A simple approximation for the nuclear density matrix

V01ume 738, num6er 3

PHY51C5 LE77ER5

27 Fe6ruary 1978

A 51MPLE APPR0X1MA710N F 0 R 7HE NUCLEAR DEN517Y MA7R1X

X. CAMP1 and A. 8 0 U Y 5 5 Y

1n5t1tut de Phy514ueNuc1da1re, D1v1510nde Phy514ue 7hd0r14ue 1, 91406 0r5ay Cedex, France Rece1ved 5 June 1977 Rev15ed manu5cr1pt rece1ved 25 N0vem6er 1977

we pr0p05e a new appr0x1mat10n f0r the nuc1ear den51ty matr1x 6a5ed 0n the Den51tyMatr1x Expan510n (DME) 0f Ne9e1e and vauther1n. When app11ed t0 a num6er 0f 51mp1epr061em51t 91ve56etter re5u1t5 than the 51ater and the truncated DME appr0x1mat10n5. 1n many phy51ca1 51tuat10n5 1t 15 0ften de51red t0 eva1uate 4uant1t1e5 1nv01v1n9 1nte9ra15 0ver the nuc1ear den51ty matr1x p(r, r•) = 2ac~a(r)~(r~ ). Ca1cu1at10n5 0f matr1x e1ement5 0f tw0-60dy 0perat0r5 are examp1e5. 7he5e 4uant1t1e5 are cum6er50me t0 ca1cu1ate, 1n part1cu1ar 6ecau5e 0f the c0mp11cated an9u1ar dependence 0n r and r•. 1n the pa5t, a num6er 0f appr0x1mat10n5 f0r p(r, r•) have 6een pr0p05ed. 7he m05t c0mm0n1y u5ed 0ne 15 the 51ater appr0x1mat10n (5L). 1n th15 appr0x1mat10n 0ne a55ume5 f1r5t, that the 51n91e-part1c1e funct10n5 are p1ane wave5, and 5ec0nd that f0r a f1n1te 5y5tem the Ferm1 m0mentum 15 91ven 6y the 10ca1 den51ty appr0x1mat10n

p(R +5/2, R - 5 / 2 ) - p ( R , 5 ) ~. p(R)]1(kF5),

(1)

where r(R) = ~a1V¢~(R)12 and k 15 50me avera9e va1ue 0f the re1at1ve m0mentum 0f tw0 1nd1v1dua1 1nteract1n9 part1c1e5. A1th0u9h the 0r191na1 1nf1n1te expan510n 5h0u1d 6e 1ndependent 0f the part1cu1ar ch01ce 0f k, th15 ch01ce 15 cruc1a1 f0r the truncated expan510n (2) 6ecau5e 1t determ1ne5 the c0nver9ence rate 0f the 5er1e5. Ne9e1e and Vauther1n ch005e k(R) = kF(R). 7hen e4. (2) 6ec0me5 the 51ater m1xed den51ty (1) p1u5 a c0rrect10n term. 7he va11d1ty 0f th15 appr0x1mat10n ha5 6een w1de1y d15cu55ed 1n ref5. [ 2 - 4 ] . 1t 15 the purp05e 0f th15 n0te t0 5h0w that 1t 15 p055161e t0 1mpr0ve up0n the appr0x1mat10n (2) wh11e keep1n9 the 51mp11c1ty 0f the 51ater appr0x1mat10n (1). We pr0p05e t0 def1ne the avera9e re1at1ve m0mentum 6y

where k F = (37r2p) 1/3 (p 15 e1ther Pneutr0n5 0r Ppr0t0n5),

R = (r + r•)/2, 5 = r - r• and ~(x) = (21 + 1)•• 1t(x)/x 1. 7h15 15 0f c0ur5e exact 0n1y 1n a h0m09ene0u5 1nf1n1te 5y5tem. M0re 5ucce55fu1 15 the Den51ty Matr1x Expan510n (DME) pr0p05ed 6y Ne9e1e and Vauther1n [1,2]. 7he den51ty matr1x 15 expanded 1n a 7ay10r 5er1e5 0f the re1at1ve 5eparat10n d15tance 5 a60ut the centre-0fma55 p01nt R. Perf0rm1n9 the an9u1ar 1nte9ra1 0ver the d1rect10n5 0f5 and after 50me rearran9ement 0f the 5er1e5 reta1n1n9 0n1y der1vat1ve thr0u9h 5ec0nd 0rder, they 06ta1n p(R, $) = p(R)/1

