Aleksandra Petković

Benjamin Reichert

Zoran Ristivojevic

Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity

We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interactioninduced localization. The quantum contribution to the boson density gives rise to small corrections of the bound state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.

I. INTRODUCTION

Boundaries play an important role in one-dimensional quantum liquids affecting, e.g., correlation and response functions, as well as the ground-state energy [START_REF] Eggert | Magnetic impurities in half-integerspin Heisenberg antiferromagnetic chains[END_REF][START_REF] Eggert | Impurities in S = 1 / 2 Heisenberg Antiferromagnetic Chains: Consequences for Neutron Scattering and Knight Shift[END_REF][START_REF] Fabrizio | Interacting one-dimensional electron gas with open boundaries[END_REF][START_REF] Eggert | Boundary Effects on Spectral Properties of Interacting Electrons in One Dimension[END_REF][START_REF] Gaudin | Boundary Energy of a Bose Gas in One Dimension[END_REF][START_REF] Blöte | Conformal Invariance, the Central Charge, and Universal Finite-Size Amplitudes at Criticality[END_REF][START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF][START_REF] Reichert | Exact Results for the Boundary Energy of One-Dimensional Bosons[END_REF]. The boundary has also an impact on the density profile of particles n(x), which is suppressed at the boundary position x = 0. Away from it, the particle density in fermionic systems shows socalled Friedel oscillations [START_REF] Friedel | Metallic alloys[END_REF]. They describe an oscillatory decay of n(x) -n 0 , where n 0 is the average density. The envelop of Friedel oscillations follows 1/x K law for spinless fermions [START_REF] Fabrizio | Interacting one-dimensional electron gas with open boundaries[END_REF][START_REF] Egger | Friedel Oscillations for Interacting Fermions in One Dimension[END_REF]. Here K is the Luttinger liquid parameter, which is determined by the interaction between fermions. The periodicity of oscillations is controlled by the average fermion density.

Friedel oscillations in fermionic systems are studied within the harmonic Tomonaga-Luttinger liquid description [START_REF] Fabrizio | Interacting one-dimensional electron gas with open boundaries[END_REF][START_REF] Egger | Friedel Oscillations for Interacting Fermions in One Dimension[END_REF]. This is the low-energy theory of both, interacting fermions and bosons in one dimension [START_REF] Haldane | Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids[END_REF]. It was applied for bosons in Ref. [START_REF] Cazalilla | Low-energy properties of a one-dimensional system of interacting bosons with boundaries[END_REF] where the same pattern of density oscillations was found for separation longer than the inverse mean density of bosons, 1/n 0 . Applied to the special case of weaklyrepulsive bosons where K 1, the result of Ref. [START_REF] Cazalilla | Low-energy properties of a one-dimensional system of interacting bosons with boundaries[END_REF] implies very rapid saturation of n(x) -n 0 . Such fast recovery of the density is not expected to occur in weakly interacting superfluids where the density should change on the scale comparable to the healing length ξ ∼ K/n 0 , which is much longer than 1/n 0 . Thus, in order to describe the density profile of bosons one needs to go beyond the harmonic Tomonaga-Luttinger liquid theory.

In this paper we study weakly-interacting bosons in a semiinfinite system. We consider the equation of motion for the field operator. At the mean-field (classical) level it reduces to the Gross-Pitaevskii equation [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF]. Its solution reveals that the spatial extent of the depletion of the boson density imposed by the boundary is controlled by the healing length ξ. At distances longer than ξ, the mean-field density exponentially rapidly reaches the constant value n 0 . Accounting for the effect of quantum fluctuations around the mean-field solution within the Bogoliubov-de Gennes formalism, we find that the density reaches n 0 much slower, following 1/x 2 law.

Our approach enables us to obtain an analytic expression for the density profile at all distances.

