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The linked-cluster expansion for the deuteron optical potential
INN.ILS
OF
PHYSICS:
The
46,
93-112
(1967)
Linked-Cluster Optical W.
Laboratory
for
F.
Nuclear Science of Technology,
Expansion for Potential*
JUNKIN
F.
AND
and Physics Cambridge,
the Deuteron
VILLARS
Department, Massachusetts
Massachusetts 02139
Institute
The Goldstone-Hugenholtz linked-cluster expansion for the transit,ion amplitude is derived for elastic deuteron scattering reactions. From this expression, the deuteron optical potential is derived, and the relat,ionship between this potential and the single-nucleon optical potential is established. I. INTRODUCTION
Many properties of t’he N-body system have been studied in terms of the Goldstone-Hugenholtz linked-clust’er perturbation series. For example, the wavefunct’ion and the energy of the ground state have both been expressed in terms of such a perturbation series (I)-( 3). The exact expression for Feshbach’s generalized optical potential for single-nucleon scattering has also been given as a linked cluster perturbation series (4)-( 6). The terms of such a linked-cluster expansion are expressed as matrix elements between independent-particle states interacting through the residual two-body interaction. Therefore, it is easy to establish the connection between the linkedcluster expansion for the transition amplitude and shell-model calculations for t,he transit’ion amplitude, such as t’hose done by Lemmer and Shakin (7) and by Weidenmuller (S), (9). Furthermore, the terms in the expansion that are responsible for various features of the transition amplitude, such as sharp energy resonances and the intermediate st’ructure associated with (‘doorway states” (IO), can be identified. From the numerical calculations that have been carried out by Lemmer and Shakin in examining a specific nuclear reaction (single nucleon scattering off “0 or ‘“N) it seems evident that this approach to the many-body problem can yield results which can be compared with experimental data (7). Obviously, it would be useful to extend the technique of t.he GoldstoneHugenholtz linked cluster expansion to deal with more complicated nuclear reac* This work mission under
is supported in part cont,ract AT(30-1)2098.
through
furlds 93
provided
by the
Atomic
Energy
Com-
94
JUNKIN
AND
VILLARS
Cons, such as nuclear reactions between composite projectiles. The experimental analysis of elastic deuteron scattering data is often made in terms of a deuteron optical potential. A recent analysis by Perey and Perey (11) seems to indicat’e t’hat the deuteron optical potential has a definite relationship to the experimental single-nucleon optical potential. In this paper we will derive an exact expression for the deuteron optical potential in terms of a linked-cluster perturbation series. The leading terms of this series will be examined, and the relationship between the deuteron optical potential and the single nucleon optical potential will be established. II.
FORMULATION
OF
THE
PROBLEM
We n-ill consider the elastic scattering of deuterons off the ground state of a nucleus of N nucleons, ]fN), which has an energy, Ef, . It will be assumed that this state is a nondegenerate energy level (closed shell). The expressions will be written in the notation of second quantization and will be given in a basis of independent-particle Hartree-Fock states. The following notat,ion will be used: Let U be the self-consistent Hartree-Pock potential for a system of N nucleons. The wavefunction for an independent-particle Hartree-Fock state is written (F~( X) , and it satisfies the equation (fi = 2?7l = c =
1)
-
V&&r)
+ J d32’(2 1Vlx’)&r’)
= &&E)
(11.1)
Let af be the creation operator for such a state. For continuum states, OL refers t#o the asymptotic momentum of the incoming plane wave of the scattering state ( as well as spin and isospin). The Hartree-Fock states are normalized so that’ {ak, aa) = S((Y - p). If it is assumed that the only interaction between nucleons is the two-body interaction, the Hamiltonian for a system of any number of nucleons can be written in terms of the Hartree-Fock creation and annihilation operators as follows : H = c
0
hUa+U, + 5 5
(a@ 1dA 1?'~)U,+ULJ+U~U,
(a 1u 1fl)U,+Up
- z
(1I.a)
rX
(~4 I dA /rh) Specifically,
is th e antisymmetrized
((Yp 14.4! 7x) = 1 dxl dxZ dx3 dx4 ( 21%
two-body
interaction
between
I J / ~324)Pa*bl)(PB*(zz) . ((oy(x3h(24)
The Hartree-Fock independent particle st,at#e iNO). In this state all the orbitals (=
nucleons.
(11.3) -
Y)h(23by(24)).
ground state for N nucleons is t,he one-particle states) with energy less
LINKEWCLUSTER
EXPANSIOS
FOR
DEUTERON
OPTICAL
POTENTIAL
8.5
t,hxn ts (the Fermi energy) are filled and all &hers, unoccupied. I7 is the usual Hartree-Fork potential for N particles, defined by the equat8ion (a I C’ 18) -A5
* ”
(ax I J.4 IPX)
(11.4)
The notation c XIN, means t’hat t,he summation extends over all the orbit:& for which EA 5 q , i.e., that the summat,ion est#ends over all the occupied states of IN,,). It is convenient at this point to introduce the terminology of particle st,ates and hole states. il hole state refers to a Hartree-Foc*k orbital nith energy less than the Fermi energy, while a particle stztte refers to a Ante \vith energy greater than q . The state IN,) plays the role of the vacuum in this terminology, so that the ~nzmber of holes refers to the nuniber of orbit& of [N,) that, are rwt occupied. Thus, a one-particle stat.e refers to a st’ate of N + 1 nucleons with all t’he orbitals of IN,) and one additional Hartree-Fwk orbit,al being occupied, lvhile a one-hole state is actually a stat,e consisting of N - 1 nucleons, in lvhich one and only one of t’he stat’es of IN,) is unoccupied. ak is t,he cr&ion operat’or for a p:article if E, > q , but a, is the creation operator for a hole if cLI! 5 Ed. The number operator for a parMe state is N, = a P‘tl p , ep > ~1 , while t,he number operator for holes in Nh = ahah+, Q <= tf . The Hamiltonian can be n-ritt#en in a particle-hole representation as follows: H = EN + c cpNp P The residual interaction
c
EON,, + I’ = H, + V.
(11.5)
h
can t#hen be divided into three parts: 6’ = v, + v,, + VP), .
(KG)
T’, describes particle-part*icle scatkering, Vh describes hole-hole scattering, and I’,,, describes the remainder of the int#eraction, which involves the scattering of both particles and holes, as well as particle-hole pair creation and annihilation. As examples of the explicit Hartree-E’ock representation of some of these operat,ors, (11.6a)
The creat’ion operators for single-nucleon st’ates have been defined. Similarly, one can define the creation operator for a two-nucleon state such as the deuteron state. Thus, let /K) be a plane-wave deuteron state of momentum K and energy E, . (E, includes both the kinetic energy and the binding energy of the deuteron.) Let AK’ be the creation operabor for such a state: IK) = A,f 1O),
96
JUNKIN
AND
VILLARS
where IO) is the true vacuum state with can be written in terms of Hartree-Fock Ad The amplitudes from
= 5 CK(c@>Udq9+.
C,( c@) satisfy
a S&r&linger
HAd (CT + q3 - EK)Cz&a
no nucleons. Of course, this operator single-nucleon creation operators:
+ ; g
(11.7)
equation,
IO) = EKAd
which can be obtained
IO):
(II.Sa)
(apI 3d 1uX)CIc(uX> (IISb)
-
F
(a 1Ul U)CKbP)
- 5
(PI UI U)CKbU)
= 0.
The normalization will be chosen such that (0 1APAX’ 10) = 6(K’ - K). This operator, AK’, will now be used to generate the deuteron-nucleus scattering state I@:&), (a (N + 2)-nucleon state). In order to obtain this state, it is useful to define the operator Jd = [H, Ad] E’,A,+. Notice that (IMa) implies that JR IO) = 0. One possible representation for the scattering state jJ(r$$) ( an eigenstate of H with energy E = EjN + EK) can be obtained by applying JR to the state IfN), giving
Jd IfN) = (H By the inversion of this equation geneous equation, one obtains
(-%v + &~)AK+
and the addition
I\k:,fi) = A&~N) + EfN + &t These scattering
states are normalized (\E:,+k I\E:;;t)
amplitude,
to the homo-
(11.9)
H * irl JK+IfN)’
lxl?:&)
= 6(K
of Ad). the X-matrix
(as a consequence of the normalization In terms of these scattering states, fined as usual by
where the transition
of the solution
so that
= (9&k
SRJEC= (@;;;kt 1*;;jJ
ifs).
