Long range correlations and photo effect in nuclei

Publishing Co ., Amsterdam Nuclear Physics 22 (1961) 14---33 ; © North-Holland

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NUCLEI LONG RANGE CORRELATIONS AND PHOTO EFFECT IN W. BRENIG t of Technology . Departmer' of Physics and Laboratory for Nuclear Science, Massachusetts Institue Cambridge 39, Massachusetts tt Received 6 July 1960 Abstract : A detailed quantum mechanical treatment of the collective model for the giant dipole resonance in the low energy nuclear photo effect is presented . A relation WD = ckD for the resonance energy WD is derived from sum rule considerations, where kp is related to the nuclear radius by kD ~ n/2R and c to the structure factor s(k) of nuclear matter at small momenta k by s (k) = k/2mc. The velocity c is calculated and the semiclassical expression for c in terms of the symmetry energy K (c = (2K/m)1) is found to be valid only for extremely strong interactions. For weaker forces one obtains a larger value of c than indicated by this relation and a value of wD in better agreement with the results of shell model configuration mixing calculations .

1. Introduction In the collective model 1) for the giant resonance in the low energy nuclear photo effect the resonance state is assumed to be connected with a collective oscillation of the proton fluid against the neutron fluid. For vanishing two particle interactions this description has been shown to be identical with the singleparticle model 2). The single-particle model, however, seems to give a value of the resonance energy which is only half of the observed value. In the past the introduction of an effective mass m* ti -Iffm has been suggested $) to remove this discrepancy. This however is in disagreement with experimental results in direct reactions 4) and with the current fashion in the theory of nuclear matter, which assumes a smaller momentum dependence of i he effective two particle forces in nuclear matter and consequently an effective mass m* .. m ttt. The effect of two particle interactions has been included approximately in the collective model in a serniclassical treatment') which allowed to relate the resonance energy directly to the symmetry energy of nuclear matter. However, it has been pointed out 8) that in slightly non-ideal Fermi gases such as nuclear t On leave of absence from Max Planck Institut für Physik, München, Germany . ft This work is supported in part through AEC Contract ÁT(30-1)-2098 by Funds provided by

the U. S. Atomic Energy Commission, the Office of Naval Research and the Air Force Office of Scientific Research. ttt The equilibrium con6ition which seemed to suggest the introduction of a small effective mass 5) is now supposed to be fulfilled by means of a strong density dependence of the forces leading to rearrangement terms 6) . 14

LONG RANGE CORRELATiONS

15

matter the semiclassical treatment is not valid. The collective eigenstates resulting from a more rigorous quantum mechanical calculation agree with the classical solutions only in the limiting case of very strong interactions, whereas the actual nuclear forces effects the energy of the quantum mechanical solutions only by a few percent 9) . This is in apparent contradiction to the results of shell model configuration mixing 10) calculations which indicate a strong influence of two particle interactions on the resonance energy. These discrepancies may be considered as a motivation for a reinvestigation of the collective model. 2 . Dipole Strength and Two-Particle Correlation

The scattering of y-rays is determined by the index of refraction which for low energies where the wavelength of the light is small compared to the size of the nucleus is proportional to the dipole strength function n(co)= d (co)-ina(co)

=

1

1<0IDln>12

n vin -GO-{-2y

+(w

> -ei) .

The real part d(co) is proportional to the elastic scattering cross-section, whereas the imaginary part determines the total absorption cross-section ar(w) - wa (co) . The width y occurring in (1) is in general different: for different levels, but as far as one is not interested in the fine structure of a (co) one can replace it by an average value y. The dipole operator D in (1) is given by D

= 1 Y xi Z prot . N neutr.

