N Jahan

Mokus

G Royer
email: royer@subatech.in2p3.fr

J Jahan

N Mokus

Shapes describing the alpha decay, cluster radioactivity, fusion, fission and fragmentation phenomena

Shapes describing the alpha decay, cluster radioactivity, fusion, fission and fragmentation phenomena

Introduction

In chemistry, mechanics, astrophysics, hydrodynamics, nuclear physics,... the transition from one almost spherical system towards several spherical or ellipsoidal objects or vice versa (alpha decay, cluster radioactivity, fragmentation, fission, fusion,...) may be described in regarding the evolution of the density profile in the system during the deformation or in following the deformation of specific assumed geometric shape sequences. In this last approach to determine the macroscopic energy it is necessary to know the main characteristics of the shape: the position of the centre of inertia, the root mean square radius, the volume, the surface, the Coulomb function, the moments of inertia, the quadrupole moment,... An extensive review of the geometrical relationships used in macroscopic nuclear physics may be found in Ref. [START_REF] Hasse | Geometrical Relationships of Macroscopic Nuclear Physics[END_REF]. In a first part, most of these shape sequences used to describe the nuclear states and reactions are reviewed. In the following sections the definitions and geometric properties of different shape sequences that we have previously used and bring together recently [START_REF] Royer | [END_REF] are recalled. Within macro-microscopic models they allow the study of the fusion [3,4], fission [5,6], fragmentation [7,8] and alpha decay and cluster radioactivity processes [9,10,11]. The following shapes will be successively presented: one ellipsoid, two ellipsoids, elliptic and hyperbolic generalized lemniscatoids, prolate compact ternary shapes, bubbles and tori.

To study n α-nuclei as nuclear molecules, different configurations such as linear chain, triangle, square, tetrahedron, pentagon, square bipyramid, trigonal pyramid, hexagon, octahedron, octagon and cube have also been selected [12].

Mini-review of the main theoretical nuclear shapes

Within macroscopic models such as the liquid drop model or droplet model the total energy of a system is, mainly, the sum of the volume, surface and Coulomb energy and shape-dependent functions are used. For charge or matter distributions of constant density the relative shape-dependent surface B s and Coulomb (or gravitational) B c functions are respectively [START_REF] Hasse | Geometrical Relationships of Macroscopic Nuclear Physics[END_REF]

B s = σ dσ 4πR 0 2 , ( 1
)
B c = 15 16π 2 R 0 5 dτ dτ | r -r | , ( 2
)
where R 0 is the radius of the sphere of same volume.

When the shapes are axially symmetric, B c reduces to

B c = 0.5 v(θ i ) v 0 R(θ i ) R 0 3 sin (θ i )dθ i . (3)
The potential v(θ) is determined at the surface of the shape, v 0 being the surface potential of the sphere

v(θ i ) v 0 = 3 2πR 0 2 ρ (ρ i + ρ) dz dθ + (z i -z) dρ dθ K(k) -1 2 [(ρ i + ρ) 2 + (z i -z) 2 ] dz dθ D(k) (ρ i + ρ) 2 + (z i -z) 2
dθ (4) where ρ = R(θ) sin θ, z = R(θ) cos θ. D(k) and K(k) are calculated from elliptic integrals. These shape-dependent B s and B c functions are dimensionless and equal to one at the sphere.

For axially symmetric shapes the inverse effective moment of inertia is defined by

I ef f -1 = I -1 -I ⊥ -1 . ( 5
)
I and I ⊥ are the parallel and perpendicular moments of inertia. The dimensionless quadrupole moment Q relatively to the z axis is given by

Q = 1 R 0 5 (3z 2 -r 2 )dτ. (6)
.

The parameter β, often used in γ spectroscopy, follows

β = 0.75 √ 5π QR 0 2 r 2 -1 , ( 7
)
β = 2 π 5 I ⊥ -I I ⊥ + 0.5I . ( 8
)

Small deformations

For small oblate or prolate deformations from the sphere (nuclear ground states, planets) the spheroidal expansions have been used [13] within a small expansion of the eccentricity e. For prolate spheroids, the minor a and major c semi axes are given by

a = R 0 (1 - 1 6 e 2 - 5 72 e 4 + ...), c = R 0 (1 + 1 3 e 2 + 2 9 e 4 + ...). ( 9
)
For oblate spheroids e 2 must be replaced by -e 2 . The shape-dependent functions are:

I ⊥ =
Several harmonic expansions have been proposed [START_REF] Hasse | Geometrical Relationships of Macroscopic Nuclear Physics[END_REF]. The α n parameterization is defined by [14,15,16]

R(θ) = λ -1 R 0 [1 + N n=1 α n P n (cos θ)], (15)
N being a cutoff parameter, λ a parameter allowing the volume conservation and P n a Legendre polynomial.

The volume conservation can be included into the deformation parameters [15]

R(θ) = R 0 [1 + N n=0 a n P n (cos θ)]. ( 16
)
The β n parameterization [17] takes into account the volume conservation only up to quadratic terms in the deformation parameters

R(θ) = R 0 [1 + N n=2 (β n Y n0 (θ) - β n 2 4π )]. ( 17
)
Y n0 are the spherical harmonics. The parameter β 2 is usually denoted by β.

The relative quadrupole and hexadecapole moments read [18]

Q = 4 π/5(β 2 + 0.36β 2 2 + 0.967β 2 β 4 + 0.328β 2 4 + 0.023β 3 2 + ...), Q 4 = 4 3 √ π(β 4 + 0.725β 2 2 + 0.983β 2 β 4 + 0.411β 2 4 + 0.416β 3 2 + ...). (18)
. The ε n parameterization has also been defined to describe deformations greater than spheroidal deformations using expansions about a spheroid [19].

To define the semi-axes a, b and c of triaxial ellipsoids the parameters β H and γ H have been introduced by Hill and Wheeler [14],

a = R 0 exp[α H cos(γ H - 2π 3 )], ( 19
) b = R 0 exp[α H cos(γ H + 2π 3 )], (20)
c = R 0 exp[α H cos(γ H )]. ( 21
)
0 ≤ γ H < π/3 and α is unrestricted.

β H = 4π 5 α H . ( 22
)
When γ varies from 0 to π/3 the ellipsoid changes from axially prolate to axially oblate shapes. The dimensionless surface and Coulomb functions are given by 23) This parameterization has been widely used to describe oblate nuclear ground states.

B s = 1 + 2 5 α 2 H - 2 21 α 3 H cos(3γ H ).., B c = 1 - 1 5 α 2 H - 1 105 α 3 H cos(3γ H ).. (
For the Bohr parameterization [17] the semiaxes are defined by

a = R 0 [1 + 5/4πβ B cos(γ B - 2π 3 )], (24) b
= R 0 [1 + 5/4πβ B cos(γ B + 2π 3 )], (25)
c = R 0 [1 + 5/4πβ B cos(γ B )]. ( 26
)
so that in first order Bohr's parameters β B , γ B are equal to Hill and Wheeler's parameters β H , γ H . Recently, to describe precisely the nuclear ground-state shapes a "stretched" representation of the perturbed-spheroid parametrization has been introduced [20,21]. The stretched coordinates ξ, η and ζ are defined by

ξ = mω 0 h [1 - 2 3 ε 2 cos(γ + 2 3 π)] x, ( 27
)
η = mω 0 h [1 - 2 3 ε 2 cos(γ - 2 3 π)] y, ( 28
)
ζ = mω 0 h [1 - 2 3 ε 2 cos(γ)] z. ( 29
)
and it follows the definitions of a stretched radius vector, a stretched polar angle and a stretched azimuthal angle.