= 6(K - K’)
- K’)
for elastic scattering - %riT,l,
TWX , for elastic scattering
,
is de(11.10)
is given by
T xr+c= (f~ IJF Is:‘,k) \ B,++&T+~
‘_ H+i/~+ fN
K
f”/
(11.11)
LINKED-CLUSTER
EXPANSION
FOR DEUTERON
OPTICAL
POTENTIAL
97
This expression is exact, except that the recoil of the nucleus IfN) has been ignored. (In other words, it has been assumedthat the nucleus is infinitely heavy relative to the incident deuteron.) Since the expression is written in the notation of second cluantization, the necessary antisymmetrization of all the expressions is automatically taken into account properly. This paper will st,art with this exact expression for the elastic transition amplitude and derive 211 expression for the effective potential for two-nucleon scattering react,ions. This effective potential will be able to produce deuteron breakup as well as elastic deuteron scattering, since this potential will act on each nucleon of t,he deut,eron. The generalized deuteron optical potential will then be derived from t,his effective potential. When deut’eron scattering is studied, it is evident that there are at least two complications that would not be present if the incident projectile were a single nwleon. The first problem is the lack of orthogonality between the incident deuteron and the bound Hartree-Fock orbitals. If the incident projectile were a single nucleon, it could be identified wit’h the appropriate continuum HartreeFock orbital. Of course, such a continuum stat,e is orthogonal to all bound Hartree-Fock orhitals. However, when the incident projectile is a composite projectile (,a deut8eron), the expansion of the deuteron state in terms of Hartree-Fock single-nwleon states requires t’he use of bound Hartree-Fock orbitals below the Fermi level, that is, hole stat#es.Thus, even in the lowest approximation, where the nuclear ground state \fN) is appi&mated by the Hartree-Fock ground state, _ IN,), there is 311 overlap in phase space between the incident deuteron and the target nucleus. A second complication arises because the proton and the neutron of the incident deuteron interact. This interaction is an intrinsic part of the deut,eron and it, cannot scatter the deuteron. On Oheot#herhand, the two-body interaction between t,he proton of the deuteron and a neutron of the nucleus should produce scattering and contribute to the transition amplitude. However, all neutrons are identical, and there is no obvious way to distinguish between these two int#eractions, although any physical picture of the rea&on would indicate that’ only the second type of interact’ion will contribut’e to the transit,ion amplit,ude. In order to eliminate these complications, it is useful to define a two-nucleon eigensmte of a modified Hamiltonian, H, , given by H, = c tpNp + VP (11.12j P This Hamiltonian has been chosen so that it has eigenstates which consist only of particle states (states which are orthogonal to the orbitals occupied in IN,)). However, enough of the two-body interaction between the nucleons has been included in this Hamiltonian so that it is still possible to define a twonucleon eigenstate that satisfiesthe asymptotic boundary condition of having an
9s
JUNKIN
AND
VILLARS
incoming (or outgoing) component which is a plane wave deuteron of momentum K. Since [P, H,] # 0, such an eigenstate will be a scattering st#ate and the scattered wave will consist of scattered deuterons, free proton and neut’ron states (corresponding t#o the deuteron breakup reaction), and states with one nucleon in a bound state A, 0 > Q > er, and one nucleon in the continuum. Let jK’*‘) be the two-nucleon eigenstate of Hz . Hz (Kc*‘)
= E, lK’*‘)
(11.13)
The (&) indicates that the scattered wave consists asymptotically of radial o”tgoipg ) two nucleon states or in other words, the only (~“uctog~n”,“)part of fi”Tz a deuteron of momenkm’ K. The creation operator A,+ which creates a two-nucleon state (a plane-wave deuteron of momentum K) has already been given by
AK+= 2 wd%+a~+, AR IO)= IK) Similarly, we can define the creation state IK’*‘) to be D~cA, ,
D&c*, IO) = IK’*‘),
operator D&u
for the two-nucleon
= c zDKc*-,(a/3)&+as+. a@
However, since Dkc+, creates an eigenstate in terms of particle states,
of Hz, it can be written
D;c+t, = c DK(j-) (p1&&,& PIP2 The integral equation relating
]K’*‘)
[W.7)1
.
scatt,ering (11.14) entirely (11.15)
to IK) is given by
Dhi) 10)= (1 + (11.16) 1k”*;‘)
= (1 +
where Vz is the difference between the Hamiltonians Vz = Hz - H.
determining
IK) and I@*)), (11.17)
In nuclear reactions in which a single nucleon is scattered elastically off the nucleus lfN), the problem of solving the (N + 1 )-body problem can be reduced to the problem of solving the one-body scattering problem in the presence of a potential, the generalized single-nucleon optical potential. Our objective is to reduce the (N + 2)body problem (two-nucleons incident on ]fN)) to the problem of two bodies scattering off an effective (two-nucleon-nucleus) potential. Thus, it is our objective to write the elastic scattering transition amplitude in the form
LINKED-CLUSTER
EXPANSION
TK’K= (K’ lli, II(
FOR
+ (P
= (K’ IV2jP)
DEUTEROK
OPTICAL
JW(&) IX(‘))
+ ((Ri(-) (W(E,)
(11.1s)
1 . ( I+
E,
99
POTEiYTIAL
-
H,
-
W(EK)
+
i?j
W(EK)
jK’f’,)
\
>
states. Of course, this is the n-here I&), jK’+‘), and jli-“‘) are all two-nucleon well ltnown expression for the transition amplitude for the scatkering from the sum of t,wo potentials. Such an equation defines the effective potential for tnonucleon scat#tering reactions to he W’,ff(EK)
= vs + W(E,).
/J?‘+‘) is the eigenstate of Hz + W(E,) amplitude could also be writt’en
(11.19)
= H + W,,,(EK)
TE;tK = (K’ ( W’,fr(ER)
and t’he s&tering
/ I?‘+‘).
(11.20)
In Section III of this paper, such an expression will be derived for the transiamplit#udes. The potential I;, will be examined. ?T’(EK) gives corrections tKJ the potential V? , and the linked-cluster perturbation series for W(E,) will be derived. In this manner, a11 exact expression will be derived for t,he effective potential, Fv,fi(EK) = V2 + W(E,).’ In Section IV, t’he deuteron optical potential will be derived from this effective pot,ential by eliminating the other channels, such as t’he deuteron breakup channel. The relationship bet’\leen the deuteron optical potent’ial and t#he singlenucleon optical potential will then be examined. tkJI1
III.
DERIVATION
FOR
THE
EFFECTIVE POTENTIAL
(TWO-NUCLEOX-NUCLEUS)
One possible representat,ion for I*;$), the t’otal scattering nucleons, has already been given in t’erms of AK’. j \k:;k)
= (-4=+ +
E,, + E, ’ - H f iv
states of N + 2
JK+ iv), )I
[(II.sj]
JK+ = [H, A,+] - ICK A,+. However,
another possible representation
1 In order to make the Bruckner theory concerned with this
for such states may be obtained
such a linked-cluster expansion converge, (for example, if the residual interaction additional complication in this paper.
by
it may be necessary to apply is singtdar). We shall not be
100
JUNKIN
using the operator &+,
AND
VILLARS
[Eq. (11.14)],
I!?gk) = (Dh
+
J&w
l
EfN -I- EK - H f
= [H, D&l
iq
JL *) I fN), >
- ExD&<+).
(111.1) (111.2)
This representation for the states \%i’,k) is much more convenient when one is deriving the linked-cluster expansion for the deuteron optical potential. In terms of this representation, XKIK = (\k:,kt
[UI.lO)l
I$$.) 1 + DK’(F) EfN + EK - H + iv
and from the S-matrix
the following
T-matrix
is derived
\k:;jK >
(see Appendix
Tivir = ~FK + T$k , tK’K
T Sk
= (f.v I JKt(-I = (IN (Jgf(-)
=
(0
/ AKtV2&+,
10)
=
(K’
(111.3) I) : (111.4)
1 vz
1 Kc+‘),
(111.4a)
I\kjzk) (DL<+j
+ EjN + E ‘-
H + irl
K
JL(+)) I .fz~j- (IIIAb)
To derive the linked-cluster expansion for T$k , we make use of a relation between Tgk and the deuteron Green’s function (which will simplify the calculation) . This is based on the following. THEOREM. Define the Green’s functio?l: G(w;
K’p K) s
1DKr(-)
-&
D;c+, IfN
(111.5)
Then lim .b-K=BK’
(20 - EK - EfN)(w
- E,f
- Ef,)G(w;
K’, K) = T$k.
(111.6)
u~-rEK+E,~+it,
This theorem is essentially the nonrelativistic form of the well known fieldtheory result, i.e., that the transition amplitude is just the residue of the secondorder pole of the retarded Green’s function at the scattering energy. (A proof of this is given in Appendix II.) Our objective is to expand this expression for G(w; K’, K) in powers of the residual interaction between the “D” states and the nucleus. To do this, we define a zero-order Hamiltonian, Hn . HD = H, + PVP
= H, + PV,P
(111.7)
LINKED-CLUSTER
EXPANSION
\vhere P is the projection
FOR
operator p = $5 c
DEUTERON
OPTICAL
POTENTIAL
101
given by: &dl,
(111.8)
INOHNOI apy%q
PIP2
Inspection of the properties of Ho shows that both IN,) and DLc*) \N0) are eigenstates of Ho . Furthermore, all N + 2 nucleon states of the type 2 - 1 particles, 1 holes (1 # 0), are also eigenstates of Ho , since PVP = 0 for such states. The residual interaction in powers of which we will expand the expression for G(w; K’, K), is thus
Vn = H - HD = V - PVI’
= V - PV,P
[W.8>1
It should be noted that V, allows nucleons to interact with each other via the D cannot scatter a two-particle t#w-o-body interaction with one exception-V state into another two-particle state. To develop the perturbation series, we define the operator: (To(
x,
y)
E
e-p-u)*e-~*o,
(6/6X)~~D(X,
Uo(2, x) = 1, y) = -V,(z)Uo(z,
This leads to the well-known
V,(z)
y),
E e+HDJrDe-ZHD.
(111.9)
series expansion for UD(x, y) :
. {V,(u,)V,~uz)VD(UQ) . . . V,(u,)) (III lo) ClUl clu2. . ' nu, =%; C-1)"j Z>U1>U~>"'>ll,,>2/ . v,(u,)v,(u,) . . VD(7.4,). We no~v use ITo to establish lim Ti,( 0, - p) 1N,) p-+”
a relationship
between
IfN) and IN,).
= lim eBENeP / N,) B-m (111.11)
1.f~) = lim exp MEf, P-fUsing this expression
UD(O, -PI I No) (.fNINo) *
for If N) in the deuteron Green’s function,
- ENI1 I (fNI N4”
= lirn ev [2P(Ef,
8-l)
- EN)~
one obtains
102
JUNKIN
AND
VILLARS
(111.12)
. (No 1U,@? + 5, ~)DF-J Dp(-)
(x)ePDL+)
UD(O, -0)
1No)
(2) = eZHDDKt,-) ePD.
Using Wick’s theorem to remove all graphs in which not linked to the operator DKl(-) or D&lc+~ , (No ) U&3
+ x:, x)&e)
(2) Udx,
= (No I UD(P + 2, x)Dr-I
0)Dfrw
Co(0,
-0)
the interaction
V. is
1No)
(r) UD(z, O)D!A+, Un(0,
-p)
. (No I UD(P + 2, -PI
] NJ,
(111.13)
I No).
