Hugo Le Blevec

Rami Klaimi

Stefan Weithoffer

Abdel Charbel

Amer Nour

Baghdadi

Low Complexity Non-binary Turbo Decoding based on the Local-SOVA Algorithm

Keywords: Non-binary Turbo codes, low complexity decoding, Local SOVA

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Since their invention, binary turbo codes [START_REF] Berrou | Near Shannon limit errorcorrecting coding and decoding: Turbo-codes[END_REF] and iterative decoding have found their way into a wide range of communication standards like the third and fourth generations (3G, 4G) of mobile communications. Furthermore, they have participated to the rediscovery of low density parity check (LDPC) codes [START_REF] Mackay | Near Shannon limit performance of low density parity check codes[END_REF]. Benefiting from a direct mapping to high order constellation symbols [START_REF] Liva | Short turbo codes over high order fields[END_REF], the non-binary (NB) variants for both code classes defined over Galois Fields GF(q) have been shown to outperform their binary counterparts. However, the corresponding high decoding complexity imposes limits on the Galois Field order q and on the achievable frame sizes in practice [START_REF] Klaimi | Design of Low-Complexity Convolutional Codes over GF(q)[END_REF], [START_REF] Davey | Low-density parity check codes over GF(q)[END_REF]. For non-binary low density parity check (NB-LDPC) codes, these complexity issues have been addressed in a number of works [START_REF] Barnault | Fast decoding algorithm for LDPC over GF(2 q )[END_REF], [START_REF] Boutillon | Bubble check: a simplified algorithm for elementary check node processing in extended min-sum non-binary LDPC decoders[END_REF] paving the way for hardware implementations. On the other hand, the strong focus on the Min-Log-MAP (MLM) algorithm, which has long been the standard decoding algorithm for binary Turbo codes, has carried over to a large extent to NB Turbo decoding. In [START_REF] Klaimi | Low-complexity decoders for non-binary turbo codes[END_REF], complexity reduced MLM decoding was proposed for NB Turbo decoding, inspired by the bubble check algorithm (BC) [START_REF] Boutillon | Bubble check: a simplified algorithm for elementary check node processing in extended min-sum non-binary LDPC decoders[END_REF]. To calculate the state metric recursion and the extrinsic information, this algorithm resorts to a subset of the n m < q highly reliable transitions in the code trellis. These n m transitions are processed using a 2D-bubble sorter for the add compare select (ACS) operations. A complexity reduction ranging from 50% to 75% in terms of ACS operations was achieved at the cost of < 0.3dB in terms of frame error rate (FER) compared to the MLM.

Recently, the Local-SOVA (LSOVA) algorithm was proposed as a reduced-complexity alternative to the MLM for binary Turbo codes [START_REF] Le | Revisiting the Max-Log-Map Algorithm With SOVA Update Rules: New Simplifications for High-Radix SISO Decoders[END_REF]. Based on a path-centric view on the code trellis, LSOVA was shown to achieve a computational complexity reduction of up to 37% compared to the MLM with a penalty of less than 0.1dB in terms of FER for radix-8 decoding. Practical implementation in 22nm technology [START_REF] Weithoffer | Low-complexity Computational Units for the Local-SOVA Decoding Algorithm[END_REF] exceeded computational complexity predictions to achieve around 45% of reduction.

In this work, we apply for the first time the LSOVA algorithm to the decoding of NB-Turbo codes and propose additional complexity reductions dedicated to the NB case. We achieve a reduction of up to 80% compared to the MLM in terms of ACS operations, getting us one step forward towards efficient hardware implementations for NB Turbo decoding.

The remainder of this paper is structured as follows: Section II gives the necessary background on NB-Turbo decoding and Section III formulates the LSOVA for the decoding of NB-Turbo codes (NB-TC). The proposed method to reduce the complexity of the LSOVA is presented in Section IV, while its corresponding complexity is addressed in Section V. To assess the impact of the simplifications on the error correcting performance, simulation results for different code rates are shown in Section VI. Comparisons with state-of-the-art are performed in Section VII. Section VIII concludes the paper.

II. NON-BINARY TURBO DECODING

1) Non-Binary Turbo Codes -Structure and Constituent Codes: In the following, we consider a NB-TC consisting of two concatenated recursive systematic convolutional codes over GF(q). These latter follow the structure illustrated in Fig. 1-a. Composed of one memory element, it provides an excellent tradeoff between complexity and error correcting performance [START_REF] Klaimi | Low-complexity decoders for non-binary turbo codes[END_REF]. The code coefficients a 1 , a 2 and a 3 respect a 1 = 0 and a 2 +a 3 = 0 leading to a fully connected trellis [START_REF] Klaimi | Design of Low-Complexity Convolutional Codes over GF(q)[END_REF]. In other words, the q 2 transitions in the trellis are labeled by the q 2 possible combinations of systematic s and parity p symbols as shown in Fig. 1-b.

2) Min-Log-MAP decoding of NB convolutional codes: Standard component decoding algorithm for binary Turbo codes is the MLM algorithm [START_REF] Robertson | A Comparison of Optimal and Sub-Optimal MAP decoding Algorithms Operating in the Log-Domain[END_REF] and its symbol based variant is considered here as reference [START_REF] Berkmann | On turbo decoding of nonbinary codes[END_REF]. For a clear link to related work, we will adopt in the following the notation from [START_REF] Le | Revisiting the Max-Log-Map Algorithm With SOVA Update Rules: New Simplifications for High-Radix SISO Decoders[END_REF]. The symbol-based MLM algorithm provides for each symbol d = i, with i ∈ {0, ..., q -1} and trellis step k, an estimate on the a Posteriori Logarithmic Probability (ALP) L i k :

L i k = min (s,s )|d k =i (A k (s) + Γ k (s , s) + B k+1 (s )) (1)
In Eq. ( 1), (s, s ) refers to the transition from state s to state s and Γ k (s, s ) to the associated branch in trellis step k. A k (s) and B k+1 (s ) are the forwardand backward state metrics for states s, s ∈ {0, ..., q -1} and are calculated recursively by

A k (s) = min (s,s ) (A k-1 (s ) + Γ k-1 (s, s )) (2)
B k (s ) = min (s,s ) (B k+1 (s) + Γ k (s, s )) (3)
The complexity of computing Eqs. ( 1), ( 2) and ( 3) is dictated by the galois field order q, since for the fully connected trellis q 2 terms (for the q 2 transitions (s, s )) have to be compared.

III. THE LOCAL SOVA ALGORITHM FOR NB DECODING

The LSOVA algorithm was proposed in [START_REF] Le | Revisiting the Max-Log-Map Algorithm With SOVA Update Rules: New Simplifications for High-Radix SISO Decoders[END_REF] and has since been demonstrated to allow reduced complexity hardware implementation [START_REF] Weithoffer | Low-complexity Computational Units for the Local-SOVA Decoding Algorithm[END_REF]. In contrast to the MLM, which operates on branches (s, s ) in the trellis, the LSOVA operates on Paths along r trellis steps k, k + 1, ..., k + (r -1) for a radix order of R = 2 r . A path P is defined as

P = {M, u, L} ∈ R × {0, q -1} r × {R + } r ( 4
)
where M is the path metric computed through radix-4 paths in the trellis of Fig. 1 (b). In the following, we assume r = 1, so that a path corresponds to a single branch in the trellis. Based on the path definition from Eq. ( 4), the LSOVA requires a Merge operation to consolidate two paths

M = A k (s) + Γ k (s, s ) + B k+(r-1) (s ),
P a = {M a , u a , L a } and P b = {M b , u b , L b } into one path P c = {M c , u c , L c }.
Therefore, it has three functions to update M, u and L:

M c = f 0 (M a , M b ) = min(M a , M b ) (5)
u c = f 1 (u a , u b ) = u a , if f 0 (M a , M b ) = M a u b , if f 0 (M a , M b ) = M b (6)
L c = L c |L c = f 2 (L a , L b ) . (7)
Where f 2 corresponds to the Hagenauer rule (HR) [START_REF] Hagenauer | A Viterbi algorithm with soft-decision outputs and its applications[END_REF] or the Battail rule (BR) [START_REF] Battail | Pondération des symboles décodés par l'algorithme de Viterbi[END_REF], [START_REF] Lin | Improvements in SOVA-based decoding for turbo codes[END_REF]:

L c = f 2 (L a , L b ) (8) = min (L p , ∆ p,p ) , if u a = u b (HR) min (L p , ∆ p,p + L p ) , if u a = u b (BR).
BR is used in the case of identical symbols u a = u b , whereas for the case of differing symbols u a = u b , HR is used. In Eq. ( 8),

p = argmin a,b (M a , M b ), p = argmax a,b (M a , M b ) and ∆ p,p = M p -M p .
When merging all paths at a trellis stage k into P c , this latter can be considered as the Maximum Likelihood (ML) path for the symbol. It was proven in [START_REF] Le | Revisiting the Max-Log-Map Algorithm With SOVA Update Rules: New Simplifications for High-Radix SISO Decoders[END_REF] that the merge operation is associative and commutative, which allows for an arbitrary ordering of the necessary merge operations. On the one hand, applying an ordering that merges paths labeled by the same information symbols d k leads to the MLM [START_REF] Le | Revisiting the Max-Log-Map Algorithm With SOVA Update Rules: New Simplifications for High-Radix SISO Decoders[END_REF]. On the other hand, applying a two-phase ordering where, in a first phase (in [START_REF] Weithoffer | Low-complexity Computational Units for the Local-SOVA Decoding Algorithm[END_REF] called ACS phase), paths with identical ending state s are merged allows to avoid BR updates entirely for radix-2 (i.e. r = 1) decoding, since branches leading to the same state are always labeled by different symbols. In a second phase (SOU phase) the resulting winning paths are then merged together, minimizing the overall amount of needed BR updates.

