Jeanne Heintz

Francesca Legittimo

Sébastien Bidault
email: sebastien.bidault@espci.fr

†institut Langevin

Dimers of Plasmonic Nanocubes to Reach Single-Molecule

Keywords: strong coupling, plasmonics, nanocubes, scattering spectroscopy, silver Graphical TOC Entry

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Entangling light and matter by bringing quantum emitters in a strong coupling regime with a resonant cavity 1 is essential for the optical processing of quantum information, in particular by allowing few-photon nonlinearities. [START_REF] Birnbaum | Photon Blockade in an Optical Cavity with One Trapped Atom[END_REF][START_REF] Loo | Optical Nonlinearity for Few-Photon Pulses on a Quantum Dot-Pillar Cavity Device[END_REF][START_REF] Volz | Ultrafast All-Optical Switching by Single Photons[END_REF][START_REF] Wang | Turning a molecule into a coherent two-level quantum system[END_REF] While single-emitter strong coupling has been efficiently achieved with dielectric microcavities at liquid helium temperatures, [START_REF] Reithmaier | Strong coupling in a single quantum dot-semiconductor microcavity system[END_REF][START_REF] Yoshie | Vacuum Rabi Splitting with a Single Quantum Dot in a Photonic Crystal Nanocavity[END_REF][START_REF] Peter | Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity[END_REF] observing this phenomenon outside of a cryostat remains an open challenge. [START_REF] Palstra | Hybrid cavity-antenna systems for quantum optics outside the cryostat?[END_REF] Femtosecondscale electron-phonon coupling occurring in quantum emitters at room temperature imply that they can only efficiently interact with broadband resonators, which, as a result, must confine optical fields at scales much smaller than the wavelength of light in order to maintain a large Purcell effect and thus efficient emitter-resonator coupling strengths. [START_REF] Palstra | Hybrid cavity-antenna systems for quantum optics outside the cryostat?[END_REF][START_REF] Busson | Accelerated Single Photon Emission from Dye Molecule-Driven Nanoantennas Assembled on DNA[END_REF] This is one of the promises of plasmonic antennas, which feature a combination of low quality factors and deep sub-wavelength mode volumes. [START_REF] Koenderink | On the use of Purcell factors for plasmon antennas[END_REF][START_REF] Sauvan | Theory of the Spontaneous Optical Emission of Nanosize Photonic and Plasmon Resonators[END_REF] Recently, few-emitter strong coupling, down to a single emitter, was observed in a number of plasmonic resonators with nanoscale gaps based either on self-assembled nanostructures (nanoparticle on mirror [START_REF] Chikkaraddy | Single-Molecule Strong Coupling at Room Temperature in Plasmonic Nanocavities[END_REF][START_REF] Chikkaraddy | Mapping Nanoscale Hotspots with Single-Molecule Emitters Assembled into Plasmonic Nanocavities Using DNA Origami[END_REF][START_REF] Pelton | Strong Coupling of Emitters to Single Plasmonic Nanoparticles: Exciton-Induced Transparency and Rabi Splitting[END_REF] -NPoMor dimer geometries) [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] or top-down fabricated antennas (on substrates [START_REF] Santhosh | Vacuum Rabi Splitting in a Plasmonic Cavity at the Single Quantum Emitter Limit[END_REF] or on a scanning probe tip). [START_REF] Groß | Near-Field Strong Coupling of Single Quantum Dots[END_REF] However, these phenomena were not observed in a reproducible or scalable way and the coupled systems generally feature low emission quantum yields in order to maximize the quality factor of the resonator. In particular, while scanning probe microscopy allows an active control of the emitter-resonator coupling strength, this approach is incompatible with on-chip integration. [START_REF] Groß | Near-Field Strong Coupling of Single Quantum Dots[END_REF] For self-assembled and lithographically fabricated nanostructures, numerous issues remain: the number of emitters is not controlled a priori [START_REF] Chikkaraddy | Single-Molecule Strong Coupling at Room Temperature in Plasmonic Nanocavities[END_REF][START_REF] Santhosh | Vacuum Rabi Splitting in a Plasmonic Cavity at the Single Quantum Emitter Limit[END_REF][START_REF] Leng | Strong Coupling and Induced Transparency at Room Temperature with Single Quantum Dots and Gap Plasmons[END_REF] or the coupling strength is below the visibility criterion for strong coupling [START_REF] Chikkaraddy | Mapping Nanoscale Hotspots with Single-Molecule Emitters Assembled into Plasmonic Nanocavities Using DNA Origami[END_REF] or strong coupling is only observed in a few percent of the sample. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Leng | Strong Coupling and Induced Transparency at Room Temperature with Single Quantum Dots and Gap Plasmons[END_REF] The influence of local facets and tips in polycrystalline gold nanoparticles (AuNPs) was recently highlighted as a possible reason for the low probability in observing strong coupling in NPoM resonators and AuNP dimers. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Leng | Strong Coupling and Induced Transparency at Room Temperature with Single Quantum Dots and Gap Plasmons[END_REF] Indeed, the lightning-rod effect [START_REF] Novotny | Theory of Nanometric Optical Tweezers[END_REF] favors larger field confinements and enhancements specifically if the emitter is positioned in close proximity to a tip, while the inverse effect is observed near a crystalline facet. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] However, it is not possible to specifically position emitters on the tips of polycrystalline nanoparticles. Alternatively, DNA-based self-assembly, in particular with the use of DNA origamis, has the ability to organize anisotropic particles with an excellent control over their relative orientation [START_REF] Kuzyk | Reconfigurable 3D Plasmonic Metamolecules[END_REF][START_REF] Zhan | Reconfigurable Three-Dimensional Gold Nanorod Plasmonic Nanostructures Organized on DNA Origami Tripod[END_REF][START_REF] Lan | Programmable Supra-Assembly of a DNA Surface Adapter for Tunable Chiral Directional Self-Assembly of Gold Nanorods[END_REF] or to functionalize their tips or facets specifically. [START_REF] Niu | DNA Origami-Based Nanoprinting for the Assembly of Plasmonic Nanostructures with Single-Molecule Surface-Enhanced Raman Scattering[END_REF] It is thus interesting to investigate whether designing plasmonic resonators in which emitters are positioned between the tips of anisotropic particles would facilitate the reproducible observation of single-molecule strong coupling at room temperature.

Anisotropic plasmonic particles, such as nanorods, 25 nanobipyramids 26 or nanocubes, [START_REF] Niu | DNA Origami-Based Nanoprinting for the Assembly of Plasmonic Nanostructures with Single-Molecule Surface-Enhanced Raman Scattering[END_REF][START_REF] Ge | Hybrid Plasmonic Nano-Emitters with Controlled Single Quantum Emitter Positioning on the Local Excitation Field[END_REF] offer high local curvatures that are associated with larger field confinements and enhancements than nanospheres. Dimers of nanorods or nanobipyramids are however ill-suited to couple efficiently with fluorescent molecules that emit at visible frequencies since their resonances are redshifted to the near infrared. [START_REF] Rogobete | Design of Plasmonic Nanoantennae for Enhancing Spontaneous Emission[END_REF] This is not the case for plasmonic nanocubes, for which the plasmon resonance wavelengths are only slightly redshifted with respect to nanospheres. [START_REF] Park | Precisely Shaped, Uniformly Formed Gold Nanocubes with Ultrahigh Reproducibility in Single-Particle Scattering and Surface-Enhanced Raman Scattering[END_REF] Furthermore, monodisperse and monocrystalline gold nanocubes (AuNCs) with sharp tips were demonstrated in the literature [START_REF] Park | Precisely Shaped, Uniformly Formed Gold Nanocubes with Ultrahigh Reproducibility in Single-Particle Scattering and Surface-Enhanced Raman Scattering[END_REF] as well as silver nanocubes [START_REF] Zhou | Facile Synthesis of Silver Nanocubes with Sharp Corners and Edges in an Aqueous Solution[END_REF] (AgNCs), allowing an extra degree of freedom to optimize visible strong coupling and emission yields in the considered plasmonic resonators by exploiting the lower non-radiative decay channels in silver nanostructures. [START_REF] Baida | Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO2 Nanoparticles[END_REF] This is why we investigate here the ability of dimers of silver or gold nanocubes to reach a strong coupling regime with a single fluorescent molecule, while maintaining high emission yields. Electrodynamic simulations demonstrate that local field enhancements and Purcell factors are larger in dimers of AuNCs than in dimers of gold nanospheres (AuNSs) or of polycrystalline AuNPs, relaxing the conditions for which the visibility criterion for strong coupling can be reached when coupled to a single emitter. These calculations indicate a trade-off between emission yield and visible strong coupling in AuNC dimers, which can be tailored by tuning the size of the nanocubes. Since there is an ongoing debate in the literature on the choice of the bulk dielectric function of silver to analyze nanoparticles, [START_REF] Baida | Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO2 Nanoparticles[END_REF] we compare single AgNC scattering spectroscopy measurements to simulations performed with the most commonly used Ag dielectric functions in order to validate our model. Comparing theoretical strong coupling conditions in dimers of gold and silver nanocubes, we observe that AgNCs combine an increase in quality factor with a significant gain in emission yield. Our results therefore provide realistic plasmonic architectures in which single-molecule strong coupling can be achieved in a robust and reproducible way, while allowing photon emission yields well above 50%.

The coupling strength g between a single quantum emitter and a plasmonic resonator depends on the Purcell factor of the resonator F p for a given position and orientation of the emitter, as well as the damping rate of the resonator γ p and the radiative decay rate of the emitter without the resonator Γ R0 : 9,16,32 g = F p γ p Γ R0 /2. F p corresponds to the change in the local density of optical states (LDOS) for the emitter in the resonator, relative to a homogeneous medium, [START_REF] Bouchet | Quantum Dipole Emitters in Structured Environments: a Scattering Approach: Tutorial[END_REF] and is computed by substituting the quantum emitter for a classical dipole and inferring its total dissipated power with the resonator P tot and its total radiated power without it P R0 : F p = P tot /P R0 . [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Bharadwaj | Spectral Dependence of Single Molecule Fluorescence Enhancement[END_REF][START_REF] Bidault | Picosecond Lifetimes with High Quantum Yields from Single-Photon-Emitting Colloidal Nanostructures at Room Temperature[END_REF] By considering that the electronic resonance of the emitter ω 0 matches the plasmonic resonance ω p = ω 0 , it is possible to define a visible strong coupling figure of merit corresponding to a coupling rate between emitter and resonator faster than the sum of all loss rates: 4g/(γ p + γ 0 ) > 1, with γ 0 the dephasing rate of the emitter. 9, [START_REF] Chikkaraddy | Mapping Nanoscale Hotspots with Single-Molecule Emitters Assembled into Plasmonic Nanocavities Using DNA Origami[END_REF][START_REF] Pelton | Strong Coupling of Emitters to Single Plasmonic Nanoparticles: Exciton-Induced Transparency and Rabi Splitting[END_REF][START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Törmä | Strong Coupling between Surface Plasmon Polaritons and Emitters: A Review[END_REF][START_REF] Zengin | Realizing Strong Light-Matter Interactions between Single-Nanoparticle Plasmons and Molecular Excitons at Ambient Conditions[END_REF] Classical electrodynamic simulations also allow an estimation of the radiative yield Φ of the coupled emitter-resonator by calculating the power radiated by the dipole P R in the resonator: Φ = P R /P tot . In the weak coupling regime, for values of F p of several orders of magnitude, Φ is generally similar to the final emission quantum yield of the coupled system, unless the initial quantum yield of the emitter is close to 0. [START_REF] Bidault | Picosecond Lifetimes with High Quantum Yields from Single-Photon-Emitting Colloidal Nanostructures at Room Temperature[END_REF][START_REF] Busson | Photonic Engineering of Hybrid Metal-Organic Chromophores[END_REF] Furthermore, by exciting the plasmonic resonator with a plane wave polarized along the dimer axis with a wavelength matching the longitudinal resonance, we can estimate the maximum enhancement of the electric field in the gap of the antenna.