However,
lim (NoI UdP + z, -PI IN,) 8-fm G(w;K’,K)
=
lim Be-
Im w>O
.(No! U,(P
+ x, x)Dg,(-I
Since the expression out, and the limit p + G(w; K’, K)
= lim exp [(2p + z)(gN - E,,)] B+m ia dx exp [z(,u) - El,)] (-1) s0 (x) U,(x,
] (N,, IfN) 1’; (‘I’*“)
(111.15)
O)Dfcc+, Uo(O, -p> I iV,)L
is linked, all the integrals 00 can be taken. Therefore,
converge and can be carried
= %$ mz g .
i
No
I(
! H
VD E N
D
1 w + EN - EfN - H, We now examine this expression
n DK,(-)
1 ~JJ+ EN - EfN - Ho EN
(111.16)
1 VD mN \ - H, >I O/L
to find the terms that have the requiredsecond
LINKED-CLUSTER
EXPANSION
FOR
UEUTERON
OPTICAL
order pole at w + EJN + E, + iv. Obviously, the terms with at most a single pole in the complex w plane. For n = m = 0, lim
EI(=EK P
103
POTENTIAL
(w - Ef, - EK)(w - E!,” - E,t)C:(w; k”, K)
1 = 0 can have
= n=nz=O
,M-E,~+E~+~~
(111.17)
w + EN - EfN - H, 1
w + E, - EfN - H, >
2+1 D&c+,
/No>
I,
For n or m # 0, the intermediate state must be an eigenstate of HD with an eigenvalue greater than E, (from the linked requirement). Therefore, the terms with n or m # 0 can have at most a single pole in the complex w plane at the scattering energy. Therefore
This is t)he desired series for T$k . Each term in the power series can be represented by diagrams, and the contribution from the linked diagrams can be calculat,ed. In order to extract the effective pot’ential from this expression, we note that’ t.he initial and final state is a two-particle state. Furthermore, a two-particle state may appear as an intermediate state in the expression for T$k . Therefore, we define an irreducible part of a diagram as being any part that cannot be divided into two parts by breaking two particle lines. In other words, a twopaAicle state never appears as an intermediate state in any irreducible part of a diagram. The t’erm CY( E,) in the effective potential, We,,(E’,) = IT2 + W(E,), is then the sum of all such irreducible parts, since the complete expression for Tkyk is obviously the iteration of such irreductible parts. To obtain this result mat,hcmatiaally, we define the folio\\-ing operator:
104
JUNKIN
AND
VILLAHS
Obviously, the contribution of all possible interactions in which a two-particle state scatters into another two-particle state without forming such a state as an intermediate state is given by (NO / ~~~cz~~W(E~)u~~ca~, ( N,). Once a twoparticle state is formed as an intermediate state, it must propagate as a “D” state until it has another W(EK) type of interaction. Thus, the infinite series for !P$k can be rewritten in terms of the operator W(E,),
We now use the fact that IN,) = flh u; IO), and that both W(E,) and Ok,*, commute with all CL*+.In addition, we use the following relationship between H, operating on a two-particle state and H, operating on a two-nucleon state: HDD+
(q
uhi)
10)
=
HD
d)D+
(q
IO) (111.21)
= (p Thus, we can commute obtain Tk’3c = 2 (O 1D,J(-) =
0 j Dgrc-)
the uFT operators
WL%)
(E
R
_ ;
2
d)
(Hz - E,)D+
t,hrough the expression
+ iq W(EK))’
to
(111.22) 1 W(E&
+ iq
W(Er.)
>
Thus, the exact expression for the transition amplitude scattering has been reduced to the following expression:
+
for T$:
DL<+, (0)
W(E,)
EK - Hz -
T K’K = (K’I
IO).
DLc+) IO for elastic deuteron
vz 1K’+‘) Eg - He -
The effective potential for reactions by the nucleus IfN) is therefore Wed&)
1 W(E.y)
+ iv
where two nucleons are elastically
scattered
= V, + W(E,).
[(11.19)1
Let us now examine the nature of V, and some of the terms of the linkedcluster expansion for W(E,). V, is the difference between the modified Hamil-
LINKED-CLUSTER
EXPANSION
FOR
tonian H, and the Hamiltonian va.rious terms of lit .
DEUTERO?;
H. Therefore,
OPTICAL
105
POTENTIAL
we can work
out. explicitly
the
V, = H, - H, Hz = c
[(11.12)]
cpNp + lip,
[(11.6,2)]
H = c ~.ua+am + ; 2 (cyp 1\‘A 1yX)u,+up+axu, 01 YX
vz= -c q‘uh+cqL + u - 4I h
(at least
C’
oneholestate)
-
U, [(11.2)]
(apj
JA 1 ~h&‘?+~~~,
The first term of VT has zero matrix elemenk in the expression for TK’Iz( Ch Eh(K’ / ahtab I Kc+‘) = 0, since the “D” states have no components corresponding to hole stat’es and do not produce any scattering). It, may t,herefore be omitted, so that
Va= u- ?*
C’
iafij
dA ~$+,+~,9+~~~,
(at least one hole statr)
(111.23)
.
The first term of VP is just the Hartree-Fock potential. This is a one-body operator t.hat acts to scatt,er each of the incoming nucleons.’ The second part of VZ is a two-body operator that would not be present if the incident projectile were a single nucleon. This part of 1’2 is an expression of the effect of t’he exclusion principle on the motion of t#he deuteron as it, propagates through t’he nucleus. In free space (if no other nucleons were present’), the proton and neutron of the deuteron can interact freely with each ot.her via the two-body intera&ion and scatter each other in and out of any Harkee-Fock states. However, as the deut,eron propagates through the nucleus, certain Hartree-Fock states are occupied by the nucleus. Because of t#he exclusion principle, ttie proton and neut’ron of the deuteron can no longer interact to scatter into these occupied stnt~es. Thus, the exclusion principle “weakens” or removes part of the interaction between the proton and neukon of the incident deuteron. This “weakening” of t’he interaction that holds the deuteron together can break up the deuteron or scatter it elastically. W(E,) gives corrections to V2 . W(E,) is explicitly energy-dependent,, and it will have an imaginary (non-Hermitian) component if any inelast#ic process (other than deuteron breakup) is energetically possible. The linked-cluster 2 The Therefore,
Hartree-Fock VP includes
potential the Conlomb
gives t,he scat,terirsg
approximate and hreakup
Coulomb field of the deuteron.
of
the
nnrle~~s.
106
JUNKIN
AND
FIG.
FIG.
VILLARS
1
2
expansion for W(EK) has been given, and an examination of this perturbation series shows that W( E,) contains corrections to U, the one-body part of Vz , and to
the two-body part of Vz . Some of the diagrams contributing to the one-body part of W ( ER) are given in Fig. 1. These diagrams plus the Hartree-Fock potential are the diagrams which define the single-nucleon generalized optical potential. The only difference is that the one-body part of WrEf( E,) is the single-nucleon optical potential calculated “off the energy shell,” since the energy in the denominators of the linked-cluster expansion is E R - ~6 , rather than the energy of the nucleon being scattered. Of course, this makes no difference when the Hart’reeFock potential is being calculated, since U is not explicitly energy dependent. Thus, adding all the one-body parts of W(E,) to the Hartree-Fock potential, we find that the one-body part of W,,,(EK) corresponds to the sum of two single-nucleon optical potentials calculated off t#he energy shell by roughly half the binding energy of the deuteron. W( EK) also gives corrections to the two-body part of V, . Some of t’he diagrams that contribute to the two-body part of W( EK) are given in Fig. 2. It is possible to int,erpret the diagrams of Fig. 1 as representing reactions in which one of the incident n&eons interacts with the nucleus to produce an excited intermediate state and t’hen the same nucleon interacts to return to the nucleus to its ground state. In this interpretation some of the diagrams of Fig. 2 would represent one of the incident’ nucleons producing the excited state and the other nucleon inter-
LINKED-CLUSTER
EXPANSION
FOR
DEUTERON
OPTICAL
107
POTENTIAL
acting to return the nucleus to it.s ground &ate, while some of the other diagrams of Pig. 2 would represent, the simultaneous interaction of the incident nucleons with the nuc#leus to produce excGt,ed intermediate states. Obviously, all these terms are unique t,o reactions in whic*h incident, projectiles, composed of t’wo or more nucleons, are s;cat#tered elastically by t#he nucleus IfN), and there is no relat,ionship between the two-body part of W(E,) and the single-nucleon generalized optical potential. IV. THE A.
DERIVATION
OF
THE
IlEUTERON EXACT
OPTICAL
GENERALIZED
POTENTIAL DEUTERON
OPTICAL
I'OTENTIAL
In the previous se&ion, the effective potential for a reaction in which two incident nucleons (the proton and neut,ron of the deuteron) are scatkered elaxtitally by the nucleus IjN) n-asderived. This effect,ive potential is not the generalized deuteron optical potential, for this effective potential can produce deuteron breakup and a certain portion of the deuteron stripping reaction as well as deuteron scattering. The deut’eron optical potential must describe only the elastic deuteron channel, and the presence of the other channels must appear as an absorptive part. It is, therefore, a potential which describes the center-of-mass motion of the deuteron. This pot’ential will tu~w be obtained from the effective potential, W,,,( E,). Let jJ?‘+‘) be the two-nucleon s&tering state in the presence of the effe&ve potential W,ff(F ,= ) , with the boundary conditions of having an incoming deuteron of momentum K, energy 8, , and outgoing scattered waves.
In terms of these states,
T R,K = (K’ I W,,,(&c)
I a(+‘).