IV. THE NEAREST-NEIGHBORS APPROACH

In the following, we assume the GF(q) symbols to be mapped to constellation symbols directly as in [START_REF] Klaimi | Design of Low-Complexity Convolutional Codes over GF(q)[END_REF]. In the constellation diagram (see example for GF(64) in Fig. 3), the minimum distance d 0 between two points in the constellation is well defined as d 0 = (2/ Ēs ) 2 where Ēs is the average energy per symbol. Over an Additive White Gaussian Noise (AWGN) channel, from the Union Bound (UB) [START_REF] Berrou | Near Shannon limit errorcorrecting coding and decoding: Turbo-codes[END_REF], the probability of error decays exponentially with the coding gain jointly composed of the code rate and the accumulated Euclidean distance along the diverging-converging (DC) trellis sequences [START_REF] Klaimi | Union bound evaluation for non-binary turbo coded modulations[END_REF]. Therefore, the constellation points located close to the received symbol have an exponentially larger likelihood of being the transmitted symbol compared to the far ones. Consequently, for a considered symbol s we propose to consider only a subset of paths within a radius R s that encompasses a certain number of neighboring candidate symbols for which the update rules of Eq. ( 8) are applied. These paths are used for the update of the L-values using the distance between constellation symbols as a preliminary information about their reliabilities:

R s = s • d 0 , 1 ≤ s ≤ log 2 (q). ( 9
)
L i is updated if the symbol i verifies D(u, i) ≤ R s where u is the symbol carried by the path with the highest metric M and D is the Euclidean distance. By applying this method, the number of operations required during each merge is limited. Hence, considering the total amount of performed merges, this method can considerably reduce the overall complexity of the decoding. Note that for a certain radius R s , the number of points considered will differ with the position of the symbol in the constellation: symbols located close to the edges will contain less points within R s than points in the middle. Therefore, the achievable complexity reduction is expected to vary depending on the input frame. We refer to the proposed approach as nearest neighbor approach (NN-LSOVA) and will illustrate it with the following example.

I R 3 12 R 2 12 R 1 12 d0
Figure 3: 64-QAM with some R s configurations.

Consider the case where q = 64 and let P 1 and P 2 be two paths such that

P 1 = {M 1 , u 1 , L 1 } and P 2 = {M 2 , u 2 , L 2 } where L 1 = {L 0 1 , L 1 1 , ..., L 63 1 } and L 2 = {L 0 2 , L 1 2 , ..., L 63 2 }. Let's assume that u 1 = 12 = u 2 , M 1 ≥ M 2 and the considered radius is R 1 = d 0 = 1.
When merging the two paths with the LSOVA algorithm, the update rule is applied to all the L-values, resulting in 64 update operations. However, the NN-LSOVA will only apply the update rule on the Lvalues corresponding to symbols in the considered radius. Here symbols 12, 13, 14, 28 and 44 are to be updated, which means only computing

f 2 (L 12 1 , L 12 2 ), f 2 (L 13 1 , L 13 2 ), f 2 (L 14 1 , L 14 2 ), f 2 (L 28 1 , L 28
2 ) and f 2 (L 44 1 , L 44 2 ). For all the other symbols, the values carried by the winning path P 1 will be kept, lowering the amount of update operations to only five.

The NN-LSOVA offers then a new trade-off between complexity and performance that we will discuss hereafter.

V. COMPLEXITY OF THE NN-LSOVA

In the following, we call layer a set of 2-by-2 merge operations executed in parallel. The merging of paths in the SOU phase is carried out in the LSOVA algorithm following a binary 2-by-2 tree expressed as a sequence of layers. We denote by N l,q the amount of 2 by 2 merge operations to be performed in a layer l. This number depends on the number of symbols q and the index of the layer l and can be computed as N l,q = q 2 l . For a non-binary code over GF(q), there are log 2 (q) layers: the first one having q paths, the second q 2 , the third q 4 until choosing the winning path. Further, we evaluate the amount of L-value update operations for layer l considering neighbours within R l s , by λ l,q,s . This latter varies with the constellation order and the position of the symbol carried by the path. To illustrate this, refer again to Fig. 3 with R s = 1 • d 0 . If the winning path carries a symbol at the center of the constellation, a total of 5 L-values will have to be updated. However, if the carried symbol is located at the corner of the constellation, only 3 updates will have to be performed. Note that the L-value of the symbol carried by the winning path also needs to be updated. Having a varying level of complexity reduction depending on the location within the merge tree merge operations on one side and on the location of the symbol carried by the winning path on the other, we provide bounds on the the complexity analysis. They depend on the value of R l s (a design choice) and the 2 extreme possible position values of the winning path within the constellation as shown for the case of q = 64 in Tab. I.

Table I: Boundary values of λ q,s for different R s with q = 64. We can then bound the complexity in terms of number of updates for the decoding of one symbol as:

C tot = log 2 (q) l=1 N l,q λ l,q,s = log 2 (q) l=1 q λ l,q,s 2 l ( 10
)
Reducing the radius size R l s has a noticeable impact on the error correcting performance of the decoder. Therefore, we evaluate a number of benchmark configurations in the next section before comparing the complexity of the proposed algorithm with state of the art in Section VII.

VI. SIMULATION RESULTS

We consider NB-TCs over GF(64) mapped to a 64-QAM constellation, designed from rate-1/2 NB-CCs as in Fig. 1-(a). The coefficients a 1 , a 2 and a 3 were chosen to achieve the highest minimum cumulated Euclidean distance, and are equal to 31, 5 and 18, respectively. To construct the finite field, we used the following primitive polynomial: P GF(64) (D) = 1 + D 2 + D 3 + D 5 + D 6 . The Turbo code interleaver uses an Almost Regular Permutation (ARP) designed for the frame size of K s = 160 GF(64) symbols (960 bits) following [START_REF] Garzón-Bohórquez | Improving turbo codes for 5G with parity puncture-constrained interleavers[END_REF]- [START_REF] Garzón Bohórquez | Mitigating correlation problems in turbo decoders[END_REF]. To assess the proposed NN-LSOVA algorithm, several low-complexity configurations were evaluated (see Table II). These configurations use different radii for different LSOVA layers l, leading to a layer-dependent complexity. 1) Simulation results for rate 1/3: Fig. 4 (a) compares the FER performance at the output of a decoder with full LSOVA (≡ full MLM) on one side and NN-LSOVA using a uniform radius of 3 and 5 across all merge layers (configurations C 3 , C 6 in Table II) on the other. Clearly, the uniform reduction of R l s leads to an undesired error floor at a FER of about 3 • 10 -4 for C 6 while the decoder fails completely for C 3 . Recall that the merge operations are performed in a binary tree, which suggests that the impact of a reduced radius varies in between layers. Therefore, Fig. 4 (b) compares NN-LSOVA configurations with non-uniform reductions in R l s . Overall an improvement of the FER performance can be observed when moving from a uniform reduction of R l s to a non-uniform reduction. Performing the last two merge layers with a radius of R 6 s = 6 leads to a better performance, even when reducing the radius more in the first layers (C 1 vs. C 2 ). Comparing C 2 , C 4 and C 5 reveals that the NN-LSOVA is more sensitive to a reduction of R s in the first layer than in the middle layers. C 4 shows the best performance.

C 1 C 2 C 4 C 5 BC n m =13 BC n m =5 C 3 C 6 C 4 BC n m =13 BC n m =5 C 7 C 8 C 9 C 10 C 11
2) Simulation results for rate 4/5: As mentioned in section IV, the probability of error decreases exponentially with the coding gain. For a similar target error rate to rate 1/3, the rate 4/5 suffers from a lower coding gain mainly due to the considerable decrease in the minimum cumulated Euclidean distance due to puncturing. Hence a larger operating SNR is required or equivalently, only a lower AWGN variance can be supported. This is expected to reduce the entailed penalty from considering smaller R l s values. Simulation results shown in Fig. 4 (c) confirm this expectation. Not only does C 4 yield Full-LSOVA performance, but a further reduction as for C 8 is possible without FER loss. However, results for C 7 , C 9 , C 10 and C 11 demonstrate that a radius reduction down to 1 even for the middle layers is to be avoided.

VII. COMPLEXITY COMPARISON

With the performance obtained in the previous section in mind, we provide here a complexity analysis for the simulated configurations based on Table I and Eq. [START_REF] Weithoffer | Low-complexity Computational Units for the Local-SOVA Decoding Algorithm[END_REF]. The complexity reduction of the NN-LSOVA decoding in comparison with the full LSOVA decoding (i.e. with full radius R l s for each layer) is illustrated in Table II, for the best and worst case scenarios, i.e. when λ l,q,s is at its minimum or maximum value.

