On the temperature dependence of the level density parameter

V01ume 1648, num6er 4,5,6

PHY51C5 LE77ER5

12 Decem6er 1985

0N 7HE 7EMPERA7URE DEPENDENCE 0F 7HE LEVEL DEN517Y PARAME7ER Er1c 5 U R A U D , Peter 5 C H U C K 1n5t1tut de5 5c1ence5 Nuc1ka1re5, 53, Avenue de5 Martyr5, F-38026 6ren061e cedex, France

and Ra1ner W. H A 5 5 E 1n5t1tut Laue-Lan9ev1n, 156X centre de 7r1, F-38042 6ren061e cedex, France

Rece1ved 14 Au9u5t 1985

we 5tudy the va11d1ty0f 8ethe•5 f0rmu1a f0r the 1eve1den51ty0(E*) w1th exc1tat10n ener91e5 E* c0rre5p0nd1n9 t0 re1at1ve1y h19h temperature5 ( 7 - 4-6 Mev). F0r th15 purp05e we ca1cu1ate 1n a 5e1f-c0n515tent5em1-c1a551ca1m0de1 the dependence 0f the 1eve1den51ty parameter a 0n the temperature/exc1tat10n ener9y 0f the nuc1eu5,we 5h0w that the 1nf1uence0f temperature repre5ent5 1e55 than 5% up t0 7 - 4 - 6 Mev 0r E * - 1.5-3 Mev/A. 709ether w1th an e5t1mate 0f the ener9y dependent c0rrect10n5 th15a110w50ne t0 u5e 8ethe•5 f0rmu1a w1th the emp1r1ca12er0-temperature va1ue 0f the 1eve1den51typarameter up t0 5uch h19h temperature5.

Heavy-10n c0111510n5at 1ntermed1ate ener91e5 are n0w 91v1n9 5tr0n9 exper1menta1 ev1dence f0r the f0rmat10n 0f nuc1e1 w1th h19h exc1tat10n ener91e5 E* [1]. Am0n9 0ther n0n-ne9119161e d1ff1cu1t1e5 the de5cr1pt10n 0f the deexc1tat10n pr0ce55e5 15 very 1mp0rtant, 1n part1cu1ar t0 11nk the5e exc1ted 5y5tem5 t0 the re51dua1 nuc1e1 wh1ch are actua11y 065erved. 7h15 de5cr1pt10n ma1n1y 1nv01ve5 an evap0rat10n c0de 1n wh1ch the 1eve1 den51ty p(E*) p1ay5 a centra1 r01e. A c0mm0n1y u5ed expre5510n f0r p ( E * ) 15 91ven 6y 8ethe•5 f0rmu1a [2] 1n wh1ch the kn0w1ed9e 0f the 50-ca11ed 1eve1 den51ty parameter a 15 fundamenta1. 5tr1ct1y 5peak1n9 8ethe•5 f0rmu1a 15 a 2er0-temperature 11m1t and 1t5 exten510n t0 very h0t/exc1ted nuc1e1 15 n0t 5tra19htf0rward, a11 the m0re that p ( E * ) 15 very 5en51t1ve t0 the va1ue 0 f a . 7he purp05e 0f th15 n0te 15 t0 5h0w, thr0u9h a 5e1f-c0n515tent 5em1-c1a551ca1ca1cu1at10n at f1n1te temperature, that 8ethe•5 f0rmu1a may 1n fact 6e u5ed w1th 1t5 2er0-temperature va1ue 0 f a up t0 rather h19h temperature5 ( 7 ~ 4 - 6 MeV). 8ef0re ca1cu1at1n9 the 1eve1 den51ty 1t 15 nece55ary t0 have a rea50na61e de5cr1pt10n 0f a h0t/exc1ted nuc1eu5.7he ma1n d1ff1cu1ty ar151n9 1n a de5cr1pt10n 212