[(ILao)]
la’+‘) has outgoing waves that asymptotically are scattered deuterons, two free nucleons in Hartree-Fock continuum states (corresponding to the breakup reaction), and states where one nucleon is in a bound Hartree-Fock particle state, and the other nucleon is in a Hartree-Pock continuum state (corresponding to a certain portion of the stripping reaction). Let Id) be the internal statme (in configuration spare, the relative coordinate wavefunction) of the deuteron. Then, we use the projection operat’or Pd = 1d)(fZl
(IV.1)
to projecbt out the part of I@+‘) that has the internal st’ructure of the deuteron, IF+‘)
= Id) p$+,“> + I$+‘),
Id) IDE’)
= Pd p7’+‘).
(IV.“)
108
JUNKIN
AND VILLARH
Id) jOi;“> has an incoming deuteron of momentum K, and outgoing scattered deuterons, while ICF’) hasall the other outgoing scattered states. Since [Pd , H] = 0, the equation (H
+
Weff(&))
IK’+‘)
=
EK
IK’+‘)
can be written as two coupled equations: (H
+
Wc?ffdd
(H
+
-
Weff,,
EK)
Id)
-
EK)
pk+)>
=
Weffdc
ICk”),
/ck+‘>
=
Weffcd
Id)
(IV.3a) (IV.3b)
lDk”>.
Using the boundary condition that ICk”) has only scattered waves, we obtain 1 ck+‘) = EK - Weff,, -H+iq
W effcd
4
pk+‘>.
(IV.4)
By means of this equation, we can eliminate ICF’) from Eq. (IV.3a), and after integrating out the internal deuteron coordinate, we are left with
Uopt =
- E, + CB+ Uo,,.(E,)) IOk”> = 0, 1 - Pd + WdEK) EK _ Weff,, _ H + ir, WdEK)
(IV.5)
(HcM.
(I
d W,rr(&)
W effcc = (1 - PdWeff(&c)(l
‘/j\
(Iv*@ (IV.7)
- Pd)
(CBis the binding energy of the deuteron). If DK(X) is the wavefunction associated with the state IOk”) then (-Xv”
- ~zP)DK(+) (X) +
s
d3X’U,,t(Ex
; X, X’)Dg(+)(X’)
= 0
(IV.8)
and TK’K = (&
s
d3X d3X’ exp ( -~K.x)
This concludes the derivation Ud-W. B. THE RELATIONSHIP SINGLE-NUCLEON
t~k,~(~~ ; X, X’)Dd+~ (X’).
(IV.9)
of the generalized deuteron optical potential,
BETWEEN THE DEUTERON OPTICAL POTENTIAL
OPTICALPOTENTIAL
AND THE
It is often assumedthat the deuteron optical potential should be approximately equal to the sum of the proton and neutron optical potentials, averaged over the internal wavefunction of the deuteron (13)-(15). This assumption allows one to obtain the deuteron optical potential in terms of the single-nucleon optical potentials. Now that the exact expression for V+( EK) has been obtained [Eq. (IV.6)], it is easy to examine the exact expression and determine what terms are ignored when this approximation is made. Obviously, if this approximation is good, the second term of U,,,( EK) ,
LINKED-CLUSTER
EXPANSION
FOR
DEUTERON
OPTICAL
1 - Pd EK - H - Weff,, + it, Weff(EK)
POTENTIAL
;\ “/
109
’
must be small. However, this term includes the effect of closing out the breakup channel and a large portion of the stripping channel. Therefore, in most reactions, t(he berm i?r(d / W,,,(E,)
(1 - P,)S(E,
- H - Weff,,) W&E,)
1d)
(IV.7)
should be the dominant term in determining the imaginary part of Cr,,,,( EK). By contrast, most of the real part of U,,,t( EK) will probably come from the first term, (d 1 Weff(EK) / d). As was shown in Section III, the perturbation series for W,ff(E,) can be divided into two parts, a one-body operator (the Hartree-Fock potential Ir and all the one-body correction terms from W(E,)) and a two-body operator (part of which comes from the exclusion principle). Since the two-body operator part of Weff(EK) has no relationship to the single-nucleon opt’ical potential, the above-mentioned approximation ignores all such terms of W,ff( ER). However, as was shown in Section III, the one-body part of Weff(EK) is indeed the sum of two single-nucleon generalized optical potentials, the only difference being that these single-nucleon optical potentials are calculated “off the energy shell”. The optical potential as usually applied, is the result of an energy average of the generalized optical potential. If such an energy average is made, it is possible that the fact that the nucleons are being scattered off t’he energy shell (b) roughly half the binding energy of the deuteron) may be unimportant. In anJ event, the scattering of t’he individual nucleons by the Hartree-Fock pot#ential is taken into account properly, since the Hartree-Fock potential is not explicitl! energy-dependent. Thus, it can be seen that the approximation, which obtains the deuteron optical potential by assuming that it is the sum of the proton and neutron optical potentials averaged over the internal state of the deuteron, differs from the exact expression for lrOPt( EK) in three respects: 1. It ignures t’he second term of CTopt(EK), 1 - Pd E, - H - Weff,, + 6 This should not be a valid approximation if one is considering the imaginary part of CLdE,). 3. It ignores the fact that the individual nucleons are being scattered off the energy shell. If the Hartree-Fock potential is the dominant part of the total potential, or if an energy average is made over a snfficiently large energy interval, this may still be a very good approximat’ion.
110
JUNKIN
AND
VILLARS
3. It ignores terms which contribute to the two-body part of bV,ff(EK). However, these terms require that both incident nucleons interact with the nucleus, and, for reactions which occur at high energies or large center-of-mass angular niomentum, the contribution from these terms may be small. Therefore, it seems evident that there are certain types of deuteron reactions where this approximation will give a good approximation for the real part of the deuteron optical potential. An analysis of experimental data indicates that there are several sets of parameters for the deuteron optical potential which seem to give a good fit to the elastic scattering data. However, a recent analysis by Perey and Perey (11) indicates that (w-hen a spin-orbit part is added to the deuteron optical potential) the best fit is obtained by the set of parameters that describes a deuteron optical potential whose real part is close to the sum of the singlenucleon optical potentials averaged over the internal wavefunction of the deuteron. In this paper we have shown how to extract, from the exact expression for the deuteron optical potential, the terms which would lead to such a relationship. It will be interesting at this point to examine some of the correction terms to this approximation. It will be especially important to determine t’he correction t#erms to the imaginary part and the spin orbit part of the deuteron optical potential, since it is likely that these correction terms may be rather large. This aspect of the problem is being investigated at present. APPENDIX
I
The eigenstates of Hz can be classified by the boundary conditions satisfied by the eigenstates. The state created by D:c+, is classified by the “in” boundary conditions, since it has an incoming deuteron wave of momentum K and outgoing scattered waves. On the other hand, the state created by D&C-J is classified by the “out” boundary conditions, since it has an outgoing deuteron of momentum K and incoming scattered waves. Of course, there are other t’wo-nucleon eigenstates which satisfy other “out” boundary conditions, such as having two outgoing nucleons of momentum kl and kz or one outgoing nucleon of momentum k1 and the other nucleon in a bound state Y. The “out” states are not orthogonal to the “in” states, but one can be expanded in terms of the other.
+ c
s(k, v; K)D:,c-,
= c
S(CY;K)Dh
= SaR - 2ri6(Ea
- EK)taK,
a
ku
s(a; K) tczg = (0 ID& D;c+,
= Dkc-I
Vz A,’
IO)
- 2ri c 6(E, a
- EX)taK DLc-, .
,
(A.1)
LIKKED-CLUSTER
The S-matrix SK’K = (\k/,L . I*.;;:)
EXPBNSION
POR
DEUTEROli
for the elastic deuteron s&tering I*(‘,:- ) = <*:,;,
= (1 + El,
Using Eq. (Al)
*I
+ E
I*.;;;
) -
@l&r,
OPTICAL
111
POTENTIAL
reaction is given b? I\kj,;)
+ 6(K - K’)
K
in the equat’ion for Iqj,k> one obtains
I*L:! = (1 + E,N_ ;, _ H-ir, (H - E,, - w)D:~++
fN).
112
JUNKIN
(w - Eirr - Ef,>(w
- Ex -Ef,)G(w;
= (w - E,c - EfN) (f,v I&(-)
RECEIVED:
AND
VILLARS
K’, K) I&t+,
1~5)
June 22, 1967 REFERENCES
1. J. GOLDSTONE, Proe. Roy. Sot. London, Ser. A 239, 267 (1957). 1. C. BLOCH, Nucl. Phys. 7,451 (1958). 3. N. M. HUGENHOLTZ, Physica 23, 481 (1957). 4. J. 8. BELL, Formal theory of the optical model. In “Lectures on the Many-Body Problem” (E. R. Caianiello, Ed.), p. 91. Academic Press, New York, 1962. 5. H. FESHBACH, Ann. Rev. Nucl. Sci. 9, 49 (1958). Chap. IX. North-Holland, Amster6. G. E. BROWN, “Unified Theory of Nuclear Models,” dam, 1964. 7. R. H. LEMMER AND C.M. SHAKIN, Ann. Phys.27, 13 (1964). 8. H. A. WEIDENMULLER,~\~ZLC~. Phys. 76, 189 (1966). 9. H. A. WEIDENMULLER AND K. DIETRICH,NUCZ. Phys. 83, 332 (1966). IO. H. FESHBXH, A.K. KERMAN, AND R.H. LEMMER, Ann.Phys. 41, 230 (1967). 11. C. M. PEREY BND F. D. PEREY, Deuteron optical model analysis with spin-orbit potential. ORNL Report No. 1529, June 6, 1966. 11. F. VILLARS, Lecture Notes (Collision Theory), International Course on Nuclear Physics, Institute for Theoretical Physics, Trieste, Italy, October 3-October 15, 1966. 13. F. J. BLOORE, Nucl. Phys. 68, 298 (1965). 14. G. R. SATCHLER, Nucl. Phys. 21, 116 (1960). 15. S. WATSNABE,lV'Uc~. Phys. 8, 484 (19%).
