Y Savoye-Peysson

D Mazon

M Scholz

J Bielecki

D Dworak

A Jardin

K Król

P Malard

Denis Guibert

J Walkowiak

à la diffusion de documents scientifiques de niveau

recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

With the decision to use heavy metallic elements like W for PFCs in ITER, the in-depth understanding of the interaction between high-Z impurities and the physics of RF waves is of crucial importance. This key problem can be addressed extensively in the WEST tokamak at the ITER-relevant RF power level, since it is equipped with actively cooled tungsten PFCs, while all auxiliary heating and current drive systems are RFbased, in particular at the LH frequency. The HARMONIA project is a joint effort between French (CEA IRFM) and Polish (IFJ PAN) teams to study the interplay between heavy impurity transport and LH current drive in the WEST tokamak. It addresses many topics, which can be gathered into three categories :

1. Theory/modeling : Advanced LH wave modelling, Atomic physics and kinetic calculations 2. Data analysis : Soft X-ray tomographic inversion, Impurity concentration and electron temperature determination from multi-energy X-ray measurements

Data Analysis

• Atomic form factor F(q) : Fourier transform of the density of bound electrons • Universal approach for describing the density of bound electrons from quantum calculations whatever the type of ion and its ionization state : multi-Yukawa description (up to 4 exponentials) • Consistent description of screening effects on collisions (elastic and inelastic) in Fokker-Planck (LUKE) calculations (runaway to slide-away) and bremsstrahlung (R5-X2) whatever the plasma regime. • New variational quantum model for the mean excitation energy (inelastic scattering)

• Prediction of the LH power absorption profile is a challenging issue because of the large spectral gap in existing machine (low T, aspect ratio) : toroidal refraction is not strong enough to prevent stochasticity to develop before absorption. However, it is not observed experimentally. • It was inferred that the spectral gap is already filled in the SOL, before enter the core plasma, by edge turbulence.

• The Tail Spectral Model (TSM) is tested in extreme conditions where the propagation domain is bound and the usual toroidal upshift of the parallel refractive index N || cannot be bridged by this mechanism:  without TSM, no LH current and HXR predicted;  with TSM, modeling consistent with observations.

• Chain of codes : ALOHA + (TSM) + C3PO + LUKE + R5-X2. TSM routinely used for WEST LH discharges.

ALOHA TSM

Soft X-ray tomographic inversion

Impurity concentration and electron temperature from multi-energy X-ray measurements

• Validation the robustness of the SXR reconstruction against the noise level by Tikhonov inversion method • Possibility to include the magnetic equilibrium (anisotropic regularization) in the reconstruction.

Tomographic reconstruction test for different numbers of cameras used in the reconstruction. Good resilience up to 20% noise level.

• Experimental test on WEST with SXR and HXR combined analysis.

The

  HARMONIA project : a joint effort to study the interplay between heavy impurity transport and the current density profile driven by the lower hybrid wave in the WEST tokamak Introduction Advanced Lower Hybrid (LH) wave modelling 1 CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France. 2 Institute of Nuclear Physics Polish Academy of Sciences (IFJ PAN), PL-31-342, Krakow, Poland * http://west.cea.fr/WESTteam

3 .

 3 Diagnostic developments : Gas Electron Multiplier (GEM) techniques and experimental validation -Peysson 1 , D. Mazon 1 , M. Scholz 2 , J. Bielecki 2 , D. Dworak 2 , A. Jardin 2 , K. Król 2 , P. Malard 1 , Denis Guibert 1 , J. Walkowiak 2 and the WEST team* 𝜀 𝜂 𝑊 = η(ℎ𝑣)𝜀 𝑊 ℎ𝑣,𝑇 𝑒 𝑑ℎ𝑣 = 𝑛 𝑒 .𝑛 𝑊 .𝐿 η 𝑊 𝑇 𝑒 𝐿 η 𝑊 : filtered cooling factor for tungsten η(ℎ𝑣) : spectral range Pulse #58327 time trace evolution of GEM total number of counts measured/modelled (left), GEM total counts per channels (middle), total measured / modelled spectra (right)Diagnostic Developments• Multi-machine validation of the Tail Spectral Model for Lower Hybrid modelling.• Universal description of quantum atomic models for incorporation of screening effects in fast electron dynamics (Fokker-Planck and bremsstrahlung calculations). • Robustness of SXR emission tomographic reconstruction against data noise and anisotropic regularization.• Determination of the impurity (tungsten) concentration from X-ray multi-energy line-integrated measurements.• GEM synthetic modelling have now been benchmarked with experiments.• New WEST GEM SXR measurements (C7 campaign) have been successfully performed on long pulses (up to 60s) in a routinely automatic manner and are now validated.ConclusionsAcknowledgements. This work has been partially funded by the National Science Centre, Poland (NCN) grant HARMONIA 10 no. 2018/30/M/ST2/00799. We gratefully acknowledge Poland's high-performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2022/015994. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 -EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. This work has been published in the framework of the international project co-financed by the Polish Ministry of Education and Science, as program "PMW".

2 SXR

 2 𝑐 𝑊,0 ,𝑇 𝑒,0 -𝑓 𝑖,𝑚𝑒𝑎𝑠 RMS error map for WEST #54981@t=6.5s and result of the HXR optimization.  Energy resolution capabilities  Radiation resilience (gaseous detection volume)  Compact 𝑐 𝑊 = 2 × 10 -4 Pulse #58327 parameters (left), c W determination by SXR horizontal camera modeling (middle), radiative power modelling from SXR horizontal and vertical GEM cameras (right) • Proof-of-principle: (c W , T e ) estimation in plasma core based on synthetic X-ray measurements. SXR RMS error map for a synthetic scenario (proof-of-principle). Left : ALOHA + TSM LH power spectrum, Center : LH domain of propagation and ray-tracing (C3PO), Right : HXR emission line-integrated profile (full chain of codes) Top left : tungsten ionized fraction vs plasma temperature, Top center : density of bound electrons for tungsten by quantum relativistic codes, Top right : Multi-Yukawa fit of quantum density of bound electrons density, Bottom Left: Multi-Yukawa fit of quantum density of bound electrons linear density, Bottom center : R2 agreement between multi-Yukawa and quantum calculations, Bottom right : Mean excitation energy for tungsten by different models using multi-Yukawa description.

CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France

Institute of Nuclear Physics Polish Academy of Sciences (IFJ PAN), PL-31-342, Krakow, Poland * http://west.cea.fr/WESTteam

the WEST tokamak.


Julien Rouyer
email: julien.rouyer@univ-reims.fr

Alain Ninet Crestic

Lmr

Two New Integer Sequences Related to Crossroads and Catalan Numbers

Keywords: {1}, {2}, {3}, {4} }, { {1}, {2}, {3, 4} }, { {1}, {2, 3}, {4} }, { {1, 4}, {2}, {3} }, { {1, 2}, {3}, {4} } { {1, 2, 3, 4} }, { {1, 2, 3}, {4} }, { {1, 2, 4}, {3} }, { {1, 3, 4}, {2} }, { {1}, {2, 3, 4} }, { {1, 2}, {3, 4} }, { {1, 4}, {2, 3} }, { {1, 3}, {2}, {4} }, { {1}, {2, 4}, {3} }

The marriageable singles sequence represents the number of noncrossing partitions of the finite set {1, . . . , n} in which some pair of singleton blocks can be joined while remaining noncrossing. The lonely singles sequence represents the number of all the other noncrossing partitions of the finite set {1, . . . , n} and is the difference between the Catalan numbers sequence and the marriageable singles sequence. The 14 first terms of these sequences are given, as well as some of their properties. These sequences appear when one wants to count the number of ways to cross simultaneously certain road intersections.

Examples 5. The set [4] has 5 marriageable singles partitions which are

Introduction

The number of noncrossing partitions of the finite set {1, . . . , n} (with n any positive integer) is very well known to be the Catalan number C n . See, for example, Stanley [2, entry 159, p. 43] and Roman [4, pp. 51-60] for a quick introduction to Catalan numbers and noncrossing partitions. Noncrossing partitions have been hugely studied, since at least Becker [START_REF] Becker | Planar rhyme schemes[END_REF], where they are called planar rhyme schemes but their systematic study began with Kreweras [START_REF] Kreweras | Sur les partitions non croisées d'un cycle[END_REF] and Poupard [START_REF] Poupard | Étude et dénombrement parallèles des partitions non-croisées d'un cycle et des découpages d'un polygone convexe[END_REF]. Simion [START_REF] Simion | Noncrossing partitions[END_REF] presents a summary of related results available in 2000 and some further work can be found in McCammond [START_REF] Mccammond | Noncrossing partitions in surprising locations[END_REF], Callan [START_REF] Callan | Sets, lists and noncrossing partitions[END_REF] and Kim [START_REF] Kim | New Interpretations for Noncrossing Partitions of Classical Types[END_REF].

The study of combinatorial properties of crossroads led us to determine the number of noncrossing partitions such that no pair of singleton blocks {i} and {j} (with i = j) can be merged into the pair {i, j} while the partition remains noncrossing. These noncrossing partitions appear when one wants to determine the number of possible manners to cross simultaneously a road intersection in which entries and exits are alternated, with the constraint that U-turns are prohibited. For a quick introduction to road intersection crossing management for intelligent vehicles, see Rouyer et al. [START_REF] Rouyer | A road intersection control in urban intelligent transportation systems[END_REF] and Bai et al. [START_REF] Bai | A general framework for intersection traffic control with backpressure routing[END_REF].

