Guy Louchard
email: louchard@ulb.ac.be

Number of Singletons in Involutions of large size: a central range and a large deviation analysis

Keywords: Involutions, Singletons, Asymptotics, Saddle point method, Multiseries expansions, Analytic Combinatorics. 2010 Mathematics Subject Classification: 05A16 60C05 60F05

In this paper, we analyze the asymptotic number I(m, n) of involutions of large size n with m singletons. We consider a central region and a non-central region. In the range m = n-n α , 0 < α < 1, we analyze the dependence of I(m, n) on α. This paper fits within the framework of Analytic Combinatorics.

Introduction

During the last few years, we have been interested in asymptotic properties of some permutations parameters. For instance, in [START_REF] Louchard | The number of inversions in permutations: A saddle point approach[END_REF] (in cooperation with H.Prodinger), [START_REF] Louchard | Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach[END_REF], [START_REF] Louchard | Asymptotics of the Eulerian numbers revisited: a large deviation analysis[END_REF], we analyzed the number of inversions, of cycles (related to the Stirling numbers of the first kind), of rises (related to Eulerian numbers). We extended the Gaussian approximation with more terms and we also considered some large deviation expansions. In this paper, we turn to another property: the number of singletons in involutions: an involution is a permutation σ such that σ 2 is the identity permutation. this corresponds to cycles of size 1 and 2. See Bona [START_REF] Bóna | Combinatorics of Permutations[END_REF] for details. We will use the Saddle point method: see Flajolet and Sedgewick [2, ch. VIII] for a nice introduction.

We denote by I n the total number of involutions of size n and by I(m, n) the number of all involutions of size n with m singletons . We define a random variable J n by the relation

P(J n = m) = I(m, n) I n .
This the number of singletons in an involution chosen (uniformly) at random among all involutions of size n. In [2, ch. VIII, p.558], using the Saddle point technique, Flajolet and Sedgewick give the first terms of the asymptotic expansion of I n , also obtained by Knuth [START_REF] Knuth | The Art of Computer Programming, volume 3: Sorting and Searching[END_REF]. See also Moser and Wyman [START_REF] Moser | On the solution of x d = 1 in symmetric groups[END_REF]. In Section 2, we provide a more detailed expansion of I n . In [2, ch. VIII, p.691], the authors provide the first terms of the mean and variance of J n . In Section 2, we consider a detailed analysis of all moments of J n . In [2, ch. VIII, p.692], the authors prove the dominant Gaussian asymptotic of J n by using together the Saddle point technique and a generalized quasi-powers technique (see Sachkov [START_REF] Sachkov | Probabilistic Methods in Combinatorial Analysis[END_REF], Hwang [START_REF] Hwang | Théorèmes limites pour les structures aléatoires et les fonctions arithmétiques[END_REF], [START_REF] Hwang | On convergence rates in the central limit theorems for combinatorial structures[END_REF]). In Section 3, we give a detailed analyzis of the asymptotic distribution of J n . In Section 4, we consider a large deviation range: m = n -n α , 0 < α < 1. In this section, we will use multiseries expansions: multiseries are in effect power series (in which the powers may be non-integral but must tend to infinity) and the variables are elements of a scale. The scale is a set of variables of increasing order. The series is computed in terms of the variable of maximum order, the coefficients of which are given in terms of the next-to-maximum order, etc. This is more precise than mixing different terms.

An appendix provides a justification of some integration procedures. Note finally that our approach can be used in generalizations of the involution: we can deal with cycles of any chosen sizes and deal with singletons or other specific cycle.

The moments

We have the classical (exponential) generating functions

f 1 (z) = ∞ n=0 I n z n n! = e z+z 2 /2 , f 2 (z, y) = ∞ m=0 ∞ n=0 I(m, n) z n n! y m = e zy+z 2 /2 , f 3 (z, m) = ∞ n=0 I(m, n) z n n! = e z 2 /2 z m m! .
We note that m and n do have the same parity: n -m is even. We define m := -1 j=0 (m -j) as the th falling factorial of m. We have

E(J n ) = ∞ m=0 m P(J n = m) = S l I n , S l = ∞ m=0 m I(m, n) = n! [z n ] ∂ ∂y f 2 (z, y) y=1 = n! [z n ]z e z+z 2 /2 = n![z n-]e z+z 2 /2 , E(J n ) = n! I n I n- (n -)! .
Now we turn to an asymptotic expansion of I n .