It is the distance between the centre of mass of the protons and the neutrons. 10> is the ground state and Ink are the excited states of the system, con = En -- E0 the excitation energies . Experimentally one finds a resonance in the total absorption cross-section, i .e. a rather well defined peak at an energy of about coD %k:f 80/A I MeV (A is the mass number of the nucleus) . In the shell model the only possible excitations contributing to (1) are singleparticle excitations . For an oscillator well, for instance, only the matrix elements between neighboring shells are different from ,zero, which would give a value 40 i1lel' kF 1 .24 kF2 ~ -- 0.94 ~D = ~osc ti ^ mR AI A ~. 2m

for the resonance energy, following from the requirement that the mean square radius f y.2 p(r)d 3r of the nucleus be e4~ial to ßr02 A I. This is only half as large as the experimental value for coD . Now there are, in general, many degenerate single-particle excitations contributing to (1) and one has to solve a secular equation among them to get the eigenstates after taking the interai. ~~-7n into

w. BRRNIG mount, This has been done in a schematic model by Brown and Bolsterli and for Ott by riot and Flowers both calculations 1°} indicating that the dipole strength is pushed up to higher energies. Let `der a. qualitative argument for this effect which is independent . of eon i tion mixing calculation and which can be used for a quantitative estimate later on. Consider, for instance, as a measure for the location of the e strength the average value WD -

fwa(co)dw fa(w)dw

(1), sung over the intermediate states and introducing the state I D\ = Dr,O x one obtains for it

then usi dipole

ce)D =

--Eu .

sere are a few a~ in which the dipole state is an eigenstate of the Hamilto nian r inta e, in the shell model with an oscillator well (as pointed out by Blink 4~ ); . The. re is true in the schematic Brown-Bolsterli model 10) and in the sendclassical eecttive model ') to be discussed below. In general, however, this is not the case and in large nuclei the dipole state is far from being an `gestate. "Nevertheless the expectation -,due of its energy is a useful quantity becau:- ort its elam r.nection -s) mrith the measurable quantity (4) .

L-tudes'

&T

D in

010

ground state and dipole state (schematic).

~ ` izf1 e_m'~ O.t irsteractioT1 on (r) one mQxy compare. the average _,~ , Vaty :_r p~~~~b ~~ tiii anip u~'e for te h cr -i~fire of r -la .ss (li~;tance .i between IM(1 :trr ~ the g un em state and fe titt)ole state. They are plotted tlie clipole state the proton are he neti te2i, w1iich for ictttfcctivc~ interactions between the e ener ef the dipolc, state dip, Rou f dy speziking, `D> may e ~ - L.ukf 4 ~-',Vcitiki l stttt a u ,f il . ~~ Lc'c:;tivC' c seil~tift
LONG RANGE CORRELATIONS

17

against neutrons. It has been pointed out by Migdal, Jensen and Steinwedel 7), that in these oscillations the nucleons have to do work against the symmetry energy. Since this is known to be about twice as large experimentally as the theoretical value for a free Fermi gas one can expect the interactions to have a large effect on 01D' Let us now consider the collective proton-neutron vibrations more in detail. As long as surface contributions are unimportant the properties of these excitations can be obtained from the properties of (infinitely extended) nuclear matter as is done in the semiclassical collective model . The excitations in this case are plain waves, corresponding to a periodical fluctuation in the neutron excess (10> is now the ground state of nuclear matter), Pklo> pk

= Ik>,

(6)

being the Fourier transform of the neutron excess density Pk = ,_

Z prot .

eikxi _

N neutr.

eikxß.

The finite size of the nucleus then is only important in determining the effective wavelength which has to be used in (6). Comparing the collective state (6) with fig. 1 one expects k to be of the order of k .. :r/2R. Jenen and Steinwedel, who gave a detailed semiclassical analysis of the collective modes (6), determine k by the condition that there is no flow of protons against neutron across the boundary of the nucleus yielding k = 2.08/R .

(8)

Another determination of the k to be used in (6) may be obtained directly from (5) . If one uses the Thomas-Kuhn-Reiche sum rule to evaluate the numerator in (5), one obtains (OD

=

2m

,

where the average value of D2 in the ground state involves only the two particle correlation function p(r, r') = < 0I pr p,,.l O>, with

pr =-

= f xx'P(r, r )d3 rd3 r',

prot.

6(r - r,) - 6(r-ri) . N neutr.

(10)

The TKR dipole sum rule is only valid in the absence of exchange forces .