Large deformations

The Lawrence shapes defined, in cylindrical coordinates, by the quartic ρ 2 = az 4 + bz 2 + c has first been used to describe both the spherical, spheroidal constricted, scission and separated shapes [START_REF] Hasse | Geometrical Relationships of Macroscopic Nuclear Physics[END_REF]22]. These shapes have been generalized to asymmetric shapes [23] and given by

ρ 2 = R 3 0 λ[z 2 0 -(z + z s ) 2 ][z 2 | z 2 | +(z + z s -z 1 ) 2 ]. ( 30
)
z 0 is the half-length, z 1 is the asymmetry parameter, z 1 = 0 for symmetric shapes. z 2 is the constriction or necking-in parameter and z s is introduced to fix the centre of mass at z = 0. λ guarantees volume conservation. The particular cases: sphere, spheroids, scission and separated spheres occur respectively when

z 0 = R 0 and z 2 → ∞, z 0 = R 0 and z 2 → ∞, z 2 = 0, z 2 < 0. Using the dimensionless parameters ζ i = z i /R 0 the geometric characteristics follow Q = 8 5 πζ 5 0 λ[ 1 7 ζ 2 0 + 1 3 ζ 2 2 - 1 3 λ( 1 21 ζ 4 0 + 2 7 ζ 2 0 ζ 2 2 + ζ 4 2 )], (31)
I = ζ 5 0 λ( 1 21 ζ 4 0 + 2 7 ζ 2 0 ζ 2 2 + ζ 4 2 ), (32)
I ⊥ = 1 2 [I + ζ 5 0 λ( 3 7 ζ 2 0 + ζ 2 2 )]. (33)
These shapes lead to rather elongated configurations with shallow necks and the scission configuration of two tangent spheres is not reproduced.

To describe fusion, scattering of two nuclei or compact fission, different matched surfaces of revolution have been proposed. The two sphere parameterization for overlapping and separated fragments [24,25,26] is defined as

ρ 2 = r 2 -(z -d) 2 f or z > 0, (34)
d is the distance from the centres of the separated or overlapping spheres to the origin. r is the maximal transverse distance. Assuming volume conservation the radius of the two spheres is R = 2 -1/3 R 0 . The parameter ε = 1d/r is used. ε = 1 corresponds to a single sphere, 0 < ε < 1 to two overlapping spheres, ε = 0 to two tangent spheres and ε < 0 to two separated spheres.

To study the exchange of nucleons between two fragments or nuclei the dumbbell parameterization has been defined in adding a cylindrical neck between two portions of a sphere [26,27]. This shape sequence describes the two touching sphere configuration but the derivatives of the curve are not continuous at the contact point of the cylinder and spheres.

The most flexible parameterization is the matched quadratic surfaces [START_REF] Hasse | Geometrical Relationships of Macroscopic Nuclear Physics[END_REF]20,21,28,29]. Three smoothly joined portions of quadratic surfaces of revolution are used : spheres, spheroids, hyperboloids, cylinders and cones. They are completely specified by

ρ 2 = a 2 1 - a 2 1 c 2 1 (z -l 1 ) 2 , l 1 -c 1 ≤ z ≤ z 1 , ( 35
)
ρ 2 = a 2 2 - a 2 2 c 2 2 (z -l 2 ) 2 , z 2 ≤ z ≤ l 2 + c 2 , ( 36
)
ρ 2 = a 2 3 - a 2 3 c 2 3 (z -l 3 ) 2 , z 1 ≤ z ≤ z 2 . ( 37
)
The left-hand surface is denoted by 1, the right-hand one by 2 and the middle one by 3. Each surface is specified by the position of its center l i , its transverse semiaxis a i , and its semi-symmetry axis c i . At the left and right intersections of the middle surface with the end surfaces the value of z is z 1 and z 2 respectively. After matching the surfaces and their derivatives it remains 6 parameters:

σ 1 = (l 2 -l 1 )/u, σ 2 = a 2 3 /c 2 3 , σ 3 = 0.5[a 2 1 /c 2 1 + a 2 2 /c 2 2 ], (38)
α 1 = 0.5(l 1 + l 2 )/u, α 2 = (a 2 1 -a 2 2 )/u 2 , α 3 = a 2 1 /c 2 1 -a 2 2 /c 2 2 , ( 39
)
where

u = 0.5(a 2 1 + a 2 2 ). ( 40
)
A subset of the matched quadratic surfaces parameterization restricted to two spheres smoothly connected by a quadratic surface has also been employed [30].

A deformed two-centre shell model [31] has been developed mainly built from two ellipsoidally deformed Nilsson-type oscillators for axially symmetric shapes. The selected shape consists of two overlapping or not ellipsoids and a sphere rolling around the symmetry axis and being tangent to the two ellipsoids. Four geometric parameters define the shape: the two ratios of the ellipsoid semiaxes, the neck sphere radius and the distance between the fragment centres.

Recently [32] small and large deformations have been investigated within a Fourier expansion of deformed shapes

ρ 2 = R 2 0 ∞ n=1 [a 2n cos( (2n -1)π 2 z -z sh z 0 ) + a 2n+1 sin(nπ z -z sh z 0 )]. (41)
The left and right ends of the shape are located at z min = z shz 0 and z max = z sh + z 0 . The shift parameter z sh allows to replace the centre of the nuclear shape at the origin of the coordinate system. The parameters a 2 , a 3 and a 4 describe the quadrupole, octupole and hexadecapole deformations or, equivalently, the elongation, reflection asymmetry and neck degree of freedom.

It is worthwhile to note that even though the approach within Hartree-Fock-Bogoliubov theory and Gogny forces [33] is very different from the liquid drop model approach the idea is always to minimize the energy of the system via all the possible deformations and, mainly, the Q 2 , Q 3 and Q 4 multipolar moments.

To describe ternary fission or quasi-molecular ternary excited states several configurations have been investigated. A restricted family of three colinearly aligned or triangle-like configurations has been defined but no simple formulas are available for the parameterization or for the liquid drop energies [34]. A shape consisting of three identical and aligned intersected spherical fragments or nuclei has also been used. The distance between the centers of the side fragments is the only degree of freedom, once the parent is fixed [35].

The ellipsoidal and two separated ellipsoid shapes, the elliptic lemniscatoids [36], the cassinian ovals [37,38], the ternary shapes, the toroids [7,39] and the bubbles [8,39] that we have also used or defined will be more detailed in the following sections.

Ellipsoidal shapes

In polar coordinates the axially symmetric ellipsoids are defined by

1/R(θ) 2 = sin 2 θ/a 2 + cos 2 θ/c 2 . ( 42
)
a is the perpendicular semi-axis, c being half the elongation on the revolution axis. The relation a 2 c = R 3 0 ensures the volume conservation. R 0 is the radius of the sphere (see figure 1).

The related eccentricity is e 2 = 1s 2 for the prolate shape while for the oblate case e 2 = 1s -2 [START_REF] Chandrasekhar | Ellipsoidal figures of equilibrium[END_REF][START_REF] Schultheis | [END_REF]42]. For the prolate shape, the surface function is

B s = (1 -e 2 ) 1/3 2 1 + sin -1 (e) e(1 -e 2 ) 1/2 (43)
and for the oblate shape

B s = (1 + 2 ) 1/3 2 1 + ln( + (1 + 2 ) 1/2 ) (1 + 2 ) 1/2 2 = s 2 -1. ( 44
)
The relative Coulomb function reads in the prolate case

B c = (1 -e 2 ) 1/3 2e ln 1 + e 1 -e (45)
and for the oblate shape

B c = (1 + 2 ) 1/3 tan -1 . ( 46
)
For the prolate ellipsoidal shapes, I and Q are given by

I ⊥ = s -4/3 + s 2/3 2 , ( 47
)
I = s 2/3 , ( 48
)
Q = 8π 15 (s -4/3 -s 2/3 ). ( 49
)

Two separated ellipsoid shapes

For separated binary systems it may be necessary to introduce the ellipsoidal deformations (see figure 2) of the two objects (drops, nuclei, planets, galaxies,...). The Coulomb (or gravitational) interaction energy between two coaxial ellipsoids is given by [START_REF] Chandrasekhar | Ellipsoidal figures of equilibrium[END_REF][START_REF] Schultheis | [END_REF]43,44,45]

