’ering data (5). However, if, for some reason not’ clear to the authors, the (eC/z)’ dependence should be appropriate to the spin-orbit potential (but not to t’he static terms) this might satisfy the requirements of short range as determined by the empirical fitting of the high-energy nucleon-nucleon scattering observations. The situation with respect to t,he possibility of a spin-orbit term in the Q = 0 states is rat,her uncertain experimentally (5). Our model also generates a term but its sign and its order of magnit’ude depend strongly on the assumptions vt, , about the terms VI1 and VB1 (see previous section and Table II), and it can easily be made very small. IMPROVED
APPROXIMATION
FOR
THE
STATIC
POTEKTIAL
One can see from the previous sections that t,he atomic model in its simplified version can reproduce qualitatively many of the outstanding features of the nucleon-nucleon potential. The most interesting result seems to be t,he possibility of deducing a spin-orbit potential of the same kind as the term assumed in the GT and SM potentials. The result’s of the stat’ic calculation developed in the foregoing show, however, that is not possible to reproduce consistently all of the relevant features of the static potent,ial. In fact, it seems diflicult to obtain at the same time attraction in the triplet even states and repulsion in the singlet odd state. However, before we attribute this partial failure to the atomic model itself, we should investigate more thoroughly the effects of the approximations used in the calculation. In particular, one can be somewhat skeptical about the validity of the simplified picture, in which the ‘(atomic” structure of one of the physical nucleons is disregarded. Perhaps suc,h a simplified version suffers from its failure to take into account exchange effects between t,he two pions, which would be present in a complete atomic model computation. For this reason we have carried out a calculation of t,he stat#ic potential based on the complete application of the atomic model. We describe here briefly this calculation and give the results: The total Hamiltonian of the two-nucleon system which has now to be used is given by (1). According to the Heitler-London approximation, the unperturbed wave function corresponding to an eigenstate of given tot,al spin P, & , total iso-spin a, ma , and parity (&) can be written
e&,T,T3(1, 2) = 241, 2) + 42, 1)
(1.9
58
COSTA
AND
FELD
where the U’S are the appropriate combinations of the eigenfunctions uA(l), uB(2), UA(2), uB( 1) of the physical nucleons, as given in Part I. To the first order, the two-nucleon static potential is now given, aside from an additive constant, by the expectation value of the total Hamiltonian (1)) v
=
Hl+l
Hl2
(16)
K+A
where Hll = (U(l,2)
IHI U( 1, 2))
fJl2
=
IHI
K
= Ml,
(41,2>
42,
1))
(17)
2) I41,2))
A = (u(l,2) 14% 1)) Performing all the integrations implied by (17) and proceeding in analogy with the simplified model, we obtain the six nucleon-nucleon potentials ‘VC+, ‘VC-, “V,+, 3Vc-, 3VT+, 3VT-. These are given by the following expressions.6 --I 2 ‘vc- = 5 dTN(zhl + 8~13Pb) + (Su33+ ~31VYd ( X ) - 3VllW( +P(
z)]
--2 2 lv,+
=
5 (
>
cp(~)[(ll~,l
+
16df%)
+
(16~33+
lh)g%)
+ 15Vn W(7)h”)(x)1 3 VC.T
e-32 2 (p(~)[(llu11 + 161)13)f% + = ; ( )
(16~33+
(18)
ll~dg%(x)
+ 15Vll
wmx~)1
We reproduce in the Appendix the explicit expressions for f(z), g(z), h(z), and cp(T), U( 7)) functions of x = CYTand 7 = arc , which appear in (18). The first two terms in the brackets come from the “direct” matrix elements HU , the third one from the “exchange” matrix elements Hrz of (17). It turns out that the “exchange” interaction dominates over the “direct” one, as was the case in Part III for the pion-nucleon scattering phase-shift computation. We now compare our results with those obtained by the authors already cited. 6 Note that the ujt’s p/l37 012N 8 Mev.
are defined
as in Part
III,
and
that
the
potentials
are in units
of
MESONS
r
HAMADA’S +
(5)
AND
NUCLEONS.
59
IV
ATOMIC MODEL
POTENTIALS
POTENTIALS
I
V(x)
\
2
r
o-
0.5
I
15 (pr3
-
2
0
I
I
05
I
I X-
1.5
2
FIG. 1. Nucleon-nucleon potentials. We have plotted the dependence of the potentials on the separation, with the asymptotic (Yukawa) behavior removed, in units of pc2 S 137 Mev. (a) The potentials of Hamada (7), based on experiment plus meson field theory. (b) Results of one of the best fits of the full atomic model computations.
the sake of convenience we shall consider, in particular, the results of Hamada (7)) since these are available in analytic form. These results have been plotted in Fig. 1 (a). Let us examine the expression (18). We not’e, again, t’hat the “exchange” terms are the most important ones and that these are proportional to the potential vll of the pion-core interaction. Since this potential is fixed, for a given range, 7, by the static properties of the physical nucleon (2)) we anticipate that the essential features of the potential, insofar as they are provided by the exchange terms, are essentially fixed and that it will be rather difficult to deviate appreciably from them. As a consequence, as an inspection of the exchange term shows, it will also be difficult to reproduce all the features of the Hamada potential. Thus, keeping the exchange terms only, the potentials (Eqs. (18) ) may be For
60
COSTA
written,
to within
AND
FELD
a common factor ‘v,-
-
lv,+
-
- L!llh(1) 5vuh (1)
“VF,, 3 + VC.T -
5vnh$ -vdm
(3)
Evaluation of the factors h(x), given in the Appendix, leads to the relationships between the various potentials as shown in the last three columns of Table II, corresponding to values of x: = 1,~ = 2, and CE>> 1, respectively. Qualitatively, these results indicate a reproduction of many, but not all, of the salient features of the observed potentials. The most serious failure is for the 3VT+ term. In order to see how far we can go with the atomic model, we adopt as parameters, together with the range 7 of the pion-core potential, the three ratios among the potentials Ujt = vjt/vlr . We now try to find the best fit that can be obtained to the Hamada potential. We limit our considerations to J: > 1, as this is the only region in which we expect our computations to be reliable. As a first step, we adopt as the relationships among the terms of the Hamada potential in the region’ x cv l-2. (a)
IV,-
‘v -‘V,+(-2)
(b)
3Vc+ ‘v ->$‘V,+( (e)
+M)
(c)
“vc-
(d)
3V,+ ‘v ‘V,+(
“VT- ‘v -M’V,+(
N /l&J’vc+(
-45)
+2)
-x)
(20) The important features are, as previously discussed, the strong and attractive ‘Vc+ and 3VT+, and the strong and repulsive IV,- (relations 20(d) and (a) ) . The other features-the small and even uncertain values of the triplet-central potentials (20(b) and (c)) and the medium and repulsive 3V,--are less firmly established, both experimentally and theoretically; for example, the potentials in Table I are in complete disagreement with respect to 3VCf. Disregarding common factors in (18)) our potentials are IV,-
-
(1 + 8u,,)f”‘(z)
‘VC+ -
(11 + 16~13)f(‘)(2)
“G,,, 3 + V c.-r -
(11
+ (82~~~+ u31)gC1)(z-) - 3w(T)h(‘)(x) + (162433+ llu3Jg(‘)(z)
+ 15+)h”‘(r) (21)
(1 +
+
16~13)f&%(~) Su,,)f%(2)
+ +
(162433
(8~33
+
lW&(x)
+ u&$(4
+
15w(+&(r)
- 34&%(x)
We have tried to fit some of the above conditions (20) and to see what happens to the other terms. What we find is that, while it is possible to fit almost all of the above relationships (“almost” refers to the fact that we have to weaken some of them), the conditions for such a fit are not acceptable from the point 7 The sponding
relationships coefficients
shown correspond to z N 1; the values of lVc-+ for z z 2.