In what follows and the next section, we give definitions and quick examples. We then proove some properties of both marriageable singles and lonely singles sequences. At the end, we give the first values of those sequences and formulate several conjectures concerning them.

Definition 1. For all n ∈ N, [n] denotes the n-set {1, . . . , n}. In particular, [0] = ∅. Definitions 2. Let n be a non-negative integer and let

π = {A 1 , . . . , A k } be a partition of [n] (i.e., ∪ k i=1 A i = [n] and ∀ 1 ≤ i < j ≤ k, A i ∩ A j = ∅ and ∀ 1 ≤ i ≤ k, A i = ∅), this partition π is said to be a • crossing partition if ∃ 1 ≤ i ≤ k, ∃ 1 ≤ j ≤ k, i = j, ∃ a < b ∈ A i , ∃ c < d ∈ A j , a < c < b < d, • noncrossing partition if ∀ 1 ≤ i < j ≤ k, ∀ a < b ∈ A i , ∀ c < d ∈ A j , a < b < c < d, or c < d < a < b, or a < c < d < b, or c < a < b < d.
Definitions 3. A noncrossing partition π of [n] is called marriageable singles partition if there exists a pair of singleton blocks {i} and {j} in π that can be joined while remaining noncrossing.

More precisely, let π = { A 1 , . . . , A k } be a noncrossing partition of [n] with at least two singleton blocks A 1 = {i} and A 2 = {j}. Then π is a marriageable singles partition iif

π ′ = { {i, j}, A 3 , . . . , A k } is a noncrossing partition of [n].
Conversely, a noncrossing partition of [n] is called lonely singles partition if it is not a marriageable singles partition.

Let M n and L n be the number of marriageable singles and lonely singles partitions of [n] respectively. The marriageable singles and lonely singles sequences are (M n ) n≥0 and (L n ) n≥0 respectively. Remark 4. A partition that contains at most one singleton block is clearly a lonely singles partition. In an equivalent way, a marriageable singles partition contains at least two singleton blocks.

The first seven have no pair of singleton blocks and are clearly not marriageable singles partitions. The last two have only one pair of singleton blocks, but after merging it, they both give { {1, 3}, {2, 4} } which is a crossing partition. Lemma 6. Let C n (for n ≥ 0) denote the number of noncrossing partitions of [n]. Then, we have

C n = L n + M n .
Proof. As the set of noncrossing partitions is the disjoint union of the sets of lonely singles and marriageable singles partitions, the result follows immediately.

Remark 7. It is well known that C n = 1 n+1 2n n
is the nth Catalan number. The sequence (C n ) is referenced as A000108 in the On-Line Encyclopedia of Integer Sequences [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]. Remark 8. The unique partition of [0] = ∅ is the empty partition ∅. This partition is noncrossing (the first Catalan number is C 0 = 1) and it is a lonely singles partition.

Standard road intersection

Noncrossing complete matchings of 2n points lying on a line and noncrossing complete set of chords are two of the many standard combinatorial objects counted by the Catalan numbers. Noncrossing complete matchings and noncrossing complete set of chords are in natural bijection with the noncrossing partitions, as explained by Stanley [2, entries 59 and 61, p. 28].

We introduce here some notions on road intersections corresponding to noncrossing sets of chords. Definition 9. A road intersection with n entries and n exits alternated is called a standard road intersection of size n.

Let E 1 , . . . , E n denote the entries and X 1 , . . . , X n denote the exits of a standard road intersection of size n. These entries and exits are numbered clockwise (Figure 1 gives such a representation of a standard road intersection of size n = 4).

E 1 X 1 E 2 X 2 E 3 X 3 E 4 X 4 Figure 1: A Standard Road Intersection of size n = 4.
We represent graphically a way to cross simultaneously a standard road intersection by a bipartite graph (see Figures 2 and3 for two examples with n = 4). In these graphs, black vertices represent entries and white vertices represent exits. Each edge represents the crossing of the intersection by a vehicle going from an entry to an exit.

Each entry can be connected to each exit, unless restrictions are indicated.

Definition 10. For a given standard road intersection, any edge starting from one entry E i and ending to one exit X j is a lane. E i X j denotes such a lane.

Definition 11. For a given standard road intersection, any lane of the type E i X i , i.e., any edge starting from one entry E i and ending to exit X i is a U-turn. Definition 15. For a given standard road intersection, an MSL is said to be absolute when it does not contain two U-turns E i X i and E j X j (with i = j) that can be changed into two lanes E i X j and E j X i to give another MSL.

Lemma 16. For a given standard road intersection of size n, the set of MSL is in one-toone correspondence with the set of noncrossing partitions of [n]. The number of MSL of a standard road intersection of size n is equal to the Catalan number C n . Proof. As a MSL can be seen as a set of n nonintersecting chords joining 2n points, this result is very well known. See Stanley [2, entry 59, p. 28].

E 1 X 1 E 2 X 2 E 3 X 3 E 4 X 4
{ {1, 2, 3}, {4} }. E 1 X 1 E 2 X 2 E 3 X 3 E 4 X 4
E 1 X 1 E 2 X 2 E 3 X 3 E 4 X 4
The following three corollaries are just reinterpretations of the lonely singles and marriageable singles definitions in the language of road intersections.

Corollary 17. M n is the number of nonabsolute MSL of a standard road intersection of size n.

Corollary 18. L n is the number of absolute MSL of a standard road intersection of size n.

Corollary 19. L n is the number of MSL for a restricted standard road intersection of size n.

3 Properties Proposition 20. Both sequences (L n ) and (M n ) are increasing: L n < L n+1 for all n ≥ 2 and M n < M n+1 for all n ≥ 3.

Proof. We build a simple injective map f n from the set LS n of the lonely singles partitions of [n] to the set LS n+1 of the lonely singles partitions of [n + 1] by merging the singleton {n + 1} to the unique element A 1 of a partition π that contains the number 1 (one could equally prefer to use the number n instead of the number 1: the idea is to stick the number n + 1 to one of its two direct neighbours 1 or n):

LS n → LS n+1 f n : π = { A 1 , . . . , A k } → { A 1 ∪ {n + 1}, . . . , A k } where 1 ∈ A 1 and A 1 ∪ • • • ∪ A k = [
n] and A i ∩ A j = ∅ for all i = j. For example,

f 3 ({ {1, 2}, {3} }) = { {1, 2, 4}, {3} } f 4 ({ {1}, {2, 4}, {3} }) = { {1, 5}, {2, 4}, {3} }
The map f n is clearly injective and any lonely singles partition π gives a lonely singles partition f n (π). As L n is the cardinality of LS n and L n+1 is the cardinality of LS n+1 , we obtain L n ≤ L n+1 .

In a similar way, we build a simple injective map g n from the set MS n of the marriageable singles partitions of [n] to the set MS n+1 of the marriageable singles partitions of [n + 1] by adding the singleton {n + 1} to any partition π:

MS n → MS n+1 g n : π → π ∪ { {n + 1} } For example g 4 ({ {1}, {2, 3}, {4} }) = { {1}, {2, 3}, {4}, {5} }
The map g n is clearly injective and any pair of marriageable singletons {i} and {j} of π ∈ MS n remains marriageable as elements of g n (π) ∈ MS n+1 . As M n is the cardinality of MS n and M n+1 is the cardinality of MS n+1 , we obtain M n ≤ M n+1 .

More precisely, we have L n < L n+1 for all n ≥ 2 and M n < M n+1 for all n ≥ 3: it is easy to build a lonely singles and a marriageable singles partitions of [n+1] that are not in the images of the maps f n and g n , e.g. respectively { [n], {n+1} } and { {1}, . . . , {n-1}, {n, n+1} }.

Corollary 21. lim n→+∞ M n = lim n→+∞ L n = +∞ Proof.
It is an an immediate consequence of Proposition 20.

Proposition 22. For all n ≥ 0, we have

C n + 3M n ≤ M n+2 .
Proof. We build four simple injective maps h n , i n , j n and k n with disjoint images included in the set MS n+2 of the marriageable singles partitions of [n + 2].

• The map h n is defined on the set NC n of all noncrossing partitions of [n] by adding both the singletons {n + 1} and {n + 2} to any noncrossing partition π:

NC n → MS n+2 h n : π → π ∪ { {n + 1}, {n + 2} }
The maps i n , j n and k n are defined on the set MS n of the marriageable singles partitions of [n].

