M N Chernodub

Zero-point fluctuations in rotation: Perpetuum mobile of the fourth kind without energy transfer *

In this talk we discuss a simple Casimir-type device for which the rotational energy reaches its global minimum when the device rotates about a certain axis rather than remains static. This unusual property is a direct consequence of the fact that the moment of inertia of zero-point vacuum fluctuations is a negative quantity (the rotational vacuum effect). Moreover, the device does not produce any work despite the fact that its equilibrium ground state corresponds to a permanent rotation. Counterintuitively, the device has no internally moving mechanical parts while its very existence is consistent with the laws of thermodynamics. We point out that such devices may possibly be constructed using carbon nanotubes subjected to the strong -but experimentally feasible -magnetic field. The effect in the carbon nanotubes is enhanced due to the presence of massless charged excitations. We call this "zero-point-driven" device as the perpetuum mobile of the fourth kind.

I. INTRODUCTION

Perpetual motion: Motion that continues indefinitely without any external source of energy; impossible in practice because of friction.

Webster's online dictionary [START_REF]Webster's online dictionary, definition of perpetual motion[END_REF] In the quoted Webster's definition, the moving object is assumed to be coupled to (or, in other words, interacting with) an external environment, otherwise the perpetual nature of the object's motion is a trivial and inevitable consequence of the energy conservation law. Due to the interaction with the environment, a moving (say, rotating) object will sooner or later release all its (rotational) kinetic energy to the surrounding media. The gradual loss of the rotational energy can be described by the precense of a friction force acting on the object from the environment. As a consequence, the object will ultimately stop its rotational motion coming to rest; hence the verdict "impossibility in practice" in the Webster's definition.

In this talk, following Refs. [START_REF] Chernodub | [END_REF], we point out, that the perpetual motion may exist. The device, which we call "Perpetuum mobile of the fourth kind" [START_REF] Chernodub | [END_REF], is suggested to have the following physical properties:

• It is a large object made of the Avogadro-scale (N ∼ 10 23 ) constituent particles.

• It is a solid body which, counterintuitively, has no internally moving parts.

• It is made of a material, which possesses propagating (preferably, massless) elementary excitations.

• Its lowest energy state corresponds to a uniform rotation of the device as a whole about a certain axis with a certain optimal angular frequency (a non-rotating device has a higher energy compared to its energy in the optimally rotating state).

• The unusual non-monotonic dependence of the rotational energy on the angular frequency are due to zero-point (vacuum) fluctuations of the elementary excitations.

Thus, we define the perpetuum mobile of the fourth kind as a solid macroscopic device for which the lowest-energy state corresponds to a uniform rotation driven by the zero-point fluctuations without any energy transfer.

The energy properties of our perpetuum mobile are very counterintuitive, because in order to stop the rotation of the perpetuum mobile we should invest (not gain!) some amount of work in this process because the cessation of rotation increases (not decreases) the rotational kinetic energy of the device. In an elementary act of interaction of the rotating device with an external environment (say, with a molecule of a gas), the device • cannot loose energy and stop rotating because the rotating device is already residing in its lowest-possible energy state;

• can gain energy and stop rotating.

Below, for the sake of brevity, we will sometimes call the perpetuum mobile of the fourth kind as "the device". An alternative "time crystal" [3,4] approach to the permanent motion was inspired by the persistency of the electric currents in superconducting rings (or, alternatively, one can think of the persistently flowing superfluid Helium in a closed ring-shaped pipe). Due to specific properties of the relevant excitation spectra, the energy exchange between the superconducing (superfluid) and dissipative components of the corresponding systems is absent, so that the electric current (superfluid flow) is running forever. Mathematically, the "time crystal" is an exotic classical, semiclassical or quantum-mechanical system of moving particles which is described by either singular Hamiltonians involving multiple powers of time derivatives [3], or by Hamiltonians which involve time-nonlocal terms [4]. This suggestion is both controversial [5] and inspiring. Recently, a very interesting "space-time crystal" system was suggested to be realised at cryogenic temperatures in a form a permanently flowing current of cold ions [6].

Contrary to the "time crystal" approach, our alternative approach -which we call "the perpetuum mobile of the fourth kind" [START_REF] Chernodub | [END_REF] -is based on standard nonsingular class of local free (!) Hamiltonians of relatively simple solid state systems (and may, presumably, be realised using doped torus-shaped carbon nanotubes [START_REF] Chernodub | [END_REF]).

II. NEGATIVE MOMENT OF INERTIA OF ZERO-POINT FLUCTUATIONS

Let us assume for a moment that the perpetuum mobile of the fourth kind (described in the Introduction) does exist: this large macroscopic device rotates permanently in its ground (lowest) state without producing any work. The necessary condition of the permanent rotation is that the non-rotating state should have a higher energy compared to the optimally rotating state. What should be a driving mechanism behind this unusual energy behaviour? In the present Section we are sketching the answer to this question.

Let us consider a large macroscopic rigid body of an arbitrary shape which rotates about one of its a principal axis of inertia with the angular frequency Ω. For the sake of simplicity let us consider a slow, nonrelativistic rotation. Then the classical kinetic energy of rotation is a quadratic function of the angular frequency:

E cl (Ω) = I cl Ω 2 2 , (1)
where I cl is the corresponding classical principal moment of inertia. The energetically favourable state, E cl rot = 0, corresponds to the absence of rotation, Ω = 0. Thus, the classical formula (1) seems to imply that that the permanent rotation should not, in general, correspond to a lowest energy state. However, there is a hidden assumption which has led us to this mundane conclusion: We have implicitly assumed that the moment of inertia of the rigid body is a positive quantity:

I cl > 0 .
(

) 2
Can we question the validity of Eq. (2)? At the first sight it seems that the answer is "no" as the positiveness condition (2) is a very natural fact for any classical system. For example, consider a point mass m rotating about a certain axis at the fixed distance R (one can imagine that the point is attached to the axis by a massless rod). The rotational kinetic energy of the system is given by Eq. [START_REF]Webster's online dictionary, definition of perpetual motion[END_REF], where the moment of inertia is

I cl = mR 2 .
(

) 3
The classical mass m and the distance R are positive quantities so that the moment of inertia (3) is, naturally, a positive quantity too [START_REF] Chernodub | [END_REF]. Since any classical body can be considered as a superposition of point masses, the positivity condition (2) for all extended classical bodies is also an obvious fact.

Mathematically, the moment of inertia (3) can be a negative quantity if and only if the mass m is a negative quantity (the distance squared, R 2 , is always a positive quantity). Physically, however, the total mass of any (classical or quantum) object is a positive quantity so that the condition m < 0 is definitely unnatural. Thus, it seems that we have again arrived to the fact that Eq. ( 2) is valid.

Nevertheless, in Refs. [START_REF] Chernodub | [END_REF] we claim that there may exist certain systems which have a negative moment of inertia for, at least, small angular frequencies. These systems contain, generally, two interacting (classical and quantum) subsystems. The classical subsystem is a rigid body which has a positive moment of inertia. The quantum subsystem is described by zero-point fluctuations of quantum excitations which have, in general, a negative moment of inertia (the latter is the essence of the "rotational vacuum effect" [START_REF] Chernodub | [END_REF]). Although the total mass of the device is a positive quantity, the total moment of inertia may be a negative quantity because the quantum subsystem may couple to the rotation stronger compared to its classical counterpart.

FIG. 1. Left:

The toy device which shows that zero-point vacuum fluctuations have a negative moment of inertia. A free massless scalar field may propagate along the infinitesimally thin ring, while the cut of the ring imposes a Dirichlet boundary condition on this field (from Ref. [START_REF] Chernodub | [END_REF]). Right: in the magnetic field background B the total (classical plus zero-point) rotational energy of the ring E = E(Ω) becomes a non-monotonic discontinuous function of the angular frequency Ω. In this example the minimum of the rotational energy [with E(Ω ch ) < E(0)] is reached at the nonzero characteristic frequency Ω ch = 0, Eq. ( 27); see the discussion in the text.

In order to illustrate our idea in general, and the rotational vacuum effect in particular, we consider a very simple device; see Fig. 1

(left). This toy device contains:

A. Classical subsystem:

• We consider an infinitely thin (but, generally, massive) ring of the radius R.

• The ring may rotate about its natural symmetry axis.

• The ring has a cut.

B. Quantum subsystem:

• The ring is made of a material which possesses massless (scalar) excitations.

• The excitations are free (they do not interact with each other).

• The excitations may propagate along the ring.

• The excitations cannot propagate through the cut of the ring.

C. The classical and quantum subsystems are coupled to each other thanks to the cut.

For the sake of simplicity, we assume that the cut of the ring is infinitesimally thin while the tunnelling of excitations from one side of the cut to another side is absent.

Our next logical steps are as follows:

1. It is easy to show that the zero-point (vacuum) fluctuations of the excitations in the ring have a negative energy (called, in general, the Casimir energy [7]).

2. It turns out that zero-point fluctuations do indeed have a negative moment of inertia ("the rotational vacuum effect") as was rigorously demonstrated in Refs. [START_REF] Chernodub | [END_REF] and confirmed in Ref. [8].

3. The negative moment of inertia implies that the rotational energy of the zero-point vacuum fluctuations decreases as the angular frequency Ω of the device increases.

4. However, the rotational vacuum effect -in its simplest formulation -is very small [START_REF] Chernodub | [END_REF]: the (negative) zero-point moment of inertia constitutes a small fraction of the (positive) classical moment of inertia of the device itself.

5. Nevertheless, the rotational vacuum effect may be enhanced if (a) the scalar excitations in the ring are electrically charged; and

(b) the ring is pierced by an external magnetic flux.

The enhancement mechanism may work in doped torus-shaped nanotubes [START_REF] Chernodub | [END_REF].

6. Due to the enhancement, the negative zero-point rotational energy may overcome the positive classical rotational energy for certain angular frequencies. Then, the lowest energy state should correspond to a permanently rotating device [START_REF] Chernodub | [END_REF].

Below we will briefly discuss the rotational vacuum effect as well as the possible existence of the perpetuum mobile of the fourth kind following Refs. [START_REF] Chernodub | [END_REF].

