Pierre Heidmann
email: pierre.heidmann@cea.fr

Four-center bubbled BPS solutions with a Gibbons-Hawking base

We construct four-center bubbled BPS solutions with a Gibbons-Hawking base space. We give a systematic procedure to build scaling solutions: starting from three-supertube configurations and using generalized spectral flows and gauge transformations to extend to solutions with four Gibbons-Hawking centers. This allows us to construct very large families of smooth horizonless solutions that have the same charges and angular momentum as supersymmetric black holes with a macroscopically large horizon area. Our construction reveals that all scaling solutions with four Gibbons Hawking centers have an angular momentum at around 99% of the cosmic censorship bound. We give both an analytical and a numerical explanation for this unexpected feature.

Introduction

Multi-center BPS bubbling solutions brought about important breakthroughs in the understanding of the microstate geometries of black holes. In the context of the fuzzball proposal (see [START_REF] Mathur | The Fuzzball proposal for black holes: An Elementary review[END_REF][START_REF] Bena | Black holes, black rings and their microstates[END_REF][START_REF] Mathur | Fuzzballs and the information paradox: A Summary and conjectures[END_REF][START_REF] Balasubramanian | Black Holes as Effective Geometries[END_REF] for reviews), they allow to understand the quantum structure of black holes by resolving singularities of classical black hole solutions into smooth, horizonless geometries. The main features of the geometries which replace the singularity are topologically non-trivial cycles called "bubbles" maintained by fluxes. The physics of bubbled geometries is highly constrained by certain regularity conditions, known as the bubble equations, and the "no-CTC condition" [START_REF] Bena | Bubbling supertubes and foaming black holes[END_REF][START_REF] Denef | Supergravity flows and D-brane stability[END_REF] which make the charges, the angular momenta and the positions of the centers all interdependent. Furthermore, to obtain horizonless microstate solutions with a black-hole-like throat, the bubbles need to "scale", that is to say, to shrink in the R 3 base of the Gibbons-Hawking (GH) space [START_REF] Bena | Mergers and typical black hole microstates[END_REF][START_REF] Bena | Plumbing the Abyss: Black ring microstates[END_REF]. Consequently, it is important to investigate the relation between charges, angular momenta and positions to understand the miscrostate geometries. Bubbling BPS solutions with a Gibbons-Hawking base in five dimensions have been meticulously studied since a few years. A very large class of microstate geometries of supersymmetric black holes have been constructed from those solutions. The solutions can be written in a M2-M2-M2 frame or in other duality frames such as D1-D5-P or as D0-D4-F1 (for a review see [START_REF] Bena | Black holes, black rings and their microstates[END_REF]).

Finding specific examples of smooth horizonless multi-bubble solutions that have the same charges as a large black hole is quite non-trivial. Indeed, most solutions one can construct by putting fluxes on a multi-center Gibbons-Hawking base have angular momenta larger than the black hole cosmic censorship bound and hence can not be thought of as black hole microstate geometries. Trying to modify these fluxes to reduce the angular momenta gives very often rise to closed timelike curves. This is why in the literature, one can only find sparse specific examples of multi-center BPS solutions [START_REF] Bena | Black holes, black rings and their microstates[END_REF][START_REF] Bena | Mergers and typical black hole microstates[END_REF][START_REF] Vasilakis | Mind the Gap: Supersymmetry Breaking in Scaling, Microstate Geometries[END_REF][START_REF] Bianchi | More on microstate geometries of 4d black holes[END_REF]. For example, one seven-center BPS three-charge black hole solution was built in [START_REF] Bena | Mergers and typical black hole microstates[END_REF]. The solution is scaling and the angular momentum J is about 84% of its maximal value. However, some distances between the centers are 1000 times bigger than others.

In this article, we present a systematic construction and analysis of the largest known family of scaling four-center smooth horizonless solutions that have the same charges as large black holes. Our construction allows us to easily build scaling fourcenter BPS solutions with any aspect ratios between the centers. We study the charges and angular momentum of the solutions, as well as the entropy parameter, H, given by

H ≡ Q 1 Q 2 Q 3 -J 2 Q 1 Q 2 Q 3 . (1)
The entropy parameter H indicates how far is the solution from its corresponding maximally spinning black hole solution. Our first surprising result is that all our solutions have a fine-tuned entropy parameter around 0.01. Several arguments lead us to the idea that four Gibbons-Hawking center BPS solutions which do not have an entropy parameter close to 0 are really rare. Moreover, we will show that when there is no difference in scales between the inter-center distances, all BPS solutions with four GH centers have an entropy parameter around 0,01. The main advantage of BPS solutions with four GH centers is that they can be related by spectral flows to three supertubes in Taub-NUT (TN). Indeed, three-supertube constructions in Taub-NUT are easier to analyze than general four-GH center solutions and spectral flows are specific transformations which ensure that CTC are not introduced. Consequently, one can mostly concentrate on three-supertube constructions.

In Section two, we construct several classes of specific solutions with three supertubes in TN. We give a systematic method to build examples of such scaling solutions. In Section three, we use generalized spectral flow to construct bubbled BPS solutions with four GH centers from the three-supertube solutions. We then illustrate our method by constructing several classes of solutions with four GH centers. These solutions have an angular momentum 99% of the cosmic censorship bound. In Section four, we present a numerical and an analytical proof that all solutions with four GH centers and black hole charges have such large angular momenta.

2 Particular solutions with three BPS supertubes in Taub-NUT

Three-supertube BPS solutions in Taub-NUT

The notations and the results are taken from [START_REF] Vasilakis | Mind the Gap: Supersymmetry Breaking in Scaling, Microstate Geometries[END_REF][START_REF] Bena | Multi-Center non-BPS Black Holes: the Solution[END_REF]. We will recall the three-charge solutions for a system of three supertubes in Taub-NUT. In M-theory, the three charges correspond to three M2 branes wrapping three orthogonal 2-tori inside a 6-torus. The eleven-dimensional metric is:

ds 2 11 = -(Z 1 Z 2 Z 3 ) -2/3 (dt + k) 2 +(Z 1 Z 2 Z 3 ) 1/3 ds 2 4 +(Z 1 Z 2 Z 3 ) 1/3 3 I=1 dx 2 I + dy 2 I Z I , (2)
where dx 2 I + dy 2 I is the metric of the I th 2-Torus and ds 2 4 is the four-dimensional Gibbons-Hawking metric with one Gibbons-Hawking center:

ds 2 4 = V -1 (dψ + A) 2 + V dx 2 + dy 2 + dz 2 (3)
with

V = h + q r , ∇ × A = ∇V (4)
The solutions have a three-form potential A:

A = 3 I=1 A (I) ∧ dx 2 I ∧ dy 2 I . (5)
The one forms k and A (I) depend only on the four space coordinates transverse to the 6-Torus. The dipole field strengths Θ (I) are defined as:

Θ (I) ≡ dA (I) + d dt + k Z I . (6)
In order to have BPS solutions, the fields must satisfy the following equations [START_REF] Gutowski | General supersymmetric AdS(5) black holes[END_REF][START_REF] Bena | One ring to rule them all ... and in the darkness bind them?[END_REF],