in parentheses
are the corre-
MESONS
AND
NUCLEONR.
IV
61
of view of the atomic model. If, indeed, the values of the ratios u13 , us1 , and u33 are to be consistent with the results on pion-nucleon scattering previously derived (3)) and also with the requirement that there be only one bound stat’e, the ( 1, 1) state, the values of u13 , ual , u33 are limited both in magnitude and sign; vll of course must be taken to be attractive and strong. However, for a given solution (i.e., assumption of any three out of the five conditions (20) ) the requirements of the nucleon-nucleon potential determine t’he pion-core potentials, u, without regard to considerations of pion-nucleon scattering. Thus, some of the signs turn out to be wrong if we try to fit the conditions 20 (a) and (d) simultaneously (solutions which assume (a),(b),(d); (a),(c),(d); (a),(d),(e). Moreover for such solutions the terms u13 , ual , u33 turn out to be t,oo large. The most reasonable set of solutions seems to be the ones reproducing the central terms ‘VC*, 3Vo’ (conditions 20 (a, b, c)) . For such solutions the values and signs of the u;i’s are consistent with the requirements of the atomic model, and wit.h the values used in the pion-nucleon scattering problem (I). For instance, for t’he values 7 = 0.6 and cy = 0.8, we obtain: ~13- -0.8 vll , v31 - 1.0 v11, 2133 - 0.2 Vll ; the nucleon-nucleon potentials corresponding to this choice of parameters are shown in Fig. l(b). With this and similar solutions, one obtains the right order of magnitude for the central terms but, unfortunately, the tensor terms turn out to have the wrong signs. The situation is analogous if we try to fit the G and GT potentials (see Table I). One might attempt to improve the situation by regarding the quantity w( 7) in Eqs. (18) as a parameter rather than using the expression (A$) given in the Appendix; W(T) represents the ratio of the contributions of the exchange and ordinary part,s of the interaction (Eqs. (16) and (17) ) . The expression (A.5) given in t’he Appendix is computed for pion-core interactions, vjt , whose shape is given by a “square well.” Different forms for the shape of these interactions would be expected to yield different values of W(T). It turns out that some reasonable decrease in the values of W(T) can yield some improvements in the fit by modification, in the right direction, of some of the unacceptably large values of the Ujt (other than ull). However, even with these improvements, we have not found it possible to obt.ain a reasonable solution which can satisfy, simultaneously, both the conditions 20(a) and 20(d). SUMMARY
AND CONCLUSIONS
Unlike the previous applications to the static properties of nucleons (Part I) and to the pion-nucleon low-energy interactions (Parts II and III), the atomic model is only indifferently successful in accounting for the properties of the static nucleon-nucleon potential in the range (r 2 &/PC) over which a one meson approximation is expected to apply. Even setting aside the failure t,o reproduce the Yukawa form--the model gives an ( e-*/zr)2 dependence rather
62
COSTA
.4ND
FELD
than (e-“/x)-which can be ascribed to the deficiency of a treatment of pions as “real” particles, rather than as quanta of a field, the computations fail in predicting the wrong signs for some of the important terms in the nucleonnucleon interaction; in particular, it does not appear t’o be possible, with a reasonable set of pion-core interactions, to account simultaneously for both the central and the tensor terms in the observed static nucleon-nucleon potential. The results of the computation, in the various approximations considered in this paper, have been summarized in Table II and Figure l(b) . On the other hand, the atomic model, even in its simpler form (in which one of the “atoms” is replaced by an “effective scattering center”), provides a straight-forward and simple “explanation” of the nucleon-nucleon spin-orbit interaction as arising from the dominant (3, 3) interaction between the pion and the physical nucleon, although it is not clear that we can reproduce, in a satisfactory fashion, the requirement of short range for the spin-orbit potential. We believe that in the computations described above we have extended the atomic model to the limit of its usefulness (if not somewhat beyond). In the previous papers, and to a lesser extent in this one, it has been shown how the dominant features in the low-energy interactions between pions and nucleons arise as a consequence of the properties of the single pion plus core term in the wave function of the physical nucleons. However, when we deal with systems involving more than one nucleon and thus, perforce, more than one pion, effects of pion creation and absorption, of pion exchange and, possibly, of the pionpion interaction are expected to increase in significance. We believe it. is reasonable to conclude, from the results reported in the foregoing, that such effects play a significant role in determining the low-energy nucleon-nucleon interaction, even in the asymptotic region. The successful computation of such phenomena requires, unfortunately, the full arsenal of field-theoretic techniques. APPENDIX
1. The functions F(s) and G(x) which appear in Eqs. potential, are given by the following expressions:
(7) for the static
(A.1)
MESONS
AND
NUCLEONH.