III. THE ROTATIONAL VACUUM EFFECT IN THE VACUUM OF NEUTRAL SCALAR FIELDS

Consider a neutral massless scalar field φ = φ(t, ϕ) which is defined on a circle (infinitesimally thin ring) of radius R; see Fig. 1(left). The corresponding Lagrangian is:

L = 1 2 ∂ µ φ ∂ µ φ ≡ 1 2 ∂φ ∂t 2 - 1 R 2 ∂φ ∂ϕ 2 . ( 4
)
The scalar field is a periodic function [φ(t, ϕ + 2π) ≡ φ(t, ϕ)] of the angular coordinate ϕ ∈ [0, 2π). The cut of the ring -see Fig. 1(left) -imposes the simplest, Dirichlet boundary condition on the neutral field φ (we call the cut as the "Dirichlet cut").

First, let us consider a static, non-rotating circle. The boundary condition at the Dirichlet cut is as follows (the points ϕ = 0 and ϕ = 2π are identified):

φ(t, ϕ) ϕ=0 ≡ φ(t, ϕ) ϕ=2π = 0 . (5)
It is pretty straightforward to evaluate the zero-point (Casimir) energy associated with the vacuum fluctuations of the scalar field φ. The local energy density,

Ē(R) = 1 2πR modes m ε m (R) , (6)
is given by the infinite sum over energies ε m (R) of all individual fluctuation modes m. This sum is a divergent quantity both for the circle with a finite radius R and in a free space (R → ∞). However, the difference between these energy densities, Ēphys (R) = Ē(R) -Ē(∞), is a finite, experimentally measurable observable which is often called the Casimir energy density [7]. The total energy of the quantum fluctuations is as follows:

E phys (R) = 2πR Ēphys (R) ≡ 2πR Ē(R) -Ē(∞) , (7)
The local energy density of the quantum fluctuations E is given by the vacuum expectation value of a time-time component of the stress-energy tensor, T µν :

E(t, ϕ) ≡ T 00 (t, ϕ) = lim t →t lim ϕ →ϕ 1 2i ∂ ∂t ∂ ∂t + 1 R 2 ∂ ∂ϕ ∂ ∂ϕ G(t, t ; ϕ, ϕ ) , (8)
where we have expressed the expectation value of the stress-energy tensor T µν ,

T µν (x) = ∂ µ ∂ ν - 1 2 g µν ∂ λ ∂ λ 1 i G(x, x ) x→x , (9)
via a Feynman-type Green function [START_REF] Milton | The Casimir Effect: Physical Manifestations of Zero-Point Energy[END_REF]: G(x, x ) = i Tφ(x)φ(x ) . Here the symbol "T" stands for the time-ordering operator and x = (t, ϕ).

The Green function G(t, t ; ϕ, ϕ ) is defined by the following equation:

∂ 2 ∂t 2 - 1 R 2 ∂ 2 ∂ϕ 2 G(t, ϕ; t , ϕ )= 1 R δ(ϕ -ϕ )δ(t -t ) . (10)
The Green's function G should vanish at the position of the Dirichlet cut, ϕ, ϕ = 0, 2π. The Green function G(t, t ; ϕ, ϕ ) can be expressed via the eigenvalues λ ω,m and eigenfunctions φ ω,m of the second-order differential operator in the right hand side of Eq. ( 10):

∂ 2 ∂t 2 - 1 R 2 ∂ 2 ∂ϕ 2 φ ω,m (t, ϕ) = λ ω,m φ ω,m (t, ϕ) , (11)
where m = 1, 2, 3, . . . and ω ∈ R, and the orthonormal and complete eigensystem is

λ ω,m = m 2 4R 2 -ω 2 , φ ω,m (t, ϕ) = e -iωt χ m (ϕ) , χ m (ϕ) = 1 √ πR sin mϕ 2 . (12)
For the sake of convenience, the real-valued eigenfunctions of Eq. ( 11) with the same eigenvalues are combined into one complex function φ ≡ φ

(C) ω,m -iφ (S) ω,m .
The Green's function is given by the following formula,

G(t, t ; ϕ, ϕ ) = +∞ -∞ dω 2π ∞ m=1 φ ω,m (t, ϕ)φ † ω,m (t , ϕ ) λ ω,m -i = i π G ϕ 2 , ϕ 2 , |(t -t )| 2R , (13)
where function G is defined as follows:

G(x, y, z) = ∞ m=1 sin(mx) sin(my) m e -imz = 1 4 ln 1-e i(x+y-z) 1-e -i(x+y+z) [1-e i(x-y-z) ] [1-e i(-x+y-z) ] . (14)
The infinitesimal term i ensures that the Green function ( 13) is of the Feynman type [START_REF] Milton | The Casimir Effect: Physical Manifestations of Zero-Point Energy[END_REF].

Using the explicit representation of the Green's function ( 13) the (non-regularised and divergent) local energy density (8) can be computed straightforwardly:

T 00 (t, ϕ) = - 1 2π lim t →t 1 (t -t) 2 - 1 96πR 2 . ( 15
)
Notice that the energy density (15) depends on neither the angular variable ϕ nor time variable t. As the divergent piece in Eq. ( 15) does not depend on the radius of the ring, it is easy to get the regularised expression for the zero-point energy E phys and the mean energy density Ēphys of the scalar field at the non-rotating circle with the Dirichlet cut:

E phys ≡ 2πR Ēphys = - 1 48R , Ēphys = - 1 96πR 2 . ( 16
)
The energy vanishes in the limit of the infinitely large circle, R → ∞, as it should be. The result ( 16) is known as the Casimir energy of the string of the length l = 2πR [START_REF] Milton | The Casimir Effect: Physical Manifestations of Zero-Point Energy[END_REF][START_REF] Luscher | [END_REF].

In order to find the moment of inertia of the zero-point vacuum fluctuations let us repeat the above calculations for the case of the rotating ring, Fig. 1(left), which rotates uniformly about its axis with the angular velocity Ω. Mathematically, the rotation leads to the following time-dependent boundary condition imposed by the Dirichlet cut 1 :

φ(t, ϕ) ϕ=[Ωt] 2π = 0 , (17)
where [x] 2π = x + 2πn, n ∈ Z, 0 [x] 2π < 2π denotes the modulo operation with the base of 2π. In the static limit, Ω = 0, the boundary condition (17) reduces to Eq. ( 5). The Green function in the rotating case should satisfy the general equation ( 10) and respect the time-dependent boundary conditions (17). Similarly to Eq. ( 13) the Green function can be expressed via the orthonormal and complete eigensystem of the problem [START_REF] Milton | The Casimir Effect: Physical Manifestations of Zero-Point Energy[END_REF] and ( 17) in the laboratory frame:

φ m,ω (t, ϕ) = 1 πR sin m 2 [ϕ -tΩ] 2π exp -iω t - ΩR 2 [ϕ -tΩ] 2π 1 -Ω 2 R 2 , ( 18
)
λ ω,m = 1 -Ω 2 R 2 4R 2 m 2 - ω 2 1 -Ω 2 R 2 . ( 19
)
1 Another characteristics of the zero-point energy, the associated centripetal force, was considered in Ref. [9].

We do not calculate this quantity in our approach since the dynamics of the rotating body depends on other quantities, which are the rotational energy, moment of inertia and angular momentum.

The explicit form of the Green function of the rotating ring is as follows [START_REF] Chernodub | [END_REF]:

G Ω (t, t; ϕ, ϕ ) = i π G θ(t, ϕ) 2 , θ(t , ϕ ) 2 , |(1 -Ω 2 R 2 )(t -t ) -ΩR 2 [θ(t, ϕ) -θ(t , ϕ )]| 2R ,
where θ(t, ϕ) = [ϕ -Ωt] 2π and the function G(x, y, z) is given in Eq. ( 14). The corresponding zero-point energy density ( 8) is given by the following formula:

T 00 (t, ϕ) = - 1 2π lim t →t 1 (t -t) 2 - 1 + Ω 2 R 2 96πR 2 . ( 20
)
The energy density (20) of the rotating ring is space-time independent similarly to the energy density of the static ring [START_REF] Pendry | [END_REF]. The divergent piece does not depend on neither the radius of the ring R nor the angular frequency Ω so that it does not contribute to the physical part of the rotational energy density of zero-point (ZP) fluctuations:

E ZP Ω (t, ϕ) ≡ T 00 (t, ϕ) phys = - 1 + Ω 2 R 2 96πR 2 . ( 21
)
For a static device, Ω = 0, the energy density of the zero-point vacuum fluctuations (21) reduces to the known result (16). Finally, we arrive to the exact relativistic expression for the total zero-point energy of the rotating ring with the cut [START_REF] Chernodub | [END_REF]:

E ZP Ω ≡ R ˆ2π 0 dϕ E ZP Ω (t, ϕ) = - 1 + R 2 Ω 2 48R . (22)
Curiously, the Ω-dependent part of the rotational energy of the relativistic zero-point fluctuations (22) has a very simple form resembling the nonrelativistic classical expression (1). Result (22) was confirmed in Ref. [8] by using a different method.

The moment of inertia of the zero-point vacuum fluctuations is a negative quantity:

I ZP ≡ ∂ 2 ∂Ω 2 E ZP Ω = - R 24c , (23)
where we have restored the Planck constant and the speed of light c. The energy of the zero-point vacuum fluctuations decreases, as the angular frequency increases. Thus, the classical positivity condition (2) is not valid for the zero-point vacuum fluctuations (23). Notice that despite the fact that I ZP < 0, an isolated device cannot self-accelerate because of the conservation of the angular momentum (the device should exchanges the angular momentum with environment by, say, emitting or absorbing a photon).

Equation ( 23) provides us with the lowest bound on the moment of inertia for the ringtype devices [Fig. 1(left)] driven by the neutral scalar vacuum [8]. The zero-point moment of inertia ( 23) is an very small quantity, corresponding to a fractional (in terms of ) angular momentum [8]. Thus, the positive moment of inertia of a real massive device (3) is expected to be always greater than the negative moment of inertia of the zero-point fluctuations (23).