In order to highlight the influence of high local curvatures on the emitter-resonator coupling strength, let us first consider the gain in local field enhancement, Purcell factor and strong coupling figure of merit when going from a perfect gold nanosphere to a gold nanocube at fixed nanoparticle volume. Recent experimental realizations have shown that a sharp AuNC typically features a radius of curvature at the tip corresponding to ∼10% of the nanocube edge length, compared to ∼25% for a rounded cube. [START_REF] Park | Precisely Shaped, Uniformly Formed Gold Nanocubes with Ultrahigh Reproducibility in Single-Particle Scattering and Surface-Enhanced Raman Scattering[END_REF] We thus compare in Figure 1-a the local field enhancement observed in a 2 nm gap between two 40 nm diameter AuNSs and two rounded or sharp 30 nm AuNCs (to provide similar particle volumes and with radii of curvatures of 7.5 nm and 3 nm, respectively), using Boundary Element Method (BEM) [START_REF] García De Abajo | Retarded field calculation of electron energy loss in inhomogeneous dielectrics[END_REF][START_REF] Hohenester | MNPBEM -A Matlab Toolbox for the Simulation of Plasmonic Nanoparticles[END_REF] simulations performed with tabulated data for the dielectric constant of gold [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF] and an optical index of 1.4 for the environment as observed in recent strong coupling measurements in AuNP dimers. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] These nanoparticle sizes and gap lengths were chosen as they were recently shown, theoretically and experimentally, to provide the onset of few-molecule strong coupling for spherical AuNPs. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] BEM simulations allow the study of anisotropic plasmonic particles and are compatible with extreme nanoscale gaps, down to 1 nm. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] Simulations are not performed with interparticle spacings below 1 nm to avoid the influence of quantum non-local effects. [START_REF] Savage | Revealing the Quantum Regime in Tunnelling Plasmonics[END_REF][START_REF] Cha | Probing Quantum Plasmon Coupling Using Gold Nanoparticle Dimers with Tunable Interparticle Distances Down to the Subnanometer Range[END_REF] However, these simulations consider a point-like source dipole and neglect the dependence of the dielectric function of the metal with respect to the wavevectors of the radiated electromagnetic field. Furthermore, going beyond the point-like emitter approximation would require a quantum-mechanical treatment of the electronic wavefunctions of the molecule. [START_REF] Vukovic | Fluorescence Enhancement of Chromophores Close to Metal Nanoparticles. Optimal Setup Revealed by the Polarizable Continuum Model[END_REF][START_REF] Sakko | Dynamical coupling of plasmons and molecular excitations by hybrid quantum/classical calculations: time-domain approach[END_REF][START_REF] Neuman | Coupling of Molecular Emitters and Plasmonic Cavities beyond the Point-Dipole Approximation[END_REF][START_REF] Doppagne | Single-molecule tautomerization tracking through space-and time-resolved fluorescence spectroscopy[END_REF][START_REF] Fregoni | Strong Coupling between Localized Surface Plasmons and Molecules by Coupled Cluster Theory[END_REF] In Figure 1-a, we observe, in all cases, local field enhancements in excess of 10 5 in the center of the dimers, with a gain of nearly an order of magnitude for the sharp AuNCs compared to AuNSs. These strong local field enhancements are related to the radiative part of the Purcell factor by reciprocity [START_REF] Bharadwaj | Spectral Dependence of Single Molecule Fluorescence Enhancement[END_REF] and, as shown in Figure 1-b, dimers of 40 nm AuNSs and 30 nm AuNCs feature values of F p that exceed 10 6 for a 1 nm gap, when considering a dipolar emitter in the center of the dimer, oriented along its axis and with ω 0 matching the longi- tudinal resonance of the dimer. It is important to note that while it is possible to assemble plasmonic resonators using chemically etched monocrystalline AuNSs, [START_REF] Huh | Comparative Study of Plasmonic Resonances between the Roundest and Randomly Faceted Au Nanoparticles-on-Mirror Cavities[END_REF][START_REF] Yoon | Ideal Dimers of Gold Nanospheres for Precision Plasmonics: Synthesis and Characterization at the Single-Particle Level for Identification of Higher Order Modes[END_REF][START_REF] Nicoli | DNA-Mediated Self-Assembly of Plasmonic Antennas with a Single Quantum Dot in the Hot Spot[END_REF] recent realizations of few-molecule strong coupling were instead performed with polycrystalline faceted AuNPs, for which assembly along facets is the most probable in either NPoM or dimer geometries. [START_REF] Chikkaraddy | Single-Molecule Strong Coupling at Room Temperature in Plasmonic Nanocavities[END_REF][START_REF] Chikkaraddy | Mapping Nanoscale Hotspots with Single-Molecule Emitters Assembled into Plasmonic Nanocavities Using DNA Origami[END_REF][START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] Since polycrystalline faceted particles are well described theoretically by rounded icosahedra, [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Yoon | Surface Plasmon Coupling in Dimers of Gold Nanoparticles: Experiment and Theory for Ideal (Spherical) and Nonideal (Faceted) Building Blocks[END_REF] we also plot in Figure 1-b the values of F p for dimers of rounded icosahedra of the same volume and oriented along parallel facets. Figure 1-b also provides the computed values of Φ for all dimer geometries, which are similar and remain below 20%. This is logical as radiative decay channels are strongly dependent on the size of the AuNPs and particles with a volume equivalent to a 40 nm diameter sphere provide low emission yields. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Bidault | Picosecond Lifetimes with High Quantum Yields from Single-Photon-Emitting Colloidal Nanostructures at Room Temperature[END_REF] However, these low radiative decay rates are related to a higher resonance quality factor and, therefore, to a higher strong coupling figure of merit. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] The computed values of the damping rate of the plasmonic resonator γ p and of the coupling strength g, as a function of the interparticle distance, and which are needed to estimate the strong coupling figure of merit, are provided in Figure S1 of the supplementary information. We observe that using nanocubes instead of spheres or polycrystalline particles relaxes the conditions in which strong coupling should be visible in AuNP dimers. In particular, comparing simulations for the polycrystalline AuNPs aligned by facets, recently used in strong coupling experiments, [START_REF] Chikkaraddy | Single-Molecule Strong Coupling at Room Temperature in Plasmonic Nanocavities[END_REF][START_REF] Chikkaraddy | Mapping Nanoscale Hotspots with Single-Molecule Emitters Assembled into Plasmonic Nanocavities Using DNA Origami[END_REF][START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] and the proposed sharp AuNCs assembled along their tips, we observe an increase by nearly a factor of 3 of the strong coupling figure of merit and an increase by a factor of 2 of the minimum interparticle spacing at which single-molecule strong coupling could be observed. It is important to stress that an increase of 3 for the figure of merit corresponds to an increase of g similar to what would be observed when coupling 9 emitters identically with a given plasmonic resonator. [START_REF] Chikkaraddy | Single-Molecule Strong Coupling at Room Temperature in Plasmonic Nanocavities[END_REF][START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] Such an increase is therefore essential to go from few-molecule to single-molecule realizations of strong coupling.

While dimers of AuNCs would provide a significant gain in Purcell factor and coupling strength compared to similarly sized spherical AuNPs, Figure 1-b indicates a negligible gain in emission yield. To investigate whether increasing the size of the nanocubes could provide larger values of Φ without sacrificing the coupling strength, we compare in Figure 1-d the evolution of the figure of merit and of the emission yield as a function of the particle spacing for dimers of sharp 30 nm and 40 nm AuNCs (with radii of curvature of 3 nm and 4 nm, respectively). These simulations indicate that the faster resonator decay rate γ p , associated with larger particles, increases Φ by a factor of 2 and lowers the strong coupling figure of merit by a modest 20%. This means that, by tuning the nanoparticle size, it is possible to favor either the coupling strength or the emission yield and that dimers of AuNCs could be strongly coupled to a single molecule while maintaining a final emission yield close to 50%.

To further increase Φ while maintaining visible strong coupling, one solution is to decrease the non-radiative decay rate in the resonator, for instance by substituting gold for silver. This is another reason why nanocubes are attractive as high quality AgNCs were experimentally demonstrated [START_REF] Zhou | Facile Synthesis of Silver Nanocubes with Sharp Corners and Edges in an Aqueous Solution[END_REF] and can be exploited in self-assembled nanostructures using DNA nanotechnology. [START_REF] Lermusiaux | Flexible Synthesis of High-Purity Plasmonic Assemblies[END_REF] However, there was a recent debate on the choice of the dielectric function of silver that provides the best agreement with single particle spectroscopy. [START_REF] Baida | Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO2 Nanoparticles[END_REF] Indeed, when studying the dipolar mode of single spheres, the two most widely used dielectric functions reported by Johnson and Christy 40 or by Palik [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF] both provide good agreements with simulations performed using slightly different nanoparticle sizes. [START_REF] Baida | Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO2 Nanoparticles[END_REF] However, the ohmic losses in the Palik data set are significantly larger in the visible range than for Johnson and Christy, providing very different strong coupling figures of merit. On the other hand, more recent measurements performed on silver films indicate similar values of the real part of the dielectric function compared to Johnson and Christy but a slightly larger imaginary part. [START_REF] Yang | Optical dielectric function of silver[END_REF] To investigate which dielectric function provides the best agreement with the optical properties of silver nanocubes, we study 78 nm AgNCs, with an estimated radius of curvature of 9 nm (estimated from electron microscopy images, see Figure 2-a), using both ensemble extinction measurements in ethanol (Figure 2-b) and single particle scattering spectroscopy on particles spin-coated on a glass coverslip and covered in index-matching oil to minimize the oxidation of silver (typical examples in Figures 2-c-d). The scattering spectra of single AgNCs are simulated in BEM using the three reported dielectric constants of silver and a Γ 0 is negligible compared to γ p (unless the initial quantum yield is nearly 0) and that the two strongly-coupled eigenmodes equally share the damping rates of the emitter and of the resonator behaving as damped harmonic oscillators (as expected in the Jaynes-Cummings model), [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF][START_REF] Bouchet | Quantum Dipole Emitters in Structured Environments: a Scattering Approach: Tutorial[END_REF] this means that the final decay times of the strongly-coupled eigenmodes should be of the order of 10 fs and thus slightly lower than the considered dephasing time h/γ 0 =12 fs of the isolated molecule (with h the reduced Planck constant).

Furthermore, if the resonator dominates both the radiative and non-radiative damping rates of the coupled system, then both eigenmodes should feature emission yields equal to Φ. These results thus indicate how dimers of 30 nm silver nanocubes should allow single-molecule strong coupling at room temperature with an emission yield well above 50%. A potential limiting factor for AgNC dimers that is not considered in these simulations is the possible appearance of a thin silver oxide layer on the surface of the nanocubes, which would effectively increase the interparticle distance. However, the sharp quadrupolar modes observed experimentally on single AgNCs in Figure 2 indicate that this effect should be limited if the particles are kept from direct contact with air. For long-term stability of these hybrid nanostructures, AuNC dimers, with only slightly lower coupling strengths and emission yields, are an appealing alternative.

