Jérémy Jean
email: jeremy.jean@pequan.lip6.fr

Stef Graillat
email: stef.graillat@lip6.fr

Fast dot product over finite field

Keywords: Finite field, floating-point arithmetic, error-free transformations, FMA, RNS, GMP

Finite fields are widely used in numerous areas like cryptography, error-correcting codes or computer algebra. Dot products are ubiquitous in all computations especially when dealing with linear algebra. Developing fast libraries for computing dot products in finite fields is a key tool to tackle various problems in scientific computing. In this paper, our aim is to present several possibilities to use fast floatingpoint units for computing dot products in finite fields. The main concern is then to properly manage rounding errors that may appear during the computation. To solve this problem, we use error-free transformations (EFT). Using these EFT on recent processors (with an FMA), we show that it is possible to deal with large finite fields. We also compare our approach with Residue Number Systems (RNS), which is a modular approach. The RNS approach is presented using either integer arithmetic or floating-point arithmetic. Numerical experiments make it possible to compare the performances of these different approaches.

INTRODUCTION

Let p ≥ 3 be a prime, and (ai), (bi) two vectors of N scalars of Z/pZ. We want to compute the dot product of a and b in Z/pZ, a • b = N X i=1 ai bi (mod p).

To do so, we will be using two different approaches relying on two different arithmetics. The first one uses floatingpoint arithmetic as described in the IEEE 754 standard [1] whereas the second one extends the classic integer arithmetic into a Residue Number System (RNS) to compute with smaller values.

The paper is organized as follows. In Section 2, we define the arithmetics used in dot product computation methods that are described in Section 3. Particularly, floatingpoint arithmetic requires the knowledge of basic notions of rounding (see 2.1.3) and some results we will be using in this paper: Sterbenz's subtraction (see 2.1.4) and error-free transformations (see 2.1.5). An introduction to RNS computation is presented in Section 2.2. This is nothing than a modular computation followed by a reconstruction with the Chinese Remainder Theorem. But it makes it possible to use numbers of small size and so use the integer units of the processor. In Section 3, we describe the different algorithms used to perform a dot product on finite field. Finally, in section 4, we give experimental timing results of all presented methods. The reference algorithm we will be comparing to is based on GMP [START_REF] Granlund | GNU multiple precision arithmetic library 4[END_REF], an open-source multi-precision library.

VARIOUS ARITHMETICS

Floating-point arithmetic

We use floating-point numbers in double precision to represent Z/pZ integers. Denoting M the size of the double precision mantissa (53 bits according to the IEEE 754 standard), we limit p to p -1 < 2 M -1 .

(1)

Any integer of the finite field could then be represented exactly by a floating-point number. The term M -1 is necessary rather than just M to be able to sum exactly at least two integers in the field without introducing a rounding error. In the sequel, we will assume the rounding mode to be directed toward zero. This is needed to ensure the error to be positive in applications of error-free transformations (see 2.1.5).

Notations

We denote by F the set of all floating-point numbers in the chosen precision. The operation • ∈ {+, -, * , /} on

a, b ∈ F equals a • b in R and is rounded to fl(a • b) in F (assume b = 0 if • = /).
Application fl can thus be seen as a non-injective application on R into F. For x ∈ F, ufp(x) will be the unit in the first place of x and ulp(x) the unit in the last place of x [START_REF] Muller | On the definition of ulp(x)[END_REF]. We give proper definitions about ulp and ufp in the next section. We will refer to machine precision as u = 2 -M +1 , because we chose rounding toward zero. We then define the unit in the last place (ulp) and unit in the first place (ufp) of floating-point numbers. Unit in the last place is the gap between two very close floating-point numbers. To be exact: ulp(x) is the gap between the two finite floating-point numbers closest to the value x, even if x is one of them (Kahan's definition) [START_REF] Muller | On the definition of ulp(x)[END_REF].

Definitions

As for the unit in the first place, we define it as follows:

∀x ∈ R, ufp(x) = ( 0 if x = 0 2 ⌊log 2 |x|⌋ if x = 0 (2)
For x ∈ R, the value ufp(x) denotes the weight of the most significant bit in the representation of x.

Rounding

Floating-point numbers of F form a subset of R and because of physical limitations, one can not represent an infinite amount of numbers like R. What we can do though, is imagine F as a discrete set such that every real number x would have a faithful representation in F. Clearly, F inherits from the classical ordering of R and leaving apart underflow and overflow particular situations, we have:

∀x ∈ R, ∃(a, b) ∈ F 2 s.t. a ≤ x < b. (3)
Following the IEEE 754 standard for double precision, our floating-point arithmetic is faithful (Definition 1), so that one needs to choose between a or b for the floating-point image fl(x) of x. Action of choosing is called rounding and introduces errors in computations.

In following Figure 1, rounding mode toward zero (x) and to the nearest •(x) are shown for a positive real number x. Note that in our case, rounding toward zero on positive numbers leads to a positive round-off term.

Generally speaking, any operations on floating-point numbers may introduce rounding errors. In the two next sections, we present a case where a subtraction can be done exactly and particular methods known as error-free transformations that make it possible to exactly compute the rounding error.

Sterbenz's result

Sterbenz suggested in [START_REF] Sterbenz | Floating-point computation[END_REF] a particular case of subtraction where no error is introduced. In others cases, when numbers are too far from each other, one can not compute exactly their difference.

Error-free transformations

When it comes to multiply two floating-point numbers, a leak of precision arises naturally: the 2M -bit product can not be stored as a single M -bit floating-point number. One though may use two of them, to virtually extend the precision. The product is thus not store in one piece, but as an unevaluated sum of two floating-point numbers. We refer to so called methods as error-free transformations, which can be extended to more complex operations than the product, as we develop in the following.

The following algorithm applies when computing a product of two positive floating-point numbers. We make use of a special instruction denoted FMA (fused multiply-add) to evaluate exactly the round-off term of the floating-point product. This operation had been included in the IEEE 754 standard in 2008 and performs We have: ab = x + y, x = fl(ab), 0 ≤ x ≤ ab, 0 ≤ y < u.ufp(x) and 0 ≤ y < ux.

Proof 2. 1 Because a and b are floating-point numbers, each with M significant bits at most, their mathematical product ab requires at most 2M bits. When rounding ab down toward zero, one truncates the last M digits, so that the floating-point result fl(ab) and the exact mathematical difference ab-fl(ab) can be represented by two floating-point numbers with M digits each. Hence, the exact mathematical difference requires only M bits. Thus, abfl(ab) happens to be a floating-point number with at most M bits, which rounding to M bits leaves intact. Consequently, FMA produces the exact result abfl(ab):

y = FMA (a, b, -fl(ab)) = ab -fl(ab).
Therefore, we exactly have:

ab = fl(ab) + (ab -fl(ab)) = x + y
The chosen mode of rounding by truncation implies the floating-point result to be at most the exact result: fl(ab) ≤ ab, thus 0 ≤ x ≤ ab. Because y is the round-off error in the rounding of the floating operation x = fl(ab), the truncation imposes: y < u.ufp(x) and ufp(ab) ≤ fl(ab) ≤ ab, hence:

0 ≤ y < u.ufp(x) ≤ u.ufp(ab) ≤ ux.
We now present an other error-free transformation related to euclidean division by a power of two.

Suggested in [START_REF] Hecker | Let's get to the (floating) point[END_REF] and quoted in [START_REF] Rump | Accurate floating-point summation part I: Faithful rounding[END_REF] by S. Rump, this algorithm splits a floating-point number into two non-overlapping others.

Algorithm 2 -ExtractScalar Require: a ∈ N ∩ F, and σ = 2 k , k ∈ N, σ ≥ a Ensure: x ∈ N ∩ F, y ∈ N ∩ F such that a = x + y q ← fl(σ + a) x ← fl(q -σ) y ← fl(a -x) return (x, y)
Theorem 3. Let x and y be the result of ExtractScalar applied to a ∈ N ∩ F and σ ∈ F, σ = 2 k , k ≥ M . We have:

a = x + y, 0 ≤ y < u σ, 0 ≤ x ≤ a, x ∈ uσN.
Proof 3. The idea behind this splitting method is to use the rounding mechanism of the floating-point unit. Set to be toward zero, the rounding behaves the same way as a truncation. In terms of bits, the M -bit string a is divided in two strings s1 and s2 which do not overlap such that the concatenation s1 + s2 equals a. As subparts of a, both bitstrings s1 and s2 are in F.

If a = 0, the result x = y = 0 is obvious. We assume then 0 < a ≤ σ. Like in the algorithm, let q = fl(a + σ). We have

σ ≤ q ≤ a + σ ≤ 2σ, so that: σ 2 ≤ q ≤ 2σ.
1 This proof is largely inspired from the proof in [START_REF] Nievergelt | Scalar fused multiply-add instructions produce floating-point matrix arithmetic provably accurate to the penultimate digit[END_REF]. By Sterbenz's result (Theorem 1), we get:

fl(q -σ) = q -σ. (4)
Hence, x = q -σ exactly. Let δ be the error of truncation in the first summation:

a + σ = q + δ.
Note that in this equality δ ≥ 0 because rounding is directed toward zero.

q = fl(a + σ) ≤ a + σ. (5)
Moreover, the sum of the two positive numbers a and σ leads to an error δ, which is a floating-point number as well:

δ ∈ N ∩ F. With this, a -x = a -q + σ = δ,
and thus:

fl(a -x) = fl(δ) = δ.
This last equality means:

y = fl(a -x) = a -x ∈ N ∩ F.
All in all, a = x + y with x, y ∈ N ∩ F. With ( 4) and ( 5), we have: q -σ ≤ a, which means: x ≤ a. Since q ≥ σ, x is positive. As a ≥ x, we have: y = a -x ≥ 0. Moreover, the rounding of the first sum forces: δ < u.ufp(a + σ), which leads to δ < uσ. Then, 0 ≤ y < uσ.

Finally, we know that q ∈ u.ufp(q) N, and by q ≥ σ, we have ufp(q) ≥ σ, so that q ∈ uσN. In a same way, σ ∈ uσN, so that q -σ ∈ uσN. From (4), if follows that x ∈ uσN.