The viability of using NN-LSOVA is confirmed by a complexity reduction of up to 52% in terms of number of updates for the worst case scenario for the C 4 configuration with FER performance competitive to Full-LSOVA/MLM for rate 1/3 and 4/5 and up to almost 90% for configuration C 8 for rate 4/5. However, in order to allow a fair comparison with the MLM algorithm, we need to express the complexity in terms of ACS (Add Compare Select) operations. Note, that the NN-LSOVA only affects the complexity for the extrinsic computation. Each L-value update in the SOU of the LSOVA/NN-LSOVA is done using Eq. ( 8), requiring at most one ACS operation. Every merge therefore needs λ l,q,s ACS operations plus one comparison between the path metrics to find the winning path, which will be counted as half an ACS operation. For the decoding of one frame, we get:

ACS LSOVA = n enc • K s • n it • log 2 (q) l=1 q 2 l λ l,q,s + 1 2 (11)
where n enc is the number of component encoders in the NB-TC structure, K s is the number of GF(q) symbols in the frame and n it is the number of performed decoding iterations. As a point of comparison, we take the BC-based method from [START_REF] Klaimi | Low-complexity decoders for non-binary turbo codes[END_REF] for its different approach for complexity reduction in decoding NB-TCs. Its complexity for the extrinsic computation is:

ACS BC = n enc • K s • n it • 2n m • q (12)
Where n m is a parameter of the algorithm that denotes the size of the used sorted tables. The number of ACS operations in the extrinsic computation of the MLM is computed by

ACS MLM = n enc • K s • n it • q 2 (13)
Based on Eqs. ( 11), ( 12) and ( 13), Table III With the proposed NN-LSOVA, we achieve between around 40% and 80% of complexity reduction in comparison to the MLM (worst case, not considering C 6 which has insufficient FER performance), which corresponds to an additional complexity reduction of between around 17% and 58% compared to the BC. In particular, a complexity reduction of at least 52.06% is achieved for configuration C 4 , which is competitive in terms of FER performance compared to the MLM and the BC for the two considered rates of 1/3 and 4/5. To obtain the same low complexity with the BC, we would need to have n m = 13 for the worst case, and n m = 5 for the best case, significantly penalizing the FER performance (see Fig. 4)

VIII. CONCLUSION

In this work, we applied for the first time the LSOVA algorithm to the decoding of NB-Turbo Codes. Moreover for a direct mapping between the GF(q) code and constellation symbols, we proposed a new reduced complexity decoding algorithm. Denoted by NN-LSOVA, it achieves a complexity reduction by up to 52% in terms of performed ACS operations compared to the MLM while maintaining similar FER performance. It can even reach up to almost 80% for applications targeting only high code rates or FER larger than 10 -4 . Furthermore for the same target performance, it was shown to outperform the prior art solution based on the BC by up to more than 30%. These results show the NN-LSOVA algorithm to be a promising path towards efficient hardware implementations for NB Turbo decoding, and future works will consider combining the BC for state metric computation with the NN-LSOVA for the soft output.

Figure 1 :

 1 Figure 1: (a) Structure of the NB recursive convolutional component code considered in this work. (b) Trellis of a NB recursive convolutional code over GF(4).

Figure 2 :

 2 Figure 2: Two radix-4 paths in the trellis of Fig. 1 (b).

Figure 4 :

 4 Figure 4: FER curves after 8 decoding iterations of: (a) Full LSOVA, C 3 and C 6 (b) Full LSOVA, C 1 , C 2 , C 4 , C 5 and BC with n m = 13 and n m = 5 (c) Full LSOVA, C 4 , C 7 -C 11 , and BC with n m = 13 and n m = 5. Rate: (a) 1/3, (b) 1/3, (c) 4/5.

Table II :

 II Selected NN-LSOVA configurations, number of updates and complexity comparison for best-/worst case.

	Configuration	Updates	Compl.	Updates	Compl.
	(R 1 s R 2 s R 3 s R 4 s R 5 s R 6 s )	(bc)	red. (bc)	(wc)	red. (wc)
	C1 = (3,4,4,5,5,6)	951	76.41%	2498	38.05%
	C2 = (2,3,4,5,6,6)	713	82.32%	1700	57.84%
	C3 = (2,2,2,2,2,2)	378	90.63%	819	79.69%
	C4 = (3,3,3,3,6,6)	765	81.03%	1932	52.08%
	C5 = (2,3,3,6,6,6)	701	82.61%	1560	61.31%
	C6 = (5,5,5,5,5,5)	1638	59.38%	3969	1.56%
	C7 = (3,1,1,6,6,6)	669	83.41%	1496	62.90%
	C8 = (2,2,2,2,4,6)	429	89.36%	938	76.74%
	C9 = (2,1,2,2,4,6)	381	90.55%	810	79.91%
	C10 = (2,2,1,2,4,6)	405	89.96%	874	78.32%
	C11 = (2,2,2,1,4,6)	417	89.66%	906	77.53%

Table III :

 III compares the complexity in terms of performed ACS operations for the extrinsic computation. A value of n m = 25 is assumed for the BC, which results in a FER performance close to MLM. Furthermore, n enc = 2, K s = 160 and n it = 8. No. of ACS operations for MLM, BC and NN-LSOVA and complexity reduction compared to MLM.

	Alg.	Num. of ACS Op.	Compl. red. [%]
	MLM	10,485,760	0
	BC [8]	8,192,000	21.24
	NN-LSOVA Config.	worst	best	worst	best
	C1	6,475,520	2,515,200	38.24	76.01
	C2	4,432,640	1,905,920	57.73	81.82
	C3	2,177,280	1,048,320	79.23	90.00
	C4	5,026,560	2,039,040	52.06	80.55
	C5	4,074,240	1,875,200	61.15	82.12
	C6	10,241,280	4,273,920	2.33	59.2
	C7	3,910,400	1,793,280	62.71	82.90
	C8	2,466,560	1,163,520	76.48	88.90
	C9	2,154,240	1,056,000	79.46	89.93
	C10	2,318,080	1,117,440	77.89	89.34
	C11	2,400,000	1,148,160	77.53	89.66

ACKNOWLEDGMENT This work was partially funded by the French National Research Agency projects QCSP (ANR-19-CE25-0013-01) and TurboLEAP (ANR-20-CE25-0007).

REFERENCES


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A comparative investigation of plasmonic properties between tunable nanoobjects and metallized nanoprobes for optical spectroscopy

Keywords: plasmonics, dark-field imaging, total-internal-reflection scattering spectroscopy, tip-enhanced optical spectroscopy, atomic force microscopy, nanofabrication, nano-antennas, localized surface plasmons

In order to evaluate the optical efficiency of tip-based probes for future tip-enhanced optical spectroscopy applications, we have developed an experimental setup based on the coupling of an achromatic inverted microscope equipped with a total internal reflection objective and an atomic force microscopy (AFM) head. This spectroscopic tool has been validated using individual nanofabricated antennas (gold nanodisks/nanocones) on a glass substrate, which act as nanoresonators based on localized surface plasmons. Spectrally tunable transverse electric and magnetic plasmonic resonances are identified and are in excellent agreement with numerical calculations performed as a function of the nano-antenna geometry and size. We have investigated a series of state-of-the-art gold-coated AFM probes, which are commonly used for tip-enhanced (Raman spectroscopy) optical experiments. Their scattering spectrum consists of resonances depending on the tip sharpness or granularity superimposed on a broad emission spectrum due to semi-infinite metal layer acting as a non-resonant antenna. From the comparison between the plasmonic response of both types of optical antennas, a new generation of probes for tip-enhanced optical spectroscopy is proposed, in which single plasmonic nanoantennas are engineered at the apex of a non-metallic AFM tip. As from numerical simulation results, such tips would ensure a spectral tunability as a function of the material, size, and geometry, together with expected high enhancement factors. Such features would allow the design of spectrally tunable surface enhanced Raman spectroscopy substrates, and should be a reliable and efficient alternative to tips commonly used in tip-enhanced optical spectroscopy experiments such as tip-enhanced Raman spectroscopy.

INTRODUCTION

The investigation of optical properties of molecules or individual nanostructures with a spatial resolution less than the diffraction limit requires an electromagnetic field confined at the nanoscale-this can be achieved using adapted optical scanning tip-based probes. This leads, in the case of Raman spectroscopy for example, to the technique referred to as tip-enhanced Raman spectroscopy (TERS) [START_REF] Anderson | Locally Enhanced Raman Spectroscopy with an Atomic Force Microscope[END_REF][START_REF] Hayazawa | Metallized Tip Amplification of Near-Field Raman Scattering[END_REF][START_REF] Stöckle | Nanoscale Chemical Analysis by Tip-Enhanced Raman Spectroscopy[END_REF][START_REF] Pettinger | Nanoscale Probing of Adsorbed Species by Tip-Enhanced Raman Spectroscopy[END_REF] , which can be used today for correlated chemical and physical surface mapping with nanoscale spatial resolution [START_REF] Su | In Situ Topographical Chemical and Electrical Imaging of Carboxyl Graphene Oxide at the Nanoscale[END_REF][START_REF] Smithe | Nanoscale Heterogeneities in Monolayer MoSe 2 Revealed by Correlated Scanning Probe Microscopy and Tip-Enhanced Raman Spectroscopy[END_REF] . Scanning probes used for the optical signal amplification benefit in general from plasmonic properties, i.e. sustain propagative and/or localized surface plasmons, whose excitation results in an enhanced optical evanescent field confined in a nanometric volume at the tip apex-referred to as the 'hot-spot'-and governs the tip-enhanced spatial resolution. In the case of Raman spectroscopy, the amplification factor strongly depends on the shape of the tip, and ranges from 10 4 to 10 8 7,8 , enabling TERS mapping with a nanometric resolution [START_REF] Kumar | Tip-Enhanced Raman Spectroscopy: Principles and Applications[END_REF][START_REF] Wickramasinghe | Billion-Fold Increase in Tip-Enhanced Raman Signal[END_REF] . Tip-enhanced optical spectroscopy can also be used e.g. to map photoluminescence properties at the nanoscale [START_REF] Okuno | Probing the Nanoscale Light Emission Properties of CVD-Grown MoS2 Monolayer by Tip-Enhanced Photoluminescence[END_REF][START_REF] Kumar | Simultaneous Topographical, Electrical and Optical Microscopy of Optoelectronic Devices at the Nanoscale[END_REF] .