Let Ω denote the circle ρe iθ . By Cauchy's theorem, it follows that

I n /n! = 1 2πi Ω f 1 (z) z n+1 dz = 1 ρ n 1 2π π -π f 1 (ρe iθ )e -niθ dθ using z = ρe iθ = 1 ρ n 1 2π π -π exp ln(f 1 (ρe iθ )) -niθ dθ = 1 ρ n f 1 (ρ) 2π π -π exp iκ 1 θ -niθ - 1 2 κ 2 θ 2 - i 6 κ 3 θ 3 + • • • dθ, (1)
where

κ i (ρ) := ∂ ∂u i ln(f 1 (ρe u ))| u=0 .
Now we have κ 1 = ρ + ρ 2 and we set κ 1 -n = 0 such that the saddle point is the root (of smallest modulus) of ρ + ρ 2 -n = 0 (From now on, we only provide a few terms in our expansions, but of course we use more terms in our computations). This leads to

ρ = - 1 2 + 1 2 √ 4n + 1 = √ n -1/2 + 1/8 1 √ n - 1 128 n -3/2 + 1 1024 n -5/2 - 5 32768 n -7/2 + O 1 n 9/2 , ln(ρ) = 1/2 ln (n) -1/2 1 √ n + 1/48 n -3/2 - 3 1280 n -5/2 + 5 14336 n -7/2 - 35 589824 n -9/2 + O 1 n 5 .
See Good [START_REF] Good | Saddle-point methods for the multinomial distribution[END_REF] for a neat description of this technique.

The dominant part of (1) gives

f 1 (ρ) ρ n = exp(E 1 ), E 1 = ρ + ρ 2 /2 -n ln(ρ) = n/2 + ρ/2 -n ln(ρ),
with the substitution ρ 2 = n -ρ. (This substitution will be frequently used in the sequel.) Now we turn to the integral. We have

κ 2 = -ρ + 2n,
and more generally

κ j = -(2 j-1 -1)ρ + 2 j-1 n.
We choose a splitting value θ 0 such that κ 2 θ 2 0 → ∞, and

κ 3 θ 3 0 → 0, as n → ∞. If we choose θ 0 = n β , we must have n 2β+1 → ∞, n 3β+1 → 0.
For instance, we can use θ 0 = n -5/12 . We must prove that the integral

K n = 2π-θ 0 θ 0 exp ln(f 1 (ρe iθ )) -niθ dθ is such that |K n |
is exponentially small (tail pruning). This is done in [2, ch. VIII, p.559]. Now we use the classical trick of setting

∞ j=2 κ j (iθ) j /j! = 1 2 (n -ρ)(e 2iθ -1 -2iθ) + ρ(e iθ -1 -iθ) = -u 2 /2.
Computing θ as a series in u, this gives, by Lagrange's inversion,

θ = ∞ i=1 a i u i n i/2 , (2)
with, for instance,

a 1 = 1/2 √ 2 + 1/8 √ 2 √ n - 1 64 √ 2 n - 3 256
√ 2 n 3/2 + 11 4096 √ 2 n 2 + O 1 n 5/2 , a 2 = -1/6 i - 1/24 i √ n + 1/48 i n + 1 192 i n 3/2 - 1 192 i n 2 + O 1 n 5/2 .

This expansion is valid in the dominant integration domain

|u| ≤ √ n θ 0 a 1 = n 1/12 . Setting dθ = dθ du du, we integrate on u = [-∞, ∞].
2 √ π √ n F 1 , F 1 = 1 + 1/4 1 √ n - 19 96 n -1 - 13 384 n -3/2 + O 1 n 2 , I n n! ∼ 1 2 √ π √ n F 1 exp(E 1 ).
Now we turn to

I n- (n-)! . We successively have ρ + ρ 2 -n + = 0, ρ 2 = n --ρ is used as a next substitution, ρ = √ n -1/2 + 1/8 -1/2 √ n + - 1 128 + 1/16 -1/8 2 n -3/2 + 1 1024 - 3 256 + 3 64 2 -1/16 3 n -5/2 + - 5 32768 + 5 2048 - 15 1024 2 + 5 128 3 - 5 128 4 n -7/2 + O 1 n 9/2 , ln(ρ ) = 1/2 ln (n) -1/2 1 √ n -1/2 n + 1/48 -1/4 n 3/2 -1/4 2 n 2 + - 3 1280 + 1/32 -3/16 2 n -5/2 -
n -9/2 + O 1 n 5 , f 1 (ρ ) ρ n = exp(E 1, ), E 1, = n/2 + ρ /2 -/2 -(n -) ln(ρ ), κ j, = -(2 j-1 -1)ρ + 2 j-1 (n -),
θ is again given by (2),

a 1, = 1/2 √ 2 + 1/8 √ 2 √ n + - 1 64 √ 2 + 1/4 √ 2 n -1 + 1/8 √ 2 - 3 256 √ 2 n -3/2 + O 1 n 2 , F 1, = 1 + 1/4 1 √ n + - 19 96 + 1/2 n -1 + 1/4 - 13 384 n -3/2 + O 1 n 2 , I n- (n -)! ∼ 1 2 √ π √ n F 1, exp(E 1, ).
Note that setting = 0, we recover of course I n . The detailed expansions of ρ and ln(ρ ) are used in E 1 and E 1, .