0f nuc1e1 at f1n1te temperature 15 t0 take 1nt0 acc0unt the c0nt1nuum 5tate5 wh1ch are 0ccup1ed a5 500n a5 the temperature 15 n0n.2er0.7h15 0ccupat10n pr06a6111ty ref1ect5 the fact that a h0t nuc1eu515 un5ta61e a9a1n5t part1c1e em15510n.A p055161e (5tat1c) appr0ach, tak1n9 c0n515tent1y 1nt0 acc0unt the c0ntr16ut10n 0f the c0nt1nuum, 15 t0 de5cr16e the 5y5tem a5 a therma112ed nuc1eu51n e4u1116r1um w1th a 9a5.7h15 9a5 ha5 n0 phy51ca1 ex15tence 1n a heavy-10n c0111510n6ut exert5 a pre55ure wh1ch exact1y c0unter6a1ance5 the pre55ure 0f the evap0rat1n9 nuc1e0n5, 50 that the evap0rat10n pr0ce55 can 6e repre5ented a5 a 5er1e5 0f e4u1116r1um 5tate5.7h15 pre5cr1pt10n may 6e ea511y 1ntr0duced 1n mean-f1e1d ca1cu1at10n5 6ecau5e 0f the ex15tence, f0r 91ven temperature and chem1ca1 p0tent1a1, 0f tw0 501ut10n5 0f the Hartree-F0ck (HF) e4uat10n5, c0rre5p0nd1n9 re5pect1ve1y t0 the nuc1eu5 w1th 1t5 evap0rated nuc1e0n5 (nuc1eu5 + 9a5, N 6 ) and t0 the nuc1e0n 9a5 a10ne (9a5, 6 ) [3]. 7he5e tw0 501ut10n5 mer9e 1nt0 each 0ther at 1ar9e d15tance5 and the pr0pert1e5 0f the nuc1eu5 may 6e extracted fr0m a therm0dynam1ca1 p0tent1a1 ~ def1ned a5 the d1fference 6etween the 9rand p0tent1a15 ~2N6and 126 a550c1ated t0 the5e tw0 501ut10n5 [3]. 0370-2693[85/$ 03.30 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

V01ume 1648, num6er 4,5,6

PHY51C5 LE7rER5

A p055161e an5at2 t0 the5e HF Ca1Cu1at10n5, a110w1n9 t0 repr0duce the avera9e HF 4uant1t1e5, wh1ch 15 4u1te 5uff1c1ent f0r 0ur purp05e, 15 pr0v1ded 6y 1ntr0duc1n9 5em1-c1a551ca1meth0d5, 50 that we have perf0rmed 0ut ca1cu1at10n5 1n the framew0rk 0f a 7h0ma5--Ferm1 (7F) appr0x1mat10n 0f the pre5cr1pt10n 0f ref. [3]. When u51n9 a 5kyrme-type effect1ve 1nteract10n, a5 we a150 d0, the p0tent1a1 ~ may 6e expre55ed a5 a funct10na1 0f the 10ca1 part1c1e den51t1e5 pN6(r) and 06(r ) re5pect1ve1y 0f the N6- and 6-11ke 501ut10n5.7he actua1 nuc1eu515 then def1ned a5 a 5add1e p01nt 0f ~ [3] w1th re5pect t0 the var1at10n5 0f PN6 and 06. W1th the e4u1116r1um den51ty pr0f11e5PN6 and P6 0ne can ca1cu1ate any re1evant phy51ca1 4uant1ty 5uch a5 the entr0py, the chem1ca1 p0tent1a15 and the exc1tat10n ener9y, wh1ch 15def1ned, f0r a 91ven nuc1eu5, a5 the d1fference 0f the ener91e5 0f the nuc1eu5 at 7 and 7 = 0. We want t0 p01nt 0ut that E* 15 thu5 06ta1ned w1th0ut referr1n9 t0 any 1eve1 den51ty parameter. We have checked that 0ur re5u1t5 0n h0t nuc1e1, 1n th15 2er0-0rder 7 h 0 m a 5 Ferm1 appr0x1mat10n a5 we11 a5 1n the m0re 50ph15t1cated extended-7h0ma5-Ferm1 appr0ache5 [4], are 1n c105e a9reement (~ 1-5%) w1th the avera9e HF 4uant1t1e5 [5]. 1n add1t10n, the c0mpar150n 0f 0ur ca1cu1at10n5 w1th 50me recent exper1menta1 data c0ncern1n9 the max1mum exc1tat10n ener9y/temperature that a nuc1eu5 can 5u5ta1n w1th0ut d151nte9rat1n9 1mmed1ate1y, 15 very enc0ura91n9 [6]. 0nce we kn0w the e4u1116r1um c0nf19urat10n 0f the nuc1eu5,1 we can ca1cu1ate the 51n91e-part1c1e 1eve1 den51ty at ener9yE, wh1ch 15 def1ned 6y 9(E) =