W. BRENIG

18

Bethe and Levinger 11) have shown that exchange forces modify (9) inito WD -

1+a 2m '

(9a)

with a being of the order of 0.3 to 0.4 thus giving a higher value for con . Approximately the same result was obtained by Gell-Mann, Goldberger and Thirringl 2) using dispersion relation techniques . The physical reason behind this result is, that the dipole state has more pairs in relative p-states than the ground state. Since the exchange forces are repulsive in p-states they tend to push up the energy of ti 1e dipole state. To evaluate (10) we use the approximation P(r, r') = P(r)P(r,Mrrr,), where p(r) is the particle density normalized to 1 and v(r) the two-particle correlation function of nuclear matter. Introducing the Fourier transformation (po the density of nuclear matter) of a(r) one obtains

Pocr(r) - -

1

Po

= f s(k)e'`-rd3k 2

f~f

xp(r)eik-rd3r 1 s(k)dsk .

(12) (13)

For large nuclei the main contribution of this integral comes from small values of k (k m-_ 1 JR) for which as we shall see s (k) is proportional to k ; s (k) --- k/2mc. Therefore (13) may be approximated by =

_~- s (kD) kD 2 ' 2mckD 1

(14)

where kD is independent of the form of the two particle correlation (consequently independent of the interaction) and only determined by the size of the nucleus. Using (9) and (14) one obtains OD =

kD2(1+a) 2ms (kD)

= ck D .

15

This result now may be compared witl the assumption of the collective model, namely that the energy of the dipole st Lte is given by the energy of a collect;ve mode (6) of nuclear matter. Indeed (,ne finds (neglecting exchange forces) exactly 13) , __ kD2 Eo 2ms(kD)

(16)

The value of kD row can be determined by comparing (16) in the non-interacting case with the result (3) . Straightforward evaluation of the left hand of

LONG RANGE CORRELATIONS

(16) for an ideal Fermi gas gives OD -

kF

3 ~ kD

Therefore one obtains using (3) kD = 1 .57/R ,. a/2R.

(1 ;t)

We should mention that (17) is in disagreement with both the: Jensen-Steinwedel result and the value recently given by Brueckner and Thieberger 14) . Jensen and Steinwedel assume the relation

c- f m , 2K

where K is the symmetry energy parameter in the Bethe-Weizsäcker formula. Since for an ideal Fermi gas K = kF2/6m one obtains

whereas Brueckner and Thieberger ob ain c

F. =k m

(19a)

Jefisci: and 5tieinwedels result is smaller than (17), however, they use a value (8) of kD which is larger than (15) . We therefore, conclude that as in the collective models the energy of the dipole state can be determined from the energy of the corresponding collective mode (6) in nuclear matter. However, one obtains only rough estimates of their energy by using semiclassical pictures . In fact, 1 .. has been observed by Landau 9) that a weakly non-ideal Fermi gas can sustain collective excitations but of a nature rather different from ordinary classical type, and we shall see later on tb at these excitations contribute only part of the long range behaviour of a Fermi gas. These observations are in agreement with the success of the shell model and the optical model, which indicates a mean free path of nucleons large compared to the size of the nucleus, whereas for a stable classical density fluctuation. one would expect a mean free path small compared to the wavelength Ilk. 3. Long

a

e Correlations of a Fermi

as

In the last few years several methods have been developed to treat the long range and collective behaviour of Fermi gases. They may be classified into three apparently different but essentially equivalent groups

W. BRENIG

20

a) The most elementary method consists of the "time dependent selfconsistent field" equation for the excited states. The theory of Landau 8) mentioned above is based on a Thomas-Fermi field but can be generalized to the Hartree Fock method 15) . b) The "random phase approximation" which originally was invented for the 16), but later on has treatment of long range correlations in dense electron gases been applied also to nuclear matter 9). c) The "summation of selected diagrams" in perturbation theory which also has. been applied to the dense electron gas 17,18) . The various interrelations between these methods may be most elegantly studied using the method of Greens functions 19) . We do not want to give a new derivation of the results obtained in all these methods but restrict ourselves to a brief discussion of what is necessary for our purpose . Let us base our discussion again on the "strength function" which in the nuclear matter case in analogy to (1) may be defined as 12 +(W -> -co) n (k, w) = d(k, w) -ina (k, w) =1 1 n (wn - «>+'Y

(20)

and which is connected with the index of refraction of nuclear matter 20) . The Fourier transform of the two-particle correlation s (k) can be obtained . from the imaginary part of n (k, w) by integration over w S(k) =

Jo a (k, co)do.>.