We can also observe in Figure 3-a that the distance dependence of Φ is different for the considered AuNC and AgNC dimers. This is due to the wavelength dependence of the imaginary part of the dielectric constants of gold and silver and to the dependence of the longitudinal resonance of dimers with respect to interparticle spacings. In these simulations, we consider that the resonance of the emitter matches the longitudinal resonance wavelength of the plasmonic dimer ω p . Since the minimum imaginary part of the dielectric function of gold is around 680 nm, [START_REF] Olmon | Optical Dielectric Function of Gold[END_REF] the maximum emission yield will typically be observed when ω p reaches this wavelength, which occurs for an interparticle spacing around 2 nm for 40 nm AuNCs, compared to ∼1 nm for 30 nm gold cubes. In the case of silver, the imaginary part of the dielectric function increases with the wavelength (as can be expected for a Drude metal), [START_REF] Yang | Optical dielectric function of silver[END_REF] explaining the decrease of Φ when the particles get closer and the longitudinal resonance redshifts.

All the simulations presented in Figures 1 and3-a-b consider a perfectly centered emitting dipole with an orientation along the dimer axis. While recent self-assembly strategies, in particular using DNA, have shown an excellent position [START_REF] Busson | Accelerated Single Photon Emission from Dye Molecule-Driven Nanoantennas Assembled on DNA[END_REF][START_REF] Puchkova | DNA Origami Nanoantennas with over 5000-fold Fluorescence Enhancement and Single-Molecule Detection at 25 µM[END_REF] and orientation [START_REF] Gopinath | Absolute and Arbitrary Orientation of Single-Molecule Shapes[END_REF][START_REF] Adamczyk | DNA Self-Assembly of Single Molecules with Deterministic Position and Orientation[END_REF] control over fluorescent molecules, it is interesting to consider how robust the appearance of a strong coupling regime will be for an imperfectly centered or oriented transition dipole. Figures 3-c-d show how the strong coupling figure of merit varies as a function of the off-axis orientation and position of the source dipole, with interparticle spacings of 1 nm and 2 nm. We observe that strong coupling should remain visible for emitters with an off-axis orientation up to 45 • with a 2 nm spacing and up to 75 • with 1 nm. As observed in the local field distribution of Figure 1-a, the enhancement process is maximum at the very center of the dimer as confirmed by Figure 1-d since strong coupling will only be visible if the emitter is within 2 nm or 3 nm of the dimer axis, if the interparticle spacing is 2 nm or 1 nm, respectively.

While these calculations show how precise the self-assembly strategy will need to be in order to position a single emitter between the tips of two plasmonic nanocubes, it is important to stress that the tips of gold nanocubes were recently functionalized specifically with DNA strands. [START_REF] Niu | DNA Origami-Based Nanoprinting for the Assembly of Plasmonic Nanostructures with Single-Molecule Surface-Enhanced Raman Scattering[END_REF] Alternatively, one could consider a NPoM resonator geometry by assembling a plasmonic nanocube on a metallic mirror by one of its tips in order to avoid issues regarding the relative orientation of the two cubes in self-assembled dimers, while retaining a high local field enhancement and confinement. In both cases, an additional electric field enhancement could arise from the spontaneous appearance of atomic protrusions in the nanoscale gap. [START_REF] Carnegie | Room-Temperature Optical Picocavities below 1 nm3 Accessing Single-Atom Geometries[END_REF] In order to reach single-molecule strong coupling at room temperature with high emission yields, we therefore propose to exploit the high local curvatures offered by dimers of plasmonic nanocubes aligned along their tips, which feature higher coupling strengths than dimers of perfect or faceted gold nanospheres while maintaining resonance wavelengths in the visible range. Electrodynamic simulations indicate that the stringent experimental conditions at which visible strong coupling can be achieved at room temperature are significantly relaxed by the use of both AuNCs and AgNCs. It is possible to favor the coupling strength or the emission yield by tuning the nanocube size but the use of 40 nm AuNCs and 30 nm AgNCs should provide both single-molecule strong coupling and emission yields of nearly 50% for gold and above 60% for silver. Experimental realizations with AgNCs could even exceed these expectations as the ohmic losses, experimentally observed by scattering spectroscopy on single cubes, appear lower than those theoretically expected using typical dielectric constants for silver.

Interestingly, both gold [START_REF] Niu | DNA Origami-Based Nanoprinting for the Assembly of Plasmonic Nanostructures with Single-Molecule Surface-Enhanced Raman Scattering[END_REF] and silver 52 nanocubes are compatible with DNA-based selfassembly, which has been shown as an efficient bottom-up strategy to reach strong coupling at room temperature [START_REF] Chikkaraddy | Mapping Nanoscale Hotspots with Single-Molecule Emitters Assembled into Plasmonic Nanocavities Using DNA Origami[END_REF][START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] and which also allows a control over the orientation of dye molecules. [START_REF] Gopinath | Absolute and Arbitrary Orientation of Single-Molecule Shapes[END_REF][START_REF] Adamczyk | DNA Self-Assembly of Single Molecules with Deterministic Position and Orientation[END_REF] DNA nanotechnology thus provides all the required ingredients to produce bright strongly coupled hybrid nanostructures featuring few-photon nonlinearities at room temperature.

Methods

Simulations

Boundary Element Method simulations are performed using the MNPBEM toolbox. [START_REF] Hohenester | MNPBEM -A Matlab Toolbox for the Simulation of Plasmonic Nanoparticles[END_REF] The 2D meshing of the spheres and nanocubes are generated in Matlab while, for rounded icosahedra, the 3D meshing is first generated using COMSOL. The trapezoidal mesh for spheres is generated by longitudes and latitudes with a minimum size of 0.5 nm at the interparticle gap. The triangular mesh of nanocubes is generated as an extruded rounded polygon using dedicated functions of the MNPBEM toolbox as detailed elsewhere [START_REF] Hohenester | MNPBEM -A Matlab Toolbox for the Simulation of Plasmonic Nanoparticles[END_REF] and with a minimum mesh size of 0.2 nm at the tips. The total dissipated and radiated powers of a classical electric dipole in the resonator are compared to the power radiated without the resonator in order to compute F p and Φ for a given position, orientation and emission wavelength.

By tuning the emission wavelength for a longitudinally coupled emitter and monitoring the frequency at which the maximum total decay rate is achieved, as well as the full-width at half maximum, it is possible to infer the resonance wavelength ω p and decay rate γ p of the longitudinal mode of the dimer. The local field enhancements are generated by exciting the plasmonic resonators with a plane wave polarized along the dimer axis at the longitudinal resonance wavelength. Scattering cross-sections of single silver nanocubes are computed by exciting the AgNC with a wavelength-tunable plane wave propagating perpendicularly to one face of the cube and polarized along one edge and then integrating the radiated power on a sphere around the particle.

Convergence of these simulations with respect to the mesh size was verified in the case of spheres by comparing them to Mie theory calculations and observing differences below 5% at a 1 nm interparticle gap distance and below 1% for a 2 nm gap. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] In a general way, the data of Figure 3

Optical Measurements

The extinction spectrum of commercial silver nanocubes (Nanocomposix, USA, 78 nm edge lengths) is measured with a typical concentration of 1 mg/mL in ethanol (Duetta, Horiba).

Single AgNCs are studied in scattering spectroscopy by spin-coating 100 µL of the nanocube solution on a freshly cleaned glass coverslip and adding a few drops of index matching oil.

Darkfield spectra are measured in an inverted microscope (IX71, Olympus) coupled to a fiber-coupled (50 µm core diameter) imaging spectrometer (Acton SP300 with Pixis 100 CCD detector, Princeton Instruments). White light from a 100 W halogen lamp is focused on the sample using an oil-immersion 1.2-1.4 NA dark-field condenser. Scattered light is collected with a 100 × 0.6 NA immersion oil objective. A Single AgNC is aligned with the optical fiber using a piezoelectric stage (P-562.3CD, PI) before its raw scattering spectrum is measured with a 2 s acquisition time. A background spectrum is measured with the same acquisition time on an empty area of the sample. The background is subtracted to the measured spectra before the obtained data are divided by a smoothed background spectrum (after substracting a constant offset of the CCD detector) in order to account for the wavelength-dependent illumination and detection. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] The extinction spectrum of commercial silver nanocubes (Nanocomposix, USA, 78 nm edge lengths) is measured with a typical concentration of 1 mg/mL in ethanol (Duetta, Horiba). Single AgNCs are studied in scattering spectroscopy by spin-coating 100 µL of the nanocube solution on a freshly cleaned glass coverslip and adding a few drops of index matching oil. Darkfield spectra are measured in an inverted microscope (IX71, Olympus) coupled to a fiber-coupled (50 µm core diameter)

imaging spectrometer (Acton SP300 with Pixis 100 CCD detector, Princeton Instruments).

White light from a 100 W halogen lamp is focused on the sample using an oil-immersion 1.2-1.4 NA dark-field condenser. Scattered light is collected with a 100 × 0.6 NA immersion oil objective. A Single AgNC is aligned with the optical fiber using a piezoelectric stage (P-562.3CD, PI) before its raw scattering spectrum is measured with a 2 s acquisition time.

A background spectrum is measured with the same acquisition time on an empty area of the sample. The background is subtracted to the measured spectra before the obtained data are divided by a smoothed background spectrum (after substracting a constant offset of the CCD detector) in order to account for the wavelength-dependent illumination and detection. [START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF]

Figure 1 :

 1 Figure 1: Increased emitter-resonator coupling in gold nanocubes. (a) Simulated local field enhancements, |E/E 0 | 2 , in the center of a dimer of 40 nm AuNSs, rounded 30 nm AuNCs or sharp 30 nm AuNCs, with a 2 nm interparticle spacing (the longitudinal resonance wavelengths and the maximum enhancements are provided on the top and bottom of the figures, respectively). (b) Computed Purcell factor F p (solid lines), and radiative yield Φ (dashed lines), in dimers of 40 nm AuNSs (blue data), 40 nm rounded icosahedra (light blue), 30 nm rounded AuNCs (orange) and 30 nm sharp NCs (yellow) for a longitudinally coupled emitter positioned in the center and as a function of the interparticle spacing s. (c) Visible strong coupling figure of merit 4g/(γ p + γ 0 ) (solid line, see text for details) as a function of s in the same geometry as (b). (d) Visible strong coupling figure of merit (solid lines) and Φ (dashed lines) in dimers of 30 nm (yellow) and 40 nm (red) sharp AuNCs for a longitudinally coupled emitter positioned in the center and as a function of the interparticle spacing s.

16

 16

Figure 1 -

 1 Figure 1-c shows the evolution of the strong coupling figure of merit in dimers of 40 nm AuNSs, 40 nm AuNPs and 30 nm AuNCs as a function of the interparticle spacing when considering typical values of the radiative decay rate and dephasing time for the dipolar emitter (Γ R0 = 0.2 µeV and γ 0 = 50 meV).[START_REF] Heintz | Few-Molecule Strong Coupling with Dimers of Plasmonic Nanoparticles Assembled on DNA[END_REF] The computed values of the damping rate of

Figure 2 :

 2 Figure 2: Comparison between experiments and simulations on single silver nanocubes. (a) Typical electron microscopy image of 78 nm AgNCs with an estimated radius of curvature of 9 nm. (b) Ensemble extinction spectrum of 78 nm AgNCs in ethanol. (c-d) Typical single AgNC scattering spectra. Simulated scattering spectra of 78 nm AgNCs using the dielectric constant of Palik 53 (e), Johnson and Christy 40 (f) and Yang et al. (g).[START_REF] Yang | Optical dielectric function of silver[END_REF] Experimental and simulated scattering sepctra in (c-g) are fitted by two energy-dependent Lorentzian functions (green and red solid lines). (h) Distribution of quadrupolar quality factors measured with 14 single AgNCs, compared to the theoretical values inferred from the simulations in (e-g).