E c,int (r) = Q 1 Q 2 r [s(λ 1 ) + s(λ 2 ) -1 + S(λ 1 , λ 2 )] λ i 2 = c i 2 -a i 2 r 2 . ( 50
)
In the prolate case, s(λ i ) is

s(λ i ) = 3 4 1 λ i - 1 λ i 3 ln 1 + λ i 1 -λ i + 3 2λ i 2 , (51)
and, for the oblate configurations,

s(λ i ) = 3 2 1 ω i + 1 ω i 3 tan -1 ω i - 3 2ω i 2 ω i 2 = -λ i 2 . ( 52
)
S(λ 1 , λ 2 ) is expressed by a two-fold summation S(λ 1 , λ 2 ) = ∞ j=1 ∞ k=1 3 (2j + 1)(2j + 3) × 3 (2k + 1)(2k + 3) × (2j + 2k)! (2j)!(2k)! λ 1 2j λ 2 2k . ( 53
)

Elliptic lemniscatoids and asymmetric binary quasimolecules

Elliptic lemniscatoids

The elliptic lemniscatoid shapes allows to describe continuously the transition from to two tangent equal spheres to one sphere or vice-versa via compact and creviced shapes with almost spherical ends, assuming volume conservation. In nuclear physics this shape sequence has been used to simulate, as a first approximation, the symmetric fusion and fission [5,36].

Mathematically, the lemniscatoid is the first pedal surface of a prolate ellipsoid and the inverse of an oblate ellipsoid, the prolate ellipsoid being the reciprocal of the oblate ellipsoid (see figure 3).

The prolate ellipsoidal shape is expressed as

x 2 /a 2 + y 2 /a 2 + z 2 /c 2 = 1 ( 54
)
while its reciprocal oblate ellipsoid reads as

a 2 x 2 + a 2 y 2 + c 2 z 2 = 1. ( 55
)
The elliptic lemniscatoid is given by

a 2 x 2 + a 2 y 2 + c 2 z 2 = (x 2 + y 2 + z 2 ) 2 . ( 56
)
The ratio s = a/c of the neck radius to the half-elongation of the system defines

L E 1 E 2 H M M O Figure 3.
Relation between the elliptic lemniscatoid L and the ellipsoids E 1 and E 2 . The point H of the lemniscatoid L is the projection of the point O onto the tangential plane to the prolate ellipsoid E 1 in M. The point M' of the oblate ellipsoid E 2 is the inverse of H. The revolution axis is the horizontal axis.

completely the shape. When s increases from 0 to 1 the lemniscatoid varies continuously from two tangent spheres to a single one (see figure 4). When the perpendicular x axis is taken as axis of revolution the elliptic lemniscates generate pumpkin-like configurations (see section 9)).

In polar coordinates, the elliptic lemniscatoids are simply given by

R(θ) 2 = a 2 sin 2 θ + c 2 cos 2 θ. ( 57
)
The volume and surface area follow

V ol = 4 3 πR 0 3 = π 12 c 3 4 + 6s 2 + 3s 4 √ 1 -s 2 sinh -1 2 s 2 √ 1 -s 2 (58)
and

S = 4πR 0 2 B s = 2πc 2 1 + s 4 √ 1 -s 4 sinh -1 1 s 2 √ 1 -s 4 , ( 59
)
R 0 being the radius of the spherical object. r, the distance between the mass centres of the right and left parts of the object, is defined by

r = 2 c 0 zd 3 r c 0 d 3 r . ( 60
)
For these elliptic lemniscatoids r is simply given by

r = πc 4 1 + s 2 + s 4 3V . ( 61
)
The perpendicular and parallel moments of inertia (relatively to the moment of inertia of the equivalent sphere 2 5 mR 0 2 ) are

I ⊥,rel = c 5 s 2 512(1-s 2 )R 0 5 112 s 2 + 8 + 30s 2 -135s 4 + 120s 4 -135s 6 √ 1-s 2 sinh -1 1-s 2 s 2 , ( 62
)
I ,rel = c 5 s 2 512(1-s 2 )R 0 5 32 s 2 + 48 + 100s 2 -210s 4 + 240s 4 -210s 6 √ 1-s 2 sinh -1 1-s 2 s 2 . ( 63
)
The dimensionless quadrupole moment Q is

Q = πc 5 s 2 96(1 -s 2 )R 0 5 ⎡ ⎣ 16 s 2 -8 -14s 2 + 15s 4 - 24s 4 -15s 6 √ 1 -s 2 sinh -1 ⎛ ⎝ 1 -s 2 s 2 ⎞ ⎠ ⎤ ⎦ . ( 64
)
Figure 4. On the left, evolution of the elliptic lemniscatoids versus the ratio s = a/c. The shape varies continuously from two spheres (s = 0) to one sphere (s = 1). The neck exists for s < 0.5 √ 2. On the right, two halves of different lemniscatoids of same perpendicular distance a are linked in the transverse plane.

Asymmetric binary quasimolecules

To describe the path from one sphere to two different spheres in contact or vice versa (see figure 4) two halves of different elliptic lemniscatoids may be connected in the transverse plane assuming the same transversal radius. These shape sequence allows the study of the alpha emission or capture and cluster radioactivity [9,10,11,46], the asymmetric fusion [3,4] and the asymmetric fission through compact and creviced shapes [6].

Mathematically the shape is defined by

R(θ) 2 = a 2 sin 2 θ + c 1 2 cos 2 θ 0 ≤ θ ≤ π/2 a 2 sin 2 θ + c 2 2 cos 2 θ π/2 ≤ θ ≤ π. ( 65
)
c 1 and c 2 are the two half-elongations on the z axis. The relation R 0

3 = R 1 3 + R 2 3 ensured the volume conservation, R 0 , R 1 and R 2
being the radii of the total and different spheres. The two parameters s 1 = a/c 1 and s 2 = a/c 2 define completely the quasimolecular shape. The two radii R 1 and R 2 allows to link s 1 and s 2 :

s 2 2 = s 1 2 s 1 2 + (1 -s 1 2 )(R 2 /R 1 ) 2 . ( 66
)
When s 1 increases from 0 to 1 the shape varies from to two touching unequal spheres to one sphere. The volume conservation of the two parts is assumed. The separation plane is at a distance z v from the origin. r 1 and r 2 are calculated from the quantity z v , solution of the equation

1 3 z 3 - 1 2 a 2 z + 1 12 (2c 2 3 + 3a 2 c 2 ) + (67) 1 2 c 2 2 -a 2 D 2 sinh -1 c 2 D -D 2 sinh -1 z D -z √ z 2 + D 2 = 4 3 R 2 3 with D 2 = a 4 /4(c 2 2 -a 2 ).
The positions r 1 and r 2 are then obtained by

r 1 = 1 4 3 R 1 3 z v 4 -a 2 z v 2 4 + c 1 4 + a 2 c 1 2 + a 4 12 - ( 68
)
as 2 2 3(1 -s 2 2 ) ⎡ ⎣ z v 2 (1 -s 2 2 ) s 2 2 + a 2 4 3/2 - a 3 8 ⎤ ⎦ ⎫ ⎬ ⎭ and r 2 = 1 4 3 R 2 3 z v 4 -a 2 z v 2 4 - a 4 4 1 -s 2 2 s 2 4 + ( 69
)
as 2 2 3(1 -s 2 2 ) ⎡ ⎣ a 3 1 s 2 2 - 1 2 3 - z v 2 (1 -s 2 2 ) s 2 2 + a 2 4 3/2 ⎤ ⎦ ⎫ ⎬ ⎭ .
The distance r between the mass centres of the right and left parts of the physical object is r = r 1 + r 2 .

In nuclear physics the search for superheavy nuclei and new very asymmetric decay has renewed interest in the understanding and description of the alpha decay and cluster radioactivity. Since the pioneering work of Viola and Seaborg [47] several models have been developed and regularly improved [48,49,50,[START_REF] Poenaru | Handbook of Nuclear Properties[END_REF][START_REF] Poenaru | Nuclear Decay Modes[END_REF][START_REF] Parkhomenko | [END_REF]54,55,56,57,58,59,[START_REF] Poenaru | Clusters in Nuclei[END_REF]61,62,63,64,65,66,67,68,69,70,71,72,73,74,75] and analytic expressions have been proposed to calculate rapidly the alpha and cluster decay halflives. Comparisons between these models and their predictive capacities may be found in Ref. [68,69,73,74].