Θ (I) = 4 Θ (I) ∇2 Z I = 1 2 C IJK 4 Θ (J) ∧ Θ (K) dk + 4 dk = Z I Θ (I) . (7)
The Hodge dual 4 is the Hodge dual of the four-dimensional Gibbons-Hawking space, and C IJK is the absolute value of the constant symmetric tensor. We consider axisymmetric supertube configurations. The positions of the supertube centers are given by the distances a 1 , a 2 and a 3 on the z-axis in the following order

a 1 > a 2 > a 3 > 0. (8)
The BPS equations (7) allow to consider three types of two-charge supertubes. Each supertube carries a dipole charge k I and two electric charges Q (I) j at the points j = I. Consequently, the dipole strengths Θ (I) and the scalar fields Z I are sourced by harmonic functions which we call K I and L I

K I = α I + k I r I . L 1 = 1 + Q (1) 2
4r 2 + Q (1) 3 4r 3 , L 2 and L 3 by permutation, (9)
where r I is the distance to the I th center

r I ≡ x 2 +y 2 +(z -a I ) 2 . ( 10
)
The complete solution for the warp factors is:

Z I = L I + 1 2 C IJK K J K K V = 1 + J =I Q (I) J 4r J + C IJK h+ q r k J k K 2r J r K + α J α K 2 + α J k K r K , (11)
and

k = µ (dψ+A) + ω, (12)
with µ = 1 6 V -2 C IJK K I K J K K + 1 2 V -1 K I L I + M, ( 13
)
where M is another harmonic function which we will take to be

M = m ∞ + m 0 r + 3 j=1 m j r j , (14)
and ω is another vector which satisfies the following equation in the R 3 base of TN:

× ω = V µ -µ V -V 3 I=1 Z I K I V . (15)
Thus, µ is

µ = 1 V 2 k 1 k 2 k 3 r 1 r 2 r 3 + α 1 α 2 α 3 + C IJK 2 α I k J k K r J r K + α I α J k K r K + C IJK 4V α I + k I r I 1 + Q (I) J 4r J + Q (I) K 4r K + M. (16)

Bubble equations and regularity conditions

In addition to the BPS equations [START_REF] Bena | Mergers and typical black hole microstates[END_REF], the solutions must satisfy several regularity conditions that we recall in this section. First, divergent terms along the ψ-fiber in the metric must vanish. Second, closed timelike curves (CTC) must be absent. We organize the harmonic functions in a symplectic vector H:

H = H 0 , H I , H I , H 0 ≡ V, K I , L I , 2M = ĥ + 3 j=0 Γ j r j , (17)
with ĥ = (h, α 1 , α 2 , α 3 ; 1, 1, 1, 2m ∞ ) , Γ 0 = (q, 0, 0, 0; 0, 0, 0, 2m 0 ) , Γ j = 0, k j ; 1 4 Q j , 2m j .

(18) The conditions of regularity at the singularities r j → 0 j = {0, 1, 2, 3} with r 0 = r are summed up by the following equations [START_REF] Denef | Supergravity flows and D-brane stability[END_REF]:

3 J=0 Γ I , Γ J r IJ = -Γ I , ĥ I = {0, 1, 2, 3} , (19)
where , is a symplectic product defined as follows

A, B = A 0 B 8 -A 8 B 0 + A I B 8-I -A 8-I B I . (20)
Using the notation

Γ IJ = k I Q (I) J -k J Q (J) I , (21)
we obtain the following bubble equations

Γ 12 r 12 + Γ 13 r 13 = 8m 1 h+ q a 1 -4k 1 + Q (2) 1 α 2 + Q (3) 1 α 3 Γ 21 r 12 + Γ 23 r 23 = 8m 2 h+ q a 2 -4k 2 + Q (1) 2 α 1 + Q (3) 2 α 3 Γ 32 r 23 + Γ 31 r 13 = 8m 3 h+ q a 3 -4k 3 + Q (1) 3 α 1 + Q (2) 3 α 2 q m ∞ + 3 j=1 m j a j = m 0 h. (22)
Furthermore, all the parameters in M are constrained to avoid Dirac strings at each pole and to avoid divergences along the ψ-fiber [START_REF] Bena | Multi-Center non-BPS Black Holes: the Solution[END_REF]:

m 1 = Q (2) 1 Q (3) 1 32k 1 , m 2 = Q (1) 2 Q (3) 2 32k 2 , m 3 = Q (2) 3 Q (1) 3 32k 3 , m 0 = 0, m ∞ = - 3 j=1 m j a j . (23)
Finally, the absence of CTCs is satisfied when the quartic invariant I 4 is greater than ω 2 in the three-dimensional base space of the solution [START_REF] Berglund | Supergravity microstates for BPS black holes and black rings[END_REF]:

I 4 ≡ Z 1 Z 2 Z 3 V -µ 2 V 2 > ω 2 .
(24)

Examples of solutions

The goal of this section is to construct several examples of supertube charges and dipole charges such that the bubble equations have physical solutions. In particular, we will focus on solutions with no difference in scales between the inter-center distances. We know from [START_REF] Bena | Mergers and typical black hole microstates[END_REF] that the depth of the solution's throat is inversely related to the distance between the centers in R 3 . Thus, the centers must cluster to form a concentrated object in order for the microstate geometry to look like a black hole when seen from large distances. Several methods exist for constructing such solutions. In general solutions scale when the centers move away from axisymmetry. However, to scale axisymmetric solutions we will slightly change the value of the charges, as in [START_REF] Bena | Mergers and typical black hole microstates[END_REF].

Specific choice of parameters and "scaling conditions"

We consider the following choice of charges and dipole charges inspired from the choice made in [START_REF] Vasilakis | Mind the Gap: Supersymmetry Breaking in Scaling, Microstate Geometries[END_REF]

k 1 = k 3 = -k 2 = k > 0 Q (2) 1 = Q (2) 3 = α > 0 Q (1) 3 = Q (3) 2 = β > 0 Q (1) 2 = Q (3) 1 = γ > 0. (25)
Finding an explicit scaling condition is crucial. Indeed, one can find many solutions of the bubble equations which do not scale. However, one must be able to go to a scaling limit to build a solution which looks like a black hole from far away. For this purpose, we solve the bubble equations ( 22) by computing the charges for a given cluster of supertubes infinitely close to the Gibbons Hawking center, that is to say in the limit a i → 0. So we consider an arbitrary arrangement of distances

a j = λ d j j = {1, 2, 3} , (26)
with λ 1 , d 1 > d 2 > d 3 . (27)
We linearize the bubble equation ( 22) in powers of λ

1 λ α + γ d 1 -d 2 + β -γ d 1 -d 3 - q 4k 2 d 1 αγ = h 4k 2 αγ -4 + αα 2 + γα 3 k 1 λ - α + γ d 1 -d 2 - α + β d 2 -d 3 + q 4k 2 d 2 βγ = - h 4k 2 αγ + 4 + γα 1 + βα 3 k 1 λ γ -β d 1 -d 3 + α + β d 2 -d 3 - q 4k 2 d 3 αβ = h 4k 2 β 2 -4 + βα 1 + αα 3 k . (28)
Consequently, the left-hand sides of the previous equations must be close to 0 if we want λ to go to 0. With the specific choice of charges and dipole charges (25), analytic solutions can be found