Figure 3 -

 3 Figure 3-a shows the computed Purcell factors and emission yields in the center of 30 nm AgNC dimers (with a 3 nm radius of curvature) compared to the values obtained with 30 nm and 40 nm AuNCs. We observe that the values of F p for silver and gold nanocubes are nearly identical, indicating that the local curvature is probably the main parameter in optimizing the confinement and enhancement of the local electric field. However, since Ag-NCs feature lower ohmic losses, the emission yield is nearly 2 times larger for silver and the strong coupling figure of merit in Figure 3-b is 10% larger than for gold (values of γ p and g for these dimer geometries are provided in Figure S1). Interestingly, the simulated values of γ p in Figure S1 indicate that the damping rate of the resonator should reach values of the order of 100-150 meV for interparticle gaps compatible with strong coupling for both

Figure 3 :

 3 Figure 3: Strong coupling with high emission yields in dimers of silver nanocubes. (a) Computed Purcell factor F p (solid lines), and radiative yield Φ (dashed lines), in dimers of 30 nm AgNCs (grey data), 30 nm AuNCs (yellow) and 40 nm AuNCs (red) for a longitudinally coupled emitter positioned in the center and as a function of the interparticle spacing s. (b) Visible strong coupling figure of merit (solid line) as a function of s in the same geometry as (a). (c) Visible strong coupling figure of merit as a function of the orientation of the dipolar emitter in dimers of 30 nm AgNCs (grey data), 30 nm AuNCs (yellow) and 40 nm AuNCs (red) with 1 nm (solid solid line) and 2 nm (dotted line) spacings. (d) Visible strong coupling figure of merit for a longitudinal emitter as a function of its position with respect to the dimer axis, for dimers of 30 nm AgNCs (grey data), 30 nm AuNCs (yellow) and 40 nm AuNCs (red) with 1 nm (solid solid line) and 2 nm (dotted line) spacings.

  -d allow the observation of digital artefacts due to the mesh size when monitoring the evolution of the strong coupling figure of merit with respect to the transverse position of the source dipole. The comparison between Mie theory and BEM simulations in the case of spheres shows that a smooth and not a stepwise evolution of the strong coupling figure of merit indicates numerical convergence.

Acknowledgement

This work was supported by Agence Nationale de la Recherche via projects ANR-15-CE09-0003 and ANR-21-CE09-0010, and by LABEX WIFI (Laboratory of Excellence within the French Program "Investments for the Future") under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*.

Supporting Information Available

The Supporting Information is available free of charge on the ACS Publications website.

Complementary figure providing the computed values of the damping rate of the plasmonic resonators and of the coupling strength g.


Cécile Michel
email: cecile.michel@cnrs.fr.

Chapter 12 Weighing Units and Weights in the Context of Trade Between Upper Mesopotamia and Anatolia (Nineteenth and Eighteenth Centuries BCE)

At the very beginning of the second millennium BCE,1 inhabitants of the Aššur (Qala'at Sherquat) citystate, in Iraq on the Tigris, engaged in large-scale trade with Anatolia. This period, which covers the first three centuries of the second millennium BCE is generally called 'Old Assyrian'. Assyrians exported tin and textiles to Central Anatolia, and brought back gold and silver. 2 They established trading settlements on the Anatolian Plateau, far from their homeland. Their archives, mainly dated to the nineteenth century, have been found at three sites, with the vast majority coming from excavations in the lower town at Kaneš (modern Kültepe), near Kayseri, in Turkey. Kaneš has, up to now, provided 22 500 cuneiform tablets written in the Old Assyrian dialect. Only 40 such tablets have been found scattered within the round citadel, in and outside of the palace. Thus we do not have official archives of the Anatolian kingdom if they ever existed. In the lower city, north-east of the mount, 22 460 tablets have been excavated, with approximately 22 000 dating to the nineteenth century (phase II) and 460 to the eighteenth century (phase Ib). Note that for Aššur, the home city of the Assyrian merchants, the Old Assyrian level has not been excavated; only 25 tablets from this period have been unearthed, probably because they had been discarded in later levels. The majority of these are school texts. 3 The Old Assyrian tablets document the regular trade established by Assyrian merchants between Aššur and Kaneš. In Aššur, they bought tin originating from the Far East; textiles, made locally or imported from Southern Mesopotamia; and donkeys. The merchandise was taken to Anatolia on donkey caravans. 4 Aššur itself did not produce much, except textiles and donkey harnesses, but was an important centre in an international trading network. In Anatolia, the Assyrians sold their merchandise for cash or on credit, at times increasing their profits via the local trade of copper and wool. 5 When coming back from Kaneš, the Assyrians brought gold and silver with them. The gold was usually hoarded in Aššur or used to buy tin. The silver was refined and assayed in both Kaneš and Aššur, and was the favoured means of payment. Part of it was immediately reinvested in new caravans. In Anatolia, tin was sold at twice its price in Aššur, and each piece of textile at three times its price. 6 This significant profit was reduced by the many taxes the merchants had to pay in Aššur, en route, and in Kaneš. During their trading activities and financial operations, 7 Assyrian merchants had to carry out various computations. They needed to know the different measurement systems and to be able to calculate prices by converting an amount of a product into silver or copper according to a tariff decided in advance. 8 They used an additive decimal system to count objects, and among the different metrological systems they used, the weight system dominated because of the nature of the goods traded, predominantly metals. 9 Tin, gold, silver and iron were measured according to the Mesopotamian sexagesimal weight system in gu, mana, gin and še; one gin weighed a little more than eight grams ( Merchants' archives constitute an excellent source for analysing quantification and computation practices. The metrological list of weights found at Kaneš, the only extant example of this type of list, may be analysed in parallel to similar, contemporaneous Old Babylonian lists; it highlights how quantities were shaped in the Old Assyrian corpus. Recording the various operations made with weights enables the identification of weight standards and their verification, as well as discrepancies and corrections. 11 Texts dealing with manipulations of weights reveal the existence of the Anatolian mana, which was lighter than the Mesopotamian mana, and the use of a simplified weight system to quantify copper on the Anatolian Plateau. 12 A large collection of stone weights discovered at Kaneš may be compared with the data provided by the archives and the metrological list, and an attempt to correlate some of these weights with textual measures will be made.

A Progressive Metrological List of Weights

In Ancient Mesopotamia, scribes learned how to read, write, count and calculate with the help of various school texts. 13 The towns of Aššur and Kaneš have produced about twenty Old Assyrian school tablets. 14 Those dealing with computation are mainly round tablets bearing small conversion exercises of one product into another, usually silver. 15 Excavations in Kaneš have also provided the aforementioned metrological list. According to Proust (2007: 269, Fig. 2), who has studied material from Old Babylonian Nippur, about 40 per cent of the mathematical tablets produced in a school context consisted of metrological lists and tables. 16 The metrological lists relate to the learning of quantities and the various systems of measuring 6 Veenhof (1988). 7 Larsen (1977), Michel (2001: 303-357).

8 Michel (2008b). 9 Michel (2006b). 10 The translations usually found in the English literature for these measurement units are: gu2 'talent'; ma-na 'mina'; gin2 'shekel'; še 'grain'.

11 Veenhof (1972: 54-68); Dercksen (1996: 80-85). 12 Dercksen (1996: 86-89); Michel (1998: 258-261). 13 Veldhuis (1996;2011); Proust (2007); Robson (2008). 14 Michel (2008b). Bibliography relative to these school texts: Michel (2003: 139-140; 2006a: 445; 2011a: 432). 15 Michel (2006b). 16 Note that about 10 to 20 per cent of the Mesopotamian school texts are usually mathematical texts (Veldhuis 1997).

Table 12.1 The Mesopotamian weight system used in the Old Assyrian sources units in which they could be expressed. They consist of enumerations of measurement values, in the form of simple progressive lists, containing one or more of the different measuring systems (capacities, weights, surfaces and lengths). These lists allow us to understand the various measurement units and their corresponding numerical values, 17 as well as the relations existing between the different measurement units. Metrological tables present the same enumerations as the lists, but in the form of tables giving the equivalent number of each measurement value in the sexagesimal place-value notation. 18

An Old Assyrian Metrological List

Two fragments of this metrological list, Kt t/k 76+79, dated to the eighteenth century were found in a house in Kaneš. 19 This corresponds to the second phase (Ib) of the Old Assyrian settlement in Kültepe lower town. The large tablet (18 × 16.5 × 2 cm), partly destroyed, is quadrilateral and thick. It presents four columns on each side, covered with coarse signs. The lines are not regular and there is no alignment between the various columns. This tablet was presumably written by a student. On the obverse, the scribe registered a progressive list of measurement values, expressed in weights, from 1 gin to 60 or 100 gu, and on the reverse, a lexical list, with logograms and Akkadian words, recording metals, stones and plants.

A list of metals and metal objects makes up only nine lines (v 1-9) and lists gold, iron, tin, copper and bronze (silver is curiously lacking) and the names for various vessels and cauldrons in bronze and copper. 20 The list of stones is much longer, at least forty lines (from v 10 to possibly part of column vii).

It starts with semi-precious stones, like lapis lazuli, carnelian and rock crystal (?), and progresses to more common ones, such as grindstones. 21 The last two columns, which are much too fragmentary to be transliterated, enumerate a number of plants (Figs 12.1 and 12.2). 17 In the weight system, for example, the numerical signs used before gu2that is, aš, differ from those used before še, gin2 and mana, that is, diš. The phenomenon being systematic in the use of weight system in this corpus, we shall not indicate the type of numerical sign used in parenthesis unless otherwise specified. Thus, instead of 1(aš) gu2 3(diš) ma-na, we will write 1 gu2 3 ma-na. 18 Proust (200818 Proust ( , 2010)). 19 The two fragments of this text, discovered in 1968, were edited by Hecker (1993: pl. 47), and a photo of col. i and ii (Kt t/k 79 obverse) is published by Özgüç (1986: pl. 47, 2). For a first attempt of a reconstruction, see Michel (1998: 254). I am grateful to F. Kulakoğlu, Director of Kültepe-Kaneš excavations, who very kindly provided me with good photos of these two fragments. 20 Lines v 4-9 are cited by Dercksen (1996: 79). 21 Michel (2017).

The metrological list inscribed on the obverse of this school text, even if broken, may be reconstructed as follows (Table 12.2): The top and the bottom of the tablet are destroyed. Suggested restorations of measurement values are proposed between brackets at the beginning and at the end of each column.

The list of weights starts with 1 gin (of silver) and increases by units of 1 gin until 19 gin. But, from 11 gin onwards, the scribe added a progression in še which starts with 7 1/2 še, followed by 15 še (twice 7 1/2), then fractions of gin. The three first fractions are specific to the Old Assyrian signs list and do not exist elsewhere in the cuneiform corpus. The first sign is a hapax legomenon, the last two are to be read 1/6 and 1/4. 22 Then follow the existing cuneiform signs of fractions corresponding to 1/3, 1/2, 2/3 and 5/6. These fractions of gin are equivalent to the following amounts of še (gin): 30 (1/6), 45 (1/4), 60

(1/3), 90 (1/2), 120 (2/3), 150 (5/6). Thus, the first still unknown sign should correspond to a fraction of gin which is equivalent to an amount of še comprising between 15 and 30 še. The addition of the two previous weights, 7 1/2 še + 15 še equals to 22 1/2 še, a quantity that is quite common in the texts and always written in še; it corresponds to 1/8 gin. Thus, the unknown fraction sign must be read 1/8 (Fig. 12.3).23 After 19 5/6 gin, the scribe continues with an arithmetic progression of mana, starting with 1/3 mana (= 20 gin), using the classic sequence of fractions: 1/3, 1/2, 2/3, 5/6 (thus from 10 to 10 gin up to 1 mana).