Hyperbolic lemniscatoids

The shape of a viscous object at the scission point depends strongly on the mean free path of its constituants and its viscosity. The scission may occur through compact and creviced shapes or through shallow and elongated shapes. This last deformation valley can be simulated within hyperbolic lemniscatoids (Cassinian ovaloids) [37,38].

For the one-body configurations the hyperbolic lemniscatoids are expressed as

x 2 = -z 2 + 0.5c 2 (s 2 -1) + 0.5c 8(1 -s 2 )z 2 + c 2 (1 + s 2 ) 2 . ( 70
)
For the two-body shapes the ovals are defined by

x 2 = -z 2 -0.5c 2 (s 2 + 1) + 0.5c 8(1 + s 2 )z 2 + c 2 (1 -s 2 ) 2 . ( 71
)
These shapes are one-parameter dependent when the volume conservation is assumed.

For the one-body shapes the ratio s = a/c of the minor and major axes can still be used.

For the two-body configurations, the opposite s of the ratio of the distance between the tips of the fragments and the system elongation has been selected (see figure 5 and [38]). When s evolves from -1 to 1 the shapes varies from two infinitely separated and identical spherical objects to one sphere. For the one-body shape the volume of the system is

V ol = πc 3 12 ⎡ ⎣ -2 + 6s 2 + 3 (1 + s 2 ) 2 2(1 -s 2 ) sinh -1 ⎛ ⎝ 2 2(1 -s 2 ) 1 + s 2 ⎞ ⎠ ⎤ ⎦ ( 72
)
and for the two-body shapes

V ol = πc 3 12 ⎡ ⎣ -2(1 + s) 3 + 3 (1 -s 2 ) 2 2(1 + s 2 ) sinh -1 ⎛ ⎝ 2(1 + s) 2(1 + s 2 ) (1 -s) 2 ⎞ ⎠ ⎤ ⎦ . ( 73
)
For the one-body shapes, the surface function reads

B s = c 2 4R 0 2 × ⎡ ⎣ 4(1 + s 2 ) + 2 2(1 + s 2 ) 1 -s 2 s 2 F sin -1 √ 1 -s 2 , 1 √ 1 + s 2 (74) -2(1 + s 2 ) 2(1 + s 2 ) 1 -s 2 E sin -1 √ 1 -s 2 , 1 √ 1 + s 2 ⎤ ⎦ ,
E and F being incomplete elliptic integrals. B s is calculated numerically for the twobody configuration. The distance r between the centres of the right and left parts is for the one and two-body shapes

r = c 4 8R 0 3 × (1 + 4s 2 + s 4 ), (75)
r = c 4 8R 0 3 × (1 -s 2 ) 3 1 + s 2 . ( 76
)
For the one-body shapes the parallel and perpendicular moments of inertia are given by

I = c 5 512(1 -s 2 )R 0 5 × (77) 147 -27s 2 -15s 4 -225s 6 -15(1+s 2 ) 2 (15-34s 2 +15s 4 ) 2 √ 2(1-s 2 ) sinh -1 2 √ 2(1-s 2 ) 1+s 2 , I ⊥ = c 5 1024(1 -s 2 )R 0 5 × (78) 269 + 251s 2 -145s 4 -255s 6 -15(1+s 2 ) 2 (17-30s 2 +17s 4 ) 2 √ 2(1-s 2 ) sinh -1 2 √ 2(1-s 2 ) 1+s 2 .
The quadrupole moment is defined by

Q = πc 5 192(1 -s 2 )R 0 5 × ( 79
) ⎡ ⎣ -5 + 61s 2 -23s 4 + 39s 6 + 3(1 + s 2 ) 2 (13 -38s 2 + 13s 4 ) 2 2(1 -s 2 ) sinh -1 ⎛ ⎝ 2 2(1 -s 2 ) 1 + s 2 ⎞ ⎠ ⎤ ⎦ .
For the two-body shapes these quantities are expressed as

I = c 5 512(1 + s 2 )R 0 5 × 147 + 225s + 27s 2 -15s 3 -15s 4 + 27s 5 + 225s 6 + 147s 7 (80) - 15(1 -s 2 ) 2 (15 + 34s 2 + 15s 4 ) 2 2(1 + s 2 ) sinh -1 ⎛ ⎝ 2(1 + s) 2(1 + s 2 ) (1 -s) 2 ⎞ ⎠ ⎤ ⎦ , I ⊥ = c 5 1024(1 + s 2 )R 0 5 × 269 + 255s -251s 2 -145s 3 -145s 4 -251s 5 + 255s 6 (81) +269s 7 - 15(1 -s 2 ) 2 (17 + 30s 2 + 17s 4 ) 2 2(1 + s 2 ) sinh -1 ⎛ ⎝ 2(1 + s) 2(1 + s 2 ) (1 -s) 2 ⎞ ⎠ ⎤ ⎦ , Q = πc 5 192(1 -s 2 )R 0 5 × -5 -39s -61s 2 -23s 3 -23s 4 -61s 5 -39s 6 -5s 7 (82) + 3(1 -s 2 ) 2 (13 + 38s 2 + 13s 4 ) 2 2(1 + s 2 ) sinh -1 ⎛ ⎝ 2(1 + s) 2(1 + s 2 ) (1 -s) 2 ⎞ ⎠ ⎤ ⎦ .

Comparison between the elliptic lemniscatoid and hyperbolic lemniscatoid shape valleys

The elliptic lemniscatoid shape sequence describes the rapid transition from one sphere to two tangent spheres or vice versa and to the rapid formation and rupture of a neck in one-body systems while keeping almost spherical ends or the rapid coalescence of almost spherical separated systems. The hyperbolic lemniscatoid shape sequence describes a deformation where the elongation occurs before the constriction.

The quadrupole moment and the perpendicular moment of inertia which are the main available geometrical experimental data are similar in the two paths. In contrast the parallel and effective moment of inertia are quite different in the two valleys (see figure 6 and [38]) but their values are difficult to obtain experimentally (via the angular distribution of the fragments). So it is not easy from the experimental data to know the real path investigated in nuclear reactions at low energies. The different possible shapes taken by a fissioning nucleus were firstly explore using a development of the radius in terms of Legendre polynomials, thinking that the fission process is only governed by the balance between the repulsive Coulomb forces and the attractive surface tension forces. This method leads naturally to smooth elongated one-body configurations resembling to hyperbolic lemniscatoids. Later on, it appears that, within liquid drop models, to reproduce the fusion data and the alpha and cluster radioactivities one must introduce an additional term, the so-called proximity energy, to take into account the strong nuclear interaction between nucleons of the surfaces in regard between almost spherical nuclei or in a deep crevice in one-body shapes. This proximity energy is evidently very small for elongated shapes with shallow necks and, then, it is often neglected. In contrast this term is essential to describe the fusion or fission through compact and creviced shapes. It has been shown [5,38] that it exists a degeneracy in energy between these two deformation valleys. These two paths implicitly seem to correspond to different values of the nuclear viscosity (diabaticity or adiabaticity ?) and then different dynamics and different degrees of thermalization. The dependence of the nuclear viscosity on the temperature, angular momentum, isospin,... obtained from fission experiments may be found in [76].

Prolate symmetric and asymmetric ternary shapes

Prolate symmetric ternary shapes

Assuming volume conservation, prolate symmetric ternary shapes varying from three aligned tangent identical spheres to one sphere or vice versa can be described from elliptic lemniscatoids. Such configurations may simulate several excited molecular states of 12 C formed from three α particles during the stellar nucleosynthesis. To define these ternary quasimolecular shapes, the cartesian coordinates are preferable to the polar coordinates since the symmetry plane does not correspond to the neck planes.