(k5) (2)

t 52:13 (k5) [-~ t V2p(R) - f1R) + } k2P(R)1, + -~

1 La60rat01re a550c16au c.N.R.5.

7h15 ha5 the v1rtue t0 cance1 the c0eff1c1ent 0f the c0rrect10n term 1n the DME and thu5 we 06ta1n an expre5510n f0rma11y 1dent1ca1 t0 the 51ater appr0x1mat10n

p(R, 5) -~ p(R)]1 (/~5),

(4)

6ut w1th/~ def1ned 6y e4. (3). 7h15 can 6e v1ewed a5 a ch01ce 0f an effect1ve avera9e m0mentum that acce1erate5 the c0nver9ence 0f the truncated expan510n (2). 1n what f0110w5 the va11d1ty 0f 0ur appr0x1mat10n 15 checked 6y c0mpar1n9 w1th exact den51ty matr1ce5. We f1r5t c0n51der a un1f0rm 1nf1n1te 5y5tem. A5 1n the ca5e 0f the DME each term 0f the 7ay10r expan510n 0f the den51ty matr1x 6ey0nd the f1r5t van15he5 263

V01ume 738, num6er 3

PHY51C5 LE77ER5

27 Fe6ruary 1978

1 0 ,, -..,. d-L.2 ~ ~

.-, ~.

1~.° .......... - - • . • ~ ~.01<.~:~ "~ ,

1 ~°

"-,•-, "•,-,,

"~,,~.1 ° ~ < ~ - " ~ "

.,1 .....

",,•5,.

,

2~ 01 0

; ~x-~. . . . .

.5

1.0

1.5

2.0

"." - ~ " ~ - - .~, -1% -~- " ~ - : - , 1 2.5

3.0

3.5 4.0 5(rm)

2 0

~ •,•,

0

.5

1.0

--

1.5

-

2.0

•,,

"•"" ~ " " " 2.5

3.0

3.5 4.0 5(Fm)

F19. 1. Rat10 0f 0ff-d1a90na1t0 d1a90na1den51ty matr1x 0(R +5/2, R - 5/2)/0(R) a5 a funct10n 0f the 1nterpart1c1e d15tance 5 f0r var10u5 va1ue50f the d15tance fr0m the centre 0f the nuc1eu5R (1n fm). 7he 10wer/upper end5 0f the ••err0r 6ar5•• are the exact va1ue5 ca1cu1ated f0r 5 para11e1/perpend1cu1ar t0 R. 7he d1fferent curve5 are exp1a1ned 1n the text. 1dent1ca11y [ 1] and tak1n9 1nt0 acc0unt that 1n th15 ca5e k 15 n0th1n9 6ut k F we a150 06ta1n the exact re5u1t. A5 an examp1e 0f the 0ther extreme we c0n51der the nuc1eu5 4He. U51n9 harm0n1c 05c111at0r (H0) 51n91e-part1c1e wave funct10n5 w1th parameter 6 = (x/h/m60) 1/2 we 06ta1n fr0m e4. (4) a 51mp1e expan510n f0r 5ma11 5,f0r the 54uare 0f the den51ty matr1x (the re1evant 4uant1ty 1n m05t ca5e5):

1t 15 w0rth n0t1c1n9 that f0r any nuc1eu5 de5cr16ed 6y H 0 wave funct10n5, the a5ympt0t1c f0rm 0f/~ a5 R -+ 00 15 (5/26) 1/2, a n0n-van15h1n9 va1ue. 1n c0ntra5t, f0r wavefunct10n5 ca1cu1ated 1n a f1n1te we11 p0tent1a1 (5uch a5 a 5ax0n-W00d5 0r Hartree-F0ck p0tent1a15) we f1nd

,02(R, 8) = p2(R)(1

where e 15 the 5eparat10n ener9y 0f the 1ea5t 60und 0r61ta1.7h15 5h0u1d 6e c0mpared t0 the m0re rap1d fa110ff 0f kF,

-

52/26 2 + 2-~-9(54/64) - ...),

where we have u5ed/c = (5/262)1/2 a5 91ven 6y e4. (3). 7h15 15 t0 6e c0mpared w1th the exact re5u1t ,02(R, 5) = p2(R) e = p2(R)(1 -

52/262

52/26 2 + ~(54/6 4) - ...).

