Gabriel Capson

Tojo Lbe

Dr Gabriel Capson-Tojo
email: gabriel.capson-tojo@inrae.fr

Albie Zuo

Meng Gan

Pablo Ledezma

Damien J Batstone

Tim Hülsen
email: t.huelsen@foulingdoctors.com

Raul Muñoz

Ángel Robles
email: angel.robles@uv.es

Resource recovery using enriched purple phototrophic bacteria in an outdoor

Keywords: Resource recovery, PPB, wastewater, biofilm, single-cell protein Resource recovery, PPB, wastewater, biofilm, single-cell protein Conceptualization, Methodology, Investigation, Writing -Original Draft, Visualization, Supervision Methodology, Investigation, Visualization Conceptualization, Resources, Writing -Reviewing and Editing, Supervision, Project administration, Funding acquisition Conceptualization, Methodology, Investigation, Resources, Writing -Reviewing

Purple phototrophic bacteria (PPB) can produce single-cell protein from wastewater at high yields. Attached growth as biofilm might generate a high-quality consistent product. This study compared suspended and attached growths of enriched PPB cultures in an outdoor flat plate photobioreactor (FPPBR) treating poultry-processing wastewater. Attached growth showed lower VFA removal efficiencies (95±2.7 vs. 84±6.4%) due to light limitations and low substrate diffusion rates. Nevertheless, similar overall treatment performances and biomass productivities were achieved (16±2.2 and 18±2.4 gCOD•m-2•d-1 for attached and suspended) at loading rates of 1.2-1.5 gCOD•L-1•d-1. The biofilm had higher quality than the suspended biomass, with lower ash contents (6.9(0.6)% vs. 57(16)%) and higher PPB abundances (0.45-0.67 vs. 0.30-0.45). The biofilm (20-50% of the total biomass) might be used as feed and the suspended fraction as fertiliser, increasing the process economic feasibility. Semi-continuous PPB growth outdoors as biofilm is technically feasible, obtaining a superior product without jeopardising process performance.

Introduction

Resource recovery from wastewater using purple phototrophic bacteria (PPB) is gaining momentum. In recent years, the technology has advanced rapidly, moving from lab-scale proof-of-concept setups to pilot/demonstration scale reactors running on real wastewater (Hülsen et al., 2022c[START_REF] Hülsen | Domestic wastewater treatment with purple phototrophic bacteria using a novel continuous photo anaerobic membrane bioreactor[END_REF][START_REF] Puyol | The photo-anaerobic raceway for domestic sewage: cradle-to-cradle design in wastewater treatment[END_REF]. The main benefit of PPB lies on their high C and COD yields (up to 1 g CODbiomass•g CODremoved -1 ; potentially higher than 1 g C•g C -1 depending on the substrate redox state), paired with simultaneous COD and nutrients removal, and the potential to generate a wide range of high value-added products (e.g., polyhydroxyalkanoates (PHAs), single-cell protein (SCP), pigments or fertilisers) [START_REF] Capson-Tojo | Purple phototrophic bacteria for resource recovery: Challenges and opportunities[END_REF]. These features are enabled by their photoheterotrophic metabolism, which uses light as energy source (producing ATP via photophosphorylation) and organic compounds (preferably volatile fatty acids (VFAs)) as source of carbon and electrons [START_REF] Alloul | Dehazing redox homeostasis to foster purple bacteria biotechnology[END_REF].

Efficient photoheterotrophy relies on light availability, particularly within the near-infrared (NIR) spectrum, as the pigments mostly responsible for light harvesting in PPB (bacteriochlorophylls (BChls)) have absorption peaks at wavelengths in the 780-1,040 nm range [START_REF] Saer | Light harvesting in phototrophic bacteria: Structure and function[END_REF]. This exclusive NIR-absorption capability allows PPB enrichment from virtually any wastewater stream (Hülsen et al., 2018a[START_REF] Hülsen | Domestic wastewater treatment with purple phototrophic bacteria using a novel continuous photo anaerobic membrane bioreactor[END_REF][START_REF] López-Serna | Photobioreactors based on microalgaebacteria and purple phototrophic bacteria consortia: A promising technology to reduce the load of veterinary drugs from piggery wastewater[END_REF]. Nevertheless, the need of NIR light also imposes challenges, such as the design of reactors to provide sufficient light for effective growth. This has always been the case for microalgae systems, but stronger NIR-light attenuation compared to ultraviolet-visible (UV-VIS) wavelengths and higher biomass concentrations in PPB cultures further exacerbate the issue of light limitation in PPB reactors [START_REF] Capson-Tojo | Light attenuation in enriched purple phototrophic bacteria cultures: implications for modelling and reactor design[END_REF]Hülsen et al., 2022a), noting that PPB thrive on lower light intensities compared to algae [START_REF] Dalaei | Municipal wastewater treatment by purple phototropic bacteria at low infrared irradiances using a photo-anaerobic membrane bioreactor[END_REF][START_REF] Hülsen | Domestic wastewater treatment with purple phototrophic bacteria using a novel continuous photo anaerobic membrane bioreactor[END_REF]. The economic constraints of artificial illumination necessitate outdoor cultivation systems, using sunlight as energy source [START_REF] Capson-Tojo | Purple phototrophic bacteria for resource recovery: Challenges and opportunities[END_REF], which further aggravates the challenge of efficient light distribution, as light intensities cannot be controlled and are subject to natural cycles (e.g., day/night intervals).

Few studies have researched PPB growth outdoors. Pioneer articles evaluating this option did not focus on wastewater treatment or nutrient recovery, but on the generation of hydrogen and PHA from artificial media [START_REF] Adessi | Sustained outdoor H2 production with Rhodopseudomonas palustris cultures in a 50 L tubular photobioreactor[END_REF][START_REF] Carlozzi | Growth characteristics of Rhodopseudomonas palustris cultured outdoors, in an underwater tubular photobioreactor, and investigation on photosynthetic efficiency[END_REF][START_REF] Carlozzi | Biomass production and studies on Rhodopseudomonas palustris grown in an outdoor, temperature controlled, underwater tubular photobioreactor[END_REF]. A couple of recent articles have studied the performance of outdoor PPB flat plate PBRs (FPPBRs) treating real industrial wastewaters (Hülsen et al., 2022c(Hülsen et al., , 2022b)). Both studies have shown the technical feasibility of the approach, with high organic matter removal efficiencies (over 71% for COD in batch biofilm-reactors and over 90% for VFAs in continuous reactors). In both cases, the removal of nutrients was limited by the availability of biodegradable COD. In continuous reactors, biomass productivities of 25-84 g VS•m -2 •d -1 were achieved (with conservative estimates from COD removal rates of 6.0-24 g VS•m -2 •d -1 ; VS being volatile solids) at hydraulic retention times (HRTs) of 2.1-2.4 d, giving for the first time numbers that could be extrapolated to perform techno-economic analyses (Hülsen et al., 2022c). The produced biomass had an average crude protein (CP) content of 58% and an amino acid profile suitable for feed applications, making it a potential source of SCP.

Phototrophs (e.g., algae) are mostly grown as suspended cultures in high rate algal ponds (HRAPs), because they provide reasonable productivities (6.8-42 g•m -2 •d -1 ) at low capital costs (~40 USD per m 2 in 2014) [START_REF] Richardson | A financial assessment of two alternative cultivation systems and their contributions to algae biofuel economic viability[END_REF]Young et al., 2017). Recent results from PPB open ponds have shown that these systems can provide effective resource recovery from a variety of wastewaters [START_REF] Alloul | Operational Strategies to Selectively Produce Purple Bacteria for Microbial Protein in Raceway Reactors[END_REF][START_REF] García | A systematic comparison of the potential of microalgae-bacteria and purple phototrophic bacteria consortia for the treatment of piggery wastewater[END_REF][START_REF] López-Serna | Photobioreactors based on microalgaebacteria and purple phototrophic bacteria consortia: A promising technology to reduce the load of veterinary drugs from piggery wastewater[END_REF][START_REF] Sepúlveda-Muñoz | A systematic optimization of piggery wastewater treatment with purple phototrophic bacteria[END_REF]. Nevertheless, these studies have also underlined that the combination of poor NIR light penetration, high PPB biomass concentrations, and the need to limit oxygen diffusion (to avoid PPB out-competition (Capson-Tojo et al., 2023, 2021)), might complicate the use of open ponds for PPB growth, especially when considering high PPB abundances. Another challenge that is generally overlooked in suspended cultivations systems (e.g., open ponds or PBRs based on suspended growth) is the accumulation of undesired components within the biomass. These include particulate inerts, heavy metals, xenobiotics, or even pathogenic microorganisms that survive in the suspension. These contaminants are harvested with the generated biomass, reducing the quality of the product (Hülsen et al., 2022c).

A possibility to overcome biomass contamination is growing the biomass as biofilms in PBRs, either as granules or as biofilms attached onto submerged surfaces. As contaminants tend to remain in suspension, this is a simple solution to improve biomass quality.

Attached/biofilm growth can also help to reduce biomass harvesting/dewatering costs, as aggregates settle faster than suspended biomass, and attached biofilms can be directly harvested at much higher concentrations than suspended biomass. Recent studies have proven the feasibility of forming PPB granules under artificial illumination, with settling rates higher than 30 m•h -1 (similar to existing granular technologies) and high removal rates (e.g., 1.1 kg COD•m -3 •d -1 ) [START_REF] Cerruti | Enrichment and Aggregation of Purple Non-sulfur Bacteria in a Photobioreactor for Biological Nutrient Removal From Wastewater[END_REF]Stegman et al., 2021). [START_REF] Hülsen | Application of purple phototrophic bacteria in a biofilm photobioreactor for single cell protein production: Biofilm vs suspended growth[END_REF] grew PPB enriched biofilms onto artificially-illuminated Perspex tubes, obtaining high pollutant removal efficiencies and recovering over 60% of the biomass as biofilm, with productivities up to 15-20 g VS•m -2 •d -1 . With these biofilms, biomass was harvested at 160 g TS•L -1 (vs. below 2-3 g TS•L -1 for common suspended PBRs (Hülsen et al., 2022c); TS being total solids) and had very low ash contents (4.0% vs. 30% in the influent wastewater). Outdoors, a recent study using FPPBRs (run in batch) showed that PPB can be efficiently grown onto reactor walls from industrial wastewaters (Hülsen et al., 2022b). Biomass was consistently harvested at 90 g TS•L -1 , with CP contents of 50-65%, ash contents below 10%, and biofilm areal productivities up to 14 g TS•m -2 •d -1 . A suspended system treating the same wastewater resulted in much more diluted suspended biomass (1.2-2.5 g TS•L -1 ), with much higher ash contents (~30%) (Hülsen et al., 2022c). The main drawbacks that might limit the application of phototrophic biofilm systems are their higher capital costs compared to suspended systems [START_REF] Enzing | Microalgae-based products for the food and feed sector: an outlook for Europe[END_REF], the reduced diffusion of soluble compounds, and light-induced problems, such as photo-inhibition or shading [START_REF] Kesaano | Algal biofilm based technology for wastewater treatment[END_REF]. Despite these limitations, PPB-biofilm systems are still a promising alternative to address the aforementioned challenges, noting that a valuable product forms the basis of the economic feasibility of any PPB-mediated wastewater treatment process. Particularly for PPB-enriched reactors, biofilm systems have the potential to considerably increase the PPB proportion in the harvested biomass, as flanking microorganisms such as fermenters growing in synergy with PPB would remain in suspension (Hülsen et al., 2022c). This is crucial for a consistent biomass quality (e.g., high protein and low ash contents), and therefore for a high market value of the potential SCP product.