Focusing here on the case of TERS, a large diversity of plasmonic probes with different designs have been so far discussed in the literature. These include smooth electrochemically etched metallic wires [START_REF] Kharintsev | Etchant-Based Design of Gold Tip Apexes for Plasmon-Enhanced Raman Spectromicroscopy[END_REF][START_REF] Lopes | Fast and Reliable Fabrication of Gold Tips with Sub-50 Nm Radius of Curvature for Tip-Enhanced Raman Spectroscopy[END_REF][START_REF] Ren | Preparation of Gold Tips Suitable for Tip-Enhanced Raman Spectroscopy and Light Emission by Electrochemical Etching[END_REF] commonly used for scanning tunneling microscopy (STM) and granular metallic coatings onto silicon cantilevers commonly used for [START_REF] Yang | Rational Fabrication of a Gold-Coated AFM TERS Tip by Pulsed Electrodeposition[END_REF][START_REF] Brejna | Electroless Deposition of Silver onto Silicon as a Method of Preparation of Reproducible Surface-Enhanced Raman Spectroscopy Substrates and Tip-Enhanced Raman Spectroscopy Tips[END_REF][START_REF] Sanders | Facile Fabrication of Spherical Nanoparticle-Tipped AFM Probes for Plasmonic Applications[END_REF][START_REF] Trautmann | Plasmon Response Evaluation Based on Image-Derived Arbitrary Nanostructures[END_REF] atomic force microscopy (AFM). It has been shown that the roughness of the metallic layer deposited onto probes [START_REF] Taguchi | Optical Antennas with Multiple Plasmonic Nanoparticles for Tip-Enhanced Raman Microscopy[END_REF] but also the size of the metallic apex itself [START_REF] Maouli | Optical Antennas for Tunable Enhancement in Tip-Enhanced Raman Spectroscopy Imaging[END_REF] have a strong influence on enhancement factors, which vary both in amplitude and as a function of the excitation wavelength. Metal-coated AFM-TERS probes appear simpler to use than STM-TERS probes and allow in principle non-conductive samples to be investigated. The majority of AFM-TERS probes involve today the use of gold as a metallization layer, mainly for studies in materials sciences. Gold is interesting for its intense plasmonic activity in the visible region but also for its chemical inertness providing long probe lifetimes. However, gold layers only produce plasmonic resonances with wavelengths in the red part of the spectrum, which limits applications e.g. towards the field of semiconductors or biology that require to lower the penetration of the Raman excitation laser in the sample but also to avoid parasitic fluorescence signal. Silver [START_REF] Rasmussen | Tip-Enhanced Raman Scattering of a DNA Binding Compound[END_REF][START_REF] Yeo | Enhancement of Raman Signals with Silver-Coated Tips[END_REF] and aluminum [START_REF] Poborchii | Optimization of Tip Material and Shape for Near-UV TERS in Si Structures[END_REF][START_REF] Lipiec | Preparation of Well-Defined DNA Samples for Reproducible Nanospectroscopic Measurements[END_REF] can be used to reach lower wavelengths for such experiments-indeed, the plasmonic activity of silver takes place in the blue-green part of the spectrum, while aluminum can even reach UV [START_REF] Park | Ultraviolet Tip-Enhanced Nanoscale Raman Imaging: Ultraviolet Tip-Enhanced Nanoscale Raman Imaging[END_REF][START_REF] Yang | Deep Ultraviolet Tip-Enhanced Raman Scattering[END_REF] wavelengths. These materials suffer in practice in terms of sulfidation or oxidation in air, which limits the probe lifetime [START_REF] Opilik | Degradation of Silver Near-Field Optical Probes and Its Electrochemical Reversal[END_REF][START_REF] Opilik | Chemical Production of Thin Protective Coatings on Optical Nanotips for Tip-Enhanced Raman Spectroscopy[END_REF][START_REF] Barrios | Extending Lifetime of Plasmonic Silver Structures Designed for High-Resolution Chemical Imaging or Chemical and Biological Sensing[END_REF] . A reliable control of the spectral response of AFM-based TERS probes, i.e. their spectral tunability, is a still key aspect for TERS experiments and developments.

With the aim of improving the plasmonic response and amplification factors of TERS probes, recent work has started to go beyond metal-coated probes and design the apex of scanning probe microscopy tips [START_REF] Vasconcelos | Tuning Localized Surface Plasmon Resonance in Scanning Near-Field Optical Microscopy Probes[END_REF][START_REF] Archanjo | Plasmon 3D Electron Tomography and Local Electric-Field Enhancement of Engineered Plasmonic Nano-Antennas[END_REF] for instance by using focused ion beam etching in order to isolate the metallic part that composes the apex from the rest of the metallized tip. This breaks the conventional semi-infinite plasmonic tip model ('non-resonant' tip) by creating an electromagnetic cavity of controlled shape at the tip apex [START_REF] Novotny | Effective Wavelength Scaling for Optical Antennas[END_REF] , with spectrally well-defined resonances related to the excitation of localized surface plasmon modes (i.e. 'resonant tips'). In practice, such structured tips as well as granular metallic tips commonly used in TERS experiments may combine non-resonant effects (moderately-enhanced electromagnetic field on the whole white light spectrum) together with resonant effects (presence of spectrally defined resonances), depending on the tip granularity or apex shape. Moreover, protusion effects have been studied at the theoretical level down to the atomic scale [START_REF] Barbry | Atomistic Near-Field Nanoplasmonics: Reaching Atomic-Scale Resolution in Nanooptics[END_REF][START_REF] Trautmann | A Classical Description of Subnanometer Resolution by Atomic Features in Metallic Structures[END_REF] .

In order to experimentally investigate nano-fabricated probes for tip-enhanced optical spectroscopy applications, in view of controlling their manufacturing process and their spectral tunability, we have developed in this work a spectroscopy tool based on the coupling of an AFM head and an achromatic inverted microscope equipped with a total-internal-reflection (TIR) fluorescence objective. This setup is designed to measure the spectroscopic scattering response of engineered plasmonic nanostructures [START_REF] Sonnefraud | Nanoplasmonics: Engineering and Observation of Localized Plasmon Modes[END_REF][START_REF] Rycenga | Controlling the Synthesis and Assembly of Silver Nanostructures for Plasmonic Applications[END_REF] without vacuum constraints such as in electron energy loss spectroscopy [START_REF] García De Abajo | Optical Excitations in Electron Microscopy[END_REF][START_REF] Colliex | Electron Energy Loss Spectroscopy Imaging of Surface Plasmons at the Nanometer Scale[END_REF] . The operating principle is close to dark-field scattering 40- 43 -but using a TIR illumination [START_REF] Liu | Spatially Resolved Scattering Correlation Spectroscopy Using a Total Internal Reflection Configuration[END_REF][START_REF] Ament | Single Unlabeled Protein Detection on Individual Plasmonic Nanoparticles[END_REF][START_REF] Hoffmann | Fabrication and Characterization of Plasmonic Nanocone Antennas for Strong Spontaneous Emission Enhancement[END_REF][START_REF] Von Olshausen | Coherent Total Internal Reflection Dark-Field Microscopy: Label-Free Imaging beyond the Diffraction Limit[END_REF][START_REF] Talaga | Colocalized Dark-Field Scattering, Atomic Force and Surface-Enhanced Raman Scattering Microscopic Imaging of Single Gold Nanoparticles[END_REF][START_REF] Yang | Evanescent Wave-Assisted Symmetry Breaking of Gold Dipolar Nanoantennas[END_REF] in order to measure the AFM tip scattering spectrum under scanning conditions, i.e. when an AFM tip is brought into contact with a (transparent) substrate.

This spectroscopic tool has been validated for individual resonant nano-antennas (Au nanodisks or nanocones) on a glass substrate. Spectrally tunable and highly reproducible polarizationdependent plasmonic resonances are identified and are in excellent agreement with numerical calculations performed as a function of the nano-antenna geometry and size. In addition, we investigated a series of state-of-the-art metal-coated AFM tips commonly used for TERS experiments. Their scattering spectrum consists in resonances varying on the tip sharpness or granularity, superimposed on a broad emission spectrum due to semi-infinite metal layer acting as a non-resonant antenna. The comparison between both types of tips is striking, since lithographically-defined nano-antennas exhibit well-defined and spectroscopically tunable plasmonic resonances, while TERS probes based on Au-coated AFM tips only go beyond a non-resonant scattering response only for granular and/or smallest tip apexes. This comparison, together with additional numerical calculations, show that a new generation of AFM-TERS probes can be envisaged, in which single plasmonic nanofabricated antennas will be localized at the apex of a dielectric AFM tip, ensuring both reproducible and spectrally tunable hot-spots as a function of the plasmonic material, size, and geometry.

EXPERIMENTS AND RESULTS

The spectroscopy tool developed for the study is shown in Figure 1-based on an AFM mounted on an inverted microscope (BioScope, Brüker, Germany). The inverted microscope has been fully rebuilt as shown in Figure 1 using mirror and lenses based on silver-coated mirrors in order to ensure an achromatic behavior over the visible spectrum. The setup is based on a 100x TIR objective (IMMOIL-F30CC, Olympus, Germany) in which a white light source is introduced with a large incidence angle at the TIR objective/glass substrate interface in order to create an evanescent optical field. The TIR objective is used in dark-field mode by blocking the light directly reflected at the objective/glass substrate interface-and only collects the scattered light which is analyzed through a spectrometer (VS7000, HORIBA Scientific, France) equipped with an uncooled CCD detector. This setup enables the acquisition of scattering spectra of single plasmonic structures on a glass substrate within ~100 ms as well as the characterization of individual AFM cantilever tips brought into the evanescent field at the TIR objective/glass substrate interface in tapping or contact mode (see Methods).

We first tested our optical spectroscopy setup on arrays of individual plasmonic nano-antennas consisting either in Au nanodisks or in Au nanocones lithographically-patterned onto a glass substrate-see Figure 2. The structures have been fabricated by evaporation through a PMMA resist-based lift-off mask written by electron beam lithography. Because the opening of holes in the mask is progressively reduced during metal deposition, it is possible to fabricate either Au nanodisks with tilted sides at low metallization thickness (Figure 2a), or Au nanocones, once the holes in PMMA mask are fully closed at a critical metallization thickness (Figure 2b) [START_REF] Arscott | On Evaporation via an Inclined Rotating Circular Lift-off Shadow or Stencil Mask[END_REF][START_REF] Eschimese | Size and Shape Control of a Variety of Metallic Nanostructures Using Tilted, Rotating Evaporation and Lithographic Lift-off Techniques[END_REF] .