(21)

Let us first calculate n(k, w) in zeroth order. This has been done already by Hubbard 17) but we need his results only in the limiting case of small k values which can be obtained more easily by direct calculation . In zeroth order Pk1O> has only non-vanishing matrix elements to states with one particle above and one hole inside the Fermi sea having the total momentum k, spin zero and isospin 1 In> = (

-

prot .

)at(q+2k)a(q - 'k)IO>

neutr.

= Ik, q>,

(22)

where at and a are the creation and destruction operators of nucleons and the sum is to be taken over the two spin possibilities. The energies co. of these states are 1 k.q [(q+! wk, Q -2k)2 ~ (23) (q 2m m Therefore after introduction of the momentum distribution nk of the ideal Fermi gas n k --

1 0

(! i ;5 kF), (Ikl >k),

(24)

LONG RANGE CORRELATIONS

21

eq . (20) tales the form (assuming equal numbers of protons and neutrons) no (k,&»

_ 4

-n
4?LkF3

k- q

m

-( 0+1Y

d3q+ « «À) À) _> _> d3q+

_(0) .

Fig. 2 shows the region of integration in q-space . For small values of k one can

Fig. 2. Region of integration in q-space.

use the approximation =

(1-na+}k)n-ik

and after introducing the quantity

k~q Iqi

(26)

a(I I -kf)

kF k, m

c-Ie - --

which is the maximum possible excitation energy for a given momentum k, one obtains for the dispersive and absorbtive parts of n o (k, &))

, 3 klcol

ao (k, co) =

2

kF (ok 2 0

do (k , co ) =

23

in.

kF2

:t~ (I«)1 :

(Icvl

Z --

Ok)y

(28a)

> Ok),

~ co - cc~ k ~~ In (L)k (.0-}-LUk ,

.

(,,8b)

Fig. 3 shows do (k, co) for positive values of co . From (28a) it f(.-;llows immediately so (k) =

o

which agrees with (16) and (17) .

k ao(klco)d(o = 3 k F

(29)

the kwith par may the us matrix this ticle line an now gives doubly are be the light hole then isospin MIA element of indicated approximately take light the (k, pair the produces shaded quantum co)contribution the (0)/wk) flip first quantum In = 4 3our of interaction the Diagrams by 41 the region order Dispersive ga'k-q acase, creates the annihilation interaction scattering ng+Ik-ng-Ik replaced terms (corresponding in solid where for =fig part into a the HRENIG is lines particle {q-rv(r)d3r of 3, the by given particle-hole (or Since account diagram nv(q'-q) the pair an an asmatrix The in average function annihilation most creation tofig hole interaction to is the interaction n«+kk-na'+Ik 4, absent my of pair, first element of in operator the value by -(O+i wwhich order whose the integral and and indicated vo for y ray the P0 recreation) the ciuse A propagation small wavy is (30) graphical In scattering connectto by zeroth values comes line v(0) the of

w.

22

.

Let representation

. à, do

3

0.5

.

.

Fig .

.

represents order functions

.

dotted

Fig . the ed diagram

.

.

.

nl

.ei

--co+i my

k- q0

(30)

where v(q) is from of

e

ƒ .

(31) . .

LONG RANGE CORRELATIONS

This gives finally

n(k, co) = no(k, (o)+¢Povono2(k,

23

(32)

This result shows that even for arbitrarily small values of vo the first order term

is large compared to the zeroth order term if no becomes large. This breakdown of straight forward perturbation theory is of course to be expected because of the degeneracies involved in the zeroth order result . A finite answer can be obtained after summing the ladder diagrams of the particle hole scattering, yielding

(k, _ no 0) n (k' ~) 1- 4Povono(k, 03) .

(33)

This is indeed the result following from the theories mentioned above. If there is a hard core, in addition to the ladders introduced above, one has to sum the diagrams for particle-particle scattering, to remove the infinities of the matrix element vo . This will be discussed in the next section . Furthermore, there are the terms responsible for the energy gap. We will neglect these with the argument that the energy gap is of the order of 1 MeV and is therefore expected to effect the final result for con (~ 20 MeV for intermediate nuclei)-not very much, even if it should require a drastic change in the formalism to include those terms. Eq. (33) now provides us with the desired information about the correlation function s (k) and in addition about the perturbed eigenvalues, which are given by the poles of d(k, co) . Since the poles of n, according to (33), are given by the zeros of the denominator, one has the eigenvalue equation (34)

do (k, con) = 4 Povo

Let us compare this condition with the common secular equation method. In this method one assumes the eigenstates to be linear combinations of the states (22), (35) in> _ 1 ek, jk, qi, the coefficients ck, q being determined by the secular equation (V is the volume) 1 k-q (36) U)n Ck,Q = -k,aV v (q - q)ck,Q, . 9n