α = 4k 2 q a 3 2 a 3 + a 2 1 a 3 (2a 2 -a 3 ) + a 1 a 2 (a 2 2 -5a 2 a 3 + 2a 2 3 ) a 2 (a 1 -a 2 ) (a 1 -a 3 ) (a 2 -a 3 ) + O (λ) β = 4k 2 q a 3 2 a 3 + a 2 1 a 3 (2a 2 -a 3 ) + a 1 a 2 (a 2 2 -5a 2 a 3 + 2a 2 3 ) a 1 (a 1 -a 2 ) (a 2 -a 3 ) 2 + O (λ) γ = 4k 2 q a 3 2 a 3 + a 2 1 a 3 (2a 2 -a 3 ) + a 1 a 2 (a 2 2 -5a 2 a 3 + 2a 2 3 ) a 3 (a 1 -a 2 ) 2 (a 2 -a 3 ) + O (λ) . (29)
The main difference between our solutions and the solutions in [START_REF] Vasilakis | Mind the Gap: Supersymmetry Breaking in Scaling, Microstate Geometries[END_REF] is that the three centers and the Gibbons-Hawking center are scaling together in our solutions, while in [START_REF] Vasilakis | Mind the Gap: Supersymmetry Breaking in Scaling, Microstate Geometries[END_REF] only supertubes were scaling. This will prove to be valuable when we construct scaling solutions with four GH centers. Those formulas are useful to have an idea about the values of charges we need to choose to obtain a particular scaling solution with three supertubes in Taub-NUT. We apply those formulas in the next sections to find specific examples of solutions with no difference in scales between the inter-center distances.

Solutions with h = 0 and with no constant terms in all K I

Imposing h = 0 and q = 1 is principally motivated by having a Gibbons-Hawking metric which looks like R 4 at infinity. The first example of charges and dipole charges is

h = 0 q = 1 k 1 = k 3 = -k 2 = k = 100 Q (2) 1 = Q (2) 3 = α = 70 000 Q (1) 3 = Q (3) 2 = β = 100 000 Q (1) 2 = Q (3) 1 = γ = 233 333. (30)
We solve numerically the bubble equations (22):

a 1 = 8.46360 . . .×10 -2 , a 2 = 4.02091 . . .×10 -2 , a 3 = 1.80731 . . .×10 -2 . ( 31
)
One can check straightforwardly by plotting the quartic invariant I 4 and ω 2 in the R 3 space that the no-CTC condition (24) is satisfied. We notice that many other different scaling solutions can be built from (24).

• Charges, angular momentum and entropy of the solution

We give also the charges and the angular momentum of the solution:

Q 1 = Q 3 = 293 333 4 Q 2 = 45 000 J = 44 666 625 8 . (32)
The Bekenstein-Hawking entropy of the corresponding black hole is then

S = 2π Q 1 Q 2 Q 3 -J 2 = 9.12309 . . . × 10 7 .
(33)

We compute the entropy parameter H which informs how far the charges of the solution are from those of the corresponding maximally-spinning black hole.

H = 0.87 . . . . (34)
• Scaling solutions

Finally, we scale the solution. We fine-tune the values of the initial charges, we solve the bubble equation, and then we check the absence of CTCs. The scaling process is summed up in the following table: Therefore, when γ is converging towards a certain value, the supertube positions shrink drastically. When the centers get closer and closer, the size of the throat gets larger and larger as explained in [START_REF] Bena | Mergers and typical black hole microstates[END_REF] and depicted in Fig. 1. Because a BPS black hole has an infinite throat, the scaling process makes the bubbling solution gets more and more similar to a BPS black hole. We find a specific example of scaling microstate geometry of three BPS supertubes in R 4 corresponding to a BMPV microstate with H = 0.87.

Sol q k α β γ a 1 a 1 a 2 a 1 a 3

Solutions with h = 0

Adding a non-zero constant term in V will make the Gibbons-Hawking metric look like R 3 × S 1 at infinity. Such a configuration will be convenient when spectral flows and gauge transformations will be applied to reach general BPS solutions with four GH centers. To cope with the constant term in V , constant terms in K I must be turned on to satisfy the absence of the CTCs at infinity (24). We fix the constant terms as follows:

α 1 = -2hm ∞ α 2 = α 3 = 0. (35)
Indeed, such a choice ensures that Figure 1: A schematic description of a scaling geometry. The physical size of the bubbles remains the same whereas the throat of the solutions deepens [START_REF] Bena | Mergers and typical black hole microstates[END_REF].

µ (∞) = 0 Z I (∞) = 1. (36)
We choose the following values of charges and dipole charges as an illustration:

h = 1 q = 1 k 1 = k 3 = -k 2 = k = 100 Q (2) 1 = Q (2) 3
= α = 70 000

Q (1) 3 = Q (3) 2
= β = 100 000

Q (1) 2 = Q (3) 1
= γ = 240 000.

(37)

The bubble equations ( 22) are solvable, the no-CTC condition is satisfied and the three-center positions are a 1 = 3.04216 . . .×10 -2 , a 2 = 1.50188 . . .×10 -2 , a 3 = 6.69370 . . .×10 -3 . (38)

• Charges, angular momentum and entropy of the solution

The charges and angular momentum of the solution are :

Q 1 = Q 3 = 75 000 Q 2 = 45 000 J = 5 687 500. (39)
The Bekenstein-Hawking entropy of the corresponding black hole solution is

S = 2π Q 1 Q 2 Q 3 -J 2 = 9.33592 . . . × 10 7 . (40)
Finally, the entropy parameter H is

H = 0.87 . . . . (41)
• Scaling solutions

As the previous section, we scale the solution by adjusting only the value of γ. We find an example of scaling microstate geometry of three BPS supertubes with one Gibbons-Hawking center in Taub-NUT. Furthermore, if we compare both tables with h = 1 and with h = 0, the solutions are really similar to each other in the scaling limit. So the constant terms of the harmonic functions are irrelevant in the scaling limit.

3 Constructing BPS solutions with four Gibbons-Hawking centers

Four-GH center bubbled solutions

In this chapter, we consider BPS bubbled solutions with four GH centers. Those solutions can be entirely generated from the family of three-supertube solutions by using three generalized spectral flows and three gauge transformations. The elevendimensional metric is the same as (2). However, we consider a much more general set of harmonic functions {V, K I , L I , M }. Each function has poles at each GH center. We use the following usual notations:

V = q ∞ + 3 j=0 q j r j , M = m ∞ + 3 j=0 m j r j K I = k I ∞ + 3 j=0 k I j r j , L I = l I ∞ + 3 j=0 l I j r j , (42)
r 0 , r 1 , r 2 and r 3 are the same distances as in the first section with r 0 corresponding to r. In this framework, the regularity constraints are

l I j = - 1 2 C IJK k J j k K j q j j ∈ {0, . . . , 3} m j = 1 12 C IJK k I j k J j k K j q 2 j j ∈ {0, . . . , 3}. (43)
The equations [START_REF] Bena | Multi-Center non-BPS Black Holes: the Solution[END_REF] for the warp factors Z I , (13) for µ, [START_REF] Bena | Spectral Flow, and the Spectrum of Multi-Center Solutions[END_REF] for ω, the bubble equations ( 19) and the no-CTC condition (24) are still valid.