OF
PHYSICS:
The
46,
93-112
(1967)
Linked-Cluster Optical W.
Laboratory
for
F.
Nuclear Science of Technology,
Expansion for Potential*
JUNKIN
F.
AND
and Physics Cambridge,
the Deuteron
VILLARS
Department, Massachusetts
Massachusetts 02139
Institute
The Goldstone-Hugenholtz linked-cluster expansion for the transit,ion amplitude is derived for elastic deuteron scattering reactions. From this expression, the deuteron optical potential is derived, and the relat,ionship between this potential and the single-nucleon optical potential is established. I. INTRODUCTION
Many properties of t’he N-body system have been studied in terms of the Goldstone-Hugenholtz linked-clust’er perturbation series. For example, the wavefunct’ion and the energy of the ground state have both been expressed in terms of such a perturbation series (I)-( 3). The exact expression for Feshbach’s generalized optical potential for single-nucleon scattering has also been given as a linked cluster perturbation series (4)-( 6). The terms of such a linked-cluster expansion are expressed as matrix elements between independent-particle states interacting through the residual two-body interaction. Therefore, it is easy to establish the connection between the linkedcluster expansion for the transition amplitude and shell-model calculations for t,he transit’ion amplitude, such as t’hose done by Lemmer and Shakin (7) and by Weidenmuller (S), (9). Furthermore, the terms in the expansion that are responsible for various features of the transition amplitude, such as sharp energy resonances and the intermediate st’ructure associated with (‘doorway states” (IO), can be identified. From the numerical calculations that have been carried out by Lemmer and Shakin in examining a specific nuclear reaction (single nucleon scattering off “0 or ‘“N) it seems evident that this approach to the many-body problem can yield results which can be compared with experimental data (7). Obviously, it would be useful to extend the technique of t.he GoldstoneHugenholtz linked cluster expansion to deal with more complicated nuclear reac* This work mission under
is supported in part cont,ract AT(30-1)2098.
through
furlds 93
provided
by the
Atomic
Energy
Com-
94
JUNKIN
AND
VILLARS
Cons, such as nuclear reactions between composite projectiles. The experimental analysis of elastic deuteron scattering data is often made in terms of a deuteron optical potential. A recent analysis by Perey and Perey (11) seems to indicat’e t’hat the deuteron optical potential has a definite relationship to the experimental single-nucleon optical potential. In this paper we will derive an exact expression for the deuteron optical potential in terms of a linked-cluster perturbation series. The leading terms of this series will be examined, and the relationship between the deuteron optical potential and the single nucleon optical potential will be established. II.
FORMULATION
OF
THE
PROBLEM
We n-ill consider the elastic scattering of deuterons off the ground state of a nucleus of N nucleons, ]fN), which has an energy, Ef, . It will be assumed that this state is a nondegenerate energy level (closed shell). The expressions will be written in the notation of second quantization and will be given in a basis of independent-particle Hartree-Fock states. The following notat,ion will be used: Let U be the self-consistent Hartree-Pock potential for a system of N nucleons. The wavefunction for an independent-particle Hartree-Fock state is written (F~( X) , and it satisfies the equation (fi = 2?7l = c =
1)
-
V&&r)
+ J d32’(2 1Vlx’)&r’)
= &&E)
(11.1)
Let af be the creation operator for such a state. For continuum states, OL refers t#o the asymptotic momentum of the incoming plane wave of the scattering state ( as well as spin and isospin). The Hartree-Fock states are normalized so that’ {ak, aa) = S((Y - p). If it is assumed that the only interaction between nucleons is the two-body interaction, the Hamiltonian for a system of any number of nucleons can be written in terms of the Hartree-Fock creation and annihilation operators as follows : H = c
0
hUa+U, + 5 5
(a@ 1dA 1?'~)U,+ULJ+U~U,
(a 1u 1fl)U,+Up
- z
(1I.a)
rX
(~4 I dA /rh) Specifically,
is th e antisymmetrized
((Yp 14.4! 7x) = 1 dxl dxZ dx3 dx4 ( 21%
two-body
interaction
between
I J / ~324)Pa*bl)(PB*(zz) . ((oy(x3h(24)
The Hartree-Fock independent particle st,at#e iNO). In this state all the orbitals (=
nucleons.
(11.3) -
Y)h(23by(24)).
ground state for N nucleons is t,he one-particle states) with energy less
LINKEWCLUSTER
EXPANSIOS
FOR
DEUTERON
OPTICAL
POTENTIAL
8.5
t,hxn ts (the Fermi energy) are filled and all &hers, unoccupied. I7 is the usual Hartree-Fork potential for N particles, defined by the equat8ion (a I C’ 18) -A5
* ”
(ax I J.4 IPX)
(11.4)
The notation c XIN, means t’hat t,he summation extends over all the orbit:& for which EA 5 q , i.e., that the summat,ion est#ends over all the occupied states of IN,,). It is convenient at this point to introduce the terminology of particle st,ates and hole states. il hole state refers to a Hartree-Foc*k orbital nith energy less than the Fermi energy, while a particle stztte refers to a Ante \vith energy greater than q . The state IN,) plays the role of the vacuum in this terminology, so that the ~nzmber of holes refers to the nuniber of orbit& of [N,) that, are rwt occupied. Thus, a one-particle stat.e refers to a st’ate of N + 1 nucleons with all t’he orbitals of IN,) and one additional Hartree-Fwk orbit,al being occupied, lvhile a one-hole state is actually a stat,e consisting of N - 1 nucleons, in lvhich one and only one of t’he stat’es of IN,) is unoccupied. ak is t,he cr&ion operat’or for a p:article if E, > q , but a, is the creation operator for a hole if cLI! 5 Ed. The number operator for a parMe state is N, = a P‘tl p , ep > ~1 , while t,he number operator for holes in Nh = ahah+, Q <= tf . The Hamiltonian can be n-ritt#en in a particle-hole representation as follows: H = EN + c cpNp P The residual interaction
c
EON,, + I’ = H, + V.
(11.5)
h
can t#hen be divided into three parts: 6’ = v, + v,, + VP), .
(KG)
T’, describes particle-part*icle scatkering, Vh describes hole-hole scattering, and I’,,, describes the remainder of the int#eraction, which involves the scattering of both particles and holes, as well as particle-hole pair creation and annihilation. As examples of the explicit Hartree-E’ock representation of some of these operat,ors, (11.6a)
The creat’ion operators for single-nucleon st’ates have been defined. Similarly, one can define the creation operator for a two-nucleon state such as the deuteron state. Thus, let /K) be a plane-wave deuteron state of momentum K and energy E, . (E, includes both the kinetic energy and the binding energy of the deuteron.) Let AK’ be the creation operabor for such a state: IK) = A,f 1O),
96
JUNKIN
AND
VILLARS
where IO) is the true vacuum state with can be written in terms of Hartree-Fock Ad The amplitudes from
= 5 CK(c@>Udq9+.
C,( c@) satisfy
a S&r&linger
HAd (CT + q3 - EK)Cz&a
no nucleons. Of course, this operator single-nucleon creation operators:
+ ; g
(11.7)
equation,
IO) = EKAd
which can be obtained
IO):
(II.Sa)
(apI 3d 1uX)CIc(uX> (IISb)
-
F
(a 1Ul U)CKbP)
- 5
(PI UI U)CKbU)
= 0.
The normalization will be chosen such that (0 1APAX’ 10) = 6(K’ - K). This operator, AK’, will now be used to generate the deuteron-nucleus scattering state I@:&), (a (N + 2)-nucleon state). In order to obtain this state, it is useful to define the operator Jd = [H, Ad] E’,A,+. Notice that (IMa) implies that JR IO) = 0. One possible representation for the scattering state jJ(r$$) ( an eigenstate of H with energy E = EjN + EK) can be obtained by applying JR to the state IfN), giving
Jd IfN) = (H By the inversion of this equation geneous equation, one obtains
(-%v + &~)AK+
and the addition
I\k:,fi) = A&~N) + EfN + &t These scattering
states are normalized (\E:,+k I\E:;;t)
amplitude,
to the homo-
(11.9)
H * irl JK+IfN)’
lxl?:&)
= 6(K
of Ad). the X-matrix
(as a consequence of the normalization In terms of these scattering states, fined as usual by
where the transition
of the solution
so that
= (9&k
SRJEC= (@;;;kt 1*;;jJ
ifs).