We are now ready to compute E(J n ) = S In . We derive

exp(E 1, -E 1 ) = n /2 T 1 , T 1 = 1 -1/2 √ n -1/8 2 n + 1/48 -1/8 2 + 5 48 3 n -3/2 + - 1 96 2 + 1/16 3 + 1 4 n -2 + - 1 384 3 + 1 64 4 - 41 3840 5 n -5/2 + O 1 n 3 , F 1, F 1 = T 2 , T 2 = 1 + 1/2 n + 1/8 n 3/2 + O 1 n 2 , T 3 = T 1 T 2 = 1 -1/2 √ n + -1/8 2 + 1/2 n + 7 48 -3/8 2 + 5 48 3 n -3/2 + O n 2 .
This leads to the following theorem:

Theorem 2.1 The asymptotic expansion of the factorial moments of J n are given by

E(J n ) = n /2 1 -1/2 √ n + -1/8 2 + 1/2 n + 7 48 -3/8 2 + 5 48 3 n -3/2 + O 1 n 2 .
The first moments of J n are now immediate:

M = S 1 I n = √ n -1/2 + 3/8 1 √ n -1/8 n -1 + O 1 n 3/2 , E(J n (J n -1)) = S 2 I n , σ 2 = S 2 I n + M -M 2 = √ n -1 + 5/8 1 √ n + O 1 n , σ = 4 √ n -1/2 1 4 √ n + 3/16 n -3/4 + O 1 n .
All moments can be similarly mechanically obtained

Distribution of J n

We consider the central range M -2σ < m < M + 2σ. We now have

κ i (ρ) := ∂ ∂u i ln(f 3 (ρe u ))| u=0 ,
and m = M + xσ, x = Θ(1). This leads to

κ 1 = ρ 2 + m, ρ 2
= n -m (we use that as a substitution in the sequel) ,

κ j = (n -m)2 j-1 , ρ = √ n -1/2 -1/2 x 4 √ n + 1/8 1 √ n + -1/8 -1/8 x 2 n -1/32 x n 5/4 + O 1 n 3/2 , ln(ρ) = 1/2 ln (n) -1/2 1 √ n -1/2 x n 3/4 -1/4 x n 5/4 + - 5 48 -1/4 x 2 n -3/2 - 3 32 x n 7/4 + O 1 n .
The dominant part of (1) gives

f 3 (ρ) ρ n = exp(E 2 ), E 2 = ρ 2 /2 + m ln(ρ) -ln(m!) -n ln(ρ), we know that ln(m!) = -m + m ln (m) + 1 2 ln (2πm) + 1 12 m -1 + O 1 m 2 , hence E 2 = (-1/2 ln (n) + 1/2) n + √ n -1/4 -1/2 x 2 -1/4 ln (n) -1/2 ln (2) -1/2 ln (π) + -1/2 x + 1/6 x 3 4 √ n + 5 24 + 1/4 x 2 -1/12 x 4 1 √ n + -1/6 x -1/4 x 3 + 1/20 x 5 n 3/4 + O 1 n .
Now we turn to the integral. Proceeding as previously, we have