However, the negative moment of inertia can be drastically enhanced by an external magnetic field B if the massless particles are electrically charged [START_REF] Chernodub | [END_REF]. The corresponding zero-point energy E ZP Ω,B and the zero-point angular momentum L ZP Ω,B ,

E ZP Ω,B ≡ R ˆ2π 0 dϕ T 00 (t, ϕ) = -1 + 6M Ω,B (M Ω,B + 1) 1 + Ω 2 R 2 24R , (24)
L ZP Ω,B ≡ R ˆ2π 0 dϕ x 2 T 10 -x 1 T 20 = -1 + 6M Ω,B (M Ω,B + 1) ΩR 12 , (25)
are related to each other by a "classical" relation:

L ZP Ω,B = ∂E ZP Ω,B ∂Ω eBΩR 3 1-Ω 2 R 2 / ∈Z , where M Ω,B = eBΩR 3 1 -Ω 2 R 2 ∈ Z , (26)
provided the angular frequency Ω and the magnetic field B do not correspond to the discontinuities in the integer-valued quantity M Ω,B (here x is a largest integer which is smaller than x). In the limit of vanishing magnetic field the charged-field result (24) turns into the neutral-field expression (22) multiplied by a factor of two because the charged (complex) field has two degrees of freedom. In the nonrelativistic limit, ΩR c, the first discontinuity of the zero-point energy (24) happens at the characteristic frequency,

Ω ch (B) = c eBR 3 . ( 27
)
A global minimum of the total (classical plus zero-point) rotational energy is reached, generally, at a nonzero optimal frequency, Ω = n Ω ch with n ∈ Z (up to small relativistic corrections). The illustrative example of the total rotational energy, shown in Fig. 1(right), corresponds to Ω = ±Ω ch = 0; see Ref. [START_REF] Chernodub | [END_REF] for further details.

The magnetic field strongly enhances the zero-point energy (24) because the integer number M Ω,B may become very large. In Ref. [START_REF] Chernodub | [END_REF] a possible application of this approach to torus-shaped carbon nanotubes was discussed and it was shown that the enhancement factor may reach astronomically high values (10 15 and higher). To this end, it is enough to compute the enhanced rotational vacuum effect assuming the bosonic, and not fermionic, nature of the massless excitations, because in one spatial dimension the zero-point (Casimir) energies for a free massless scalar field and a free massless fermion field for certain boundary conditions are identical [12]. For a carbon nanotube of a finite diameter the fermionic nature of the massless excitations should affect certain features of the zero-point energy [13]. Indeed, the nanotube is a spatially two-dimensional system because it could be considered as a (twodimensional) graphene sheet -which possesses massless fermionic excitations [14] -rolled into a cylinder.

Notice, however, that in computing of Eq. ( 24) we have assumed that the charged particles are free, so that they may interact with the external magnetic field while the photon exchange between them is absent. The importance of the photon-mediated interaction is unclear for a moment.

Finally, we notice that the device may be made macroscopically large by assembling the open rings along their common axis in a metamaterial-like [START_REF] Pendry | [END_REF] fashion; see Fig. 2. FIG. 2. A macroscopic permanently rotating device is visually similar to the very first proposal of a metamaterial [START_REF] Pendry | [END_REF] made of the C-shaped split-ring resonators (from Ref. [START_REF] Chernodub | [END_REF]).

IV. PERPETUUM MOBILE OF THE FOURTH KIND AND THERMODYNAMICS

In this Section we briefly notice that the perpetuum mobile of the fourth kind does not violate the laws of thermodynamics (Ref. [START_REF] Chernodub | [END_REF] contains a more detailed discussion): I. The very existence of this device is obviously consistent with the first law of thermodynamics because no work is produced by the object which rotates in its lowest-energy ground state (otherwise it would be the perpetuum mobile of the first kind which is obviously forbidden by the energy conservation law).

II. Due to the same reason our device does not spontaneously convert thermal energy into mechanical work so that the second law is not violated as well (otherwise it would be the perpetuum mobile of the second kind which is forbidden too).

III. Again, due to the same reason this device cannot serve as a perpetual energy storage (thus our device is not the perpetuum mobile of the third kind).

IV. Our device is the perpetuum mobile of the fourth kind: it does not produce work and it cannot serve as a perpetual energy storage. Instead, it is a rigid (one-piece) macroscopic device which exhibits perpetual motion due to zero-point fluctuations.

Can a permanently rotating macroscopic body be in thermodynamic equilibrium with the external environment?2 In the equilibrium, the angular velocity Ω, angular momentum L, energy E and entropy S of the body are related to each other as follows [16]:

Ω = ∂E ∂L S . (28)
If the energy of the system is a smooth function of the angular momentum, E = E(L), then in the equilibrium the angular velocity of the body should always be zero, Ω = 0 (this statement works, obviously, at any temperature including the absolute zero). Thus, we arrive to the important conclusion [START_REF] Chernodub | [END_REF]:

the rotational energy of a permanently rotating device must be a (single-valued) discontinuous function of its angular momentum.

Remarkably, this property is always satisfied for our device due to the presence of the discontinuities in the zero-point contributions to the energy (24) and to the angular momentum (25); see Fig. 1(right) for illustration. Moreover, in Ref. [START_REF] Chernodub | [END_REF] we have explicitly demonstrated that for our device the thermodynamic relation (28) does indeed give rise to nonzero angular velocity at the equilibrium, Ω = 0. Thus, the existence of the perpetuum mobile of the fourth kind is consistent with the laws of thermodynamics.

V. CONCLUSIONS

We have demonstrated that zero-point vacuum fluctuations may possess a negative moment of inertia so that the rotational energy of the zero-point fluctuations decreases with the increase of the angular frequency ("the rotational vacuum effect" [START_REF] Chernodub | [END_REF]). The presence of a magnetic field background may drastically enhance the negative moment of inertia of zero-point fluctuations so that the negative rotational energy of the zero-point fluctuations may compensate the positive classical rotational energy of the device itself [START_REF] Chernodub | [END_REF]. In this case the device -which does not produce any work -becomes a perpetuum mobile of the fourth kind driven by the zero-point fluctuations. Technologically, such devices may be realised in carbon nanotube systems subjected to the strong, experimentally reachable, static timeindependent magnetic field.

The author thanks G. E. Volovik for rising this important question.

ACKNOWLEDGMENTS

The author is grateful to M. Asorey, E. Elizalde, K. Kirsten, K. Milton, M. Plyushchay, M. Schaden, D. Vassilevich and G. E. Volovik for interesting discussions. The work was supported by Grant No. ANR-10-JCJC-0408 HYPERMAG (France).


M Sorlin

D Stanoiu

Zs Sohler

S K Dombradi

B A Bogner

H Brown

J D Hergert

Holt

L Cáceres

A Lepailleur

O Sorlin

M Stanoiu

D Sohler

Zs Dombrádi

S K Bogner

B A Brown

H Hergert

J D Holt

A Schwenk

F Azaiez

B Bastin

C Borcea

R Borcea

C Bourgeois

Z Elekes

Zs Fülöp

S Grévy

L Gaudefroy

G F Grinyer

D Guillemaud-Mueller

F Ibrahim

A Kerek

A Krasznahorkay

M Lewitowicz

S M Lukyanov

J Mrázek

F Negoita

F De Oliveira

Yu.-E Penionzhkevich

Zs Podolyák

M G Porquet

F Rotaru

P Roussel-Chomaz

M G Saint-Laurent

H Savajols

G Sletten

J C Thomas

J Timàr

C Timis

Zs Vajta

Nuclear structure studies of 24 F

Keywords: numbers: 21.60.Cs, 23.20.Lv, 27.40.+z

I. INTRODUCTION

Nuclear forces play a decisive role in our understanding of the structure of atomic nuclei, driving the creation and evolution of shell gaps, the onset of deformation, development of halo structures, and determining the limits of particle stability. The nuclear shell model provides a framework to determine the properties of nuclei from a set of single-particle energies (SPEs) and two-body matrix elements (TBMEs) defined in a given valence space outside some assumed inert core. When based only on two-nucleon (NN) forces within the valence space, the SPEs and TBMEs need suitable renormalization to experimental data to provide a precise description of nuclear structure. These "effective" SPEs and TBMEs implicitly capture the effects of many-body processes, such as core polarization, as well as neglected three-nucleon (3N) forces [1][2][3][4] and coupling to the particle continuum.

As the standard shell-model approach typically uses TBMEs that are independent of mass number A or sim-ply scaled, it may become insufficient near the limits of stability, where the last nucleons are only loosely bound and have radial wavefunctions which extend to larger radii and couple to unbound states [5]. Reductions of the neutron-neutron TBME by 25% were required to model the structure of the neutron-rich C isotopes, which are a factor of two less bound than the O isotopes [6,7]. More recently, the study of the weakly bound 26 F nucleus, which can be viewed as an 24 O core plus a deeply bound d 5/2 proton and an unbound d 3/2 neutron, has shown that a reduction of the πd 5/2 -νd 3/2 TBME by about 20% better reproduced the energies of its J = 1 + , 2 + , 4 + states [8]. In addition, recent theoretical calculations for states close to the neutron-separation threshold show that an increased role of coupling to the particle continuum may in part account for the modification of shell structure of dripline nuclei [9,10].

The fluorine isotopic chain is well suited to study the evolution of valence-space interactions with increased valence neutron-to-proton asymmetry towards the dripline.

Moreover, many F isotopes are located near the doubly magic 16 O, 22 O and 24 O systems. Considering these O isotopes as almost inert cores, fluorine wavefunctions should then be weakly mixed, making possible the search for subtle effects related to their weak binding and proximity to the continuum. Located between 22 O and 24 O, 24 F is an excellent candidate for such a study, because its spectroscopy is expected to be relatively simple. As in 26 F, states located near the neutron separation energy, S n = 3.840 (10) MeV, could be influenced by effects arising from asymmetric proton-to-neutron binding.

In a simple shell-model picture, the lowest-lying states in 24 F with spin-parity 2 + and 3 + can be considered pure πd 5/2 ⊗ νs 1/2 configurations on top of an 22 O core, while 0 + and 1 + states are expected from πs 1/2 ⊗ νs 1/2 configurations. Close to the dripline, the excitation of one neutron to the d 3/2 orbits gives rise to the I π = 1 + -4 + multiplet due to the πd 5/2 ⊗ νd 3/2 coupling observed at low energy in 26 F [8]. Configurations originating from neutron core excitations are also found below the neutronseparation threshold.

While previous β-decay studies [11,12] agreed on the 24 O half-life (about 65 ms), deduced delayed-neutron emission probabilities differed significantly: P n = 58(12)% in Ref. [11] and P n = 18(6)% in Ref. [12]. From the latter work, it was expected that 82(6)% of the β strength would decay to bound states in 24 F [12]. Because only 60% of the β-decay strength was observed, it was proposed that this missing strength may feed higherlying 1 + excited state(s), likely of πd 5/2 ⊗ νd 3/2 origin, that could not be observed experimentally due to the lack of statistics. Three γ transitions associated with the decay of 24 O were observed [12], but the statistics were not sufficient to unambiguously establish a 24 F level scheme. No other spectroscopic information was known on 24 F before the present study.