~ ~(E - e1) = 7r(E - £ 0 . (1) 5tate5 1 7he 2er0-0rder 5em1-c1a551ca1appr0x1mat10n 0f9 15 ea511y 06ta1ned 6y rep1ac1n9 the trace 6y an 1nte9rat10n 0ver the pha5e 5pace and the 51n91e-part1c1e ham11t0n1an 0perat0r/1 6y 1t5 c1a551ca1c0unterpart H c: 7r~

f d3rd3p

171~Hc=p2/2m + V(r),

(2)

900d a9reement w1th the exact ca1cu1at10n 1n the ca5e 0f harm0n1c 05c111at0r p0tent1a15 [7]. 1n the framew0rk 0f 0ur 5u6tract10n pr0cedure we have t0 take 1nt0 acc0unt the effect5 0fthe 9a5, 50 that after hav1n9 perf0rmed the 1nte9rat10n5 0ver the m0mentum p, the 1eve1 den51ty 9(E) 15 06ta1ned a5 a d1fference 0f tw0 0ne-d1men510na1 1nte9ra15 0ver the rad1u5,2.

1 ( f4~rr2(h2/2m,)~36/2(E_VN6)1/2 dr

9(E) = 2rr2

- f47rr2(h2/2m*)~3/2(E-

,1 F0r the 5ake 0f 51mp11c1tywe 5ha11wr1te the f0rmu1ae 1n the ca5e 0f a 5ymmetr1cnuc1eu5 w1thA nuc1e0n5.

V6)1/2 d r ) ,

(3)

where VN6 and V6 re5pect1ve1y den0te the 5e1fc0n515tent 51n91e-part1c1e p0tent1a15 f0r the nuc1eu5 + 9a5 and 9a5-11ke 501ut10n5 and where m* = m * ( N 6 / 6 ) 15 the effect1ve ma55 0f the 1nteract10n u5ed (wh1ch 15 a 11near funct10n 0f the part1c1e den51ty f0r 5kyrme f0rce5). N0t1ce at 1a5t that 1n e4. ( 3 ) E 15 the ener9y eva1uated fr0m the 60tt0m 0f the nuc1eu5 + 9a5 p0tent1a1 we11. Kn0w1n9 the 1eve1 den51ty 9(E), 1t 15 ea5y t0 def1ne a Ferm1 ener9y e F ( 7 ) at temperature 7 6y (5ee f00tn0te 2). A = f 9(e) {1 + exp[(e - eF(7))/7] }--1 d e , (4) 0 and the a550c1ated 1eve1 den51ty parameter a(7) = ~rr29(eF). N0t1ce that th15 deffm1t10n 0 f a take5 c0mp1ete1y 1nt0 acc0unt the dependence 0f the 51n91epart1c1e ener91e5 0n the exc1tat10n ener9y and that th15 temperature dependent a 15 a pr10r1 n0t t0 6e u5ed 1n 8ethe•5 f0rmu1a. An0ther 4uant1ty 0f 9reat 1ntere5t 15 the 51n91epart1c1e exc1tat10n ener9y E~p wh1ch read5: E~p(7~) -- E5p(7 ) - E5p(7 = 0) =• 0

e9(e) {1 + exp[(e -- eF(7))/7]}-1 de F(7=0)

(2~h)3 • where V(r) 15 the 51n91e-part1c1e p0tent1a1.7h15 5em1c1a551ca1 appr0x1mat10n 0 f 9 ha5 6een pr0ved t0 6e 1n

12 Decem6er 1985

-

e9(e)

de,

(5)

0 ,2 At 2er0 temperature the 5ec0nd 1nte9ra11n e4. (3) 51mp1y van15he5and the Ferm1 fact0r 1n e4. (4) reduce5 t0 the 5tep funct10n 0(e F - e). 213