This work aimed at systematically studying the differences between suspended and attached PPB growth strategies for wastewater treatment and resource recovery under realistic working conditions. For this purpose, an outdoor demonstration-scale PPB FPPBR (the first of its kind) was operated in continuous regime, fed with real industrial wastewater in an on-site pilot. This article addresses the following questions: (i) how does the biofilm formation impact the FPPBR performance? (ii) are the biomass productivities reduced due to biofilm formation? (iii) is the biomass quality (as SCP source) higher in attached systems compared to suspended? (iv) is NIR-light availability across the FPPBR affected by the presence of biofilm?

Materials and methods

Wastewater source, pretreatment and characteristics

Raw poultry-processing wastewater, produced on-site at a poultry-processing facility in Brisbane (Australia), was used as substrate. The wastewater was a mixture of effluents resulting from feather removal, bird degutting and general cleaning. The wastewater was drawn from an existing grit trap (barely functional) and fed into a 1,000 L intermediate bulk container (IBC), which acted as equalization tank to allow for continuous feeding (the wastewater was only produced during certain hours of the day). From the intermediate IBC, the wastewater was pumped into a second IBC, which functioned as pre-fermentation tank.

The pre-fermented wastewater (FWW), with concentrations of 2,501-2,926 mg TCOD•L -1 , 558-851 mg COD-VFA•L -1 , 137-188 mg N•L -1 and 29-36 mg P•L -1 was used to feed the FPPBR. The wastewater characteristics of the raw wastewater and the FWW during different operational periods (described below, see Table 1) are shown in Table S1 (see Figure S1 (raw wastewater) and Figure S2 (FWW) for the corresponding daily data). A considerable increase in VFAs, NH4 + -N, and PO4 3--P was observed after fermentation.

Flat plate photobioreactor (FPPBR) overview

FPPBR set-up

The systemschematically represented in Figure S3 was set up in Brisbane (Australia), at a poultry processing facility. The raw wastewater from the grit trap was pumped into the intermediary IBC for equalization via a submersible pump (Ryobi 750W Stainless Steel Submersible Pump), working between 11 am and 4 pm (to exclude cleaning events) when water was available (automatic on/off was facilitated via a float switch integrated in the pump). This allowed to obtain a representative sample of the wastewater that could be fed when no wastewater was produced at the facility. The intermediary IBC was equipped with overflows at the bottom (with a U-pipe, to reduce solids accumulation) and the top, directing excess wastewater back into the grit trap. This equalization IBC was necessary to allow a consistent/continuous flow into the pre-fermentation IBC, which was located around 150 m away at an elevation of ~7 m, close to the FPPBR. To feed the pre-fermentation IBC, raw wastewater from the equalization IBC was pumped (using a Mono CP25, NOV Australia Pty Ltd., Victoria, Australia; 25 L•min -1 ) in pulse/pause intervals of 15/75 min between 11am and 6pm (due to head loss caused by height differences, the actual flow rate was ~13 L•min -1 ). This operational approach resulted in a daily flow rate of 910 L•d -1 , obtaining an HRT of 1.0 d in the fermenter. The level in the fermenter was maintained via overflows at the top and the bottom of the container (via a standpipe). The fermenter-IBC was continuously mixed with a top entry agitator (DM-10 Series, FluidPro Pty Ltd, Australia) to avoid the formation of fat layers and the accumulation of sediments. Turbulence at the surface was minimized to reduce oxygen transfer. The FPPBR was fed with FWW (via a pump Mono CP11, NOV Australia Pty Ltd., Victoria, Australia; 11 L•min -1 ) in pulse/pause intervals of 5/85 min for 12 h per day (during daytime).

The FPPBR consisted of 8 modules (1,220 mm x 1,220 mm x 80 mm (L x H x W)) made from 15 mm clear acrylic joined by o-rings. The total length of the FPPBR was 9.8 m, with a working volume of 953 L and an illuminated surface to volume ratio of 21 m 2 •m -3 . The reactor walls and bottom were covered with an UV-VIS absorbing foil (Lee filter ND 1.2 299) to limit light input from non-NIR wavelengths. A polystyrene foam-sheet cover was used. This cover did not provide a gastight seal, and hence there was some atmospheric exchange.

Influent FWW was provided via valves at the bottom of the first two modules. Mixing was provided via a hydro-mechanical device (protected intellectual property), which served also to remove the biofilm when required. The effluent from the FPPBR was collected in a third 1.0 m 3 IBC to allow harvesting of the suspended biomass (via a centrifuge IC45-M, Interfil, Australia). A more detailed description of the FPPBR can be found in Hülsen et al. (2022c) (see also Figure S3 for a schematic representation of the treatment train).

FPPBR operation

No inoculum was needed, as the plant was already running when the experiments started (continuation of the work presented in Hülsen et al. (2022c)). The FPPBR was fed between 6.30 am and 6.30 pm (approximate daytime hours), obtaining an overall influent flow rate of 440 L•d -1 and an HRT of 2.1 d (or 1.0 d considering only the time when light was available).

To study the differences between suspended and attached PPB growth, the FPPBR was operated for 78 days, divided into two operational periods. During Period I (duration of 36 d) the system was operated in suspended growth mode, mechanically resuspending the biofilm every 30 min to avoid attachment of PPB biofilm on the inner walls. As for the mixing system, a precise description of the device used for biofilm wiping cannot be provided due to protected intellectual property. During Period II (duration of 39 d), mixing was provided as in Period I, but the PPB biofilm on the reactor walls was not wiped off. Thus, PPB were allowed to grow attached to the FPPBR inner walls. For biomass collection and to avoid light limitations, the biofilm was harvested via wiping every 3-4 days, which was sufficient to allow a proper biofilm development (Hülsen et al., 2022b). Other than uncontrollable parameters such as weather, irradiance and wastewater characteristics, all operational parameters were equal between periods. It is important to note that due to variabilities in the FWW composition, the organic loading rate (OLR), and the nitrogen and phosphorus loading rates (NLR and PLR) varied (even if the HRT was maintained). The operational conditions for both periods are shown in Table 1, as well as average solar exposures, T, and pH (data over time can be found in Figure S4).

Sample collection for reactor follow-up and light attenuation assessment

Samples of raw wastewater, FWW, FPPBR effluent and attached biofilm were generally taken twice a week (Monday and Thursday), between 10 am and 12 am (usually during zenith time). Biofilm samples were collected using a plastic scraper (before biofilm removal mechanically). The samples were immediately placed in a cooling box with ice and afterwards stored at -20 ºC, prior to further analysis. Samples were analysed for concentrations of total COD (TCOD), soluble COD (SCOD), VFAs, NO2 --N, NO3 --N, NH4 + -N, PO4 3--P, total Kjeldahl nitrogen (TKN; both soluble and total), total phosphorus (TP; both soluble and total), TS, and VS. A detailed description of analytical methods and data analysis can be found in Supporting Information S1.

Two daily cycle studies with a duration of 24 h were carried out at the end of each operational period (days 36-37 and 71-72) to evaluate the dynamic behaviour of the FPPBR during daynight natural cycles. This was performed using an auto sampler (ISCO 3700 C Portable Automatic Sampler, RS Hydro, Stoke Prior, England), taking samples of 200 mL every 1.5 hours for 24 hours. The auto sampler was filled with ice to inhibit further activity before collecting all the samples early on the next morning. The samples were collected using a syringe to take a 50 mL grab sample of the well-mixed 200 mL sample. Afterwards, the samples were transported on ice and stored at -20 ºC before analysis. The same analyses as above were performed.

Photon flux densities (PFD) were measured during Phase II (days 57, 61, 71 and 78) to determine the light attenuation effect of the biofilm layer (and the acrylic wall and UV-VIS absorbing foil; by measuring the PFD at a given distance from an artificial light source, and placing in between either an acrylic plate or the foil). For this purpose, PFDs (expressed in µmol•m -2 •s -1 ) were recorded with a SQ-620 extended range PFD sensor (with a resolution of 0.1 μmol•m -2 •s -1 ), measuring PFD from 340-1,050 nm, coupled to an AT-100 µcache Bluetooth micro-logger (Apogee Instruments, Inc., USA). PFDs were measured before and after wiping the biofilm from the reactor walls (covered with UV-VIS absorbing foil, which absorbs over 90% of light below 790 nm). The sensor was attached to a 1 m metal rod, which allowed its submersion to 50 cm (around half of the reactor depth). Once in place, the sensor was pointed towards the inner reactor wall, and PFDs were registered (carefully placing the sensor right onto the biofilm, without wiping it) and with the sensor resting on the opposite inner wall (8 cm distance between sensor and wall).

Microbial community analysis

Samples of suspended biomass and attached biofilm were collected for analysis of the microbial communities. Samples were taken at days 7, 18, 34 (Period I, suspended only), and 57, 61, 75, 78 (Period II, both suspended and biofilm). These samples were preserved at -80ºC and given to the Australian Centre for Ecogenomics for DNA extraction and 16S Amplicon sequencing, using Illumina Miseq Platform. The universal primer pair 926F (50-AAACTYAAAKGAATTGACGG-30) and the 1392wR (50-ACGGGCGGTGWGTRC-30) primer sets were used [START_REF] Engelbrektson | Experimental factors affecting PCR-based estimates of microbial species richness and evenness[END_REF]. Trimmomatic was used to trim raw paired reads and to remove reads shorter that 190 bp and/or with low quality (with a Phred-33 lower than 20) [START_REF] Bolger | Trimmomatic: A flexible trimmer for Illumina sequence data[END_REF]. The resulting trimmed paired reads were assembled using PANDAseq, with default parameters [START_REF] Masella | PANDAseq: PAired-eND Assembler for Illumina sequences[END_REF]. The removal of adapter sequences was carried out using the FASTQ Clipper from the FASTX-Toolkit [START_REF] Pearson | Comparison of DNA sequences with protein sequences[END_REF].

The generated high-quality joined sequences were analysed via QIIME v1.8.0 [START_REF] Caporaso | QIIME allows analysis of high-throughput community sequencing data Intensity normalization improves color calling in SOLiD sequencing[END_REF], using the open-reference picking strategy for amplicon sequence variants (ASVs)

given by uclust [START_REF] Edgar | Search and clustering orders of magnitude faster than BLAST[END_REF], at 3% phylogenetic distance. Taxonomy was assigned by uclust against the SILVA rRNA database (128_release) [START_REF] Quast | The SILVA ribosomal RNA gene database project: Improved data processing and web-based tools[END_REF]. ASVs with one or two reads were filtered from the ASVs table by command filter_-asvs_from_asv_table.py in QIIME. ASVs were processed according to Hülsen et al. (2018b).