Nanodisks or nanocones are obtained by choosing the PMMA mask aspect ratio (resist thickness/resist opening width) and the metallization thickness (see Methods for details). We start with a series of Au nanodisks fabricated on a glass substrate, with a thickness of about 50 nm and diameters in the 100-160 nm range. Both transverse electric (TE, incident electric field parallel to the substrate) and transverse magnetic (TM, incident magnetic field parallel to the substrate) linear polarizations (see Figure 1c) have been probed under TIR conditions and are hereafter labelled TIR-TE and TIR-TM. For each single nano-antenna, we have recorded the associated TIR scattering spectrum in the 400-900 nm wavelength range. Background noise (recorded on the glass substrate) has been subtracted from experimental TIR spectra, and a normalization by the reference spectrum of the excitation white light (measured by collecting the incident directly reflected at the glass/air interface, without mask) has been applied. For sake of comparison with experiments, theoretical scattering spectra have been computed together with three-dimensional representations of the far-field radiation for each TE or TM polarization, using finite element simulations (COMSOL software, see Methods).

Experimentally-obtained and theoretically-computed data are shown in Figure 3a and 3b to allow a comparison. We observe in the TIR-TE polarization a single resonance, which exhibits a redshift for increasing Au nanodisks diameters. This mode corresponds to a plasmon associated with a horizontal dipole (HD, see illustration in Figure 3b), for which a spectral redshift with larger diameters is predicted by the plasmon theory and is linked to the lowering of the restoring force exerted on free electrons at the surface of the particle [START_REF] Langhammer | Localized Surface Plasmon Resonances in Aluminum Nanodisks[END_REF] . The experimental data are in excellent agreement with theoretical scattering spectra. Experimental far-field images (see Figure 3c) also show similarities with the calculated radiation patterns. However, the situation is more complex for the TIR-TM polarization. Experimental scattering spectra (Figure 3d) exhibit two resonances-these are well accounted for by numerical modelling (Figure 3d). The resonance at larger wavelength corresponds to the same horizontal dipolar mode as in the TIR-TE polarization and exhibits the same redshift with increasing nanodisk diameter. This reminiscence of the HD mode is due to the tilt angle of the optical excitation in the TIR objective (see schematic diagram of the TIR-TE and TIR-TM polarizations in Figure 1). In contrast, the shorter-wavelength resonance is almost insensitive to the nanodisk diameter. This resonance is well accounted for in numerical calculations and corresponds to a vertical dipolar (VD) mode [START_REF] Marae-Djouda | Angular Plasmon Response of Gold Nanoparticles Arrays: Approaching the Rayleigh Limit[END_REF] (see inset of Figure 3e for an illustration). The experimental evolution of the wavelength and amplitude of the two resonances as a function of the nanodisk diameter is accurately predicted by numerical modeling, taking the incidence angle of the illumination equal to 44°. The calculated far-field radiation patterns of the VD mode, shown in Figure 3f, presents a typical 'donut' shape radiation pattern, observed in the experiments, and which is due to the cancellation of far-field radiation in the direction of the induced dipole. SI-1). TIR-TM scattering spectra and numerical calculations both reveal a dipolar and a higher-order vertical plasmonic mode, as shown in the inset of Figure 4. The first series of nanocones (NC1 to NC5), with heights ranging from 105 to 205 nm, exhibits a pronounced dipolar resonance in the visible range (550-800 nm) except for NC5. In the latter case, both experimental and theoretical results show the emergence of a higher order mode in the green region, of quadrupole type, since the dipolar resonance falls in the infrared. For larger nanocones (here, labeled from NC6 to NC9, with heights from 195 to 245 nm), the quadrupolar plasmon resonance dominates in the 500-650 nm wavelength range. As in the case of the Au nanodisks, the evolution of experimental TIR spectra are consistent with the numerical analysis conducted as a function of the Au nanocone dimensions. Altogether, the results obtained both on Au nanodisks and nanocones show the relevance of our TIR scattering spectroscopy setup to capture the plasmonic properties of nano-resonators as a function of their size and geometry. They also promote the ability of finite element calculations used in this work to quantitatively account for the detailed optical properties of individual metallic nano-resonators as a function of their geometry. We also characterized individual cantilever-based AFM-TERS probes fabricated by metal evaporation on standard commercial AFM tips (ACCESS-NC probes, Applied NanoStructures, Inc., USA). We describe in the following the acquisition of TIR spectra on the example of a Au-coated tip, and then compare TIR scattering spectra for a set of three Au-coated probes.

Cantilevers based probes (see Figure 5a) are brought into the evanescent optical field of the TIR objective covered with a glass substrate using the AFM head of the original AFM setup covering our TIR imaging setup (see Figure 5b and5c). AFM-TERS probes are first engaged in tapping mode on the glass substrate above the center of the TIR objective. The position of the tip is finely adjusted in a second step, using the AFM head horizontal piezoelectric displacements, in order to bring the tip to the center of the optical excitation spot. The acquired TIR spectra are subtracted from the optical setup background noise, which is recorded as a reference in absence of any tip, and then normalized with respect to the excitation light spectrum. The TIR spectra for an Au-coated AFM-TERS probe is presented in Figure 5c. The green curve corresponds to the TIR spectrum acquired with the TERS probes retracted to ~200 nm away from the substrate surface, and exhibits almost no response-as would be expected for a probe out of the evanescent optical field from the substrate. The increased noise level at the edges of the recorded spectrum is due to the cut-off of the optical setup (mainly, of the TIR objective) out of the visible range, which reduces the amount of light detected at such wavelengths. A TIR scattering spectrum of the TERS tip is first shown (blue curve in Figure 5c) when the tip is engaged on the surface in tapping mode (average oscillation amplitude of ~40 nm); also corresponding to a bright spot in the dark-field image (see insets in Figure 5c).

The TIR scattering spectrum shows a well-defined resonance at a wavelength of about 600 nm, while a background develops at larger wavelengths (> 800 nm). When the tip approaches the surface by switching the AFM regulation from tapping mode to contact mode (orange curve in Figure 5c), the TIR spectrum shows an enhanced signal and preserved structure, with an observable redshift of the resonance now occurring at a wavelength of ~650 nm. This redshift in the scattering spectrum can be explained by the interaction of the dipolar localized plasmon mode supported by the tip and its image in the glass substrate [START_REF] Saison-Francioso | Dependence between the Refractive-Index Sensitivity of Metallic Nanoparticles and the Spectral Position of Their Localized Surface Plasmon Band: A Numerical and Analytical Study[END_REF] .

We also performed a comparison between a set of three Au-coated AFM-TERS probes (see Figure 6) using the same protocol as in the experiments shown in Figure 5. Probes appear almost identical in individual SEM images (see Figure 6a). They exhibit a tip apex radius of curvature between ~35 nm and ~45 nmthis radius mainly depends on the metal coating local roughness at the tip apex. TIR scattering spectra are shown in Figure 6b. The systematic sharp peak observed at a wavelength of 685 nm corresponds to a parasitic detection of the AFM head laser and needs to be disregarded. The probes present a 'white' scattering background response (see Figure 6b), which is typical of a semi-infinite metallic (non-resonant) antenna. In addition, the scattering spectra also show a plasmon resonance around 630 nm and 850 nm for the tips with radius of curvature about 35nm and 40 nm respectively. For a larger radius of curvature (45 nm), the main resonance reaches the infrared region. We thus observe as a trend that sharper tips produce localized surface plasmon resonances with high oscillator strength and occur at shorter wavelengths. This spectral dependence related to the variation of the probe radii of curvature of only 5 nm (most certainly due to the roughness at the tip vicinity) is strong and appears difficult to control during the thermal evaporation of the gold layer on the AFM tip.

When put in the general context of tip-enhanced optical spectroscopy measurements, the work presented in this study shows that an important difference exists between metallic nanostructures fabricated by nanolithography (e.g. the nanodisks or nanocones presented and measured here) and the AFM based probes commonly used in TERS experiments (here, metalcoated tips). It is striking that the former nanostructures reproducibly give rise to clear-cut localized surface plasmon resonances with a tunable and controllable wavelength, while the latter tips show a variability of localized surface plasmon resonance wavelength from tip to tip.

One reason for such a difference is that the localized surface plasmon resonance of metal-plated probes is related to specific tip apex radius size and/or granularity and thus strongly size dependent, while the wavelength of the LSPR of individual metallic nanostructures is controlled by a mesoscopic size parameter such as their height in the case of nanocones of the Figure 4.

Nanocones exhibit additionally a reproducible apex radius of 9.9 ± 1.3 nm (the error bar corresponds to the standard deviation for a series of 168 nanocones), contributing to a narrowing of the electromagnetic field lines at the tip apex [START_REF] Stockman | Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides[END_REF] as required for intense field enhancement required for TERS experiments [START_REF] Cui | Simulation Study of Near-Field Enhancement on a Laser-Irradiated AFM Metal Probe[END_REF][START_REF] Geshev | Calculation of Tip Enhanced Raman Scattering Caused by Nanoparticle Plasmons Acting on a Molecule Placed near a Metallic Film[END_REF] . We therefore suggest that a new generation of AFM-TERS probes based on single nanostructures (such as the nanodisks or nanocones lithographically patterned at the apex of a cantilever) and engineered with a specific design or geometry according to an optimized numerical modelling. Such cantilevers should exhibit localized surface plasmon resonances with tunable wavelength and adjustable enhancement factors (e.g. from the control of the plasmonic dipolar resonance), and are expected -at least in principle-to offer superior performances compared to current metal-coated AFM-TERS probes. Such tips should ensure high field enhancement factors together with a controllable spectral tunability as a function of the plasmonic material, size, and geometry.