Residue Number System arithmetic

We will now refer to Residue Number System as RNS. A RNS can be defined as a set of n constant integers {m1, m2, . . . , mn} referred to as moduli. We assume the n integers mi, with i = 1, . . . , n to be coprime. Let us denote M = m1 • • • mn the product of all the moduli. Any arbitrary number X smaller than M can be uniquely represented by n integers {x1, x2, . . . , xn} such that xi = X (mod mi), for i = 1, . . . n.

We will sometimes note xi = |X|m i . It is then quite easy to compute addition, subtraction and multiplication with RNS numbers. Indeed, for addition, Z = X ± Y (mod M ) can be done as

zi = xi ± yi (mod mi), for i = 1, . . . n.
For multiplication, it is accomplished in a similar manner by computing Z = XY (mod M ) with zi = xiyi (mod mi), for i = 1, . . . n.

Given an integer X less than M , it is easy to construct its RNS counterpart using euclidean division. We will also need the converse : given a RNS number, find its integer counterpart. This can be done via the following formula (known as Chinese Remainder Theorem or CRT),

X = n X i=1 aiMi|X|m i , (6)
where Mi = M/mi and ai are such that aiMi = 1 (mod mi), for i = 1, . . . n.

The ai are computed using the Euclid's extended algorithm. This way of computing X from its RNS counterpart is not the more efficient way but is sufficient if needed only once during the computation.

DOT PRODUCT COMPUTATION

Delayed-division

Theorem 4. Strengthening the general hypothesis (1) on p and assuming there exists λ ≥ 1 such that:

λ(p -1) < 2 M -1 , (7)
there exists an algorithm computing the dot product of two vectors of Z/pZ of size N using only N/λ reductions in Z/pZ. We will refer to this algorithm as λ-algorithm.

Principle of the algorithm. The actual algorithm we implemented is available in pseudo code in the Appendix of this paper. The basic idea of this algorithm relies on avoiding reductions. By introducing the λ parameter, we may accumulate λ times elements of Z/pZ without invoking a costly reduction. For a double product ai bi, the use of the error-free transformation TwoProduct gives us an unevaluated sum hi + ri, which equals exactly this double product (Figure 2). With considerations on ulp and ufp, we introduce a variable l = ⌊log 2 (p)⌋, such that p < 2 l+1 and:

∀x ∈ [0, 2 l+1 -1] ∩ F, ( 0 ≤ x ≤ 2 l =⇒ x ∈ Z/pZ 2 l < x < 2 l+1 =⇒ x -2 l ∈ Z/pZ
which gives us a way to characterize elements of Z/pZ (Figure 3). Once we only have elements of this field, we can sum them together using the property resulting from λ.

x - The main interest in assumption of Lemma (4) lies in delaying reductions modulo p, which are heavy computations for the processor. Under hypothesis [START_REF] Hecker | Let's get to the (floating) point[END_REF], one can add λ integers of Z/pZ without exceeding 2 M . So, rather than dividing by p at each step in the dot product, one can delay the reduction and divide only N/λ times.

2 l ∈ Z/pZ x ∈ Z/pZ 0 ≤ x ≤ 2 l 2 l ≤ x < 2 l+1 2 l 0 2 l+1
Let us set:

l = ⌊log 2 (p)⌋ and u = 2 -M +1 .
Note that by definition of l, we have: l = log 2 (ufp(p)). Then, 2 l = ufp(p). Since p is a prime greater than 3, we have

2 l < p < 2 l+1 , (8)
which means, in terms of bits: ufp(p) < p < 2.ufp(p).

In the field of prime characteristic p, the upper bound for elements is p -1, and from the previous inequality, we have (p is odd):

2 2l < (p -1) 2 < 2 l+2 (2 l -1).
We will know distinguish two cases regarding the value on the modulo p, whether p 2 can be represented in one floatingpoint number or not. Let i such that 1 ≤ i ≤ N .

Case 1: 2l > M The product ai bi may exceed M bits because one float is not enough to represent p 2 , but two would. Applying the errorfree transformation TwoProduct, one can deduce (Theorem 2): ai bi = hi + ri, with hi = fl(ai bi), 0 ≤ hi ≤ ai bi, 0 ≤ ri < u.ufp(hi).

Note that because rounding mode had been chosen to be toward zero, all numbers are positive. That's why hi is less than (or equal to) the exact value ai bi.

Without loss, assume ri > 0. If ri is null, then the product has been done without error and the following holds. Hence, from previous results, 0 < hi ≤ ai bi ≤ (p -1) 2 .

Since hi < 2 2l+2 , we get ufp(hi) ≤ 2l + 1, so that:

0 < hi < 2 l+2 (2 l -1), ( 9
) 0 < ri < 2 2l-M +2 . (10)
The general hypothesis (1) on p sets l < M -1, and then 2l < l + M -1. This leads to the necessary condition to apply ExtractScalar to hi with parameter σ = 2 l+M :

2l + 2 ≤ l + M. (11)
We then have: hi ≤ σ and ExtractScalar on (σ, hi) gives hi = αi + βi with:

0 ≤ βi < 2 l+1 , 0 ≤ αi < 2 l+2 (2 l -1), αi ∈ 2 l+1 N. (12) So: 0 ≤ αi 2 -(l+1) < 2 l+1 -2, ( 13
) and: 0 ≤ βi < 2 l+1 . ( 14
)
Splitting interval [0, 2 l+1 ] into two halves, [0, 2 l -1] and [2 l , 2 l+1 -1], (13) leads to two cases for αi:

• If 2 l ≤ αi 2 -(l+1) < 2 l+1 -2, then : 0 ≤ αi 2 -(l+1) -2 l < 2 l -2 < 2 l ,
so that (8) leads to: αi 2 -(l+1) -2 l ∈ Z/pZ.

• Otherwise, if αi 2 -(l+1) lies in the first half, that is to say: 0 ≤ αi 2 -(l+1) < 2 l , we have: αi 2 -(l+1) ∈ Z/pZ.

Equally for βi, (14) gives:

• If βi lies in the second half, 2 l ≤ βi < 2 l+1 , we then have: 0 ≤ βi -2 l < 2 l , so that: βi -2 l ∈ Z/pZ.

• Otherwise 0 ≤ βi < 2 l , and (8) gives: βi ∈ Z/pZ.

In the same way with ri, (10) and [START_REF] Sterbenz | Floating-point computation[END_REF], it leads to: 0 < ri < 2 l . Thus, the result is directly given by ( 8): ri ∈ Z/pZ.

All in all, we get integers in Z/pZ one can sum together exactly λ times without reducing modulo p, thanks to hypothesis [START_REF] Hecker | Let's get to the (floating) point[END_REF]. Let nα and n β the number of 2 l corrections needed for α and β respectively. These values are defined by: nα = #{αi / 2 2l+1 ≤ αi < 2 2l+2 }, and:

n β = #{βi / 2 l ≤ βi ≤ 2 l+1 }.
In the finite field Z/pZ, we then have the dot product:

a • b = N X i=1 αi + N X i=1 βi + N X i=1 ri = X nα (αi -2 l ) + X N -nα αi + X n β (βi -2 l ) + X N -n β βi + X N ri + (nα + n β ) 2 l
Each of first five sums can be computed exactly by groups of size λ. One reduction is needed once the number of terms accumulated in each sum had reached λ.

Case 2: 2l ≤ M Here, the result of the product ai bi does not need more than M bits to be computed exactly. The previous case where 2l ≥ M still holds, even if the error-free transformation for the product is useless, since it produces ri = 0. This remark may be the starting point for a slightly different usage of ExtractScalar: when called at most three times with correct parameters, we can show that, for some size of vectors, there exists an algorithm computing the dot product using no reduction in the main loop.

Binary RNS basis with integers

The aim is here to use a RNS basis to compute the dot product a • b. We assume that the coefficients ai, bi are integers in [0, p -1]. As a consequence, the dot product a • b considered as an integer satisfies

0 ≤ a • b ≤ N (p -1) 2 .
We choose a RNS basis {m1, m2, m3, m4} such that

N (p -1) 2 < m1m2m3m4.
We can compute a • b in the RNS basis. We will in fact get four numbers r1, r2, r3, r4 such that

a • b = r1 (mod m1) = r2 (mod m2) = r3 (mod m3) = r4 (mod m4).
Using formula [START_REF] Granlund | GNU multiple precision arithmetic library 4[END_REF], we can find 0 ≤ r < m1m2m3m4 such that

a • b = r (mod m1m2m3m4). ( 15
)
Because of the choice of the mi, we have 0 ≤ r < m1m2m3m4,

and 0 ≤ a • b < m1m2m3m4
As a consequence, we have

|a • b -r| < m1m2m3m4.
From (15), it follows that m1m2m3m4 divides a • b -r. As a consequence, a • b = r in Z. It is then easy to obtain the result in Z/pZ. Indeed, we have

a • b = r (mod p).
To perform the computation of the dot product in the most effective way, we use the four coprime integers

B = {2 n , 2 n + 1, 2 n -1, 2 n-1 -1} (16)
as basis for RNS, with n = 32. Considering this word-size, internal computations are achieved in 64-bit integer precision. Reductions in the four-component basis use following equalities:

|2 n |2n+1 = |2 n + 1 -1|2n+1 = -1, ( 17
)
|2 n |2n-1 = |2 n -1 + 1|2n-1 = +1, ( 18
)
|2 n | 2 n-1 -1 = |2(2 n-1 -1) + 2| 2 n-1 -1 = +2. ( 19
)
Using those formulas, one performs really fast reductions using logical operations embedded in the processor. For example, let us consider the forward conversion of an integer X into its residues in B defined by ( 16). The first residue modulo 2 32 is simply obtained by keeping the 32 less significant bits of X, which is exactly |X| 2 32 . Writing the Euclidean division of X by 2 32 , one gets:

X = q × 2 32 + r, with: r = |X| 2 32 .
The three others residues derive from (17), ( 18) and ( 19):

|X|2n+1 = |q 2 32 + r|2n+1 = |r -q|2n+1, (20)
|X|2n-1 = |q 2 32 + r|2n-1 = |r + q|2n-1, (21)
|X| 2 n-1 -1 = |q 2 32 + r| 2 n-1 -1 = |2q + r| 2 n-1 -1 . (22)
This particular basis enables us to reach good speedups in comparison to classical bases where four primes are used. In this ordinary case, one would then have to perform divisions each time a forward conversion is needed.

To reconstruct the result using the CRT, we use the GMP library. This is necessary because in the formula (6), we need to compute the product of three numbers of 32 bits, which may lead to a number of 96 bits. But the use of GMP has a negligible cost in the algorithm since it is used once at the end.