1 2 (n -m)(e 2iθ -1 -2iθ) = - u 2 2 ,
θ is again given by ( 2),

a 1 = 1/2 √ 2 + 1/4 √ 2 √ n + 1/4 √ 2x n 3/4 + 1/16 √ 2 n + 1/4 √ 2x n 5/4 + -1/16 √ 2 + 1/2 √ 2 -1/8 -1/8 x 2 -1/4 x 2 √ 2 -8 -1/16 x 2 - 3 128 √ 2 n -3/2 + 7 64 √ 2x n 7/4 + O 1 n 2 a 2 = -1/6 i - 1/6 i √ n - 1/6 ix n 3/4 - 1/12 i n - 1/4 ix n 5/4 + -1/6 ix 2 -1/16 i n 3/2 - 17 96 ix n 7/4 + O 1 n 2 This integration gives 1 2 √ π √ n F 2 , F 2 = 1 + 1/2 1 √ n + 1/2 x n 3/4 -1/24 n -1 + 1/2 x n 5/4 + 1 96 -12 √ π + 36 √ πx 2 √ πn 3/2 -1/32 x n 7/4 + O 1 n 2 , I(m, n) n! ∼ 1 2 √ π √ n F 2 exp(E 2 ). So, finally I(m, n) I n = F 2 /F 1 exp(E 2 -E 1 ), exp(E 2 -E 1 ) = 1 √ 2πn 1/4 e -x 2 /2 F 3 , F 3 = 1 + -1/2 x + 1/6 x 3 4 √ n + 1/6 + 3/8 x 2 -1/6 x 4 + 1 72 x 6 1 √ n + -1/4 x - 53 144 x 3 + 37 240 x 5 -1/48 x 7 + 1 1296 x 9 n -3/4 + O 1 n , I(m, n) I n = 1 √ 2πn 1/4 e -x 2 /2 F 4 , F 4 = F 2 /F 1 F 3 = 1 + -1/2 x + 1/6 x 3 4 √ n + 5 12 + 3/8 x 2 -1/6 x 4 + 1 72 x 6 1 √ n + 1/8 x - 47 144 x 3 + 37 240 x 5 -1/48 x 7 + 1 1296 x 9 n -3/4 + O 1 n .
This leads to the local limit theorem:

Theorem 3.1 The asymptotic distribution of J n in the central range is given by the local limit theorem:

P(J n = m) = 2 1 √ 2πn 1/4 e -x 2 /2 1 + -1/2 x + 1/6 x 3 4 √ n + 5 12 + 3/8 x 2 -1/6 x 4 + 1 72 x 6 1 √ n + 1/8 x - 47 144 x 3 + 37 240 x 5 -1/48 x 7 + 1 1296 x 9 n -3/4 + O 1 n . (3)
Of course more terms can be mechanically computed, but the expressions become much more intricate. Note carefully the factor 2 in front of our expression: this is justified in the Appendix. This is also justified by probabilistic reasoning: as only one over 2 values of m leads to a non-zero I(m, n) expression, the probability must be multiplied by 2. The tail pruning is also considered in the Appendix: the choice of θ 0 is the same as for I n .

To check the quality of our asymptotics, we have chosen n = 2000. this gives M = 44.22968229 . . . , σ = 6.613262555 . . ., a range m ∈ [30, 58]. 4 Large deviation m = n -n α , 0 < α < 1

We use again f 3 (z, m). The multiseries's scale is here n n

α 1/ε if α > 1/2 and n 1/ε α if α > 1/2. We set ε = n α-1 , m = n(1 -ε), L := ln(n).

This leads to

κ 1 = ρ 2 + m, ρ 2 = n -m = n α (
we use that relation as a substitution in the sequel) ,

κ j = n α 2 j-1 , ρ = n α/2 , ln(ρ) = α 2 L, ln(m) = L -ε -1/2 ε 2 -1/3 ε 3 -1/4 ε 4 + O(ε 5 ).
The dominant part of (1) gives

f 3 (ρ) ρ n = exp(E 4 ), E 4 = ρ 2 /2 + m ln(ρ) -ln(m!) -n ln(ρ) = n α /2 + (n -n α ) α 2 L -ln(m!) -n α 2 L, ln(m!) = -n + n α + n(1 -ε) ln(m) + 1 2 ln (2πm) + 1 12 m -1 - 1 360 m -2 + O 1 m 3 .
Now we use the substitution 1) , this leads to

nε j = n jα-(j-
n(1 -ε) ln(m) = nL -n α + 1/2 n 2α-1 + 1/6 n 3α-2 + 1/12 n 4α-3 -n α L + 1/4 n 5α-4 + O(n 6α-5 ),
exp(E 4 ) = F 5 exp(E 5 ),

F 5 = 1 2πn(1 -ε) 1 -1/12 1 (1 -ε) n + 1 160 1 (1 -ε) 2 n 2 + O( 1 n 3 ) , E 5 = (1/2 -1/2 α L + L) n α + (-L + 1) n -1/2 n 2α-1 -1/6 n 3α-2 -1/12 n 4α-3 -1/4 , n 5α-4 + O(n 6α-5
).

Now we turn to the integral. Proceeding as previously, we have

1 2 (n -m)(e 2iθ -1 -2iθ) = - u 2 2 , θ = ∞ i=1 a i u i n αi/2 , a 1 = √ 2 2 , a 2 = - i 6
This integration gives