To search for members of the d 5/2 -d 3/2 multiplet, the spectroscopy of 24 F has been studied using two complementary experimental methods in this work. First, 1 + states were accessed from the β decay of 24 O, which has an I π = 0 + ground state. In a separate experiment, higher-spin states were produced in the fragmentation of projectile nuclei leading to 24 F. The results were then compared to shell-model calculations based on the benchmark USDA and USDB empirical Hamiltonians [13] as well as ab initio valence-space Hamiltonians derived from NN+3N forces [14]. The excellent agreement between these two calculations results in a robust description of the newly measured states.

II. β-DECAY OF 24 O

A. Experimental set-up

The 24 O nucleus was produced via fragmentation of a 36 S 16+ primary beam delivered by the GANIL facility in a 237 mg/cm 2 Be target placed at the entrance of the LISE spectrometer [15]. The energy and the average intensity of the primary beam were 77.6 MeV/u and 2 eµA, respectively. The projectile-like fragments were separated by the LISE achromatic spectrometer. A 9 Be wedge-shaped degrader of 1066 mg/cm 2 was placed at the dispersive focal plane of LISE to improve the ion selection. As shown in Fig. 1 the selected nuclei were identified at the end of the spectrometer by means of their energy loss (∆E) in two silicon detectors of 500 µm thickness and their time-of-flight (TOF) referenced to the cyclotron radio-frequency. An Al foil of adjustable inclination was placed after the two Si detectors to allow the implantation depth of the 24 O ions into a 1 mm double-sided-silicon-strip detector (DSSSD) of 5×5 cm 2 with 16×16 strips. The pixels in the DSSSD were used to establish spatial and time correlations between the βparticles and the 24 O parent. Energy threshold of the individual strips were set to ∼80 keV. A 5 mm-thick Si detector was placed after the DSSSD to control the implantation depth of 24 O.

Four segmented Ge Clover detectors of the EXOGAM array [16] were placed around the DSSSD detector to provide βγ coincidences. A γ-ray efficiency ε γ of 6.5% at 1 MeV was extracted from the β-decay of 28 Ne which was transmitted in the same set-up and for which the intensities of the γ transitions were known [17]. For each implanted nucleus, a β efficiency (ε β ) of 63(3)% was extracted from the intensity ratio between an identified γ-line gated or ungated on the β-correlation condition. The relative γ-ray intensities were obtained from the ε β and ε γ values.

B. Results

The β-gated γ-ray spectrum following the implantation of a precursor 24 O nucleus between 0-250 ms is shown in Fig. 2 a). This time condition favors the observation of γ rays associated to the decay of 24 F while it suppresses all transitions belonging to contaminant nuclei or daughter activity. The previously reported γ-lines [12] at 521 keV, 1309 keV and 1830 keV are clearly visible. All other observed transitions are not attributed to the decay of 24 O.

A total of ∼10 5 24 O nuclei were implanted in the DSSSD detector, which is a factor of 10 larger than in Ref. [12]. The improved statistics obtained in the present work permitted a study of γγ coincidences (Fig. 2 b), from which it was deduced that the 1 + level at 1830 keV is fed with I β =57(4) % and decays by a cascade of 521 keV and 1309 keV γ-rays. The ordering of these two transitions could not be determined unambiguously from the β-decay study as they have the same relative intensity (Table I). The in-beam experiment presented in Section III B shows that the 1309 keV γ-ray feeds a level at 521 keV. The 1830 keV level decays by a competing branch directly to the ground state, as well. The γ-ray energies, relative intensities and branching ratios of the observed transitions are listed in Table I.

The background subtracted summed time-distribution of the 521 keV, 1309 keV and 1830 keV γ-rays with respect to the 24 O implantation is shown in Fig. 3. A fit with a single exponential decay-curve yields a half-life of T 1/2 =80(5) ms, in agreement with the 61 +32 -19 ms value of Ref. [11] within the statistical uncertainties but longer than the value of 65(5) ms reported in Ref. [12]. Using the β-decay Q-value and the absolute feeding intensity of the 1830 keV state a logf t value of 4.25 (6) has been obtained, which is consistent with an allowed Gamow-Teller transition.

The observational limit for the population of other 1 + states has been measured to be 1.0(4)%. The β-delayed neutron emission probability has been extracted to be P n =43(4)% from the yields of the 1830 keV and 1309 keV transitions normalized to the total number of 24 O de-TABLE I. Experimental energies, spin and parity assignments, transition energies, relative intensities per 100 decays. γ branching ratio (BR) for the excited states of 24 F observed in the β-decay of 24 O experiment. tected. This value is in agreement with P n = 58(12)% previously reported in Ref. [11] but larger than the value of 18(6)% measured in Ref. [12].

Ei [keV] I

π i → I π f Eγ [keV] Iγ[\ 100 decays] BR[%] 521(1) 2 + 1 →3 + 1 521(1) 21(2) 1830(1) 1 + 1 →3 + 1 1830(1) 39(3) 68(5) 1 + 1 →2 + 1 1309(1) 18(2) 32 (3)
In a separate setting of the LISE spectrometer, the βdecay of 24 F was studied as well. From this data set the direct feeding to the 2 + state at 1981 keV and the 4 + state at 3963 keV in 24 Ne were observed [18]. It establishes that the spin and parity of the ground state of 24 F is 3 + . It follows that the 521 keV state is located in between the 3 + ground state and the 1 + excited state at 1830 keV. As the 521 keV state is not populated directly in the 24 O β-decay its spin and parity must therefore be 2 + .

III. IN-BEAM SPECTROSCOPY OF THE 24 F NUCLEUS A. Experimental set-up

The 24 F nucleus was produced in a two-step reaction. A 36 S primary beam of 77.5 A MeV with an average intensity of 6.5 eµA underwent fragmentation in a 398 mg/cm 2 C target placed between the two superconducting solenoids of the SISSI [19] device. The reaction products were separated and selected trough the ALPHA spectrometer by means of the Bρ-∆E-Bρ method [15,20]. The transmitted cocktail beam was composed of 25,26 Ne, 27,28 Na and 29,30 Mg nuclei with energies ranging between 54 and 65 MeV/u. The identification of the beam ions was performed by the combined measurement of their time-of-flight (TOF) over a flight path of 80 m using two micro-channel plates and their energy loss in a plastic scintillator of 103.5 mg/cm 2 thickness located at the entrance of the SPEG spectrometer [21]. Two C foils of 51 mg/cm 2 thickness were placed before and after the plastic scintillator constituting a secondary "active" target. The 24 F nuclei were produced in this secondary target through the fragmentation of 27 Na. Once produced, the 24 F nuclei were separated from other reaction residues in the SPEG spectrometer. The identification of the ions was performed on an event-by-event basis at the final focal plane of the SPEG by measuring their time-offlight (TOF) with a plastic scintillator, their energy loss (∆E) and position in an ionization and two drift chambers, respectively. In addition, 74 BaF 2 detectors of the Château de Cristal array surrounded the secondary target at an average distance of 30 cm. Prompt γ-ray emission was measured in coincidence with the nuclei identified at the final focal plane of SPEG. The photopeak efficiency of the array was 24 %, 42 % and 29 % for γ-ray energies of 100 keV, 600 keV and 1300 keV, respectively.

B. Results

Prompt γ rays observed in coincidence with the 24 F nuclei identified in SPEG are listed in Table II. The singles γ-ray spectrum of 24 F is shown in Fig. 4. The three γ rays at 527 (10) keV, 1309 (22) keV and 1827 (11) keV correspond, within the experimental uncertainties, to those observed in the β-decay of 24 O at 521(1) keV, 1309(1) keV and 1830(1) keV, respectively. The excitation energy of the 1829 (26) keV state has been extracted by the weighted mean of the energies of the two decay branches. It is in agreement with the value obtained in the β-decay data set. The γγ-coincidence between the 527 keV and 1309 keV transitions (Fig. 5 a) is confirmed. The ordering of the two transitions is obtained from their relative intensity since the 527 keV γ-ray intensity is much larger than that of the 1309 keV transition (Tab. II) the 527 keV state is placed below. It can be fed directly in the reaction and/or by other higher-lying excited states. In addition, four new transitions have been observed between 2200 keV and 4000 keV. To get a reasonable line shape, the response function of the BaF 2 array was simulated using the Geant4 package. In the simulation the energy dependence of the peak width, the cut-off energy, the Doppler shift and the Doppler broadening were taken into account. The energy-dependent width of the γ ray peak has been extracted from the spectroscopy of other nuclei produced in similar experimental conditions [22]. For γ-rays with energies greater than ∼1.5 MeV, in addition to the photo-absorption and Compton effects pair creation also starts to play a significant role. Since an add-back procedure was used in the analysis of the γ-ray spectra, the escape of the annihilation and Compton scattered γ rays are suppressed. As a consequence, the line shape is well described using a Gaussian plus a long, low-energy tail. Using the simulated line shapes the γ-ray spectrum between 2.2 and 4.0 MeV could be described by four γ-rays at 2384(64), 2739 (14), 3118 (33) and 3562 (22) keV (inset of Fig. 4).

A clear coincidence between the 527 keV and 3118 keV γ-rays is observed (Fig. 5 b)), establishing a state at 3639 keV excitation energy, considering that the 527 keV transition corresponds to the 521 keV γ-line observed in the β-decay experiment. No other γγ-coincidences were observed, indicating that the 2384 keV, 2739 keV and 3562 keV transitions decay directly to the ground state. Therefore, three new excited states are proposed at 2384(64) keV, 2739 (14) keV and 3562 (22) keV.

IV. DISCUSSION

The experimental level scheme deduced in the discussed experiments up to the neutron separation energy of 3.84 MeV is shown in Fig. 6. The spin and parity of the previously reported 1830 keV state [12] can be firmly established to be 1 + from the β-decay of 24 O. The level scheme has been completed by four new states at higher excitation energy. In order to clarify their spin and parity assignment, the experimental excitation energies and branching ratios were compared to predictions of different theoretical calculations (Fig. 6).