Spectral flows and gauge transformations

For BPS solutions, generalized spectral flows and gauge transformations are wellunderstood [START_REF] Bena | Spectral Flow, and the Spectrum of Multi-Center Solutions[END_REF][START_REF] Dall'agata | U-duality and non-BPS solutions[END_REF]. They mix the harmonic functions leaving the bubble equations and the entropy function invariant. Furthermore, they send the family of three-supertube solutions to the family of solutions with four GH centers. The harmonic functions {V, K I , L i , M } of BPS solutions transform under three gauge transformations parameterized by three constants g I as [START_REF] Bena | Black rings in Taub-NUT[END_REF]:

V → V, K I → K I + g I V, L I → L I -C IJK g J K K - 1 2 C IJK g J g K V, M → M - 1 2 g I L I + 1 4 C IJK g I g J K K + 1 12 C IJK g I g J g K V, (44)
and under a family of generalized spectral flows parameterized by three constants γ I as [START_REF] Bena | Spectral Flow, and the Spectrum of Multi-Center Solutions[END_REF]:

M → M, L I → L I -2 γ I M, K I → K I -C IJK γ J L K + C IJK γ J γ K M, V → V + γ I K I - 1 2 C IJK γ I γ J L K + 1 3 C IJK γ I γ J γ K M.
(45)

One particular example of solutions

We start from the particular solution of three supertubes in Taub-NUT background of section 2.3.3 with h = 0 and γ = 233 400. We perform three independent spectral flows then three independent gauge transformations. To obtain physical solutions and a Gibbons-Hawking metric which looks like R 4 at infinity, the transformations are constrained such that in the resulting solution:

• The constant term in V is 0.

• The sum of all charges in V is 1.

• The charges in V are integers.

• The constant terms in all K I are 0.

• The values of the charges and dipole charges are rational [START_REF] Balasubramanian | Four Dimensional Black Hole Microstates: From D-branes to Spacetime Foam[END_REF].

To satisfy the conditions above, we first leave a non-zero constant term in V in order to remove the constant terms in each K I with three gauge transformations. Then we fix one spectral flow parameter to satisfy the second condition. The third condition cannot be easily satisfied. So, we fix the two remaining spectral flow parameters to have the closest values to integers. Afterwards, all six transformation parameters are fixed. We remove by hand the constant term in V . Because the bubble equations are changing significantly, we have to solve them accordingly. For this purpose, two options exist :

• We solve the bubble equations by considering the positions of the centers as variables and all the charges are fixed.

• We solve the bubble equation by considering the charges as variables and the positions are fixed. In the scaling limit, one can see straightforwardly that an infinitesimal change of three dipole charges (k 1 0 , k 2 0 and k 3 0 for instance) can annihilate a change of order one on the right hand side of the bubble equations (19). Consequently, one can find a solution to the bubble equations only perturbing infinitesimally the values of the charges which ensures that no closed timelike curves will appear.

In general, we apply the first method. However, when it was not possible to do, we apply the second one. At that point, the values of charges and dipole charges are real numbers with an infinite number of digits. The last step is to round the values taking into account the necessary regularity conditions (43). Once again, the solutions are not invariant, we have to check that the bubble equations are still solvable and that the no-CTC condition is still satisfied. We present the result of applying the above procedures on the three-supertube configuration with h = 0 and γ = 233 400: (

V = 1 r + 1 r 1 + 12 r 2 - 13 r 3 K 1 = - 2087
) 46
The bubble equations are solved by the following positions:

a 1 = 2.67495 . . .×10 -2 , a 2 = 1.27076 . . .×10 -2 , a 3 = 5.70977 . . .×10 -3 . ( 47
)
• Charges, angular momentum and entropy

The charges and angular momentum of the solution are:

Q 1 = 99858458954459 1300000000 Q 2 = 83964235108323 2600000000
Q 3 = 856306630373655247 7800000000 J = 705272590995929898902049 1352000000000000 . ( 48
)
The Bekenstein-Hawking entropy of the corresponding black hole solution is

S = 9.12620 . . . × 10 7 . (49)
Finally, the entropy parameter H is

H = 0.000775. ( 50
)
The angular momentum is extremely close to the maximally spinning value. This is not a coincidence and this is one of the key points of our paper. Picking random threesupertube configuration also gives a near-maximally spinning solution after spectral flows and gauge transformations. In order to prevent this, a very fine tuning between the parameters of the three-supertube solution is required. The main reason is that the charges Q 1 , Q 2 and Q 3 strongly increase under spectral flows if all the conditions imposed for V are satisfied. Consequently, the angular momentum is forced to reach a value close to √ Q 1 Q 2 Q 3 to leave the entropy function invariant. This will be fully described in section 4.

Scaling solutions

We check whether the new solution can still scale. We only change k 1 2 , but any other K I coefficient would have worked. Furthermore, we take into account that any change of k 1 2 produces a change in the L I 's charges and in M according to (43). We give the steps in the following table: Thus, we built asymptotically R 4 solutions with a shrinking cluster of four bubbled GH centers with no difference in scales. These are microstates of a black hole with a macroscopically large horizon. In the next section we investigate how systematic the large J is.

Sol k 1 2 a 1 a 1 a 2 a 1 a 3 1

Discussion

Solutions with four GH centers with no difference in scales are necessarily near-maximally spinning

Understanding the impact of spectral flows on a cluster of three supertubes in Taub-NUT by an analytical approach is complicated due to the number of parameters and the complexity of the equations. However, by a numerical approach, it appears that performing three constrained spectral flows on a three-supertube solution with no difference in scales between the inter-center distances (that is to say a 3 , a 2 -a 3 and a 1 -a 2 are of the same order of magnitude) will result in an entropy parameter extremely close to 0. By "constrained spectral flows" one means that the resulting harmonic function V must have integer charges whose sum is equal to one. We reached this conclusion after having studied many different examples which have led to a value of H around 0.01 each time. Insights of our approach are given in the next two sections and in appendix A.

Numerical analysis

First, we investigate all the solutions with four GH centers where the centers scale as follows:

a 1 -a 2 a 3 ≈ 1 a 2 -a 3 a 3 ≈ 1. (51)
We would like to scan all the solutions of this kind, to compute their entropy parameters, and to find out if they are indeed all of the order of 0.01.

Let us list all the free parameters. The starting configuration is a three-supertube solution with six positive charges Q (J) I , three dipole charges k I and one Gibbons-Hawking charge q. The constant terms do not enter in the formula for the entropy parameter and hence are irrelevant. The scaling conditions fix three charges. Furthermore, we notice that changes of q can be reabsorbed into changes of dipole charges. So without loss of generality we can consider that q is equal to one. We can also restrict the sign changes of dipole charges by symmetry. Then, all six parameters of the spectral flows and the gauge transformations are constrained to cancel the constant terms in all K I and to have a Gibbons-Hawking harmonic function of the form:

V = 1 r - 2 r 1 + 1 r 2 + 1 r 3 . ( 52
)
This constitutes the best choice to maximize the value of the entropy parameter as shown at the end of appendix A. The process to obtain such solutions is the same as the one given in section 3.3. Finally, one needs to vary only three charge ratios, the norms and some signs of the dipole charges to scan all the solutions satisfying (51) and (52). We built a huge number of such four-center solutions using a loop algorithm.