1 2 √ πn α F 6 , F 6 = 1 -1/6 1 n α + 1 72 1 n 2α + O 1 n 3α . Finally I(m, n) n! = exp(E 5 )F 7 , F 7 = 1 2 √ πn α F 6 F 5 = 1 2 √ πn α 1 -1/6 n α-1 + 1 72 1 n 2α + O 1 n 3α × × 1 2πn(1 -ε) 1 -1/12 1 (1 -ε) n + 1 160 1 (1 -ε) 2 n 2 + O( 1 n 3 ) , E 5 = (1/2 -1/2 α L + L) n α + (-L + 1) n -1/2 n 2α-1 -1/6 n 3α-2 -1/12 n 4α-3 -1/4 n 5α-4 + O(n 6α-5 ).
The choice of θ 0 is here n -5α/12 , (κ 1 , κ 2 = O(n α )). The tail pruning is the same as for I(m, n). Finally, we obtain the following asymptotic result:

Theorem 4.1 The asymptotic expression of the I(m, n) for large deviation m = n -n α , 0 < α < 1 is given by

I(m, n) n! = 2 exp(E 5 )F 7 , = 2 1 2 √ πn α 1 -1/6 n α-1 + 1 72 1 n 2α + O 1 n 3α × × 1 2πn(1 -ε) 1 -1/12 1 (1 -ε) n + 1 160 1 (1 -ε) 2 n 2 + O( 1 n 3 ) × × exp [(1/2 -1/2 α L + L) n α + (-L + 1) n -1/2 n 2α-1 -1/6 n 3α-2 -1/12 n 4α-3 -1/4 n 5α-4 + O(n 6α-5 ) . (4)
Note that we prefer to keep two separate factors in (4): one in powers of n α and one in powers of n instead of mixing them.

Let us analyze the importance of the terms in E 5 . We have two sets: the set A of dominant terms, which stay in the exponent and the set B of small terms, leading to a coefficient of type (1 + ∆), with ∆ small. The property of each term may depend on α. In E 5 , each term n jα-(j-1) is in A if j > 1 1-α and in B otherwize. We finally mention that our non-central range is not sacred: other types of ranges can be analyzed with similar methods.

To check the quality of our asymptotics, we have first chosen n = 2000 and a range α ∈ (0.125, 0.45). This corresponds to the range m ∈ (1968, 1998).

Figure 4 gives ln(I(n, m)/n!) (circle) and ln(Equ.( 4))(line) with the substitutions

n α = n -m, ε = (n -m)/n, α = ln(n -m)/ln(n), n 2α-1 = (n -m) 2 /n, . . ..
Figure 5 gives the quotient of ln(I(n, m)/n!) and ln(Equ.( 4)) (circle). Another way for checking the quality is to fix α. We choose α = 1/4 and n ∈ (1950,2000). Of course, we must use an integer value for m: m = n -n α + 1 . Hence, in (4), we use α as the root of n -n α -m = 0.

Figure 6 gives ln(I(n, m)/n!) (circle) and ln(Equ.( 4)) (line). The relative error is of order 5.10 -4 . Figure 7 gives the quotient of ln(I(n, m)/n!) and ln(Equ.( 4)) (circle). which has two dominant peaks at 0 and π. So we must be more precise, we have, with n -m even, e 1/2 ρ 2 e 2 iθ +i(m-n)θ = e 1/2 ρ 2 cos(2 θ) cos 1/2 ρ 2 sin (2 θ) + θ (m -n) , and, indeed, if we set θ = π + δ, we recover the same expression. Hence a factor 2 in (3). The large deviation case leads to the same analysis.

Figure 1 2 √Figure 1 : 2 √Figure 2

 12122 Figure 1 shows I(m, n)/I n (circle) and a first asymptotic 2 e -x 2 /2 √ 2πσ (line).

Figure 3 2 √

 32 Figure 3 gives the quotient of I(m, n)/I n and the asymptotic 2 e -x 2 /2 √ 2πσ (box) as well as the quotient of I(m, n)/I n and the asymptotic Equ. (3) (circle).

Figure 2 :Figure 3 : 2 √

 232 Figure 2: I(m, n)/I n (circle) and the asymptotic Equ. (3) (line)

Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 4567 Figure 4: n = 2000, ln(I(n, m)/n!) (circle) and ln(Equ.(4)) (line)