A slightly different approach we present now consists in using floating-point numbers to represent the four residues.

Binary RNS basis for floating-point arithmetic

The idea behind a RNS basis in floating-point arithmetic is the performance. Indeed, the use of floating-point rather than integer makes it possible to use BLAS (Basic Linear Algebra Subprograms 2 ) routines. These routines are highly efficient but work only with floating-point numbers (and not integers). Even if it is not very important for dot products (with a mild size), the use of BLAS is very important when dealing with matrix-vector products or matrix-matrix products.

We use the same basis as (16) but with n = ⌊M/2⌋, where M is the size of floating-point mantissa in double precision. This is needed to allow exact floating-point multiplication. An important difference with the integer case relies on reductions: one can not here use logical instructions to perform division modulo 2 n . As in the prime basis, one has to divide four times at each step to compute residues in RNS basis.

EXPERIMENTAL RESULTS

In this section, we present our experimental results for the previously detailed algorithms. Both take two input parameters: the size N of vectors and the prime characteristic p of the field. Measured performances in all the following refer to time of computation and do not take into account memory-related operations.

Environment used for benchmarks was an Intel Itanium2 1.5 GHz processor with a FMA instruction in its floatingpoint unit. Along with algorithm comparisons, we compiled all twice, using both gcc (the GNU Compiler Collection) and icc (Intel C Compiler), since the latter is well-suited for the Intel Itanium family.

First off, concerning the floating-point λ-algorithm, we show how the value λ affects the performances. Numerically, λ represents where p lies in the floating-point range: λ is small for big p and big for small p. Computationally, it traduces how many iterations we can make without reducing partial results. On Figure 4 (linear scale) and Figure 5 (logarithmic scale), one can clearly see the values for λ when p is increasing in the interval [0, 2 52 ], whereas GMP-based algorithm runs in constant time for a fixed N . Timing comparisons between GMP (steady) and λ-algorithm (increasing steps for decreasing λ) for p ∈ [0, 2 52 ] and N = 100000. We present results for two very different sizes of input vectors, N ∈ {512, 40000}, when the prime characteristic p lies in both float or double ranges, that is [0, 2 23 ] and [0, 2 52 ] respectively. We distinguish those two ranges for RNS double method to be applicable. This method requires indeed a floating-point product to be exact.

Values in tables are timing ratios between reference and measured algorithms: if ratio is greater than 1, then the measured one is faster; and slower if less than 1. When p lies in the interval [0, 2 52 ], ratios are really different whether p is small or big, that is whether λ is big or small, respectively. The best performances are reached when p is small, since many divisions are avoided. In tables, we show minimal and maximal values (min/max) of the staircase (similar to Figure 4). There is no need to make this distinction when p lies in the smaller interval [0, 2 23 ] since λ is not small enough regarding the size N of vectors. In this range, λ would be bigger than 2 29 , so that differences in timings are negligible. Minimal ratios for λ-algorithm are generally smaller than 1, since they are reached for λ = 1, which leads to a reduction at each step (Table 1). For λ ≥ 2, reductions generally do not occur that often, so that a speedup occurs.

Huge differences between RNS with binary basis and RNS with prime basis are explained by reductions. In the binary case, moduli are almost power of two so that we use logical instruction AND to perform all reductions in 32 bits (section 3.2). These instructions speed up a lot the implementation in comparison with the slow division instruction of the prime moduli in the prime basis. This is even more true when working on a dedicated machine with a special compiler.

Here, icc makes it run a lot faster than gcc (Table 2 In Table 2, we also compared λ-algorithm and RNS working with double floating-point numbers. In that case, residue arithmetic is not efficient, even with the binary basis, since there is no logical instruction in floating-point units (FPU). Consequently, reductions cost a lot and make the whole computation relatively slow.

During benchmarking, we also evaluate results on different sizes of vectors. For N = 2048 for instance, results were not that different from the 512 case. For very large vectors, with 40000, 100000 or 500000 elements in particular, our presented algorithms get genuine speedups. In double precision when computing with 40000-element vectors (Table 3), one gets a speedup of 2 for the λ-algorithm and more than 7 for the RNS one with the binary basis. This difference holds when computations are performed in the smaller field (Table 4). In that case, again and for the same reasons as detailed before, λ-algorithm is three times faster than the RNS one using floating-point unit.

Reference

Reference

CONCLUSIONS AND FUTURE WORK

We have shown different ways to compute dot product in a prime finite field making use of floating-point and Residue Number System (RNS) arithmetics. For that matter, critical operations are reductions in the field. The first idea relies on the λ parameter of section 3.1, which allows us to avoid systematic reduction. Great performances are reached for big λ when the prime characteristic p of the field may be up to 2 52 . The second idea introduces the RNS to perform computation on smaller values on a binary basis. In that special case, reductions are almost costless since moduli are power of two.

The conclusion of this work forks into two main lines regarding which arithmetic one considers. Floating-point numbers happen to be a widely studied area and we showed in this paper a way of using them to perform integer computations. Although this is not the major area of application, one can manage them correctly to compute in high order fields. The other side of the conclusion is directly linked to integer numbers, since it introduces residue arithmetic. We use two bases to perform computations: the classic one with prime numbers, and a particular one with four coprimes quasi power of two. This latter RNS usage happens to be very fast since reductions are almost costless.

Definition 1 .

 1 For floating-point numbers a and b and operation • ∈ {+, -, ×, /}, let c = a•b exactly (assuming b = 0 if • = /). Let x and y be consecutive floating-point numbers with the same sign as c such that: |x| ≤ |c| < |y|. Then, the floating-point arithmetic has the faithful property if fl(a • b) = x whenever c = x and fl(a • b) is either x or y whenever c = x.

xFigure 1 :

 1 Figure 1: Rounding modes for x ∈ R + .

Theorem 1 .

 1 If a and b are floating-point numbers of F such that: b 2 ≤ a ≤ 2b, then the result computed by fl(a -b) is exact. That is: fl(a -b) = a -b. Proof 1. Without loss of generality, assume a > b > 0. This implies ulp(a) ≥ ulp(b), so both a and b are in ulp(b)N, and so is a -b. The hypothesis of the theorem states a ≤ 2b so that a-b ≤ b, hence a-b is in ulp(b)N but not larger than b. As a consequence, a -b happens to be a floating-point number and fl(a -b) = a -b.

FMATheorem 2 .

 2 (a, b, c) = a × b + c with only one rounding. Algorithm 1 -TwoProduct Require: a, b ∈ F such that a, b ≥ 0 Ensure: x ∈ F and y ∈ F such that ab = x + y x ← fl(ab) y ← FMA(a, b, -x) return (x, y) Let x and y be the result of TwoProduct applied to a and b.

Figure 2 :

 2 Figure 2: TwoProduct on ai and bi

Figure 3 :

 3 Figure 3: TwoProduct on ai and bi

Figure 4 :

 4 Figure 4:Timing comparisons between GMP (steady) and λ-algorithm (increasing steps for decreasing λ) for p ∈ [0, 2 52 ] and N = 100000.

2

  http://www.netlib.org/blas/

Figure 5 :

 5 Figure 5: Timing comparisons between GMP (steady) and λ-algorithm (increasing curve) for log 2 (p) ∈ [1, 52] and N = 100000.

Table 1 :

 1 Ratio of time computations for prime characteristic p ∈ [0, 2 52 ] and size of input vectors N = 512.

	Reference Algorithm	gcc	icc
		λ-algorithm	0.30/1.23 0.28/1.17
	GMP	RNS (binary basis) RNS (prime basis)	1.90 0.14	4.13 0.97
		RNS (double)	-	-

Table 2 :

 2 ). Ratio of time computations for prime characteristic p ∈ [0, 2 23 ] and size of input vectors N = 512.

	Reference	Algorithm	gcc	icc
		λ-algorithm	1.15 1.05
	GMP	RNS (binary basis) 1.77 3.83 RNS (prime basis) 0.14 0.90
		RNS (double)	0.31 0.33
	RNS (double)	λ-algorithm	3.73 3.16

Table 3 :

 3 Ratio of time computations for prime characteristic p ∈ [0, 2 52 ] and size of input vectors N = 40000.

		Algorithm	gcc	icc
		λ-algorithm	0.54/2.01 0.49/1.67
	GMP	RNS (binary basis) RNS (prime basis)	3.42 0.26	7.30 1.63
		RNS (double)	-	-

Table 4 :

 4 Ratio of time computations for prime characteristic p ∈ [0, 2 23 ] and size of input vectors N = 40000.

		Algorithm	gcc	icc
		λ-algorithm	1.92 1.81
	GMP	RNS (binary basis) 3.57 8.10 RNS (prime basis) 0.26 1.81
		RNS (double)	0.60 0.66
	RNS (double)	λ-algorithm	3.21 2.74

We are currently working on a port of our algorihms on GPU (Graphics Processing Unit). With the λ-algorithm, this implies a pretreatment of data to get them distribute in groups of proper size to be able to sum them together exactly. As for RNS computations, we might compute the four components at the same time. The more serious drawback of GPU is the lack of efficiency in double precision. It would then be interesting to adapt RNS with floating-point arithmetic and compare performances.