= 6(K - K’)
- K’)
for elastic scattering - %riT,l,
TWX , for elastic scattering
,
is de(11.10)
is given by
T xr+c= (f~ IJF Is:‘,k) \ B,++&T+~
‘_ H+i/~+ fN
K
f”/
(11.11)
LINKED-CLUSTER
EXPANSION
FOR DEUTERON
OPTICAL
POTENTIAL
97
This expression is exact, except that the recoil of the nucleus IfN) has been ignored. (In other words, it has been assumedthat the nucleus is infinitely heavy relative to the incident deuteron.) Since the expression is written in the notation of second cluantization, the necessary antisymmetrization of all the expressions is automatically taken into account properly. This paper will st,art with this exact expression for the elastic transition amplitude and derive 211 expression for the effective potential for two-nucleon scattering react,ions. This effective potential will be able to produce deuteron breakup as well as elastic deuteron scattering, since this potential will act on each nucleon of t,he deut,eron. The generalized deuteron optical potential will then be derived from t,his effective potential. When deut’eron scattering is studied, it is evident that there are at least two complications that would not be present if the incident projectile were a single nwleon. The first problem is the lack of orthogonality between the incident deuteron and the bound Hartree-Fock orbitals. If the incident projectile were a single nucleon, it could be identified wit’h the appropriate continuum HartreeFock orbital. Of course, such a continuum stat,e is orthogonal to all bound Hartree-Fock orhitals. However, when the incident projectile is a composite projectile (,a deut8eron), the expansion of the deuteron state in terms of Hartree-Fock single-nwleon states requires t’he use of bound Hartree-Fock orbitals below the Fermi level, that is, hole stat#es.Thus, even in the lowest approximation, where the nuclear ground state \fN) is appi&mated by the Hartree-Fock ground state, _ IN,), there is 311 overlap in phase space between the incident deuteron and the target nucleus. A second complication arises because the proton and the neutron of the incident deuteron interact. This interaction is an intrinsic part of the deut,eron and it, cannot scatter the deuteron. On Oheot#herhand, the two-body interaction between t,he proton of the deuteron and a neutron of the nucleus should produce scattering and contribute to the transition amplitude. However, all neutrons are identical, and there is no obvious way to distinguish between these two int#eractions, although any physical picture of the rea&on would indicate that’ only the second type of interact’ion will contribut’e to the transit,ion amplit,ude. In order to eliminate these complications, it is useful to define a two-nucleon eigensmte of a modified Hamiltonian, H, , given by H, = c tpNp + VP (11.12j P This Hamiltonian has been chosen so that it has eigenstates which consist only of particle states (states which are orthogonal to the orbitals occupied in IN,)). However, enough of the two-body interaction between the nucleons has been included in this Hamiltonian so that it is still possible to define a twonucleon eigenstate that satisfiesthe asymptotic boundary condition of having an
9s
JUNKIN
AND
VILLARS
incoming (or outgoing) component which is a plane wave deuteron of momentum K. Since [P, H,] # 0, such an eigenstate will be a scattering st#ate and the scattered wave will consist of scattered deuterons, free proton and neut’ron states (corresponding t#o the deuteron breakup reaction), and states with one nucleon in a bound state A, 0 > Q > er, and one nucleon in the continuum. Let jK’*‘) be the two-nucleon eigenstate of Hz . Hz (Kc*‘)
= E, lK’*‘)
(11.13)
The (&) indicates that the scattered wave consists asymptotically of radial o”tgoipg ) two nucleon states or in other words, the only (~“uctog~n”,“)part of fi”Tz a deuteron of momenkm’ K. The creation operator A,+ which creates a two-nucleon state (a plane-wave deuteron of momentum K) has already been given by
AK+= 2 wd%+a~+, AR IO)= IK) Similarly, we can define the creation state IK’*‘) to be D~cA, ,
D&c*, IO) = IK’*‘),
operator D&u
for the two-nucleon
= c zDKc*-,(a/3)&+as+. a@
However, since Dkc+, creates an eigenstate in terms of particle states,
of Hz, it can be written
D;c+t, = c DK(j-) (p1&&,& PIP2 The integral equation relating
]K’*‘)
[W.7)1
.
scatt,ering (11.14) entirely (11.15)
to IK) is given by
Dhi) 10)= (1 + (11.16) 1k”*;‘)
= (1 +
where Vz is the difference between the Hamiltonians Vz = Hz - H.
determining
IK) and I@*)), (11.17)
In nuclear reactions in which a single nucleon is scattered elastically off the nucleus lfN), the problem of solving the (N + 1 )-body problem can be reduced to the problem of solving the one-body scattering problem in the presence of a potential, the generalized single-nucleon optical potential. Our objective is to reduce the (N + 2)body problem (two-nucleons incident on ]fN)) to the problem of two bodies scattering off an effective (two-nucleon-nucleus) potential. Thus, it is our objective to write the elastic scattering transition amplitude in the form
LINKED-CLUSTER
EXPANSION
TK’K= (K’ lli, II(
FOR
+ (P
= (K’ IV2jP)
DEUTEROK
OPTICAL
JW(&) IX(‘))
+ ((Ri(-) (W(E,)
(11.1s)
1 . ( I+
E,
99
POTEiYTIAL
-
H,
-
W(EK)
+
i?j
W(EK)
jK’f’,)
\
>
states. Of course, this is the n-here I&), jK’+‘), and jli-“‘) are all two-nucleon well ltnown expression for the transition amplitude for the scatkering from the sum of t,wo potentials. Such an equation defines the effective potential for tnonucleon scat#tering reactions to he W’,ff(EK)
= vs + W(E,).
/J?‘+‘) is the eigenstate of Hz + W(E,) amplitude could also be writt’en
(11.19)
= H + W,,,(EK)
TE;tK = (K’ ( W’,fr(ER)
and t’he s&tering
/ I?‘+‘).
(11.20)
In Section III of this paper, such an expression will be derived for the transiamplit#udes. The potential I;, will be examined. ?T’(EK) gives corrections tKJ the potential V? , and the linked-cluster perturbation series for W(E,) will be derived. In this manner, a11 exact expression will be derived for t,he effective potential, Fv,fi(EK) = V2 + W(E,).’ In Section IV, t’he deuteron optical potential will be derived from this effective pot,ential by eliminating the other channels, such as t’he deuteron breakup channel. The relationship bet’\leen the deuteron optical potent’ial and t#he singlenucleon optical potential will then be examined. tkJI1
III.
DERIVATION
FOR
THE
EFFECTIVE POTENTIAL
(TWO-NUCLEOX-NUCLEUS)
One possible representat,ion for I*;$), the t’otal scattering nucleons, has already been given in t’erms of AK’. j \k:;k)
= (-4=+ +
E,, + E, ’ - H f iv
states of N + 2
JK+ iv), )I
[(II.sj]
JK+ = [H, A,+] - ICK A,+. However,
another possible representation
1 In order to make the Bruckner theory concerned with this
for such states may be obtained
such a linked-cluster expansion converge, (for example, if the residual interaction additional complication in this paper.
by
it may be necessary to apply is singtdar). We shall not be
100
JUNKIN
using the operator &+,
AND
VILLARS
[Eq. (11.14)],
I!?gk) = (Dh
+
J&w
l
EfN -I- EK - H f
= [H, D&l
iq
JL *) I fN), >
- ExD&<+).
(111.1) (111.2)
This representation for the states \%i’,k) is much more convenient when one is deriving the linked-cluster expansion for the deuteron optical potential. In terms of this representation, XKIK = (\k:,kt
[UI.lO)l
I$$.) 1 + DK’(F) EfN + EK - H + iv
and from the S-matrix
the following
T-matrix
is derived
\k:;jK >
(see Appendix
Tivir = ~FK + T$k , tK’K
T Sk
= (f.v I JKt(-I = (IN (Jgf(-)
=
(0
/ AKtV2&+,
10)
=
(K’
(111.3) I) : (111.4)
1 vz
1 Kc+‘),
(111.4a)
I\kjzk) (DL<+j
+ EjN + E ‘-
H + irl
K
JL(+)) I .fz~j- (IIIAb)
To derive the linked-cluster expansion for T$k , we make use of a relation between Tgk and the deuteron Green’s function (which will simplify the calculation) . This is based on the following. THEOREM. Define the Green’s functio?l: G(w;
K’p K) s
1DKr(-)
-&
D;c+, IfN
(111.5)
Then lim .b-K=BK’
(20 - EK - EfN)(w
- E,f
- Ef,)G(w;
K’, K) = T$k.
(111.6)
u~-rEK+E,~+it,
This theorem is essentially the nonrelativistic form of the well known fieldtheory result, i.e., that the transition amplitude is just the residue of the secondorder pole of the retarded Green’s function at the scattering energy. (A proof of this is given in Appendix II.) Our objective is to expand this expression for G(w; K’, K) in powers of the residual interaction between the “D” states and the nucleus. To do this, we define a zero-order Hamiltonian, Hn . HD = H, + PVP
= H, + PV,P
(111.7)
LINKED-CLUSTER
EXPANSION
\vhere P is the projection
FOR
operator p = $5 c
DEUTERON
OPTICAL
POTENTIAL
101
given by: &dl,
(111.8)
INOHNOI apy%q
PIP2
Inspection of the properties of Ho shows that both IN,) and DLc*) \N0) are eigenstates of Ho . Furthermore, all N + 2 nucleon states of the type 2 - 1 particles, 1 holes (1 # 0), are also eigenstates of Ho , since PVP = 0 for such states. The residual interaction in powers of which we will expand the expression for G(w; K’, K), is thus
Vn = H - HD = V - PVI’
= V - PV,P
[W.8>1
It should be noted that V, allows nucleons to interact with each other via the D cannot scatter a two-particle t#w-o-body interaction with one exception-V state into another two-particle state. To develop the perturbation series, we define the operator: (To(
x,
y)
E
e-p-u)*e-~*o,
(6/6X)~~D(X,
Uo(2, x) = 1, y) = -V,(z)Uo(z,
This leads to the well-known
V,(z)
y),
E e+HDJrDe-ZHD.
(111.9)
series expansion for UD(x, y) :
. {V,(u,)V,~uz)VD(UQ) . . . V,(u,)) (III lo) ClUl clu2. . ' nu, =%; C-1)"j Z>U1>U~>"'>ll,,>2/ . v,(u,)v,(u,) . . VD(7.4,). We no~v use ITo to establish lim Ti,( 0, - p) 1N,) p-+”
a relationship
between
IfN) and IN,).
= lim eBENeP / N,) B-m (111.11)
1.f~) = lim exp MEf, P-fUsing this expression
UD(O, -PI I No) (.fNINo) *
for If N) in the deuteron Green’s function,
- ENI1 I (fNI N4”
= lirn ev [2P(Ef,
8-l)
- EN)~
one obtains
102
JUNKIN
AND
VILLARS
(111.12)
. (No 1U,@? + 5, ~)DF-J Dp(-)
(x)ePDL+)
UD(O, -0)
1No)
(2) = eZHDDKt,-) ePD.
Using Wick’s theorem to remove all graphs in which not linked to the operator DKl(-) or D&lc+~ , (No ) U&3
+ x:, x)&e)
(2) Udx,
= (No I UD(P + 2, x)Dr-I
0)Dfrw
Co(0,
-0)
the interaction
V. is
1No)
(r) UD(z, O)D!A+, Un(0,
-p)
. (No I UD(P + 2, -PI
] NJ,
(111.13)
I No).