PHY51C5 LE77ER5

V01ume 1648, num6er 4,5,6

and Wh1Ch 15 preC15e1y the eXC1tat10n ener9y 1nV01Ved 1n 8ethe•5 f0rmU1a. Let U5 n0te that we d0 n0t U5e the C1a551Ca1def1n1t10n 0f the 51n91e-part1C1e ener9y 1n Wh1Ch a fact0r 0f 1/2 15 1ntr0dUCed f0r C0mpar150n w1th HF 61nd1n9 ener91e5 [8]. 0Ur def1n1t10n 15 51mp1y the C1a551Ca1C0Unterpart 0f the 4Uantum 51n91e-part1C1e ener9y and hence d0e5 n0t 1nV01Veth15 fact0r. A5 Can 6e 5een 1n ta61e5 1--3, E~p 15 Very C105e t0 the ••t0ta1•• eXC1tat10n ener9y E* (1E* -- E5*p1~< 10%). 7h15 p01nt 15 fundamenta1 6eCaU5e 1f 0ne a55Ume5 that E* 15 a rea50na61e e5t1mate 0f an exper1menta1 exc1tat10n ener9y E~xp, 1t 15 p055161e t0 rep1ace, 1n 8ethe•5 f0rmu1a, E~p 6y E~x p w1th0ut mak1n9 a 1ar9e err0r, whatever the va1ue 0fE~xp 15. Let u5 a150 n0te that the ••51n91e-part1c1e•• 1eve1 den51ty parameter a5p 7a61e 1 C0mpar150n 0f the t0ta1 exc1tat10n ener91e5E* w1th 51n91epart1c1e ener91e5E~p at var10u5 temperature5 7 f0r 20ap6, w1th the 5KM f0rce. A150are 1nd1catedthe va1ue50f the ••51n91e-part1c1e•• 1eve1den51ty parameter def1ned a5 a5p = * 2 E5p/7 . 7he5e va1ue50fa5p are t0 6e c0mpared w1th the va1ue 0fa ca1cu1atedat 2er0 temperature, wh1ch 15a = 16.82 MeV-1 f0r 208 P6.

E* (MeV) E*n (MeV) -21 a5p (MeV-)

7 = 2 MeV

4

6

8

66.5 65.5 16.38

253.3 236.4 14.78

553.6 504.3 14.01

977.6 897.6 14.00

7a61e 2 5ame a5 ta61e 1, f0r 902r. 7he 2er0-temperature va1ue 0f a 15a = 7.87 MeV-1 .

E* (MeV) E~p (61eV) a5p (MeV-1 )

7 = 2 MeV

4

6

8

30.8 31.0 7.75

119.1 117.6 7.35

256.3 246.6 6.85

438.2 414.1 6.47

7a61e 3 5ame a5 ta61e 1, f0r 4°Ca. 7he 2er0-temperature va1ue.0fa 15a = 3.93 MeV-1 .

E* (MeV) E~p (MeV) a5p (MeV-1 )