We have computed their entropy parameter to establish the variation of the entropy parameter as a function of the free parameters. We concluded that in any situation the entropy parameter is at most of order of 0.01. Then, we extended to solutions which do not necessarily satisfy (51) and (52). We observed the same feature. All the details are presented in appendix A.

An analytical argument

We investigate the near-maximally spinning feature in the context of the specific choice of charges and dipole charges made in 2.3.1. With this specific choice it is easier to understand analytically why the conditions imposed to the spectral flows produce an important increase in the charges Q 1 , Q 2 and Q 3 which requires an increase in J to leave the entropy invariant. We sum up how we proceed in the following steps:

• We express the charges α, β and γ using the scaling condition relation (29) to reduce the number of independent parameters. We obtain approximations as F a 1 a 3 , a 2 a 3 4k 2 for each charge where the F 's are rational fractions.

• Assuming no difference in scales (for instance 1.2 a 3 < 1.1 a 2 < a 1 < 10 a 3 ) allows to restrict the functions F a 1 a 3 , a 2 a 3

to certain ranges of values for each charge.

• We analytically express how Q 1 , Q 2 and Q 3 transform under three generalized spectral flows one of which is fixed by requiring the sum of the charges in V to be equal to one.

• For each domain of F a 1 a 3 , a 2 a 3

we show numerically that if Q 1 , Q 2 and Q 3 do not double after spectral flows, the spectral flow parameters must be in the same order of magnitude as k α at most. • Finally, by studying the transformation of V under spectral flows (45) in the specific domains of parameters found above, this gives rise to a function V with charges smaller than one which is non-physical.

With those two analyses, we conclude that any BPS bubbled scaling solution with four GH centers with no difference in scales is necessarily near-maximally spinning.

By the same kind of arguments, if no assumption about the positions is made, building a solution which is not a near-maximally spinning solution is also difficult but not impossible [19].

Conclusion

We have built several examples of four-center BPS bubbled solutions. Starting with three supertubes in Taub-NUT, we gave a systematic protocol and we built several scaling solutions. Secondly, using generalized spectral flows we generated BPS solutions with four Gibbons-Hawking centers. We gave a numerical protocol to obtain such solutions from three-supertube scaling solutions and we built specific examples. We reached one important conclusion: BPS solutions with four GH centers that do not have an entropy parameter close to 0 are rare. Furthermore, when such solutions have no difference in scales between the inter-center distances, they are necessarily near-maximally spinning.

There are several interesting directions for future research. The first is to find BPS solutions with four GH centers which are not near-maximally spinning. We expect them to display a difference in scales between the inter-center distances, as in the examples built in [START_REF] Bena | Mergers and typical black hole microstates[END_REF]. Using algorithms to fine-tune the initial three-supertube parameters, one can plan to find such solutions [19]. One can also investigate different four-center constructions with one supertube and three Gibbons-Hawking centers. Those solutions are smooth in the D1-D5-P duality frame and can be generated from three-supertube solutions by two spectral flows. Thus, one may expect them not to necessarily be near-maximally spinning solutions even when the centers have no difference in scale. Secondly, one can extend our work to almost-BPS four-center solutions [START_REF] Goldstein | Almost BPS black holes[END_REF] following the same ideas. Thirdly one can add non-Abelian hair to the solutions as it was done in [START_REF] Ramirez | Non-Abelian bubbles in microstate geometries[END_REF]. Finally, one can also investigate whether solutions with more than four Gibbons-Hawking centers without difference in scale are also near-maximally spinning and, more generally, how universal is the feature we found. q, k 1 , -k 2 and k 3 equal to 1. Each graph has been built with 10000 scaling solutions which are not necessarily CTC-free and the result has been interpolated and smoothed for readability. Thus, it is important to point out that a point of one curve does not necessarily correspond to a CTC-free solution with four GH centers. Nevertheless, any CTC-free four-center solution meeting the conditions above has an entropy parameter near or below the curve's prediction.

In the domains studied, the entropy parameter is indeed of order 0.01. One also has to check the regions where H increases. For this purpose, we have investigated the evolution of the entropy parameter as a function of one of the charge ratios for a larger range of values. We observed that the evolutions for any value of the two other charge ratios are similar to the four evolutions shown in Fig. 3 and that H is bounded between 0 and 0.09. However it is important to point out that in the domains of very large or very small charge ratios the solutions can possibly have closed timelike curves. Indeed, if we want the four centers to scale in one block, the bubble equations (19) imply that one supertube at least must have a negative dipole charge. Consequently, if some charges are small compared to the value of the dipole charge, the warp factor [START_REF] Bena | Multi-Center non-BPS Black Holes: the Solution[END_REF] of the corresponding supertube may be negative around other poles. Thus, the upper bound 0.09 of the entropy parameter can be thought of as a generous upper bound for CTC-free solutions.

Similarily, we studied the entropy parameter as a function of the dipole charges and their signs. We found in all configurations the same kind of evolutions as in Fig. 2. Consequently, any entropy parameter of solutions with four Gibbons-Hawking centers satisfying (51) and (52) are of the order of 0.01. The last step is to ensure that solutions which do not satisfy (51) and (52) have the same property. The graph Fig. 4 illustrates that if the GH function V has higher charges the entropy parameter is getting closer to 0. So the condition (52) gives the solutions which have the highest values of H. We vary also the condition (51) taking into account that the inter-center distances must remain of the same order. The graphs Fig. 5 show that changing the aspect ratio a 1 a 2 does not produce a significant change of the entropy parameter. This exhaustive numerical analysis shows that any solution with four GH centers with no difference in scales is necessarily near-maximally spinning.

Q 2 (3) / Q 3 (1)
ℋ (a)

Q (1) 2 Q (3) 1
Q 2 (3) / Q 3 (1)
{q 2 ,q 3 }={1,-2} {q 2 ,q 3 }={2,-3} {q 2 ,q 3 }={3,-4}

12 Figure 3 : 1 , -k 2 and k 3 equal to 1

 12311 Figure 3: The entropy parameter H as a function of one of the charge ratios with q, k 1 , -k 2 and k 3 equal to 1.

Figure 4 :

 4 Figure 4: The entropy parameter H as a function of one of the charge ratios for three different GH functionsV with q = 1, k 1 = -k 2 = k 3 = 1, Q (1) 2 Q (3) 1 = 1 and Q (2) 3 Q (2) 1 = 0.2.

(a) a 1 a 2 ≈ 1 . 1 (b) a 1 a 2 ≈ 3 . 1 (c) a 1 a 2 ≈ 5 . 6 (d) a 1 a 2 ≈

 2112312562 10.1

Figure 5 :

 5 Figure 5: The entropy parameter H as a function of two charge ratios and one distance ratio with q, k 1 , -k 2 and k 3 equal to 1.