The end of column i and the beginning of column ii are destroyed, but may be easily restored: the sequence of weights follows a progression of units of 1 gin (as in column i).

In column ii, line 12, 1 mana 19 gin is expressed in two different ways, by the additive notation 1 mana + 19 gin and the subtractive: 1 1/3 ma-na la2 (minus) 1 gin2. Then, the development again progresses by 10 gin using the four existing fractions of mana (which correspond to sixths): 1/3, 1/2, 2/3, 5/6. The reconstitution of the end of column ii is not certain, but a development of weights in mana is obvious at the beginning of the third column, thus, a progression of gin is probably added to 1 5/6 mana in order to reach 2 mana. Further, to the progression of mana units, it seems quite obvious that the scribe added a progression of še units as shown by the end of the sequence in column iii, lines 14-17 up to 19 mana 19 še. Then, for the second time, the scribe uses a subtractive notation 20 la2 (minus) 1 ma-na to express this time 19 mana. But curiously enough, since the development of mana was made in parallel to a progression of še, this amount is smaller than the previous weight by 19 še: this is unique in such a metrological list.

From 20 mana there is an arithmetic progression by ten mana -except for 55 mana -up to 1 and 2 gu, and by units of 1 gu up to 19 gu. Here, for the third time, an equivalent of this last weight is also given in a subtractive notation: 20 la2 (minus) 1 gu2. The list ends with weights of gu developed by 10 units up to 60 (presumably written with six wedges). The last line, which is broken, may perhaps contain:

1 me-at gu2: 1 hundred gu.

Whether we consider gin or mana, all the fraction sequences in this list correspond to a more or less regular progression of the inferior unit. The use of fractions becomes less frequent while the measurement units increase: the scribe does not use fractions of gu but multiples of mana. However, the progression of weights has no methodical character: for example, as indicated in italics in the following table (Table 12.3), quantities in še are added to progressive integers of gin and mana. Similarly, the subtractive notations indicated in bold in Table 12.3, mark breaks in the progression. Are the irregularities found in this list specific to this particular text, or can we find parallels in the contemporaneous Old Babylonian corpus?

Comparison with Contemporaneous Old Babylonian Lists and Tables of Weights

Metrological lists and tables are attested throughout the Ancient Near East, from Kaneš in Anatolia to Susa in Elam; for the great majority, they date to the Old Babylonian period, which is more or less contemporaneous with the Old Assyrian period. 24 Christine Proust (2010) estimates that, of a total of a little less than six hundred metrological lists and tables examined so far, 80 per cent were found in the Old Babylonian Nippur, south of Mesopotamia. The Old Babylonian metrological lists and tables more or less follow the same progression, with a few local variations.

In order to compare the Old Assyrian weight metrological list to Old Babylonian ones, we choose five samples originating from various places. The first is made of the combination of several manuscripts from Nippur as reconstructed by Christine Proust. 25 The second and third documents are preserved at Yale and their provenience remains unknown; one is a list, 26 the second is a table inscribed on a prism. 27 The fourth sample is a list from Kiš, 28 and the last, a list from Mari written on a cylinder (Table 12.4). 29

24 Neugebauer (1935: 88-94); Nemet-Nejat (1993: 17-19); Proust (2007: 97-117, 311-315). 25 Proust 2007: 311-315 (composite text). ). See also Proust (2009). 26 Neugebauer (1935: 92-94).

27 Chambon and Robson (2011:136-143).

28 Robson (2004: no. 19). 29 Chambon (2002: 497-499).

Table 12.4 Comparative table of Old Babylonian and Old Assyrian metrological lists and tables for weights

The table presenting these different documents shows that the Old Assyrian metrological list of weights differs from all the texts dated to the Old Babylonian period that seem to have had the same function.

These usually start with the quantity 1/2 še and present a long development in še units, while the Kaneš list begins with 1 gin. In the Kaneš list, the progression both in še and fractions of gin is added to the arithmetic progression of gin (beginning from 11 gin) as if they were disconnected. Rather than še units, the smaller amounts added increase from 7 1/2 še up to 1/6 gin (7 1/2 -15 -22 1/2 -30), and from 15 up to 1/3 gin and from 30 up to 5/6 gin. A second progression of še units increases in parallel to a development of mana units (in italics on Table 12.2). The Old Babylonian documents, after a progression of units of še up to 29, show an arithmetic development by 5 or 10 še after each fraction of gin; the most regular example being the list of Kiš (5 -10 -15 -20 še). It is interesting to note that the Mari cylinder adds the quantity 22 1/2 še between 22 and 23 še, a quantity attested by the fractional notation 1/8 gin in the Kaneš list. The Old Babylonian lists and tables use the classic Mesopotamian expressions for 1/6 and 1/4: igi-6-gal2 and igi-4-gal2, while the Kaneš lists bears the specific Old Assyrian signs for these two fractions which are, in both cases, used only for gin and not for higher weight units.

The development of mana units is quite similar in all the documents, except that some Old Babylonian tablets progress in units up to 30 mana while others, as well as the Kaneš list, stop at 20 mana, and then progress by 10 mana up to 1 gu. The Kaneš sample repeats three times the previous quantity using successively the additive then the subtractive notation 19 gin (1/3 la2 1 gin2), 19 mana (20 la2 1 ma-na) and 19 gu (20 la2 1 gu2). Most of the texts end with 1(geš2) gu2 = 60 gu2. 30 We can summarize the remarkable characteristics of the Kaneš list as follows:

 No quantity inferior to 1 gin (in contrast to some Old Babylonian samples which start with 1/2 še)  Two different arithmetic progressions of še, one with gin and the other one with mana. The first increases by 7 1/2 še -not attested in the Old Babylonian texts -, and uses symbols for fractions of gin; the sign 1/8 is a hapax legomenum.  Use of subtractive notations creating steps in the progression of the list.

None of these features are found in the Old Babylonian lists and tables. Thus, the Kaneš metrological list of weights bears witness to a culture of quantification different from the similar, contemporaneous culture evidenced by documents produced by the Old Babylonian schools of Southern Mesopotamia or Mari, which are quite homogeneous. When writing the list, the Kaneš scribe's intention was not only to memorize the weight system that was most widely used by the merchants, but also to practice the different notations of integers and fractions. This is confirmed by the use, three times, of subtractive notation (with 'la2') which, in one case, gives a quantity smaller than the previous one. This metrological list is the only text of this category found at Kaneš, and the site has not provided any numerical tables. 31 Nippur schools produced metrological lists and tables in about equal quantities. Christine Proust noted that metrological lists were often written on the other side of lexical lists -which is the case for the Old Assyrian sample -while metrological tables were more linked to numerical tables, indicating a higher level in mathematical education. 32 Thus, metrological lists and tables did not have the same function.

For their daily accounts, the Assyrian merchants only needed to know the notation of the weight measures and the relationships between the different weight units.

How Quantities Were Shaped: Fractions Linked to Weight Units

The Old Assyrian metrological list of weights analysed above shows that the Assyrian merchants used, with sexagesimal weight measurement units, measurement values expressed by a combination of signs for numbers from 1 to 9 (diš, aš) and tens (u) in an additive decimal notation; they gave a special importance to fractions in the notation of quantities. These fractions are mainly found in connection with gin and mana units, but 1/2 is also associated with še. By contrast, fractions are never linked to the larger unit of weight, the gu.

Beside the use of the Mesopotamian signs for 1/3, 1/2, 2/3 and 5/6 for gin and mana -and ignoring the classic expression igi-n-gal2 for the writing of inverses -the Assyrians invented signs for 1/8, 1/6 and 1/4 which are only found with the gin unit in the list. 33 The subtractive notation 1/3 mana minus 1 gin for 19 gin allows the fraction 1/3 (for 19 and 20 gin) to be used twice (Table 12.5). All these fractions -except 1/8 -are attested many times in the merchant accounts. 34 The quantities of še occurring in the Old Assyrian texts are the followings: 5 še (rare), 35 7 1/2 še, 10 še, 15 še, 20 še (very rare), 36 22 1/2 še. 37 This last weight is common, which means that the sign for 1/8 (gin) attested in the metrological list seems to have been created on purpose by its scribe.

Then quantities are expressed in fractions of gin, as for example 1/6 gin (=30 še), 1/4 gin (=45 še), etc. Signs for 1/6 and 1/4 are associated exclusively with the gin unit. But they are also very occasionally found to quantify realia, such as textiles: 1/6 tug2 (1/6 of a textile) 38 or 1/4 tug2 (1/4 of a textile). 39 Subtractive notation allows 3/4 gin (1 la2 1/4 gin2) to be expressed for which there was no specific sign. 40 But, curiously enough, it is also used sometimes to write fractions for which there was a specific sign, for example 2/3 gin (1 la2 1/3 gin2) or 5/6 gin (1 la2 1/6 gin2). 41 On the other hand, the fractions 1/3 gin and 1/6 gin are never written by way of a subtractive notation. Only inverses are subtracted from a quantity, and this quantity being either an integer (from 1 to 20) or a fraction of the upper unit, or of the unit itself: 1/3 mana minus (la2) 1/6 gin. 42 According to their archives, merchants preferred to use fractions of the superior unit rather than integers of the inferior unit, for example, they used 1/3 of mana more than 20 gin. This certainly had an impact on their computing practices, and it is also possible that the use of fractions of the superior unit reflects specific weighing practices.

Weighing and Weight Standards

The principal goods traded by Assyrian merchants: metals (gold, silver, tin, copper), textiles, wool, precious stones (lapis lazuli, carnelian), and donkeys, were quantified by weight or by number. Weighing was done with the help of a twin-pan balance. Weighing scales have been found in pairs in merchant 33 Michel (2006b). 34 Michel (1992). Thus, they lacked the fraction 1/5 which could have been useful in some computations. 35 JCS 14, 9:4; TPAK 1, 204 1; KTS 1, 52b:1. 36 JCS 14, 9:9. 37 Note that we find one example of 27 1/2 še ku3-babbar '27 1/2 še of silver' in AKT 3, 29:9 (neither copy nor photo of the tablet is available). This could correspond to 22 1/2 + 5 še, both quantities attested in the Old Assyrian documentation. 38 graves at Kültepe dating from the nineteenth and eighteenth centuries. They are made in copper, are round and convex, and pierced by four holes. Their relatively small size -diameter between 5.7 and 8.3 cm -implies that, if they correspond to real scales, they were used only to weigh small quantities. 43 Their discovery in merchant graves naturally bears witness to the importance of weighing for this profession (Fig. 12.4). The words na4 in Sumerian and abnum in Akkadian, which mean 'stone', often refer to weights, implying that weights were regularly made of stone. The Old Assyrian tablets document weight standards, manipulations with weights and weight discrepancies, while archaeology has provided dozens of weights of various shapes.

Weight Standards

Texts refer to weight standards used in the weighing process; their owner, either a town or an individual, may be mentioned as well as their weight. Weights could belong to individuals, to town authorities and other institutions:

'2 gu 1/3 mana of good quality copper according to the stone-weight of [the town of] Tuhpia and 50 mana of copper in pieces according to our weights 45 or 43 Kulakoğlu and Kangal 2010, No. 404 (Kt 96/k 108) and 405 (Kt 01/k 11), and personal communication by F. Kulakoğlu. 44 For a recent description of this seal, see Michel (2014). For the weighing practices at Mari at the same period, see Joannès (1989).