Ch Theisen

Spectroscopy of proton-odd transfermium nuclei *

Keywords: numbers: 27.90.+b, 23.60.+e, 23.20.Lv, 23.20.Nx, 21.10.Re

Introduction

Mendelevium (Z=101) and Lawrencium (Z=103) isotopes have been discovered nearly 50 years ago at Berkeley and Dubna. However, very scarce information on their structure has been obtained up to now. For instance, the ground-state spin and parity of most Md and Lr isotopes remains uncertain. The nuclear structure of transfermium isotopes and the identification of single-particle states is of great importance to understand the stability of the heaviest elements, which is entirely due to shell effects. Proton orbitals that are thought to be responsible for the spherical super-heavy shell closure are found at the Fermi surface of the deformed nuclei around Z=102. Detailed spectroscopy of the transfermium nuclei is therefore essential to move toward a better understanding of the limits of stability. In-beam gamma spectroscopy has become possible in this mass region over the past ten years with the coupling of efficient recoil separators and arrays of germanium detectors at the target position. The pioneering prompt γ-ray spectroscopy of 254 No performed in parallel works at Argonne [1] and at the University of Jyväskylä (JYFL) [2] has triggered experimental and theoretical works in the transfermium region (for a review of recent works, see e.g. [3,4]; see also [5] in these proceedings). Rotational bands in the neighbouring even-even 252 No and 250 Fm have been observed. Recently, isomeric states have been studied in 254 No, giving insight into the single-particle structure through particle-hole excitations [6,7]. Odd nuclei are crucial since they also provide information on the single-particle structure. Figure 1 shows the single-particle spectrum obtained from an Hartree-Fock-Bogoliubov calculation [8]. Information on the next magic proton number can come from the location of the [521]1/2 -orbital, which originates from the 2f 5/2 shell above the predicted shell gap at Z=114. This orbital becomes active in the Z ∼ 102 region where the quadrupole deformation β 2 is found to be ∼ 0.3. Only a few odd isotopes have been studied so far using prompt gammaray spectroscopy: the neutron-odd 253 No [9] and the proton-odd 251 Md and 255 Lr that will be presented in section 3.

Complementary to prompt spectroscopy, combined gamma and conversion electron spectroscopy after α decay provides details of the singleparticle structure. Alpha-decay is indeed very sensitive to the details of the initial and final wave functions. Alpha-electron and α-gamma coincidences provide additional information on the spin and parity of the populated states. This technique has been applied in the spectroscopy of 255 Lr, 251 Md and 247 Es and will be presented section 2.

Spectroscopy after α decay of 255 Lr

Two experiments aimed at the study of the 255 Lr decay have been performed at JYFL and at GANIL (E375 experiment). In both experiments, the 255 Lr nuclei were populated using the fusion-evaporation reaction 209 Bi( 48 Ca,2n) 255 Lr. At GANIL and JYFL, fusion-evaporation products were separated from the beam and the large background of parasitic reactions such as fission, transfer and elastic scattering by the LISE Wien filter and RITU separator, respectively. The detection setup at the focal plane consisted in both cases of a time-of-flight detector, a Double Sided Silicon Striped Detector for implanted recoils and α particles, a tunnel of silicon detectors for electron spectroscopy, and Ge detectors for gamma-ray spectroscopy. Details of the experimental devices can be found in [8].

The level schemes resulting from the considerations presented in this section are shown in figure 2.

New alpha lines from the 255 Lr and 251 Md decays were observed. The al-Fig. 1. Single-particle spectra of 250 Fm for protons (top) and neutrons (bottom) obtained with the SLy4 interaction. The vertical grey bar indicates the range of ground-state deformations predicted for this and neighbouring nuclei [8].

pha decay of 251 Md is found to be in coincidence with two gamma-rays of 293 keV and 243 keV, while no coincidences with electrons could be observed. This allows us to deduce the multipolarity of the transitions and hence the spin and parity of 251 Md ground-state to be 7/2 -. The 50 keV transition from 9/2 + to 7/2 + in 247 Es is deduced to have M1 character. This almost fully converted transition is emitted in coincidence with the 7550 keV alpha transition, resulting in a partial summing in the silicon detector and hence in a peak at 7590 keV. GEANT 4 [10] simulations including atomic processes were performed and reproduce within 4 keV the experimental data. Fluorescence, Coster-Kronig and Auger electron emission yields taken from [START_REF] Firestone | Table of isotopes[END_REF] were used. The observed summing effect is therefore the indirect probe of atomic effects which could so far not be studied in this mass region. The large uncertainties in these atomic yields (for instance the uncertainty in the L1 fluorescence yield of 20 %) may account for the small discrepancies between our simulations and experimental data. New atomic calculations are in progress and preliminary results [START_REF] Ch | [END_REF] indicate that the L1 fluorescence yield taken from [START_REF] Firestone | Table of isotopes[END_REF] is overestimated by about 10%. More detailed calculations are in progress. It is important to note that a 3/2 -state is expected at low excitation energy in 247 Es. Since selection rules do not favour the feeding of this state, it could not be observed. The ground-state spin and parity of 247 Es remains therefore uncertain.

In 255 Lr, an isomeric state having an half-life of 2.5 s has been observed. Using the intensities and half-lives of four alpha-lines allows us to deduce a 1/2 -ground-state and 7/2 -isomeric state for 255 Lr.