In the first quadrant the equation defining these shapes is (see figure 7 and [77])

x 2 = 0.5 a 2 -2(z -d) 2 + a 4 + 4(z -d) 2 (c 2 -a 2 ) . ( 83
)
a and c are respectively the neck radius and half the elongation of the corresponding binary shape. The parameter s = a/c is always sufficient to define the shape. d is the distance between the transverse x axis and the position of the crevice, namely

d = ⎧ ⎨ ⎩ 0.5c 1-2s 2 1-s 2 for 0 ≤ s < 0.5 √ 2 0 f o r0 .5 √ 2 ≤ s ≤ 1. (84)
The maximal perpendicular distance is given by

h max = 0.5c/(1 -s 2 ) 1/2 when 0 ≤ s < 0.5 √ 2 a when 0.5 √ 2 ≤ s ≤ 1. ( 85
)
The volume is given by

V ol = 4 3 πR 0 3 = πc 3 12 4 + 6s 2 + g(α) + 3s 4 √ 1 -s 2 ln 2 -s 2 + 2 √ 1 -s 2 h(α) , ( 86
)
R 0 being the radius of the total sphere, α = d/c and

g(α) = 6α + 6αs 2 -8α 3 for 0 ≤ s < 0.5 √ 2 0 f o r0 .5 √ 2 ≤ s ≤ 1 (87)
and

h(α) = -2α √ 1 -s 2 + 1 -s 2 when 0 ≤ s ≤ 0.5 √ 2 s 2 when 0.5 √ 2 ≤ s ≤ 1. ( 88
)
The dimensionless surface function is

B s = ⎧ ⎪ ⎨ ⎪ ⎩ c 2 2R 2 0 1 + √ 1-2s 2 2 + s 4 √ 1-s 4 ln √ 2(1+ √ 1-s 4 ) √ 1-s 2 - √ (1+s 2 )(1-2s 2 ) for 0 ≤ s ≤ 0.5 √ 2 c 2 2R 2 0 1 + s 4 √ 1-s 4 ln 1+ √ 1-s 4 s 2 for 0.5 √ 2 ≤ s < 1.
(89)

The distance between the right and left parts of the system is expressed as

r = ⎧ ⎨ ⎩ c 2α + πc 3 (11-8s 2 )
48V (1-s 2 ) 2 when 0 ≤ s < 0.5 √ 2

πc 4
3V (1 + s 2 + s 4 ) when 0.5

√ 2 ≤ s ≤ 1. ( 90
)
When the three spheres are separated, the total length of the shape is

l = 1.5r + 1.625R 1 , ( 91
)
the radius of a small sphere being R 1 = 3 -1/3 R 0 .

Figure 7. On the left, prolate and symmetric ternary quasimolecular shapes. The shape varies from three touching equal and aligned spheres (s = 0) to one sphere (s = 1). The neck disappears when s > √ 2/2. On the right, quasimolecular shapes varying from one sphere to two equal spheres in contact with a central smaller sphere. The ratio between the radii is 0.5.

Prolate asymmetric ternary shapes

To describe more asymmetric ternary configurations (see figure 7) and [78,79]) from the asymmetric binary quasimolecular shapes (see section 5) it is sufficient to cut the smallest fragment at its maximal perpendicular distance by a plane perpendicular to the axis of revolution and to introduce a mirror symmetry. Then, the two parameters s 1 = a/c 1 and s 2 = a/c 2 define completely the shape mathematically. a is the neck radius. c 1 and c 2 are respectively the elongations of the external and central fragments. The one sphere configuration correspond to s 1 = s 2 = 1. For s 1 = s 2 = 0 two equal spheres of radius R 1 touch a central smaller sphere of radius R 2 . s 1 and s 2 may be linked by the expression

s 2 2 = s 1 2 s 1 2 + (1 -s 1 2 )(R 2 /R 1 ) 2 (92)
and consequently

c 2 2 = c 1 2 [s 1 2 + (1 -s 1 2 )(R 2 /R 1 ) 2 ]. (93)
In the first quadrant the shape is defined by

x 2 = -(z -d) 2 + 0.5s i 2 c i 2 + 0.5c i 4(1 -s i 2 )(z -d) 2 + s i 4 c i 2 (94)
(i = 1 for z > d and i = 2 for z < d).

The volume is given by

V ol = πc 1 3 12 4 + 6s 1 2 + 3s 1 4 √ 1 -s 1 2 sinh -1 2 √ 1 -s 1 2 s 1 2 (95) + πc 2 3 12 6α + 6αs 2 2 -8α 3 + 3s 2 4 √ 1 -s 2 2 sinh -1 2α √ 1 -s 2 2 s 2 2
where α = d/c 2 .

The surface S reads

S = 2πc 1 2 1 + s 1 4 √ 1 -s 1 4 sinh -1 √ 1 -s 1 4 s 1 2 (96) +2πc 2 2 ⎡ ⎣ α 1 -s 2 2 + s 2 4 √ 1 -s 2 4 sinh -1 ⎛ ⎝ α 2(1 -s 2 4 ) s 2 2 ⎞ ⎠ ⎤ ⎦ .
The distance between the mass centres of the right and left parts is

r = πc 1 4 3V (1 + s 1 2 + s 1 4 ) + c 2 2α + πc 2 3 α 4 3V . -5 + 8s 2 2 + 16s 2 6 -16s 2 8 (1 -2s 2 2 ) 2 . ( 97
)
When the three spheres are separated :

B s = 2 + (R 2 /R 1 ) 2 (2 + (R 2 /R 1 ) 3 ) 2/3 (98) r = 3 2 + (R 2 /R 1 ) 3 (R 2 /R 1 ) 4 R 1 4 + 4 3 D . ( 99
)
2D is the distance between the centres of the two equal external spheres.

More exotic shapes

Experimentally, very distorted configurations have been observed in nuclear physics, astrophysics... and some of them can be roughly approximated by toroidal shapes and bubbles. For example, in heavy-ion reactions at Fermi energies, complicated shapes intermediate between toroids and bubbles have been indirectly detected [80,81]. These systems are often transient states formed after a violent collision and evolving towards pumpkin-like shapes and, later on, tori and n-spheres emitted in the same plane or evolving to bubbles splitting in the whole space.

A pumpkin-like shape may be generated by elliptic lemniscates (see figure 4) in taking the vertical axis as axis of revolution (see figure 8). The dimensionless parameter s = a/c is always sufficient to characterize the shape. When s diminishes from 1 to 0, an hollow progressively appears. The shape varies from a sphere to a ring torus for which the upper and lower hollows are connected together. Later on, this initial configuration evolves to a torus with large radius r t . The dimensionless parameter s t allows to follow this evolution [7,8] s t = (r tr s )/2r s , (100) r s and r t being the sausage and torus radii.

Pumpkin-like shapes

The volume, moment of inertia, mean square radius and surface function are given by (109)

V ol = 4πR 0 3 3 = 4πc 3 3 ⎡ ⎣ s 3 4 + 3 8 ⎛ ⎝ s + sin -1 √ 1 -s 2 √ 1 -s 2 ⎞ ⎠ ⎤ ⎦ , ( 101
)
I ⊥,rel = 3c 5 2R 0 5 (1 -s 2 ) ⎛ ⎝ -

Bubbles

Bubbles of thick skin are often formed within violent reactions where the out of equilibrium effects play an essential role. Calculations within bubbles of constant density can give a first rough approach of the more complex reality [8]. Assuming volume conservation, the bubble characteristics can be calculated from the ratio p = r 1 /r 2 of the inner and outer radii r 1 and r 2 .