• The map i n adds the pair {n + 1, n + 2} to a marriageable singles partition π:

MS n → MS n+2 i n : π → π ∪ { {n + 1, n + 2} }
• The map j n merges the singleton {n + 2} with the unique element A 1 of a marriageable singles partition π that contains the number 1 and adds the singleton {n + 1} to π:

MS n → MS n+2 j n : π = { A 1 , . . . , A k } → { A 1 ∪ {n + 2}, A 2 , . . . , A k , {n + 1} } where 1 ∈ A 1 , A 1 ∪ • • • ∪ A k = [n] and A i ∩ A j = ∅ for all i = j.
• the map k n merges the singleton {n + 1} with the unique element A 1 of a marriageable singles partition π that contains the number n and adds the singleton {n + 2} to π:

MS n → MS n+2 k n : π = { A 1 , . . . , A k } → { A 1 ∪ {n + 1}, A 2 , . . . , A k , {n + 2} } where n ∈ A 1 , A 1 ∪ • • • ∪ A k = [n] and A i ∩ A j = ∅ for all i = j.
For example,

h 4 ({ {1, 2, 3}, {4} }) = { {1, 2, 3}, {4}, {5}, {6} } i 4 ({ {1, 2}, {3}, {4} }) = { {1, 2}, {3}, {4}, {5, 6} } j 4 ({ {1, 2}, {3}, {4} }) = { {1, 2, 6}, {3}, {4}, {5} } k 4 ({ {1, 2}, {3}, {4} }) = { {1, 2}, {3}, {4, 5}, {6} }
We have the immediate following properties:

• h n (π) is a marriageable singles partition of [n + 2] for all noncrossing partition π of [n],
• i n (π), j n (π) and k n (π) are marriageable singles partitions of [n + 2] for all marriageable singles partition π of [n] (the marriageable pairs can change depending on the map selected),

• h n , i n , j n and k n are injective maps; thus, -NC n and h n (NC n ) have the same cardinality C n , -MS n and i n (MS n ) and j n (MS n ) and k n (MS n ) have the same cardinality M n ,

• the sets

h n (NC n ), i n (MS n ), j n (MS n ) and k n (MS n ) are disjoint; thus, h n (NC n ) ⊔ i n (MS n ) ⊔ j n (MS n ) ⊔ k n (MS n ) ⊂ MS n+2 ,
• as M n+2 is the cardinality of MS n+2 , we obtain

C n + 3M n ≤ M n+2 .
Definition 23. For all nonnegative integers n, m and k, let NC n,m,k be the number of noncrossing partitions of [n] in m classes with k singleton blocks.

Proposition 24. For all nonnegative integers n and m with (n, m) = (0, 0), the number of noncrossing partitions of [n] in m classes with no singleton block is

NC n,m,0 = 1 n -m + 1 n m n -m -1 m -1 .
When (n, m) = (1, 1), the number of noncrossing partitions of [n] in m classes with exactly one singleton block is

NC n,m,1 = n m -1 n -m -1 m -2 .
Proof. Poupard [START_REF] Poupard | Étude et dénombrement parallèles des partitions non-croisées d'un cycle et des découpages d'un polygone convexe[END_REF] proved that, when n ≥ 1 and m ≥ 1, the number of noncrossing partitions of [n] in m classes with no singleton block is NC n,m,0 =

1 n-m+1 n m n-m-1
m-1 , and n ≥ 2m. When n = 0 and m ≥ 1 or n ≥ 1 and m = 0 or m ≥ ⌊ n 2 ⌋ + 1, this equality stands and gives a number of such partitions equal to 0.

When n ≥ 1 and m ≥ 1, the set of noncrossing partitions of [n] in m classes with exactly one singleton block is clearly in one-to-one correspondence with the set of couples (π, i), where π is any noncrossing partition of [n -1] in m -1 classes with no singleton block and 1 ≤ i ≤ n. Then, when (n, m) = (1, 1), we obtain that number of noncrossing partitions of [n] in m classes with exactly one singleton block is

NC n,m,1 = n×NC n-1,m-1,0 = n n-m+1 n-1 m-1 n-m-1 m-2 = n m-1 n-m-1 m-2 .
Remark 25. Here are a few details on specific cases:

• The empty partition is the unique noncrossing partition of the empty set [0] in 0 class with no singleton block. Thus, NC 0,0,0 = 1.

• The set [START_REF] Rouyer | A road intersection control in urban intelligent transportation systems[END_REF] has a unique noncrossing partition. This partition has m = 1 class and it is a singleton block. Thus, NC 1,1,1 = 1 and NC 1,m,k = 0 when (m, k) = (1, 1) and, in particular, NC 1,1,0 = 0.

• For all positive integer n, [n] has no noncrossing partition in m = 0 class. Thus, for all nonnegative integer k, NC n,0,k = 0 and in particular NC n,0,0 = 0.

Proposition 26. For all n ≥ 2,

L n ≥ ⌊ n 2 ⌋ m=1 1 n -m + 1 n m n -m -1 m -1 + ⌊ n+1 2 ⌋ m=2 n m -1 n -m -1 m -2 .
Proof. As any noncrossing partition with at most one singleton block is a lonely singles partition, we have

L n ≥ ⌊ n 2 ⌋ m=1 NC n,m,0 + ⌊ n+1 2 ⌋
m=1 NC n,m,1 and the result follows, using Proposition 24.

Proposition 27. For all n ≥ 3,

M n ≥ n-1 i=1 n j=i+1   ⌊ n+i-j-1 2 ⌋ m=0 1 n + i -j -m n + i -j -1 m n + i -j -m -2 m -1 × ⌊ j-i-1 2 ⌋ m=0 1 j -i -m j -i -1 m j -i -m -2 m -1   Proof.
Let π be a marriageable singles partition of [n] with exactly two singleton blocks {i} and {j} (with i < j). The set of such partitions π is clearly in one-to-one correspondence with the set of triplets ({i, j}, π 1 , π 2 ) where 1 ≤ i < j ≤ n and π 1 and π 2 are noncrossing partitions of the sets [n + ij -1] and [ji -1] respectively, with no singleton blocks. Thus,

M n ≥ n-1 i=1 n j=i+1   ⌊ n+i-j-1 2 ⌋ m=0 NC n+i-j-1,m,0 ⌊ j-i-1 2 ⌋ m=0 NC j-i-1,m,0   .
Remark 28. The previous inequality can be improved, considering three or more singleton blocks that can all be joined.

Known values and conjectures

Very few values of the sequences L n and M n are known for the moment. They are given in Table 1. L n and M n are respectively sequences A363448 and A363449 in the Online Encyclopedia of Integer Sequences (OEIS) [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]. The values of M n have been obtained with the algorithms available in the attached files, provided in pdf and ipynb formats. The complexity of these algorithms is exponential: they need to be improved and to be run with a more powerful computer to obtain values of L n and M n for n ≥ 15. Calculations were performed under Python 3.7.3 using Jupyter Notebook 5.7.6 with a 3.19 GHz i7-8700 processor and 32 GB RAM.

n L n L n /L n-1 M n M n /M n-1 C n M n /L n M n /C n Computing time for M n 0 1 0 1 0 0 0s 1 1 1 0 1 0 0 0s 2 1 1 1
Conjectures 29. We conjecture the five following propositions:

∀n ∈ N, n ≥ 9 ⇒ M n > L n , (1)
lim n→+∞ M n L n = +∞, (2)
lim n→+∞ M n C n = 1, (3)
lim n→+∞ L n C n = 0, (4)
lim n→+∞ M n+1 M n = lim n→+∞ L n+1 L n = 4. ( 5
)
Remark 30. Obviously, conjecture (2) implies conjecture (1). Table 1 shows that Mn Ln grows slowly and conjecture 2 may stand. More clearly, Table 1 shows that Mn Cn seems to grow quite quickly and let think that conjecture 3 stands. As C n = M n + L n , conjectures (2), ( 3) and ( 4) are clearly equivalent. It is well known and easy to prove that lim n→+∞ C n+1 Cn = 4. Table 1 shows that the similar limits given in conjecture (5) look very realistic as well.

Figure 2 :

 2 Figure 2: A bipartite graph associated with the intersection represented in Figure 1, with an example of absolute MSL corresponding to the noncrossing partition { {1, 2, 3}, {4} }.

Figure 3 :

 3 Figure 3: A bipartite graph associated with the intersection represented in Figure 1, with an example of a nonabsolute MSL corresponding to the noncrossing partition { {1, 2}, {3}, {4} }.

Figure 4 :

 4 Figure 4: Simplified graph of Figure 2, showing explicitly the noncrossing lonely singles partition { {1, 2, 3}, {4} }.

Figure 5 :

 5 Figure 5: Simplified graph of Figure 3, showing explicitly the noncrossing marriageable singles partition { {1, 2}, {3}, {4} }.

Figure 6 :

 6 Figure 6: Modification of Figure 3 to obtain an absolute MSL corresponding to the noncrossing partition { {1, 2}, {3, 4} }.

Figure 7 :

 7 Figure 7: Simplified graph of Figure 6, showing explicitly the noncrossing lonely singles partition { {1, 2}, {3, 4} }.

Table 1 :

 1 Values of L n , M n , C n , L n /L n-1 , M n /M n-1 , M n /L nand M n /C n (given with 2 digits), with computing time of M n , for all n ≤ 14.

	2	1	0.5	0s

Acknowledgements

We would like to thank Frédéric Blanchard and Rupert Wei Tze Yu for their constant help. Our deepest gratitude goes to the anonymous referee who gave us directions to improve this paper. Thank you Jeffrey Shallit for your great patience during the reviewing process of this paper.