APPENDIX

] ← TwoProduct(h, nα + n β ) q ← q (mod p) r ← r (mod p) q ← fl(q + r) q ← q (mod p) res ← fl(res + q) res ← res (mod p)


Gilles Montavon

Christos Apostolidis

Frank Bruchertseifer

Urska Repinc

Alfred Morgenstern
email: alfred.morgenstern@ec.europa.eu

Spectroscopic study of the interaction of U(VI) with transferrin and albumin for speciation of U(VI) under blood serum conditions

Keywords: 230 U, targeted alpha therapy, speciation, albumin, transferrin

The quantitative description of the interactions of uranium with blood serum components is of high relevance for a rational design of molecules suitable for in vivo chelation of uranium. We have determined the stability constants for the complexation of U(VI) with human serum transferrin and albumin by Time Resolved Laser Induced Fluorescence Spectroscopy and Difference Ultraviolet Spectroscopy. Both proteins interact strongly with U(VI), forming ternary complexes with carbonate acting as a synergistic anion. Together with literature data describing the interaction of U(VI) with low molecular weight inorganic and organic serum components, the speciation of U(VI) in blood serum was calculated. In agreement with published experimental data, the model calculation shows that complexation with proteins and carbonate ion governs U(VI) speciation; 35 % of U(VI) is bound to proteins and 65 % to carbonate. Among the protein pool, albumin is the main protein interacting with U(VI). In

Introduction

The principle of targeted alpha therapy (TAT) is based on the stable binding of alpha emitting radionuclides to cancer selective carrier molecules, such as antibodies or peptides, via bifunctional chelating agents (BCAs). Due to the short range (< 100µm) and the high linear energy transfer (≈100 keV/µm) of alpha radiation in human tissue, TAT allows to selectively deliver a highly cytotoxic radiation dose to targeted cells while sparing surrounding healthy tissue [1]. The alpha emitter 230 U (t 1/2 = 20.8 d) is a promising novel radionuclide for application in targeted alpha therapy of cancer [2][3][4]. For safe therapeutic application of 230 U in targeted therapy, a chelating agent is required to link the radiometal to biological carrier molecules in a stable manner, since release of the alpha emitter from the radioconjugate in vivo might cause toxicity to normal organs. Due to the relatively long halflive of 230 U of 20.8 days, the radioconjugate should show high stability over extended time periods. Ideally a suitable chelating agent should form uranium complexes of higher stability than ligands competing for uranium complexation under physiological conditions, such as carbonate, phosphate and proteins [5][6][7]. In order to evaluate the potential of candidate molecules for uranium chelation, in particular with respect to the stability of their uranium complexes in vivo, a full thermodynamic description of the interaction of uranium with competing ligands present in human body fluids, and in particular in blood serum, would be highly desirable. Certainly a quantitative speciation of uranium under physiological conditions is also of high relevance for the understanding of toxicological effects of uranium as well as for the development of effective decorporation agents for uranium [5,8].

Hexavalent U(VI) is the most stable oxidation state of uranium under physiological conditions [8]. It has been reported that U(VI) in the blood stream is forming complexes with carbonate and serum proteins, in particular with albumin and transferrin [7,[9][10][11]. The thermodynamic parameters describing the interaction between U(VI) and relevant low molecular weight inorganic and organic ligands have been extensively studied and are rather well known [START_REF] Guillaumont | Update on the chemical thermodynamics of uranium, neptunium, plutonium, americium and technetium[END_REF][START_REF] Smith | NIST Critically Selected Stability Constants of Metal Complexes Database, Version 8.0 for Windows[END_REF]. However, the interaction between U(VI) and serum proteins is still a subject of discussion with respect to the nature of the interacting proteins [11,[START_REF] Huang | [END_REF], the quantitative description of the interaction [5] and the coordination environment of U(VI) [15], in particular with respect to the possible presence of the carbonate ion in the coordination sphere [7,[START_REF] Huang | [END_REF]15].

Human serum albumin (HSA) is the most abundant protein in human blood plasma at a concentration of 30 to 50 g/L and comprises about half of blood serum protein. The interaction of uranium and HSA has been studied as early as 1948 by Guzman Barron et al. [16] using ultrafiltration. The authors report the reversibility of uranium binding to HSA and the influence of carbonate and citrate as competing ligands. No further thermodynamic analysis was performed. Chevari et al. [17] report a conditional stability constant of log K=10 for the binding of uranium to HSA at pH=6 using a Shubert-type method [18], while Duff et al. [19] have determined conditional stability constants of log K = 1.6*10 7 M -1 and log K = 2.8*10 5 M -1 for the binding to high-affinity and low-affinity sites at pH 5.5 using isothermal titration calorimetry.

Human serum transferrin (HSTF) is an iron binding single-chain glycoprotein with a molecular mass of 79570 Da, containing 679 amino acids and consisting of two structurally related lobes referred to as N and C lobe. Each lobe is further divided into two subdomains by a cleft, with each cleft housing a metal binding site. Metal binding to HSTF is generally facilitated through concomitant binding of carbonate as a synergistic anion [20]. The concentration of HSTF in serum is ca. 2.5 g/L, where about 30% are saturated with ferric ion.

HSTF is also the primary serum transport agent of a large number of tri-and tetravalent metal ions [21]. Stevens et al. [9] reported that U(VI) in blood plasma of beagles injected with uranium citrate was present as 60% carbonate complex and 40% protein bound, with most of the protein-bound fraction presumably being associated with transferrin. The binding of uranium(VI) to HSTF has been studied by Scapolan et al. [10] using time-resolved laserinduced fluorescence spectroscopy (TRLFS) at physiological pH. The formation of a 2:1 complex of U(VI):HSTF has been reported, with an overall conditional stability constant of log K = 16. Sun et al. [START_REF] Sun | Metal sites in proteins and models[END_REF] have developed a correlation of the overall metal-binding constant of HSTF with the first metal hydrolysis constant, leading to a value of log K = 14.7 for the U(VI)-HSTF complex. This correlation was further refined by Ansoborlo et al. [6] through a critical assessment of the metal ion hydrolysis constants and the consideration of two individual binding sites, predicting values of log K = 14.1 and 12.6 for the C and N lobe, respectively. Vidaud et al. [15] have studied the structural properties of HSTF complexes of U(VI), and propose the existence of two binding sites, where U(VI) is coordinated under participation of two tyrosine residues.

To our knowledge, thermodynamic data describing the complexation of uranium by HSA or HSTF applicable to a wide variety of body fluid compositions have not been reported to date. In this work we have determined the stability constants for the complexes of U(VI) with HSA and HSTF using TRLFS and Difference Ultraviolet Spectroscopy (DUS). Based on these parameters and including literature data of thermodynamic constants of U(VI) complexation with low molecular weight organic and inorganic ligands, a model calculation was performed to describe U(VI) interactions in human blood serum in a bottom-up approach.

The reliability of the simulation was assessed by comparison with published experimental in vitro speciation results [9,11,17]. To identify promising molecules for in vivo chelation of 230 U, the serum stability of several U(VI) chelate complexes was simulated.

Materials and methods

Reagents

Human serum transferrin (apo-transferrin human ≥ 98 %, Sigma) was used as received or purified by size-exclusion chromatography on a PD-10 column using 0.1 M NaCl/0.05 M HEPES at pH 7.4 as an eluent. No difference in complexation was observed between purified and non-purified HSTF and no distinction was made in the following. HSTF concentration was determined spectrophotometrically at λ = 280 nm using 93,000 cm -1 M -1 as a molar extinction coefficient [START_REF] Chasteen | [END_REF]. Human serum albumin (albumin from human serum ≥ 96 %, Sigma, mol wt 66,478 Da by calculation) was used as received. U(VI) stock solution was prepared by dilution of a standard solution (SPEX CertiPrep® Single-element Solution Standard, [U] = 1000 μg/mL, 2 % HNO 3 ) with addition of sodium hydroxide to obtain a final concentration of 1 × 10 -3 M at pH ~ 3. All other chemicals were reagent grade. HSTF, HSA and carbonate stock solutions were freshly prepared before use. Solutions were prepared with ultrapure water (MilliQ, 18 MΩ . cm) and pH measurements were performed using a Lab850 pH meter combined with a micro electrode (Schott, Germany).

Sample preparation

All experiments were performed in solution of pH 7.4 ± 0.1 and ionic strength 0.1 (NaCl) buffered with 2 × 10 -3 M HEPES at room temperature (23 ± 2 °C). Experiments in the absence of carbonate were performed in a glove box under 100% nitrogen atmosphere. Samples in the carbonate system were freshly prepared and kept closed to prevent equilibration with atmospheric CO 2 . Irrespective of the system studied, equilibrium was shown to be established rapidly, i.e. in less than 20 minutes. During this time period, the concentration of carbonate in the closed system is considered to be constant. The experimental conditions are summarized in Table 1.

Time Resolved Laser induced Fluorescence Spectroscopy (TRLFS)

Details concerning the spectroscopic device as well as details on how spectroscopic data were obtained are described in [24]. U(VI) was excited at 430 nm with a laser intensity of about 3 mJ. As shown in Figure 1A, complexation of U(VI) by HSTF reaches equilibrium in less than 20 minutes and leads to a complete extinction of the fluorescence signal. Under given experimental conditions, the percentage of U(VI) not bound to HSTF can thus be determined according to Eq. ( 1):

tot FI FI )) VI ( U %( = (1)
where FI tot and FI correspond to the fluorescence intensity measured before and after addition of HSTF, respectively. The applicability of the method is limited in the presence of high carbonate concentrations, since carbonate complexes of U(VI) also do not emit fluorescence [10].

Difference ultraviolet spectroscopy (DUS)

DUS is the most widely used technique to evaluate metal binding to HSTF [20]. UVspectra were recorded using an Ultrospec™ 2100 pro UV/Visible spectrophotometer. Metal complexation was evaluated from the absorbance measurements after addition of the metal using the difference spectra of the metal-protein complexes vs. unmetallated apo-protein.

Either the peak height or the peak area in the range 235-271 nm were used to describe and quantify U(VI)-HSTF interaction. The main advantage of the method is that is the absorbance signal of HSTF is not influenced by the presence of other ligands which may interact with U(VI) (e.g. carbonate). The interaction of HSA with U(VI) was studied using HSTF as a competitive agent [25].

Modelling the binding between U(VI) and proteins

Several models can be used to describe the interactions between U(VI) and proteins. The simplest model (model 1) considers the metal ion interaction with the protein as a global reaction, without considering the possibility that several binding sites of different affinity may exist. Furthermore, the metal ion speciation in solution, i.e. the species distribution between U(VI) and low molecular weight ligands is not taken into account. Equilibrium constants derived from this model represent conditional constants and can only be applied to the experimental conditions from which they have been obtained (i.e. pH, medium composition, protein concentration range). This model was used by Scapolan et al. [10] to describe U(VI) interaction with HSTF according to Eq.( 2):

HSTF U HSTF U 2 2 → ← + (2)
The obtained constant cannot be extrapolated to blood serum conditions, since the constant was determined in the presence of a lower concentration of carbonate (2 × 10 -4 M) compared to the carbonate concentration of blood serum (2.5 × 10 -2 M). This simple model was also used by Chevari et al. [17] to describe the interaction of U(VI) and HSA considering the formation of a 1:1 complex at pH 6, consequently also not allowing to apply the constant to serum conditions.