214

7 = 2 MeV

4

6

8

15.2 15.9 3.98

58.6 59.9 3.75

125.8 125.7 3.49

211.0 204.1 3.19

12 Decem6et 1985

def1ned a5 a5p = E~p/7 2 15, f0r temperature5 up t0 7 ~ 2 - 3 MeV, very c105e t0 the va1ue 0 f a ca1cu1ated at 2er0/10w temperature 6y mean5 0f e4. (3). 7h15 a9reement 15 a 900d te5t 0f the accuracy 0f 0ur ca1cu1at10n 0fE~p and hence 0f9(E), 6ecau5e at 10w temperature5 E~p 15 typ1ca11y 0f the 0rder 0f 0.5% 0f the 51n91e-part1c1e ener91e5 E5p [e4. (5)]. F0r h19her temperature5 ( 7 ~ 4 MeV), e5t1mat1n9 the 1eve1 den51ty parameter a5 a5p d0e5 n0t h01d anym0re and we have t0 u5e e4. (3). 1n f19.1 we have p10tted the va1ue5 0f the 1eve1 den51ty parameter a ( 7 ) = ~n29(eF) [e4. (3) f0r E = eF(7)] a5 a funct10n 0f the temperature/exc1tat10n ener9y f0r 3 nuc1e1: 40Ca, 902r and 208p6. N0t1ce that the 11nk 6etween temperature and exc1tat10n ener9y 15 nuc1eu5 dependent. 7he 5tr1k1n9 feature 0f th15 f19ure 15 that up t0 7••" 4 MeV f0r 1ead, 7 ~ 6 MeV f0r 21rc0n and 7 "~ 8 MeV f0r ca1c1um the va1ue 0 f a 15 c0n5tant w1th1n 1e55 than 2%. 7h15 mean5 that up t0 an upper 11m1t depend1n9 0n the nuc1eu5 the 51n91e-part1c1e ener91e5 ar0und the Ferm1 ener9y d0 n0t depend 0n the exc1tat10n ener9y 0f the nuc1eu5, wh1ch can 6e 4ua11tat1ve1y checked 6y c0mpar1n9 0ur re5u1t5 w1th HF 5pectra [3]. 8ey0nd 7 ~ 4 - 6 MeV the effect5 0f f1n1te temperature are 6ec0m1n9 1ncrea51n91y 1mp0rtant. 1n part1cu1ar, f0r 7>~ 8 MeV 0ne f1nd5 a 11m1t1n9 temperature, depend1n9 0n the nuc1eu5, 6ey0nd wh1ch the 1atter d0e5 n0t ex15t anym0re 0f C0u10m6 1n5ta6111t1e5 [3]. 7he rap1d 1ncrea5e 0 f a 1n th15 h19h temperature re910n 15 pr06a61y re1ated t0 th15 1n5ta6111ty. A5 we have ca1cu1ated a 6y u51n9 0n1y the 51n91epart1c1e 5pectrum, we f1nd the0ret1ca1 va1ue5 4u1te d1fferent (at 2er0 and then f0r 7 ~ 4 - 6 MeV) fr0m the exper1menta1 va1ue5.1t ha5 h0wever 6een 5h0wn [9] that the 1nc1u510n 0f dynam1ca1 c0rre1at10n5 5uch a5 the c0up11n9 0f 51n91e-part1c1e 0f tw0-part1c1e 0neh01e 0r tw0-h01e 0ne-part1c1e 1ntermed1ate 5tate5 enhance5 the 1eve1 den51ty at the Ferm1 ener9y 6y up t0 40%, 6r1n91n9 1t 1n c105e a9reement w1th exper1menta1 data. 7he dynam1ca1 c0rrect10n5 91ve r15e t0 an ener9y dependent c0rrect1ve term V" t0 the Hartree-F0ck p0tent1a1 VHF , and V 15 u5ua11y 1nc0rp0rated1nt0 an effect1ve ma55 m/m* = 1 + d(VHF + V)/dE, wh1ch w0u1d enter d1rect1y the 1eve1 den51ty [e4. (3)]. 8ef0re draw1n9 any def1n1te c0nc1u510n a60ut the actua1 dependence 0f a rea115t1c 1eve1 den51ty parameter a 0n the temperature/exc1ta-

V01ume 1648, num6er 4,5,6

PHY51C5 LE77ER5

0 E*(MeV)

100

250

12 Decem6er 1985

500

1000

20

19

208p6

18

17

16 1

1

1

1

1

0

1

1

E*

100

1

1

7

5

10

250

500

902r a

~ 1

8

(MeV-1) 7 1

1

t

1

1

1

5

E*

50

1

10

7

100

200

40Ca

a

..J 3 1

0

J

h

1

1

1

5

7(MeV)

1

,0

F19.1. Va1ue5 0f the 1eve1den51ty parameter a = ~1r29(eF(7)) [e4. (3)] a5 a funct10n 0f temperature (10wer 5ca1e5)/exc1tat10n ener9y ,(upper 5ca1e5)f0r 208p6 (fu1111ne), 902r (da5hed 11ne)and 4°Ca (da5h-d0tted 11ne), w1th the 5KM f0rce.

t10n ener9y 1t 15 hence nece55ary t0 e5t1mate the temperature dependence 0f F . F0r th15 purp05e we have perf0rmed a phen0men01091ca1 5em1-c1a551ca1 ca1cu1at10n u51n9 the m0de1 recent1y deve10ped 6y Ha55e et a1. [1 0]. 7h15 m0de1 15 n0t tru1y c0n515tent w1th 0ur 5e1f-c0n515tent prev10u5 ca1cu1at10n 0f a [e4. (3)] 6ut appear5 a5 5uff1c1ent f0r e5t1mat1n9 the 0rder 0f ma9n1tude 0f the effect5 0f temperature 0n V. 7he 1ma91nary part 0f the ma55 0perat0r 15 ca1cu1ated