If one uses the same average value of the matrix element as in (32), one obtains from (36) 4 ( 1- ng+ik)na-ik - 4 (37) A

k - q

- 019

Povo

contained which is the same as (35) except for the term (con -~ - (o,a ) which is in (34) because of (25) but missing in (37) .

24

Wt. BRENIG

from the self consistency These terms arise in method a) quoted above treatment of ground state requirement, in method b) from an approximate (37) correspond correlations in the secular equation . In fact the missing terms in Fermi sea which can to contributions of particles inside and holes outside the in method c) since not occur in the absence of correlations in the ground state, all graphs are even functions of co ("crossing symmetry" .) see if the Let us consider the simple example of Brown and Bolsterli to secular difference between (34) and (37) is quantitatively relevant. The simple equation corresponding to (37) gives

Y

1+1

a w-- £O a

_ 1(

d

(3s)

where d is a measure of the interaction strength and supopsed to be of the order of d - pk~ wosc The equation corresponding to (34) is obtained from (38) by adding the term ((o -->. --co) yielding 2(l+1)wa - j(l+1) W2--~~2 d

(39)

After replacing to, by its average ti coos, one obtains the two solutions (D

= (Oosc+d 2

wD =

ose -}- 2dcoose .

(40a) (40b)

then (40a) gives WD = 2coosc whereas (40b) gives only con = V3coosc . This seems to indicate, that the difference between both methods is not very large although not entirely negligible t. It seems to be desirable to have a clearer picture of the physical origin of the additional terms in (34) apart ,from their half formal occurrence in the three methods quoted above. Lei: us now consider the imaginary part of n. According to (33) one obtains If one assumes d

= t~oSC

a

_

~

(1 ._

a0

._4Povodo)2+(4~Povoao)2

( 41 )

Let us introduce the dimensionless measure A -- 3mpovof4kF2 of the interaction strength . In the next chapter a relation A --- KIK o-1 will be derived, where KO is the symmetry energy of the free Fermi gas . Experimentally íî is close to one. In fig . 5, a is plotted as a function of co for fixed k, and for A -= 1 . For small values of coJk one finds a value a - ao ( 1 +A) -2. If co Jk increases this value is exceeded because of a zero of 1-,pa vodo occurring close to -. . t Compare, however, the discussion after eq . (45a).

25

LONG RANGE CORRELATIONS

There is another zero co, of 1--Ipo vo do for co, > cok . This zero corresponds to an isolated discrete eigenvalue above the zero order continuum of particle-hole states and therefore to a kind of bound state of the ,particle-hole pair. This state has been discussed in great detail in the papers on collective states in Fermi gases quoted above. Since ao is zero for co > cok the value of a is not defined at cap . But taking the limiting value of ao --> 0 one obtains a(k, co)-_

4

Povo

(42)

ó(1-°4Povodo(kico) .

Wk

41c

w

Fig. 5. Absorptive part of n as compared with ao . The dotted line indicates the area of the ó-f ur`ction at coy.

Using the approximation

one finds

do

-

co Wn 11 2+ln ` ~ --11 12-In 2kF 2 k

a -

12 k k e - 2+21a)a(03 ( 0c ) 12 F % == £O

k(1+2e-(2+21 ?, ))

(co > Ok), (45)

which for A - 1 gives a - so (k) - 0 .27 - b(co-co,), where so(k) ~ 4k,'k F . The integral foka(k, co) dco has been evaluated numerically and found to be 0 .31 s.(k) . Therefore together with (44) one obtains a value of 06) s (k) -- 0.58 so (k), (OD -- 1 .73 (1+(x)coose, it rile which shows the importance of long range correlations in pushing up the strength to higher energies .