Analytical methods and data analysis

Concentrations of TCOD and SCOD were determined using COD cell tests (Merck, 1.14541.0001, Darmstadt, Germany), and those of NH4 + -N, NO3 --N, NO2 --N and PO4 3--P were measured via flow injection analysis (FIA; QuikChem8000, Hach Company, Loveland, USA). TS and VS concentrations were determined according to APHA 26 . TKN and TP contents were measured by digestion with sulphuric acid, potassium sulphate and copper sulphate as catalysts in a block digester (Lachat BD-46, Hach Company, Loveland, CO, USA), according to [START_REF] Patton | Methods of Analysis By the U.S. Geological Survey National Water Quality Laboratory-Determination of Total Phosphorus By a Kjeldahl Digestion Method and an Automated Colorimetric Finish That Includes Dialysis[END_REF]. Concentrations of VFAs were measured by gas chromatography (GC; Agilent Technologies 7890A GC System, Santa Clara, CA, USA), using a polar capillary column (DB-FFAP Column, Agilent) and a flame ionisation detector (GC/FID). Soluble compounds were determined after filtration through a 0.45 mm membrane filter (Millipore, Millex®-HP, Merck Group, Darmstadt, Germany). CP contents in the biomass were estimated from the NH4 + -N and TKN contents, according to [START_REF] Eding | Design and operation of nitrifying trickling filters in recirculating aquaculture: A review[END_REF].

Daily solar irradiance values (expressed in MJ•m -2 ) were collected from a local weather station operated by the Australian Bureau of Meteorology, located 2.0 km away from the industrial site (http://www.bom.gov.au/jsp/ncc/cdio/weatherData/av?p_nccObsCode=193&p_display_type= dailyDataFile&p_startYear=&p_c=&p_stn_num=040917).

Values corresponding to measurements are presented as averages and their variability as standard deviation, represented as 𝑋 ̅ (𝑠 𝑋 𝑖 ), where 𝑋 ̅ is the average value for the data Xi, and 𝑠 𝑋 𝑖 is the corresponding standard deviation. Outputs and calculated parameters are presented as average value, with uncertainty expressed as uncertainty in mean based on a two-tailed t-test (95% confidence, 5% significance threshold), represented as 𝑋 ̅ ±𝐸 𝑋 ̅ , where 𝐸 𝑋 ̅ is the 95% confidence interval. Statistical differences were assessed via ANOVA (after verifying that data were normally distributed and homoscedasticity), carried out using the software R (The R Foundation, version 4.1.2).

Results and discussion

Reactor performance: suspended vs. attached growth

The average removal efficiencies, removal rates and biomass productivities in the FPPBR for each operational period are shown in Table 2 (dynamic effluent characteristics can be found in Figure S5). During Period I (suspended growth), no net removal of organics or nitrogen occurred, as the assimilation into suspended biomass dominated. As previously reported for similar HRTs (Hülsen et al., 2022c), VFA removal was almost complete, with average removal efficiencies of 95±2.7% during Period I (Table 2). Also as previously seen (Hülsen et al., 2022c), a residual fraction of SCOD (non-degradable) remained in the effluent (300-600 mg•L -1 ), limiting the overall SCOD removal (average of 65±7.0%). Again in agreement with previous results, nutrient removal was limited by the availability of biodegradable COD (with removal efficiencies of 56% and 42% for TN and TP; calculated as difference between the total concentration in the influent vs. the soluble nutrient concentration in the effluent (Hülsen et al., 2022c)), which explains the relatively high levels of NH4 + -N and PO4 3--P in the effluent (Figure S5).

When the growth strategy was changed from suspended to attached growth (day 37; Period II), the levels of TCOD, TKN and TP in the effluent decreased (despite an increase in the FWW input solids, see Figures S2 andS5). This was caused by the formation of biofilm, which resulted in the overall removal of TCOD (average of 29±9.7%, see Table 2). Figure 1A shows the absence of TCOD removal during Period I and the TCOD removal due to biofilm growth and attachment during Period II. The almost instantaneous change between both periods suggests that operational changes (i.e., stopping biofilm wiping every 30 min) were responsible for the observed changes (instead of shifts in process response or biological changes). Three-four days were sufficient for the formation of an active biofilm (see Figure S6 for biofilm pictures).

While N and P removal efficiencies were similar between periods, the VFA removal was slightly lower (significantly, p-value of 0.002) in Period II, despite the lower overall OLR during this period (1.5±0.2 and 1.2±0.2 g COD•L -1 •d -1 in Periods I and II, respectively).

Average VFA removal efficiencies (84±6.4% in Period II) and time trends confirm this, identifying that a lower VFA removal was maintained during Period II (Figure 1B). The slightly decreased performance in Period II is confirmed by comparing the SCOD, TKN and TP removal rates, all lower compared to Period I (see Table 2). The lower OLR in Period II could partially explain the reduced removal rates and biomass productivities, but definitely not the lower VFA removal efficiencies. Different hypotheses can explain this observation: (i) the diffusion rate of soluble substrates into the biofilm limited VFA availability and reduced the overall uptake rates, which were close to optimal values due to the low HRT applied (1.0 d during daytime); (ii) the formation of biofilm reduced light availability in the bulk liquid, reducing VFA removal rates due to lack of light in the suspended phase; (iii) a combination of (i) and (ii). The light availability might have been affected by generally lower light intensities during the last days of Period II (Figure S4). These hypotheses will be further discussed in the coming sections.

Results from the cycle studies further confirm that VFA removal was limited during attached growth (Figure 2). While VFA accumulation during night-time was barely detectable during Period I (suspended growth), during attached growth a VFA baseline remained in the reactor even during daytime (50-60 mg COD•L -1 •d -1 ). This obviously resulted in higher VFA concentrations during night-time due to hydrolysis and fermentation of residual organics taking place, while no light was available to consume the produced VFAs. This VFA accumulation impacted the pH in the reactor, up to 7.9 in Period I and always below 7.7 during Period II (see Figure 2).

The measured biomass productivities between both periods were similar (see Table 2 andFigure 3). Note that during attached growth (Period II) the productivities were calculated by measuring the biomass concentrations in the liquid after wiping, with most of the biomass in suspension. We tried to measure/estimate the amounts of produced biofilm, but due to technical complications related to an incomplete biofilm harvesting, we could not do it. In addition, limitations of the wiping system led to underestimating the amounts of produced biomass, as some biofilm was not completely wiped and residual biomass remained on the wiping system. Furthermore, floating biofilm aggregates formed after wiping caused heterogeneities when sampling. This was further confirmed by mass balances over the reactor, which closed accurately for Period I (suspended) and confirmed non-negligible losses of TCOD, TN and TP for Period II (see Table S2). Therefore, the measured productivities could be expected to be slightly higher than those obtained (10-30% according to mass balances).

Regardless of these uncertainties, the measured values do not differentiate between the solids corresponding to biomass and those already present in the influent that remained undegraded.

Therefore, these values overestimate the obtained productivities. To solve this, biomass productivities were also estimated from the amounts of SCOD removed, assuming a biomass yield of 1 g CODbiomass•g CODremoved -1 , common for PPB growing photoheterotrophically [START_REF] Capson-Tojo | Purple phototrophic bacteria for resource recovery: Challenges and opportunities[END_REF]. The estimated productivities were statistically similar between both periods, as the amounts of SCOD removed were also similar (the OLRs were slightly lower in Period II but the SCOD removal efficiencies were indeed higher, due to slightly lower SCOD undegradable fractions in the influent, see Figure S5). The high productivities obtained (measured values of 42±8.8 and 39±9.0 g VS•m -2 •d -1 for each period), even for the estimated productivities (averages of 18±2.4 and 16±2.2 g COD•m -2 •d -1 ), which are underestimations that only consider SCOD removal, are in agreement with previous results from other studies, confirming that PPB can achieve high productivities in outdoor reactors (Hülsen et al., 2022c(Hülsen et al., , 2022b)). The relatively consistent trend in the estimated productivities (up to 25 g COD•m -2 •d -1 ), shown in Figure 3B, confirms the consistency and robustness of the system, keeping a stable performance despite the variable influent characteristics and environmental conditions.

Impact of biofilm on light availability through the reactor

To confirm if the slightly lower performances during Period II (attached growth) were caused by light attenuation due to biofilm formation, the photon flux densities (PFDs) at the inner reactor wall were measured before and after biofilm wiping, at different dates (around noon on days 57, 61, 71 and 78; see Table S3). The results showed that the biofilm absorbs, on average, 61±40% of the incident NIR-light, with variations depending on the value of the incident PFD. This obviously increased the light attenuation effect throughout the reactor, and might explain the lower removal efficiencies and rates during attached growth. To confirm this, the PFDs behind the reactor inner wall (i.e., right behind the biofilm) and at the opposite side were measured before and after biofilm wiping (see Figure 4). The PFDs at the opposite side of the wall were similar before and after biofilm wiping (9.7-12 vs. 11-12 µmol•m -2 •s -1 ; or 1.3-1.6 vs. 1.6-1.7 W•m -2 assuming an average NIR wavelength of 900 nm). Both with and without biofilm, these light intensities would have limited effective PPB growth, reducing the growth kinetics. While this effect is reduced due to the light input from both reactor sides in a FPPBR, and due to refraction, the impact of attenuation through the culture is clear, especially considering that light half saturation constants (KI; value at which kinetics are halved) of 4.6 W•m -2 have been determined for PPB enriched-cultures [START_REF] Capson-Tojo | Light attenuation in enriched purple phototrophic bacteria cultures: implications for modelling and reactor design[END_REF]. Efficient growth at NIR levels of 1.4-3.0 W•m -2 has been reported for enriched cultures adapted to low light intensities, but with slower growth kinetics compared to higher irradiances and at solid retention times (SRTs) of 5 d [START_REF] Dalaei | Municipal wastewater treatment by purple phototropic bacteria at low infrared irradiances using a photo-anaerobic membrane bioreactor[END_REF][START_REF] Puyol | A mechanistic model for anaerobic phototrophs in domestic wastewater applications: Photo-anaerobic model (PAnM)[END_REF]. The main difference between the suspended and the attached system concerns mostly the incident light intensity available right after the wall, which impacts the suspended PPB fraction behind the biofilm. Taking the most extreme case (day 57, Figure 4A), the PFD before biofilm wiping was 23 µmol•m -2 •s -1 (3.8 W•m -2 ) and the value after wiping was 81 µmol•m -2 •s -1 (11 W•m -2 ). 3.8 W•m -2 is already lower than the reported KI values, meaning that right behind the biofilm, the available light was already limiting the PPB uptake rates (but still allowed growth). Due to attenuation by water and suspended biomass, this effect was more pronounced at higher distances from the FPPBR wall (i.e., approaching the FPPBR centre at 4 cm from each wall, which was the darkest point). Considering this, it can be concluded that the proportions of dark volumes in the bulk liquid (along the reactor) were higher during attached operation, which limited the uptake kinetics in suspension, resulting in an incomplete VFA consumption at the applied HRT (2.1 d) and in lower pollutant removal rates (Table 2). This assertion (and the above values) are only applicable under the particular environmental conditions, biomass concentrations, and reactor geometry used in this study. The microbial communities, which are discussed in the coming section, further confirmed light limitation in the bulk liquid (suspended biomass).

Biomass characteristics

The characteristics of the harvested biomass during both periods are shown in Table 3. The benefits of attached growth appear evident when comparing the suspended and attached samples. The concentration of biomass was 50 times higher in the biofilm (90.5(1.3) g TS•L -1 vs. 1.80(0.60) g TS•L -1 ), easing the post-processing requirements of dewatering and drying, or avoiding them entirely, for example when utilising PPB paste as feed base after heat sterilisation. In addition, the biofilm had much lower ash content (6.9% vs. 43%). The latter is a crucial parameter for feeding applications, and, in the case of fish meal substitution, it must be below 20% (personal communication with fish feed manufacturers). Therefore, while the harvested biofilm could be used directly as feed, the suspended biomass from Period I could only be used as fertiliser, a product with a much lower value (i.e., 0.42-0.47 USD•kg -1 vs. 1.5-1.6 USD•kg -1 [START_REF] Capson-Tojo | Purple phototrophic bacteria for resource recovery: Challenges and opportunities[END_REF]).