A more complex model (model 2) takes into account the metal ion speciation and the possible presence of different binding sites at the protein surface:

i) te protein(si - U i) te protein(si U → ← + (3)
The description of the interaction remains however incomplete as the functional groups implied in the complexation are not explicitly differentiated. Given that the pH and ionic strength are kept constant, the model can be used as a predictive tool to describe the interaction in different media. This is the model generally used to describe metal-HSTF interactions. Metal ion speciation in solution is often simplified and taken into account in the calculation by introducing a strong complexing agent having a known equilibrium complexation constant for U(VI) (see [20] and quoted references).

When a microscopic model (model 3) is used, interactions at the molecular level are converted to reactions according to the law of mass action and thermodynamic parameters can be derived. To our knowledge such a model has never been used to describe metal protein interactions considering the lack of spectroscopic data characterizing the interaction and considering the complexity to describe metal ion interactions with organic polyelectrolytes (e.g., see the case of natural organic matter [START_REF] Hummel | Modelling in Aquatic Chemistry[END_REF]).

For the purpose of our study, only models 2 and 3 were applicable and were used to describe the interaction of U(VI) with HSA and HSTF, respectively. The interaction of U(VI) with proteins was described using a surface complexation model in analogy to studies investigating the interactions of metal ions with natural organic polyelectrolytes (see e.g. [START_REF] Fein | [END_REF]) as detailed below.

Modelling the binding of U(VI) to HSTF

HSTF was shown to display two structurally related but slightly different binding sites ( S HSTF 1

≡ and S HSTF 2 ≡ ), each binding site interacting with U(VI) under the participation of two tyrosine groups [15]. The binding sites react with water according to Equation (4):

+ - → ← - + - → ← + ≡ + ≡ + ≡ + ≡ O H S HSTF O H SH HSTF O H SH HSTF O H SH HSTF 3 2 1,2 2 1,2 3 1,2 2 2 1,2 (4)
The pKa value associated to the Tyr188 residue was recently measured by 2D NMR-pH titration as 6.9 (0.5 M KCl) [28]. Based on the Davies equation we recalculated a value of 7.2 for zero ionic strength [START_REF] Davies | Ion association[END_REF] for introduction into the calculation code. For the other tyrosine residue Tyr95, no data is available and a pKa value of 10 (10.2 at zero ionic strength), associated with a "normal" tyrosine residue, was used [START_REF] Snyder | [END_REF]. We assume that the species interacting with the binding sites in the absence of carbonate is UO 2 2+ , i.e. there are no ternary complexes formed with other anions present in the medium (OH -, Cl -, NO 3 -). The reaction is then described by Eq.( 5):

2 2 , 1 2 2 2 2 , 1 SUO HSTF UO S HSTF ≡ + ≡ → ← + - (5)
Each lobe of transferrin contains two tyrosine Fe(III) binding sites that are remarkably similar ( [20] and quoted references). This was also observed to a certain extent with U(VI): Vidaud et al. demonstrated that both tyrosines at the iron binding site are involved in uranium binding [15]. Irrespective of the metal ion considered [20], the stability constants for interaction with the binding sites differ by approximately one unit in their log K value, the difference being probably related to outer-sphere effects [25]. This ratio of 10 was set as a fixed parameter in the fitting of our experimental data. In the presence of carbonate, a ternary complex may be formed [7,15] according to Eq.( 6):

n 2 n 3 2 2 , 1 n 2 3 2 2 2 2 , 1 ) CO ( SUO HSTF ) CO ( UO S HSTF - → ← - + - ≡ + + ≡ (6)
No electrostatic effects were considered in the calculation and the parameters must be considered as operational.

Modelling the binding of U(VI) to HSA

Metal binding to HSA is of complex nature, characterized by multiple binding sites whose affinity and binding capacity are varying (e.g. [31][32][33]). The microscopic model 3 is therefore not applicable and model 2 was used instead. To describe the interaction of U(VI) with HSA in blood, Chevari et al. [34] proposed an exchange process between Ca(II) and U(VI).

According to [35], about 70% of calcium in the blood serum is complexed with low molecular weight molecules, while the rest is bound to proteins. The protein bound fraction of calcium can be considered as attached mainly to HSA: Ca is known to interact with HSA [31,32] and Ca(II) interaction with HSTF is expected to be negligible; its interaction is weak [20] and protein-bound Ca concentration represents about 8 % of the total HSTF concentration. Based on a typical serum composition [6], this leads to an average of 0.7 Ca atoms bound per HSA molecule. The total number of binding sites (N tot ) reported in the literature for Ca on HSA varies [32] with the highest value reported as 30 [31]. The value depends on the HSA/metal ratio: for a given data set, N tot reflects the total molar binding ratio of the predominant calcium binding class and not the maximal molar binding ratio [32]. Anderson showed, however, that Ca interaction under physiological conditions is governed by the interaction with a strong site [31]. A site of stronger affinity was as well considered for other divalent metal ions like VO 2+ [33] and Ni 2+ [36]. Glennon & Sarkar showed that both Cu 2+ and Ni 2+ , although presenting different coordination properties, interact with the same HSA binding site [36]. Based on these considerations, our model considers the existence of only one binding site at the HSA surface which interacts with both U(VI) and Ca(II) cations, according to Eq.( 7):

2n 2 n 3 2 n 2 3 2 2 2 2 ) (CO SUO HSA ) (CO UO S HSA SCa HSA Ca S HSA - → ← - + + → ← + ≡ + + ≡ ≡ + ≡ (7)
Contrary to the equilibrium generally considered in model 2 (Eq. ( 3)), the possible formation of ternary complexes of U(VI) with HSA and carbonate must be considered under serum conditions (see comments above). All calculations were performed using the simulation code PHREEQC, a geochemical modelling code for aqueous systems [START_REF] Parkhurst | User's guide to PHREEQC-a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF]. Stability constants for metal ion complexation in solution were taken from NEA-OECD [START_REF] Guillaumont | Update on the chemical thermodynamics of uranium, neptunium, plutonium, americium and technetium[END_REF] (complexation of U(VI) with inorganic ligands), NIST [START_REF] Smith | NIST Critically Selected Stability Constants of Metal Complexes Database, Version 8.0 for Windows[END_REF] (U(VI) and citrate) and Llnl.dat [START_REF] Parkhurst | User's guide to PHREEQC-a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] (Ca complexation) thermodynamic databases.

Results and discussion

3.1. Binding of U(VI) to proteins

3.1.1.Binding of U(VI) to HSTF.

TRLFS experiments performed in the absence of carbonate show a significant complexation of U(VI) by HSTF at pH 7.4, i.e. 76 % of U(VI) were found to be bound to HSTF in the presence of a 4-fold excess of HSTF over U(VI). The stability constants describing the interaction (Eq. ( 5)) were found to be log K = 12.4 and log K = 11.4 (I=0) for the high-affinity and low-affinity site at zero ionic strength, respectively (Table 2). This set of constants was used to calibrate the UV-spectroscopy method, i.e. to relate the area of the absorption peak at 242 nm with the amount of the complexed species U(VI)-HSTF. As this absorption band is linked to the interaction of the metal ions to the phenolic groups of the tyrosine residues [10,20], we assume that the extinction coefficient is not affected by the nature of other ligands coordinated in the first coordination sphere of U(VI), i.e. the calibration parameters are applicable also when the synergistic carbonate ion is bound (Eq. ( 6)).

The experimental data obtained in the presence of carbonate are presented in Figure 1B.

The values obtained by TRLFS and DUS are in good agreement. The fraction of U(VI) bound to HSTF increases with increasing carbonate concentration up to a total carbonate concentration of ca. 2 × 10 -4 M, indicating the formation of a ternary complex in solution, with carbonate acting as synergistic anion according to Eq. ( 6). At higher concentrations of total carbonate, the fraction of HSTF-bound U(VI) decreases, due to increasing competition of carbonate with HSTF for U(VI) complexation. The complete set of experimental data could be well described considering one carbonate ion bound to U(VI) with stability constants of log K = 20.8 and log K = 19.8 (I=0) for the ternary complexes formed at the high-affinity and low-affinity site (Table 2). The data could not be fitted without considering the formation of ternary complexes. Under blood serum conditions, the ternary complexes will be the dominant species, i.e. the binary U(VI)-HSTF complex formed according to Eq.( 5) will not be formed.

The results of a titration of HSTF with U(VI) are presented in Figure 2A. Increasing metallation of HSTF leads to an increase of absorbance signal that could be extrapolated by non-linear regression to a plateau at Δε = (30 ± 2) × 10 3 M -1 cm -1 at 242 nm associated with the formation of a 2:1 U(VI):HSTF complex. This value is in excellent agreement with the values previously reported for U(VI) [15] and for other metal ions [20,[START_REF] Pecoraro | [END_REF] According to the literature, there is a good correlation between the strength of metal binding to transferrin and the stability constant for hydroxide binding to the same metal ion [6,20]. Compared to the values of log K = 14.1 and 12.6 derived from this correlation for strong and weak sites, respectively [6], the values determined in this work are approximately one order of magnitude lower (see Table 2). This might be explained by the fact that the binding sites of HSTF are ideally pre-organized to complex spherical cations in three dimensions like Fe(III), whereas complexation of the linear uranyl cation must account for its preferred equatorial coordination. This is in agreement with the recent work of Vidaud et al.

[15] who showed that the coordination with U(VI) cannot be compared with that of Fe(III)

and that the participation of the His249 ligand in the coordination sphere of uranyl can be excluded.

3.1.2.Binding of U(VI) to HSA.

The binding between U(VI) and HSA was studied using a displacement method in the presence of HSTF [25]. This method is applicable because no significant absorption of the U(VI)-HSA complex occurs in the HSA concentration range explored (< 4×10 -5 M) that could interfere with the measurement of the absorbance signal of the U(VI)-HSTF complex. This observation indicates that HSA does not interact with U(VI) via tyrosine groups, in agreement with the nature of the interaction of HSA with Cu(II) and Ni(II) at the strong binding site [36].

The system with HSTF and HSA competing for U(VI) complexation is also a good representation of the situation in blood serum. The concentration of carbonate ion in the system was varied to assess its role for the coordination of U(VI) with HSA. It was found that the data at equilibrium are not affected by the addition order of the components, indicating that the interaction of U(VI) with the two proteins is reversible. The experimental results are shown in Figure 3. In order to obtain a good fitting of the experimental data, the formation of a ternary HSA-U(VI)-carbonate complex needs to be taken into account at higher concentrations of carbonate (Fig. 3B). The stability constants were determined as log K = 10.8 (I = 0) for U(VI)-HSA and log K = 17.7 (I = 0) for U(VI)-HSA-carbonate as summarized in Table 2.