63
IV
T = (YT~defines the “range” of the interaction, as discussed in Part III. 2. An approximation (valid for T < 1) for the function F&Z, T) which pears in the spin-orbit terms, Eqs. (13) and (14)) is F&X,
T) CT% 1 + ;+ (
; + ;
>
T5
(A.21
3. Finally, we record expressions for the different functions Eqs. (18) for the static potential (improved approximation) : f(l)(X)
= f
g(l)(x)
= &
(9 + ; 18 + ;
+ 9
+ $
+ 2
p(7)
h’“(x)
‘v ;
t’P(x)
= f
#(CT)
= f
= 10e2’ 27+
w(T).(p(T)
= -
sinh27+
10’:J?t3’e-r{2
(
x + 5 + ; + i2
‘(1 7
+ F
in
>
>
(9 + $ + g + ; 18 + ;
appearing
+ g)
+ 228 + $ x3
(
ap-
+ $)
+ $4
+ $2
>
(A.3)
- cosh27)
*
sin f cash 7 + +p
T2 cost - ~sin .$ r2 + p t
(A.5)
6-l
COST.1
AXD
FELD
where
RECEIVED:
January
3, IcJcj2 REFERENCES
1. G. COSTA AND B. T. FELD, dnn. Phys. (SY) 9,354 (1960). 2. B. T. FELD, dnn. Phys. (XI-) 1,58 (1957). 3. B. T. FELD, An?&. Phys. (XV) 4, 189 (1958). 4. J. M. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics.” Wiley, New York, 1952. 5. J.L. GAMMELAND R.M. THALER, “Progress in Elementary Particle and Cosmic Ray Physics,” Vol. V, Chap. II. North-Holland, Amsterdam, 1960. 6. M. TAKETANI, S. MACHIDA, AND S. ONLTMA, Progr. Theoret. Phys. (Kyoto) 6, 638 (1951); ibid. 7, 45 (1952). 7. T. HAIXADA, Progr. Theoret. Phys. (Kyoto) 24,1033 (1960). 8. R. A. BRYAN, Wuovo cimento 16,895 (1960). 9. Y. GARTENHAUS, Phys. Rev. 100, 900 (1955); ibid. 107, 291 (1957). 10. J. L. GAMMEL AND R. THALER, Phys. Rev. 107, 291 (1957). if. K. A. BRUECKNERAND K. M.WATSOK, Phys.Rev.92,1023 (1953). 12. M. SUGAWARA AND S. OKOBO, Phys. Rev. 11’7,605 (1960). 13. H. FESHBACH, E. LOMON AND A. TUBIS, Phys. Rev. Letters 6, 635 (1961). 14. P. SIGNELL AND R. E. MARSHAEL, Phys. Rev. 109, 1229 (1958). 15. S.OKUBO AND R.E. MARSHAK,:Inn. Phys. (.yP) 4,166 (1958). 16. R. CIRELLI AND G. STABILINI, S~ppl.,\'~ovo cirnento 20,157 (1961).
18, 47-64
OF PHYSICS:
Mesons
and
(1962)
the Structure Nucleon-Nucleon
of Nucleons. Potential*
Part
IV. The
G. COSTAR AND B. T. FELD Physics
Department
and Laboratory Technology,
for Nuclear Science, Massachusetts Cambridge, Massachusetts
Institute
of
The atomic model, in which physical nucleons are described as compound systems of a nucleon core plus a single pion, has been used as the basis for a computation of the low-energy interaction between two physical nucleons. Advantage is taken of previous work for the specification of the interaction between the pion and the nucleon core in the different possible states of t.otal angular momentum and isotopic spin. An initial simplification, where we replace one of the “atoms” by an “effective scattering center” is only moderately successful in reproducing the gross features of the observed nucleonnucleon interaction. The full computation (approached in the spirit of the Heitler-London approximation to the H-H interaction) improves on the qualitative fit, but it turns out to be impossible to reproduce simultaneously the asymptotic behavior of both the central and tensor terms in the nucleonnucleon potential using a reasonable set of pion-core interactions. A significant result is the derivation, using the simplified model, of a nucleon-nucleon spin-orbit interaction whose range is shorter than that of the other terms in the potential. Our computation thus provides a strong indication that the spin-orbit interaction may arise from the same features of the pionnucleon interaction as give rise to the resonant pion-nucleon scattering in the state of T = J = 3/t?. INTRODUCTION
In an earlier communication (1) the “atomic” model of the physical nucleons, previously developed (9, 3)) has been applied to the problem of pion-nucleon scattering.’ The model is a semiclassical one, in which the physical (observed) nucleon wave function is approximated by the combination of a nucleon LLcore” and a single (real) pion, bound in a p-state with total angular momentum -9; ( pljz) and isotopic spin -J/z. The results of the previous investigations demonstrate the usefulness of the atomic model in providing a reasonable and intuitive picture of the main proc* This work was supported by a joint program of the Atomic Energy Office of Naval Research and the Air Force Office of Scientific Research. t Present address: Division Theorique, CERN, Geneva 23, Switzerland. 1 References (1-9) will be referred to as Parts III, I, and II, respectively. 47
Commission,
the
48
COSTA
ASD
FELD
esses bet,wcen pions and nucleons based on the generally accepted Yukawa interaction. In order to investigat’e further t,his feature, we have ext’ended the applicat,ion of the at’omic model to the problem of t,he nucleon-nucleon interaction. One would hope that the model might provide an understanding of the static interactions between two nucleons, and possibly give a simple picture of the origin of the spin-orbit roupling. The present communication is devoted t’o t’he investigat,ion of these questions. Neglect’ing any pion-pion interaction (as we have also in Part III), a straightforward application of the atomic model reduces the interaction between two nucleons to the interaction between the (two) pions and the (t’wo) nucleon “cores” in different states of the two (physical) nucleon system. The problem is analogous to the problem of the H? molecule, a problem which has been treated by an approximation of the Heitler-London type. We have maintained here the description of the P-wave interaction between a pion and the nucleon-core in terms of a set of square-well potentials, with strengths vjt depending on the angular momentum and isotopic spin of the (j, t) st.ate in question and with a common ra,nge r. (1). The resulting expression for the potential is not expected to have physical meaning at distances less than TO(assumed considerably less than the pion Compton wavelength) ; in other words we hope to obtain the long-range part of the nucleon-nucleon potential. This, however, corresponds to that part of t’he potential about which most is known, both experimentally and theoretically. In fact, both the conventional field-theoretic and the double-dispersion investigations, based on an analysis of the forces according to t’heir ranges, are limited to the long-range potential. However, we are not primarily concerned with the derivation of a two-nucleon potential which will provide a precise fit to the experimental data; instead, we limit ourselves to an attempt at understanding the qualitative features of the potential. Towards this end, we have further simplified the comput8ation by adopting an approximation in which the “atomic” description (i.e., the assumption of a compound ( 1, 1) bound state of a nucleon core and a pion) has been used for only one of the two physical nucleons; the effect of the other (nucleon “core” plus pion) is replaced by that of a simple scattering center. In this approximation, then, only one pion is explicitly present at any time; t#his pion is assumed to intera,ct with the core to which it is bound through the afore-mentioned potential vjt and with the other (physical) nucleon through a P-state potential, V, whose strength VJT depends on the total angular momentum J and isotopic spin T of the pion plus physical nucleon system. From an empirical point of view, we might define these potentials V,, in terms of the observed P-wave pion-nucleon scattering phase shifts. However, we shall not strictly adopt this approach but, rather, in the following we consider the VJT as parameters, which must still be consistent with the requirements of the observed pion-nucleon scattering.