The experimental data have been compared to new Hartree-Fock-Bogoliubov calculations using the Skyrme interaction SLy4. The same formalism as in [START_REF] Bender | [END_REF] is used. Theoretical calculations were performed for 247 Es, 251 Md and 255 Lr, but also for large isotopic chains. It is indeed instructive to make comparisons with all available experimental data, and to observe the evolution of single-particle states as a function of the neutron number.

The experimental data clearly indicate a deformed shell closure at N=152. As an example, the distance between the up-sloping 7/2 -and down-sloping 7/2 + reaches a maximum at N=152 [8,14]. HFB calculations favour however N=150 as a magic deformed number as shown in figure 1, though the distance between the two corresponding spherical shells seems correct. Experimentally, the 1/2 -and 7/2 -states are always observed close in energy, which suggests that the spherical 2f 5/2 should be slightly pushed up. The position of the 3/2 -state (suspected at low excitation energy in 243-249 Es isotopes) would be better reproduced if the spherical 2f 7/2 shell shell was also located slightly higher. This could be understand as a consequence of a known overestimation of the spin-orbit splitting by HFB calculations using the SLy4 interaction [15]. Combining all these considerations results in a Fig. 2. Level scheme of 247 Es, 251 Md and 255 Lr deduced from the two experiments performed at GANIL and JYFL [8].

slight underestimation of the Z=114 spherical gap, which remains however too small to establish a major spherical shell closure.

Prompt gamma-ray spectroscopy of 255 Lr and 251 Md

The prompt gamma-ray spectroscopy of 251 Md and 255 Lr was performed at JYFL using the RITU separator and the Jurogam germanium array. Details of the experimental technique can be found in e.g. [16]. The excitation function for the fusion-evaporation reaction 205 Tl( 48 Ca,2n) 251 Md was measured for the first time and shows a maximum of ∼ 760 nb at at center-of-target energy of 214 ± 2 MeV [17]. 255 Lr was populated using the fusion-evaporation reaction 209 Bi( 48 Ca,2n) 255 Lr.

The 251 Md spectrum was obtained using the Recoil Tagging (RT) tech-nique. Indeed, the small ∼ 10 % α-branching ratio does not allow the Recoil Decay Tagging (RDT) technique to be used and the excitation-function measurement shows that the contribution from other channels is negligible.

The spectrum shows complicated structures (see figure 3) which could be identified using gamma-gamma coincidences. A rotational band consisting of 8 transitions could be unambiguously observed. This sequence is observed without a signature partner, which is usually a hint for a K=1/2 configuration. As shown in the previous section, the [521]1/2 orbital from the 2f 5/2 spherical shell is the only possible candidate. We have calculated within the HFB formalism the moment of inertia of rotational bands built on the low-lying single-particle states. The similarity of the results for the different configurations does not allow us to draw any conclusions.

We have therefore calculated the electromagnetic properties of rotational bands. From the calculated wave functions, it is possible to deduce the reduced B(E2) intra-band and B(M1) inter-band transition probabilities. Details of the formalism can be found in [19,20]. Using the calculated transition energies and the electron conversion coefficients, we have calculated the transition ratio T(M1)/T(E2) which can be compared to experimental data. Our calculations show that the structure built on the [514]7/2 -bandhead should decay mainly via two E2 cascades with almost equal intensity, while the structure built on the [633]7/2 + band-head should be dominated by M1 inter-band transitions. In the special case of the [521]1/2 orbital, the α = -1/2 signature partner is shifted up compared to the favored α = +1/2 signature band, the signature splitting being the dominant contribution in the deduced T(M1)/T(E2) ratio. Our calculations shows that the [521]1/2 configuration has a decoupling parameter a ∼ 1, and only the decay pattern for the band built on this configuration is compatible with the experimental data.

Since the cross section for the fusion evaporation reaction 209 Bi( 48 Ca,2n) 255 Lr is only about 300 nb, gamma-gamma coincidences could not be used for an unambiguous identification of rotational structures. However, the spectrum shown in figure 3 suggest the presence of two rotational structures, one without and the other with a signature partner. The αbranching ratio of 255 Lr is large enough for using the RDT technique. The presence of the 7/2 -2.5 s isomer (see previous section) allows us to select gamma-rays feeding this state. The spectrum gated on the 7/2 -isomer decay is compatible with two signature partner bands, in agreement with the decay pattern calculated for 251 Md (similar electromagnetic properties are expected for both isotopes). The spectrum gated on the 1/2 -ground-state decay displays peaks compatible with the 1/2 -and 7/2 -rotational bands. Indeed, part of the isomeric decay proceeds via the ground state.

Results obtained in 251 Md, 255 Lr experiments are therefore consistent with the observation of 1/2 -and 7/2 -states with rotational bands built on top of these quasi-particle states.