w1th a phen0men01091ca1 mean-f1e1d p0tent1a1, and the c0rre5p0nd1n9 rea1 part V" 15 then e5t1mated v1a the u5ua1 d15per510n re1at10n. A fuU acc0unt 0f th15 ca1cu1at10n w1116e 91ven e15ewhere [11 ] 6ut we th1nk 5uff1c1ent f0r 0ur purp05e t0 ca1cu1ate the 510pe dV/dE1E=~F f0r part1c1e5 w1th 2er0 k1net1c ener9y. 7h15 1atter re5tr1ct10n mean5 that we have avera9ed 0ver the n0n-10ca11ty and c0n51dered the nuc1ear 1nter10r 0n1y. 0ur re5u1t5 are 5h0wn 1n f19.2 where 215

V01ume 1648, num6er 4,5,6

PHY51C5 LE77ER5

0.3

12 Decem6er 1985

1

Cd~=~F

0.22

0.2

1

~

1

1

1

[

•

1

1

1

7 (MeV)

1 10

F19.2. 510pe 0f the dynam1ca1e0rre1at10n ener9y at the Ferm1 ener9y ver5u5temperature. N0te the 1ar9e 5ca1e 0n the 0rd1nate. 0ne can 5ee that the 1nf1uence 0 f 715 weak at 10w temperature5. N0te 1n part1cu1ar the 1ar9e 5ca1e u5ed 0n the 0rd1nate 0 f f19.2 and the fact that d V~/dE var1e5 0n1y 6etween 20% and 25% 0f the effect1ve ma55 f0r 7 ~ 5 MeV. A5 a c0n5e4uence, 1t 5eem5 rea50na61e t0 c0n51der that up t0 7 ~ 4 - 6 MeV a c0mp1ete ca1cu1at10n 0 f a , tak1n9 c0n515tent1y 1nt0 acc0unt the c0ntr16ut10n5 0f the 51n91e-part1c1e 5tate5 p1u5 the c0rre1at10n5, w0u1d n0t depend 0n the temperature very much, 91v1n9 r15e t0 a chan9e 0f n0t m0re than a few percent. 709ether w1th the fact that the 51n91e.part1c1e exc1tat10n ener9y E~p 15 very c105e t0 the t0ta1 exc1tat10n ener9y E*, th15 temperature 1ndependence 0 f a a110w5 0ne t0 u5e 8ethe•5 f0rmu1a w1th exper1menta1 va1ue5 0f the 1eve1 den51ty parameter (06ta1ned at 7 = 0) and w1th E* = E~x p, up t0 7 ~ 4 - 6 MeV. 1t 5h0u1d 6e n0ted that 1f the effect5 0 f f1n1te temperature had 6een 1mp0rtant we c0u1d n0t have drawn any p051t1ve c0nc1u510n 6ecau5e, 0ut51de the 10w temperature re910n, 1t w0u1d n0t have had any 5en5e t0 u5e 8ethe•5 f0rmu1a w1th any va1ue 0f the 1eve1 den51ty parameter. 1n th15 n0te we have ca1cu1ated the 51n91e-part1c1e 1eve1 den51ty 9 1n the framew0rk 0f a 5e1f-c0n515tent 7h0ma5-Ferm1 m0de1 at f1n1te temperature. We have 5h0wn that the 51n91e-part1c1e exc1tat10n ener9y, wh1ch can 6e ea511y deduced fr0m 9,15 very c105e (up t0 "••10%) t0 the t0ta1 exc1tat10n ener9y and that the effect5 0f temperature 0n the 1eve1 den51ty parameter, 1.e. 0n the 5pectrum ar0und the Ferm1 ener9y, 15 216

ne9119161e up t0 7 ~ 4 - 6 MeV. A5 a c0n5e4uence the 5tandard 8ethe f0rmu1a can 6e u5ed, 1n f1r5t appr0x1mat10n, w1th the exper1menta1 va1ue5 0 f the exc1tat10n ener9y and 0f the (2er0-temperature) 1eve1 den51ty parameter up t0 temperature5 7••- 4 - 6 MeV (0r E*/A ~ 1 . 5 - 3 MeV, depend1n9 0n the nuc1eu5), w1th 900d accuracy. We w0u1d 11ke t0 thank M. Durand and D. Vauther1n f0r fru1tfu1 d15cu5510n5 dur1n9 the rea112at10n 0f th15 w0rk.

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