Henrik H Kristensen

Lorenz Kranabetter

Constant A Schouder

Jacqueline Arlt

Frank Jensen

Henrik Stapelfeldt

Editors' Suggestion Laser-induced Coulomb explosion imaging of alkali-metal dimers on helium nanodroplets

Alkali-metal dimers, Ak 2 , residing on the surface of He nanodroplets, are doubly ionized due to multiphoton absorption from an intense 50-fs laser pulse leading to fragmentation into a pair of alkali-metal cations. Based on the measured kinetic energy distributions P(E kin ) of the Ak + fragment ions, we retrieve the distribution of internuclear distances P(R) via the Ak 2+ 2 potential curve. Results are obtained for Na 2 , K 2 , Rb 2 , and Cs 2 in both the 1 1 g + ground state and the lowest-lying triplet state 1 3 u + and for Li 2 in the 1 3 u + state. For Li 2 , K 2 , and Rb 2 , the center of the measured P(R) is close to the center of the wave function (R) of the vibrational ground state in the 1 1 g + and 1 3 u + states, whereas for Na 2 and Cs 2 small shifts are observed. For all the Ak 2 , the width of the measured P(R) is broader than | (R)| 2 by a factor of 2-4. We discuss that resonance effects in the multiphoton ionization and interaction of the Ak + ion with the He droplet give rise to the observed deviations of P(R) from | (R)| 2 . Despite these deviations, we deem that timed Coulomb explosion will allow imaging of vibrational wave packets in alkali-metal dimers on He droplets surfaces.

I. INTRODUCTION

When a molecule in the gas phase is irradiated by a sufficiently intense femtosecond laser pulse it undergoes multiple ionization. The resulting multiply charged molecular cation is typically unstable and breaks apart into ionic fragments, a process termed Coulomb explosion [1]. The spatial orientation and structure of the molecule, at the instant the laser pulse arrives, are imprinted on the emission direction and on the kinetic energy of the ionic fragments. Therefore, measurements of these experimental observables provide an opportunity for extracting information about how the molecule is turned in space and about its structure at the instant the laser pulse ionizes the molecule [2,3]. In particular, for a two-body system such as a diatomic molecule, the distribution of internuclear distances P(R) can be determined from the measured distribution of kinetic energies P(E kin ) of the atomic fragment ions provided there is a unique correspondence between R and E kin . This is fulfilled if the interaction between the fragment ions is characterized by a single potential curve like in the case of two H + ions (He + ions) produced from double ionization of a hydrogen molecule [4,5] (He dimer [START_REF] Zeller | Proc. Natl. Acad. Sci. USA[END_REF]) or for two atomic ions resulting from double ionization of a dissociating molecule, such as I 2 , where the internuclear distance is so large that there is no molecular bonding left [START_REF] Petersen | [END_REF]. In many other cases, the multiply charged ions can however fragment via several different potential curves, which precludes an accurate retrieval of P(R) in the parent diatomic molecule or molecular dimer.

* henriks@chem.au.dk Recently, this was illustrated for double ionization of (CS 2 ) 2 and Ar 2 [8].

Here we study femtosecond laser-induced Coulomb explosion of alkali-metal homodimers, i.e., Ak 2 , where Ak is either Li, Na, K, Rb, or Cs. Upon double ionization, Ak 2 will be stripped of its two valence electrons and the resulting closed-shell structure of Ak 2+ 2 gives rise to only a single repulsive potential curve. As such, Coulomb explosion of alkali-metal dimers appears well suited for determining the distribution of their internuclear distances. The purpose of the present work is to explore if this is the case and how closely the determined P(R) resembles the internuclear wave function. The alkali-metal dimers are formed at the surface of nanometer-sized droplets of liquid helium, which offers an opportunity for studying Ak 2 in both the 1 1 g + ground state and in the lowest-lying triplet state 1 3 u + (often termed the X state and the a state, respectively) [9][10][11][12][13][14]. Our motivation for the work is the fact that if Coulomb explosion is capable of retrieving P(R), and thereby the internuclear wave function, it opens opportunities for directly imaging the time evolution of vibrational wave packets. In particular, given the presence of the He droplet, it should allow real-time observations of how coupling between the He environment and the dimer perturbs the coherence and perhaps the eigenstate populations of vibrational wave packets [15][16][17].

II. PRINCIPLE OF COULOMB EXPLOSION IMAGING OF Ak 2

The starting point of our method is double ionization of an alkali-metal dimer Ak 2 in either the 1 1 g + or the 1 3 u + state. Assuming that the dimers have equilibrated to the 0.37 K temperature of the droplets [18], then only the vibrational ground state is populated in either of the two electronic states. The double ionization happens as a result of absorption of N photons from an intense femtosecond laser pulse as illustrated in Fig. 1(a) for the case of Na 2 , i.e., Na 2 + Nhν → Na 2+ 2 + 2e -. With a central wavelength of the laser pulse of 800 nm, absorption of at least nine (ten) photons is required to double ionize Na 2 in the 1 potential curve of Na 2+ 2 . The repulsive character of this curve causes the molecular dication to subsequently break apart into a pair of Na + ions, i.e., Coulomb explosion due to the electrostatic repulsion between the two Na + ions. During the Coulomb explosion, the potential energy of Na 2+ 2 is converted into kinetic energy of the two atomic fragments. Thus, a Na 2+ 2 ion with an internuclear separation of R will produce two Na + ions each with E kin = 1 2 V (R), where V (R) is the potential curve of Na 2+ 2 . In the experiment, we measure the distribution of kinetic energies P(E kin ). Due to the unique correspondence between R and V (R), we can determine the distribution of R, P(R), by a standard probability distribution transformation with the appropriate Jacobian. This procedure requires an explicit expression for V (R). We use the result V QC (R) illustrated in Fig. 1(a), from a quantum chemistry calculation, although the simple Coulomb potential V Coul (R) = 14.4 eV/R[Å] provides a very good approximation in the pertinent R interval around the equilibrium distance R eq at 3.08 and 5.17 Å for the

III. EXPERIMENTAL SETUP

A. Helium droplet machine

Figure 2 shows a schematic drawing of the helium droplet machine used for the experiment. The machine consists of four vacuum chambers connected with pneumatic gate valves. The helium droplet experiments utilize three of these chambers: the helium droplet source chamber (I), the doping chamber (II), and the target chamber (III). The fourth chamber (IV) is a secondary source chamber containing a pulsed supersonic Even-Lavie valve (EL-7-4-2015-HRR, HT, 1 kHz).

A continuous helium droplet beam is created in the main source chamber (I). To achieve this, high-purity helium gas is first led through 2 m of thin ( 1 16 in.) copper pipe wrapped around the first and the second stage of the coldhead of a closed cycle cryostat (Sumitomo Heavy Industries, RDK-415D), precooling the helium to around 4 K. Next the helium gas is sent through a small nozzle assembly mounted on the tip of the coldhead. The nozzle assembly can be counter heated via a pair of heating resistors (Farnell, MHP35 470F) for effective temperature control. These are operated via a temperature controller (Lake Shore 335), which also monitors the temperature via a silicon diode (Lake Shore, DT-670B-CU). The nozzle assembly is clad with copper to shield it from heat radiation to ensure stable operation. In the present work the nozzle is held at a fixed temperature T nozzle between 11 and 16 K. Ultimately, the helium gas is subcritically expanded through a 5-μm-diameter nozzle (Platinum, 0.6 mm thick and 2 mm diameter, PLANO) at a stagnation pressure of 25 bars into the source chamber. This expansion produces He droplets with a mean number of He atoms between ∼4000 and ∼15 000, determined by T nozzle [22].

The helium droplet beam formed passes through a 1-mm skimmer (Beam Dynamics, Model 2) into the doping chamber (II). The doping chamber can be equipped with up to two different pickup cells to allow sequential doping of the He droplets with atoms or molecules. Each pickup cell consists of a crucible capable of holding a solid sample, a lid for the crucible, and a socket the crucible can be placed in. The crucibles and sockets have 3-mm holes in the sides to allow the He droplet beam to pass through. During the passage the He droplets can pick up gas-phase atoms, clusters, or molecules that they collide with [22,23]. Kapton wires wrapped around the sockets can be resistively heated by sending current through in order to control the vapor pressure and thereby the doping probability in the pickup cells. Proportional integral derivative (PID) controllers are used to monitor and control the heating via a set of thermocouple wires. To enable flexible and consistent positioning of the pickup cells, the sockets are mounted on rails in the chamber. In this work, only the first pickup cell is installed and loaded with a solid sample of either Li, Na, K, Rb, or Cs. The vapor pressure is regulated to a value where some of the droplets become doped with two Ak atoms, which can then form a dimer on the surface [11,24]. Alternatively, pickup cells connected to an external gas reservoir can be used. In this way, gas-phase atoms or molecules from the external reservoir are added through an ultrafine needle valve (Kurt Lesker, VZLVM267) to the region where they can be picked up by the He droplets [25].

After passing through the pickup cell, the now-doped helium droplet beam passes through two cold traps separated by a 2-mm skimmer (Beam Dynamics, Model 2). The cold traps are filled with liquid N 2 to reduce the effusive signal from undoped atoms that escape from the pickup cell. The beam is once more skimmed by a 2-mm skimmer (Beam Dynamics, Model 2), before reaching the target chamber (III). Here the doped droplet beam is intersected by a pulsed laser beam in the center of a two-dimensional (2D) velocity map imaging (VMI) spectrometer.