W . BRENIG

26

valid if A is sufficiently The approximations (43), (44) and (45) are only 1 one obtains small, so that co o ~v wt . In the other extreme of A' > do =

m

k122

w7

2

co )

(43a)

,

(44a)

a = k 2mc coe = ck = V

2K k. m

(45a)

Furthermore, the part of a at W < Wk has vanished . This is exactly the assumption of the semiclassical model. We should mention, that the result (45a)

k

Fig. 6. Velocity c as function of interaction strength . Dotted line : present theory including exchange ; solid line : without exchange . B. T. : Brueckner-Thieberger. J. S. : semiclassical model of Jensen-Steinwedel. + : Experiment .

would not follow from the secular eq. method since the difference between (37) and (34) becomes larger with increasing interaction strength . The same can be seen from (40) . Whereas (40b) gives a dependence of ~ n ~ Vv, in accordance with (45a) the simple secular equation method would give a linear dependence of wv on vo . In fig. 6 the results of various theories for c as a function of A are plotted. Let us now estimate the value of d in (38) to compare our result with (40b) . According to Brown and Bolsterli lo) we have

® = ~1 (l + 1)G,

G = vo 4n

< fo

R i Ra+i R R+i r

2 dr.

(47)

G is some average value of the matrix element between two particle-hole states. One can get an upper limit for G by taking the radial wave function to be constant, Ri e;e V 3l ro3 A,

LONG RANGE CORRELATIONS

yielding

27

a - 3 vol (4.zro3 A) .

In general one will get a smaller value because of destructive; interference and concentration of the integrand in (47) at smaller r-values . Let us assume the lower limit to be about half of the upper limit. If one furthermore approximates Y(1+ 1) by 2 (f'A )I following from the number of states in the highest oscillator shell then using (3) one obtains finally 0.7íîcoos0 S A < 1 AAco osc and for A-, 1 'according to (40b) one finds (48)

"55coosc S OD < 1 .95cc)osc

which agrees quite well with (46) . 4. Collective Excitations and Symmetry Energy We now come to the determination of the parameter vo representing the strength of the two particle interaction. For this purpose we consider the Hartree-Fock ground state energy of nuclear matter including small deviations from equilibrium densities of protons and neutrons --) p -po 2+ (kF2 - povo) ( 1V - Z~ 2. - 3 kF2 + 3 povo+ kF2 + 3 (49) 6m 8 A 5 3m 8 (3m 4 povo ( p The first term is the average binding energy 8b -

-16 MeV,

(50)

the second one contains the reciprocal compressibility 1 - = 11 to 17 MeV,

(51)

K

and the third one the symmetry energy K ,- 24 MeV.

(5')

F2 Tpo 3 v o - -16 Me~', 2m +

(53)

Using these three different experimental values and in addition the singleparticle energy at the Fermi surface

which because of the saturation condition is equal to

Eb,

one obtains with

28

W. BRLNIG

kF2 j2m = 38 MeV four different values for puvu, =104 = MeV; -Po vo ste 72 15 90

binding energy, single-part . energy, from compressibility, symmetry energy.

(54)

The considerable variation of these values reflects the well-known failure of the weak zero range force model to produce the right saturation properties of nuclear matter. If one wants to take into account the hard core to get saturation one has to replace the potential v in (30) by the appropriate pseudopotential w which we have to determine now. It is well-known that the pseudopotential appropriate for the ground state energy is the Brueckner K-matrix. Glassgold, Heckrotte and Watson '2) therefore replace v in (30) by K. We shall see, however, that this is a poor approximation being not valid even in second order perturbation theory . The pseudopotential depends on the states the properties of which it is used to describe . In particular, it depends on the density and the neutron excess. Indeed (54) indidates that there is a strong density dependence and less strong dependence on the neutron excess . As we shall show the appropriate pseudopotential for the collective excitations responsible for the dipole resonance should be determined from the symmetry energy according to - Puyo = K8

4F2

6m

(55)

Since the experimental value of K is close to twice the theoretical value of an ideal Fermi gas we arrive at the relation -PU v0 'le

4

kF2

3 nt

56

which was used in the calculations above. In doing this we still have neglected the spin-, isospin- and momentum-dependence of the force. We hope that this approximation does not introduce large errors, since all these effects enter in a similar way in the dipole state and the symmetry energy and are therefore taken into account partly by using the experimental value of the symmetry energy. Similarly the effective potential for the regular density fluctuations (sound waves, resp . Landau's e) wroth sound) is conrie0ed with the compressibility . Because of the strong density dependence of the pseudopotential poinuA, out above this is quantitatively very different from what one would obtain using the binding eneiáy . Q therefore important in the discussion of the imaginary roots of the eigenvalue eq . (34) considereui : (-iw!sgold, ;ï Heckrotte and Watson °) . In the case of density fluctuations the states ,22) have to be repia-:e~by ik, 4i = j( v~ -t- E , prot.

neuer.