V ol = 4πR 0 3 3 = 4π 3 (r 2 3r 1 3 ), ( 110)

r 1 = R 0 p(1 -p 3 ) -1/3 , ( 111
)
r 2 = R 0 (1 -p 3 ) -1/3 . ( 112
)
The relative root mean square radius reads

Summary

In a first part, most of the geometrical shapes defined to model the nuclear states and reactions (alpha decay, cluster radioactivity, fusion, fission and fragmentation) are briefly reviewed. Later on, the shape sequences that we have used previously, mainly formed from generalized lemniscatoids, are more extensively detailed. Their geometric characteristics are given: root mean square radius, volume, surface and Coulomb energies as well as their moments of inertia and quadrupole moments. These shapes allow to describe the transitions from one compact configuration to several ones or vice versa and may be perhaps useful also to describe physical objects such as drops, droplets and fluids in chemistry, mechanics, astrophysics, hydrodynamics,...

Figure 1 .

 1 Figure 1.Ellipsoidal shapes versus the ratio s = a/c. The common revolution axis is the horizontal axis. s is respectively: 1/3, 2/3, 1, 4/3 and 5/3. The ellipsoid is oblate when s > 1 and prolate when s < 1.

Figure 2 .

 2 Figure 2. Two coaxial ellipsoid shapes.

Figure 5 .

 5 Figure 5. Cassinian ovals versus the parameter s. At the scission point, the configuration is the Bernoulli lemniscate. The development of a neck begins when s < √ 3/3.

Figure 6 .

 6 Figure 6. Relative parallel, perpendicular and inverse effective moments of inertia and quadrupole moment for the hyperbolic (path I) and elliptic (path II) lemniscatoid shape paths.

Figure 8 .

 8 Figure 8. Pumpkin-like configuration versus the ratio s of the semi-axes and torus defined by the sausage r s and torus r t radii. The axis of revolution is the vertical axis.

1 / 3 .

 13

= ( 1 -

 1 p 5 ) 1/2 (1p 3 ) -5/6 . (113)The surface, Coulomb and moment of inertia functions areB s = 1 + p 2 (1p 3 ) 2/3 , (114) B c = 1 -2.5p 3 + 1.5p 5 (1p 3 ) 5/3 , (115)I ⊥,rel = (1p 5 )(1p 3 ) -5/3 . (116)


Céline Rosticher

Bruno Viana

Guillaume Laurent

Patrick Le Griel

Corinne Chanéac
email: corinne.chaneac@upmc.fr

] C Rosticher

Insight into CaMgSi2O6:Eu2+,Mn2+,Dy3+ Nanoprobes: Influence of Chemical Composition and Crystallinity on Persistent Red Luminescence

come   L'archive ouverte pluridisciplinaire

Introduction

At the nanoscale, luminescent solids are suitable for in vivo imaging that implies some problems due to the biological environment, such as the tissues autofluorescence and absorption [1][2]. Therefore we need to develop luminescent probes that emit in the red-near infrared range: in the biological tissues wavelength therapeutic window (600-1350 nm). Indeed, biological tissues absorb a great range of UV and visible light and only a red-near infrared emission can go through the tissues [2][3]. Three main groups of luminescent inorganic nanomaterials are currently reported. Two of them are related to (i) semiconductor nanocrystals with direct or indirect gap or metal nanocrystals [4][5], and (ii) metal clusters [6][7][8] for which optical properties originate from quantum confinement effect and electronic transitions between occupied d bands and states above the Fermi level. In this paper, we focus on a third group of compounds for which the luminescence originates from the presence of luminescent centers (dopants) in the solid structure [9][10][11][12]. In this case, the luminescence closely depends on matrix or ligand-dopant association and can be adjusted to specific applications (telecommunications, biosciences, solar energy conversion) [13]. Such materials are usually prepared using conventional solid state chemistry [14][15][16][17]. More recently, promising novel luminescent nanoparticles based on lanthanide or transition metal dopedoxide composition have been synthesized combining sol-gel and hydrothermal syntheses, opening new field of research in the domain of the in vivo optical imaging systems and new imaging tools [18][19][20]. These materials are more difficult to investigate as one first required to obtain such matrices at the nanometric scale with acceptable intensity of fluorescence [21][22][23].

A first generation of inorganic luminescent nanoparticles (PLNPs), Ca0.2Zn0.9Mg0.9Si2O6 doped Eu2+, Mn2+, Dy3+, was developed by Le Masne de Chermont et al. [START_REF] Le Masne De Chermont | Proc. Natl. Acad. Sci[END_REF]. These new long lasting luminescent nanoparticles were excitable with UV light and emitted during several minutes in the red-near infrared range, therefore avoiding the tissues absorption during the light detection. Indeed PLNPs were excited before the injection in the animal that allows to prevent the tissues autofluorescence and to improve the signal-to-noise ratio. The development of these probes is initially based on the red persistent luminescence of MgSiO3 bulk compound, described by Wang et al. [START_REF] Wang | [END_REF] in which the emission corresponds to the Mn2+ radiative 4T1-6A1 transition. More recently a second generation with optimized rare earth cation (RE= Ce, Nd, Dy, Pr, Ho, and Er were evaluated) [26] and a mechanism has been proposed to explain the persistent luminescence of these PLNPs where RE3+ and oxygen vacancies appears as electron traps, and Mn2+ is the recombination luminescent center (with emission in the red range at 680 nm) [27]. This allowed the validation of a simpler chemical composition for diopside structure [26,28]. In these silicate-based families, there are still open questions on the influence of the chemical composition and of the crystalline structure on the persistent luminescence intensity.

In order to obtain nanoprobes, a "soft chemistry" synthesis (sol-gel method and acid catalysis) using molecular precursors in a silica matrix was used [START_REF] Le Masne De Chermont | Proc. Natl. Acad. Sci[END_REF]. It consists in growing luminescent nanocrystals inside a silica matrix which plays the role of a silica reservoir to obtain a doped silicate and limits the crystal growth during thermal treatment at high temperature, mandatory to crystallize a pyroxene phase. However by this technique probe sizes were still too high for an in vivo imaging application. Furthermore the marked aggregation of particles and their low luminescence intensity represent another important disadvantage for their in vivo application.

In the present work, persistent luminescence nanoparticles were synthesized using a sol-gel method inspired by Stöber process. This method enables to obtain silica nanoparticles instead of a silica matrix and thus to have a better control of the crystal growth using these nanoreactors. Core-shell type nanoparticles with a crystalline silicate core and an amorphous silica shell are obtained. Such process implies an efficient step of leaching of particles surface to dissolve the residual silica layer. In the aim of improving the persistent luminescence and studying the chemical composition influence on the persistent luminescence properties, we modified Ca/Mg/Si molar ratio. In addition, to compare our different results, the native Ca0.2Zn0.9Mg0.9Si2O6 doped Eu2+, Mn2+, Dy3+ host is used as reference and RE=Dy is kept for sake of comparison [START_REF] Le Masne De Chermont | Proc. Natl. Acad. Sci[END_REF].

Results and Discussion

Influence of chemical composition and crystallinity on persistent luminescence. XRD data presented on figure1.a show a strong evolution of crystalline structure from diopside phase for C1M1S2 composition to clinoenstatite phase corresponding to Mg2Si2O6 composition with the decrease of Ca/Mg molar ratio. The normalized persistent luminescence spectra at room temperature recorded after switching off the UV excitation (120 sec at 365 nm) exhibit three emission bands for all compounds: at 468, 600 and 670 nm (figure 1.b). The band at 468 nm is attributed to the allowed 4f65d1→4f7 transition of Eu2+, whereas the bands at 600 and 670 nm correspond to the transition of Mn2+ in Ca2+ and Mg2+ sites respectively. No characteristic Dy3+ emission has been observed. At low temperatures (>300K) as reported elsewhere Eu2+ emission can also be observed but the persistent luminescence occurs from the Mn2+ center which is at room temperature the recombination center.

With the increase of Ca/Mg molar ratio, considering a calibrated measurement with the same amount of compound, luminescence intensity of the three bands decreases and the relative intensities of the two bands at 600 nm and at 670 nm are inversed for lower calcium content.