Overall our experimental data on the interaction of U(VI) with HSTF and HSA obtained in the presence of carbonates could be well described by considering the formation of ternary complexes between the protein, one carbonate ion and U(VI). The occurrence of ternary complexes is in agreement with general understanding of metal binding to HSTF [20,21], and agrees well with the reports of Van Horn et al. and Huang et al. [7,[START_REF] Huang | [END_REF] on the binding of U(VI) to proteins. However, they propose the involvement of two carbonate ions in the U(VI) coordination sphere when bound to proteins or peptides. To the best of our knowledge, there is no experimental proof to support this statement, but it is rather an assumption based on the presence of the U(VI)-bis-carbonato complexes in blood serum. Our data show that both HSA and HSTF have a strong but similar affinity for U(VI). Indeed, if the experimental data are analysed based on model 2 (Eq. ( 3)), the stability constants obtained for U(VI) complexation are identical for both proteins (see Table 2).

Simulation of U(VI) speciation in the blood serum and comparison with in vitro speciation results

Serum composition.

The aim of our model calculation was to simulate the speciation of trace concentrations of U(VI) under serum conditions, where U(VI) may interact with all types of complexing agents present in the serum, i.e. inorganic ligands, organic ligands and proteins. Considering the complexity of the biological medium, simplifications were made considering that the interaction strength of given serum components with U(VI) will depend on their concentrations as well as the equilibrium complexation constants. All relevant inorganic ligands were taken into account (hydroxide, carbonate, phosphate, chloride, sulphate) using the constants given in [START_REF] Guillaumont | Update on the chemical thermodynamics of uranium, neptunium, plutonium, americium and technetium[END_REF]. Concerning low molecular weight organic components, only the citrate ion was considered as it is present in human serum at a relatively high concentration (1.6 × 10 -4 M) and has a relatively high stability constant for U(VI) complexation [6].

Although recent work of Vidaud et al. showed that a number of proteins can bind U(VI) [11],

only HSA and HSTF were considered in the present study as they constitute the predominant metallo-proteins present in high concentrations in human blood serum [6].

In addition, a possible competition with other metal ions present in blood serum has to be taken into consideration. Also in this case simplifications are necessary. Given the high difference in log K values (about 10 orders of magnitudes), we can safely consider that trace concentrations of U(VI) can not compete with Fe(III) for HSTF binding sites. The number of available HSTF binding sites was therefore set to 70 % of the total capacity, corresponding to the fraction of HSTF not loaded with ferric ion [20]. Another cation that has to be taken into account is Ca(II) as it is present in blood serum in relatively high concentrations [35] and it is known to strongly interact with HSA [32].

Ca interaction with HSA

In previous studies, model 1 was used to describe the interaction of calcium with HSA [32]. As discussed in the modelling part, this model is not applicable for the purpose of the present study, i.e. the competition between Ca(II) and U(VI) for the strong site must be explicitly described. For this purpose, experimental data obtained by Besarab et al. [32] in conditions similar to those found in blood serum were re-evaluated according to Eq.( 7). To characterize Ca(II) interaction with the strong site on HSA, only the experimental data where less than one Ca(II) is bound per HSA molecule were taken into account, resulting in a value for log K = 4.2 (see SM-1). Based on this stability constant together with thermodynamic data for calcium complexation taken from Llnl.dat [START_REF] Parkhurst | User's guide to PHREEQC-a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF], the species distribution of calcium in blood serum [6] was calculated. The calculation results in 34 % of Ca bound to HSA and 66 % "non-protein bound", i.e. 49 % in the ionic form, 8 % complexed with citrate, 5 % as a carbonato complex, 2 % bound to phosphates and the remaining 2 % being distributed between other species. This indicates that the stability constant for Ca(II)-HSA complexation is reliable since the calculation is in agreement with experimental data reported in [35], where 67 ± 2 % of Ca is in ionic form or complexed with low molecular weight organic and inorganic molecules, while the remaining part is bound to the protein pool.

Simulation of U(VI) speciation in human serum

Based on the stability constants for U(VI) complexation by HSTF and HSA determined in this study, together with known thermodynamic parameters for complexation of U(VI) with low molecular weight organic and inorganic ligands, the species distribution of U(VI) in human blood serum was calculated. [9,34]. The agreement is less satisfactory when comparing our model calculation with the data obtained by Vidaud et al. [11]. From in vitro experiments they concluded that about 20 % of uranyl in serum is associated with the protein pool, whereas our simulation leads to a value of 35 %.

In agreement with Chevari et al. [34], our simulation showed that the role of HSTF is minor and that binding of U(VI) to HSA is predominant. As both proteins display a similar strength for binding of U(VI), but HSA concentration in serum is one order of magnitude higher than HSTF concentration, 80% of protein bound U(VI) is associated with HSA compared to 20% of U(VI)-HSTF. This is in contrast to the predominance of HSTF binding of U(VI) reported in [9], however, the authors do not provide detailed support of their statement.

Chelating agents for application of 230 U in targeted alpha therapy

The modelling of the speciation of U(VI) in blood serum can be used to make predictions on the stability of potential U(VI)-chelate complexes under serum conditions and to select promising ligands for further testing and experimental validation of their stability in vitro and in vivo. For a simplified estimation, assuming the injection of 1 mg of 230 U labelled antibody carrying one chelate per antibody molecule, the concentration of chelate in the blood pool will be initially in the range of 10 [8] and antibody based assays [39]. Based on their high stability constants for complexation of U(VI), the calixarene family [40], in particular calix [6]-and calix [8]-arene [41,42], appear promising.

Calixarenes can easily be bi-functionalised [43] to establish the link to biological carrier molecules without modifying the functional groups available for the complexation. According to the work of Mullen et al. [44], the siderophore desferrioxamine B (DFO), that is already used for nuclear imaging when labelled with 89 Zr(IV) [45], may as well be a good candidate.

The authors report an equilibrium constant between UO 2 2+ and the deprotonated ligand of 10 17.1 M -1 for an ionic strength of 0.1 M. Another promising molecule is the versatile ligand 1,10-phenanthroline-2,9-dicarboylic acid (DCP) that has been shown to efficiently complex U(VI) in biological media [39]. The synthesis of a bifunctional derivative has been reported and a complexation constant between the deprotonated ligand and UO 2 2+ exceeding 13 has been reported [46].

Conclusions

The 2 considering (solid line) or not-considering (dashed line) the formation of a ternary complex (Eq.( 6)). concentrations. In Figure 2A, r represents the ratio between total U(VI) and HSTF concentrations. Δε is calculated from the absorbance of U(VI)-HSTF complex measured at 240 nm divided by HSTF concentration. The lines are calculated with the parameters given in Table 2. For Figure 2B, an extinction coefficient of 3 × 10 4 M -1 cm -1 characterizing the 2:1 U(VI):HSTF complex is used. Experimental conditions are summarized in Table 1. and 2 × 10 -3 M (B) of total carbonate. The system is pre-equilibrated before addition of HSTF (circle) or HSA (squares). Experimental conditions are summarized in Table 1. The lines are calculated with the parameters given in Table 2 considering (solid line) or not-considering (dotted lines) the formation of a ternary complex (Eq.( 7)).

4 M

 4 . The increase of absorption observed above the theoretical saturation level (2 U(VI) for one HSTF molecule, r=2) is due to the presence of uncomplexed U(VI) as the chosen experimental conditions do not allow for complete complexation of U(VI) by HSTF. The reliability of the determined stability constants describing the formation of the ternary complexes, derived from the experimental data presented in Figure 1B, relies on the reliability of the constants characterizing the carbonate complexation of U(VI) taken from [gui03].Therefore, to test the obtained constants, they were used to calculate U(VI) speciation in a simplified system simulating blood serum, i.e. containing HSTF and carbonate ion (4 × 10 -) as a function of concentration of citrate, a well known strong complexing agent for U(VI)[6] and the calculated results were compared to experimental data. Considering the relatively large size of the citrate molecule, we assume that the synergistic carbonate ion bound to U(VI) with HSTF will not be exchanged by citrate. The experimental data, representing the percentage of U(VI) bound to HSTF determined by spectrophotometry as a function of citrate concentration, are presented in Figure2B. As shown by the solid line, a good agreement between the prediction and the experiment is obtained showing the reliability of our stability constants for HSTF complexation of U(VI).

  stability constants determined in this work for the complexation of U(VI) by HSA and HSTF allow to predict the speciation of U(VI) in human body fluids. The model calculations could be validated through experimental data obtained in blood serum and provide a realistic, qualitative and quantitative description of U(VI) behaviour. The model initially composed of 43 equilibrium equations could be simplified to only a few, describing the formation of bis-and tris-carbonato complexes of U(VI), the complexation of U(VI) with HSA and HSTF and taking into account the competition between Ca and U(VI) for binding to the strong site of HSA.

Figures captionsFigure 1 :

 1 Figures captions

Figure 2 :

 2 Figure 2: Complexation of U(VI) by HSTF as a function of U(VI) (A) and citrate (B)

Figure 3 :

 3 Figure 3: Competition between HSTF and HSA for U(VI) in the presence of 2 × 10 -4 M (A)

Figure 1 [

 1 Figure 1

  This is also supported by the experimental data reported by Chevari et al. and Stevens et al. showing a ratio of carbonate vs. protein bound U(VI) in plasma of 64:36 and of 60:40, respectively

	The model calculation shows that U(VI) speciation is
	governed by both the carbonate ion and the proteins. 65 % of U(VI) are complexed with CO 3	2-
	as the bis-(3 %) and tris-carbonato (59 %) complexes, while the remaining fraction is bound
	to the protein pool, 28 % and 7 % with HSA and HSTF, respectively. Calcium has a
	significant effect on U(VI) speciation. Without considering the competition of Ca(II) for
	binding sites on HSA, approximately 70 % of U(VI) would be predicted to be bound to the
	protein fraction. In agreement with what is generally stated in the literature [6,7], both
	proteins (HSTF and HSA) and carbonate govern U(VI) speciation in blood serum at
	equilibrium.