MESONS
AS-D
NJCLEONS.
49
IV
The following sections summarize the results obtained for the static potential and for the spin-orbit term. As will be clear from the next section, a number of features of the observed static potential are inadequately reproduced by this calculation. In order to decide whether t,his situation is due to the approximations used, or whether it is connect’ed with t,he intrinsic failings of the atomic model itself, we have attempted a more accurate evaluation of the static potent’ial, taking into account explicitfly the “atomic” structure of both physical nucleons. We discuss the details of this calculation in a separate section. THE
STATIC
NUCLEON-KUCLEON
POTENTIAL
We start from a static Heitler-London approximation, Hamiltonian of the system of two physical nucleons is H = -VI’
in which the total
- vse + VA(l) + UB(2) + U4(2) + V,(l),
(1)
where vx(;) represents t’he interact’ion between the nucleon core X and the pion i, as given in Part III. However, instead of solving the scattering problem with the Hamiltonian (1) , we adopt a simplification which ignores the structure of one of the two nucleons, as not.ed in the preceding. Thus, the int,eraction between the two nucleons is described in terms of a potential V acting directly between t,he pion associated with an “atomic” nucleon and a simple scattering center describing the other pion-nucleon combination. In analogy with the expression used in Part III for t,he potential 8, we assume li = 681fqrr,)
c liJTOJT , J,T
(2)
in which B(r-ro) represents the radial dependence of the pion-nucleon core interaction and the OJT stand for “projection operators” which single out the appropriate states of a pion and physical nucleon with the total angular momentum J and isotopic spin T. Equation (2) can be rewritten in terms of explicit expressionsfor the project.ion operators OJT : v = [a + b( d . 1) (< . t,) + c( d * 1) + d(7 . t,)]s,, B(r-0)
(3)
where a = $gv,,
+ 4V33 + 2V13 + 2VS’,,l
b = gs’[vl, + v,, -
VI3 - v311
(3a) (3b) (3c) (34
The symbols n, 7 represent the spin and isotopic spin of the physical nucleon; t, is the isotopic spin of the pion; 1 is the relative angular momentum between t.he pion and the scattering (physical) nucleon center.
50
COSTA
The Hamiltonian
AND
FELD
(1) then reduces to the form H = -v12 + 2)8(l) + V,(l)
(4) or to the alternate one obtained by interchanging either A and B or 1 and 2, or both. The choice of the indices is, in this approximation, immaterial. We have now to evaluate the expectation values of the Hamiltonian (4) for given states of a, ab, , a, llcS, where 0 and PCare the total spin and isotopic spin of the two physical nucleons. This is performed by writing down the corresponding eigenfunctions for the two physical nucleon states, and replacing the eigenfunction of one of the nucleons by the one appropriate to t,he atomic model (see ref. 1). Taking into account the fact that the Hamiltonian exhibits only cylindrical symmetry, one obtains different interaction energies in the substates a5, = 0 and $3, = fl of the triplet spin state. We shall denot’e by 3Vl , 3Vo the expectation values of the potential corresponding to .&52= fl and 0, = 0, respectively. This dependence of the potential on the relative orientation of the total spin of the two physical nucleons implies a tensor force. If we apply to the triplet potential the standard decomposition into a central and a tensor term (4)) 3v = “v, + s,, “VT XAB = f (da.r)(dB.r)
(5)
- (dA.de)
(5’)
we can deduce the two terms 3Vc and 3VT from the expressions for 3V1 and “VO “v,
= $$(2”V, + 3vo) ;
3v, = ;,fg”v,
- 3vo).
(f-3) The even and odd character of the potential is connected with the eigenvalues of the isotopic spin IIc = 0, 1 through the Pauli principle. Hence, by evaluating separately the matrix elements for the two values of the isotopic spin, one obtains the six functions denoted in the literature by ‘Vc+, IV,-, 3Vc+, 3Vc-, 3vT+, 3VT- which describe completely the static interaction between two nucleans. The six functions can be summarized by the following expressions” ’ [V,, F”‘(z)
* The potentials
+
V,,G(‘)(x)]
are in units of r/137 = I Mev (P is the pion mass).
MESONS
ASD
NUCLEONS.
51
IV
where x = LYTand LYis the “range” parameter which characterizes the length scale of the pion bound state eigenfunction as used in Part III. The subscripts C and T, which go, respectively, with the signs + and -, refer to the triplet central and tensor interactions, respectively. Expressions for the functions F(xj and G(x) are given in the Appendix as polynomials in increasing powers of (l/x), In the asymptotic region (x >> 1), FT and GT decrease like (l/x) while the others approach constants, with the following relationships among them: Fc3’(x) C
‘v G”‘(x).
F”‘(x)
> G”‘(x)
F$3’(x) =
9 ;
Gg’(x)
‘v F”‘(x)
(84
FA3’(x)
< Gd3’(x)
(8b)
(8c)
G?‘(x)
The asymptotic form of the space dependence of the potentials is given, for this model, by (e-“/x)‘, which is the square of the usual Yukawa form characteristic of the predictions of meson theories in the asymptotic region. This feature is an inevit’able failing of the atomic model, in which the pions are treated as real particles rather than as quanta of a field; it arises, we believe, from the difference in normalization of the pion wave functions in the two approaches. Since it is our purpose, however, to investigate the qualitative features (i.e., attractive vs. repulsive, strong vs. weak) of the potential, we propose to ignore this deficiency of the atomic model. Specifically, we compare the product (e-“lx)-“V predicted by our model with (e-“lx)-‘V for any of the models based on meson field theory. In this qualitative comparison, we lean on the collection of experimental and TABLE COMPARISON Term
1vc+ vc+ VT+ ‘VcycClVT-
OF
“THEORETICAL"
TM0 (5) Attractive strong* Repulsive medium Attractive strong Repulsive strong Uncertain weak Repulsive weak
Q See also, Feshbach, *Strong, medium, (!io-0).