We 5ee that appr0x1mat10n (4) 15 exact up t0 term5 1n 52 and that the err0r 1n the term 54 15 a60ut 10%. 1n 9enera1 f0r m0re c0mp11cated 5y5tem5 0n1y numer1ca1 c0mpar150n5 w1th exact re5u1t5 are fea5161e. H0wever f0r 119ht nuc1e1 u51n9 H 0 wave funct10n5 we 5t111have 51mp1e ana1yt1ca1 expre5510n5 f0r k. F0r examp1e: /~(R)= [ 5 { 9/2 + 322]] 1/2 L36-2~ 1 + 2 ~ ] 1

f0r

1¢(R) =,[ 5~ (39/2+1222+624)] 1 +/4242 5

f0r 40Ca,

where 2 = R/6. 264

160,

/~-+ [(513) x / ~ 1 ~

k F ~ (1]R 2/3)

2 (11R)1 m ,

exp[-213

-,~-mmef112R].

We w1115ee 1n what f0110w5 that when the den51ty matr1x 15 appr0x1mated 6y 0(R)f1(k5), [c15 a much 6etter appr0x1mat10n than k F at the 5urface and 1n the ta11 0f the nuc1eu5. 1n f19. 1 the va11d1ty 0f the appr0x1mat10n (4) 15 dem0n5trated 6y c0mpar1n9 the exact n0rma112ed den51ty matr1x p(R, 5)/#(R) w1th f1 (/~5) a5 a funct10n 0f 5 f0r a ran9e 0f va1ue5 0 f R . We u5ed H 0 wavefunct10n5 f0r 40Ca (f19. 1a) and HF neutr0n wavefunct10n5 f0r 208p6 (f19. 16). 7he exact va1ue5 were ca1cu1ated f0r tw0 d1rect10n5 0f5:para11e1 and perpend1cu1ar t0 R, repre5ented 1n the f19ure 6y the end5 0f the ••err0r 6ar5••. 7he appr0x1mat10n (4), 5h0wn a5 a 5011d 11ne (PW), 15 5een t0 6e very 5ucce55fu1 f0r 5 1e55 than a few fm and f0r a11 d15tance5 R. C0mparat1ve1y the DME (5h0wn a5 a da5hed 11ne) 15 5119ht1y6etter 1n the 1nter10r 0f the

V01ume 738, num6er 3

PHY51C5 LE77ER5

nUC1eU5 and W0r5e at the 5UrfaCe. A150 5h0Wn (da5hed--d0tted 11ne) 15 the 51ater appr0X1mat10n. 1t 15 def1n1te1y 1nfer10r 60th 1n the 1nter10r and at the 5UrfaCe 0f the nUC1eU5. F r 0 m the C0mpar150n We c0nc1ude that appr0x1mat10n (4) pr0v1de5 519n1f1cant1y 6etter re5U1t5, Wh11e keep1n9 the 51mp11c1ty 0f the 51ater appr0x1mat10n. 1n 0rder t0 check m0re th0r0u9h1y the va11d1ty 0f the var10u5 appr0x1mat10n5, we c0mpute the exchan9e matr1x e1ement fp2(R, 5)0(5) dR d5 f0r c0mm0n1y u5ed tw0-60dy 1nteract10n5 0(5), that 15 ca1cu1at1n9 avera9e5 0f the 54uare 0f the den51ty matr1x w1th the appr0pr1ate we19ht5 0ver R and 5.1n f19. 2 we 5h0w 100(E~ xact - --xEaPPr~/FexactJ,-x, the percent err0r 6etween the exact and the appr0x1mate ener91e5 ca1cu1ated a5 a funct10n 0f the ran9e/~ 0f a 9au551an tw0-60dy f0rce. We 5ee 1n f19. 2 that appr0x1mat10n (4) (5011d 11ne) 91ve5 rea50na61y 5ma11 err0r5 w1th1n the ran9e 0f 5tandard nuc1ear f0rce5.7he 51ater re5u1t5 ( d a 5 h e d - d 0 t t e d 11ne) are much w0r5e even f0r very 5h0rt-ran9e f0rce5. 7he DME 0f Ne9e1eVauther1n (da5hed 11ne), 06ta1ned 6y ne91ect1n9 the 12(k5) term 1n the 54uare 0 f e 4 . (2), (a5 1n ref. [1]), 15 n0t a5 900d a5 0ur appr0x1mat10n. F0r m0derate 5h0rt-ran9e Yukawa f0rce5 we 06ta1n re5u1t5 51m11ar t0 th05e 0f f19.2, the d1fference 6etween the DME and appr0x1mat10n (4) 1ncrea51n9 w1th the ran9e/~. 1n the 11m1t/a ~ ¢0 0ne 9et5 the C0u10m6 f0rce. 1n ta61e 1 are 91ven the C0u10m6 exchan9e ener91e5 E c ca1cu1ated f0r d1fferent 119ht nu-

*/*

/./ /~/,~,:/~/~ ..,:" // /"

~

35~ ....... --

f"

,•"

J"

/J

f"

,•

J"

/./

].