  -9 M. Taking into account the concentrations and stability constants of the competing ligands carbonate, HSA and HSTF, the model calculation shows serum conditions within the residence time of the radioconjugate in vivo. A large number of potential complexing agents for U(VI) have been reported in literature for a variety of applications, including the development of decorporation agents

	that a chelate suitable for stable binding of 230 U(VI) under serum conditions is required to
	have a stability constant for UO 2	2+ of 10 19 M -1 at physiological pH if >90 % of U(VI) should

remain bound at equilibrium. However, chelating agents with lower stability constants could be useful in case their U(VI) complexes display a sufficiently high kinetic stability against dissociation under

Table 1 :

 1 Interaction between U(VI) and proteins; methods and experimental conditions.

	Method Principle	U(VI)	HSTF	carbonate concentration	experiment	Figure
	TRLFS see Eq.( 1)	5x10 -6 M		0 0 -2x10 -3 M 0 -3x10 -2 M	kinetic carbonate titration	1A 1B 1B
					2x10 -4 M	U(VI) titration	2A
	DUS	absorption of U(VI)-HSTF complex at 242 nm	2x10 -5 M	2x10 -5 M	4x10 -4 M 2x10 -4 M	competition with citrate ion (2x10 -4 -4x10 -2 M) competition with	2B 3A
					2x10 -3 M	HSA (3.10 -6 -5.10 -5 M)	3B

Table 2 :

 2 Quantitative description of U(VI) with HSTF and HSA and comparison with published data.

	Protein Equation Model Site / interacting group log K Ref.	Remarks
			Weak site	12 13	this work
				14.3*	[22]
			strong site			carbonate
	Eq. (3)	2		14.1*		concentration at 25 mM
					[6]
			Weak site	12.6*
	HSTF
						carbonate
	Eq. (2)	1		16	[10]	concentration at 0.1
						mM
	Eq. (4)		Tyr188 (both sites) Tyr95 (both sites)	-7.2 -10.2	[28] [30]
	Eq. (5) Eq. (6)	3	strong site weak site strong site weak site n=1 (CO 3 2-) 19.8 12.4 11.4 20.8	this work	see experimental conditions in Table 1
	Eq. (4)	3	strong site weak site	7.2 5.4	[19]	pH 5.5, normal CO 2 atmosphere
	Eq. (2)	1	_	10	[17]	pH=6, ionic strength fixed at 0.1 M
	HSA					carbonate
	Eq. (3)		_	13		concentration at 25
	Eq. (7)	2	n=0 (CO 3 2-) n=1 (CO 3 2-)	10.8 17.7	this work	mM see experimental conditions in Table 1
	* extrapolated values

Acknowledgment

The authors are grateful to Prof. E. Simoni and G. Lagarde (Institut de Physique Nucléaire, Orsay) for allowing us to use the TRLFS device.

Abbreviations

Targeted alpha therapy TAT

Bifunctional chelating agents BCAs

Human Serum TransFerrin HSTF Human Serum Albumin; HSA Time Resolved Laser Fluorescence Spectroscopy TRLFS Difference Ultraviolet Spectroscopy DUS

N-(2-HydroxyEthyl)Piperazine-N'-2-EthaneSulfonic acid HEPES

SUPPLEMENTARY MATERIAL

SM-1:

The Figure reports the experimental data given in [32] used to determine log K of Ca-HSA.

They include the range of albumin concentration found in the blood serum and reflect the interaction of Ca with the strong site, where at equilibrium less than one Ca is bound per albumin molecule. The line was calculated using the parameters given in the main text.


B Jurado
email: jurado@cenbg.in2p3.fr

G Barreau

C.-O Bacri

Radioactive target needs for nuclear reactor physics and nuclear astrophysics

Keywords: 29.25.-t 26.30.-k 26.20.Kn 28.20.-v 24.87.+y Radioactive Targets, Nuclear Data, Element Nucleosynthesis, Fission Reactors

Nuclear reaction cross sections of short-lived nuclei are key inputs for new generation nuclear reactor simulations and for models describing the nucleosynthesis of elements. After discussing various topics of nuclear astrophysics and reactor physics where the demand of nuclear data on unstable nuclei is strong, we describe the general characteristics of the targets needed to measure the requested data. In some cases the halflife of the nucleus of interest is so short that it is not possible to produce a target and perform the measurement. However, some alternative methods have been developed that allow one to obtain neutron-induced cross sections of highly radioactive nuclei. One of these methods is the surrogate reaction technique. We explain the principle of the surrogate method and describe the characteristics of the targets used in surrogate experiments.

Introduction

Even though nuclear astrophysics and reactor physics may appear to be rather distant topics at a first view, they have important aspects in common. Both topics are strongly based on computer simulations or models for which the availability of reliable nuclear reaction data is of crucial importance. Reactor physics uses transport codes, e.g. MCNP [1], where the history of the interaction of single neutrons with the different nuclei present in the reactor core is simulated for their complete life cycle ranging from their generation in a nuclear fission process until they diminish due to nuclear reactions. One of the primary goals of Nuclear Astrophysics is to obtain a detailed understanding of the synthesis of all the elements which make up our world. Theories of nucleosynthesis are tested by calculating isotope abundances with rather complex stellar models and comparing them with observed results. Both topics have also in common that neutron-induced reactions play an essential role. Although in nuclear astrophysics proton-, alpha-and gamma-induced reactions are also important. Typical paths for the formation of nuclei in a reactor or in a star are one or more neutron captures followed by β -decay. When the nuclei are heavy enough, the interaction with neutrons can lead to fission where the initial nucleus splits into two lighter nuclei. Thus, neutron-induced capture and fission reactions are highly interesting for both topics. Of course, the characteristics (temperature, neutron density, etc.) of a reactor core and those of a star may be very different. Therefore, the neutron-energy domains of interest in astrophysics and in reactor physics can be rather distinct. Finally, another similarity between both topics is the need of cross section measurements of unstable nuclei. Very often the production of a radioactive target as well as the realisation of the measurement in an adapted facility is extremely challenging.

Radioactive nuclei of interest in nuclear astrophysics

Elements heavier than 56 Fe can be assembled within stars by a slow neutron capture process known as the s-process or in explosive environments, such as supernovae, by the so-called rprocess which involves rapid neutron captures and the rp-process which involves rapid proton captures [START_REF] Rolfs | Cauldrons in the cosmos[END_REF]. In the rp-process consecutive proton captures onto seed nuclei followed by β + decay lead to heavier elements. This process occurs in an environment with a high rate of proton captures and high temperatures. The nuclei formed are localised in the neutrondeficient side of the chart of nuclei, see figure 1. The s-process occurs at relatively low neutron density and intermediate temperature conditions in stars. As illustrated in figure 1, this process produces stable isotopes up to Bi by moving along the valley of beta stability. Heavier nuclei such as Thorium or Uranium are produced thanks to the r-process. Immediately after a core-collapse supernova, there is an extremely high neutron flux and temperature, so that neutron captures occur much faster than β -decays, meaning that the rprocess evolves along the neutron drip line, see figure 1. Consequently, nucleosynthesis models require proton-and neutron-induced capture and fission cross sections of radioactive nuclei situated in the three regions depicted in figure 1. Clearly, there is no way to have targets of nuclei directly involved in the r-process and the heavier part of the rp-process. For this reason the experiments aiming at measuring data close to the proton and the neutron drip lines are rather based on the use of radioactive beams in inverse kinematics. On the contrary, for the s-process, there exist very specific demands where the radioactive targets required are still feasible. Indeed, reliable neutron-induced capture cross sections on branching points are crucial for testing and constraining stellar models in terms of the basic physical s-process parameters, i.e., neutron density, temperature and pressure. Branching points are nuclei with half-lives comparable to the neutron capture time scale which give rise to branchings in the reaction path. According to the review article by Käppeler and Mengoni [START_REF] Käppeler | [END_REF], the main requests are 79 Se, 85 Kr, 147 Pm, 151 Sm, 163 Ho, 170 Tm, 171 Tm, 179 Ta, 204 Tl, and 205 Pb. The half-lives of these nuclei vary from 129 days for 170 Tm to 1.5•10 7 years for 205 Pb. These nuclei are β + or β -emitters, which in some cases emit gamma rays subsequent to β decay. The different possible s-process temperatures require the experimental neutron capture cross sections to cover a wide energy range, from about 300 eV to several hundred keV. Besides the branching points, other neutron capture cross sections of unstable nuclei are also important. The study of 60 Fe(n,γ) has been identified to be of great interest for at least two reasons: (i) 60 Fe has a half-life of 1.5 million years and was found in deep layers of oceanic crust sediments. It is assumed that a nearby supernova which exploded 2.8 million years ago is responsible for the excess of 60 Fe. In order to determine the locus of this event in the solar system it is necessary to investigate the production (via 59 Fe(n,γ) 60 Fe) and destruction mechanism (via 60 Fe(n,γ) 61 Fe) of 60 Fe. Furthermore, (ii) 60 Fe is one of the few, long-lived γ-ray emitters which could be observed with γ-ray telescopes like INTEGRAL after supernovae explosions. Therefore, the production and destruction of 60 Fe is important to compare the observed amount of 60 Fe and the calculated values from supernovae models. The preparation of a 60 Fe target is treated in ref. [START_REF] Schumann | these Proceedings[END_REF].