However,
lim (NoI UdP + z, -PI IN,) 8-fm G(w;K’,K)
=
lim Be-
Im w>O
.(No! U,(P
+ x, x)Dg,(-I
Since the expression out, and the limit p + G(w; K’, K)
= lim exp [(2p + z)(gN - E,,)] B+m ia dx exp [z(,u) - El,)] (-1) s0 (x) U,(x,
] (N,, IfN) 1’; (‘I’*“)
(111.15)
O)Dfcc+, Uo(O, -p> I iV,)L
is linked, all the integrals 00 can be taken. Therefore,
converge and can be carried
= %$ mz g .
i
No
I(
! H
VD E N
D
1 w + EN - EfN - H, We now examine this expression
n DK,(-)
1 ~JJ+ EN - EfN - Ho EN
(111.16)
1 VD mN \ - H, >I O/L
to find the terms that have the requiredsecond
LINKED-CLUSTER
EXPANSION
FOR
UEUTERON
OPTICAL
order pole at w + EJN + E, + iv. Obviously, the terms with at most a single pole in the complex w plane. For n = m = 0, lim
EI(=EK P
103
POTENTIAL
(w - Ef, - EK)(w - E!,” - E,t)C:(w; k”, K)
1 = 0 can have
= n=nz=O
,M-E,~+E~+~~
(111.17)
w + EN - EfN - H, 1
w + E, - EfN - H, >
2+1 D&c+,
/No>
I,
For n or m # 0, the intermediate state must be an eigenstate of HD with an eigenvalue greater than E, (from the linked requirement). Therefore, the terms with n or m # 0 can have at most a single pole in the complex w plane at the scattering energy. Therefore
This is t)he desired series for T$k . Each term in the power series can be represented by diagrams, and the contribution from the linked diagrams can be calculat,ed. In order to extract the effective pot’ential from this expression, we note that’ t.he initial and final state is a two-particle state. Furthermore, a two-particle state may appear as an intermediate state in the expression for T$k . Therefore, we define an irreducible part of a diagram as being any part that cannot be divided into two parts by breaking two particle lines. In other words, a twopaAicle state never appears as an intermediate state in any irreducible part of a diagram. The t’erm CY( E,) in the effective potential, We,,(E’,) = IT2 + W(E,), is then the sum of all such irreducible parts, since the complete expression for Tkyk is obviously the iteration of such irreductible parts. To obtain this result mat,hcmatiaally, we define the folio\\-ing operator:
104
JUNKIN
AND
VILLAHS
Obviously, the contribution of all possible interactions in which a two-particle state scatters into another two-particle state without forming such a state as an intermediate state is given by (NO / ~~~cz~~W(E~)u~~ca~, ( N,). Once a twoparticle state is formed as an intermediate state, it must propagate as a “D” state until it has another W(EK) type of interaction. Thus, the infinite series for !P$k can be rewritten in terms of the operator W(E,),
We now use the fact that IN,) = flh u; IO), and that both W(E,) and Ok,*, commute with all CL*+.In addition, we use the following relationship between H, operating on a two-particle state and H, operating on a two-nucleon state: HDD+
(q
uhi)
10)
=
HD
d)D+
(q
IO) (111.21)
= (p Thus, we can commute obtain Tk’3c = 2 (O 1D,J(-) =
0 j Dgrc-)
the uFT operators
WL%)
(E
R
_ ;
2
d)
(Hz - E,)D+
t,hrough the expression
+ iq W(EK))’
to
(111.22) 1 W(E&
+ iq
W(Er.)
>
Thus, the exact expression for the transition amplitude scattering has been reduced to the following expression:
+
for T$:
DL<+, (0)
W(E,)
EK - Hz -
T K’K = (K’I
IO).
DLc+) IO for elastic deuteron
vz 1K’+‘) Eg - He -
The effective potential for reactions by the nucleus IfN) is therefore Wed&)
1 W(E.y)
+ iv
where two nucleons are elastically
scattered
= V, + W(E,).
[(11.19)1
Let us now examine the nature of V, and some of the terms of the linkedcluster expansion for W(E,). V, is the difference between the modified Hamil-
LINKED-CLUSTER
EXPANSION
FOR
tonian H, and the Hamiltonian va.rious terms of lit .
DEUTERO?;
H. Therefore,
OPTICAL
105
POTENTIAL
we can work
out. explicitly
the
V, = H, - H, Hz = c
[(11.12)]
cpNp + lip,
[(11.6,2)]
H = c ~.ua+am + ; 2 (cyp 1\‘A 1yX)u,+up+axu, 01 YX
vz= -c q‘uh+cqL + u - 4I h
(at least
C’
oneholestate)
-
U, [(11.2)]
(apj
JA 1 ~h&‘?+~~~,
The first term of VT has zero matrix elemenk in the expression for TK’Iz( Ch Eh(K’ / ahtab I Kc+‘) = 0, since the “D” states have no components corresponding to hole stat’es and do not produce any scattering). It, may t,herefore be omitted, so that
Va= u- ?*
C’
iafij
dA ~$+,+~,9+~~~,
(at least one hole statr)
(111.23)
.
The first term of VP is just the Hartree-Fock potential. This is a one-body operator t.hat acts to scatt,er each of the incoming nucleons.’ The second part of VZ is a two-body operator that would not be present if the incident projectile were a single nucleon. This part of 1’2 is an expression of the effect of t’he exclusion principle on the motion of t#he deuteron as it, propagates through t’he nucleus. In free space (if no other nucleons were present’), the proton and neutron of the deuteron can interact freely with each ot.her via the two-body intera&ion and scatter each other in and out of any Harkee-Fock states. However, as the deut,eron propagates through the nucleus, certain Hartree-Fock states are occupied by the nucleus. Because of t#he exclusion principle, ttie proton and neut’ron of the deuteron can no longer interact to scatter into these occupied stnt~es. Thus, the exclusion principle “weakens” or removes part of the interaction between the proton and neukon of the incident deuteron. This “weakening” of t’he interaction that holds the deuteron together can break up the deuteron or scatter it elastically. W(E,) gives corrections to V2 . W(E,) is explicitly energy-dependent,, and it will have an imaginary (non-Hermitian) component if any inelast#ic process (other than deuteron breakup) is energetically possible. The linked-cluster 2 The Therefore,
Hartree-Fock VP includes
potential the Conlomb
gives t,he scat,terirsg
approximate and hreakup
Coulomb field of the deuteron.
of
the
nnrle~~s.
106
JUNKIN
AND
FIG.
FIG.
VILLARS
1
2
expansion for W(EK) has been given, and an examination of this perturbation series shows that W( E,) contains corrections to U, the one-body part of Vz , and to
the two-body part of Vz . Some of the diagrams contributing to the one-body part of W ( ER) are given in Fig. 1. These diagrams plus the Hartree-Fock potential are the diagrams which define the single-nucleon generalized optical potential. The only difference is that the one-body part of WrEf( E,) is the single-nucleon optical potential calculated “off the energy shell,” since the energy in the denominators of the linked-cluster expansion is E R - ~6 , rather than the energy of the nucleon being scattered. Of course, this makes no difference when the Hart’reeFock potential is being calculated, since U is not explicitly energy dependent. Thus, adding all the one-body parts of W(E,) to the Hartree-Fock potential, we find that the one-body part of W,,,(EK) corresponds to the sum of two single-nucleon optical potentials calculated off t#he energy shell by roughly half the binding energy of the deuteron. W( EK) also gives corrections to the two-body part of V, . Some of t’he diagrams that contribute to the two-body part of W( EK) are given in Fig. 2. It is possible to int,erpret the diagrams of Fig. 1 as representing reactions in which one of the incident n&eons interacts with the nucleus to produce an excited intermediate state and t’hen the same nucleon interacts to return to the nucleus to its ground state. In this interpretation some of the diagrams of Fig. 2 would represent one of the incident’ nucleons producing the excited state and the other nucleon inter-
LINKED-CLUSTER
EXPANSION
FOR
DEUTERON
OPTICAL
107
POTENTIAL
acting to return the nucleus to it.s ground &ate, while some of the other diagrams of Pig. 2 would represent, the simultaneous interaction of the incident nucleons with the nuc#leus to produce excGt,ed intermediate states. Obviously, all these terms are unique t,o reactions in whic*h incident, projectiles, composed of t’wo or more nucleons, are s;cat#tered elastically by t#he nucleus IfN), and there is no relat,ionship between the two-body part of W(E,) and the single-nucleon generalized optical potential. IV. THE A.
DERIVATION
OF
THE
IlEUTERON EXACT
OPTICAL
GENERALIZED
POTENTIAL DEUTERON
OPTICAL
I'OTENTIAL
In the previous se&ion, the effective potential for a reaction in which two incident nucleons (the proton and neut,ron of the deuteron) are scatkered elaxtitally by the nucleus IjN) n-asderived. This effect,ive potential is not the generalized deuteron optical potential, for this effective potential can produce deuteron breakup and a certain portion of the deuteron stripping reaction as well as deuteron scattering. The deut’eron optical potential must describe only the elastic deuteron channel, and the presence of the other channels must appear as an absorptive part. It is, therefore, a potential which describes the center-of-mass motion of the deuteron. This pot’ential will tu~w be obtained from the effective potential, W,,,( E,). Let jJ?‘+‘) be the two-nucleon s&tering state in the presence of the effe&ve potential W,ff(F ,= ) , with the boundary conditions of having an incoming deuteron of momentum K, energy 8, , and outgoing scattered waves.
In terms of these states,
T R,K = (K’ I W,,,(&c)
I a(+‘).