)aQQ+Ikaa-ik 10>,

29

LONG RANGE CORRELA?IONS

and the eigenvalue eq. (34) takes the form: 4 (Z --n (F+$k)"Q-}k E + (u) -~ A a w _ k- q m

_cU) --

3Po 00

The change in sign of the right-hand side compared to (34) is a consequence of the annihilation diagram of fig. 4 which contributes in the case of spin (and isospin) zero states like sound waves. Introducing again the approximation (26) and 3mpo vo/4kF2 -= -A, one obtains h

f

t1dyl

2

OJI(Ok -2

1

3A

which for negative í1 has no real solutions but may have imaginary roots, i.e. solutions of 1 ƒ1

q2 di7

2 2

= 2; 3A, -1 (icvicv k) +~ with ito real . Since the left-hand side of this equation is always smaller than one, these imaginary N roots can only occur if íî > 1, i.e. for sufficiently strong forces, a case which could not be excluded using the ground state K-matrix as the effective particle hole interaction matrix, yielding A  1 whereas the íî determined from the compressibility is smaller by a factor of about seven. This seems to indicate, that in the case of nuclear matter the unstable solutions of Glassgold, Heckrotte and Watson are not likely tó occur, although they cannot be excluded in general for a many body system . 2

To justify our use of the symmetry energy above, let us first determine the pseudopotentíal in second order. This can be done in the rlost elementary way using the time independent (Rayleigh-Schrödinger) perturbation theory. In this method the excited states in zeroth order are linear combinations of the particle hole excitations (spin and isospin omitted in the following considerations) : T,,,, =

aktaa 10 i,

(5î)

V

Ck, l CPk, l

(58)

=1

k, l

The higher order terms then may be obtained in general from s

.1 C

k, l g)k, t -

H

P

where Ho is the kinetic energy, v the interaction potential, E the energy of ?P and P the proj ection operator P á 1-

k, t

i (Pk, l> <9~k, 1 1 *

The solution of (59) can be written as

(so)

W. SRENIG

where

Vk,

a

is the solution of Vk, a

= Tk, 17

-

P ~

~ _, o

k, a

(6 )

(v--E+ek, a)~k, a

2

single particle energy. a = eo+sk -e, is the zeroth order energy of the The Schroedinger equation then finally gives a secular equation for Ck, a : ek,

ECk, a ^ (~k,

which in first order

t+<92 k, viv1Vk, a»Ck, a+

Vk, a

k',

CV. V 0 .(r a' O k, a

(62 )

= Tk, a is equivalent to (36) . In second order

a k,lp',

=

'9'k"-

H

4`

~

t

(63)

V92 k, a

one obtains for the effective non-diagonal matrix element <97 k, =

.

awIVk", I'>

vkb',ak'+ 1_

9, a

4-2

ykE', 9a y9a, Ik' 1-no) (1-ra)- Y, 9,a Ek+Et' -e9 -EQ Ek +ELl89-ee y ka', 9vy9a, Ik'

V

yk9,

(

k'9 ~v t, 9., a9

9',9 E a'+ E9' - Ea - ~9

(1-n9 )n9-1-2 '

np n

(64)

9',9

ytoyk9,k'9' %,(1 Ek+E9--Ek'--EIP,

The first of the four second order terms is just the second order term in the expansion of the Brueckner K-matrix Kka', ak'

VN', gay9a, Ik' = vka', ak'- 1 Ep~Ea_`~t9,a

gk .