AND
H (7) and B (8) Attractive strong Uncertain weak Attractive strong Repulsive strong Uncertain w-eak Repulsive medium Lomon and xveak
I
“EMPIRICAL" G (9) and GT (IO) Attractive strong Attractive medium Attractive strong Repulsive medium Uncertain weak Repulsive weak
NUCLEON-NUCLEON
POTENTIALS~
BW (11) Attractive strong Attractive weak Attractive medium Repulsive strong Attractive weak Repulsive weak
so (I?) Attractive strong Repulsive weak Attractive strong Repulsive strong Repulsive weak Repulsive weak
and Tubis (13) and Cirelli and Stabilini (16). are intended to stand, roughly, in the ratio
l:($$-Sg):
52
COSTA
AND
FELD
theoretical information on the nucleon-nucleon interaction contained in the review by Cammel and Thaler (5) and on some unpublished notes by H. Feshbath and A. de-Shalit.3 Table I presents a summary of the properties of the static nucleon-nucleon potentials in the asymptotic region (X >> 1) as derived by a number of investigators from meson field theoretic computations and from empirical fitting of the available data. The entries in column 2 (TMO) represent the results of a meson-theoretic computation by Taketani, Machida, and Onumu (6) taking account of both one-meson and two-meson exchange terms (the so-called OMEP plus TMEP potentials). (These ent’ries are identical with t,hose of column 2 of Table II, in which we give the results of our computation of the asymptotic nucleon-nucleon potentials.) The rest of Table I exhibits the results of a number of attempts t’o fit the observed nucleon-nucleon interactions with potentials which, in the asymptotic limit, are assumed to join smoothly onto the meson-theoretic potentials (although the different attempts do not always use the same version of the asymptotic meson-theoretic potentials). In particular, these fits make different assumptions concerning the size and nature of the “hard core” in the region of small 12 5 0.5; they also involve different assumptions concerning the spinorbit interaction, not shown in Table I. Table I indicates common agreement on the singlet potentials: strongly attractive and repulsive for ‘Vo+ and ‘lice, respectively; and on the tensor potentials: strongly attractive for the 3VT’ and weakly repulsive for “V,-. Both of the triplet central potentials appear to be relatively weak, but there is no general agreement on their signs. For the comparison of our results with Table I, it is necessary to make specific assumptions about the parameters VJ, . We assume first that Vz3is the dominant term of the potential V, in accordance with the observations on low-energy pion-nucleon scattering, and disregard completely the interactions in the other (J, T) states. The nonvanishing terms of (7) then reduce to lvc+ ‘v (3
(; v34 G’Yx)
vc- s (;>’ (; v33)G%?$ 3v*- ‘v We note that the dominant rise to a static nucleon-nucleon
43
(attractive) interaction
3 We are indebted to these authors, especially material available and for clarifying discussions.
(9)
(; Va3) G&3’(2).
V33 pion-nucleon interaction gives in only three states; the properties to Professor
Feshbach,
for
making
this
-
I.
Attractive strong” Repulsive medium Attractive strong Repulsive strong Uncertain weak Repulsive weak
Meson theory (TMO)O
a See Table
Term
-
-
-
TABLE
II
Attractive medium Repulsive weak
Attractive strong
V33 only
-
-
_
I
Uncertain weak Repulsive weak Repulsive strong
-
-
Repulsive strong Attractive weak Repulsive strong
-
-
-
I
Repulsive medium At.tractive weak Repulsive strong
-
1
Attractive strong Repulsive weak Repulsive weak Repulsive medium Attractive weak Attractive weak
X=1
-
model term only)
Attractive strong Repulsive weak Repulsive weak Repulsive medium Attractive weak Attractive weak
x=2
Improved (exchange
COMPARISON OF RESULTS FOR THE ASYMPTOTIC STATIC POTENTIAL
Attractive strong Repulsive weak Attractive weak Repulsive medium Attractive medium Repulsive weak
.x >> 1
+q
z
3 Ei
m + i3
2
E
54
C’OSTA
AND
FELD
of these potent’ials are summarized in column 3 of Table II. Although these potentials show some of the main features of the corresponding meson-theoretic potentials, as given in Table I, they leave much to be desired. In particular, we observe that the dominant I/a, interaction is not at all involved in determining the properties of the potentials in three of the nucleon-nucleon configurations, of which especially the 3VTf and ‘V,- are both strong. This may be a special feature of the simplification which we have introduced into this computationi.e., that of replacing one of the physical (pion plus core) nucleons by a simple scattering center; this question will be examined in a section to follow. On the other hand, the deficiencies of the above approximation may indicate the importance of the other, small pion-nucleon potentials for the nucleonnucleon interaction. In this case, the low-energy pion-nucleon scattering provides little guidance in the choice of the other Ir,, , since the small P-wave phase shifts are only very poorly known (1). At the same time, the atomic model, as developed in Part III, is quite ambiguous concerning the properties of the V.,, other than VZ3 . Nevertheless, we shall examine the dependence of the nucleon-nucleon potentials, Eqs. (7)) on the small V.,r , to determine whether these may be brought into closer accord with the observations by the choice of an appropriate set of VJT . Let us consider, then, those terms in the nucleon-nucleon potential (Eq. 7) which do not contain V33 . Of these, we would like to obtain a strongly repulsive 3VT+. Taking into account the ‘Vc-, a weak 3VC+, and a strongly attractive asymptotic relations for x >> 1 (Eqs. (8) ) , it is easy to demonstrate that these requirements cannot be met simultaneously.4 Thus, attainment of the required sign for ‘V,- and weak 3Vcf 1s * possible by choosing VII and V3, of the same order of magnitude but with opposite signs, i.e., VI, > 0, Vsl < 0, (column 4, Table II). However, this choice leads to a repulsive 3VT+. Other choices, for instance, the assumption of the dominance of either VII or Val , can lead to the same sign for ‘V,- and 3Vc+, although, if the dominating term were V3131 , we could thereby obtain the opposite sign for 3VT+. These conclusions are summarized in columns 4 and 5 of Table II. In any event, the reasonable fitting of the IV,+, 3Vc-, and 3VT- potentials need not be disturbed if we maintain the predominance of the V33 term; indeed, the agreement shown in column 3 of Table II can even be improved by an appropriate choice of VI3 . But there does not appear to be any choice of the pa4 However, in until rather ships apply With Table
these II.
the asymptotic large values
values,
relations among of .z. For example,
the functions F and in the range z = l-2,
G, Eqs. (S), do not set the following relation-
F(” ‘v GU’ y $‘fi’cW N 2Gc’V k 30 (FTC31 = &(a)). the choice of 1’ 31 _V - 2Vll > 0 leads to the results
shown
in column
(8’) 6,
MESONS
AND
NUCLEONS.