45 Kt 93/k 278:1-6, 2 gu2 13 1/3 ma-na, urudu sig5: i-na a-ba-an, Tu3-uh3-pi2-a, u3 50 ma-na urudu, i2 ! (KAM)-kum, i-na ab2na-ti2-ni.

7 mana of copper according to our weights. 46 The various weight standards attested in the Old Assyrian texts are the following:  10 gin and 1/2 mana: '(the weight of) the silver has been verified by means of my 1/2-mana and 10-gin weights' 47  1/2 mana: 'a 1/2-mana weight' 48  1 mana: 'His own 1-mana weight'; 49 'By the 1-mana weight of the trade bureau'; 50 'By the 1mana weight of Šū-Ištar'; 51 'By my own 1-mana weight'. 52  10 mana: 'By your own ten-mana weight'. 53  To these, we may add those attested in the texts quoted below (Table 12.6): 1/4 gin -2/3 gin -1/2 gin -1 gin -1/3 mana.

The Verification of Weight Standards, Discrepancies and Corrections

As there were many weight standards, it was important to verify them. Both official and private weights were regularly verified and could then be said 'certified' (takittum):

From it (previous amount), we gave Uṣur-ša-Ištar 16 mana of silver (weighed) by the 1-mana weight of the trade colony, certified (weight). 54 He transferred 14 gu 30 mana of (copper or wool?) to me (weighed) by your weights, certified (weights). (Fig. 12.6) 55 8 mana of silver (weighed) by my 1-mana weight, certified (weight). 56 When checked, many weights were found to be too light: 'Your 1-mana weight is deficient'. 57 A correction had to be carried out by the addition of a small weight to the weights on the set of scales: 58 In this example, the 10-mana weight being used is lighter by 10-gin, a discrepancy of 1.66 per cent.

The following text shows a smaller discrepancy of 0.83 per cent, since the 1-mana trade colony weight is lighter by 1/2-gin (Fig. 12.7): 59 Table 12.6 Weight discrepancies attested by Old Assyrian texts 59 Kt 93/k 462:7-13 (unpublished), 15 ma-na ku3-babbar ṣa-ru-pa2-am, a-ṣe2-er 1 ma-na-im, ša ka3-ri-im 1/2 gin2-ta, ta-ur-ma ku3-babbar, u2-du-u2 i-ṣe2-er, ka3-ri-im, A-šur3-tak2-la2-ku, i-šu.

The discrepancies found in the Old Assyrian texts range from 0.83 per cent to 2.1 per cent for a 1-mana weight standard (Table 12.6). One may observe that a 10-mana weight is used to weigh quantities expressed in gu, while a 1-mana weight is placed on the scale to weigh quantities expressed in mana (even 40 mana).

Two Cultures of Quantification

As they were trading in a different country, some 1 000 km away from their home, the Assyrian merchants had to deal with a different culture of quantification in Anatolia, even though the country was using the Mesopotamian weight system.

The Mesopotamian and the Anatolian Mana

The Anatolian standard mana, called aban mātim 'weight of the country', was a little lighter than the Mesopotamian mana. A letter dated to the eighteenth century gives a metrological correlation between both standards as follows (Fig. 12.8) :60 1 a-na Šu-Ištar qi2-bi2-ma um-ma A-šur3-gal-ma i-na 16 tug2 ša ik-ri-bi-ka3 a-na 2 tug2 2 me-at urudu a-ba-an ma-tim Tu3-ra-me-el : i-di2-na-am a-na 2 tug2 5 1 me-at 80 ma-na urudu a-ba-an ma-tim Šu-Hu-bur i-di2-na-am a-na 2 ! tug2 1 me-at 80 ma-na urudu a-ba-an ma-tim A-šur3-ba-ni i-di2-na-am 2 me-at 80 ma-na urudu a-ba-an ma-tim ša 3 ? tug2 A-la-hu-um 10 i-di2-na-am ši2-ti2 ṣu2-ba-ti2 3 tug2 ma-aṭ-u2-ti2šu-nu e-zi-ba-ma 16 gin2-ta ta-ad-nu 1 tug2 ša qa2-tim ša i-a-ti2 i-di2-na-ni a-na 10 gin2ku3-babbar ta-di2-in a-na 2 tug2 2/3 ma-na /ku3-babbar Lu-lu i-di2-na-am šu-nigin2 urudu ša i-ta-di2-nu 15 8 me-at 40 ma-na a-ba-an ma-tim a-na ab2-na-ti2-ni i-tu3-ur-ma 7 me-at 60 ma-/na 12.7). Thus, the Mesopotamian mana was about 10 per cent heavier than the Anatolian mana. All the people involved in this letter were Assyrians which explains why they needed to convert the amount of copper obtained into their own standard. The Anatolian mana is often referred to in the texts by both Anatolians and Assyrians. For example, the following letter was sent by an Assyrian to an Assyrian and mentions the use of the weight of the country: 61

The remainder of your sû-stones, 6 pieces weighing 8 gu minus 10 mana, has been sold for 4 hundred 40 mana [of copper in pieces] according to the weight of the country.

When the two standards were in a text written by an Assyrian merchant, only the 'weight of the country' was specified; the Mesopotamian weight standard was then implicit as in the following text: 62

Your 20 textiles of current quality have been taken to the palace; (at a price of) 40 mana of copper each (piece), according to the 'weight of the country': you are registered in the trade bureau (for) 4 hundred 87 mana of copper for two-thirds of your textiles, you can take it when the copper becomes available.

For the (last) third of your textiles, at 9 gin (of silver) each, you can take within 10 days 1 mana 1/6 gin of silver.

The price in copper of two-thirds of the textiles (13 1/3 textiles) is equivalent to 533 1/3 mana of copper according to the 'weight of the country'. The amount of copper given in the text, 487 mana, is about nine per cent less, and would thus correspond to the Mesopotamian mana. The price in silver of onethird of the textiles (6 2/3 textiles) should be 1 mana, but the scribe computed a little more: 1 mana 1/6 gin. 63 The Anatolian mana was mostly used to weigh copper. However, it is attested once to weigh silver, 64 and once to weigh tin, in the following small text: 65 61 KTS 1, 7a:19-22, ši2-ti2 su2-a-ti2-ka3, 6 su2-a-tim 8 gu2 la2 10 ma-na, a-na 4 me-at 40 ma-na, a-ba-an ma-tim ta-ad-na. To the texts cited by Veenhof (1972: 54-57) and Dercksen (1996: 86-89) mentioning the 'Anatolian mana', one may now add AKT 5, 45:15'; AKT 5, 50:10; Prag I 520:18; Sadberg 16:4; VS 26 72:12-15. Note that the text AKT 6, 251:11-12 mentions the 'weight of the [local] palace' (a-ba-an, e2-gal lim ) which might be another way to express the Anatolian weight. 62 CCT 2, 24:23-27, 20 tug2 tu 3-ka3 ša qa2-/tim, i-na e2-gal lim i-le-qe2-u2 ! , 4[0] ma-na-ta urudu i-na a-ba-an, ma-tim ša še2-ni-pe3-e ša tug2 ti 2-ka3, 4 me-at 87 ma-na urudu, e2 ka3-ri-im la2-ap2-ta ti 2, i-wa-ṣa urudu ta-la2-qe2 ša šal2-ša-at, tug2 ti 2-ka3 9 gin2ta 1 ma-na 1/6 /gin2, ku3-babbar a-di2 10 u4-me : ta-la2-qe2. 63 Dercksen (1996: 86, 88). 64 'Send me with the first caravan 10 gin of silver under your seal for the 8 1/3 gin of silver according to the weight of the country', VS 26, 72:12-15, a-na 8 ! 1/3 gin2 ku3-babbar /i-na a-ba-an ma-/tim, 10 gin2 ku3-babbar ku-nu-ka3-ka3, iš-ti2 a-li-kiim, pa2-ni-e-ma še2-bi4-la2-nim; text cited by Dercksen (1996: 88). The ratio between the two standard mana would be here 16.67 per cent. 65 Kt 94/k 240 (complete text), 10 ma-na, an-na i-na, a-ba-an ma-tim, a-na I-šu-hi-im, a-di2-in.

I gave 10 mana of tin according to the weight of the country to Išuhum.

Weighing Tin in Mesopotamia, Copper in Anatolia

Among the commodities traded by the Assyrian merchants in Anatolia, tin, gold, silver, and iron were measured according to the Mesopotamian sexagesimal weight system in še, gin, mana and gu, as in the following example: 66

In all: 4 hundred 10 gu 11 mana of tin, (is) the value (in tin) of Imdīlum's caravan

The principal units of this weight system were the mana and the gin; this can be implied from the frequent use of fractions for these two units. Such a focus led perhaps, in Anatolia, to the parallel development of a specific way of quantifying copper in mana. 67 The Assyrians traded copper only within Anatolia, where it was abundant and therefore cheap. 68 The metal circulated in the form of ingots, which could be counted with integers or weighed; large volumes of copper were exchanged, often many gu (300 kg): '73 gu 10 mana of copper in pieces (2.195 tonnes)' 69 or '98 gu 50 mana of copper in pieces (2.965 tonnes).' 70 Copper could also be quantified in hundreds of mana, in a system ignoring the other units in the Mesopotamian weight system, and perhaps influenced by the local decimal system. 71 Thus, in the text documenting the difference from the Anatolian mana quoted above (Kt u/k 3), the scribe wrote:

Herewith, 7 1/2 gu of good quality <copper> and a hundred [mana] of copper in pieces. 79 Note that inside Anatolia, there were specific weights (abnum) to quantify wool, which corresponded to unknown units:

65 stones of wool. 80 This weight could be implied in the following examples:

1 hundred 50 units of wool. 81 As price of 3 hundred units of wool, I received 1 hundred 37 mana of copper as his share. 82 The use of two different methods to express weights of copper, based on a mixture of the decimal and sexagesimal systems or only on a decimal base reflects two cultures of quantification, Mesopotamian and Anatolian; the latter borrowed the system of the former but adapted it. As they settled in forty or so towns on the Anatolian Plateau, the Assyrian merchants introduced their writing and their metrology, but might have been influenced by a local decimal tradition that they partially adopted. Should we then consider that the quantification of copper using only the mana always referred to the Anatolian mana standard? A text proves that the use of the Anatolian weight did not necessarily imply a count in hundreds of mana: 83 2 gu 10 mana copper in pieces according to the weight of the country.

Such different traditions for quantifying metal weights are also attested by texts from Upper Mesopotamia and Middle Euphrates. 84 A small Mari administrative tablet (eighteenth century) gives the conversion of an amount of copper offered by the king of Yamhad (an area of Aleppo) to Mari, on the Middle Euphrates, as follows: [START_REF] Michel | 1 gal 6 li-mi 7 me 40 gin2 urudu kur, up-pu-uš2-ma, 4 gu2 39 ma[END_REF] 1 ten thousand 6 thousand 7 hundred 40 gin of mountain copper, it is computed as 4 gu 39 mana of mountain copper.

Thus:

10000 + 6000 + 700 + 40 (16740) gin of mountain copper, it is computed as 4 gu 39 mana of mountain copper.

A tablet from Qal'at al-Hadi, which may correspond to the ancient city of Šurnat, south-east of Šubat-Enlil (Tell Leilān), 86 proposes such a conversion, but applied to silver: '58 mana of silver, it is computed as 3 thousand 5 hundreds gin of silver.' 87 Both samples highlight the use of the gin instead of the use of mana in Old Assyrian texts.