  The scaling process is summed up in the following table:

	Solutions	γ	a 1	a 1 a 2	a 1 a 3
	1	240 000 3.04216 × 10 -2 2.02557 4.54481
	2	234 000 2.97358 × 10 -3 2.09593 4.66672
	3	233 400 2.96688 × 10 -4 2.10405 4.68150
	4	233 340 2.96621 × 10 -5 2.10481 4.68283
	5	233 333,4 2.96614 × 10 -7 2.10490 4.68299

Acknowledgements

I would like to thank Iosif Bena and Pedro Ramirez for the interesting discussions. This work was supported by an ENS Lyon doctoral grant.

A Numerical analysis of the entropy parameter of four-GH center solutions

In this section, we detail how we scanned all the different values of the entropy parameter we can get from scaling solutions with four Gibbons-Hawking centers. As explained in section 4.1.1, we first scanned all the scaling solutions which satisfy

by varying the free parameters of the constructions: three charge ratios (

(2) 1

and

for instance) and the three dipole charges k 1 , k 2 and k 3 . The graphs Fig. 2 give the variations of the entropy parameter according to three charge ratios from 0.1 to 4 with (a)

Figure 2: The entropy parameter H as a function of the charge ratios with q, k 1 , -k 2 and k 3 equal to 1.


F Duc

X Tonon

Julien Billette

B Rollet

William Knafo

F Bourdarot

Jérome Béard

F Mantegazza

B Longuet

J E Lorenzo

E Lelièvre-Berna

P Frings

L.-P Regnault

40-tesla pulsed-field cryomagnet for single crystal neutron diffraction

Keywords: numbers, Valid appear here Keywords, Suggested keywords

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I. INTRODUCTION

Limited for a long time to 15 T and finally to 17 T produced by superconducting coils 1,2 , the magnetic field strength of DC magnets for neutron experiments has been raised very recently to 26 T at the Helmholtz-Zentrum Berlin (HZB) 3,4 . A new series-connected hybrid magnet with a horizontal field orientation was designed and constructed by the HZB and the National High Magnetic Field Laboratory (NHMFL), Tallahassee (Florida, USA) 5 . Combined with the Extreme Environment Diffractometer (EXED) 6 , the High Field Magnet (HFM) enables neutron studies in static fields up to 26 T and temperatures down to 0.65 K [7][8][9] .

Encouraged by the development of x-ray diffraction and spectroscopy in pulsed field up to 40 T at synchrotron sources [10][11][12][13][14] , significant activity aiming at increasing the magnetic field available at neutron facilities has arisen all over the world. Whereas the duty cycle [START_REF]time to cool down the magnet between two pulses[END_REF] is rather low in pulsed field set-up, the combination of pulsed field and neutron scattering environments is an alternative choice, of moderate cost, whose relevance has been proven over the past decade. Several pulsed field devices using mini-coil systems developed by the Nojiri group from the Institute for Materials Research (Tohoku University, Japan) have been used at different neutron facilities: a 30 T solenoid magnet with pulse durations of the order of 1 to 10 ms and a 25 kJ capacitor bank have been first installed in the research reactor JRR3 (JAEA, Tokai, Japan) [START_REF] Ohoyama | 17th International Conference on Mag-netismThe International Conference on Magnetism[END_REF] and later at the japanese spallation neutron sources J-PARC [START_REF] Maekawa | [END_REF]18 and at the Spallation Neutron Source (SNS), Oak Ridge National Laboratory (USA) 19 . Currently, a 40-tesla horizontal coil system and a 250 kJ compact power supply are being developed by the same group and planned to be used at J-PARC and at ISIS in United Kingdom [START_REF] Stone | Report from the workshop on neutron scattering and high magnetic fields[END_REF] . Meanwhile, two successive experiments performed on the IN22 spectrometer from the Commissariat à l'Energie Atomique (CEA) at the highflux neutron source of the Institut Laue Langevin (ILL, Grenoble) have demonstrated the feasibility of the technique: combining a mini-coil system of the Nojiri group and a transportable pulsed field generator 12 developed by the Laboratoire National des Champs Magnétiques Intenses (LNCMI, Toulouse), they led to the first successful experiments associating pulsed magnetic fields up to 30 T and neutron diffraction [START_REF] Yoshii | [END_REF]22 . These experiments have also clearly demonstrated the necessity of improving the duty cycle of the coils, in particular by increasing the duration of the field pulse by at least an order of magnitude.

In this paper, we present the first long-duration and high duty cycle pulsed-field cryomagnet producing a maximum horizontal field of 40 T designed for neutron diffraction. Compared to mini-coils, this magnet offers improved experimental conditions for single-crystal neutron diffraction studies. Remarkably better counting statistics with fewer number of pulses, i.e. a ten to fifteen times superior duty cycle, result from an increased pulse duration combined with a faster cooling of the magnet. A twice as wide angular access in the scattering plane allows probing larger parts of the reciprocal space. In addition, longer rising times greatly reduce heating by eddy currents so that lower temperatures and accurate control of the sample temperature can be achieved. Finally, the larger bore provides not only a larger volume for the sample but also allows for a better and more sophisticated sample environment than the mini-coil system. A sapphire cold finger connected to a heat exchanger inside the cone of the magnet allows to cool the sample down to 2 K whilst decoupling mechanically and thermally the sample from the magnet.

II. CRYOMAGNET DESIGN AND PERFORMANCES

The coil has been designed and built by the LNCMI-Toulouse. Its construction is based on the combination of a "standard" rapid-cooling technique 23 around a conical bore, as was already performed for a coil developed for x-ray scattering 24 . The rapid-cooling technique consists in inserting liquid nitrogen cooling channels directly into the winding to increase the cooling efficiency of the coil and reach a high duty cycle. The major technological novelty is the winding of the coil around a double-cone constituting a main part of the cryostat since the cryomagnet should have conical openings on two sides for maximising neutron beam access. For that purpose, a highly non-magnetic 1 mm thick double-cone of stainless steel X2CrNi19-11 (EN 1.4306 / AISI 304L) was machined to provide opening angles of ±15 • and ±30 • in the scattering plane for respectively the incident and scattered beams (see Fig. 1).

The coil was wound of CuAg wire (CuAg 0.08%; ultimate tensile strength of 384 MPa at room temperature and 515 MPa at 77 K) reinforced with Zylon using the optimized reinforcement density technique 25 . It is composed of 32 layers of 4 to 40 turns. The inner diameter (winding diameter) is Ø24.5 mm (Ø16.5 mm inside the stainless steel double-cone). Its external diameter and height are Ø280 mm and 210 mm, respectively. To ensure high mechanical strengths, the wire was wound onto a coil-body of glass fiber-epoxy composite (FR4/G11).