Variation in Ca/Mg molar ratio has therefore an influence both on crystalline structure of particles and on Mn2+ distribution in the two cationic sites. Consequently, the persistent luminescence intensity is enhanced when promoting the emission band at 670 nm observed for the lowest calcium content. Concerning the influence of crystalline structure, such silicate compounds belong to the pyroxenes group for which the general formula is M2M1(SiO3)2, with the small cation M1 surrounded by 6 oxygen atoms and the larger M2 surrounded by 8 oxygen atoms. Small ions, like Mg2+, Fe2+ or Zn2+ always occupy M1 sites while larger ions like Ca2+ or Na+, and rare earth necessarily occupy M2 cationic sites [29,30].

CaMgSi2O6 composition follows the general structure of the diopside with a monoclinic symmetry. When small ions are the only ions in the matrix, they also occupy M2 site and form an orthopyroxene phase (enstatite Mg2Si2O6) exhibiting an orthorhombic symmetry. In figure 1.a, the phase modification from diopside to enstatite is clearly detected from a calcium substoichiometry greater than 20 percent. Below this calcium content, clinoenstatite remains the main crystalline phase. In previous study enstatite has been reported to be more efficient than diopside structure for optimized luminescent properties in the case of dysprosium doping [26]. Indeed, dysprosium in enstatite acts as a better electron trap according to the conduction band position between diopside and enstatite structures compared to depth of electron trap related to dysprosium. The crystallization of low calcium silicate composition into enstatite structure then leads to an improved LLP according to figure1.b.

In such crystalline structures, as the ionic radius of Mn2+ (in six-fold coordination) is 0.83 Å whereas the radii of Mg2+ and Ca2+ are 0.72 Å and 1.00 Å respectively [31], Ca2+ ions can occupy only the M2 sites while Mg2+ and Mn2+ cations are distributed in both sites, M1 and M2, as demonstrated by optical spectroscopy [32]. Therefore, the decrease in the amount of calcium contributes to vary the host cationic sites for Mn2+ in the crystalline structure. The persistent luminescence spectra clearly show that the quantity of Mn2+ luminescent centers in Mg2+ sites increases with the decrease of Ca/Mg molar ratio according to the relative intensities of Mn(Ca) and Mn(Mg) at 600 nm and 670 nm respectively (figure 1b). The substitution of Mg2+ ions by Mn2+ ions inside M1 sites is statically favorable when Mg2+ ions are also distributed into M2 sites. As Mn2+ in Mg2+ site is the luminescent recombination center in the persistent luminescence mechanism; regarding CaMgSi2O6 type of compounds [33], an enhancement of the red afterglow is obtained for Ca0.3Mg1.7Si2O6 composition. Nevertheless, Ca2+ inside the structure is required since a drastic decrease of LLP is observed for Mg2Si2O6 composition, probably related to a high band gap modification. Indeed, the shift of the bottom of the conduction band has been recently reported for Al doped ZnGaO4 phosphors [34].

Interestingly, the LLP also increases with the doping rate without modification of Mn2+ site distribution as shown on LLP decay curves of Ca0.6Mg1.4Si2O6 sample for two different doping rate, namely Eu(0.5%), Mn(2.5%), Dy(1%) called "doped C1" and Eu1% Mn5% Dy2% called "doped C2" on figure 2.a. The loss of persistent luminescence yield versus time appears less important for higher doping rate, and this leads to a significant intensity difference after one-hour term. EPR results (figure 2.b) confirmed that Mn2+ content is two times increased from doped C1 to doped C2 samples. EPR spectra exhibit a fine structure in the central part corresponding to well-ordered Mn2+ and a weaker structure in the wings [31] characteristic of Mn2+ cations in interaction.

For the two doping rates, the shape of the spectra is similar indicating that the Mn2+ distribution in crystalline structure is the same and there is no evidence of aggregate for highest content.

The influence of initial concentration of TEOS was then investigated for low Ca/Mg ratio and for both pyroxene based stoichiometric silica content (C0.4M1.6S2) and with an excess in silica content (C0.4M1.6S4). When the initial concentration of TEOS is higher than the one required to obtain the stoichiometry, XRD diagrams show only a poorly crystallized clinoenstatite phase (figure 3.a). However, when the initial concentration of silica is just enough to obtain pyroxene phase (C0.4M1.6S2), XRD diagram shows also the presence of diopside phase. For this silicate, 30% additional calcium in the Mg2Si2O6 initial formula are sufficient to obtain the two phases: clinoenstatite and diopside as previously reported [30]. With the increase of silica content, the nucleation of clinoenstatite can be explained considering the better diffusion of Mg2+ ions inside the silica glass during the thermal crystallization compared to bigger Ca2+ ions which probably remain trapped in the silica matrix. For lower silica content, the proximity of Ca2+ and Mg2+ ions in the matrix promotes the crystallization of the diopside phase, since the ions diffusion distance for the nucleation of this phase is shorter. Persistent luminescence spectra of series B (figure 3.b) showed the same three characteristic emission bands (480 nm, 580 nm, 680 nm) than for A serie. When the initial concentration of silicon is higher than the one required to obtain the stoichiometry, persistent luminescence intensity increases due to better crystallinity of compound. This suggests that thermal treatment need to be optimized for the different concentrations.

Such study has been done for the Ca0.3Mg1.7Si3O8 doped (C2 composition). After drying at 90°C for 30h, samples were annealed from 900°C to 1400°C for 10 h under reductive atmosphere. XRD (figure 4.a) show that diopside phase is obtained for a thermal treatment at 1150°C, an annealing temperature higher than the one required for the first generation of compounds [START_REF] Le Masne De Chermont | Proc. Natl. Acad. Sci[END_REF]. This result well agrees with a higher initial concentration of TEOS and a more limited diffusion of ions. At this temperature, the amorphous silica ratio, deduced from 29Si quantitative NMR spectra, is the lowest, agreeing with a better crystallization of the compound (table 1). Since the particle size depends on thermal treatment temperature, the NPs size was determined here by Debye-Scherrer method. An increase of the size with the increase of the temperature was previously observed [START_REF] Le Masne De Chermont | Proc. Natl. Acad. Sci[END_REF]. This increase was not observed when the nanoparticles are synthetized by a sol-gel method inspired by Stöber process (table 1).

Table 1. Amorphous silica ratio of Ca0.3Mg1.7Si3O8: Eu1%Mn5%Dy2% compound depend on thermal treatment temperature. Amorphous and crystalline silica contents have been deduced from integration of NMR peaks between -210 ppm and -120 ppm and between -70ppm and -95 ppm respectively. Particle sizes have been deduced from XRD patterns using Debye-Scherrer method. The particles have 50 nm in diameter whatever the thermal treatment temperature in the range of 900-1150°C. This is probably related to the presence of amorphous silica shell which prevents the sintering of nanoparticles. According to the crystallization degree, the sample annealed at 900°C is not persistent whereas the one annealed at 1150°C exhibit more intense emission and persistent luminescence (Figure 4.b). The persistent luminescence spectrum for Ca0.3Mg1.7Si3O8: Eu1% Mn5% Dy2% shows only emission from one crystallographic site Mn2+Mg, leading to an emission centered at 680 nm. This point is very important as we aim to develop red/near infrared nanosensors for bioimaging applications. CaF2 and MgF2 phases. This means that this treatment is too strong and destructive and leads to both silica shell and crystalline core dissolution. The sample which present the closest crystallization to the raw material is the sample treated with NH5F2 0.5M. This treatment seems to be less aggressive for the silicate nanoparticles.

As can be seen on the TEM images of figure 6, a thick layer of amorphous silica is still visible for the sample treated with NaOH 2M (see table 2). This decrease of the amount of silica is confirmed by EDX analysis while the Ca / Mg ratio remains the same (see table S.I.1). With intense for that sample. A fundamental chemical analysis confirmed that the sample treated with NH5F2 0.5M indeed contains less silicon (table 2). The treatment with NH5F2 0.5M seems to be really the most efficient treatment and produces spherical silicates nanoparticles coated by a thin (2.7 nm) layer of amorphous silica without any change on chemical composition of crystalline core (table S.I.1). The homogeneity of both chemical composition and grain size of as treated particles is effective as reported in supplementary information (table S.I.2 and figure S.I.1). EDX analysis on small zones noted the low variability in chemical composition from zone to zone (< 2% of variability) and that the resulting mean composition is in very good accordance with that of larger zone. In addition, SEM images show regular and homogeneous grain size for both as-prepared and treated sample suggesting that the step of silica dissolution did not affect crystalline particles.