The comparison of the gamma-ray spectra displayed in figure 3 shows striking similarities. Indeed, the bands interpreted as built on the [521]1/2 orbital in 251 Md and 255 Lr have similar energies within 2-3 keV. It remains to be understood whether these similarities are accidental, or whether there are similarities with the identical band phenomenon observed in many superdeformed bands [21].

Heavy-element studies at GANIL

The LISE spectrometer in its FULIS configuration has been used to perform gamma and electron spectroscopy after α decay of 255 Lr: see section 2. This facility has also been used for the search of superheavy elements [START_REF] Ch | Proc. Intern. Symposium Exotic Nuclei[END_REF]. The heavy-element program at FULIS will continue with the formation studies of super-heavy nuclei with nearly symmetric reactions [23]. Recent theoretical work suggest that the fusion-evaporation cross section is reduced for nearly symmetric systems leading to Z=104-106. However, the reduction gets smaller for heavier Z, especially if more neutron-rich systems are formed and partners with closed shells are used. The predictions differ by large factors and need to be clarified experimentally.

Since 2004, we have investigated the possibility of performing prompt spectroscopy of heavy elements with the EXOGAM germanium array and the VAMOS spectrometer. The performances of these two devices are well adapted to the spectroscopy of transfermium elements. EXOGAM is well adapted to the detection of low multiplicity cascades with an efficiency of about 25% in the energy range of interest. For a cascade of 6 emitted gamma-rays, the probability to detect at least one gamma-ray is about 80%. The VAMOS spectrometer is also well adapted to the study of very asymmetric reactions. The recoil angle and velocity dispersion for such reaction is large and the transmission of other existing spectrometers or separators coupled to a germanium array are usually below 10 %. VAMOS has a large angular acceptance and the transmission is expected to be larger than 50 %. A crucial point in these experiments is the suppression of the primary beam. A first test was performed using the reaction 208 Pb( 18 O,4n) 222 Th using the VAMOS Wien filter. Excellent performances were obtained and the feasibility of the RT technique was demonstrated. The dispersive mode of VAMOS should be tested for very asymmetric reactions using e.g. a 22 Ne beam on 197 Au, 208 Pb and 238 U targets. A good separation of the fusion-evaporation residues from the primary beam is expected since their magnetic rigidities differs by a factor 3.

Using the RT technique requires an highly segmented focal plane setup. Such a Si wall called MUSETT is being developed and will be installed at the focal plane of VAMOS in 2008. Since recoil energies are very low for very asymmetric reactions, window-less detectors will be used. The 1024 strips of the Si wall will be equipped with ASICs electronics.

A limitation of EXOGAM arises from its counting-rate capability due to the analogue electronics. It it foreseen to upgrade the array, called EX-OGAM 2, with digital electronics. It has been proved that a counting rate of ∼100 kHz can be used without major deterioration of the energy resolution. Pulse shape analysis of the signals will also be performed to determine the position of the first interaction of the gamma-rays for a better Doppler correction.

Prospects with the radioactive and high-intensity stable beams from SPIRAL 2

The SPIRAL 2 facility will deliver neutron-rich fission fragments and high-intensity stable beams from the LINAG driver. Stable beam intensities of more than 10 14 particles per second (pps) will allow detailed spectroscopy after alpha decay of transactinide elements. This corresponds, as an example, to the detection of about 1000 atoms of elements 114 per week. Dealing with such high beam intensities requires a new separator with high performances. The project S 3 aims at building for the first LINAG beams in 2011 a separator followed by a spectrometer with its associated focal plane detection system.

Neutron-rich fission fragments will be delivered by SPIRAL 2 with intensities up to 10 11 pps. Symmetric reactions using beams in the Kr or Xe region are promising although detailed reaction mechanism studies have to be performed. Besides decay spectroscopy studies, we foresee to perform prompt gamma-ray spectroscopy studies using the VAMOS spectrometer. The AGATA gamma tracking array will be available during campaigns at GANIL and will provide unprecedented efficiency. The preliminary phase of AGATA, the "demonstrator" should be installed at GANIL in 2009 and coupled to the VAMOS and EXOGAM arrays.

Details of the physics program with SPIRAL 2 can be found in [START_REF]Spiral 2, The Scientific Objectives[END_REF].

Fig. 3 .

 3 Fig. 3. Comparison between the prompt gamma-ray spectra of 255 Lr [18] (top) and 251 Md [17, 19] (bottom).

Acknowledgments

We are grateful to the University of Jyväskylä and GANIL for their excellent technical support during the experiments. Paul Greenlees and Audrey Chatillon are acknowledged for providing results before publication.