METHODS

Coupled optical and atomic force microscopy setup.

The device setup shown in Figure 1 is based on an inverted microscope (Olympus IX71, Germany) equipped with a high numerical aperture objective (NA 1.49 -UAPON 1000XOTIRF -Olympus). A broadband white light source (LDLS EQ-99XFC, ENERGETIQ, US) is used as an intense and unpolarized excitation light. To avoid chromatic aberrations, silver coated parabolic, concave and plane mirrors are used in the setup. The incident light from the fiber is firstly collimated and focused into a 50 µm pinhole before being collimated again to cross an iris and a linear polarizer-three parabolic mirrors are involved here. The filtered and polarized pinhole point source is then reflected by a long-focal concave mirror and enters the microscope through a periscope. The light is reflected onto a beam splitter and finally focused on one side of the TIR objective back focal plane to allow total internal reflection excitation at the cover slip/air interface. The focal image of the pinhole source has a size of about a quarter of millimeter and can be finely positioned in the TIR objective back focal plane using XY micrometric screws (see Figure 1b). The incident beam gets reflected on the glass-air interface under TIR conditions, goes back into the objective and is blocked by a mask on the other half of the TIR objective back focal plane. This mask can also be finely positioned in the TIR objective back focal plane using XY micrometric screws.

The substrate thickness is in the 150-200 µm range (Schott D 263® M cover glass). An immersion oil (IMMOIL-F30CC, Olympus, Germany) is used to ensure the lens-substrate contact. The first tunable iris allows us to control the excitation spot on the substrate interface.

The light scattered in the evanescent field at the coverslip/air interface is collected in the center of the TIR objective, goes through the beam splitter and is focused by the tube lens of the microscope onto a CCD camera to obtain scattered far-field imaging. To acquire TIR spectra, the scattered light is focused on a second tunable iris in order to control the collection area on the substrate interface with a spatial magnification of 0.01. It is then collimated and focused onto the entrance slit of the spectrometer (VS7000 -HORIBA Scientific, France) equipped with an uncooled CMOS camera. For an incident light source spectral power density between 40 and 85 µW/nm in the visible range, this setup performs the scattering spectrum analysis of single gold nano-antennas with a ~100 ms acquisition time and a large signal-noise ratio.

Numerical modelling. In order to compare experimental results with theory, we performed finite element simulations with the radiofrequency module of COMSOL Multiphysics. Dielectric parameters of gold used for the calculation have been extracted by interpolation of Johnson and Christy's data [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF] . To model our experimental configuration, nanoparticles or tips have been placed on or close to a glass substrate with refractive index 1.5. The excitation field is the combination of the incident (with an angle larger than the critical angle of 41.8°), the reflected planewave propagating in glass, and the evanescent wave transmitted along the interface in air. The outer layer of the computation domain is set as a perfectly matched layer where all the outgoing scattered waves are absorbed, in order to prevent unwanted reflection onto boundaries.

Nanofabrication. Electron beam lithography (Raith EBPG 500 Plus at 100kV) and thermal evaporation have been used to create arrays of gold nanostructures [START_REF] Arscott | On Evaporation via an Inclined Rotating Circular Lift-off Shadow or Stencil Mask[END_REF][START_REF] Eschimese | Size and Shape Control of a Variety of Metallic Nanostructures Using Tilted, Rotating Evaporation and Lithographic Lift-off Techniques[END_REF] . A bi-layer positive resist is deposited on the glass substrate by spin coating, followed by a deposition of a 5 nm thick layer of Ge to prevent charging effect from the electron beam (400 µC cm -2 dose, 12 nA beam current). After the writing process, the Ge layer is removed (H2O2/H2O: vol/vol = 1/1) before the resist development. Following resist development, thermal evaporation of gold is performed onto the wafer and metallic nanostructures are obtained via lift-off (using a commercial resist remover at 70°C). The key parameter to fabricate nano-disks or nano-cones are the aspect ratio of exposed cylindrical patterns and the amount of evaporated metal. To create nano-disks, only 50 nm of Au is evaporated onto a 200 nm thick resist layer, with hole diameters between 100 and 200 nm. The evaporation tends to close the hole apertures, accounting for the observed tilted edges of the fabricated nanodisks (see Figure 2a). For conical structures, 250 nm of Au is evaporated on a 500 nm thick resist layer with hole diameters between 50 and 100 nm. In this case, the holes in the PMMA layer become fully closed during the evaporation process (see Figure 2b). The Au nanocones exhibit cone half-angles of ~12°, a height ranging from 100 to 230 nm (which depends on the hole diameter of the patterns), and apex radii falling below 10 nm. All structures have been fabricated in arrays with a 10 µm pitch on the glass substrate to allow an easy access to individual structures in optical measurements.

Figure 1 :

 1 Figure 1: (a) Schematic diagram of the inverted microscope used to measure TIR scattering spectra of nano-resonators fabricated on a glass coverslip (Au nanodisks or nanocones, see Methods) or AFM tips brought into the optical evanescent field at the coverslip/air interface. The AFM head enabling the approach of cantilever tips is not represented here (see Figure 5). Orange optical path: the setup is excited with a white light introduced on the side of the TIR objective back focal plane (see b); the specularly reflected beam is blocked in the objective back focal plane in order to fulfill dark field conditions; a linear polarizer is introduced in the optical path in order to excite nano-antennas either in TE or TM modes (see c). Green optical path: the scattered light is collected via the TIR objective, enabling either the dark-field imaging (see d) or the spectral analysis of the scattered light via a visible spectrometer coupled to a CCD detector. (b) TIR objective with a cross-section in its back-focal plane, showing the incident pinhole image (orange spot), the specular reflection blocked by the dark-field mask, and the collected scattered signal. (c) Schematics of the polarizations used to probe TE and TM modes. (d) Example of acquired far-field images on arrays or metallic nano-antennas lithographically designed on a glass coverslip (1 µm or 5 µm pitch).

Figure 2 :

 2 Figure 2: Fabrication sequence for (a) Au nanodisks and (b) Au nanocones. Left: schematic diagram of the fabrication process, in which a PMMA resist layer is exposed by electron beam lithography; Au is then evaporated through the obtained mask, followed by a lift-off process leading either to nanodisks or nanocones at large evaporation thickness. Right: corresponding scanning electron micrographs for Au nanodisks of nanocones.

Figure 3 :

 3 Figure 3: (a) Experimental TIR-TE scattering spectra for a series of individual Au nanodisks as a function of their base diameter (colors are used in the rest of the figure to label for the nanodisks diameters). (b) Corresponding calculated TIR-TE spectra, showing a horizontal dipole (HD) resonance sketched in the inset. (c) Far-field image of a single Au nanodisk (top) of 130 nm diameter, and radiation patterns of the HD mode in the plane normal to the air/glass interface (middle), and in a three-dimensional representation (bottom). (d) Experimental TIR-TM spectra for the same series of Au nanodisks. (e) Corresponding calculated TIR-TM spectra, showing a resurgence of the HD mode at larger wavelength (see the black-dashed arrow as a guide to the eye) due to the tilt angle of the incident light, and a second resonance at lower wavelength, corresponding to a vertical dipole (VD) mode, mostly insensitive to the nanodisk diameter. The insets show schematics of the two dipolar modes. (f) Calculated radiation patterns associated with the VD mode (bottom) showing a donut-like shape, together with an experimental far-field image of the same Au nanodisk used in (c).

Figure 4 :

 4 Figure 4: Experimental TIR scattering spectra recorded (top) and calculated (bottom) for a TM optical

Figure 5 :

 5 Figure 5: (a) Scanning electron micrograph of the apex of the AFM-TERS cantilever-based tip (here, a Au-coated cantilever with 40 nm apex radiusscale bar 100nm). (b) The cantilever is mounted on the AFM head located above the TIR microscope. (c) Schematics of the measurement configuration, in which the TERS tip is brought in the evanescent field of a glass coverslip above the TIR objective using the regulation of the AFM head. (d) TIR scattering spectra for the Au-coated tip shown in (a). The green, blue, and orange curves are the normalized TIR scattering spectra recorded respectively with the tip located 200 nm above the glass coverslip, with the tip engaged on the surface in tapping-mode with a 40 nm oscillation amplitude, and with the tip engaged in contact mode. Far-field optical images are shown as insets.

Figure 6 :

 6 Figure 6: (a) Scanning electron micrograph provided for a set of three commercial Au-coated AFM tips. The scale bar is 100 nm. Apex radii are highlighted. (b) Corresponding TIR scattering spectra. The tip apex radii are shown in insets. The line or discontinuity at 685 nm corresponds to a parasitic detection of the laser line of the AFM head and should be disregarded.To support our statement, we present in the Supporting Information a numerical study based on

AUTHOR INFORMATION

Corresponding Author * thierry.melin@iemn.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


Damien Campello
email: damien.campello@insa-lyon.fr

Marie-Christine C Baietto
email: marie-christine.baietto@insa-lyon.fr

Michel Coret

Nicolas Tardif
email: nicolas.tardif@insa-lyon.fr

Jean Desquines

J Réthoré

A Maynadier

Thermo mechanical behavior of fresh zircaloy-4 under LOCA conditions

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

EXPERIMENTAL SETUP

The experimental setup initially designed by N. Tardif et al. [START_REF] Tardif | Experimental study of the fracture kinetics of a tubular 16MnNiMo5 steel specimen under biaxial loading at 900 and 1000°C. Application to the rupture of a vessel bottom head during a core meltdown accident in a pressurized water reactor[END_REF] has been suited for testing zircaloy cladding. Figure 1.a shows the device. The samples are 90 mm long, 0.57 mm thick and their outer radius is 4.75 mm. Argon gas is used for the test atmosphere and pressurizing the claddings. Due to the bottoming effect, a compressive force is set in order to produce a uniaxial stress state in the hoop direction. Induction device heats up the sample. The thermal distribution over the height is not uniform, even in the region of interest, between the two coils (Figure 1.a), which is 20 mm long.