The Anatolian Gin and Other Regional Standards

If the Anatolian mana was ten per cent lighter than the Mesopotamian mana, it is tempting to assume that the Anatolian gin was also ten per cent lighter than the Mesopotamian gin. In almost all of the occurrences, the expression aban mātim is applied to mana weights. But there is at least one text where it concerns a quantity given in gin. In this letter addressed by two merchants to two others bearing Assyrian names, we read the following: 88

For 8 ? 1/3 gin silver according to the weight of the country: 10 gin of silver under your seal.

If we assume that the corresponding 10 gin are weighed with a Mesopotamian standard, this would mean that the Anatolian gin was more than fifteen per cent heavier than the Mesopotamian gin. E. Ascalone and L. Peyronel have carried out a detailed study of the weight artefacts found in the Near East and dated to the third and second millennia BCE. 89 According to their study (Ascalone and Peyronel 2006: 409-422), the equivalencies between the Mesopotamian and Anatolian weight systems are ( 12.8 Equivalencies between the Mesopotamian and Anatolian weight systems Thus, the Anatolian gin is heavier than the Mesopotamian gin and there are forty gin in an Anatolian mana and not sixty as in the Mesopotamian metrological weight system. According to these authors, other standards in use in other areas are also represented among the Kültepe artefacts: 90  The Levantine gin which would weight 9.4 g (a mana of 50 gin)  The Syrian / Karkemiš gin which would weight 7.8 g (a mana of 60 gin)  The Aegean gin which weight 6.6 g.

A Collection of Weights from Kültepe

We shall now address the culture of quantification from the viewpoint of the material culture.

Excavations in Kültepe started in 1948 under the direction of T. Özgüç, and are still going on today, under the direction of F. Kulakoğlu (since 2006). Up to now, more than a hundred and sixty weights had been found in the lower city, dating from the first three centuries of the second millennium BCE. About ninety have been recorded in publications, eighty are still unpublished. These weights are predominantly made in hematite, but there are also a few weights in rock crystal and lead. They are often ovoid, in the shape of a seed; a barrel, with long-cylindrical body and flattened ends; triangular or cylindrical prisms; or zoomorphic (ducks, frogs or bovine heads, Fig. 12.9). 91

Fig. 12.9 Samples of stone weights found at Kültepe, Kulakoğlu and Kangal 2010: 319-320 (nos. 377-385, 388-389) (©Kültepe Archaeological Mission) 88 VS 26, 72: 12-13, a-na 8 ! 1/3 gin2 ku3-babbar i-na a-ba-an ma-tim, 10 gin2 ku3-babbar ku-nu-ka3-ka3. 89 Ascalone and Peyronel 2006: 41-422 (Kültepe), 423-430 (Aššur). 90 Ascalone and Peyronel (2006: 494-514). 91 Özgüç (1986: 78-81, appendix 5); Dercksen (1996: 80-86); Kulakoğlu and Kangal (2010: 319-322).

Among all these many weights, only two could form a pair corresponding to the Mesopotamian mana and to the Anatolian mana; both date to level II (nineteenth century) of the lower city (Fig. 12.10). 92 Note that Ascalone and Peyronel (2006: Tables 8.22, KT 76) give the wrong weight of 475 g for Kt f/k 103. 93 Michel (2003: 76-77); Michel (2011b: 100, 108). 94 There are some references to bronze duck-shaped figures whose use is not clear: Kt h/k 87: 3 (2 duck-shaped figures with lamp wicks); Kt c/k 1517:1 with other bronze unknown objects. -šu 13 2/3 ma-na.

If this copper object in the shape of a duck really is a weight, then this text attests the existence of weights other than those included in the Old Babylonian lexical lists (see below). The smallest stone weight attested at Kültepe weighs 0.63 g (Kt p/k 144) and could correspond to a Mesopotamian weight of 15 še (which, based on the Mesopotamian mana of 494 g, should weight 0.68 g). Old Babylonian lexical lists provide a list of stone weights (ur5-ra, Sect. 4) which may give an idea of the smallest weight in use in that period. One of the most complete lists, found at Nippur, mentions the following stone weights corresponding to še units (Table 12.9): 5 še -10 še -15 še -20 še -22 1/2 še -1/6 gin -1/4 gin -1/3 gin -1/2 gin -2/3 gin -[5/6] gin. 96 Fifty-four Kültepe stone weights are lighter than 8 g ranging from 0.63 g to 7.95 g. The weights of these stones vary considerably and there is not always clear correspondence between attested weights in the texts and these stones, if we rely on the identification of the weight of the mana proposed above. However, it is possible to make tentative identifications with the following weights for both Mesopotamian and Anatolian standards:  Mesopotamian: 15 še -22 1/2 še -25 še -1/6 gin -(1/6 gin + 7 1/2 še) -1/4 gin -1/3 gin -(1/4+1/6 gin) -(1/3 gin + 22 1/2 še) -1/2 gin -(1/2 gin + 22 1/2 še) -5/6 gin  Anatolian: 10 še -22 1/2 še -1/6 gin -1/4 gin -(1/6 gin + 22 1/2 še) -1/3 gin -1/2 gin -(1/4+1/3 gin) -2/3 gin, and in the following weights: 3/4 gin -5/6 gin We can also presumably find some Syrian, Levantine and even Aegean weights  Syrian: 15 še -1/6 gin -1/4 gin -1/3 gin -1/2 gin -5/6 gin -1 gin  Levantine: 15 še -22 1/2 še -1/6 gin -1/4 gin -1/3 gin -1/2 gin -2/3 gin  Aegean: 1/2 gin -1 gin If the identifications for Mesopotamian and Anatolian weights are correct, several weights show discrepancies with the supposed correct weights, and might have been inaccurate when small quantities of gold and silver were weighed. Twenty-one weights, thus more than ten per cent of the total number of weights, correspond more or less to the Mesopotamian gin with discrepancies ranging from -2 per cent to +2 per cent. By contrast, the Anatolian gin does not seem to be represented in the collection of weights, and only fractions of the Anatolian gin have been found. Twenty-one weights are between 1 and 2 Mesopotamian gin (exclusive). There are three weights which could correspond to the Mesopotamian 2 gin (Kt 85/k 193, Kt 10/t, 23, Kt n/t 255), and there is one weight that can be identified with the Anatolian 2 gin (Kt 10/t 57). Forty-one weights are between 2 and 12 Mesopotamian gin (exclusive). For the Mesopotamian standards, there are weights corresponding to:

3 gin -4 gin -5 gin -6 gin -7 1/2 gin -10 gin. For the Anatolian standards, there are stone weights for:

1 1/2 gin -2 gin -2 1/6 gin -2 1/4 gin -3 gin -4 gin -5 1/2 gin -6 gin, and in the following weights: 8 gin -10 gin. There are only fourteen weights between 12 Mesopotamian gin and 1 Mesopotamian mana (exclusive): 15 gin -1/3 mana -1/2 mana -5/6 mana, and very few weights over a mana: only five weights ranging from 575 g to 5 250 g. The correspondences of these could be the following:



Conclusion

Old Assyrian weights and weighing practices prove to be a fruitful topic to highlight the peculiarities of the merchants' culture of quantification. The metrological list of weights from Kaneš shows many different features compared to similar Old Babylonian texts, stressing the use of fractions of the higher units and subtractive notations; these could be indications about the way the merchants computed and weighed as they carried out their trade. Accounts documents reveal two different cultures for the use of weighing units according to the commodities quantified and the country. In Upper Mesopotamia, tin and silver were measured with the help of the four Mesopotamian weight units: gu, mana, gin and še, while in Anatolia, quantities of locally traded copper could be counted with the mana unit alone, switching to a decimal system. Thus, large quantities of metal were occasionally expressed in hundreds, even thousands, of mana. During their trading activities in Anatolia, the Assyrian merchants had to deal with different weight standards: The Anatolian mana weighed ten per cent less than the Mesopotamian mana. The multiplication of weight standards, linked to the existence of many official and private collections of weights, following both the Mesopotamian and Anatolian standards, but also those from elsewhere (Syrian, Levantine, and the Aegean), explains the variety in the hundred and sixty stone weights excavated at Kültepe; these can be linked to the weight values expressed in the merchants' accounts and in the metrological list. The measurement values obtained for some stone weights do not correspond to attested weights in Old Babylonian lexical lists. The large number of weight standards also confirms the recurrent necessity, attested by the written documentation, to regularly verify the weights and to correct discrepancies. Each owner of a set of weights would have had to know how to correct their incorrect weights, and we can imagine that merchants engaged in double weighing.

Primary Sources:

Old Assyrian Texts References

Fig. 12. 1

 1 Fig. 12.1 Photo of the obverse of Kt t/k 76+79 (©Kültepe Archaeological Mission) Fig. 12.2 Copy of Kt t/k 76+79 by Hecker 1993, pl. 47

Fig. 12 . 3

 123 Fig. 12.3 Sign created for 1/8

  Each text gives a sequence of measurement values; the following texts are used in this reconstitution (note that igi-4-gal2 means 1/4): Ni 3742 (1/2 še to 20 še), Ni 3515 (21 še to igi-4-gal2 gin2), Ni 5196 (1/3 gin2 to 5/6 gin2 25 še), Ni 3909 (1 gin2 to 15 gin2), Ni 3885 (16 gin2 to 5/6 ma-na), Ni 3515 (5/6 ma-na to 2 ma-na), Ni 1082 (10 ma-na to 14 ma-na), Ni 5196 (14 ma-na to 19 ma-na), Ni 3909 (30 ma-na to 1 gu2), Ni 5108 (3 gu2 to 16 gu2

Fig. 12 . 4

 124 Fig. 12.4 Scale pans from Kulakoğlu & Kangal 2010, nos. 404-405 (©Kültepe Archaeological Mission)

Fig. 12 . 5

 125 Fig. 12.5 Seal imprint Ka 82, seal A on the envelope KTS 1, 45a

Fig. 12 . 6

 126 Fig. 12.6 Copy of VS 26, 134 by K. R. Veenhof

Fig. 12 . 7

 127 Fig. 12.7 Photo of Kt 93/k 462 by C. Michel

Fig. 12 . 8

 128 Fig. 12.8 Photo of Kt u/k 3 in Özgüç 1986, pl. 60.1 (©Kültepe Archaeological Mission)

Fig. 12 .

 12 Fig. 12.10 The presumed Assyrian and Anatolian mana (©Kültepe Archaeological Mission).On the left: Kt 02/k 140 (level II); h: 5.5 cm, l: 10.4 cm, 493.95 g. On the right: Kt f/k 103 (level II); h: 5.9 cm, w: 7.97 cm, 444.26 g

Fig. 12 .