s-process

Radioactive nuclei of interest in nuclear reactor physics

In this work we concentrate on two topics that demand strong efforts in terms of nuclear data measurements: minor actinides incineration and the development of the Th-U cycle. Minor actinides (mainly Np, Am and Cm isotopes) are produced by successive neutron captures, beta and alpha decays starting from 238 U in the current U-Pu cycle. Because of their very high radiotoxicity, these nuclei represent one of the most harmful types of nuclear waste. At present, two different strategic approaches are proposed for minor actinides waste disposal: direct disposal without any reprocessing and spent fuel reprocessing with the aim to optimise the extraction of minor actinides and to then incinerate them. Incineration amounts for the transmutation of minor actinides into less radiotoxic or short-lived species obtained by neutron-induced fission reactions. The reliable design of reactors for incineration requires an accurate knowledge of minor actinides cross sections in a fast neutron flux. However, the available data are rather scarce. The reason for this lack of data is the short half-life of many of these nuclei which makes it very difficult to produce and to manipulate targets of these isotopes. Reactor simulations require principally neutron-induced fission (n,f), capture (n,γ), elastic (n,n) and inelastic (n,n') cross sections. However, since the neutron-energy regime of interest extends from few eV to about 20 MeV, also two-neutron emission (n,2n) cross sections are needed. Moreover, other data such as fission fragment isotopic yields, average number of neutrons per fission or number of delayed neutrons are highly interesting because these quantities strongly affect the reactor neutron flux. For most minor actinides the existing fission data are of limited quality. These fission cross sections are often measured with uncertainties greater than 20% and important discrepancies between different sets of data are observed. The situation is even worse for other decay channels such as (n,γ), (n,n') or (n,2n) where no data at all are available. Some of the minor actinides of interest are 237 Np, 240,241,242,242m,243 Am and 242,243,244,245 Cm. The half-lives of these nuclides vary from 16 hours for 242 Am to 2.14•10 6 years for 237 Np. Among these nuclei we find several β -emitters but the majority of them are strong alpha emitters and some may even fission spontaneously, which implies the emission of neutrons. Also γand x-rays can be emitted. This illustrates how difficult it can be to make and handle targets of these nuclei. Another issue of interest in reactor physics is the development of the Th-U fuel cycle. The underlying main idea is to replace the 238 U of the current fuel by 232 Th. Similarly to 238 U, 232 Th is a fertile nucleus. Indeed, a neutron capture on 232 Th followed by two β -decays leads to the formation of the fissile nucleus 233 U. The use of 232 Th as a fuel has the advantage that it increases the resources (Th is more abundant than U by a factor 3 or 4). Moreover, the radiotoxicity of the generated nuclear waste is reduced. Since Th is lighter than U one needs more neutron captures to produce minor actinides. The nuclei of interest for the Th-U cycle are 232 Th and 233 U and several neighbouring isotopes as 231 Th, 231,233 Pa, 232,233,234,235 U. They are mainly β -and alpha emitters with half-lives varying from 25.52 hours for 231 Th to 1.4•10 10 years for 232 Th. Whereas for the current U fuel cycle an enormous effort has been undertaken concerning experimental nuclear data, this is not at all the case for the Th-U cycle for which the data are not precise enough or simply missing. Even for the long-lived 232 Th the fission cross section needs to be considerably improved between few keV and 1 MeV neutron energy.

Target characteristics

The target characteristics depend strongly on the type of measurement, therefore we distinguish between three main types of experiments: neutron-induced capture, neutroninduced fission and indirect measurements with light charged particle beams based on the surrogate reaction method.

Neutron-induced capture cross sections measurements

Due to the low neutron fluxes and the reduced gamma detection efficiency, samples of few grams are generally required for this type of experiments. However, the radioactivity of unstable nuclei may limit considerably the sample quantity which has to be compensated with a particularly efficient γ-detection set-up and an intense neutron source. For example recently, the (n,γ) cross section of 241 Am(T 1/2 =432.6y) has been measured at LANSCE in Los Alamos National Laboratory using the DANCE γ-ray array and a sample of only 219±1 µg [START_REF] Jandel | [END_REF]. Very often the sample is in form of powder pressed into a container made of Al or Ti [7]. However, in some cases thin targets are used piled-up in a compact manner inside a container [START_REF] Jandel | [END_REF]8]. A precise knowledge of the isotopical composition of the sample is very important in order to correct for the background generated by possible contaminants.

Neutron-induced fission cross sections measurements

Fission fragments are heavy nuclei that may lose important amounts of energy when traversing a material. In addition, due to momentum conservation, the two fragments are emitted back-to-back in opposite directions. Therefore, in fission experiments targets should be thin enough to allow at least one of the fission fragments to reach the fission detector. Typical target thicknesses vary between 200 and 500 µg/cm 2 [START_REF] Aiche | Proceedings of the International Conference on Nuclear Data for Science and Technology[END_REF]. The sample is evaporated or electrodeposited onto a support. Typical support materials are Al, Ni or Ti. The thickness of the support ranges between few µm to 1 mm. Thin supports of few µm are used if the experiment requires the two fission fragments to be detected. Special care should be made to avoid heavy contaminants that fission, in particular neighbouring isotopes that may have not been completely separated from the raw radioactive material. The size of the targets is very variable with diameters ranging from few mm to various cm [START_REF] Audouin | Proceedings of the International Conference on Nuclear Data for Science and Technology[END_REF]. The size depends mainly on the diameter of the neutron beam. For small samples an homogeneity better than 90% is required, for large targets one needs to measure how the matter is distributed all over the target surface. For many actinides this can be done via alpha cartography or radiographic imaging [START_REF] Liebe | [END_REF].

Indirect measurements via the surrogate reaction method

In some cases the activity of the targets is such that the fabrication and the manipulation of the sample are impossible due to a lack of availability of the target material or its short halflife. Alternative indirect methods have been developed to overcome these difficulties. One of these methods is the surrogate reaction technique. This method was developed in the 70's by Cramer and Britt [12]. Given a desired neutron-induced reaction n + A that proceeds through a compound nucleus (A+1)*, the surrogate reaction method consists of producing compound nucleus (A+1)* via an alternative (surrogate) reaction, e.g., a few-nucleon transfer reaction y + X (A+1)* + b and measuring the decay (e.g. fission or radiative capture) of the compound nucleus (A+1)* in coincidence or not with the ejectile b. The ratio between the number of decays detected in coincidence with the ejectile (corrected for the decay-detector efficiency) and the total number of ejectiles gives the decay probability of the compound nucleus (A+1)*. The neutron-induced cross section for the corresponding decay channel is then deduced from the product of the decay probability measured in the surrogate reaction and the compound nucleus cross section for the neutron-induced reaction. The latter cross section is obtained from optical model calculations. The principle of the surrogate method is illustrated in figure 2. The conditions under which the surrogate method can be applied to infer neutron-induced cross sections have been investigated in refs. [13,14]. The surrogate reaction method presents several advantages: -The most important: in some cases the target X of the surrogate reaction is less radioactive and easier to produce than the target A required in the neutron-induced measurement. We will see some examples below.

Neutron-induced reaction n + A (A+1)* ( ) ( ) ( ) exp ,
-In charged particle-induced transfer reactions different combinations of nucleons can be exchanged between projectile and target leading to the production of various residues. In contrast to neutron-induced experiments, the simultaneous access to several transfer channels allows one to determine cross sections of various nuclei from just one projectile-target combination.

-Since there are two bodies in the outgoing reaction channel, the excitation energy of the compound nucleus E* follows a broad probability distribution. The E* is obtained by measuring the kinetic energy and the angle of the ejectile. The compound nucleus excitation energy can be translated into neutron energy E n via the relation E*=B n +A.E n /(A+1), where B n is the neutron binding energy of the compound nucleus. Therefore, with fixed beam energy, the surrogate method allows the determination of cross sections over a wide range of neutron energy.

This technique has been applied to the measurement of the neutron-induced fission [15] and capture [16] cross sections of 233 Pa (half-life of 27 days) via the transfer reaction 232 Th( 3 He, p) 234 Pa. The 232 Th(1.4•10 10 y) target was produced at the SIDONIE separator of the CSNSM. Recently, the surrogate reaction method has been used to determine the neutron-induced fission cross sections of 242 Cm(163 d), 243 Cm(29.1 y) and 241 Am(432.2 y) with a 243 Am(7370 y) target and a 3 He beam [START_REF] Jurado | AIP conference proceedings of the 2007 International Workshop on Compound-Nuclear Reactions and Related Topics (CNR*2007)[END_REF]. Two 243 Am targets of approximately 106 µg/cm 2 were prepared at the Argonne National Laboratory. Each target was electrodeposited on a 75 µg/cm 2 carbon backing. The surrogate method has been shown to be very successful for extracting (n,f) cross sections of unstable actinides. This technique can in principle be also used infer neutron capture cross sections. However, this application of the surrogate technique has still to be validated. In reference [START_REF] Escher | [END_REF] the possibility of using the surrogate method to infer capture cross sections of s-process branching nuclei has been discussed. The target thickness is limited by the sample activity and by the high elastic scattering cross section of the beam particle on the target nucleus that may cause detector saturation. In fission measurements one may need to detect the two fission fragments in coincidence with the ejectile, this requirement also constrains the target thickness. However, the reduced sample thickness can be largely compensated by the high intensity of the charged particle beam. Typical target thicknesses range between 100 and 250 µg/cm 2 . The sample is evaporated/electrodeposited onto a thin C support (50-100 µg/cm 2 ) [19]. The thickness of the support determines the energy straggling of the light ejectiles and thus the energy resolution associated to the surrogate experiment. A thin support is also required to limit the background generated by reactions between the projectile and the nuclei of the backing. The isotopic purity of the sample is extremely important. Neither heavy nor light contaminants should be present since reactions on the contaminants can falsify the number of coincidences decay particle-ejectile and the total number of ejectiles that are used for determining the decay probability. This background is extremely difficult if not impossible to remove.

Summary and conclusions

Neutron-induced cross sections of short-lived nuclei are of great importance in reactor physics and nuclear astrophysics. In this work we have outlined some of the regions of the chart of nuclei of interest for both topics. The main target characteristics for neutron-induced capture and fission measurements have been given. The surrogate reaction method, an indirect technique to determine neutron-induced cross sections of unstable nuclei, has been presented together with a description of the required target characteristics. The further development of research domains such as nuclear waste transmutation or the understanding of element nucleosynthesis is strongly linked to the availability of radioactive targets. Therefore, an effort should be made to maintain the current expertise and improve the existing techniques for radioactive target preparation.

Figure 1 :

 1 Figure 1: (Color online) Chart of nuclei representing the different element nucleosynthesis processes. The nuclei located on the valley of beta stability are represented by the black squares. The rapid proton capture process (rp-process) on the neutron-deficient side of the valley of stability is represented by the light blue line, the slow neutron capture process (sprocess) on the stability valley is illustrated by the pink arrow and the rapid neutron capture process (r-process) on the neutron-rich side of the stability valley by the blue line. Figure adapted from ref. [3].

Figure 2 :

 2 Figure2: Schematic representation of the surrogate method compared with the direct neutron-induced reaction. In this case the surrogate reaction is a transfer reaction. Two possible decay modes "i" (fission and gamma emission) of nucleus (A+1)* are also represented. The neutron-induced cross section σ n,i for decay channel "i" is obtained by multiplying the calculated cross section for the formation of nucleus (A+1)* through neutron absorption cal CN σ with the measured decay probability exp i P .