The resulting magnet has an inductance of 60 mH and a resistance of 259 and 1632 mΩ at liquid nitrogen and room temperature, respectively. Using a 1.15 MJ (C = 4 mF, V = 24 kV) transportable power supply 26 developed at the LNCMI-Toulouse 27 , it produces a maximum pulsed-field of 40 T with a rise time of 23 ms and a total duration of more than 100 ms every 9 min (Fig. 2(a)). At lower fields, the time between pulses is reduced: every 7 and 5 min at 36 and 31 T, respectively. For fields below 20 T, the duty cycle is limited by the time necessary to charge the capacitors, i.e., about a minute. The duty cycle of this new cryomagnet is of 28 s per day at 30 T, and of 16 s per day at 40 T 28 , with t 90% 15.5 ms (t 90% is defined as the time per pulse during which the field is greater than 90% of the maximum field), meaning that we have a field higher than 36 T for 2.5 s per day. These performances remain far more superior than The coil is immersed in liquid nitrogen and the gas evaporated after a pulse is evacuated through a chimney. The sample is under vacuum and fixed at the end of a cold finger made from sapphire. This finger is cooled down with a helium flow controlled with a cold-valve. The temperature is stabilised with a heater mounted on the finger.

those of mini-coil systems, which store less energy but have shorter pulse duration. Neutron scattering experiments [START_REF] Yoshii | [END_REF]22,29 performed with miniature coils producing field pulses of less than 10 ms duration had indeed an overall duty cycle of less than 2 s per day for a maximum field value of 30 T, with t 90% 1.9 ms.

At the centre of the magnet, the field factor B/I is equal to 8.31 ± 0.02 T/kA (i.e. I/B = 120.3 ± 0.3 A/T) 30 and shows less than 1% of variation over a sphere of radius 4 mm (see Fig. 2(b)). This indicates that the magnetic field remains relatively constant over a large volume.

The coil is installed horizontally inside a cryostat developed by the ILL and constructed by AS Scientific (UK). To allow easy replacement and efficient cooling of the coil, the liquid nitrogen bath is central, large and surrounded by a liquid helium jacket (see Fig. 1). The initial cooldown time is 2.5 hours. The maximum liquid nitrogen consumption to cool down the magnet is 1000 litres per day at 40 T. To cope with this demand, a vacuum insulated pipeline has been installed to automate the refills. It is connected to a 10.000 litres tanks shared with other instruments, which allows to perform experiments for several days.

The vertical access for the neutron beam is limited by the presence of the heat exchanger to ±7 • . This opening matches the height of the 315 mm ×100 mm window of the scattered beam. To minimise background issues, the incident and outgoing windows are both made from 2 mm thick single crystalline silicon.

III. SAMPLE TEMPERATURE CONTROL

The main current leads being fixed on one side of the coil, we have installed the heat exchanger on the opposite side along the top surface of the cone onto which the coil is wound (scattered beam side) (see Fig. 3). In order to reach 2 K, a standard ILL cold-valve has been installed at the bottom of the liquid 4 He bath on the same side to reduce the length of the capillary driving liquid 4 He to the heat exchanger. The Joule-Thomson thermal expansion is performed by pumping 4 He in the heat exchanger with a pump connected to a Ø30 mm tube located at the top of the cryostat. The sample is placed in the central Ø16.5 mm cylinder where the magnetic field is maximal without touching the double-cone cooled down to liquid nitrogen temperature.

To avoid eddy currents and minimise neutron background, the heat exchanger (developed at the ILL) was made in Torlon and sapphire (Fig. 4). Torlon is insensitive to magnetic fields and rigid enough to build flanges tight to 4 He. Two holes were drilled into the Torlon bar: one of Ø3 mm for injecting the liquid 4 He coming from the cold-valve and one of Ø4.5 mm for pumping it out. The injection is performed with a Ø0.6×1 mm capillary brazed to the cold-valve flange and inserted in the Ø3 mm hole up to its end. The thermal expansion is thus realised at the end of the Torlon bar against a piece of single crystalline sapphire called cold-finger and shown in Figure 4(b). It consists of a bar wound with a Manganin wire for regulating the temperature (heater) and an octagonal hole in which the sample holder is placed. This shape prevents the sample from rotating around the axis of the coil. With the provided sample holders, it also allows a quick rotation of the single crystalline sample by 30 • or 45 • without realignment and regluing 31 .

The length and shape of the cold finger was designed to reduce as much as possible the presence of sapphire in the beam and to avoid neutron scattering by the Torlon bar. Two thermometers are installed: one near the heater (T exch ) and one on the sample holder (T samp ). To prevent heat production by eddy currents, we have chosen to glue compact Cernox CX-1050-SD sensors packed inside a sapphire base with alumina body and lid 32 .

The sample is glued on a holder also made from single crystalline sapphire and sits in the vacuum of the cryostat. The maximum available volume is 8 × 6 × 6 mm 3 . The incident monochromatic beam hits the sample and then crosses 24 mm of sapphire as shown in Figure 4. The transmission is about 96% at λ i = 2.36 Å (or k i = 2.662 Å-1 ) and almost 98% at λ i = 1.52 Å (or

k i = 4.1 Å-1 ).
The base temperature measured near the sample is slightly less than 2 K. This is a little more than the 1.5 K expected with a helium flow system because of the radiation of the surrounding screens thermalised at 77 K. The sample temperature can be stabilised at any value up to room temperature and the cool down time does not exceed 30 min. The helium boil-off rate is of ≈ 1 litre/hour at base temperature which leads to an autonomy of 1.5 day.

IV. APPLICATION TO NEUTRON DIFFRACTION

A. Data acquisition scheme and analysis

The efficiency of this new 40-tesla cryomagnet was tested and commissioned on the IN22 spectrometer at the ILL in elastic mode without any modification of the spec- trometer. The coil was energised by the transportable 1.15 MJ pulsed field generator of the LNCMI Toulouse (Fig. 5). A schematic view of the neutron scattering geometry is shown Figure 6.

The IN22 spectrometer is operated either in doubleaxis or triple-axis mode depending on the sample used. In this later mode, the signal-to-noise ratio can be improved by using a pyrolytic graphite (PG) analyser crystal. The incident neutron wavelength is chosen between λ i = 1 Å and λ i = 2.36 Å with a PG monochromator in order to avoid the blind zone of the magnet. The sample is usually oriented with a crystallographic axis parallel or slightly misaligned with the horizontal magnetic field to reach the most interesting part of the reciprocal space. Neutron counts are measured with a 3 He fast singledetector starting saturating above 18000 neutrons per second. The detector is oriented horizontally in the double-axis mode but vertically in the triple-axis mode to take advantage of the horizontal focalisation. Neutron pulses are recorded with a digital data recorder (HIOKI Memory HiCorder model 8860) measuring the generator voltage and current, and the voltage on a pick-up coil. After correction for the neutron time of flight, the field dependence of the neutron intensities at each momentum transfer is extracted by summing data accumulated over a few identical pulsed-field shots, with either constant time-or constant field-integration windows (rises and falls of the pulsed field are analysed separately).

B. Application to the frustrated antiferromagnet TbB4

To evaluate the performances of this coil and setup, the frustrated antiferromagnet TbB 4 was reinvestigated. In this system, two successive antiferromagnetic (AF) transitions occur at T N 1 44 K and T N 2 24 K 33 , leading to a quasi-planar non-collinear AF structure described by the propagation vector k = (0 0 0) 34 . Upon the application of a magnetic field along the c-axis, TbB 4 is known for its devil-staircase-like magnetization process, stabilising successive magnetization plateaux for fields µ 0 H > 16 T 35 . Because of the high magnetic moment of Tb 3+ ions, TbB 4 has been one of the first compounds studied by neutron diffraction combined with pulsed field miniature coils [START_REF] Yoshii | [END_REF] . A model of magnetic structure consisting of Ising spins and XY spins has been proposed to explain the half-magnetization state [START_REF] Yoshii | [END_REF]36 .