~ (;. 0.

0/

.J

"1 J"

,,,,°"

1.

2. )3(frn) 3.

F19. 2. Percent err0r 6etween the exact and the appr0x1mate exchan9e ener91e5 f0r a 9au551an tw0-60dy f0rce ca1cu1ated a5 a funct10n 0f the ran9e u.

27 Fe6ruary 1978

7a61e 1 Exact (ex) and appr0x1mate C0u10m6 exchan9e ener91e5E c ca1cu1ated w1th harm0n1c 05c111at0r wave funct10n5 w1th f1xed parameter5 6 = 1.33 f0r 4He, 1.83 f0r 160, 2.08 f0r 40Ca. A150 5h0wn are the t0ta1 ener91e5E t and the c0rre5p0nd1n9 rm5 rad11 06ta1ned 6y tak1n9 the m1n1mum ener9y w1th re5pect t0 6.7he ener91e5 are 1n MeV and the 1en9th51n fm. 7he d1fferent appr0x1mat10n5 are exp1a1ned 1n the text. ex

PW

5L

DME

Ec 4He E t

-0.86 28.20 1.72

-0.78 25.55 1.79

-0.74 17.45 2.00

-0.47 23.62 1.86

Ec

-2.98 106.40 2.65

-2.75 101.70 2.70

-2.75 87.56 2.79

-2.31 100.05 2.72

-7.46 323.40 3.36

-7.03 314.25 3.41

-7.05 290.80 3.45

-6.42 312.50 3.41

(r2)1/2

160 Et

~r~)1/2

Ec 4°Ca Et

(r2)1/2

c1e1, u51n9 H 0 wave funct10n5.51m11ar re5u1t5 were 06ta1ned u51n9 rea115t1c HF wave funct10n5. We 5ee that 1n the ca5e 0f the C0u10m6 f0rce 5L and appr0x1mat10n (4) 91ve pract1ca11y the 5ame re5u1t5.7h15 mean5 that, a1th0u9h the appr0x1mat10n5 are 6a5ed 0n a 5ma11 n0n10ca11ty expan510n, they 5t11191ve avera9e rea50na61e re5u1t5 f0r the 05c111at1n9 ta11 0f the den51ty matr1x at 1ar9e d15tance5 5 . 7 h e c0rrect10n term t0 5L 1n the DME 15 t00 1ar9e. F0110w1n9 Ne9e1e and Vauther1n, we a1way5 ne91ect the j 2 - t e r m 1n the 54uare 0f the den51ty matr1x 1n the DME appr0x1mat10n. We have a150 perf0rmed ca1cu1at10n5 u51n9 the fu11 54uare 0f expre5510n (2), (a1th0u9h we 5h0u1d ment10n that 1t 15 rather ar61trary t0 reta1n 1t 1n the truncated expan510n). 7h15 1a5t appr0x1mat10n 91ve5 5119ht1y 6etter re5u1t5 than the 5tandard DME f0r m0derate1y 5h0rtran9e f0rce5 6ut much w0r5e f0r very 10n9-ran9e f0rce5.1n part1cu1ar the exchan9e 1nte9ra15 d1ver9e f0r the C0u10m6 f0rce. 1t 5h0u1d 6e empha512ed that 1t 15 m05t 9rat1fy1n9 that 0ur appr0x1mat10n w0rk5 5t111 rea50na61y we11 f0r th15 1nf1n1te-ran9e f0rce a110w1n9 1n many ca5e5 t0 treat the exchan9e term5 0f the C0u10m6 and nuc1ear f0rce5 0n the 5ame f00t1n9. We n0w app1y 0ur appr0x1mat10n t0 c0n5truct the H a r t r e e - F 0 c k ener9y den51ty. F0r 51mp11c1ty we pre5ent here the re5u1t5 f0r a den51ty-1ndependent tw060dy f0rce and f0r 5pher1ca1 and 5p1n 5aturated nuc1e1 w1thN = 2 and n0 C0u10m6 f0rce. Re5u1t5 f0r the 265