The shell-model calculations were performed using the standard USDA and USDB interactions [13,24] in the full sd valence space. The data used in the USDA/USDB fit comprises ground states and low-lying excited states of the sd-shell nuclei from A = 16 to A = 40, therefore the experimental binding energy of the ground state of 24 F is in agreement with the theoretical predictions.

In order to assign the spins of the observed states, the experimental branching ratios were compared to the theoretical ones extracted from the E2 to M1 decay branch of each state, re-scaled in energy to match the experimental value. Starting from spin assignments of the low-energy levels, the higher-energy ones can be deduced. In the calculation of the M1 reduced transition probabilities, the effective g-factors of the proton (neutron) g sp = 5.0 (g sn = -3.44), g lp = 1.174 (g sn = -0.110) and g tp = 0.24 (g tn = -0.16) for the spin, orbital and tensor components of the M1 operator are taken from Ref. [24]. The E2 reduced transition probabilities are calculated using effective charges of 1.36e and 0.45e for protons and neutrons, respectively.

The spin and parity assignment of the experimental state at 527 keV is 2 + 1 . The USDA interaction underestimates the excitation energy by ∼200 keV while better agreement is found in the USDB calculations. This 2 + 1 level decays 100% to the 3 + ground state of 24 F. Both levels belong to the πd 5/2 ⊗ νs 1/2 multiplet with an almost pure wave function (∼70%).

The excitation energy of the 1 + 1 state at 1830 keV, which originates mainly (∼50%) from a mixed πs 1/2 ⊗ νs 1/2 configuration, is underestimated in the USDA and USDB calculations. A somehow similar shift in energy between experiment and theory is found for the 1/2 + state in 25 F [23], that is due to an s 1/2 proton excitation. The 1 + 1 level decays by a 1830 keV transition to the ground state (77(10)%) and by a parallel branch to the 521 keV level (23(5)%), which agrees with the USDB calculations of 72% and 28%, respectively. The USDA calculations result in 95% decay to the ground state and 5% to the 2 + 1 level. The reduced M1 matrix element is 0.11 µ N with USDA and 0.17 µ N with USDB. This difference is consistent with the results shown in Figure 2 of Ref. [24] where M1 matrix elements for USDA and USDB differ in a random way with a root-mean square (r.m.s.) difference of about 0.10 µ N being about the same for large and small values. As suggested in [24], the r.m.s. difference between experimental and theoretical M1 matrix elements might be reduced if some of the M1 data could be used to constrain the Hamiltonian.

The level at 2384 keV decays exclusively to the ground state. The only theoretical counterpart having the same decay pattern is the 4 + 1 state, which has a 73% pure

πd 5/2 ⊗ ν(d 5/2 ) -1 (s 1/2 ) 2 configuration.
energy spacing of the experimental levels between 2400 keV and 3700 keV is better reproduced in the calculations performed with the USDA interaction. The results obtained with the USDB interaction show a spectrum with states having a regular spacing. Both calculations predict a high density of levels at high excitation energy close to the neutron separation energy, among them the 1 + 2 and 4 + 2 states belonging to the πd 5/2 ⊗ νd 3/2 multiplet. These levels are of the same origin as the 1 + 1 and 4 + 1 states in the low-energy spectrum of the weakly bound 26 F nucleus.

The spin and parity of the 2739 keV state is tentatively assigned to be 3 + 2 because any other possibility would imply a deviation between the experimental and theoretical excitation energy of more than 400 keV. Experimentally, this state decays by more than 95% to the 3 + 1 ground state in contrast to the calculations that give 3, 84, 12 % to the 3 + 1 , 2 + 1 , 4 + 1 states for USDA, respectively, and 15, 38, 47 % for USDB. The difference between experiment and theory might come from the fact that the B(M1) value to the ground state is very small: 0.0010 µ 2 N for USDA and 0.0017 µ 2 N for USDB. A B(M1) value of 0.010 µ 2 N would give 90 % branch to the ground state. The corresponding M1 matrix elements are 0.083 µ N (USDA) and 0.110 µ N (USDB) and 0.26 µ N for B(M1)=0.010 µ 2 N . This range of matrix elements is within the overall best-fit r.m.s. deviation between experimental and theoretical M1 matrix elements of 0.20 µ N (Table I in [24]). The 3 + 2 level belongs to the same multiplet as the 4 + 1 state with a ∼53% pure configuration. At higher excitation energies there is a large density of calculated levels which make the spin and parity assignments to the experimental 3562 keV and 3639 keV states difficult. The experimental branching ratios were compared to the calculated decay pattern of the 4 assigned to be 4 + 2 or 2 + 3 , that of the 3639 keV state 2 + 2 or 1 + 2 . The 3562 keV state is a good candidate for the 4 + 2 which belongs to the πd 5/2 ⊗ νd 3/2 multiplet with ∼64% pure configuration. It would be favorably populated in the reaction due to being an Yrast state. As no evidence of the existence of a 1 + 2 state was found in the β-decay experiment, the 3639 keV state is likely to be I = 2 + 2 . The 2 + 2 and 2 + 3 levels have very mixed and complex wave functions.

In addition, theoretical predictions from firstprinciples valence-space calculations, where the manybody processes and 3N forces are included are explored. Effective shell-model Hamiltonians based on two-nucleon (NN) and 3N forces were first derived for the sd-shell region within the context of many-body perturbation theory [3,[26][27][28][29], where excitations outside the valence space were calculated to third order. There it was found that both 3N forces as well as an extended valence space (i.e., including orbitals beyond the standard sd shell), were essential to describe semi-magic isotopic/isotonic chains on top of a 16 O core. The need for an extended space in the perturbative calculation of the valence-space Hamiltonian suggests that the extended orbitals need to be included nonperturbatively.

Therefore, it has been considered in this paper a novel nonperturbative method for constructing valence-space Hamiltonians: the in-medium similarity renormalization group (IM-SRG) [14,[30][31][32]. In addition to the IM-SRG other ab initio methods have now successfully treated the oxygen chain and select fluorine and neon isotopes with NN+3N forces [33][34][35][36]. In the IM-SRG, a continuous unitary transformation, parameterized by the flow parameter s, is applied to the initial normal-ordered A-body Hamiltonian such that undesirable off-diagonal couplings are driven to zero as s → ∞:

H(s) = U † (s)HU (s) = H d (s) + H od (s) , (1)
H od (s → ∞) = 0 . (2)
Taking the uncorrelated ground state of doubly magic 16 O, and defining H od to be all n-particle-n-hole excitations, H(s → ∞) will flow to the fully correlated (i.e., exact) ground-state energy as H od (s) → 0. Including excitations that connect valence-space to non-valence-space particle states in the definition of H od , the sd valence space will decouple from the core and higher shells as s → ∞. The resulting Hamiltonian H(∞) will then consist of renormalized sd-shell SPEs and TBMEs, to be used as input in a standard shell-model calculation, in addition to the 16 O core energy [14,31].

The starting point for these calculations are nuclear forces derived from chiral effective field theory [37,38]. We use the 500 MeV-cutoff N 3 LO NN potential of Ref. [39] and the local N 2 LO 400 MeV-cutoff 3N interaction of Ref. [40], evolved with the free-space SRG [41] to a lower momentum scale, λ SRG = 1.88 fm -1 . IM-SRG sd-shell Hamiltonians are then calculated following the procedure outlined above, based on SRG-evolved NN forces with 3N forces induced by the SRG evolution (NN+3N-induced) as well as with initial 3N forces (NN+3N-full). The latter are included through normal ordering with respect to the 16 O Hartree-Fock reference state, truncated at the two-body level [42,43]. For complete details, see Ref. [14]. Finally, the resulting shellmodel Hamiltonians are diagonalized to obtain the spectrum of 24 F, studying for the first time IM-SRG protonneutron valence interactions.

Without initial 3N forces, the spectrum (not shown) is much too compressed and the ordering of levels is incorrect: The first eight excited states lie below 2.0 MeV, in clear contrast to experiment and the NN+3N-full results shown in Fig. 6. While the ground-state energy of 24 F is overbound by 7.7 MeV in the NN+3N-full calculation, the predicted excited-state spectrum is in remarkably good agreement with the new experimental measurements. In particular, all excited states below the one-neutron separation threshold are less than 200 keV away from corresponding experimental levels. The only exception is a 0 + 1 state at 2640 keV, also predicted with USDA,B, which is likely not seen experimentally due to the difficulty of the fragmentation method in populating low-J states. Since coupling to the continuum is currently neglected, when included, a modest lowering of the 2 + 2 , 4 + 2 , 1 + 1 , and 5 + 1 states would be expected near threshold. Furthermore the wavefunctions and γ transitions involving these states are very similar to those discussed above for USDA,B, strengthening the proposed identifications made for the 3 + 2 and 4 + 1 states. Specifically the 3 + 2 → 3 + 1 branch is 26%, in moderately better agreement with experiment. The probable identification of the 3639 keV level as the 2 + 2 state suggests an incorrect 2 + 2 -4 + 2 ordering is seen in these calculations. This can be understood in terms of neglected continuum effects: The νd 3/2 component of the 4 + 2 state is twice as large as that in 2 + 2 , hence one would naively expect that, when added, the continuum would then lower the 4 + 2 by a greater amount than the 2 + 2 , possibly resulting in the correct ordering. This hypothesis needs to be confirmed by an unambiguous location of the 2 + 2 and 4 + 2 states in 24 F and more detailed theoretical calculations. Recent coupled-cluster calculations based on optimized N 2 LO chiral NN and 3N forces have been performed for 24 F [36], which exhibit reasonable agreement with this new experimental picture.