Region of interest (1.a), and entire experimental setup with cameras and LED rings (1.b).

Experimental setup has been designed in order to get local temperature and kinematics data. As a matter of fact, the temperature acts a lot on the creep behaviour of alloys (especially at high temperature). An infrared camera gets the thermal field which is calibrated from the thermocouples data.

The kinematics fields are calculated by 2-D Digital Image Correlation [START_REF] Sutton | Determination of Displacement Using an Improved Digital Correlation Method[END_REF] from optical full field measurements over the height of the specimen with an internal software developed at the LaMCoS. Image contours detection method [START_REF] Réthoré | Curve and boundaries measurement using B-splines and virtual images[END_REF] can also be used when viscoplastic hoop strains are too large for 2D-DIC. To avoid radiation of the alloy and errors in the kinematics fields calculations [START_REF] Pan | High-temperature digital image correlation method for full-field deformation measurement at 1200 °C[END_REF], two blue LED rings (λ=470nm) are combined with pass band filters centred on λ=470 nm. A black undercoating and a white speckle pattern are sprayed for the 2D-DIC. A black area is kept on the height in front of the infrared camera (high emissivity).

PRELIMINARY RESULTS

From the first tests, the temperature's distribution can be interrelate with the hoop strain state over the axial direction. Four cameras are distributed over the circumference and record pictures simultaneously: two of them are devoted to 2D-DIC, 1 to contours detection method and the last is consecrated to record what happens in the near infrared (Figure 2).

Thermal and kinematics fields are synchronized what leads to the identification of a multiplicative power-law describing the creep behavior of as-received zircaloy-4 fuel cladding:

m+1 creep n VM t ε =A.σ . m+1 (1)
Where creep ε is the creep strain and A, m, n are three parameters depending on the temperature. Results are compared to the literature [START_REF] Rosinger | The steady state creep of zircaloy-4 from 940 K to 1873 K[END_REF] in a validation purpose.

Infrared camera will provide a temperature field and the characterization of the multiplicative law could be effective for several temperature with a unique test, for a low strain state. These results will be presented in May with the characterization of the multiplicative law for an 800°C temperature.

3.

Figure 1

 1 Figure 1.a Figure 1.b

Figure 2

 2 Figure 2.a Figure 2.b Figure 2.c


V Busser

J Desquines

C Duriez

M C Baietto-Dubourg

J P Mardon

Zircaloy-4 cladding hydride reorientation under ring compression test conditions

Temperature increase induces partial or total dissolution of Zirconium hydrides. After this dissolution phase, a cool down under applied stresses can promote radial precipitation of hydrides. Radial hydrides are known to have a deleterious impact on the cladding strength and ductility. The objective of this study is to evaluate, using ring compression tests (RCT), the required stress level leading to the formation of radial hydrides within the cladding during a thermal-mechanical transient. The obtained results are in good agreement with literature data obtained using different specimen geometry. The tests provide quantitative information on the radial hydride induced embrittlement. Ring compression test appears to be an efficient tool for hydride reorientation analysis.

I.

INTRODUCTION

The French "Institut de Radioprotection et de Sûreté Nucléaire" (IRSN) is involved in research activities addressing the main safety issues. The radial hydride precipitation in zirconium cladding alloys has potentially a deleterious impact on cladding integrity. It is thus of primary interest for IRSN to determine the key parameters controlling radial hydride precipitation.

Under normal PWR operating conditions, the cladding is oxidized by the water, leading to the formation of an oxide layer on the external surface. This oxidation process also induces hydrogen diffusion towards the cladding [START_REF] Ells | Hydride precipitates in zirconium alloys[END_REF]. When exceeding the hydrogen solubility limit in the zirconium alloy matrix, hydride platelets precipitate [START_REF] Mcminn | The thermal solubility of hydrogen in zirconium alloys, Zirconium in the Nuclear Industry[END_REF]. After PWR irradiation, mainly circumferentially-oriented hydrides are observed in Stress Relieved Annealed (SRA) Zircaloy 4 [START_REF] Marshall | Control of hydride orientation in Zircaloy-4 by fabrication process[END_REF].

Depending on the hydrogen content, temperature increase induces partial or total dissolution of these hydrides [START_REF] Racine | Influence de l'orientation des hydrures sur les modes de déformation, d'endommagement et de rupture du Zircaloy-4 hydruré[END_REF]. After this heating phase, a cool down under applied stresses can promote radial precipitation of hydrides [START_REF] Chu | Hydride reorientation in Zircaloy-4 cladding[END_REF]. These radial hydrides have potentially a deleterious impact on the cladding strength and ductility [START_REF] Lee | Zirconium hydrides and their effect on the circumferential mechanical properties of Zr-Sn-Fe-Nb tubes[END_REF].

Hydride reorientation tests are usually performed on pressurized tubes. This technique requires a test at each stress level tested and thus induces significant material consumption. Recently, hoop tensile tests with gage section, loaded with inserted mandrels, enabled the determination of the influence of a wide range of stresses on hydride reorientation using a single specimen [START_REF] Daum | Mechanical Testing of High Burn-up Zircaloy-4 Fuel Cladding under Conditions Relevant to Drying Operations and Dry-Cask Storage[END_REF]. But these tests are affected by mandrel-sample friction, that is not accurately quantified and subjected to variations. On the opposite, ring compression test is not influenced by friction occurring at the contact interface between the sample and the loading device as it is of a rolling type [START_REF] Reddy | Phenomena associated with the crushing of metal tubes between rigid plates[END_REF][START_REF] Avalle | Static lateral compression of aluminum tubes: strain gauge measurements and discussion of theoretical models[END_REF].

The objectives of this study are: i) to evaluate the stress conditions leading to the formation of radial hydrides within the cladding during thermal-mechanical transient using ring compression tests (RCT), and ii) to examine the consequences on specimen embrittlement.

II. PRELIMINARY CALCULATIONS

Stress and strain fields used for hydride reorientation testing were numerically computed using the Finite Element Method (FEM). A hydride reorientation treatment is characterised by three steps: 1. the ring is heated to the specified maximum temperature, 2. the temperature being stabilized, the mechanical load is applied to the ring, 3. the ring is cooled down until room temperature.

The complex hoop stress (the hoop direction corresponds to the θ direction shown in Figure 1) response during a RCT to the thermal mechanical loading was obtained. Because the aim of the test is to use this hoop stress distribution to characterize radial hydride precipitation, it is necessary to keep the stress field constant during the entire thermal-mechanical loading. A strategy to obtain such a constant stress field was thus developed.

Since the ring sample is submitted to plane strain loading conditions [START_REF] Busser | Mechanical response of oxidized Zircaloy-4 cladding material submitted to a ring compression test[END_REF], the calculations are conducted using two-dimensional approach. The ring mesh is constituted of 2160 4-node elements (linear interpolation). The ring material is supposed to behave isotropically and to be well modeled with elastic-plastic constitutive relations. The isotropic strain-hardening tensile law is a power-law :

( ) n p K ε σ =
, where σ and ε p are the equivalent Von- Mises stress and the corresponding equivalent plastic strain respectively, and K and n are two material parameters determined in a previous study [START_REF] Busser | Mechanical response of oxidized Zircaloy-4 cladding material submitted to a ring compression test[END_REF]. The mechanical properties of the ring and loading device are specified in Table I. Many thermal transients have been modeled. The maximum temperature ranged between 20°C and 400°C. The predicted maximum hoop stress occurs at the equatorial location and at the outer diameter. It is labeled σ A (see Fig. 1). The correlation between the numerical stress value σ A (MPa) and the applied force F (N) is independent of the maximum temperature as long as the load is smaller than a critical value, here 30 N/mm (see Fig. 2). All the curves are superimposed whatever the temperatures considered up to a force equal to 30 N/mm. The outer diameter equatorial region of the ring is characterized by a tensile stress field. In the hoop direction, the tensile stress ranges between 0 up to the maximum stress σ A .

III. EXPERIMENTAL PROCEDURE 1. Hydride reorientation treatment

Hydrided SRA Zircaloy-4 cladding segments have been prepared. The chemical composition of the alloy is mentioned in Table II. The tubes were loaded with gaseous hydrogen by AREVA at elevated temperature. It leads to uniform circumferentially-oriented hydride distribution. Two hydrogen contents were investigated, 100 and 500 wppm.

These tests were performed using a mechanical testing device INSTRON 5566 equipped with a heated chamber. Twenty-millimeter-long samples were used for hydride reorientation treatment. It induces stress gradients depending on both circumferential and radial locations in the ring. The loading conditions of the different tests performed are reported in table III. samples were tested simultaneously. The procedure was as follows: i) sample one: first the temperature is increased up to about 350°C with an heating rate of 5°C/min, secondly a dwell time is imposed, and thirdly the temperature is decreased with a controlled cooling rate down to room temperature (see Fig. 3) while a mechanical loading is simultaneously applied since the beginning of the test, ii) sample two charged with same hydrogen content, called the reference sample, was submitted only to the same thermal transient (see Fig. 4). Finite-Element calculations provided a correlation between the maximum hoop stress level and the applied load (see Eq. 1 and Table IV). Under controlled loading conditions, the stress field in the cladding was demonstrated to be independent on cladding temperature. It means that, at a given location, hoop stress remains constant during the entire thermal-mechanical transient. After the hydride reorientation test, each sample was then cut in two 10-mm-long rings. One was used for metallographic analysis, while the second was used to evaluate the loss of ductility induced by the precipitation of radial hydrides within the cladding.

Ring compression test after reorientation treatment

Ring compression tests were systematically performed at room temperature on 10-mm-long ring sample, previously submitted to a hydride reorientation treatment. Displacement-controlled tests were performed with a 1mm/min load line displacement rate. The ring orientation is the same as for hydride reorientation tests.