The characteristics of the suspended biomass after centrifugation are also shown in Table 3.

The contents of ash (10%) were much lower than in the suspended samples because, as in Hülsen et al. (2022c), only the top fraction of the centrifuged biomass was collected.

Therefore, the inert particles (ash) remained in the bottom of the centrifuge, allowing the manual separation of both fractions. This was done because impurity-free biomass was needed for animal feed trials, but obviously this approach would not be feasible in continuous, scaled, systems.

It is important to consider that the differences between the suspended and attached biomasses are particularly noticeable in this study due to the high contents of impurities (e.g., sand, other inerts, metals, undegradable COD, etc.) in the wastewater (see Table S1 and Hülsen et al. (2022c)). The differences in ash contents would be much lower if "cleaner" industrial wastewaters, commonly used for SCP production (e.g., food-processing effluents) had been used as influent to the FPPBR.

Although we could not determine the precise proportion of biomass growing as biofilm due to technical limitations causing incomplete harvesting, results from the mass balances (Table S2) suggest that the biofilm represented at least 20-50% of the total COD. This range is in agreement with data from previous studies, reporting up to 60% of the biomass growing as biofilm (in artificially illuminated lab-scale reactors) [START_REF] Hülsen | Application of purple phototrophic bacteria in a biofilm photobioreactor for single cell protein production: Biofilm vs suspended growth[END_REF] and around 30-75% (in outdoor flat plate batch reactors (Hülsen et al., 2022b)).

The PPB relative abundances in the biomass samples are shown in Figure 5. During Period I, the PPB abundances in suspended samples were of 0.30-0.45. Compared to these values, PPB in the biofilm samples (Period II) accounted for 0.48-0.68 of the total reads. This confirms that PPB attached preferentially, while flanking communities such as fermenters remained in suspension (see fermenter proportions in the suspended samples from Period II, much higher than in the biofilm samples; Figure 5B). Therefore, biofilm growth can be used effectively to increase PPB proportions in the harvested product. The PPB proportions in suspended samples from Period II (0.30-0.32) were lower than in Period I, which can be explained by the partial growth of PPB as biofilm, reducing the amounts of suspended PPB. In addition, this observation further suggest that the lack of light was limiting PPB growth in the bulk liquid during attached growth.

Predominant PPB genera (i.e., Blastochloris sp., Rhodopseudomonas sp. and Rhodobacter sp.) were the same in Period I and Period II and in the suspended and attached fractions, confirming that abiotic factors (i.e., mechanical biomass wiping) were responsible for the performance differences, not biotic alterations. Interestingly, the appearance of other PPB genera in the biofilms, such as Rubribrivax sp. and Thiobaca sp., suggests that some PPB might have more tendency to form biofilm than others. Another explanation could be that these particular genera have different light affinity compared with other PPB.

It must be commented that the presence of methanogens in the samples (particularly in the suspended ones) was a consequence of momentary mixing issues in both the FPPBR and the fermenter IBC, which lead to some extent of solids accumulation. These issues were solved during the operational period, which is why the proportions of methanogens decreased over time as the accumulated solids were washed out.

Implications for practical implementation

This study shows that biofilm-based PPB outdoor reactors can be operated efficiently, with similar performances to those obtained with suspended growth. The increased biomass quality of the attached fraction (20-50% of the total biomass) resulted in a higher value-added product directly applicable, for example, as bulk fish feed [START_REF] Delamare-Deboutteville | Mixed culture purple phototrophic bacteria is an effective fishmeal replacement in aquaculture[END_REF]. This is enabled by the low content of ash in the biofilm, by a sufficient protein content (58% of CP), and by a suitable amino acid profile (Hülsen et al., 2022c). The improved product consistency obtained in biofilm systems is another main advantage, as its quality will not depend on the amounts of undesirables present in the influent, which will remain mostly suspended. In addition, pre-concentration steps are avoided, as the concentration of solids in the biofilm is much higher than in the suspended biomass (90.5 vs. 1.8 g•kg -1 ). Considering that harvesting of suspended phototrophic biomass, such as microalgae, accounts for 20-30% of the total costs, higher TS contents in the biomass are a clear advantage of biofilm reactors [START_REF] Acién | Economics of microalgae production[END_REF][START_REF] Singh | Microalgae harvesting techniques : A review[END_REF].

It is important to consider that in biofilm reactors the product quality will be much less compromised by the composition of the wastewater stream than in suspended systems, as most ash and other undesirable compounds will remain in suspension. Therefore, this approach could be applied with virtually any waste stream, showing an obvious benefit when valorising streams high in minerals (ash), such as piggery or poultry-processing wastewaters.

Recent studies have also shown that both biomass growth and biofilm formation might be affected by nutrient deficiency and thus by the wastewater composition (Shaikh et al., 2023a(Shaikh et al., , 2023b)). Finding wastewater streams with optimal characteristics and allowing to generate a marketable product will be essential for the effective implementation of PPB-based technologies. Obviously, "cleaner" wastewaters will always be better suited to generate clean products, minimizing hazards (e.g., pathogens or toxins) and impurities. For streams with no ash/undesirables, suspended systems might be preferable due to their simplicity and faster kinetics. Dedicated comparisons should be performed.

Efficient PPB biofilm formation in an outdoor semicontinuous FPPBR was achieved here.

Effective growth outdoors is crucial, as artificial illuminations costs are prohibitive [START_REF] Capson-Tojo | Purple phototrophic bacteria for resource recovery: Challenges and opportunities[END_REF]. The performance was not affected by the regular biofilm wiping, which implies that continuous operation is feasible, coupled to the sequential biofilm recovery every three-four days (time required for biofilm development). This process resulted in reasonably high productivities (measured values of 39±9.0 g VS•m -2 •d -1 and estimated values of 16±2.2 g COD•m -2 •d -1 ), higher than those reported for indoors PPB biofilm reactors (7.0-10 g VS•m -2 •d - 1 [START_REF] Hülsen | Application of purple phototrophic bacteria in a biofilm photobioreactor for single cell protein production: Biofilm vs suspended growth[END_REF]) and those from algal biofilm processes (up to 7.6 g TS•m -2 •d -1 [START_REF] Fica | Algae-based biofilm productivity utilizing dairy waste-water: effects of temperature and organic carbon concentration[END_REF]). The formation of a light-driven biofilm (as in Hülsen et al. (2022b)) is another main advantage of this system, as providing constant shear to develop a biofilm would result in excessive energy consumption for shear/mixing. The results presented here also show that abiotic factors were responsible for the differences observed between attached and suspended growth.

Despite its advantages, biofilm PPB systems face challenges. The poor light availability in the bulk liquid due to absorption by the biofilm leads to slower removal rates and lower VFA removal efficiencies, which are also affected by the rates of substrate diffusion through the biofilm. This must be taken into account when designing and operating these systems, as parameters such as HRT, wiping frequency (related to biofilm thickness), reactor width, and OLR will need to be optimised to maximise treatment rates. Another limitation is that biofilm growth only partially removes organics and nutrients, which imposes the need of harvesting the suspended biomass to provide acceptable removal efficiencies (Hülsen et al., 2022c). The collected suspended fraction could be used as slow-release fertiliser [START_REF] Sakarika | Purple non-sulphur bacteria and plant production: benefits for fertilization, stress resistance and the environment[END_REF], and research in this area is ongoing [START_REF] Sakarika | Purple non-sulphur bacteria and plant production: benefits for fertilization, stress resistance and the environment[END_REF]Wu et al., 2020). Another possibility would be to couple PPB biofilm processes with other technologies. In this case, PPB could produce SCP upstream, removing 20-50% of the pollutant loads. This could be used to retrofit existent plants to generate a high value-added product upstream. Post-processing via aerobic polishing, filtration or centrifugation would be done as conventionally applied to such wastewaters, with a reduced load.

Conclusions

Despite the lower VFA removal efficiencies during attached growth, this approach achieved similar overall treatment performances and biomass productivities compared to suspended growth. The harvested biofilm had higher quality than the suspended biomass, with much lower ash contents and higher PPB relative abundances. The collected biofilm (representing 20-50% of the total produced biomass) can be used as feed, resulting in a high value product.

The suspended fraction might be used as fertiliser. Semi-continuous attached growth of PPB in outdoor FPPBRs is technically feasible, generating a product with a high quality without jeopardising process performance. Stegman, S., Batstone, D.J., Rozendal, R., Jensen, P.D., [START_REF] Capson-Tojo | Purple phototrophic bacteria are outcompeted by aerobic heterotrophs in the presence of oxygen[END_REF] 1.

Table 3. Average characteristics of the harvested biomass. Roman numbers refer to operational periods described in Table 1. PPB stands for purple phototrophic bacteria, AB for aerobic bacteria and Metha. for methanogens. 12±6.6 10±2.0 T stands for temperature, HRT for hydraulic retention time, SRT for solids retention time, OLR for organic loading rate, NLR for nitrogen loading rate and PLR for phosphorus loading rate. Numbers in brackets are standard deviations. ± is 95% confidence interval. 1. Not determined due to incomplete recovery of the attached biofilm. TCOD stands for total COD, SCOD for soluble COD, VFA for volatile fatty acids, TKN for total Kjeldahl nitrogen, TP for total phosphorus, TS for total solids and VS for volatile solids. Numbers in brackets are standard deviations. ± is 95% confidence interval. 1. Measured including solids in the influent (overestimation). 2. Values estimated from the SCOD removed, assuming PPB photoheterotrophic biomass yields of 1.0 g COD•g COD -1 (underestimation).

Tables and figures

3. Calculated considering the difference between the total nutrient concentration in the influent and the soluble nutrient concentration in the effluent.

Figure 1 .

 1 Figure 1. Removal efficiencies over time in the flat plate photobioreactor for: (A) TCOD, (B)

Figure 2 .

 2 Figure 2. SCOD and VFA concentrations and pH values from the daily cycle studies carried

Figure 3 .

 3 Figure 3. Biomass productivities in the flat plate photobioreactor overtime. The values

Figure 4 .

 4 Figure 4. Photon flux densities (PFDs) measured on the inner reactor wall and on the opposite

Figure 5 .

 5 Figure 5. Relative abundances of different microorganisms in the flat plate photobioreactor

Figure 1 .

 1 Figure 1. Removal efficiencies over time in the flat plate photobioreactor for: (A) TCOD, (B)

Figure 2 .

 2 Figure 2. SCOD and VFA concentrations and pH values from the daily cycle studies carried

Figure 3 .

 3 Figure 3. Biomass productivities in the flat plate photobioreactor overtime. The values

Figure 4 .

 4 Figure 4. Photon flux densities (PFDs) measured on the inner reactor wall and on the

Table 1 .

 1 . Purple phototrophic bacteria granules under high and low upflow velocities. WaterRes. 190, 116760. https://doi.org/10.1016/j.watres.2020.116760 Wu, P., Zhang, X., Niu, T., Wang, Y., Liu, R., Zhang, Y., 2020. The imidacloprid remediation, soil fertility enhancement and microbial community change in soil by Rhodopseudomonas capsulata using effluent as carbon source. Environ. Pollut. 114254. https://doi.org/10.1016/j.envpol.2020.114254 Young, P., Taylor, M., Fallowfield, H.J., 2017. Mini-review: high rate algal ponds, flexible systems for sustainable wastewater treatment. World J. Microbiol. Biotechnol. 33, 1-13. Operational and environmental conditions in the flat plate photobioreactor during Phase I (suspended growth) and Phase II (attached growth).

	https://doi.org/10.1007/s11274-017-2282-x
	Figure and table captions

Table 2 .