55
IV
rameters which will reproduce all of the observed (and meson-theoretic) erties of the asymptotic nucleon-nucleon potentials.5 THE
SPIN-ORBIT
prop-
POTENTIAL
In this section we consider the effects of the relative motion of the two nucleans. We maintain the simplified picture used in the previous section for the evaluation of the static potential, and use Eqs. (3) and (4) to describe the potential between a pion and a physical nucleon. To avoid ambiguity, we denote by A the physical nucleon whose “atomic” structure is explicitly taken into account, and by B the other. It is clear that the relative motion will affect only those terms containing the operator (dB . l,,). We adopt the following approximation: the frame of reference will be fixed on the nucleon B, with respect to which A is moving with the velocity v. For the sake of simplicity, we shall neglect the motion of the core of the nucleon A with respect to the center of mass of A itself. The momenta of the core and of the entire physical nucleon A have then the same direction, and their ratio is equal to MC/M N l(M, = mass of the nucleon core, M = mass of the physical nucleon). It is easily shown that the static value of the expectation value of cdB . lBK)St is modified to become
(dB.lBs)drIl =
(&-1Bn)st
+
5
ds-(L
+rAa
x
p>
(10)
where p is the pion mass, p is the relative linear momentum and L = R X p is the relative angular momentum of the two physical nucleons; rA,, is the corepion distance. The added dynamic term is proportional to the ratio P/M of the pion mass to the nucleon mass. The neglected corrections are of second order with respect to this ratio. We have now to re-evaluate the expectation value of (4)) taking into account that the terms (d . 1) are to be replaced in V by the dynamic expression (10). The two-nucleon eigenfunctions will be written here as the product of two parts: one describing the static di-nucleon system A-B, and the relative orientation of the spin of A and B; the other depending on the orientation of the dinucleon system with respect to a fixed direction, (the distance between A and B is kept fixed). This last part depends on the total angular momentum J = P + 85 of the two nucleons. We have to consider only the triplet state (0 and X are not necessarily constants of the motion, but owing to the fact that the symmetry of the eigenfunction is definite, sb is a good quantum number). 5 But ‘6. Table
we may II.
do better
if we look
at the intermediately-large
range
of z N 2; see column
Fixing our attention on a given lralue of 3, az = X, and parity, the angular eigenfunction of the two uucleon systems cm be written, for t.he case x =a.
In the case of opposite parity, we have to take into account the two possible values of P = JJ f 1. In (11) there appear the spherical harmonics YfL corresponding to the orientation of the di-nucleon syst#em, and TJ~ which stand for the eigenfunct’ions of the triplet state as used in the static case. We have performed the calculation for the QC = 1 state, corresponding to odd angular momentum X, and for Q = 0 corresponding to even X. The result’s are that the expect,ation value of t’he “dynamic” part of (4) cm be writ’ten as: (wdYn
= S(x) (XL . 6)
(12)
where (P . &) represents t’he expectation value of i . 6 corresponding t’o the particular choice of J and X. This proves that the resulting potentia,l has a spin-orbit character. The complete expression for this potential can then be written as:
The superscripts ( -) , ( +) stand for odd and even parit’y and correspond to the pl; = 1 and QC = 0 states, respectively. The explicit form of the function FLS(z,~) is given in the Appendix. Let us consider Eq. (13) : A phenomenological spin-orbit potential was added by Signell and Marsha.k (&V) (14) to t#he Gartenhaus potential in order to fit the 310 Mev p-p date. The spin-orbit term suggested by them (in the QC = 1 state) is intrinsically attractive; it behaves asymptotically like (l/s) (e-“/z). Okubo and Marshak (15) showed t,hat meson field theory could provide a spin orbit term of this kind, proportional to the ratio p/M, but much smaller than what was needed to fit the experiments. Spin-orbit terms were introduced also by Gammel and Thaler (5, 10)) and by other authors (7, 8) who considered also second order terms in X. The expression (13) for KS reproduces rather well some of the main features of the SM spin-orbit potential. In fact, if one considers, as we did in the case of the static potential, that the term V,, dominates, VZs is attractive; moreover, it falls off faster tha.n the static (central) part by a factor l/z. In agreement with the meson field-theoretic result, it is proportional to the mass ratio p/M;
MESONS
AND
NUCLEON&
57
IV
its magnitude is correspondingly smaller than the main terms in the static potential by this factor. However, even though the range of the spin-orbit pot’ential is predicted by our model to be less than that of the static potential (by the aforementioned l/x factor), this does still not seem to be sufficiently short-range to fit the requirements of the nucleon-nucleon scatt’ering data (5). However, if, for some reason not’ clear to the authors, the (eC/z)’ dependence should be appropriate to the spin-orbit potential (but not to t’he static terms) this might satisfy the requirements of short range as determined by the empirical fitting of the high-energy nucleon-nucleon scattering observations. The situation with respect to t,he possibility of a spin-orbit term in the Q = 0 states is rat,her uncertain experimentally (5). Our model also generates a term but its sign and its order of magnit’ude depend strongly on the assumptions vt, , about the terms VI1 and VB1 (see previous section and Table II), and it can easily be made very small. IMPROVED
APPROXIMATION
FOR
THE
STATIC
POTEKTIAL
One can see from the previous sections that t,he atomic model in its simplified version can reproduce qualitatively many of the outstanding features of the nucleon-nucleon potential. The most interesting result seems to be t,he possibility of deducing a spin-orbit potential of the same kind as the term assumed in the GT and SM potentials. The result’s of the stat’ic calculation developed in the foregoing show, however, that is not possible to reproduce consistently all of the relevant features of the static potent,ial. In fact, it seems diflicult to obtain at the same time attraction in the triplet even states and repulsion in the singlet odd state. However, before we attribute this partial failure to the atomic model itself, we should investigate more thoroughly the effects of the approximations used in the calculation. In particular, one can be somewhat skeptical about the validity of the simplified picture, in which the ‘(atomic” structure of one of the physical nucleons is disregarded. Perhaps suc,h a simplified version suffers from its failure to take into account exchange effects between t,he two pions, which would be present in a complete atomic model computation. For this reason we have carried out a calculation of t,he stat#ic potential based on the complete application of the atomic model. We describe here briefly this calculation and give the results: The total Hamiltonian of the two-nucleon system which has now to be used is given by (1). According to the Heitler-London approximation, the unperturbed wave function corresponding to an eigenstate of given tot,al spin P, & , total iso-spin a, ma , and parity (&) can be written
e&,T,T3(1, 2) = 241, 2) + 42, 1)
(1.9
58
COSTA
AND
FELD
where the U’S are the appropriate combinations of the eigenfunctions uA(l), uB(2), UA(2), uB( 1) of the physical nucleons, as given in Part I. To the first order, the two-nucleon static potential is now given, aside from an additive constant, by the expectation value of the total Hamiltonian (1)) v
=
Hl+l
Hl2
(16)
K+A
where Hll = (U(l,2)
IHI U( 1, 2))
fJl2
=
IHI
K
= Ml,
(41,2>
42,
1))
(17)
2) I41,2))
A = (u(l,2) 14% 1)) Performing all the integrations implied by (17) and proceeding in analogy with the simplified model, we obtain the six nucleon-nucleon potentials ‘VC+, ‘VC-, “V,+, 3Vc-, 3VT+, 3VT-. These are given by the following expressions.6 --I 2 ‘vc- = 5 dTN(zhl + 8~13Pb) + (Su33+ ~31VYd ( X ) - 3VllW( +P(
z)]
--2 2 lv,+
=
5 (
>
cp(~)[(ll~,l
+
16df%)
+
(16~33+
lh)g%)
+ 15Vn W(7)h”)(x)1 3 VC.T
e-32 2 (p(~)[(llu11 + 161)13)f% + = ; ( )
(16~33+
(18)
ll~dg%(x)
+ 15Vll
wmx~)1
We reproduce in the Appendix the explicit expressions for f(z), g(z), h(z), and cp(T), U( 7)) functions of x = CYTand 7 = arc , which appear in (18). The first two terms in the brackets come from the “direct” matrix elements HU , the third one from the “exchange” matrix elements Hrz of (17). It turns out that the “exchange” interaction dominates over the “direct” one, as was the case in Part III for the pion-nucleon scattering phase-shift computation. We now compare our results with those obtained by the authors already cited. 6 Note that the ujt’s p/l37 012N 8 Mev.