For the present work, single crystals of TbB 4 enriched with 11 B were kindly provided by Iga (Ibaraki University) and Michimura (Saitama University). A single crystal (∼ 2 × 2 × 1 mm 3 , m 30 mg) was mounted on the sapphire sample holder with the scattering plane defined by [1 0 0] and [0 0 1]. Magnetic field was applied along the c-axis. In Figure 7, the field dependence of the diffracted intensity measured at the momentum transfer Q = (1 0 0) in fields up to 35 T and at temperatures T = 2, 30 and 50 K is reported. At T = 2 K, these data exhibit stepwise variations of the diffracted intensity, highlighting at least three transitions and three AF phases. When the magnetic field is applied, the intensity shows a step-like increase at µ 0 H = 16/14 T (rising/falling field) followed by a step-like decrease at µ 0 H 18 T and finally vanishes at µ 0 H = 30 T. At T = 30 K > T N 2 , the intensity decreases continuously up to µ 0 H c = 25 T, indicating a single AF phase in fields up to µ 0 H c , in agreement with the H-T phase diagram determined by magnetization measurements 35 . At T = 50 K > T N 1 , the intensity is at the background level, attesting the absence of magnetic diffraction.

In Figures 8(a) and 8(b), a comparison of the field dependences of the diffracted intensity measured on TbB 4 at T = 4.2 K and Q = (1 0 0) using the new cryomagnet and the mini-coil system [START_REF] Yoshii | [END_REF] , respectively, illustrates the improved sensitivity of our cryomagnet to detect fine changes of magnetic structures. We note that, with the mini-coil system, the magnetic field was applied horizontally with a tilt of 5 • from the c-axis (dimensions and weight of the single crystal used in Ref. [START_REF] Yoshii | [END_REF] are 4×4×3 mm 3 and m 296 mg). Similar features to the ones measured at T = 2 K are detected at T = 4.2 K with the new 40tesla cryomagnet whereas they were not observed in the previous experiment. We note that even with a smaller number of pulses, the new data show better counting statistics.

The detailed study of TbB 4 by neutron diffraction under pulsed magnetic fields using our new cryomagnet will be reported in a forthcoming paper.

V. CONCLUSION

We have developed a 40-tesla pulsed-field cryomagnet for neutron diffraction featuring an unprecedented duty cycle of 28 s per day at 30 T and 16 s per day at 40 T. The sample temperature is regulated between 2 and 300 K independently from the pulse fields generation. We illustrate the performances of this magnet with the reinvestigation of the high field magnetic structures of the frustrated antiferromagnet TbB 4 . At low temperature, the field dependence of the diffracted intensity of the antiferromagnetic reflection (1 0 0) shows step-like variations which reveal subtle changes in the magnetic structures of the different magnetization plateau phases. These new features were observed thanks to higher counting statistics and a precise control of the sample temperature resulting both from the combination of a longer pulse duration, a higher duty cycle and a longer rising time (much less heating by eddy currents) than mini-coil systems.

More than four thousand field pulses, with 65% of them at more than 30 T, have already been generated with this magnet. They have allowed us to investigate various magnetic systems such as heavy fermions materials 28,37 and quantum spins systems 38 .

This equipment is available to the ILL users through a scientific collaboration with the LNCMI.

Several improvements are planned for the near future. First we will implement a high counting rate, high efficiency position sensitive detector, having good-enough spatial and time resolutions. Then, modifications of the cryogenic environment will be made, with the target to extend the temperature range down to 1.4 K.

FIG. 1 .

 1 FIG. 1. Three-dimensional views of the magnet for neutron diffraction. (a) The coil is wound around a stainless steel doublecone. The nitrogen circulates through cooling channels and exits through the exhaust pipe highlighted in blue colour in view (b). (c) Overview of the 40-tesla cryomagnet showing the coil at the bottom of the liquid nitrogen bath and the liquid He jacket used to cool the sample down to 2 K.

FIG. 2 .

 2 FIG. 2. (a) Time profile of the field pulsed up to 40 T. (b) Magnetic field homogeneity calculated from the centre of the magnet. The longitudinal (axial) and radial distribution are denoted by B(z) and B(r), respectively.

FIG. 3 .

 3 FIG.3. Overview of the tail of the cryomagnet. The coil is immersed in liquid nitrogen and the gas evaporated after a pulse is evacuated through a chimney. The sample is under vacuum and fixed at the end of a cold finger made from sapphire. This finger is cooled down with a helium flow controlled with a cold-valve. The temperature is stabilised with a heater mounted on the finger.

FIG. 4 .

 4 FIG. 4. Photos of the heat exchanger: (a) the Joule-Thomson thermal expansion of 4 He is performed against the flange of sapphire screwed at the end of the Torlon tube in which 4 He circulates. Liquid 4 He is injected into the capillary that can be seen at the right bottom of the image. The flexible pipe through which 4 He is pumped is also shown. (b) Detailed assembly of the sapphire cold finger and sample holder.

FIG. 5 .

 5 FIG. 5. IN22 triple-axis spectrometer equipped with the mobile pulsed magnet system consisting of the mobile 1.15 MJ pulsed field generator and the 40-tesla cryomagnet.

FIG. 7 .

 7 FIG. 7. Field dependence of the diffracted intensity at the momentum transfer Q = (1 0 0) in fields up to 35 T at T = 2, 30 and 50 K, measured with the 40-tesla cryomagnet. The magnetic field was applied along the c-axis. Data are integrated within field steps ∆(µ0H )= 0.75 T. The error bars ∆I are given by the square root of the neutron counts (∆I = I/τ where τ is the total accumulation time over the number of pulses indicated for each temperature). Open and full symbols correspond to rising and falling fields, respectively.

FIG. 8 .

 8 FIG. 8. (a) Field dependence of the diffracted intensity at the momentum transfer Q = (1 0 0) in fields up to 35 T at T = 4.2 K, measured with the new cryomagnet. The magnetic field was applied along the c-axis. Data are integrated within field steps ∆(µ0H )= 0.75 T. The error bars ∆I are given by the square root of the neutron counts (∆I = I/τ where τ is the total accumulation time over the number of pulses indicated for each temperature). (b) Field dependence of the diffracted intensity at the same momentum transfer in fields up to 30 T measured with the miniature coil system (Ref. 21 ). The magnetic field was applied horizontally with a tilt of 5 • from the c-axis of the crystal. Neutron counts are summed over 120 identical pulsed-field shots, with constant time-integration windows ∆t = 40 µs. For both data sets, open and full symbols correspond to rising and falling fields, respectively.

ACKNOWLEDGMENTS

The authors are very grateful to F. Lecouturier and N. Ferreira who performed the characterization of the wire conductor (resistivity and wire strength measurements) and to J.-M. Lagarrigue who machined several pieces required for the building of the coil. The authors would like to thank H. Nojiri for very helpful technical and scientific discussions. The CEA-CRG Grenoble and the ILL are greatly acknowledged for granting the beam time for these developments and experiments. This work was financially supported by the French National Research Agency (ANR project MAGFINS: Grant N•. ANR-10-BLN-0431) and by the program Investissements d'Avenir ANR-11-IDEX-0002-02 (reference ANR-10-LABX-0037-NEXT).