 2 Average removal efficiencies, biomass productivities and removal rates in the flat plate photobioreactor for each operational period. Roman numbers refer to operational periods described in Table

Table 1 .

 1 Operational and environmental conditions in the flat plate photobioreactor during Phase I (suspended growth) and Phase II (attached growth).

	Period	I	II
	Duration (d)	36	39
	Growth strategy	Suspended	Attached
	Average daily irradiance (MJ•m -2 )	23 (4.0)	20 (4.3)
	T inside (ºC)	31 (4.0)	29 (4.3)
	pH (-)	7.3 (0.4)	7.5 (0.4)
	HRT (d)	2.1±0.12	2.1±0.04
	SRT(d)	2.1±0.12	n.d. 1
	OLR (g COD•L -1 •d -1 )	1.5±0.2	1.2±0.2
	NLR (mg TN•L -1 •d -1 )	59±22	52±12
	PLR (

mg TP•L -1 •d -1 )

Table 2 .

 2 Average removal efficiencies, biomass productivities and removal rates in the flat plate photobioreactor for each operational period. Roman numbers refer to operational periods described in Table1.

	Period	I	II
	Duration (d)	36	39
	Growth strategy	Suspended	Attached
	TCOD removal (%)	-2.7±10	29±9.7
	SCOD removal (%)	65±7.0	74±5.9
	VFA removal (%)	95±2.7	84±6.4
	TKN removal (%) 3	56±6.7	50±9.6
	TP removal (%) 3	42±8.8	39±9.0
	Biomass productivities (g VS•m -2 •d -1 ) 1	42±8.8	39±9.0
	Estimated biomass
	productivities	18±2.4	16±2.2
	(g COD•m -2 •d -1 )
	TS content (g•L -1 )	1.8 (0.6)	1.6 (0.8)
	VS content (g•L -1 )	1.1 (0.5)	1.1 (0.6)
	SCOD removal rate (mg•L -1 •d -1 )	379±85	364±48
	TKN removal rate (mg N•L -1 •d -1 ) 3	25±9.9	17±4.4
	TP removal rate (mg P•L -1 •d -1 ) 3	3.1±2.3	1.7±1.2

Table 3 .

 3 Average characteristics of the harvested biomass. Roman numbers refer to operational periods described in Table1. TS stands for total solids, VS for volatile solids, TKN for total Kjeldahl nitrogen, TP for total phosphorus and CP for crude protein.Numbers in brackets are standard deviations.

		I	I	II
	Period	(suspended	(after	(biofilm
		sample)	centrifugation)	sample)
	Duration (d)	36	36	39
	Growth strategy	Suspended	Suspended	Attached
	TS (g•kg -1 )	1.80 (0.60)	911 (2.8)	90.5 (1.3)
	VS (g•kg -1 )	1.57 (0.81)	820 (2.7)	84.3 (1.4)
	VS/TS (%)	57 (16)	90 (1.0)	93 (1.0)
	Ash (%)	43 (19)	10 (0.54)	6.9 (0.6)
	TKN (g N•kg -1 )	-	80.4 (6.80)	92.0 (7.9)
	TP (g P•kg -1 )	-	12.4 (0.38)	13.0 (0.42)
	CP (% dry weight)	-	50.3 (4.25)	57.5 (3.95)

Acknowledgments

Gabriel Capson-Tojo is grateful to the Xunta de Galicia for his post-doctoral fellowship (ED481B-2018/017). Tim Hülsen wants to acknowledge The Queensland Government, GHD, Ridley, Aquatec Maxcon and Ingham for their financial support as part of an Advanced Queensland Industry Fellowship. The authors are grateful for the support of Meat and Livestock Australia through their funding from the Australian Government Department of Agriculture, Water and the Environment, as part of its Rural R&D for Profit program.

Declaration of interests

☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☒ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

The authors declare that Tim Hülsen contributed to this study while working at Fouling Doctors, who intends to file a patent concerning the mixing/wiping technology mentioned here, with intent to commercialise.


François Fillaux
email: fillaux@glvt-cnrs.fr

Alain Cousson

Matthias J Gutmann

Evidence of macroscopically entangled protons in a mixed isotope crystal of KH p D 1-p CO 3

Keywords: numbers: 61.12.-9, 03.65.Ud, 63.10.+a, 82.30.Hk Quantum entanglement, Single crystal, Neutron scattering, Raman scattering, Hydrogen bonding

We examine whether protons and deuterons in the crystal of KH 0.76 D 0.24 CO 3 at 300 K are particles or matter waves. The neutron scattering function measured over a broad range of reciprocal space reveals the enhanced diffraction pattern anticipated for antisymmetrized macroscopic states for protons (fermions). These features exclude a statistical distribution of protons and deuterons. Raman spectra are consistent with a mixture of KHCO 3 and KDCO 3 sublattices whose isomorphous structures are independent of the isotope content. We propose a theoretical framework for decoherence-free proton and deuteron states.

Introduction

A defect-free crystal is a macroscopic quantum system with spatial periodicity. In principle, only nonlocal observables are probed by plane waves, eg. with diffraction or spectroscopy techniques. This is attested by the marked localization in reciprocal space of Bragg peaks as well as discrete phonon states. However, since the seminal work of Slater [START_REF] Slater | Theory of the transition in KH 2 PO 4[END_REF], the nonlocal nature of protons is commonly overlooked in crystals containing bistable O-H• • •O hydrogen bonds, when the coexistence of two configurations, say "L" for O1 -H • • • O2 and "R" for O1 • • • H -O2, is conceived of as classical disorder. This leads to a conflict of interpretation consisting in that a statistical distribution of protons should destroy the translation periodicity of the crystal, whereas infrared and Raman spectra evidence extended states at the Brillouin-zone center (BZC) obeying the symmetry-related selection rules of the crystal. This conflict has been tolerated because the nonlocal nature of protons cannot be evidenced by x-ray diffraction. Even with neutrons, a Bragg peak analysis is not conclusive, for statics and dynamics cannot be unravelled effectively from the unit-cell structure factor [START_REF] Nelmes | Recent structural studies of the KDP-type transition: A review[END_REF].

The same conflict of interpretation arises for crystals containing hydrogen bonded centrosymmetric dimers, such as carboxylic acids [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF][START_REF] Skinner | Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments[END_REF], or potassium hydrogen carbonate (or bicarbonate, KHCO 3 ) [START_REF] Thomas | Hydrogen-bond studies. LXXXIV. An X-ray diffraction study of the structures of KHCO 3 and KDCO 3 at 298[END_REF][START_REF] Thomas | Hydrogen bond studies. XCII. Disorder in (HCO 3 ) 2-2 and (DCO 3 ) 2-2 dimers: A neutron diffraction study of KHCO 3 and KDCO 3[END_REF][START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF][START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF]. The thermal interconversion between tautomeric configurations, LL and RR, is supposed to be correlated with a reorganization of the single and double CO bonds, along a reaction path across a multidimensional potential. This is commonly rationalized with rigid proton-pairs in an asymmetric double-well coupled to an incoherent thermal bath mimicking a liquidlike environment. It is supposed that "phonon-induced incoherent tunnelling" should prevail at low temperatures, whereas stochastic jumps should take the lead at elevated temperatures. By contrast, vibrational spectra are consistent with nonlocal observables represented by normal coordinates. There is no evidence of any symmetry breaking due to local disorder and the density-of-states remains discrete at any temperature [START_REF] Lucazeau | Low temperature Raman spectra of KHCO 3 single crystal[END_REF][START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunnelling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF].

Nowadays, the local or nonlocal behaviour of bistable hydrogen bonds can be distinguished at advanced pulsed neutron sources, with a time-of-flight Laue diffractometer measuring a much broader range of reciprocal space than any reactorbased four-circle instrument. The scattering function measured at large momentum transfer values (Q = k ik f , where k i and k f are the initial and final wave vectors, respectively) contains information on the scale of spacial correlations between L and R sites (∼ 0.5 Å). Moreover, such an instrument is unique to observing diffuse scattering, because signals in addition to Bragg peaks are effectively measured across the whole range of reciprocal space for each neutron pulse. It is thus possible to determine whether or not coherent scattering is affected by the coexistence of L and R configurations at various temperatures.

This has been exemplified with the crystal of KHCO 3 made of hydrogen bonded dimers, (HCO - 3 ) 2 [START_REF] François Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF][START_REF] Fillaux | Macroscopic quantum entanglement and "super-rigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF][START_REF] Fillaux | A neutron diffraction study of macroscopically entangled proton states in the high temperature phase of the KHCO 3 crystal at 340 K[END_REF][START_REF] Fillaux | The macroscopic quantum behavior of protons in the KHCO 3 crystal: Theory and experiments[END_REF]. Below 150 K, only those proton sites (L), corresponding to a unique configuration for dimers (LL), are occupied. The neutron scattering function reveals enhanced Bragg peaks exclusively at those particular Qvalues corresponding to the reciprocal sublattice of protons. Furthermore, the enhanced peaks are superposed to a prominent continuum of incoherent scattering, cigar-like shaped along Q-vectors perpendicular to dimer planes. By contrast, there is no evidence of similar enhanced features for the isomorphous derivative KDCO 3 . To the best of our knowledge, the dramatic increase of both coherent and incoherent scattering intensities for the sublattice of protons, not observed for deuterons, cannot be explained with any conventional model based upon Slater's localized particles. Alternatively, a quantum theory leads to different macroscopic quantum correlations (entanglement) for indistinguishable protons (fermions) or deuterons (bosons). The enhanced features can be thus explained by the spin symmetry of the "super-rigid" fermion states for which phonons are forbidden. By contrast, deuteron states are represented by phonons with no spin-symmetry.

Above 150 K, the less favored sites along the hydrogen bonds (R), at ∆ x ≈ 0.6 Å from the L sites, are progressively populated, while the space group symmetry and the unit cell parameters remain unchanged. At 300 K, the population ratio is ≈ 18(RR) : 82(LL). The enhanced features observed continuously from cryogenic to room temperatures show that the quantum correlations are temperature independent. The coexistence of LL and RR configurations is consistent with a quantum superposition of proton states perfectly insulated from decoherence by the other degrees of freedom [START_REF] Fillaux | Macroscopic quantum entanglement and "super-rigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF][START_REF] Fillaux | A neutron diffraction study of macroscopically entangled proton states in the high temperature phase of the KHCO 3 crystal at 340 K[END_REF]. Then, scattering functions, vibrational spectra, and quasi-elastic neutron scattering (QENS) data, can be reinterpreted consistently with nonlocal observables and quantum interference arising from the superposition of macroscopic tunnelling states.

Structural transitions of second order (P 2 1 /a ←→ C2/m) occur at T cH ≈ 318 K (KHCO 3 ) or T cD ≈ 353 K (KDCO 3 ). Above T c , dimers split into two centrosymmetric entities, symmetric with respect to the plane (a, c). It is quite remarkable that the enhanced features of the scattering function, as well as proton dynamics and quantum interference, do not show any discontinuity at T c [START_REF] Fillaux | A neutron diffraction study of macroscopically entangled proton states in the high temperature phase of the KHCO 3 crystal at 340 K[END_REF]. There is no evidence of classical disorder above T c . It transpires that quantum correlations arise from the space periodicity of centrosymmetric dimers, irrespective of the space group symmetry.