The secondary source chamber (IV) is placed beneath the target chamber. This chamber is equipped with an Even-Lavie valve that can be used to create a pulsed molecular beam for gas-phase experiments via a supersonic expansion [26]. The beam is directed through a skimmer (Beam Dynamics, Model 50.8) into the target chamber, where it enters the VMI spectrometer from below. In this work, a molecular beam of gas-phase I 2 is used for the energy calibration of the 2D imaging spectrometer.

The source chamber (I) is pumped by a 1900-l/s turbo molecular pump (Pfeiffer HiPace 2300), assisted by a 400-m 3 /h backing pump setup (Edward Booster EH500 Hyd supported by an Edward E2M80 pump). The doping chamber is pumped via a 520-l/s (Pfeiffer TMU 521) and a 400-l/s (Leybold Turbovac 361) turbo molecular pump, both supported by a 16-m 3 /h backing pump (Leybold Trivac D16B). The target chamber (III) and the supersonic chamber (IV) are each pumped by a 685-l/s (Pfeiffer HiPace 700) turbo molecular pump. A 70-m 3 /h backing pump (Pfeiffer Duo 65) assists the supersonic chamber pump, while the target chamber pump is assisted by the same backing pump (Leybold Trivac D16B) supporting the doping chamber pumps.

B. VMI spectrometer and laser

A standard 2D VMI spectrometer [27,28], located in the target chamber (III), is used to measure the velocity of ionic fragments created in our experiments. The VMI setup consists of an open three electrode electrostatic lens in combination with two stacked MCPs (microchannel plates-El Mul Technologies B050V, ∼40-mm active diameter) and a phosphor screen (El Mul Technologies Scintimax P47). The electrostatic lens is located inside a cylinder of μ metal. The MCP and the phosphor screen are mounted on a flight tube located on top of the target chamber. The flight tube improves the velocity resolution of low-kinetic-energy ions and helps separate ions with close mass-to-charge ratios. The chamber is equipped with multiple high-voltage feedthroughs for the three electrodes, respectively labeled repeller, extractor, and ground; the MCP (for both the front and the backside); and the phosphor screen. The phosphor screen is imaged with a CCD camera (Allied Vision Prosilica GE680) recording frames at 100 Hz, i.e., each frame contains data from ten laser shots. A high-voltage switch is used to gate the MCP such that only ions with a single mass-to-charge ratio are recorded at any given time.

The doped helium droplet beam is intersected perpendicularly by a pulsed, linearly polarized laser beam (from a Solstice Ace, Spectra-Physics, 1-kHz laser system) in the interaction region between the repeller and the extractor electrodes. The laser beam is focused by a 30-cm plano convex lens located in front of the target chamber, leading to a focal spot size of about ω 0 = 85 μm (130 μm for the Cs 2 measurements). The duration of the pulses is ∼50 fs and their central wavelengths and intensities are given in Figs. 3(aii)-3(eii).

IV. CALCULATION OF Ak 2 2+ POTENTIAL CURVES

The V QC (R) potential energy curves for the Ak 2+ 2 species were determined at the Coupled Cluster single, double and perturbative triple level with a sequence of basis sets with increasing cardinal number in order to judge the basis-set convergence. A counterpoise correction [29,30] was calculated for each internuclear distance, and the raw and counterpoise corrected energies for the two basis sets with largest cardinal number were extrapolated to the basis-set limit by an L -3 formula [31]. The final V QC (R) potential energy curves were obtained by averaging the raw and counterpoise corrected extrapolated results [32,33].

The employed basis sets were aug-cc-pCVXZ (X=D,T,Q) for Li with all electrons correlated [34], aug-cc-pCVXZ (X=D,T,Q,5) for Na with the 1s orbital treated as a frozen core [34], ANO-RCC-VXZP (X=D,T,Q) for K with five orbitals frozen [35], ANO-RCC-VXZP (X=D,T,Q) for Rb with nine orbitals frozen [35], Sapporo AXZP (X=D,T,Q) in their DKH3 contracted version [36] for Cs with 18 core orbitals frozen, and relativistic effects included by the Douglas-Kroll-Hess procedure to second order [37,38]. All calculations have been done using the GAUSSIAN 16 program package [39].

The V QC (R) potential energy curves differ from the pure Coulombic by including exchange-repulsion and dispersion effects between the Ak + ions. The V QC (R) are consequently slightly more repulsive at short distances and slightly less repulsive at long distances, compared to the pure Coulombic expression.

V. RESULTS

A. Ion images and kinetic energy distributions

Figure 3(ai)-3(ei) show the 2D velocity images recorded for different alkali-metal ions. Each image is obtained by stacking tens of thousands of frames. All images contain radial stripes and a center with no signal [see Fig. 3(di)]. The missing signal arises from a centrally positioned metal disk in front of the MCP and small mounting rods fixing it in place [8,40]. The metal disk is used to block the many Ak + ions, stemming from ionization of Ak atoms in the effusive beam or on helium droplets, from reaching the detector. The center of the images in Figs. 3(ai)-3(ci) and 3(ei) have further been cut digitally to remove the few unwanted Ak + ions passing by the edge of the central metal disk. This was done to improve the visual contrast of the outer features of the images.

The kinetic energy distributions of the ions can be determined from the 2D velocity images. To do so, we apply an Abel inversion to the ion images to extract the radial velocity distribution. We specifically employ the polar onion peeling (POP) algorithm for the Abel inversion [41]. This algorithm retrieves the radial velocity distribution starting from the outer edges of the image and then moving to its center by subtracting the contribution from the high radii onto the lower ones. Therefore, the missing signal in the center of the images does not affect the retrieved velocity distributions of the data at higher radii. We use the polarization axis of the laser pulses as the axis of symmetry for the POP algorithm [see the white arrows annotated in Figs. 3(ai)-3(ei)]. Finally, the kinetic energy distributions P(E kin ) are retrieved from the radial velocity distributions by applying the found energy calibration and the appropriate Jacobian transformation [8]. Figures 3(aii)-3(eii) show P(E kin ) for the corresponding ion images.

As discussed in our previous Letter [14], each of the P(E kin ) distributions contains two peaks that can be assigned as the Ak + ions stemming from Coulomb explosion of Ak 2 in the 1 1 g + state or the 1 3 u + state. In Figs. 3(aii)-3(eii) these peaks are labeled accordingly. The assignment is based on the close match between the central position of the observed peaks and the kinetic energy Ak + ions would obtain through Coulomb explosion starting from the equilibrium position of either the 1 1 g + state or the 1 3 u + state. Note that for Li 2 only the 1 3 u + state is observed, which is consistent with results from spectroscopic studies [10,13]. To support the interpretation that the ions in the two peaks are indeed produced by Coulomb explosion, we determined the angular covariance map of the Ak + ions within the corresponding radial ranges in the images, i.e., between the dashed and dot-dashed circles for the 1 1 g + state and between the solid and dashed circles for the 1 3 u + state. An angular covariance map makes it possible to identify possible correlations in the emission direction of the ions when they originate both from gas-phase molecules [42][43][44][45] and from molecules inside He nanodroplets [25,46,47]. In each covariance map, displayed in Figs. 3(biii)-3(eiii) and 3(aiv)-3(eiv), two diagonal lines centered at θ 2 = θ 1 ± 180 • stand out and show that the emission direction of an Ak + ion is correlated with another Ak + ion departing in the opposite direction. Such a correlation identifies the ions as originating from the Coulomb explosion channel into an Ak + -Ak + pair.

In the experiments with K 2 , Rb 2 , and Cs 2 another channel is visible outside the dot-dashed rings marking the edge of the 1 1

g + channel [see Figs. 3(ci)-3(ei)].
We have found that these ions emanate from Coulomb explosion of equilaterally shaped K, Rb, and Cs trimers into three Ak + ions [48]. These ions make up the high-energy peaks centered at 2.86 eV for K + , 2.62 eV for Rb + , and 2.40 eV for Cs + , as obtained by Gaussian fits [see Figs. 3(cii)-3(eii)]. If the trimers are only doubly ionized, they may produce Ak + ions through the following dissociative ionization channels: Ak 2+ 3 → Ak + 2 + Ak + and Ak 2+ 3 → Ak + + Ak + + Ak. Based on the equilibrium geometries of the trimers, the E kin of the Ak + ions from these channels fall outside the ranges of the Ak + ions from Coulomb explosion of the singlet and triplet dimers. Thus, we expect no pollution of the dimer signals from ionization of trimers. Experimentally, we confirmed this by recording data at lower vapor pressures in the doping cell. Under these conditions the peaks originating from Coulomb explosion of the trimers are strongly reduced, while no changes in the shape of P(E kin ) for the 1 1 g + state or for the 1 3 u + state were observed.

B. Angular distributions

The ion images in Fig. 3 exhibit angular anisotropy. This is more clearly seen in the angular distributions P(θ ) obtained by integrating the ion images along the radial axis for the regions corresponding to the 1 1 g + and the 1 3 u + channel, respectively. Figure 4 shows that the emission directions of the Li + , Na + , and Cs + (K + and Rb + ) ions are localized along (perpendicular to) the laser pulse polarization. We believe the angular anisotropy reflects the influence of electronically excited states in the multiphoton absorption process. In particular, the electronic structure of all alkali-metal dimers is such that absorption of the first photon (λ = 800 or 400 nm) is resonant or nearly resonant with an electronically excited state. This can give rise to an alignment-dependent absorption probability depending on the transition dipole moment of the electronic states involved and thereby an anisotropic angular distribution of the Ak + fragments.