The yield of usable small nanoparticles for in vivo optical imaging after etching increased from 15% for a treatment with NaOH 2M up to 65% for a treatment with NH5F2 0.5M.

Therefore a great improvement was made in terms of silica etching to obtain small nanoparticles with satisfactory persistent luminescence properties. This lead to a better dispersion of nanoparticles in suspension as reported to figure S.I.2. The persistent luminescence spectra and persistent decay curves within 30 min of Ca0.3Mg1.7Si3O8: Eu1% Mn5% Dy2% compound (figure 7) shows that persistent luminescence is more intense for the raw material before etching.

This could be attributed to some surface effects when the layer of amorphous silica is thinner

Conclusions

CaxMgySi2O6 doped Eu2+, Mn2+, Dy3+ nanoparticles were prepared by a sol-gel method inspired by Stöber process in order to develop red-near infrared persistent luminescence nanoparticles as sensors for optical imaging. In the aim of improving the persistent luminescence and studying the chemical composition influence on the persistent luminescence properties, materials with various Ca/Mg/Si molar ratio have been prepared.

Composition that favors the enstatite structure with good distribution of Mn2+ into the two coordinated sites of this structure exhibits an enhancement of long lasting phosphorescence in red-near infrared range. Crystallinity can also be optimized adjusting temperature of heating treatment. Nanoparticles with 50 nm size were treated in NH5F2 0.5M aqueous solution after thermal treatment in order to reduce the thickness of the amorphous silica coating. Acton SpectraPro 2150i spectrometer for spectral analysis. The resolution of the emission is about 10 nm (/a = 10 nm/mm) according to the experimental spectrophotometer (f=125mm, 300tr/mm blazed at 500nm). In order to compare the LLP spectra, a normalization of their intensities was done systematically. Electron paramagnetic resonance (EPR) measurements were performed with a Bruker Elexsys E500 spectrometer equipped with a Super-High-Q cavity and working at a microwave frequency of 9.4 GHz. The spectra were recorded at room temperature with a microwave power of 2 mW and a modulation field of 1 mT.

Silica shell dissolution has been studied using solid state 29Si Nuclear Magnetic Resonance (NMR). Spectra were recorded with a standard 7 mm double resonance probe on a 7 T Bruker Avance III spectrometer operating at 300 MHz for 1H and 59 MHz for 29Si. Spinning speed was set to 5 kHz and sample was packed into a 7 mm zirconia rotor. The used pulse sequence was a single 1.5 µs π/6 pulse with 45 kHz 1H decoupling during acquisition. A relaxation delay of 60 s was applied without decoupling.

Figure 1 .

 1 Figure 1. (a) XRD patterns and (b) LLP spectra of CxMyS2O6: Eu0.5% Mn2.5% Dy1% samples corresponding to series A

Figure 2 .

 2 Figure 2. (a) Persistent luminescence decay curves (exc = 365 nm, texc = 120s) and (b) EPR spectra of Ca0.6Mg1.4Si2O6 doped C1 and doped C2.

Figure 3 .

 3 Figure 3. (a) XRD patterns of serie B with various silica contents (arrows point out diopside phase) and (b) corresponding LLP spectra.

Figure 4 .

 4 Figure 4. (a) XRD patterns and (b) LLp spectra of C0.3M1.7S3O8: Eu1% Mn5% Dy2% compound depending on thermal treatment temperature

NH4F 10 M

 10 , spherical nanoparticles coated by a less thick layer of amorphous silica are obtained. The amorphous coating is the thinnest after a treatment of the sample with NH5F2 0.5M. Spherical and monodisperse nanoparticles coated by a layer of amorphous silica of 2.7 nm are then obtained.

Figure 5 .

 5 Figure 5. (a) XRD patterns and (b) NMR 29Si spectra of C0.3M1.7S3O8: Eu1% Mn5% Dy2% composition (T = 1150°C) as raw material and depending on post-annealing treatment: NaOH 2M, NH4F 10M and NH5F2 0.5M.

Figure 6 .

 6 Figure 6. TEM images of C0.3M1.7S3O8: Eu1% Mn5% Dy2% compound depending on post-annealing treatment: (a) NaOH 2M, (b) NH4F 10M and (c) NH5F2 0.5M.

  and almost removed. Indeed silica layer play a protective role preventing surface related nonradiative phenomena. Comparing the surface treatments, it appears that the sample treated with NH5F2 0.5M is the most luminescent and persistent whereas the sample treated with NH4F 10M exhibits less intense persistent luminescence. The surface treatment with NH5F2 0.5M indeed is the most efficient treatment because it reduces the aggregation of the NPs, produces spherical silicates nanoparticles coated by a thin layer of amorphous silica that still have good persistent luminescence properties.

Figure 7 .

 7 Figure 7. (a) Persistent luminescence spectra of C0.3M1.7S3O9: Eu1% Mn5% Dy2% depending on post-annealing treatment: NaOH 2M, NH4F 10M and NH5F2 0.5M and (b) luminescence decay curves (exc = 365 nmtexc = 120s).

  Silica leaching. As obtained PLNPs are coated by a layer of amorphous silica and are strongly aggregated. Samples have been thus treated in order to reduce the thickness of the amorphous silica coating using the following procedure. PLNPs powder is added in an aqueous solution of NaOH 2M, NH4F 10M or NH5F2 0.5M and stirred at room temperature for 24h with NaOH and 1h for NH4F and NH5F2. The product is then thoroughly washed with distilled water and dried under air flux. Structural and optical characterizations. Powder X-ray diffraction (XRD) study was performed on a Bruker D8 Advance diffractometer using Cu-K radiation ( = 1.5418 Å) at 45 kV and 40 mA, 0.0046°step size and 0.5s step time over a range of 10°2 to 80°2. Transmission electron microscopy (TEM) images were acquired with a FEI Tecnai 120 Twin microscope operating at 120 kV and equipped with a high resolution Gatan Orius CCD 4 k 64 k numeric camera. Excitation and emission spectra were measured on powdered samples with a Varian Cary Eclipse Fluorescence spectrophotometer at room temperature. Long lasting phosphorescence (LLP) spectra were recorded after 2 minutes irradiation with UV light, and the resulting signal was collected via an optical fiber by Roper Scientific Pixis 100 CCD camera cooled at -68°C coupled with an

Table 2 .

 2 Thickness

	Post-	Thickness of	Amorphous	Total
	annealing	amorphous silica	silica ratio	silicon
	treatment	shell		ratio
	As prepared	/	/	30.0%
	NaOH 2M	20 nm	37.3%	29.3%
	NH4F 10M	16 nm	47%	26.8%
	NH5F2 0.5M	2.7 nm	22%	24.7%

of amorphous silica layer depending on post-annealing treatment: NaOH 2M, NH4F 10M and NH5F2 0.5M. Content of silica shell compared to total silicon.

Acknowledgements

This work was supported by the French National Research Agency (ANR) in the frame of its Nanosciences and Nanotechnologies program (NATLURIM-ANR-08-NANO-025). Keywords: Diopside silicate, nanoprobes, Persistent luminescence, Long Lasting Phosphorescence, Optical Imaging

Supplementary Information

New insight out CaMgSi2O6:Eu 2+ , Mn 2+ , Dy 3+ nanoprobes: Influence of chemical composition and crystallinity on persistent red luminescence Céline Rosticher, [1] Bruno Viana, [2] Guillaume Laurent, [1] Patrick Le Griel, [1] Corinne Chanéac,* [1] [a] C. 5%, Dy 3+ 2% compound using NH5F2, deduced from EDX analysis on ten small zones, SZ, (mapping mode: local chemical composition on 500nm x 500nm surface) and on two larger zones, LZ, (average chemical composition on 30µm x 30µm surface).