Here, we examine whether the enhanced features survive, or not, in a mixed isotope crystal of KH p D 1-p CO 3 . In this case, static or dynamical disorder are clearly differentiated and the existence of distinct crystallographic domains of KHCO 3 or KDCO 3 is excluded. A static distribution of mutually exclusive particles should destroy the quantum entanglement and the enhanced features should vanish. Alternatively, if isotopes merge into separable macroscopic states, such that quantum correlations across the sublattice of protons survive, the enhanced features should be observed, although with lower contrast compared to KHCO 3 . To the best of our knowledge, such a nonlocal approach has never been examined in depth, in this context.

In a preliminary report [START_REF] Fillaux | Nonlocal protons and deuterons opposed to disorder: a single-crystal neutron diffraction study of KH 0.76 D 0.24 CO 3 and a theoretical framework[END_REF], the structure of KH 0.76 D 0.24 CO 3 at 300 K was refined from Bragg peaks measured with the four-circle diffractometer 5C4 at the Orphée reactor of the Laboratoire Léon Brillouin [17]. There was no evidence of any symmetry breaking: the space group (P 2 1 /a) is identical to that of KHCO 3 or KDCO 3 at the same temperature (Fig. 1). A structural model consistent with the proposed quantum theory was represented by four sublattices, H LL , H RR , D LL , D RR , with weights p H q LLH , p H q RRH , p D q LLD , p D q RRD , respectively:

p H [q LLH H LL + q RRH H RR ] + p D [q LLD D LL + q RRD D RR ]. (1)
The bound coherent scattering lengths are bH = -3.741 fm or bD = 6.671 fm. The refined positional and thermal parameters (R ≈ 3%) were identical for protons and deuterons whose sites are indiscernible, to within experimental precision and thermal ellipsoids. The estimated concentrations were p H = 0.76, p D = 0.24, and the relative weights were such as S = p H q LLH +p H q RRH +p D q LLD +p D q RRD = 0.99 [START_REF] Skinner | Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments[END_REF]. There was no evidence of any significant decrease of Bragg peak intensities into Laue monotonic diffuse scattering due to a statistical distribution of isotopes. Alternatively, for a sublattice of paired up nonseparable protons and deuterons, the averaged scattering length should be: b = 0.76 bH + 0.24 bD ≈ -1.242 fm,

The model is more artificial because p is no longer a free parameter and it is less satisfactory (R ≈ 4%). However, it cannot be definitely excluded. Consequently, the question as to whether the sublattices of protons and deuterons are separable entities remains largely open. The conclusions of the present work are based upon the neutron scattering function of the mixed isotope crystal presented in Sec. 2. The enhanced features show that the quantum correlations of the pure crystal are not destroyed by partial deuteration and Raman spectra are consistent with separable proton and deuteron states. In Sec. 3, the quantum theory [START_REF] Fillaux | A neutron diffraction study of macroscopically entangled proton states in the high temperature phase of the KHCO 3 crystal at 340 K[END_REF][START_REF] Fillaux | The macroscopic quantum behavior of protons in the KHCO 3 crystal: Theory and experiments[END_REF] applied to the mixed crystal accounts for macroscopic proton and deuteron states, each of which being a superposition of LL and RR states. A detailed interpretation of the enhanced features is proposed. In Sec. 4, we show that alternative explanations are inappropriate.

Experimental

The neutron scattering function

The mixed isotope crystal (≈ 30 mm 3 ) measured in the previous diffraction work [START_REF] Fillaux | Nonlocal protons and deuterons opposed to disorder: a single-crystal neutron diffraction study of KH 0.76 D 0.24 CO 3 and a theoretical framework[END_REF] was cut from a twinning-free parent crystal (≈ 20 × 15 × 3 mm 3 ) obtained from a water solution (≈ 75% H 2 O, 25% D 2 O). This parent crystal was loaded in the vacuum chamber of the time-of-flight Laue diffractometer SXD, at the ISIS pulsed neutron source [18,[START_REF] Keen | SXD -the single-crystal diffractometer at the isis spallation neutron source[END_REF]. The temperature was (300 ± 1) K. We measured 8 orientations, each for ≈ 8 hours, with (b) (b * ) either parallel or perpendicular to the equatorial plane of the detector bank. The reduced data were concatenated in order to compute the scattering function (Fig. 2). Previous studies of crystals with similar sizes have shown that multiple-scattering effects are not important. However, Bragg peaks show severe extinction. A standard analysis is consistent with the structure and isotope contents previously determined, but the modest precision (R ≈ 10%) shows that measurements conducted at different sources are complementary.

The interpretation of the scattering function is based upon the crystal structure made of centrosymmetric dimers lying in (103) planes with O• • •O bonds parallel to each other. Proton/deuteron coordinates are x, y, z, (Fig. 1), Q x , Q y , Q z , are projections of Q, U xx , U yy , U zz , are the dominant temperature factors. Dotted lines in Fig. 1 enhance the network of double lines of proton/deuteron sites, either parallel to (b), separated by ∆ x ≈ 0.6 Å, or parallel to x, with ∆ y ≈ 2.2 Å. The spatial periodicity of these gratinglike structures is D x ≈ a/ cos 42 • ≈ 20.39 Å or D y = b. The distance between (x, y) planes is D z ≈ 3.28 Å. A previous analysis of rigid body motions has shown that there is no short-range correlation between protons and carbonate entities [START_REF] François Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF] and inelastic neutron scattering (INS) spectra show that proton dynamics are separated from heavy atoms and virtually dispersion-free [START_REF] Fillaux | Proton dynamics in the hydrogen bond. The inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5 K[END_REF][START_REF] Ikeda | Inelastic neutron scattering study of the localized dynamics of protons in KHCO 3 single crystals[END_REF].

In Fig. 2, cuts of the scattering function parallel to (a * , c * ) show spots of intensity, corresponding to Bragg peaks, and, at large Q-values, enhanced streaks along Q z perpendicular to the trace of dimer planes parallel to (Q x , Q y ), with Q y ||(b * ). These streaks are not an artefact of the instrument because: (i) the detectors were calibrated with vanadium; (ii) the streaks are measured by different detectors for each of the 8 orientations of the crystal; (iii) spurious signals eventually due to an anomalous detector should appear along different trajectories.

The maps in Fig. 2 are similar to those of KHCO 3 at the same temperature [START_REF] Fillaux | Macroscopic quantum entanglement and "super-rigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF], apart from the weaker intensity of diffuse scattering due to the smaller concentration of protons. On the other hand, they are markedly different from those of KDCO 3 for which enhanced features are totally extinguished [START_REF] François Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF]. The enhanced diffraction peaks correspond to nodes of the reciprocal sublattice of protons/deuterons and there is no obvious alternative sublattice with similar parameters. They occur when k = Q y /b * = ±2πn y /(b * ∆ y ) ≈ ±n y × 2.57. For k = 0, in-phase scattering by double lines of sites parallel to x gives peaks at Q x = ±2π/∆ x = ±(10.00 ± 0.25) Å-1 and

Q z ≈ n z × 1.92 Å-1 . At k = ±3, n y ≈ 1, anti-phase scattering along Q y and Q x give peaks at Q x = ±(5.00 ± 0.25) Å-1 , Q z ≈ n z × 1.92 Å-1 . For k = ±5 (n y ≈ 2) and k = ±8 (n y ≈ 3)
, the patterns of enhanced peaks are similar to those at k = 0 and k = ±3, respectively.

A cut along the streak at Q x = (10.00 ± 0.25) Å-1 in the plane k = 0 (Fig. 3) shows enhanced Bragg peaks on the top of a continuum along Q z . Because there is no evidence of disorder along z, this continuum is essentially due to incoherent scattering [START_REF] François Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF]. It compares favorably with the Gaussian profile I 10 exp -Q 2 z U zz , with U zz ≈ 0.04 Å2 [START_REF] Fillaux | Nonlocal protons and deuterons opposed to disorder: a single-crystal neutron diffraction study of KH 0.76 D 0.24 CO 3 and a theoretical framework[END_REF]. Compared to the intensity I 0 at Q x = 0, Q z = 0, the ratio I 0 /I 10 ≈ 2, with substantial uncertainty, is much greater than expected from conventional models according to which incoherent scattering by protons should be proportional to the Debye-

Waller factor DWF = exp -(Q 2 x U xx + Q 2 y U yy + Q 2 z U zz )
, while incoherent scattering by other nuclei should be negligible. With U xx ≈ 0.035 Å2 [START_REF] Fillaux | Nonlocal protons and deuterons opposed to disorder: a single-crystal neutron diffraction study of KH 0.76 D 0.24 CO 3 and a theoretical framework[END_REF], the expected intensity ratio should be I 0 /I ′ 10 ≈ 30. The observed intensity (I 10 ) suggests that U xx is dramatically reduced for these Q x , Q y -values.

Fig. 4 (top) shows a cut at k = 0, parallel to a * , through the reciprocal plane at -(5.0 ± 0.2) Å-1 off-a * . The enhanced Bragg peak observed at -(8.5 ± 0.2) Å-1 (spotted by an arrow) hides the underlying incoherent scattering signal. Alternatively, a cut at -(4.2 ± 0.2) Å-1 off-a * is free of Bragg peaks (Fig. 4, bottom). It reveals the sharp profile of incoherent scattering centered at -(9.0 ± 0.2) Å-1 (spotted by an arrow). Compared to the enhanced Bragg peak, the shift in position by ≈ -0.5 Å-1 is consistent with the slope of the Q z axis. The half-width at half-maximum, ∆ a * ≈ 0.4 Å-1 , gives the half-width along Q x , ∆ Qx ≈ ∆ a * / cos 42 • ≈ 0.3 Å-1 , similar to that of the Bragg peak (≈ 0.2 Å-1 ). Consequently, U xx vanishes exclusively if Q x matches a node of the reciprocal sublattice of protons (see below). Otherwise, incoherent scattering is not observed, presumably because it is dramatically depressed as exp -(Q 2

x U xx ). Similarly, cuts around k = 0 (not shown) suggest that U yy vanishes sharply outside the plane.

Separation of proton and deuteron states: Raman scattering

In Fig. 5, the Raman profile of the stretching mode νOD for the pure crystal is redshifted by ≈ 600 cm -1 with respect to the νOH profile of KHCO 3 and the spectrum of the isotope mixture is a mere superposition of these profiles. There is no visible frequency shift for each profile and, therefore, there is no significant dynamical correlation between isotopes. By contrast, if protons and deuterons were paired up, a single νO(H p D 1-p ) profile should appear at an intermediate frequency, shifting smoothly from νOH (p = 1) to νOD (p = 0). This is clearly excluded. The spectra accord with separable proton and deuteron extended states consistent with the mixture of sublattices [START_REF] Slater | Theory of the transition in KH 2 PO 4[END_REF].