To illustrate the situation, we consider the case of Na 2 . The potential energy diagram in Fig. 5 depicts a number of the lowest-lying electronic states. After absorption of the first photon (λ = 800 nm), Na 2 initially residing in the 1 1 g + state will be close to the 1 1

u + state. The 1 1 g + → 1 1 u +
transition is an allowed parallel transition, meaning that the photon absorption probability is largest for the dimers with their internuclear axis parallel to the laser polarization axis. Upon further photon absorption, these parallel dimers will Coulomb explode and produce Ak + fragments emitted along the laser polarization because the Ak + fragments recoil along the direction defined by the internuclear axis of their parent dimer. Thus, the parallel character of the transition will resonantly enhance the multiphoton process of Na 2 (1 1 g + ) leading to double ionization and thereby it can explain why P(θ ) for the 1 1 g + state is localized around 0 • and 180 • . Turning to sodium dimers in the 1 3 u + state, Fig. 5 shows that absorption of the first photon brings them close to the FIG. 5. Selected potential energy curves for the Na dimer. Blue curves are singlet states, while orange curves are triplet states [19,20]. The green curve shows the 1 2 g + Na + 2 potential curve. The black solid curve shows V QC (R) representing Na 2+ 2 . The red shapes show the square of the wave function of the vibrational ground state in the 1 1 g + and 1 3 u + potentials. The green arrows indicate laser photons (800 nm) in the multiphoton ionization process drawn to scale. The blue arrows illustrate the remaining laser photons involved in the double-ionization process, not drawn to scale.

1 3 g + state. Since 1 3 u + → 1 3
g + is also an allowed parallel transition, the same arguments apply to explain why multiphoton-induced double ionization of Na 2 (1 3 u + ) produces Na + fragments emitted along the laser polarization. Besides the low-lying states discussed here, it is possible that higher-lying states in Na 2 and/or states in Na + 2 can also influence the Na + angular distributions.

Similar considerations were applied to the other alkalimetal dimers and we found that the angular distributions of the Ak + observed can also be accounted for by the parallel or perpendicular character of one-photon transitions from the 1 1 g + state or from the 1 3 u + state. Notably for K 2 , where the wavelength of the laser pulse was 400 nm, the potential energy diagram [49] shows that absorption of the first photon can occur through both parallel and perpendicular transitions. A contribution to double ionization from these two transitions will lead to a reduced anisotropy in the angular distributions, a behavior consistent with the experimental measurements. Furthermore, for the K 2 measurement, a very high intensity 3 × 10 14 W/cm 2 was used. That may have saturated the multiphoton process and thereby be part of the reason for the almost isotropic angular distribution of the K + ions. We also performed measurements at other wavelengths of the ionizing laser pulse and observed significant changes of the angular anisotropy. For instance, when a laser pulse with λ = 400 nm is used to doubly ionize Na 2 , the angular distribution of the Na + ions for both the 1 1 g + and the 1 3 u + state exhibits only a mild anisotropy and is now localized perpendicular to the laser polarization. This contrasts the parallel character of the ion emission observed for λ = 800 nm and thus corroborates that resonance effects due to electronically excited states influence the multiphoton absorption process. Finally, we note that laser-induced alignment [50,51] plays no role for the observed angular distributions since the rapid Coulomb explosion arrests any rotation of the dimer acquired during the early part of the laser pulse before double ionization is complete.

C. Internuclear distributions

As mentioned in Sec. II, the kinetic energy of an Ak + ion from Coulomb explosion of a bare Ak 2 is given by E kin = 1 2 V (R). Since V (R) is a strictly decreasing function on the R interval from 0 to ∞, we can determine the internuclear distribution P(R) by the following transformation of probability distributions:

P(R) = P(E kin )| dE kin dR | = P(E kin )| 1 2 dV (R)
dR |. Here P(E kin ) comes from the measurement (Fig. 3) and for V (R) we use V QC (R), illustrated in Fig. 1(a). The black curves in Fig. 6 show the resulting P(R) for the five different alkali-metal dimers. As references we display the calculated | (R)| 2 for the vibrational ground state in both the 1 1 g + and the 1 3 u + state. The calculations were done by solving the vibrational stationary Schrödinger equation for the isolated dimers using internuclear potentials from the literature [19,20,49,52,53]. To ease the comparison between P(R) and | (R)| 2 , the latter were normalized to the experimental data. In practice, this was obtained by fitting P(R) to the sum of two Gaussians, representing the probability distribution for the singlet and the triplet state, to ensure agreement in the overlapping region. The | (R)| 2 distributions were then individually normalized to the area of the corresponding P(R) peak extracted from the fit. For Li 2 only a single Gaussian was used for the fit.

Figure 6 shows that P(R) is broader than | (R)| 2 for all Ak 2 in both the singlet and the triplet state. From Table I, summarizing the full width at half maximum (FWHM) of the experimental and calculated R distributions, it can be seen that the FWHM of P(R) for Li 2 , Na 2 , K 2 , and Rb 2 in the 1 3 u + state is about a factor of 2 (1.8-2.4) times larger than the FWHM of the corresponding | (R)| 2 . For Na 2 , K 2 , and Rb 2 in the 1 1 g + state the ratio between the FWHM of the experimental and calculated distributions is about 4 (3.9-4.4). For the 1 3 u + (1 1 g + ) state of Cs 2 the factor is 3.7 (4.4). Table 1

VI. DISCUSSION

We believe there are three factors in the experiment that can cause the experimentally retrieved P(R) to deviate from the theoretical | (R)| 2 . The first is the double-ionization process. As explained in Sec. V B, electronically excited states of Ak 2 , and possibly also electronic states of Ak + 2 , can be transiently excited during the multiphoton absorption. The 50-fs duration of the laser pulse is long enough that the lighter dimers may undergo some internuclear motion on the resonant potential curves. Using again Na 2 as an example, Fig. 5 shows that at the equilibrium distance of the initial 1 1 g + (1 3 u + ) state, the electronically excited singlet (triplet) states have a slope that will cause an increase (decrease) of R upon internuclear motion. The same is true if Na + 2 (1 2 g + ) is transiently populated. This is consistent with the experimental observation that the CP of P(R) is larger (smaller) for the 1 ). The larger masses of K 2 , Rb 2 , and Cs 2 are expected to reduce the internuclear motion on the intermediate potential curves, an expectation consistent with the very small deviation of the CP of the measured P(R) from | (R)| 2 observed for K 2 and Rb 2 . For Cs 2 the expectation does not hold, which we comment on below. We note that the internuclear motion may also contribute to a broadening of P(R) with respect to | (R)| 2 and not just a shift. Finally, we point out that the passage through resonant states may also lead to a slight skew of the internuclear distribution due to Franck-Condon overlaps and could be partly responsible for, e.g., the small offset of P(R) from | (R)| 2 for the 1 3 u + state of Na 2 . The second factor that can cause P(R) to deviate from | (R)| 2 is the interaction between the recoiling Ak + fragment ions and the He atoms on the droplet surface. Ongoing simulations based on time-dependent density functional theory show that the interaction induces a slight bending of the Ak + ion trajectories and possibly a small loss of their kinetic energy. For a given dimer, like Rb 2 , the magnitudes of these effects are expected to be different for the singlet and the triplet state due to the different angular distributions of the dimer on the surface [54]. If the dimer lies at an angle to the surface, one Ak + fragment might interact strongly with the helium, slowing it down. The partner fragment would then receive more than half the available Coulomb energy, leading to a broadening of P(R). Furthermore, the effects are expected to be alkali-metal dependent. In particular, the observed shift of P(R) for Cs 2 may be a result of the Cs + -He interaction because the low recoil velocity of the Cs + ions, due to the large mass and small Coulomb energy, extends the interaction time compared to that of the fragments from the lighter Ak 2 .

The third factor is the kinetic energy resolution of the VMI spectrometer. We estimate that the resolution is about 100 meV. This will broaden the kinetic energy spectra and thus lead to wider P(R) distributions. Notably, such broadening will be largest for dimers with large internuclear distances, due to the shape of the Ak 2+ 2 potential curves. In particular for triplet state Rb 2 and Cs 2 , the broadening can account for up to about 15% of the measured width of the P(R) distributions.

VII. CONCLUSION AND OUTLOOK

We used Coulomb explosion, induced through double ionization by an intense 50-fs laser pulse, of alkali-metal homodimers on the surface of He nanodroplets to determine the distribution of internuclear distances P(R). The agreement of the measured P(R) with the theoretically expected internuclear distribution, i. One approach to creating vibrational wave packets is to use a moderately intense femtosecond pump laser pulse to form a coherent superposition of vibrational eigenstates in the 1 3 u + state through the dynamic Stark effect [55][56][57][58]. An advantage of this nonlinear excitation scheme is its high excitation efficiency, which eliminates the need for subtraction of signal from unexcited dimers as is otherwise necessary when one-photon excitation schemes are used [59,60]. The large magnitude and pronounced R dependence of the polarizability of alkali-metal dimers [21] make them particularly suited for dynamic Stark excitation. Results from ongoing experiments in our laboratory along these lines are very promising.