V. SUMMARY

Detailed spectroscopy of the 24 F nucleus has been obtained at GANIL using two complementary experimental techniques: β decay and in-beam γ-ray spectroscopy from projectile fragmentation. Previously reported tran-sitions (521, 1309 and 1830 keV) have been confirmed, and in addition four new γ rays have been observed for the first time. The γ-ray ordering was established from relative intensity arguments and the large statistics of the present data allowed to perform a γγ-coincidences analysis. Gathering all the available information on 24 F, a level scheme has been proposed up to the neutron separation energy. The ground state spin and parity of 24 F is unambiguously determined to be 3 + . Excitation energies and branching ratios are compared to two shell-model calculations (using the standard USDA and USDB interactions) as well as to ab initio shell-model calculations, using interactions derived from chiral NN+3N forces by means of the IM-SRG. From this comparison a clear identification of almost all measured states has been obtained. It is suggested that the 3 + 1 ground state and the 2 + 1 levels belong to the πd 5/2 ⊗ νs 1/2 multiplet with more than 70% pure wavefunctions. The 1 + 1 state has a predominant πs 1/2 ⊗ νs 1/2 configuration, while the 4 + 1 and 3 + 2 states have predominant πd 5/2 ⊗ ν(d 5/2 ) -1 (s 1/2 ) 2 configurations. At higher excitation energy, the large density of observed and calculated states makes the identification of the experimental levels more ambiguous. Tentative 2 + 2 and 4 + 2 spin parity values are proposed for the 3639 keV and 3562 keV levels, respectively. Their ordering is only reproduced by the shell model using the USDA and USDB interactions, while the calculations performed with the IM-SRG predicts an inversion of the two states. As being significantly less mixed than the 2 + 2 state, the 4 + 2 is expected to be more sensitive to continuum effects, and therefore the explicit treatment of such effects should lower its energy by a greater amount than the 2 + 2 . This would possibly result in the correct ordering in the IM-SRG model. Further spectroscopic information of 24 F at high excitation energy and more detailed theoretical calculations will be needed to elucidate the role of the continuum in the high-energy part of the level scheme of this nucleus, but the overall agreement between IM-SRG theory and experiment is globally extremely satisfactory.

FIG. 1 .

 1 FIG. 1. (Color online) Energy loss versus time-of-flight identification matrix. The relative yields of the implanted ions are shown by different shades.

FIG. 2

 2 FIG. 2. a) β-gated γ-ray spectrum following the implantation of 24 O nuclei obtained in the 0-250 ms range. b) γγcoincidence spectrum gated by the 1309 keV γ-ray.

FIG. 3 .

 3 FIG. 3. (Color online) Background-subtracted summed timedistribution gated on the 521 keV, 1309 keV and 1830 keV γ-rays following the implantation of 24 O nuclei. The fit with a single exponential decay-curve yields T 1/2 =80(5) ms.

FIG. 4 .

 4 FIG. 4. (Color online) Singles γ-ray spectrum obtained in coincidence with the 24 F nuclei produced in the in-beam γ-ray spectroscopy experiment. The inset presents a zoom on the high-energy part of the spectrum. The lines show the result of the fit with line shapes obtained from Geant simulation.

FIG. 5 .

 5 FIG. 5. γγ coincidence-spectra gated by the 1309 keV (a) and 3118 keV (b) transitions.

FIG. 6 .

 6 FIG.6. (Color on line) Experimental level scheme of24 F obtained in the in-beam and β-decay data set compared to shell-model calculations performed with the USDA and USDB interactions as well as ab-initio valence-space Hamiltonians calculated from the in-medium similarity renormalization group.

TABLE II .

 II Experimental energies, tentative spin and parity assignments, transition energies, relative feeding intensities and γ branching-ratios (BR) for the excited states in 24 F obtained in the in-beam spectroscopy experiment.

	Ei [keV] 527(10) 1829(26) 2384(64) (4 + I π i → I π f 2 + 1 →3 + 1 1 + 1 →3 + 1 1 + 1 →2 + 1 1 )→3 + 1 2739(14) (3 + 2 )→3 + 1 3562(22) (2 + 3 , 4 + 2 )→3 + 1 3562(22) 47(5) Eγ [keV] Iγ[%] BR[%] 527(10) 71(3) 1827(11) 17(2) 77(10) 1309(22) 5(1) 23(5) 2384(64) 7(3) 2739(14) 100(5) 3639(42) (1 + 2 , 2 + 2 )→2 + 1 3118(33) 34(3)

ACKNOWLEDGMENTS

The authors are thankful to the GANIL and LPC staffs and the EXOGAM collaboration. This work has been supported by the European Community contract no. RII3-CT-2004-506065, by OTKA K100835 and NN104543, by the NSF PHY-1068217 grants, by a grant of the Romanian National Authority for Scientific Research, CNCS -UEFISCDI, project no. PN-II-ID-PCE-2011-3-0487. We thank S. Binder, A. Calci, J. Langhammer, and R. Roth for providing us with chiral 3N matrix elements. Computing resources for the IM-SRG calculations were provided by the Ohio Supercomputer Center (OSC).


Post-mining management in a major French mining areaexample of the Lorraine iron ore basin

The Lorraine iron ore basin, located in the north-east of France, near the border with Luxembourg, extends over about 1700 km², in a largely urban region.

The mines were closed in 1997. In a little more than one century, 40,000 km of galleries were excavated, creating more than one billion residual empty cubic meters. Modern mining was conducted by the room and pillar method, sometimes followed by complete removal of the pillars.

At the end of the 1990s, a series of major problems (subsidence, sink holes) occurred, affecting safety for persons and property in some cases. After these events, France developed a mining risk management policy and implemented various tools to first make sure that the right expertise and research was in place, then followed that with operational tasks.

The considerable extent of the Lorraine iron ore basin, the importance and multiplicity of the identified hazards and the surface challenges meant that a specific hazard assessment methodology had to be defined and put in place, and also a method for identifying and dealing with risks. This is how an approach based on the retro-analysis of past events, and common to all of the one hundred and twenty towns in the basin, was defined and validated by a Committee of Experts. The importance of surface challenges meant that risks were prioritized and specific studies were conducted in a second phase on some sectors. The results of this work, conducted over the last 10 years, have taken concrete form through the publication of hazard maps to which municipalities refer to conduct their town planning development projects. All of the documents used for these studies and the results are in a database called LorFer.

In parallel, risk management solutions for surface challenges are used. Given the specific nature of the iron ore basin, special monitoring is planned for underground voids (below ground or surface inspections, microseismic monitoring, levelling, etc.). When the safety of persons and property can no longer be ensured by monitoring, measures are taken: treatments (filling, betterment, etc.) or expropriation.

The global approach applied to the iron ore basin is a concrete example of the policy implemented for managing former mine sites in France.

Currently, the administrative and political control is managed by State Departments and in particular the Ministry of Ecology, Sustainable Development and Energy and its Regional Divisions. The French government charges BRGM's Department of Prevention and Mine Safety with work on safety and operational management of facilities and safety equipment. Mining expertise is outsourced to an independent entity, GEODERIS, (a French "Groupement d'Intérêt Publique [Public Interest Group]" between BRGM and INERIS). To complete this approach a scientific interest group (GISOS) with several partners researches in this area.

Introduction

The Lorraine iron ore basin, located in the north-east of France, near the border with Luxembourg, extends over about 1700 km², in a largely urban region.

Major problems (subsidence, sink holes) have occurred in the basin, affecting the safety of persons and property in some cases. Following these events, France developed a mining risk management organization and implemented various tools to determine the origin of the phenomena, to assess how they progressed and their effects, to lead to ad hoc research programmes and to act operationally to prevent risks by both monitoring and work to make areas safe.

2

Lorraine iron ore basin

Geographic and mining contexts

Geography

With a surface area of about 115,000 hectares, the Lorraine iron ore basin extends from the border with Luxembourg in the north to south of Nancy, i.e. almost 120 km long and about 30 km wide. It is divided into two sub-basins: the Briey-Longwy basin in the north, and the Nancy basin in the south, covering 1,300 km 2 and 380 km 2 respectively (Figure 1).

Geology

Iron formations are found in oolitic sedimentary deposits dating from the Aalenian period (about 150 million years ago).

The deposit is alternating mineral-rich seams containing 30-35% iron, separated by less rich areas called interseams. The formation ranges from a few meters to about sixty meters thick. The various seams are found over the entire basin but with variable contents and thicknesses. Depending on the area, one to six seams were exploited in a single vertical operation.

The Lorraine deposit lies underneath the Bajocian marl-limestone series

The deposit outcrops in the north, at the borders with Belgium and Luxembourg, and in the east along the Moselle valley. It sinks with an average dip of 3% towards the west to reach a depth of 280 meters at the western limit of the exploitable, managed area.

Mining methods

Ore has been extracted since the middle ages, but industrial mining only began in the 19 th century after the Thomas-Gilchrist 1 process was invented. Iron ore was first removed manually (extraction and loading).

When it was necessary, timber was put in place to support the ceiling.

The first mining operations were conducted where the seams outcrop and became deeper gradually, leaving support pillars of diverse shapes and sizes. The galleries were partially filled in with stacked blocks of barren material intended to support the hanging wall (Figure 2). This mining work was generally poorly understood. Its age and the work's amateur nature means there was no written or mapped trace of the work. Most of the mining work was underground. In some very small areas close to the outcrop, some open-pit mines were made. After the Second World War, mechanization developed progressively. Miners could bolt the roof. The last working mine in Lorraine closed in 1993.

Modern mining methods for Lorraine iron ore can be classified into two categories, depending on whether or not they leave underground voids:

○ The most commonly used method in iron mines was the "abandoned room and pillar method" (Figure 3). This consists in digging galleries or rooms, usually 4-7 m long, separated by pillars of ore left in place. The pillars of ore left in place are various shapes (square, rectangular, trapezoid, etc.). They are generally between 2-7 meters tall, the same height as the thickness, and between 5-30 m wide. The abandoned pillars support everything above the pillars. The rooms made by extracting the material are left empty after mining. They sometimes represent a high volume. By using precise, correctly sized pillars, this technique guarantees, for a varying time period, ground stability, protecting the area above the mining structure.

○ The method of complete pillar extraction, also called "caving room and pillar" mining, is similar to the "abandoned room and pillar" method in its first phase: rooms are dug and ore pillars are left. However, in a second phase, the pillars are made narrower then deliberately caved in (since narrower pillars are only stable for the short term) or by triggering (shooting pillars by blasting). This caving deliberately creates a subsidence bowl on the surface whose effects extend out from the area where the pillars are removed underneath along an angle compared to the vertical. This technique was used in areas that presented no major challenges for protection on the surface.

More than three billions of tonnes of ore have been extracted from iron mines in Lorraine, which represents about 40,000 kilometres of galleries.

Phenomena feared for the Lorraine iron ore basin

The different mining methods used in the Lorraine iron ore basin may cause instability phenomena whose characteristics depend not only on the method itself but also on the depth of the work, the nature of the ore and the characteristics of the overburden.

Sink holes

For mining work near the surface, instability phenomena that can appear are sink holes, conical collapses a few meters in diameter. They result from progressive degradation of the hanging wall of a gallery that rises little by little in overlying strata until it breaks through the surface suddenly. Under the conditions of the Lorraine iron ore basin, sink holes cannot appear for voids located at a depth greater than 50 m. This limit value may be reduced in some cases by particular study of the area.