are defined
as in Part
III,
and
that
the
potentials
are in units
of
MESONS
r
HAMADA’S +
(5)
AND
NUCLEONS.
59
IV
ATOMIC MODEL
POTENTIALS
POTENTIALS
I
V(x)
\
2
r
o-
0.5
I
15 (pr3
-
2
0
I
I
05
I
I X-
1.5
2
FIG. 1. Nucleon-nucleon potentials. We have plotted the dependence of the potentials on the separation, with the asymptotic (Yukawa) behavior removed, in units of pc2 S 137 Mev. (a) The potentials of Hamada (7), based on experiment plus meson field theory. (b) Results of one of the best fits of the full atomic model computations.
the sake of convenience we shall consider, in particular, the results of Hamada (7)) since these are available in analytic form. These results have been plotted in Fig. 1 (a). Let us examine the expression (18). We not’e, again, t’hat the “exchange” terms are the most important ones and that these are proportional to the potential vll of the pion-core interaction. Since this potential is fixed, for a given range, 7, by the static properties of the physical nucleon (2)) we anticipate that the essential features of the potential, insofar as they are provided by the exchange terms, are essentially fixed and that it will be rather difficult to deviate appreciably from them. As a consequence, as an inspection of the exchange term shows, it will also be difficult to reproduce all the features of the Hamada potential. Thus, keeping the exchange terms only, the potentials (Eqs. (18) ) may be For
60
COSTA
written,
to within
AND
FELD
a common factor ‘v,-
-
lv,+
-
- L!llh(1) 5vuh (1)
“VF,, 3 + VC.T -
5vnh$ -vdm
(3)
Evaluation of the factors h(x), given in the Appendix, leads to the relationships between the various potentials as shown in the last three columns of Table II, corresponding to values of x: = 1,~ = 2, and CE>> 1, respectively. Qualitatively, these results indicate a reproduction of many, but not all, of the salient features of the observed potentials. The most serious failure is for the 3VT+ term. In order to see how far we can go with the atomic model, we adopt as parameters, together with the range 7 of the pion-core potential, the three ratios among the potentials Ujt = vjt/vlr . We now try to find the best fit that can be obtained to the Hamada potential. We limit our considerations to J: > 1, as this is the only region in which we expect our computations to be reliable. As a first step, we adopt as the relationships among the terms of the Hamada potential in the region’ x cv l-2. (a)
IV,-
‘v -‘V,+(-2)
(b)
3Vc+ ‘v ->$‘V,+( (e)
+M)
(c)
“vc-
(d)
3V,+ ‘v ‘V,+(
“VT- ‘v -M’V,+(
N /l&J’vc+(
-45)
+2)
-x)
(20) The important features are, as previously discussed, the strong and attractive ‘Vc+ and 3VT+, and the strong and repulsive IV,- (relations 20(d) and (a) ) . The other features-the small and even uncertain values of the triplet-central potentials (20(b) and (c)) and the medium and repulsive 3V,--are less firmly established, both experimentally and theoretically; for example, the potentials in Table I are in complete disagreement with respect to 3VCf. Disregarding common factors in (18)) our potentials are IV,-
-
(1 + 8u,,)f”‘(z)
‘VC+ -
(11 + 16~13)f(‘)(2)
“G,,, 3 + V c.-r -
(11
+ (82~~~+ u31)gC1)(z-) - 3w(T)h(‘)(x) + (162433+ llu3Jg(‘)(z)
+ 15+)h”‘(r) (21)
(1 +
+
16~13)f&%(~) Su,,)f%(2)
+ +
(162433
(8~33
+
lW&(x)
+ u&$(4
+
15w(+&(r)
- 34&%(x)
We have tried to fit some of the above conditions (20) and to see what happens to the other terms. What we find is that, while it is possible to fit almost all of the above relationships (“almost” refers to the fact that we have to weaken some of them), the conditions for such a fit are not acceptable from the point 7 The sponding
relationships coefficients
shown correspond to z N 1; the values of lVc-+ for z z 2.
in parentheses
are the corre-
MESONS
AND
NUCLEONR.