[(ILao)]
la’+‘) has outgoing waves that asymptotically are scattered deuterons, two free nucleons in Hartree-Fock continuum states (corresponding to the breakup reaction), and states where one nucleon is in a bound Hartree-Fock particle state, and the other nucleon is in a Hartree-Pock continuum state (corresponding to a certain portion of the stripping reaction). Let Id) be the internal statme (in configuration spare, the relative coordinate wavefunction) of the deuteron. Then, we use the projection operat’or Pd = 1d)(fZl
(IV.1)
to projecbt out the part of I@+‘) that has the internal st’ructure of the deuteron, IF+‘)
= Id) p$+,“> + I$+‘),
Id) IDE’)
= Pd p7’+‘).
(IV.“)
108
JUNKIN
AND VILLARH
Id) jOi;“> has an incoming deuteron of momentum K, and outgoing scattered deuterons, while ICF’) hasall the other outgoing scattered states. Since [Pd , H] = 0, the equation (H
+
Weff(&))
IK’+‘)
=
EK
IK’+‘)
can be written as two coupled equations: (H
+
Wc?ffdd
(H
+
-
Weff,,
EK)
Id)
-
EK)
pk+)>
=
Weffdc
ICk”),
/ck+‘>
=
Weffcd
Id)
(IV.3a) (IV.3b)
lDk”>.
Using the boundary condition that ICk”) has only scattered waves, we obtain 1 ck+‘) = EK - Weff,, -H+iq
W effcd
4
pk+‘>.
(IV.4)
By means of this equation, we can eliminate ICF’) from Eq. (IV.3a), and after integrating out the internal deuteron coordinate, we are left with
Uopt =
- E, + CB+ Uo,,.(E,)) IOk”> = 0, 1 - Pd + WdEK) EK _ Weff,, _ H + ir, WdEK)
(IV.5)
(HcM.
(I
d W,rr(&)
W effcc = (1 - PdWeff(&c)(l
‘/j\
(Iv*@ (IV.7)
- Pd)
(CBis the binding energy of the deuteron). If DK(X) is the wavefunction associated with the state IOk”) then (-Xv”
- ~zP)DK(+) (X) +
s
d3X’U,,t(Ex
; X, X’)Dg(+)(X’)
= 0
(IV.8)
and TK’K = (&
s
d3X d3X’ exp ( -~K.x)
This concludes the derivation Ud-W. B. THE RELATIONSHIP SINGLE-NUCLEON
t~k,~(~~ ; X, X’)Dd+~ (X’).
(IV.9)
of the generalized deuteron optical potential,
BETWEEN THE DEUTERON OPTICAL POTENTIAL
OPTICALPOTENTIAL
AND THE
It is often assumedthat the deuteron optical potential should be approximately equal to the sum of the proton and neutron optical potentials, averaged over the internal wavefunction of the deuteron (13)-(15). This assumption allows one to obtain the deuteron optical potential in terms of the single-nucleon optical potentials. Now that the exact expression for V+( EK) has been obtained [Eq. (IV.6)], it is easy to examine the exact expression and determine what terms are ignored when this approximation is made. Obviously, if this approximation is good, the second term of U,,,( EK) ,
LINKED-CLUSTER
EXPANSION
FOR
DEUTERON
OPTICAL
1 - Pd EK - H - Weff,, + it, Weff(EK)
POTENTIAL
;\ “/
109
’
must be small. However, this term includes the effect of closing out the breakup channel and a large portion of the stripping channel. Therefore, in most reactions, t(he berm i?r(d / W,,,(E,)
(1 - P,)S(E,
- H - Weff,,) W&E,)
1d)
(IV.7)
should be the dominant term in determining the imaginary part of Cr,,,,( EK). By contrast, most of the real part of U,,,t( EK) will probably come from the first term, (d 1 Weff(EK) / d). As was shown in Section III, the perturbation series for W,ff(E,) can be divided into two parts, a one-body operator (the Hartree-Fock potential Ir and all the one-body correction terms from W(E,)) and a two-body operator (part of which comes from the exclusion principle). Since the two-body operator part of Weff(EK) has no relationship to the single-nucleon opt’ical potential, the above-mentioned approximation ignores all such terms of W,ff( ER). However, as was shown in Section III, the one-body part of Weff(EK) is indeed the sum of two single-nucleon generalized optical potentials, the only difference being that these single-nucleon optical potentials are calculated “off the energy shell”. The optical potential as usually applied, is the result of an energy average of the generalized optical potential. If such an energy average is made, it is possible that the fact that the nucleons are being scattered off t’he energy shell (b) roughly half the binding energy of the deuteron) may be unimportant. In anJ event, the scattering of t’he individual nucleons by the Hartree-Fock pot#ential is taken into account properly, since the Hartree-Fock potential is not explicitl! energy-dependent. Thus, it can be seen that the approximation, which obtains the deuteron optical potential by assuming that it is the sum of the proton and neutron optical potentials averaged over the internal state of the deuteron, differs from the exact expression for lrOPt( EK) in three respects: 1. It ignures t’he second term of CTopt(EK), 1 - Pd E, - H - Weff,, + 6 This should not be a valid approximation if one is considering the imaginary part of CLdE,). 3. It ignores the fact that the individual nucleons are being scattered off the energy shell. If the Hartree-Fock potential is the dominant part of the total potential, or if an energy average is made over a snfficiently large energy interval, this may still be a very good approximat’ion.
110
JUNKIN
AND
VILLARS
3. It ignores terms which contribute to the two-body part of bV,ff(EK). However, these terms require that both incident nucleons interact with the nucleus, and, for reactions which occur at high energies or large center-of-mass angular niomentum, the contribution from these terms may be small. Therefore, it seems evident that there are certain types of deuteron reactions where this approximation will give a good approximation for the real part of the deuteron optical potential. An analysis of experimental data indicates that there are several sets of parameters for the deuteron optical potential which seem to give a good fit to the elastic scattering data. However, a recent analysis by Perey and Perey (11) indicates that (w-hen a spin-orbit part is added to the deuteron optical potential) the best fit is obtained by the set of parameters that describes a deuteron optical potential whose real part is close to the sum of the singlenucleon optical potentials averaged over the internal wavefunction of the deuteron. In this paper we have shown how to extract, from the exact expression for the deuteron optical potential, the terms which would lead to such a relationship. It will be interesting at this point to examine some of the correction terms to this approximation. It will be especially important to determine t’he correction t#erms to the imaginary part and the spin orbit part of the deuteron optical potential, since it is likely that these correction terms may be rather large. This aspect of the problem is being investigated at present. APPENDIX
I
The eigenstates of Hz can be classified by the boundary conditions satisfied by the eigenstates. The state created by D:c+, is classified by the “in” boundary conditions, since it has an incoming deuteron wave of momentum K and outgoing scattered waves. On the other hand, the state created by D&C-J is classified by the “out” boundary conditions, since it has an outgoing deuteron of momentum K and incoming scattered waves. Of course, there are other t’wo-nucleon eigenstates which satisfy other “out” boundary conditions, such as having two outgoing nucleons of momentum kl and kz or one outgoing nucleon of momentum k1 and the other nucleon in a bound state Y. The “out” states are not orthogonal to the “in” states, but one can be expanded in terms of the other.
+ c
s(k, v; K)D:,c-,
= c
S(CY;K)Dh
= SaR - 2ri6(Ea
- EK)taK,
a
ku
s(a; K) tczg = (0 ID& D;c+,
= Dkc-I
Vz A,’
IO)
- 2ri c 6(E, a
- EX)taK DLc-, .
,
(A.1)
LIKKED-CLUSTER
The S-matrix SK’K = (\k/,L . I*.;;:)
EXPBNSION
POR
DEUTEROli
for the elastic deuteron s&tering I*(‘,:- ) = <*:,;,
= (1 + El,
Using Eq. (Al)
*I
+ E
I*.;;;
) -
@l&r,
OPTICAL
111
POTENTIAL
reaction is given b? I\kj,;)
+ 6(K - K’)
K
in the equat’ion for Iqj,k> one obtains
I*L:! = (1 + E,N_ ;, _ H-ir, (H - E,, - w)D:~++
fN).
112
JUNKIN
(w - Eirr - Ef,>(w
- Ex -Ef,)G(w;
= (w - E,c - EfN) (f,v I&(-)
RECEIVED:
AND
VILLARS
K’, K) I&t+,
1~5)
June 22, 1967 REFERENCES
1. J. GOLDSTONE, Proe. Roy. Sot. London, Ser. A 239, 267 (1957). 1. C. BLOCH, Nucl. Phys. 7,451 (1958). 3. N. M. HUGENHOLTZ, Physica 23, 481 (1957). 4. J. 8. BELL, Formal theory of the optical model. In “Lectures on the Many-Body Problem” (E. R. Caianiello, Ed.), p. 91. Academic Press, New York, 1962. 5. H. FESHBACH, Ann. Rev. Nucl. Sci. 9, 49 (1958). Chap. IX. North-Holland, Amster6. G. E. BROWN, “Unified Theory of Nuclear Models,” dam, 1964. 7. R. H. LEMMER AND C.M. SHAKIN, Ann. Phys.27, 13 (1964). 8. H. A. WEIDENMULLER,~\~ZLC~. Phys. 76, 189 (1966). 9. H. A. WEIDENMULLER AND K. DIETRICH,NUCZ. Phys. 83, 332 (1966). IO. H. FESHBXH, A.K. KERMAN, AND R.H. LEMMER, Ann.Phys. 41, 230 (1967). 11. C. M. PEREY BND F. D. PEREY, Deuteron optical model analysis with spin-orbit potential. ORNL Report No. 1529, June 6, 1966. 11. F. VILLARS, Lecture Notes (Collision Theory), International Course on Nuclear Physics, Institute for Theoretical Physics, Trieste, Italy, October 3-October 15, 1966. 13. F. J. BLOORE, Nucl. Phys. 68, 298 (1965). 14. G. R. SATCHLER, Nucl. Phys. 21, 116 (1960). 15. S. WATSNABE,lV'Uc~. Phys. 8, 484 (19%).