We believe Coulomb explosion imaging of alkali-metal dimers on He droplets may be improved by the following measures. The first is to use shorter midinfrared laser pulses to induce ionization in the tunneling regime [61] and thus reduce the influence of resonances. The second is to implement a VMI spectrometer with a better energy resolution to eliminate any possible instrument broadening of the peaks in the measured kinetic energy distributions. Thereby, the resolution of the shape of the wave functions could be significantly enhanced. Finally, it will be necessary to develop a theoretical understanding of how the interaction between the He droplet and the alkali-metal fragment ions influences the kinetic energy of the latter. Ongoing simulations are addressing this point [62].

FIG. 1 . 1 g+ [ 19 ] and 1 3 u+ 1 g + and 1 3 u+ 1 g+ and 1 3 u

 111931313 FIG. 1. (a) Energy diagram showing the potential curves for Na 2 in the 1 1 g + [19] and 1 3 u+ states[20] and V QC (R) for Na2+ 2 . The square of the vibrational ground state wave functions for the 1 1 g +

g + ( 3 u+FIG. 2 .

 32 FIG.2. Schematic drawing of the helium droplet machine, which consists of four vacuum chambers: (I) the helium droplet source chamber, (II) the doping chamber, (III) the target chamber, and (IV) another source chamber equipped with a pulsed Even-Lavie valve. The target chamber contains a 2D velocity map imaging spectrometer.

1 g + ( 3 u+

 13 ) state, respectively [see Figs. 1(b) and 1(c)].

FIG. 3 .

 3 FIG. 3. (ai)-(ei) Two-dimensional velocity images of Ak + ions. The white arrows in the bottom right corners indicate the polarization axis of the laser pulses. The annotated white rings designate the regions of the 1 3 u + and 1 1 g + channels (see the text). (aii)-(eii) Kinetic energy spectra of the respective Ak + ions. Angular covariance maps are shown for the (biii)-(eiii) 1 1 g + and (aiv)-(eiv) 1 3 u + channels. The T nozzle was 11 K for the data in (a) and (b) and 12 K for the data in (c)-(e).

FIG. 4 . 1 g+ 3 u+

 413 FIG. 4. (a) Definition of the angle θ in the 2D velocity images. The white arrow in the lower right corner shows the laser polarization axis. (b) Plot of the P(θ ) distribution of the Li + ions from the 1 3 u + region. Also shown are the P(θ ) distributions of Na + , K + , Rb + , and Cs + ions from the (ci)-(fi) 1 1 g + and (cii)-(fii) 1 3 u + regions (see the text).

3 u+ 1 g+ (1 3 u+

 313 FIG. 6. Plots of the P(R) distributions for the Ak dimers. The red (blue) shapes indicate the theoretical | (R)| 2 distributions for isolated 1 1 g + (1 3 u + ) states. displaced to higher values compared to the CP of | (R)| 2 for both the 1 1 g + state (0.38 Å) and the 1 3 u + state (0.58 Å).

1 g+ state or the 1 3 u+

 13 e., | (R)| 2 for the vibrational ground state in either the 1 state, is best for Li 2 , K 2 , and Rb 2 in the triplet state. For these three dimers, the center of P(R) lies within 0.04 Å of the center of | (R)| 2 and the FWHM of P(R) is about twice as large as the FWHM of | (R)| 2 . As such, we deem our Coulomb explosion method capable of measuring time-dependent wave functions resulting from the creation of vibrational wave packets in Li 2 (1 3 u + ), K 2 (1 3 u + ), or Rb 2 (1 3 u + ) with a precision of 0.04 Å on the central position and a 1-Å resolution of the shape.

  Fig.6). Likewise, for Li 2 , the shoulder of P(R) towards smaller R values than those spanned by | (R)| 2 is consistent with internuclear motion on potential curves of

	1	g	+ (1 3	u	+ )
	state (see

TABLE I .

 I Central position and full width at half maximum for the peaks in the measured P(R) and the calculated | (R)| 2 , obtained via Gaussian fits. Up to about 15% of the measured P(R) FWHM can be explained by the experimental energy resolution of 100 meV (see the text).

									Central position (Å)	FWHM (Å)
	State						P(R)	| (R)| 2	P(R)	| (R)| 2
	Na 2	1		g	+	3.32	3.10	0.7	0.18
	K 2	1		g	+	3.96	3.94	0.8	0.18
	Rb 2	1		g	+	4.25	4.21	0.8	0.19
	Cs 2	1		g	+	4.97	4.59	0.8	0.18
	Li 2	3	u	+	4.22	4.22	1.2	0.65
	Na 2	3		u	+	5.13	5.22	1.4	0.58
	K 2	3		u	+	5.73	5.77	0.9	0.48
	Rb 2	3		u		+	6.09	6.09	0.9	0.41
	Cs 2	3		u	+	6.90	6.32	1.3	0.35
	excited Li 2 states or on the potential curve for Li + 2 (1 2	g	+

ACKNOWLEDGMENTS

We thank J. Thøgersen for expert help on keeping the laser system in optimal condition. H.S. acknowledges support from Villum Fonden through Villum Investigator Grant No. 25886. The numerical results presented in this work were obtained at the Centre for Scientific Computing, Aarhus.


M Michalet
email: morgan.michalet@icm.unicancer.fr.

R Tétreau

J-L Pasquié

O Chabre

D Azria

Stereotactic magnetic resonance imaging-guided radiotherapy for intracardiac metastases: A case report

Keywords: Adrenocortical carcinoma, Carcinome corticosurrénalien, Cardiac metastases, MR-guided radiotherapy, Métastases cardiaques, Radiothérapie guidée par IRM Radiothérapie guidée par IRM Carcinome corticosurrénalien Métastases cardiaques

Adrenocortical carcinoma is a malignant tumor with a poor prognosis and a frequent metastatic extension. In very rare cases, a cardiac metastatic disease may occur, and surgical resection is essential for its management. MR-guided stereotactic radiotherapy is an attractive radiotherapy modality for the treatment of mobile thoracic tumors, enabling the target to be monitored continuously during irradiation, while the dosimetric plan can be adapted daily if necessary. We report here the case of a patient with intracardiac metastasis secondary to malignant adrenocortical carcinoma, treated with magnetic resonance imaging-guided stereotactic radiotherapy.

R é s u m é

Le corticosurrénalome est une tumeur maligne de pronostic défavorable avec une extension métastatique fréquente. Dans de très rares cas, il est possible d'observer une atteinte métastatique cardiaque, dont la prise en charge repose avant tout sur la résection chirurgicale. La radiothérapie les, permettant de suivre la cible de fac¸ on continue pendant l'irradiation, tout en adaptant quotidiennement le plan dosimétrique selon le besoin. Nous rapportons ici le cas d'une patiente atteinte d'une métastase intracardiaque secondaire à un corticosurrénalome malin, traitée par irradiation stéréotaxique guidée par IRM.

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Introduction

Adrenocortical carcinoma is a rare endocrine malignancy arising from the adrenal gland. Adrenocortical carcinoma can secrete different cortical hormones (causing hypercorticism, hyperandrogenism, or primary aldosteronism), leading to different morbidity [START_REF] Vaidya | Treatment of adrenocortical carcinoma[END_REF]. The prognosis is bad, with most of cases metastatic or locally advanced at time of diagnosis [START_REF] Ahmed | Adrenal cortical carcinoma: pathology, genomics, prognosis, imaging features, and mimics with impact on management[END_REF]. Surgery is the cornerstone of localized diseases treatment, whereas chemotherapy is used for metastatic diseases (mitotane for indolent diseases or more aggressive platinum-based regimens for others) [START_REF] Jasim | Management of adrenocortical carcinoma[END_REF].

Cardiac tumors are rare and most of them are benign [START_REF] Taguchi | Comprehensive review of the epidemiology and treatments for malignant adult cardiac tumors[END_REF]. Among malign tumors, metastatic tumors are 100 to 1000-fold more com-mon than primary cardiac tumors [START_REF] Sarjeant | Cancer of the heart[END_REF]. Treatment of cardiac tumors is classically based on surgical resection. Radiation therapy is a limited option due to limited heart constraint doses, even in old literature on this topic [START_REF] Cham | Radiation therapy of cardiac and pericardial metastases[END_REF].

Magnetic resonance imaging (MRI)-guided radiotherapy is an interesting option for such treatment, thanks to three advantages in comparison to "standard" radiotherapy. It offers a good contrast for cardiac tumors using MR images contrast and the possibility to follow the target during irradiation with continuous cine-MR images. Moreover, it is possible to daily adapt the plan to take into account the possible volume modifications [START_REF] Mutic | The ViewRay System: magnetic resonance-guided and controlled radiotherapy[END_REF]8]. We present in this article the case of a patient treated in our institution with MRI-guided radiation for an intracardiac metastasis from a primitive adrenocortical carcinoma.