In addition, the O• • •O bond lengths are significantly different for KHCO 3 (R H = 2.58 Å) or KDCO 3 (R D = 2.61 Å) and R 0.76 = 2.59 Å for KH 0.76 D 0.24 CO 3 is seemingly a weighted average [START_REF] Fillaux | Nonlocal protons and deuterons opposed to disorder: a single-crystal neutron diffraction study of KH 0.76 D 0.24 CO 3 and a theoretical framework[END_REF]. Isomorphous sublattices of protons and deuterons exclude a statistical distribution of O• • •O distances. Otherwise, the νOH frequency should vary significantly with p, for ∆νOH/∆R p ≈ 5000 cm -1 Å-1 [START_REF] Novak | Hydrogen bonding in solids. Correlation of spectroscopic and crystallographic data[END_REF]. The absence of frequency shift, or band broadening, suggests that a mixed crystal can be represented by isomorphous sublattices p[KHCO 3 ] and (1-p)[KDCO 3 ], whose structures and dynamics are independent of p. These sublattices are not resolved by diffraction.

Theory

Admittedly the enhanced features are an evidence of quantum correlations, but different mechanisms have been envisaged: symmetry related entanglement [START_REF] François Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF][START_REF] Fillaux | Macroscopic quantum entanglement and "super-rigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF][START_REF] Fillaux | A neutron diffraction study of macroscopically entangled proton states in the high temperature phase of the KHCO 3 crystal at 340 K[END_REF][START_REF] Fillaux | The macroscopic quantum behavior of protons in the KHCO 3 crystal: Theory and experiments[END_REF]; quantum exchange [START_REF] Keen | Quantum correlation between protons in potassium bicarbonate[END_REF]; spin-spin interaction [START_REF] Sugimoto | Effects of entanglement on elastic and inelastic scattering functions for neutron scattering from a pair of nuclei in solids[END_REF]. They will be compared.

The Raman spectra suggest that the Hamiltonian for a mixed crystal can be partitioned as [START_REF] Keen | Quantum correlation between protons in potassium bicarbonate[END_REF][START_REF] Sugimoto | Effects of entanglement on elastic and inelastic scattering functions for neutron scattering from a pair of nuclei in solids[END_REF][START_REF] Hirata | On the validity of the Born-Oppenheimer separation and the accuracy of diagonal corrections in anharmonic molecular vibrations[END_REF]. The separation of proton and deuteron states is, therefore, consistent with the "super selection rule" for fermions and bosons [START_REF] Wick | The intrinsic parity of elementary particles[END_REF]. The mixed crystal is described with N a , N b , N c (N = N a N b N c ) unit cells labelled j, k, l, along crystal axes (a), (b), (c), respectively. The sublattices are described with the same unit cell containing 4p KHCO 3 and 4(1-p) KDCO 3 entities. Proton/deuteron sites are organized in two dimer entities indexed as j, k, l and j ′ , k, l, respectively, with j = j ′ . For each dimer, dynamics are represented by normal coordinates, either antisymmetric {α ar } or symmetric {α sr } (r = 1, 2, 3), which are linear combinations of displacements along x, y, z, respectively. The coordinates of the centers of symmetry, {α 0r }, {α ′ 0r }, are (x 0 , y 0 , z 0 ), (x ′ 0 , y ′ 0 , z ′ 0 ). For planar dimers, z 0 = z ′ 0 = 0. Configurations LL and RR correspond to opposite signs for x 0 and x ′ 0 . Although proton and deuteron sites may be not strictly identical, their dynamics are formally represented by the same set of normal coordinates. The wave functions in the ground state are represented by gaussian functions Ψ a 0jkl (α ar ) and Ψ s 0jklC (α sr -√ 2α 0rC ), with C = "LL" or "RR". A key issue of the theory is to clearly differentiate sites and states. Sites refer to local positional parameters, whereas each single state is extended over all indistinguishable sites. Conversely, the probability density (occupancy) at any site comprises contributions from every single state. The nonlocal observables (normal coordinates and conjugated momentums) are not bound to any particular unit cell. We should emphasize that nonlocality due to the lattice periodicity is not related to any physical interaction. It is energy free. This is totally different in nature from the long-range quantum exchange in a supersolid, like the crystal of He [START_REF] Balibar | Supersolidity and disorder[END_REF]. Obviousy, the proposal by Keen and Lovesey [START_REF] Keen | Quantum correlation between protons in potassium bicarbonate[END_REF] that nonlocality could arise from non vanishing exchange integrals is incorrect, even for the nearest sites separated by ≈ 2.2 Å [START_REF] Fillaux | Comment on "quantum correlations between protons in potassium bicarbonate[END_REF].

H p = p(H H + H KCO 3H + C H,KCO 3H ) + (1 -p)(H D + H KCO 3D + C D,KCO 3D ) + H ep . (3)

Macroscopic states

It is customary to represent lattice dynamics with plane waves (phonons). For deuterons in the ground state, this can be written as:

|D + kς C = 1 √ N Nc l=1 N b k=1 Na j=1 |Ξ 0DjklςC exp(ik • L jkl ); ( 4
)
where k is the wave vector; L jkl = ja + kb + lc is a lattice vector; a, b, c, are the unit cell vectors. The wave functions for a unit cell, Ξ 0DjklςC = Θ 0DjklC + ςΘ 0Dj ′ klC with ς = "+" or "-", is a linear combination of the wave functions for centrosymmetric sites:

Θ 0DjklC ({α 0r }) = r Ψ a 0Djkl (α ar )Ψ s 0DjklC (α sr - √ 2α 0rC
) and likewise for Θ 0Dj ′ klC ({α ′ 0r }). For centrosymmetric dimers, there is no permanent dipole, so long-range correlations through inter-dimer interaction are negligible. Phonons are virtually dispersion-free [START_REF] Fillaux | Proton dynamics in the hydrogen bond. The inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5 K[END_REF][START_REF] Ikeda | Inelastic neutron scattering study of the localized dynamics of protons in KHCO 3 single crystals[END_REF].

|D + kς C is a single-deuteron state with an occupancy number (4N ) -1 per site. This state cannot be factored with eigen states for local coordinates (x 1jkl , x 2jkl . . .). It is entangled in position. (By definition, a pure state of an arbitrary number of particles is said to be entangled when, because of its indivisible nature, it cannot be described by treating each particle separately.) Likewise, the state in momentum space is also entangled. In addition, this state meets the symmetry postulate for bosons stipulating that the sign should not change when any two dimer sites, say j, k, l, and j ′′ , k ′′ , l ′′ , or any two sites within dimers, are interchanged [START_REF] Cohen-Tannoudji | Mécanique Quantique[END_REF].

Alternatively, eigen states for fermions should be antisymmetric. An interchange of lattice sites j, k, l, and j ′′ , k ′′ , l ′′ , should transform a plane wave into the anti-phase wave and superposition of these waves should annihilate the amplitude. Hence, phonons are forbidden, except those matching the lattice periodicity, for which any interchange is neutral. Such states represent in-phase dynamics in each unit cell, namely ς = "+", and in-phase dynamics for all unit cells, namely k • L H ≡ 0 modulo 2π, where L H is any vector of the sublattice of protons. This macroscopic quantum effect excluding elastic distortions has been named "super-rigidity" [START_REF] Fillaux | Macroscopic quantum entanglement and "super-rigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF]. In addition, eigen states must be antisymmetrized with respect to interchange of proton sites within dimers, what is equivalent to changing the signs of the coordinates for the center of symmetry, say {α 0r } -→ -{α 0r }. To this end, the wave function of a dimer, supposed to be non planar for the ease of notation, is rewritten as a linear combination of the wave functions for permuted sites as: Θ ′ 0HjklCτ = 2 -1/2 [Θ 0HjklC ({α 0r }) + τ Θ 0HjklC (-{α 0r })] (and likewise for j ′ ), with τ = "+" or "-". Then, antisymmetrization requires a spin-symmetry, either singlet-like (S for τ = +) or triplet-like (T for τ = -). Finally, with the translation invariant wave function of the unit cell, Ξ ′ 0Hjkl+Cτ = Θ ′ 0HjklCτ +Θ ′ 0Hj ′ klCτ , protons states can be written as:

|H + + C = 1 √ N Nc l=1 N b k=1 Na j=1 |Ξ ′ 0Hjkl+C+ |S ; |H + -C = 1 √ N Nc l=1 N b k=1 Na j=1 |Ξ ′ 0Hjkl+C-|T . (5)
For each configuration C, the ground states are entangled in position, momentum, and spin. The singlet-like and triplet-like states are degenerate because the spin symmetry is energy-free. The proposal by Sugimoto et al. [START_REF] Sugimoto | Effects of entanglement on elastic and inelastic scattering functions for neutron scattering from a pair of nuclei in solids[END_REF] that the spin symmetry could be due to spin-spin interaction is incorrect because the actual coupling between nearest spin sites (∼ 10 4 Hz ∼ 10 -6 K) is far too weak to give rise to long-lived correlations.

For an isolated crystal, symmetry-related quantum-correlations are decoherencefree. Thanks to the adiabatic separation, there is no phonon-induced decoherence and, therefore, no transition to the classical regime, before melting or decomposition. In practice, some states can be transitorily disentangled by external perturbations, such as thermal radiations, but this is counterbalanced by spontaneous re-entanglement via decay to the ground state. In a standard gaseous atmosphere, the density of the surroundings is ∼ 1000-fold smaller than that of the crystal, so the impact of environment-induced decoherence to measurements is invisible. A source of disentanglement is the thermal population of excited proton states (for example γ a OH and γ s OH at ≈ 1000 cm -1 ), as the as splitting destroys the spin-symmetry, but this is marginal.

Because the spin symmetry holds for both LL and RR ground states, for which there is no as splitting, the sublattice at 300 K can be represented by a mixture comprising a coherent superposition of super-rigid states for protons [START_REF] Thomas | Hydrogen-bond studies. LXXXIV. An X-ray diffraction study of the structures of KHCO 3 and KDCO 3 at 298[END_REF], on the one hand, and, on the other, a coherent superposition of phonon states for deuterons (4):

|H + (T ) = τ [q ′ LLH (T )|H + τ LL + q ′ RRH (T )|H + τ RR ]; |D + k (T ) = ς [q ′ LLD (T )|D + kς LL + q ′ RRD (T )|D + kς RR ]; (6)
with |q ′ LLH (T )| 2 = q LLH (T ), etc, in accordance with (1).

Neutron scattering

The partial differential nuclear cross-section can be written as [START_REF] Lovesey | Nuclear scattering, Theory of Neutron Scattered from Condensed Matter[END_REF]

d 2 σ dΩdE = k f k i i,f p i | k f | f | V (r)|i |k i | 2 δ( ω + E i -E f ). ( 7
)
The summation runs over initial states |i , E i , weight p i , and final states |f , E f . The energy transfer is ω and dΩ is an element of solid angle. The interaction potential between the incident neutron and the target sample, V (r), comprises the interaction potential at every site. For diffraction, |i ≡ |f , E i ≡ E f , ω ≡ 0. Neutrons are diffracted by each single state. Because of the adiabatic separation, neutrons diffracted by proton or deuteron states are free of any contribution from heavy atoms. Moreover, neutrons diffracted by proton states probe exclusively the interaction potential at the proton sites and likewise for deuteron states, irrespective of whether or not these sites are identical. There is no Laue monotonic diffuse scattering. In general, V (r) does not have the periodicity of the lattice, for the potential will depend upon the nuclear position and spin orientation. However, this is not the case for the super-rigid states which match exactly the periodicity of the crystal. Upon momentum transfer to the target proton states (6), the final state is, in general, such that k f •L H = 0 modulo 2π, so elastic distortions are generated. They are represented by ellipsoids at H/D sites in Fig. 1. The super-rigid states are probed without induced distortions when

Q • L H = 0 modulo 2π. (8)
Then, neutrons probe the spin-symmetry and the coherent cross-section is the total nuclear cross-section, σ H ≈ 82.0 b, that is ≈ 45 times greater than the coherent crosssection for regular Bragg peaks, σ Hc ≈ 1.8 b, or ≈ 15 times greater than σ Dc ≈ 5.59 b, or ≈ 3-fold greater than σ KCO 3 c ≈ 27.7 b for heavy atoms. These scattering events emerge from the diffraction pattern with spectacular contrast, even at large |Q|-values as they are not attenuated by the Debye-Waller factor. Besides, they are temperature independent.