(1- n9) (1 °na)-

(65)

The other three terms therefore are missing in the effective particle-hole force, if one takes it to be just Kka', ale, More elegantly these terms are obtained in the method c) quoted above. In this case the effective particle-hole force consists of all diagrams which cannot be written as ladders with one particle and one hole line (fig. 7). Therefore in second order one has essentially the three graphs of fig. 7 (b). The first one corresponds to an intermediate particle-particle scattering and to the second order term of the K-matrix expansion of fig. 7(c) and the second and third graphs of fig. 7 (b) may be identified with the intermediate hole-hole and holeparticle scattering terms of (64) . One can now find a connection between the pseudopotential and the symmetry energy in comparing (64) with Landau's ansatz for the effective particlehole force. He considers the energy E of excited states as a function (resp. functional in infinite nuclear matter) of the momentum distribution nk)t and t In second order for the pseudopotential we may use the zeroth order nk since there is no first order change in nk .

LONG RANGE CORRELATIONS

31

postulates wkk', kk' - <99kk'IVIVkk'> - -

a2 E

(66)

k ank.'

a,4

the derivatives being taken at the ground state values of tire nk's. ( ci )

t

r

Fig. 7 . (a) Particle-hole ladder, (b) second order graphs for the pseudopotential, and (c) K-matrix expansion .

Now taking the second order expression for E, 2

E

_ 2m

v . n .np,_

I vp~ .9, a

Ey~-~-Ea,

2 ~ _o- Ea

_

(1-n  )(1--na,)rrpna, (6r)

one obtains Iykk',oe12 wkk', kk' ` vkk', kk' - 1 C(1-np)(1-na)-npnQ] o' a Eo+8a -Ek-Ek' 2 IvkDp'I +41 -.E P . P "~k'+Eo-Eí . 

(1

(68)

These terms are easily identified with the corresponding ones in (64) in the limiting case t -} k, l' -} k' i .e. long wavelengths of the collective modes. Only in this case the Landau theory can be expected . to be vaiid. This indicates another interpretation of the higher order corrections : If one writes E as _ E --

~

2

2

n,+_1 2

K'.  , n, n ,

(69)

with K,,,,,, ,p ,,, given by (65), the second and third graphs in fig. 7 (b) correspond

to derivatives of the K-matrix, i.e. rearrangement terms following from the depr- idence of the propagator

(1-- np) ( 1- #Q) on the momentum distribution of the state for which it is used. If one goes beyond the second order, the number of graphs increases rapidly and it is only possible to take into account selected subsets of the whole series . The Landau method suggests those graphs which at least should be summed for the particle hole force if the graphs are known which are important for the ground state energy. For instance, if the Brueckner approximation is valid for the ground state, and (66) is assumed to hold, then one obtains not only the diagrams of fig. 7 (c) of the K-matrix but also rearrangement terms like those in fig. 8. Assuming (66) it is easy to establish a relation between the pseudopoten-

V

r

Fig. 8. Rearrangement terms in the particle-hole force.

tial and the symmetry energy, resp. the compressibility . For this purpose consider the change in energy by replacing a certain amount 6N = Gneutr .bnk of neutrons by 6Z = Xprot.6nk = -6N protons with 6nk distributed uniformly over the Fermi surface . Then on the one hand one obtains, using (66), óE

=

k 2

211

8N-SZ 2

the last term coming from the change in the Fermi momenta for the neutrons and protons . On the other hand the definition of the symmetry energy gives

SE -K

(óN óZ) 2 A

( 71 )

Therefore, instead of the simple relation (55) one obtains -

8Po

k 2

w (Q)CIQ14n -- K- F , 6m

(72)

where the angle integration should be understood to run over the angle between k and ' the absolute magnitude of them being equal to ( ( - (' _-__ I kF .

33

LONG RANGE CORRELATIONS

A similar consideration for the density fluctuations gives a relation between the effective force w - and the compressibility namely Zpo

w (I2)dQ14z = ~1 - kF , 2

K

3m

This result differs from Landau's formula 8) by a factor z, only because of the inclusion of isospin. The author takes pleasure in thanking Professor V . F . Weisskopf for many stimulating discussions. He is also indebted to Professor K. Gottfried and Professor A. Kerman for valuable comments . He wishes to express his gratitude to the members of the M. I . T. theoretical physics group for their hospitality. The Bundesministerium fdr Atomkernenergie, Germany, supported his visit to U.S.A. by a travel grant. eferences 1) 2) 3) 4) 5) 6) 7) "') 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

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