Moreover, iron ore mining caused more than 150 shafts to be sink. The end of mining meant a large number of these sites were abandoned. The wide variations in characteristics (diameter, depth or casing), and the record of sites that close over time, make shafts a potential risk for property and people.

Depending on the context, various scenarios may lead to collapse of the shaft head:

○ either the shaft has not been back-filled and the casing ruptures then the shaft head collapses;

○ or the shaft has been back-filled at the end of mining and cleaned i.e. back-filling materials are washed in the galleries communicating with the shaft. Now that it is void it has no more support for its casing, which breaks and will lead to the shaft head collapsing.

These instabilities may be related to the effects of a sink hole.

Progressive subsidence

Progressive subsidence results from the progressive ruin of pillars underground, sufficiently widespread for the effects to rise to the surface. It leads to a bowl appearing similar to those caused by deliberate pillar extraction. In the centre of the bowl, the ground drops vertically. On the inner edges of the bowl, signs of BRGM compression appear (banks). The outer edges show signs of extension (cracks appearing). The kinetics of the bowl appearing on the surface vary from one subsidence to another but classically spread in one day for the fastest (Auboué-Coinville in 1996, Moutiers in 1997) to several months for the slowest (Roncourt in 1998, Angevillers in 2009). After this active phase, settling type movements can continue for several years, like those seen above pillar extraction.

Sudden collapses

In some cases, the mining structure is not destroyed progressively but we observe the sudden collapse of the entire ground between the underground and the surface. The surface then collapses dynamically in a few seconds. It is frequently accompanied by an earth tremor. Some have been recorded in Strasbourg and even one in Moscow (Roncourt in 1959). The suddenness of the underground collapse is shown by the air draft, whose effects can be devastating for the entire mine. Warning signs (cracks, signs of pressure and convergence) sometimes having allowed prior evacuation of sites, have often been obvious during the days before the collapse. However, the sites remain accessible, without major degradation before the phenomenon is triggered. On the surface open crevasses, up to 1 m wide according to witnesses, appear immediately around the collapsed area. By contrast with subsidence, sudden collapse is therefore a discontinuous phenomenon over time and space.

For a brutal collapse to occur, at least two conditions must be met:

○ the underground structure must be very fragile (high levels of ore extraction, slender pillars): this constitutes the geometric criterion;

○ a thick and strong bank must exist in the overlying strata. This bank, which protects the pillars from the weight of the ground breaking, triggers the collapse process. This is the geological criterion.

The phenomenon of sudden collapse may also occur in the presence of sub-surface cavities, characterized by large ceiling spans and small pillars subject to high stress.

Because of how fast it occurs (a few days) and the sudden manifestation, sudden collapse presents particular danger both during mining and after the mine has closed. This explains the substantial attention paid to developing a method, described below, that allows geological contexts to be classified to determine which mines would be likely to experience such a phenomenon.

Subsidence and settling on stoping

Until the start of the 2000s, pillar extractions were considered to lead to subsidence in ground that rose to the surface in the months that followed and that after a period of 5 years, no perceptible movement could be recorded (confirmed measurements taken at the surface). However, experience acquired in recent years questions this assumption: today it has been proven that in some circumstances pillar extraction does not lead to surface subsidence during mining operations, but that it may cause earth movements many years after the mining work.

Underground reconnaissance has shown that the old work that was manual or barely mechanized and indicated as having pillars removed on the plans, mining was not systematically continued up to shooting of the residual pillars. The potential effects of pillar extraction not having led to systematic ceiling subsidence is a function of the depth of the mine. For areas near the surface, the existence of a residual void may lead to the formation of a sink hole; the same is true for the rupture of an isolated residual pillar. For deeper areas, sink hole type phenomena are excluded, because the voids cannot reach the surface. However we cannot exclude the possibility that a set of close residual pillars, even confined by the ceiling immediately surrounding them collapsing, do not break progressively. On the surface this would mean slow, small subsidence.

For modern pillar extractions (mechanized mining) we assume that they did lead to caving and subsidence of the entire site. However the subsidence volume at the surface remains lower than the volume removed underground. The maximum subsidence in the centre of the bowl is only half of the void's height. This leads to disbondments between banks that do not close completely apart from very deep mines. For modern, shallow pillar extractions, a large overload on the surface (large building, etc.) can therefore give rise to settlings of tens of meters.

Instability in open-pit mine faces

After open-cast iron ore mining, unstable cliff faces may subsist. Phenomena such as falling rock, rock slides and slippage may occur. Nevertheless, these phenomena remain marginal in the Lorraine iron ore basin, because there are few open cast mines and they are small.

General context for post-mining in France

In France, since the first mining laws in 1791 that set the principle than mines are the property of the nation, then the law in 1810 that founded current law, many evolutions have taken place, in particular for personnel safety, environmental monitoring, abandonment of mines, etc. With the end of mining in the large French mining basins that began in 1990 (coal mines in Nord Pas-de-Calais and the iron mines in Lorraine) the "Post-mining" era began.

Major problems (subsidence, sink holes) that occurred in the Lorraine iron ore basin and affected the safety of persons and property in some cases led France to develop a mining risk management policy.

The law that structures former mine sites was enacted in 1999. It integrated the notion of mine abandonment and mining risk management. It initiated the possibility of a Mining Risk Prevention Plan for town planning, information methods and recognizes that major risks may occur. The state becomes a guarantor for damage caused by the activity of the mine operator when it stops mining or defaults. The Mining Code therefore gives French authorities major operational responsibilities.

Currently, administrative and sovereign control is managed by State Departments and in particular the Ministry of Ecology, Sustainable Development and Energy and its Regional Division (DREAL). Mining expertise is outsourced to an independent entity, GEODERIS, (a French "Groupement d'Intérêt Publique [Public Interest Group]" between BRGM and INERIS). BRGM's Department of Prevention and Mine Safety is responsible for safety and operational management of facilities and safety equipment. To complete this approach a scientific interest group (GISOS) with several partners researches in this area.

Managing future risk

Subsidence occurring in the iron ore basin between 1996 and 1998 made the authorities take note that these former mines could cause major problems and that it was necessary to assess potential public safety risks. This task was given to GEODERIS.

The considerable extent of the Lorraine iron ore basin, the importance and multiplicity of the identified hazards and the surface challenges meant that a specific hazard assessment methodology had to be defined and put in place. This is how an approach based on the retro-analysis of past events was defined, validated by a Committee of Experts and common to all of the one hundred and twenty towns in the basin.

The hazard maps established are used as base documents for town planning authorities.

Assessing mining hazards

Information phase

A first step in assessing mining hazards consists in an information phase. As much data as possible (reports, maps, etc.) is collected on mining activity. This step is crucial since it shapes the rest of the studies. The goal of this research phase is to inventory the state of understanding about the site at the start of the study. It is carried out in two phases.

Document search: in a first phase, the data was collected from concession abandonment or end of work dossiers (DADT) filed at DREAL Lorraine by the mining companies and that contained drawings of the mines that were precise enough (scale 1/5000) to undertake the hazard analysis. However, after having discovered incoherence between the information on the drawings and observations made on the ground, extra research was conducted in departmental archives and mining company archives to find additional information.

Field research: surface reconnaissance to find a maximum of sites that reached the surface (extraction or aeration shafts, access galleries, etc.). Geographic coordinates are taken using GPS and are used to match the underground drawings with the surface features. If the worked areas cannot be found, for shafts that have been filled without traces remaining on the surface, or because of an infrastructure or building that has been constructed on the site, the coordinates stated on the abandonment dossiers are recorded. This step has a twofold objective, since firstly it means that existing mines can be inventoried to assess the associated hazard, and secondly that reference points can be determined, common to the underground drawings and mapping supports (Scan25, Bd Ortho), that are then used to position mining works in relation with surface data.

This information-gathering phase leads to an information map being created that includes all the important information necessary for understanding mining work and assessing hazards.

Determining hazards

The hazard assessment used a global method to analyse the underground drawings collected during the information phase. This was developed by a committee of experts based on a retro-analysis of past events that occurred in the iron ore basin. The main steps in this method are listed here:

Determining and characterizing "homogeneous" areas:

In this step, areas called "homogeneous" in light of a certain number of geological, geotechnical and exploitation criteria have been identified. This analysis was conducted for each seam mined and each interseam.

The natural and mining characteristics of each area identified were then determined from mine drawings and documents on each mine. For each homogeneous area, the following parameters were recorded and/or calculated: gallery size, ore removal percentage, minimum and maximum depths in the area, the size of the seam mined, the presence of adjacent faults or pillar extractions, etc.

Out of the 24,000 hectares concerned, more than 4000 homogeneous areas were identified. All of the data collected is in a database (LorFer) that is integrated in a geographic information system.

Hazard assessment for rupture in the mining structure

For each homogeneous area, characteristics are analysed with a view to estimating the predisposition of the mining structure for ruin. According to the nature of the work and the type of hazard that could occur, several criteria are selected.

For room and pillar mines, the stability of pillars is assessed according to the tributary area method. In the past, the long-term resistance of iron ore was assessed. It uses data acquired throughout the mining period and feedback on experience from previous subsidence. It also takes into account ore ageing. From this the following were selected as areas having risk of progressive subsidence: for a single-seam mine, areas with stress greater than 75 bars and for each interseam in a multiple-seam mine, thicknesses less than 7 m.

In a second phase, subsidence and collapse were separated. Analysis of feedback from experience with configurations that lead to subsidence or collapse occurring confirms a difference in the progression mechanisms for these two phenomena. The purpose of this approach was therefore, on the basis of the available information, to define quantitative criteria that exclude a sudden collapse being triggered for a mined area that is judged to be potentially unstable. Therefore two criteria have been defined: one geometric criterion (Figure 4), which illustrates the fragility of mining sites (high level of ore removal, slender pillars, poor mine superimposition, etc.) and one geological criterion, which characterizes the presence of a steep, thick and strong bank in the overburden.

In a first stage, all of the areas less than 50 m from the surface were selected as sink hole hazards.

Areas with caving prior to 1945 (start of mechanisation in the iron ore basin) were selected as sink hole or subsidence hazards depending on the depth.

A sink hole hazard was identified on shafts that have not been made safe according to the rules of the art.

Finally, open cast mines were selected as having unstable mine faces.