 12 Fig. 12.11 Two weights in the shape of ducks (©Kültepe Archaeological Mission). Kt 74/k 8 (level Ib): 8.25 g, and Kt y/k 93 (level Ib): 1.9 g

Table 12

 12

	.1). 10

Table 12 .2 Transliteration of the Old Assyrian metrological list

 12

* kù-babbar = silver

Table 12 .3 A synthetic

 12

and modern presentation of this Old Assyrian metrological list (In italics: Simultaneous progression of two units. In bold: use of subtractive notation)

Table 12

 12

	Fractions of še		1/2
	Fractions
	of gin2
	1/8	1/6	1/4	1/3	1/2	2/3	5/6
	Fractions of ma-na			1/3	1/2	2/3	5/6

.5 Fractions linked to weight measurement units

  Data given by the text Kt u/k 3 According to this text, 840 mana weighed with the 'weight of the country' corresponded to 760 Mesopotamian mana (Table

			8 hundred 40 mana of copper according to the
			weight of the country in our own weights turned to
			7 hundred 60 mana.
		Textiles	Price in copper	Price in silver
			(mana)
	Total amount of textiles	16
	Textiles sold	2	200
		2	180
		2	180
		3	280
		3		3 × 16 gin
		1		10 gin
		2		2/3 mana
	total amount		840 (aban mātim) =	1 1/2 mana 8 gin
			760 (abnātīni)
	Table 12.7
			(successively)

Say to Šū-Ištar: thus (says) Aššur-rabi. Out of the 16 textiles of your votive offerings, for 2 textiles: 2 hundred (mana) of copper according to the weight of the country Turam-El gave me. For 2 textiles: 1 hundred 80 mana of copper according to the weight of the country Šū-Hubur gave me. For 2 textiles: 1 hundred 80 mana of copper according to the weight of the country Aššur-bāni gave me. 2 hundred 80 mana of copper according to the weight of the country for 3 textiles Alāhum gave me. The remainder of the textiles, their 3 textiles of bad quality, he left with me, and at 16 gin a piece they were sold. 1 textile of current quality which he gave me, for 10 gin of silver was sold. For 2 textiles, 2/3 mana of silver Lulu gave me. The total amount of copper they gave

Table 12

 12

	.8):

  Now, if we consider the amounts of še (added to gin) in the list of weights from Kültepe -which correspond to quantities attested in the accounts -we have the following sequence: 7 1/2 še -15 še -1/8 gin (22 1/2 še) -1/6 gin (30 še) -1/4 gin (45 še) -1/3 gin (60 še) -1/2 gin (90 še) -2/3 gin (120 še) -5/6 gin (150 še) Except 7 1/2 še, which weighs about 0.35 g, all of these quantities would be attested by weights found during the lower town excavations (computed by the Mesopotamian standard except 2/3 gin, or computed by the Anatolian standard except 15 še). The lexical thematic list of weights quoted above ends with the following sequence of weights: 22 1/2 še -20 še -15 še -10 še -5 še, a sequence which is somewhat different. About a third of the weights were found in graves located under the houses of the lower city, which means that merchants were regularly buried with sets of weights. An analysis of the material grave by grave could perhaps allow a reconstruction of some sets of weights. For example, four weights excavated in 1995 could possibly belong to the same grave dated to the beginning of the seventeenth century (level Ia). The weight stones recorded could amount, according to the Anatolian standard, respectively to: The study of this collection of weights together with the exact archaeological context could give much more information about weighing practices.

	Kt 95/k 107	8.5 g 2/3 gin 20 še (?)
	Kt 95/k 49	18.5 g 1 2/3 gin
	Kt 95/k 47	44 g 4 gin
	Kt 95/k 48	88 g 8 gin
	575(+) g	→ 1 Mesopotamian mana + 10 gin
	 1 012(+) g	→ 2 Mesopotamian mana
	 2 600 g	→ 6 Anatolian mana
	 4 250 g	→ 9 1/2 Anatolian mana
	 5 250 g	→ 12 Anatolian mana.

  Texts Preserved in Collections Outside of Ankara

	Text (collection)	Inventory	Publication and Edition	CDLI
		number		number
	Anatolica 12, 154 (Istanbul)	Ka 975	Donbaz and Veenhof (1985: 142-146, 154)	P361042
	AnOr 6, 15 (Oxford, Asmoleum)	1914, 425	Driver (1932: pl. IV-V); Michel (2001: no.61)	P361042
	Adana 237e (Museum of Adana)		Matouš (1965: 175-182)
	ATHE 35		Kienast (1960: 50-52)
	BIN 4, 61 (New Haven)	NBC 1899	Clay (1927: pl. XXIX-XXX); Larsen (1967: 122);
			Michel (2001: no. 146)
	BIN 4, 160 (New Haven)	NBC 1669	Clay (1927: pl. LIX-LX); Larsen (1967: 193);
			Ulshöfer (1995: no. 547)
	BIN 4, 199 (New Haven)	NBC 4058	Clay (1927: pl. LXIX-LXX); Ulshöfer (1995: no.
			9)
	BIN 6, 205 (New Haven)	NBC 1701	Stephens (1944: pl. LXV-LXVI)
	BIN 6, 237 (New Haven)	NBC 3727	Stephens (1944: pl. LXXII); Veenhof (1972: 58);
			Rosen (1977: 175)
	BIN 6, 238 (New Haven)	NBC 3792	Stephens (1944: pl. LXXII-LXXIII); Rosen (1977:
			175)
	BIN 6, 242 (New Haven)	NBC 3977	Stephens (1944: pl. LXXIII)
	CCT 1, 2 (British Museum)	BM 113259	Smith (1921: pl. 2-3); Eisser and Lewy (1930: no.
			228)
	CCT 1, 30a (British Museum)	BM 113598	Smith (1921: pl. 30); Ulshöfer (1995: no. 591);
			Michel (1991: no. 271)
	CCT 2, 13 (British Museum)	BM 113458	Smith (1924: pl. 13); Michel (1991: no. 24)	P358514
	CCT 2, 24 (British Museum)	BM 113413	Smith (1924: pl. 24); Michel (1991: no. 24);
			Michel (2001: no. 344)
	Rosenberg 1		Eisser and Lewy, J. (1930: No. 278)
	HUCA 39, 8-10	L 29-557	Lewy, H. (1968:8-10); Gwaltney (1983: no. 4)
	ICK 1, 124		Hrozný (1952: pl. XCIII)
	JCS 14, 9 (Oxford, Asmoleum)	1933.1044	Kennedy and Garelli (1960: 13-14); Ulshöfer	P361051
			(1995: no. 239)
	KTS 1, 7a	Ka. 139	Lewy (1926: pl. 7)
	KTS 1, 52b	Ka. 2	Lewy (1926: pl. 52); Ulshöfer (1995: no. 242)
	KUG 7	Giessen 3-1	Hecker (1966: 10-12); Eisser and Lewy (1930: no.
			146)
	OIP 27, 21	Alişar no.	Gelb (1935: pl. 15)
		d2200i
	Prag I 471	I 471	Hecker, Kryszat and Matouš (1998: 59-60)	P359081
	Prag I 520	I 520	Hecker, Kryszat and Matouš (1998: 113-115)	P359128
	Prag I 608	I 608	Hecker, Kryszat and Matouš (1998: 195)	P359210
	Sadberk 16 (Istanbul)	ARK. 164-	Donbaz (1999: 91-92)
		9472
	UF 7, 327		Mayer and Wilhelm (1975: 319-320, 327, no. 5)
	TC 1, 80	AO 7081	Contenau (1920: pl. LI); Ulshöfer (1995: no. 258);	P357415
			Larsen (2002: no. 167)
	TC 1, 84	AO 7085	Contenau (1920: pl. LV); Ulshöfer (1995: no. 535)	P357419
	TC 1, 102	AO 7315	Contenau (1920: pl. LXIII); Ulshöfer (1995: no.	P357437
			536)
	TC 2, 52	AO 8729	Thureau-Dangin (1928: pl. XXIX); Ulshöfer	P357542
			(1995: no. 604)
	TC 3, 113	AO 9273	Lewy, J. (1936: pl. CX)	P357679
	TC 3, 171	AO 8632	Lewy, J. (1936: pl. CXLIII); Ulshöfer (1995: no.	P357737
			190)
	TC 3, 183	AO 9348	Lewy, J. (1936: pl. XCLIX); Ulshöfer (1995: no.
			489)
	TC 3, 213	AO 8286	Lewy, J. (1936: pl. CLXXI-CLXXII)	P357779
	VS 26, 119	VAT 13513	Veenhof and Klengel-Brandt (1992: pl. XXX);	P358258
			Eisser and Lewy (1930: no. 226)
	VS 26, 134	VAT 13526	Veenhof and Klengel-Brandt (1992: pl. XXXII)	P358270

All the dates mentioned in this chapter, except bibliographical references and archaeological discoveries, are to be understood as BCE.

For a general overview of the Old Assyrian trade in Anatolia, see[START_REF] Michel | Correspondance des marchands de Kaniš au début du II e millénaire av[END_REF]Michel ( , 2008a) ) andVeenhof (2010a).

For a general bibliography of the Old Assyrian period and a catalogue of the Old Assyrian sources, see[START_REF] Michel | Old Assyrian Bibliography of Cuneiform Texts, Bullae, Seals and the Results of the Excavations at Assur, Kültepe/Kanis, Acemhöyük, Alishar and Bogazköy. Book series: Old Assyrian Archives Studies[END_REF]Michel ( , 2006aMichel ( , 2011a)).

[START_REF] Larsen | Old Assyrian Caravan Procedures[END_REF],Michel (2001: 171-301).

[START_REF] Dercksen | On the financing of Old Assyrian merchants[END_REF];[START_REF] Veenhof | Silver and credit in Old Assyrian trade[END_REF].

[START_REF] Michel | Les fractions dans les tablettes économiques du début du second millénaire en Assyrie et en Babylonie[END_REF].

[START_REF] Michel | Les marchands et les nombres : l'exemple des Assyriens à Kaniš[END_REF].

It is not impossible that the Kaneš sample included a line for 100 gu written literally 1 me-at gu2.

Another fragment of a school text from Kaneš, k/-129, bought on the antiquities market in 1959 by Ankara Museum, contains a list of multiplication formulae written linearly and without the results (a-ra2-1-kam, a-ra2-2-kam, etc.). This is part of a lexical list bearing names of stones, animals, aromatic plants and multiplication formulae(Hecker 1993: 290-291; photo of the obverse: pl. 48-2).

Proust (2008: 145-153).

gu of good quality copper, 10-gin brought back to (each) 10-mana weight, (its) silver (weight) established, and furthermore 14 gu of copper in pieces, 10-gin brought back to (each) 10-mana weight, (its) silver (weight) established.

Kt u/k 3:1-16; text published in photo byÖzgüç (1986: pl. 60.1); edited byDercksen (1996: 87). We know that there were different weight standards linked to the various kingdoms of Upper Mesopotamia, Syria and the Levant; see[START_REF] Ascalone | I pesi da bilancia del bronzo antico e del bronzo medio[END_REF],Chambon (2011: 82-99), and below Sect. 12.3.3.

hundred 40 mana of copper according to the weight of the country (Anatolian weight) 7 hundred 60 mana of copper according to our weight (Mesopotamian weight) instead of using the higher gu unit. A conversion into gu would correspond respectively to:14 gu of copper according to the weight of the country 12 gu and 40 mana of copper according to our weight.Quantities of copper expressed in a decimal system using the mana as unit could reach hundreds, if not thousands of mana. This way of writing was frequently applied to copper in pieces and bad quality copper:2 hundred 60 mana of bad (copper). 72 4 hundred 30 mana of copper.73

OIP 11, 179. For heavier weights corresponding to mana, the list mentions the following weights: 1 -2 -3 -5 -10 -15 - 20 -30 -40 -50.

The research leading to these results was supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No. 269804 'Mathematical Sciences in the Ancient World (SAW)'. I warmly thank F. Kulakoğlu, Director of Kültepe-Kaneš excavations, who gave me all his data concerning the weights that he gathered

Weight

The 10-mana weight attested by the texts has not yet been found during the excavations. The fact that the three heaviest weights correspond to the Anatolian standard fits perfectly well with the need to weigh copper in large quantities in Anatolia.

No weight is attested for the upper unit, the gu, apart from inscriptions of King Erišum: 97 I constructed two beer vats, twins ... I placed two duck [weights] of 1 gu each inside them. 97 Grayson 1987: 20, text A.0.33.1 :12-14, 2 hu-bu-re-en tu3-i-me-e[n], [x (x)] e-pu-uš 2 u2-us2-e-en ša 1 gu2-ta, i-n[a] qar-bešu-nu a-di2.