Metallographic analysis

Metallographic analysis was performed on the four tested samples. Cross sectional specimens were embedded in a resin and carefully prepared with an automatic grinding and polishing machine. The embedded samples were also chemically attacked with a solution of HNO 3 , HF and water. They were observed with a LEICA optical microscope.

IV. RESULTS AND DISCUSSION

Metallographic analysis highlighted significant hydride reorientation in locations correlated to tangential tensile stress levels exceeding a threshold (see Fig. 5 and6): the two preferred sites are situated at the outer equatorial surface and at the inner polar surface of the sample. A methodology was developed in order to interpret the results of these tests. The aim being to assess in a good agreement the critical stress levels inducing radial hydride precipitation, an analytical evaluation of the stress distribution within the cladding was conducted.

Due to geometrical and loading symmetries, the modeling of the ring compression test can be reduced to one forth as shown in Fig. 7. ( )

θ θ π θ σ θθ cos 2 ) ( cos 2 6 , 3 Le F R r Le F R r m m - -       - - = (2)
where R m , L, e are respectively the average radius, the axial length and the thickness of the ring.

It is possible to normalize the tangential stress distribution using a shape function f θθ . f θθ is the ratio of the tangential stress at any point and the maximum stress occurring at equatorial location, previously mentioned.

( ) ( ) θ σ θ σ θθ θθ , . , r f r A = (3)
Combining Eq. 2 and 3 leads to the shape function:

( )

2 1 1 2 3 cos 2 1 2 cos 2 3 +       - + -       - = π θ θ π θθ e R e R r e R f m m m (4)
The metallographic data analysis for phenomena controlled by stress levels can be easily assessed using separately, on one side, a shape function threshold and, on the other side, a tangential stress value (σ A ) (see Fig. 8). In the vicinity of the contact region between the loading device and the sample, the analytical assessment of stress distribution is far from accurate. Optical micrograph analysis was thus only performed in the region around the outer equatorial location. Many angular sectors (θ) were examined. For each angular sector studied on the micrographs, three regions are evidenced along the radial direction from the outer to the inner radius: 1. a region with only radial hydrides, 2. a region with mixed circumferential and radial hydrides, 3. a region with only circumferential hydrides, characterized by a maximum radius. The two thresholds defining the three zones allow the estimation of the 100% radial hydride threshold and the incipient radial hydride precipitation threshold (see Fig. 9).

(2)

(3) (1) (2) (3)
The fraction X of the cladding thickness in which radial hydrides are observed is defined as:

e r R X ext - = (5)
Considering Eq. 4 and 5 leads to:

                -         -       +       - - = e m e m R s e R R s f e R X cos 2 3 cos 2 1 2 1 1 2 3 2 1 2 1 π π θθ (6)
For specimens with hydrogen content of 100ppm, there is a full dissolution of hydrides during the heating phase of the thermal transient and full hydrogen content precipitation during the cooling phase. It is also possible to observe the region 1 characterised by 100% radial hydrides (see Fig. 10). For the H500_01specimen, with 500 ppm, the results are different because circumferential hydrides are not entirely dissolved during the heating phase of the heat treatment (see Fig. 11). Some circumferential hydrides are still observed everywhere in the sample. Therefore, the specimen analysis with large hydride contents can be conducted only for two regions.

Distance from equator (s-s equateur) (mm) X -Fraction of Cladding Wall Thickness

f θθ =0.1 f θθ =0.2 f θθ =0.3 f θθ =0.4 f θθ =0.5 f θθ =0.6 f θθ =0.7 f θθ =0.8 f θθ =0.9 f θθ =0.1 f θθ =0.2 f θθ =0.3 f θθ =0.4 f θθ =0.5 f θθ =0.6 f θθ =0.7 f θθ =0.8 f θθ =0.9
Circumferential hydride from X to 1 Radial hydride only from 0 to X Mixed hydride orientation en X Circumferential hydride from X to 1 Radial hydride only from 0 to X Mixed hydride orientation en X Distance from equator (s-s equateur) (mm) X -Fraction of Cladding Wall Thickness

f θθ =0.1 f θθ =0.2 f θθ =0.3 f θθ =0.4 f θθ =0.5 f θθ =0.6 f θθ =0.7 f θθ =0.8 f θθ =0.9 f θθ =0.1 f θθ =0.2 f θθ =0.3 f θθ =0.4 f θθ =0.5 f θθ =0.6 f θθ =0.7 f θθ =0.8 f θθ =0.9
Circumferential hydride from X to 1 Mixed hydride orientation en X Circumferential hydride from X to 1 Mixed hydride orientation en X Fig. 11. Optical micrograph analysis of the H500_01 specimen

The stress field, determined by finite element calculations, is consistent with the observed distribution of hydrides over the entire sample cross-section. Two hoop stress thresholds are derived, for zero hydride reorientation and full radial hydride orientation. Hoop stress threshold for 0% radial hydride precipitation deduced from other specimen geometry are in general in the range of 70-120MPa for Zircaloy-4 alloys and for a maximum temperature of the thermal transient of about 400°C [START_REF] Daum | Mechanical Testing of High Burn-up Zircaloy-4 Fuel Cladding under Conditions Relevant to Drying Operations and Dry-Cask Storage[END_REF][START_REF] Bai | Hydride embrittlement in Zircaloy-4 plates. Part II : Interaction between the tensile stress and hydride morphology[END_REF][START_REF] Alam | Cladding Tube Deformation Test for Stress Reorientation of Hydrides[END_REF]. The threshold increasing with hydrogen content is consistent with literature results [START_REF] Racine | Influence de l'orientation des hydrures sur les modes de déformation, d'endommagement et de rupture du Zircaloy-4 hydruré[END_REF]. The value deduced from this study appears to be consistent with literature, perhaps a little bit lower than expected. However, a very low cooling rate was used here, promoting lower hydride reorientation thresholds [START_REF] Bradbrook | The precipitation of zirconium hydride in zirconium and Zircaloy-2[END_REF].

After hydride reorientation treatment, samples were mechanically loaded using ring compression tests at room temperature in order to assess the influence of radial hydride precipitation on the sample ductility. The obtained load-displacement curves are illustrated in fig 12 (for 100 wppm hydride content) and 13 (for 500 wppm hydride content). The H100_04 specimen, with extended radial hydrides and no circumferential hydrides close to the outer diameter, has a significantly lower ductility than the H100_05 (without hydride reorientation). A similar result is observed on the 500 wppm samples. An outer diameter crack is nucleated at the most deformed locations. The failure load of a specimen with reoriented hydrides is 60% lower than the one of a specimen without radial hydrides. The figure 14 shows the crack pattern which can be easily linked to precipitated radial hydrides. These results are consistent with similar mechanical tests performed on Zircaloy tubing with radial hydrides [START_REF] Daum | Mechanical Testing of High Burn-up Zircaloy-4 Fuel Cladding under Conditions Relevant to Drying Operations and Dry-Cask Storage[END_REF].

VI. CONCLUSIONS

Ring compression test is an efficient tool to produce stress-controlled hydride reorientation in nuclear cladding. In this paper, analytical approach and numerical simulations, based on Finite Element Method, were performed to improve the understanding of the test.

Hydrided Zircaloy-4 (100 and 500ppm) were tested. The combined experimental-numerical analysis approach allowed for an assessment of a value of the stress threshold inducing radial precipitation of hydrides during a thermal transient. Criterion deduced from only two tests is not so far from the results presented in literature. Further modeling is required at polar locations.

Moreover, a ring compression test provides quantitative information on the radial hydride induced embrittlement. The results of the ring compression tests on samples with hydride reorientation using the RCT approach yielded embrittlement behavior consistent with literature data.

Fig. 1 .Fig. 2 .

 12 Fig.1. Definition of the equatorial and polar ring locations (axial direction view)[START_REF] Busser | Mechanical response of oxidized Zircaloy-4 cladding material submitted to a ring compression test[END_REF]

Fig. 3 .Fig. 4 .

 34 Fig. 3. Thermal-mechanical transient for specimens H100_04

Fig. 5 .

 5 Fig. 5. Optical micrograph after a hydride reorientation test for the H100_04 specimen

Fig. 6 .

 6 Fig. 6. Optical micrograph after a hydride reorientation test for the H100_04 (polar location), H100_05 (polar location), H500_1 (equatorial location) specimens (from left to right).

Fig. 7 .

 7 Fig. 7. Geometry of the problem Under the thin shell assumption, the hoop stress σ θθ can be calculated at any point G as a function of the applied load F.

Fig. 8 .

 8 Fig. 8. f θθ function mapping within the cladding

Fig. 9 .

 9 Fig. 9. Definition of both thresholdsEach optical analysis is localized by the polar coordinates s and r. s is the distance of the angular sector with respect to the equatorial location and r (comprised between inner radius and outer radius of the cladding) is the radius of the first radial hydride within the thickness e of the cladding.

Fig. 10 .

 10 Fig. 10. Optical micrograph analysis of the H100_04 specimen

  Fig 10 and 11 contribute to the threshold assessments. The results are summarized in Tables V and VI.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Load-displacement curve of the specimen H100_04 and H100_05 submitted to a RCT at room temperature

TABLE II Chemical

 II composition of the alloy (in wt%)

		Sn	Fe	Cr	O
	Low-tin SRA Zy-4	1.32	0.21	0.11	0.141

TABLE III Hydride

 III

			reorientation test matrix
	Sample I.D.	Heating rate (°C/min)	Tmax (°C)	Maximum stress σ A (MPa)	Dwell time at Tmax	Cooling rate (°C/min)
					(min)
	H100_04	5	342	229	100	0.4
	H100_05	5	342	0	100	0.4
	H500_01	5	344	225	180	0.4
	H500_02	5	344	0	180	0.4