Case presentation

A 65-year-old female patient was addressed in our department; she had a history of adrenocortical carcinoma with a first treatment 10 years before, which consisted of a left adrenalectomy. She later developed local recurrence treated by partial diaphragmatic resection 7 years later and hepatic metastases treated with chemoembolization. Six years after this, she was diagnosed with an asymptomatic cardiac metastasis located in the right ventricle. A close follow-up was proposed, and the patient had been treated with radiofrequency for pulmonary metastases 5 years and 2 years before admission in our department. The cardiac mass did not progress since first radiofrequency treatment.

One month following the second radiofrequency treatment, the patient presented with electrocardiogram modifications, without clinical symptoms. A cardiac MRI was performed and found a 41-mm heterogeneous right ventricular mass, probably with septoapical myocardial origin, with intralesional necrotic-haemorrhagic changes, confirming the progression of this cardiac metastasis (Fig. 1). The CT-scan did not find any other progression except this cardiac lesion. The case was discussed in multidisciplinary staff meeting. The surgical resection was contra indicated due to technical impossibility. The patient was proposed for a radiotherapy.

Clinically, the patient was asymptomatic, with a performance status of 0. The cardiac report was good, with a left ventricular ejection fraction of 61% on echocardiography and no kinetic troubles.

We discussed this case in our department and proposed a stereotactic MR-guided radiotherapy. We chose to use the MRIdian (Viewray®), which is a MR-linear accelerator (6-MV linear accelerator combined with a 0.35-T MR-imaging device). We prescribed30 Gy in five fractions every other day, over two weeks, with normalization on D50%, ensuring a target coverage (planning target volume [PTV]) of 95% within the 95% isodose. PTV was built with a3-mm isotropic expansion from GTV, which was delineated using MRIdian simulation images, as well as a 1.5-T diagnostic MRI after registration.

Volumes and dose constraints used are presented in Table 1.A 98.1% target coverage was obtained for the PTV V95%. The dose distribution is shown in Fig. 2.

A cardiological assessment with electrocardiogram and echocardiography was proposed before, during treatment (after the third fraction) and after the last fraction. A continuous electrocardiogram monitoring was performed during irradiation using an amagnetic scopus. The treatment was well tolerated with no evidence of cardiac dysfunction during and after treatment, and no other toxicity.

A close follow-up was performed with a clinical assessment at1, 3 and 6 months and a CTscan assessment at 3 and 6 months after treatment. We observed a progressive partial response as shown in Fig. 3. At 6 months post-treatment, the lesion measured 31 × 34 mm compared to 36 × 41 mm before treatment. The other already known secondary pulmonary lesions were stable. The patient was asymptomatic and the cardiological evaluation was perfectly stable.3.

Discussion

As cardiac metastases are rare situations, only few data are avail-able concerning the use of radiotherapy in this situation. On the other hand, some data are emerging concerning the use of stereo-tactic radiotherapy for refractory ventricular tachycardia [START_REF] Benali | Cardiac stereotactic ablative radiotherapy for refractory ventricular arrhythmias: a radical alternative? A narrative review of rationale and cardiological aspects[END_REF][START_REF] Cuculich | Non invasive cardiac radiation for ablation of ventricular tachycardia[END_REF].MR-guided radiotherapy is an interesting option for the treatment of cardiac masses. Corradini et al. described their experience of MRI-guided radiotherapy for primary cardiac sarcomas [START_REF] Corradini | MRguided stereotactic body radiation therapy for primary cardiac sarcomas[END_REF] All patients were irradiated in a salvage situation after surgery with a prescription of 30 to 35 Gy in five fractions. All patients completed the treatment with a good tolerance.

Other authors described their experience for the treatment of intracardiac and pericardial metastases using MRI-guided radiation [START_REF] Sim | MR-guided stereotactic body radiation therapy for intracardiac and pericardial metastases[END_REF]. They treated five patients with a median dose of 40 Gy in five fractions. After a median follow-up of 4.7 months, all patients presented a stable disease, partial or complete response. One patient presented with an atrial fibrillation 6 months after treatment.

Due to the larger lesion size than those described in the two previous articles, we chose to deliver a dose of 30 Gy in five fractions for this patient. We obtained a good partial response, with no other progressive lesion and no toxicity observed after a follow-up of 6months. This result is encouraging, and we hope to obtain a better response in the following months along with a good quality of life.4.

Conclusion

MRI-guided radiotherapy appears to be a potential good option for inoperable cardiac tumors. A longer follow-up is required to assess late toxicity and survival outcomes.

Disclosure of interests

The authors have not declared their activities/relationships/interests.


Alexey Grigoriev

Alexey Shakun

Nikolay Ignatiev

Boris Moshkin

Dmitry Patsaev

Alexander Zharkov

Igor Maslov

Dmitry Gorinov

Andrey Kungurov

Aleksandr Santos-Skripko

Nikolay

Occultation results by ACS TIRVIM at ExoMars TGO: aerosols and gases

TIRVIM is the first Fourier-spectrometer able to observe Sun occultations at Mars giving an access to a broad spectral range from near-IR through thermal IR: 1.7-17 µm. A dedicated solar port with limited aperture, or the full 2" nadir aperture could be used to observe occultations. In either case the FOV diameter is 2.5°, while the spectral resolution changes from 0.8 cm -1 when observing through the solar port (the "climatology" observation mode) to 0.13 cm -1 when observing with full nadir aperture. This latter "sensitive" mode requires dedicated spacecraft pointing, but allows for sensitive measurements of trace atmospheric gases. At the limb the FOV encompasses a circle of 75 km while the Sun disk diameter is 10.5 km. In the shortwavelength (SW) part of the spectrum the Sun radiation dominates and the Sun diameter mostly determines the effective FOV. In the longwavelength (LW) part the infrared emission from Mars and the atmosphere is significant or dominates and the full FOV works. The vertical sampling in the "climatology" mode is as low as 0.4 km.

The first Sun occultations showed that the following gaseous bands are well visible in the TIRVIM spectra: multiple CO 2 bands, CO at 4.7 µm, H 2 O at 6.3 µm. All these gases are routinely observed in "climatology" mode allowing for vertical profiling. The CO 2 in the 15-µm band is detectable up to the altitude of ~200 km, with a signature of non-LTE effects.

The "climatology" spectra reveal vertical structure of aerosols, sometimes layered. The spectra show H 2 O ice features at around 3 µm and 12 µm. An interesting optical phenomenon is observed around the 12-µm ice and 9-µm silicate dust bands. When the Sun approaches the horizon, a sort of emission with a particularly strong and sharp peak around 9-10 µm appears. The intensity increases by an order of magnitude in 10 s and remains at this level even when the Sun falls ~16 km below the horizon. Our preliminary interpretation is that the large TIRVIM FOV acquires the Sun IR light forward-scattered by the silicate dust particles. Just the same but timereversed picture is visible at the sunrise. We call this effect "Silicate Dawn-Dusk" (SDD). SDD is observed in both hemispheres, at different places. A detailed interpretation of SDD effect would involve comprehensive modeling (Luginin et al., EPSC 2018) but should deliver a lot of information about aerosols. TIRVIM occultations observed with the "sensitive" pointing allow to profile trace components. So far CO 2 isotopic bands including 628 were observed. Less pronounced but detectable is O 3 at 9.6 µm; for NH 3 an existing upper limit of 8 ppb [START_REF] Maguire | Martian isotopic ratios and upper limits for possible minor constituents as derived from Mariner 9 infrared spectrometer data[END_REF] could be improved.

The TIRVIM occultation results available will be reviewed, and the progress of their interpretation summarized.

Figure 1 :

 1 Figure 1: A sketch of TIRVIM observation geometry in occultation.

  Occultation results by ACS TIRVIM at ExoMars TGO: aerosols and gases The primary goal of TIRVIM is the long-term monitoring of atmospheric temperature profiles and aerosol state in nadir (see Ignatiev et al. EPSC 2018).

	Alexey Grigoriev (1), Alexey Shakun (1), Nikolay Ignatiev (1), Boris Moshkin (1), Dmitry Patsaev (1), Alexander
	Zharkov (1), Igor Maslov (1), Dmitry Gorinov (1), Andrey Kungurov (1), Aleksandr Santos-Skripko (1), Viktor Shashkin (1),
	Fedor Martynovich (1), Oleg Sazonov (1), Igor Stupin (1), Dmitry Merzlyakov (1), Yury Nikolskiy (1), Mikhail Luginin (1),
	Alexander Trokhimovskiy (1), Franck Montmessin (2), and Oleg Korablev (1)
	(1) Space Research Institute (IKI), Moscow, Russia, (2) LATMOS-CNRS, Guyancourt, France (avshakun@iki.rssi.ru)
	Abstract
	ACS is a set of three spectrometers (NIR, MIR, and
	TIRVIM observing the Mars atmosphere in solar
	occultations, nadir and limb geometry. It was built by
		Research Institute (IKI) in Moscow (Russia)
	[1]. ACS TIRVIM is a Fourier-spectrometer built
	around a 2-inch double-pendulum interferometer
	with	cryogenically-cooled	HgCdTe	detector,
	allowing operation in nadir and in solar occultation.

Acknowledgements ExoMars is the space mission of ESA and Roscosmos. The ACS experiment is led by IKI Space Research Institute in Moscow. The project acknowledges funding by Roscosmos and CNES. Science operations of ACS are funded by Roscosmos and ESA.