For the P 2 1 /a structure, z is not parallel to an axis of the sublattice of protons (Fig. 1), so the matching condition ( 8) is never realized along Q z . The super-rigidity along z is necessarily destroyed by measurements and intensities are proportional to exp -U zz Q 2 z . On the other hand, since z 0 = z ′ 0 = 0 for planar dimers, the spin-symmetry holds only in the (x, y) planes, so the differential cross-section for enhanced diffraction can be written as [START_REF] Fillaux | A neutron diffraction study of macroscopically entangled proton states in the high temperature phase of the KHCO 3 crystal at 340 K[END_REF]

dσ c dΩ ∝ τ i τ f N ′ a j=1 N b k=1 Nc l=1 exp iQ z lD z {[exp iQ y (kD y -∆ y /2) + τ i τ f exp iQ y (kD y + ∆ y /2)] × [exp iQ x (jD x -∆ x /2) + τ i τ f exp iQ x (jD x + ∆ x /2)]} 2 2 × exp(-U zz Q 2 z ) × δ(ω), (9)
where {• • •} 2 takes into account indistinguishable scattering events, either in-phase (|S -→ |S and |T -→ |T ), or anti-phase (|S -→ |T and |T -→ |S ). Enhanced diffraction by (x, y) planes is anticipated at Q Hy = ±2πn y /∆ y , Q Hx = ±2πn x /∆ x for n y even (in-phase scattering) and Q Hx = ±2π(2n x + 1)/∆ x , for n y odd (anti-phase scattering). Then, enhanced Bragg peaks occur at

Q z = 2πn z /D z , with intensities proportional to exp -U zz Q 2 z . Otherwise, if Q z = 2πn z /D z
, the super-rigid (x, y) lattice is probed without distortion for the same (Q Hx , Q Hy )-values, but the spin-symmetry is destroyed. Incoherent scattering is observed and the intensity is proportional to

σ Hi exp -(Q 2 z U zz )δ(Q x -Q Hx + Q y -Q Hy ), (10)
hence the cigar-like shape. Compared to a nonrigid lattice, the enhancement is merely due to U xx = U yy = 0. The observed intensity ratio I 0 /I 10 ≈ 2 is at variance with the expected ratio of ≈ 1.5, but this discrepancy could be representative of errors in the background correction. Figures 234can be thus rationalized with three different kinds of scattering events: (i) regular Bragg diffraction by the crystal lattice; (ii) enhanced diffraction by the fully entangled sublattice of protons; (iii) incoherent scattering by the super-rigid planes of protons.

Discussion

The proposed theory is based upon fundamental laws of quantum mechanics applied to the crystal in question: the structure is periodic; dimers are centrosymmetric; indistinguishable protons are fermions; indistinguishable deuterons are bosons. It leads to macroscopically entangled states and, in the special case of protons, to super-rigidity and spin-symmetry with observable consequences. This theory is consistent with a large set of experimental data (neutron diffraction, QENS, INS, infrared and Raman) and, to the best of our knowledge, there is no conflict with any observation. There is, therefore, every reason to conclude that the crystal is a macroscopic quantum system for which only nonlocal observables are relevant.

Because quantum entanglement emerges naturally from the crystal structure, there is no reason to introduce any further interaction. From the viewpoint of economy (Ockham's razor), alternative theories based upon quantum exchange [START_REF] Keen | Quantum correlation between protons in potassium bicarbonate[END_REF] or spin-spin interaction [START_REF] Sugimoto | Effects of entanglement on elastic and inelastic scattering functions for neutron scattering from a pair of nuclei in solids[END_REF] are less satisfactory. Moreover, these correlations are very unlikely to be significant and it is doubtful that these short-range interactions, essentially confined to dimers, could account for macroscopic correlations, especially at elevated temperatures.

Protons are unique to demonstrating quantum entanglement, because they are fermions and because the very large incoherent cross-section can merge into the total coherent cross-section. No other nucleus can manifest such an increase of its coherent cross-section. The enhanced features can be, therefore, unambiguously assigned to protons, in accordance with their positions in reciprocal space. They are evidences of macroscopic quantum correlations which have no counterpart in classical physics.

According to the theory, the pair correlation function for protons represented by ellipsoids in Fig. 1 are due to measurement induced phonons. The structure free of induced distortions should comprise Dirac functions for protons and ellipsoids for deuterons. The spatial wave functions for protons [START_REF] Thomas | Hydrogen-bond studies. LXXXIV. An X-ray diffraction study of the structures of KHCO 3 and KDCO 3 at 298[END_REF] describe the distribution of the center of mass of the proton states with respect to that for all other atoms and the uncertainty principle holds at each site of the perfectly correlated sublattice.

It is widely believed that macroscopic objects are, in principle, governed by the laws of quantum mechanics and classical physics is an emergent phenomenon at an appropriate limit when the system interacts with an environment that might as well consist of its internal degrees of freedom [START_REF] Zurek | Decoherence, einselection and the quantum origin of the classical[END_REF]. The crystal of bicarbonate is a counterexample that is never amenable to classical physics, because (i) quantum entanglement is as robust as the crystal; (ii) the adiabatic separation insulates the proton states from every decoherence mechanism; (iii) the energy gap for excited proton states is much greater than kT .

A different approach places the emergence of classicality on the impossibility of distinguishing individual quantum levels with coarse-grained measurements [START_REF] Kofler | Classical world arising out of quantum physics under the restriction of coarse-grained measurements[END_REF]. Transposed to diffraction, the interaction potential in [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF] is replaced by the mean potential, V (r) = i p i i| V (r)|i , that is then Fourier transformed into the unit-cell structure factor [START_REF] Lovesey | Nuclear scattering, Theory of Neutron Scattered from Condensed Matter[END_REF]. Eigen states and nonlocal observables are thus translated into classical variables for particles located at definite sites and neutrons scattered by various nuclei can interfere, irrespective of whether or not these nuclei are involved in the same eigen states. Because of the opposite signs of the scattering lengths for protons and deuterons, interferences could build up in such a way that some Bragg peaks would be more intense for KHCO 3 than for KDCO 3 , or vice versa. However, these enhanced peaks are not expected to single out the reciprocal sublattice of protons and the structure factor does not account for the enhanced incoherent scattering along Q z in any way. In addition, the rigid-body analysis does not provide any evidence of correlation between protons and heavy atoms [START_REF] François Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF]. There is, therefore, experimental support to conclude that the assumption of classicality emerging from coarse-grained measurements is only a convenient approximation hiding the quantum nature of the crystal. Nevertheless, quantum effects embedded in the Bragg peaks transpire as the mean scattering length (2) is less satisfactory than the mixture of distinct sublattices (1), although the difference is not large enough to be firmly conclusive.

Conclusion

Neutron diffraction and Raman spectra show that the mixed isotope crystal is not a statistical mixture of KHCO 3 and KDCO 3 entities. The crystal is a macroscopic quantum system made of separable matter waves over the isomorphous sublattices of KHCO 3 and KDCO 3 . The quantum theory shows that deuteron (boson) states can be represented by phonons whereas antisymmetrized proton (fermion) states demonstrate super-rigidity and spin-symmetry. This macroscopic quantum entanglement is energy free and decoherence free. It is as robust as the crystal structure and the emergence of the classical regime at elevated temperatures is forbidden. The main outcome of the present work is that macroscopic quantum effects for protons are not destroyed by deuteron states. Despite their different masses, isotopes are not defects leading to localization, possibly because they are isoelectronic.

The quantum theory of diffraction by proton states allows us to distinguish different events. First, in the most general case, momentum transfer destroys the quantum entanglement and regular Bragg peaks are observed. Second, for those particular Q-values matching the nodes of the reciprocal sublattice, the super-rigidity and the spin-symmetry are probed without any measurement induced symmetry breaking. The incoherent cross-section merges into the total coherent cross-section and the corresponding peaks show markedly enhanced intensities featuring macroscopic quantum entanglement. Because of the anomalous Debye-Waller factor, these peaks are observed at large Q-values and they are temperature independent. Third, for those Qvalues matching only the nodes of the reciprocal planes of protons, the spin-symmetry is destroyed and cigar-like shaped rods of incoherent scattering by the super-rigid planes are observed in between the enhanced peaks. Clearly, the unit-cell structure factor is not appropriate to account for these enhanced features.

This work presents one single case of macroscopically entangled states on the scale of Avogadro's constant in a mixed isotope crystal at room temperature. The quantum theory suggests that such macroscopic quantum effects should be of significance for many hydrogen bonded crystals. Hopefully, unforeseeable consequences of the quantum view advocated in this work will appear in the future.

Figure 1 .

 1 Figure 1. (Color on line) Schematic view of the crystalline structure of KH 0.76 D 0.24 CO 3 at 300 K. Top: projection onto (a, b) planes. Bottom: projection onto (a, c) planes. The solid lines represent the unit cell. Proton and deuteron sites are indiscernible within thermal ellipsoids. The dashed lines emphasize the sublattice of protons/deuterons. The arrows point to the sites populated at low temperature.

1 )Figure 2 . 1 )Figure 3 .

 1213 Figure 2.Diffraction patterns of KH 0.76 D 0.24 CO 3 at 300 K in (a * , c * ) planes. The arrows point to the ridges of intensity for momentum transfer parallel to (z) and perpendicular to the trace of dimer planes (dashed lines), which contain the (x) direction, as defined in Fig.1. The (y) direction parallel to (b * ) is perpendicular to (a * , c * ) planes. k = Q y /b * .

1 )Figure 4 .

 14 Figure 4. Cuts of the scattering function along a * in the plane k = 0, at -(5.0 ± 0.2) Å-1 (top) and -(4.5 ± 0.2) Å-1 (bottom) off-a * , after background correction. The arrows point to the enhanced Bragg peak (top) and to the diffuse scattering (bottom).

3 Figure 5 .

 35 Figure 5. Raman spectra of powdered crystal of KHCO 3 (top), KH 0.5 D 0.5 CO 3 (middle), and KDCO 3 (bottom) at 300 K. The dashed lines are guides for the eyes. Spectra were recorded with a triple-monochromator DILOR XY equipped with an Ar + laser emitting at 4880 Å. The spectral resolution was ≈ 2 cm -1 . The samples were sealed in glass tubes.

  H H , H D , represent the sublattices of H + , D + ; H KCO 3H , H KCO 3D , those of heavy nuclei; C H,KCO 3H , C D,KCO 3D , couple light and heavy nuclei. The sublattices are bound through the electronic term H ep . There is no dynamical coupling between the KHCO 3 and KDCO 3 subsystems which are supposed to be separable: any state vector of H p is a tensor product of state vectors of the subsystems. Furthermore, the adiabatic separation of H H and H KCO 3H , or H D and H KCO 3D , leads to factorable state vectors for the subsystem of bare protons (fermions), or deuterons (bosons)