V01ume 738, num6er 3

PHY51C5 LE77ER5

9enera1 ca5e w1116e pre5ented e15ewhere. 7he ener9y den51ty can 6e wr1tten a5

H(R) = (h2/m) r(R) +Hd(R) -- 27rp2(R)

f52d5j•2•1(k5) Vex(x),

(5)

where 0 = 0n = 0p, r = r . = rp and Vex =• [V 11 • 3V13 • 3V31 + 9V33]. We keep f0r the d1rect term Hd(R) the exact expre5510n 6ecau5e, a5 5h0wn 1n ref5. [2,3] the DME expan510n pr0p05ed 1n re1•. [1 ] 15 n0t accurate en0u9h. Vary1n9 expre5510n (5) w1th re5pect t0 a part1cu1ar wave funct10n ~a(R) = (un1/r) y[n and 1t5 der1vat1ve5 t 1t ¢~(R) and ~ ( R ) (and 1nte9rat1n9 6y part5 when nece55ary), a 5y5tem 0f 5ec0nd-0rder d1fferent1a1 e4uat10n5, w1th an effect1ve ma55, 15 06ta1ned:

h2 (u•• 1(1+1)u)+ 8•u• 2m*

r2

+ ( V d +A + C+•8••

-8•/2r)u = eu,

27 Fe6ruary 1978

preferred 1n th15 pre11m1nary 111U5trat10n t0 te5t the Va11d1ty 0f expre5510n (5) 6y 51mp1y perf0rm1n9 a re5tr1cted var1at10na1 ca1cu1at10n. We C0mputed the ener9y den51ty w1th H 0 wave funct10n5 and m1n1m12ed the ener9y w1th re5pect t0 6, the 05c1Uat0r parameter. 7he exact ener91e5 were ca1cu1ated very accurate1y u51n9 ana1yt1ca1 expre5510n5 f0r the 7a1m1 1nte9ra15 [5]. We u5ed the 8r1nk and 80eker 81 f0rce [6]. 7he m1n1ma1 t0ta1 ener91e5 E t a5 we11 a5 the c0rre5p0nd1n9 rm5 rad11 are 5h0wn 1n ta61e 1. We 5ee that the pre5ent appr0x1mat10n a5 expected fr0m the prev10u5 d15cu5510n w0rk5 6etter than 5L and DME 91v1n9 h0wever an 0veraU under61nd1n9 wh1ch decrea5e5 w1th the num6er 0f part1c1e5. We c105e 6y remark1n9 that 6ecau5e 0f 1t5 51mp11c1ty th15 appr0x1mat10n can 6e ea511y app11ed t0 c0mp1ex pr061em5 5uch a5 Hartree--F0ck ca1cu1at10n5 w1th rea115t1c f0rce5 0r der1vat10n 0f n0n-10ca1 0pt1ca1 p0tent1a15 [7]. W0rk 1n the5e d1rect10n515 1n pr09re55. We are Very 1nde6ted t0 N. AUer6aCh f0r a CarefU1 read1n9 0f the manu5Cr1pt.

where h2/2m * = h2/2m - 8, 8 = 5 (C/k2),

ReferenCe5

cA-

9np f d 5 Vex(5) J1 (k5)(k5) j2(1c5), 12 18~rp

k2

fd5 vex(5)21(k5),

and Vd 15 the exact expre5510n f0r the d1rect HF p0tent1a1. 7he5e e4uat10n5 are 51mp1er than th05e 06ta1ned w1th the DME [1]. 1n5tead 0f 501v1n9 numer1ca11y the5e e4uat10n5, we

266

[1] J.W. Ne9e1e and D. Vauther1n, Phy5. Rev. C5 (1972) 1472. [2] J.W. Ne9e1e and D. Vauther1n, Phy5. Rev. C11 (1975) 1031. [3] D.W.L. 5prun9, M. Va111~re5,X. Camp1 and C.M. K0, Nuc1. Phy5. A253 (1975) 1. [4] J. 7re1ner and H. Kr1v1ne,J. Phy5.62 (1976) 285. [5] R. 7h1e6er9er, Nuc1. Phy5. 2 (1956-57) 533. [6] D.M. 8r1nk and E. 80eker, Nuc1. Phy5. A91 (1967) 1. [7] N. V1nh Mau, t0 6e pu6115hed.