Characterizing the hazard

After having identified the potential hazard areas, the hazard in each area is characterized: this is principally to determine the parameters of subsidence bowls, to assess the risk of sudden collapse for the room and pillar areas selected and to determine the safety radius around the shafts. No specific studies are done first, particularly for areas that can be affected by sink holes. For sudden collapse, only the geometric criterion is studied in the first phase. The geological criteria were assessed in a second phase, only on the areas that presented surface concerns.

Summarizing and mapping hazards

After the analysis conducted as described previously, a hazards map can be created for the sector. (Figure 5)

Managing future risk: the plan for preventing mining risks

The hazards map localizes and ranks the areas affected by potential phenomena. The hazards are classified by several levels, taking account of the nature of the phenomena, if possible the predisposition that they may occur, and their intensity. It does not include the nature of the surface occupation. It converts the forward potential for hazards or harm that the former mining activity may cause objectively, in the sector studied.

The hazard maps should be taken into account for town planning documents to limit new risks caused by new challenges appearing. More restrictive than for developing local town planning documents, mining risk prevention plans (PPRM) have been instituted by the legislator to, among other things, be able to make building permits subject to requirements to prevent damage that could affect construction in case of accidents or mine subsidence. This being the case, above this size, the PPRM and town planning documents make the areas in which no reasonable requirements can be set unsuitable for construction to ensure the safety of persons and property.

Mining risk prevention plans (PPRM) implemented by French authorities pursuant to the law dated 30 March 1999 (art. 94 of the Mining Code), define the areas exposed to mining hazards, taking account of the nature and intensity of the risks. They also establish land-use and town-planning rules.

The phenomena taken into account in them are in particular ground movements, flooding, hazardous gas fumes, land and water pollution.

When mining hazards are known and classified, a PPRM is developed in five phases:

○ the Prefecture sets a perimeter where the mining hazards extend or are repeated (since this perimeter may include several towns);

○ the instructing department creates it (French Direction Départemental des Territoires DDT, supported by DREAL under the prior definition, and which makes the hazard maps available);

○ consultation with the towns affected and other departments; ○ plan approved by Prefectural decree.

In the Lorraine region, 165 towns are affected by mining hazards and for these, 21 PPRM have been issued for 71 towns.

5

Managing current risk

Risk assessment

After having characterized and mapped the hazards, surface risks are analysed to define at-risk areas. This involves identifying the risks located in line with hazard areas. The risk level is obtained by factoring the hazard level and the vulnerability of the challenge, i.e. the severity of the consequences should the hazard occur.

For progressive subsidence hazards, a multi-criteria analysis was conducted, with a view to ranking all of the areas having vulnerable sites determined previously. This ranking takes into account the characteristics of the subsidence bowl (deformation, subsidence amplitude, etc.) and the vulnerability of the site. The site's vulnerability was estimated as a first analysis by a qualitative criterion with five classes (town, village or with area with long buildings and apartment buildings, terraced houses, housing developments or isolated houses, commercial areas) taking account of both population density and sensitivity of buildings to subsidence. From this methodology risk areas can be classified into four classes. Out of the thousands of progressive subsidence hazard areas identified in the iron ore basin, about 370 have buildings on the surface.

For areas at risk of sink holes, taking account of the large extent of shallow areas, the first decision was to define a hazard for all sites at less than fifty meters deep. However, for the sectors where the surface is built up, it was necessary to specify the sink hole hazard. To do this, we had to understand the mines' precise characteristics (height and width of galleries, height and nature of the overlying strata, exact position of shafts). During the first phase of study, this information was not always available. An "underground" reconnaissance was then conducted. Two sorts of investigations were conducted, depending on whether the sites were accessible or not. For the accessible areas, each gallery was visited, the site's state and the geometric characteristics of the galleries (width, height) were recorded as were any instabilities that were observed (start of sink holes, ceiling falling, cracked pillars, etc.). The sites were also surveyed and precise positioning compared to the surface determined.

For inaccessible areas, either because of submergence or because of access problems (walled-off galleries, ceiling falls blocking access, pillar extraction, presence of harmful gases, etc.), sites were drilled into from the surface. This can provide two types of information: when drilling led to a cavity, a tube was inserted to allow a laser or sonar analysis instrument to pass through. Sometimes drilling did not lead to a clearly identified void. In this case, analysis of parameters recorded confirmed that the sector has had its pillars extracted. The operating method to determine the drilling location enables for sure not to drill into a pillar and therefore removes the risk of not detecting the searched cavity.

Once all these operations are complete, the sink hole risk can be studied more carefully and the risk can be assessed on the basis of the vulnerability of whatever is present at the surface. Of about 1800 areas of sink hole hazards identified, 292 have a built up surface (2013 data).

Risk management

The goal of the current risk management policy in France is to preserve the safety of persons.

Risks can be treated by three different means:

○ eliminate the challenges;

○ remove the hazard by strengthening or filling the cavity;

○ monitoring.

France has delegated operational management for risks related to former mines to BRGM (monitoring and treatment).

Monitoring

The evaluation of mining hazards, particularly ground movement phenomena, does not enable an occurrence date for instability to be defined. The purpose of implementing monitoring suited to the risk is to monitor change in a phenomenon and to be warned if it is imminent. Therefore monitoring reduces the risk by taking account of the requirements to limit the effects of the phenomenon at the surface and to ensure protection of persons and property.

Monitoring is therefore put in place either while waiting for the risk to be actively dealt with or as a substitute for dealing with the risk. It also means that the phenomenon can be followed to help plan active treatments over time.

Given the large number of hazard areas identified in the Lorraine iron ore basin, it seems difficult to eliminate all the challenges i.e. all the hazards. For hazards that present no imminent danger for people, monitoring is preferred.

In the case of progressive subsidence, microseismic monitoring probes have been developed. This is a microseismic listening technique for noise emitted while mining work is stopped. The purpose is to detect precursor signs for subsidence before it becomes visible. This technique (Figure 6) is suitable for monitoring inaccessible or hazardous mine sites. Currently, about fifty of the built up areas are covered by microseismic monitoring. About 370 progressive subsidence risk areas are identified with built up areas above them on the Lorraine iron ore basin, representing a little more than 1700 hectares. On the surface sink holes are characterized by the sudden creation of a crater. But this appears and changes slowly underground. The phenomenon can be anticipated by regular monitoring of underground work. When mining work is accessible from underground, it is monitored by visual inspection of the risk area, with particular care taken in sectors located under housing (Figure 7). The frequency of inspections is defined by the degradation of mine sites and generally varies between every one and three years. If change is observed, the frequency may be increased.

When the areas to monitor are not accessible underground, 3D imaging of galleries is used, by laser or sonar via drilling. In addition, video footage and photographs are taken. Monitoring frequencies are set according to the same criteria as for underground inspections. For the risk of sudden collapse, which is a real danger and difficult for people to predict, monitoring cannot be the long-term solution. Two solutions are therefore being considered: cavity treatment or population expropriation. Regardless of the solution chosen, the period before implementation can be long. And so as not to leave the populations under the thread of such instability, provisional monitoring is considered. This takes account of the depth of the mine, the state of pillar fracturation and mine accessibility. Microseismic monitoring, acoustic auscultation or video monitoring may be used.

Removing risk: expropriation of challenges or treatment of mine voids

When monitoring does not make it possible to control the risk, solutions to treat the risk must be considered.

Article L. 174-6 of the Mining Code sets out that "if mining risks seriously threaten the safety of persons, property exposed to this risk can be expropriated by the State [French authorities], under the conditions set out by the expropriation code for the cause of public utility, when means of safeguarding and protecting populations are more expensive than expropriation".

Accordingly, in such cases, a technical and economic study on filling and in parallel an assessment of the cost of expropriation are conducted. The State then decides on the best solution on the basis of the assessments.

In the Lorraine iron ore basin, six areas at risk of sudden collapse have been expropriated, regarding 113 families in two towns. Only one risk area was filled. It is located in Thil. Mine sites were filled by a flow of fly ash and cement over an area of 26 hectares and affecting 26 families.

Currently, no area at risk of sink holes has been expropriated. A dozen areas at risk of sink holes have been filled, representing about 0.5 hectares. Generally, filling is from the surface from drill holes by injecting concrete (Figure 8). Beforehand, barriers are set up to make the area inaccessible. They are made underground (concrete walls) or by injection from the surface. One area was treated by sand injections (Cité Curel in Moyeuvre-Grande in 1999). This method was chosen because the mine sites were underwater.

Conclusions

Following the stoppage of iron mining operations in Lorraine and the appearance of very serious problems putting people's safety at risk, France has implemented a set of regulatory, technical and organizational measures including:

○ developing a method for identifying and assessing hazards and risks suited to the iron ore basin;

○ implementing a policy for preventing mining risks integrating requirements for town planning, including, if need be, creation of a PPRM.

○ Developing monitoring methods suited to each risk type ○ implementing protective measures affecting the safety of persons and property in some cases, including treatment and expropriation.

To implement this global policy, France has relied on its mining expert body GEODERIS, and BRGM, which has operational management in former mine sites.

Figure 1 :

 1 Figure 1: location of the Lorraine iron ore basin

Figure 2 :

 2 Figure 2: old mining methods (Dalstein, 1994; BRGM)

Figure 3 :

 3 Figure 3: room and pillar mines

Figure 4 :

 4 Figure 4: showing a purely geometric criterion, f, discriminant analysis of 16 separate progressive subsidence events that occurred in Lorraine (on the left) and sudden collapses (on the right). X-axis: the discriminant function f is graduated from -6 to +6. Y-axis: the probabilities of the value of the discriminant function f (CGP and CGB are the centres of gravity for the two families) (Josien, 2010). On the left the risk of sudden collapse is dismissed, on the right the risk of sudden collapse cannot be dismissed.

Figure 5 :

 5 Figure 5: extract from the hazardous ground movement map for the town of Ottange (57) (GEODERIS, 2009)

○

  public survey in the towns affected;

Figure 6 :

 6 Figure6: principle of microseismic monitoring[START_REF] Bennani | Surveillance des risques d'effondrement dans l'Après-mine, Besoins, Méthodes : apport de la microsismique[END_REF]

Figure 7 :

 7 Figure 7: visit underground of an area at risk for sink holes (BRGM)

Figure 8 :

 8 Figure 8: injection into the cavity in Franchepré in Moyeuvre-Grande (57)