IV
61
of view of the atomic model. If, indeed, the values of the ratios u13 , us1 , and u33 are to be consistent with the results on pion-nucleon scattering previously derived (3)) and also with the requirement that there be only one bound stat’e, the ( 1, 1) state, the values of u13 , ual , u33 are limited both in magnitude and sign; vll of course must be taken to be attractive and strong. However, for a given solution (i.e., assumption of any three out of the five conditions (20) ) the requirements of the nucleon-nucleon potential determine t’he pion-core potentials, u, without regard to considerations of pion-nucleon scattering. Thus, some of the signs turn out to be wrong if we try to fit the conditions 20 (a) and (d) simultaneously (solutions which assume (a),(b),(d); (a),(c),(d); (a),(d),(e). Moreover for such solutions the terms u13 , ual , u33 turn out to be t,oo large. The most reasonable set of solutions seems to be the ones reproducing the central terms ‘VC*, 3Vo’ (conditions 20 (a, b, c)) . For such solutions the values and signs of the u;i’s are consistent with the requirements of the atomic model, and wit.h the values used in the pion-nucleon scattering problem (I). For instance, for t’he values 7 = 0.6 and cy = 0.8, we obtain: ~13- -0.8 vll , v31 - 1.0 v11, 2133 - 0.2 Vll ; the nucleon-nucleon potentials corresponding to this choice of parameters are shown in Fig. l(b). With this and similar solutions, one obtains the right order of magnitude for the central terms but, unfortunately, the tensor terms turn out to have the wrong signs. The situation is analogous if we try to fit the G and GT potentials (see Table I). One might attempt to improve the situation by regarding the quantity w( 7) in Eqs. (18) as a parameter rather than using the expression (A$) given in the Appendix; W(T) represents the ratio of the contributions of the exchange and ordinary part,s of the interaction (Eqs. (16) and (17) ) . The expression (A.5) given in t’he Appendix is computed for pion-core interactions, vjt , whose shape is given by a “square well.” Different forms for the shape of these interactions would be expected to yield different values of W(T). It turns out that some reasonable decrease in the values of W(T) can yield some improvements in the fit by modification, in the right direction, of some of the unacceptably large values of the Ujt (other than ull). However, even with these improvements, we have not found it possible to obt.ain a reasonable solution which can satisfy, simultaneously, both the conditions 20(a) and 20(d). SUMMARY
AND CONCLUSIONS
Unlike the previous applications to the static properties of nucleons (Part I) and to the pion-nucleon low-energy interactions (Parts II and III), the atomic model is only indifferently successful in accounting for the properties of the static nucleon-nucleon potential in the range (r 2 &/PC) over which a one meson approximation is expected to apply. Even setting aside the failure t,o reproduce the Yukawa form--the model gives an ( e-*/zr)2 dependence rather
62
COSTA
.4ND
FELD
than (e-“/x)-which can be ascribed to the deficiency of a treatment of pions as “real” particles, rather than as quanta of a field, the computations fail in predicting the wrong signs for some of the important terms in the nucleonnucleon interaction; in particular, it does not appear t’o be possible, with a reasonable set of pion-core interactions, to account simultaneously for both the central and the tensor terms in the observed static nucleon-nucleon potential. The results of the computation, in the various approximations considered in this paper, have been summarized in Table II and Figure l(b) . On the other hand, the atomic model, even in its simpler form (in which one of the “atoms” is replaced by an “effective scattering center”), provides a straight-forward and simple “explanation” of the nucleon-nucleon spin-orbit interaction as arising from the dominant (3, 3) interaction between the pion and the physical nucleon, although it is not clear that we can reproduce, in a satisfactory fashion, the requirement of short range for the spin-orbit potential. We believe that in the computations described above we have extended the atomic model to the limit of its usefulness (if not somewhat beyond). In the previous papers, and to a lesser extent in this one, it has been shown how the dominant features in the low-energy interactions between pions and nucleons arise as a consequence of the properties of the single pion plus core term in the wave function of the physical nucleons. However, when we deal with systems involving more than one nucleon and thus, perforce, more than one pion, effects of pion creation and absorption, of pion exchange and, possibly, of the pionpion interaction are expected to increase in significance. We believe it. is reasonable to conclude, from the results reported in the foregoing, that such effects play a significant role in determining the low-energy nucleon-nucleon interaction, even in the asymptotic region. The successful computation of such phenomena requires, unfortunately, the full arsenal of field-theoretic techniques. APPENDIX
1. The functions F(s) and G(x) which appear in Eqs. potential, are given by the following expressions:
(7) for the static
(A.1)
MESONS
AND
NUCLEONH.
63
IV
T = (YT~defines the “range” of the interaction, as discussed in Part III. 2. An approximation (valid for T < 1) for the function F&Z, T) which pears in the spin-orbit terms, Eqs. (13) and (14)) is F&X,
T) CT% 1 + ;+ (
; + ;
>
T5
(A.21
3. Finally, we record expressions for the different functions Eqs. (18) for the static potential (improved approximation) : f(l)(X)
= f
g(l)(x)
= &
(9 + ; 18 + ;
+ 9
+ $
+ 2
p(7)
h’“(x)
‘v ;
t’P(x)
= f
#(CT)
= f
= 10e2’ 27+
w(T).(p(T)
= -
sinh27+
10’:J?t3’e-r{2
(
x + 5 + ; + i2
‘(1 7
+ F
in
>
>
(9 + $ + g + ; 18 + ;
appearing
+ g)
+ 228 + $ x3
(
ap-
+ $)
+ $4
+ $2
>
(A.3)
- cosh27)
*
sin f cash 7 + +p
T2 cost - ~sin .$ r2 + p t
(A.5)
6-l
COST.1
AXD
FELD
where
RECEIVED:
January
3, IcJcj2 REFERENCES
1. G. COSTA AND B. T. FELD, dnn. Phys. (SY) 9,354 (1960). 2. B. T. FELD, dnn. Phys. (XI-) 1,58 (1957). 3. B. T. FELD, An?&. Phys. (XV) 4, 189 (1958). 4. J. M. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics.” Wiley, New York, 1952. 5. J.L. GAMMELAND R.M. THALER, “Progress in Elementary Particle and Cosmic Ray Physics,” Vol. V, Chap. II. North-Holland, Amsterdam, 1960. 6. M. TAKETANI, S. MACHIDA, AND S. ONLTMA, Progr. Theoret. Phys. (Kyoto) 6, 638 (1951); ibid. 7, 45 (1952). 7. T. HAIXADA, Progr. Theoret. Phys. (Kyoto) 24,1033 (1960). 8. R. A. BRYAN, Wuovo cimento 16,895 (1960). 9. Y. GARTENHAUS, Phys. Rev. 100, 900 (1955); ibid. 107, 291 (1957). 10. J. L. GAMMEL AND R. THALER, Phys. Rev. 107, 291 (1957). if. K. A. BRUECKNERAND K. M.WATSOK, Phys.Rev.92,1023 (1953). 12. M. SUGAWARA AND S. OKOBO, Phys. Rev. 11’7,605 (1960). 13. H. FESHBACH, E. LOMON AND A. TUBIS, Phys. Rev. Letters 6, 635 (1961). 14. P. SIGNELL AND R. E. MARSHAEL, Phys. Rev. 109, 1229 (1958). 15. S.OKUBO AND R.E. MARSHAK,:Inn. Phys. (.yP) 4,166 (1958). 16. R. CIRELLI AND G. STABILINI, S~ppl.,\'~ovo cirnento 20,157 (1961).