The nonuniform density profile of the Bose gas determines the potential for a weakly-coupled quantum impurity introduced in the system. For repulsive interaction between the impurity and the bosons, the impurity can be localized. We solve the Schrödinger equation and characterize the impurity by the energy spectrum of the bound states, their wave functions, and the mean position. We find the condition for the appearance of the bound states, which is a threshold for a single dimensionless parameter that involves the masses of the impurity and of the particles of the Bose gas as well as the interaction strengths. We note that a related phenomenon of selftrapping, i.e., the localization of a single impurity in homogeneous Bose-Einstein condensates was studied in Refs. [START_REF] Lee | Polarons and Bose decondensation: A self-trapping approach[END_REF][START_REF] Cucchietti | Strong-Coupling Polarons in Dilute Gas Bose-Einstein Condensates[END_REF][START_REF] Sacha | Self-localized impurities embedded in a one-dimensional Bose-Einstein condensate and their quantum fluctuations[END_REF][START_REF] Kalas | Interaction-induced localization of an impurity in a trapped Bose-Einstein condensate[END_REF][START_REF] Bruderer | Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling[END_REF][START_REF] Ardila | Strong coupling Bose polarons in a two-dimensional gas[END_REF]. Contrary to the case of the self-trapping where a bound state is created due to the significant distortion of the density of the host system due to the coupling with the impurity, in our case it is the boundary that critically modifies the density. Other related phenomena include the localization of bosonic atoms by fermionic ones in attractive Bose-Fermi mixtures examined in Ref. [START_REF] Lühmann | Self-Trapping of Bosons and Fermions in Optical Lattices[END_REF]. The formation of bounds states of two impurities immersed in one-dimensional liquid has been studied recently in Refs. [START_REF] Dehkharghani | Coalescence of Two Impurities in a Trapped One-Dimensional Bose Gas[END_REF][START_REF] Pasek | Induced pairing of fermionic impurities in a one-dimensional strongly correlated Bose gas[END_REF][START_REF] Huber | In-medium bound states of two bosonic impurities in a one-dimensional Fermi gas[END_REF], while the higher-dimensional cases are studied, e.g., in Ref. [START_REF] Naidon | Two Impurities in a Bose-Einstein Condensate: From Yukawa to Efimov Attracted Polarons[END_REF][START_REF] Panochko | Two-body effective potential between impurities in ideal BEC[END_REF][START_REF] Camacho-Guardian | Bipolarons in a Bose-Einstein Condensate[END_REF]. We eventually mention the study of a ionic impurity in a condensate [START_REF] Astrakharchik | Ionic polaron in a Bose-Einstein condensate[END_REF].

The paper is organized as follows. In Sec. II we introduce the model of interacting bosons in a semi-infinite geometry. In Sec. III we solve the equation of motion for the single-particle field operator. We find the mean-field solution and the first two quantum corrections. This enables us to evaluate the density of bosons, including its quantum contribution, in Sec. IV. In Sec. V we consider the problem of a mobile quantum impurity interacting with the system. We solve the Schrödinger equation for the impurity and study its properties. Section VI is devoted to discussions, while in Sec. VII we summarize our results. In Appendix A we present a simplified procedure that leads to the density in the regime of large separations from the boundary.

II. THE SEMI-INFINITE BOSE GAS

We are interested in the influence of the boundary at x = 0 on the physical quantities in a one-dimensional system of interacting bosons. We thus study a long system described by the Hamiltonian

H = L 0 dx -Ψ † 2 ∂ 2 x 2m Ψ + g 2 Ψ † Ψ † Ψ Ψ . ( 1
)
Here m is the mass of bosons, while the coupling constant g > 0 describes the repulsive contact interaction between them.

The system size is L; however we will eventually consider the thermodynamic limit. The bosonic single particle operators Ψ and Ψ † satisfy the usual equal time commutation relations

[ Ψ(x, t), Ψ † (x , t)] = δ(x -x )
, while the other commutators vanish. The model ( 1) is characterized by the dimensionless parameter [START_REF] Lieb | Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State[END_REF] γ = mg/ 2 n 0 , where n 0 is the mean particle density. The boundary of the system imposes the nullification of the single-particle operator at its position,

Ψ(x = 0, t) = 0. (2)
The condition [START_REF] Eggert | Impurities in S = 1 / 2 Heisenberg Antiferromagnetic Chains: Consequences for Neutron Scattering and Knight Shift[END_REF] implies that the boson density also vanishes at x = 0. Our first goal in this work is to calculate the density profile of the bosons

n(x) = Ψ † (x) Ψ(x) , (3)
where the average is with respect to the ground state of the Hamiltonian [START_REF] Eggert | Magnetic impurities in half-integerspin Heisenberg antiferromagnetic chains[END_REF]. For simplicity, we introduce the dimensionless coordinates for the position and the time, respectively, defined as

X = x ξ µ , T = t /µ . ( 4
)
Here ξ µ = / √ mµ denotes the healing length, while µ is the chemical potential. Assuming the single particle operator in the form

Ψ(x, t) = µ g ψ(X, T )e -iT , (5)
its equation of motion i ∂ t Ψ = [ Ψ, H] in the dimensionless units becomes i∂ T ψ(X, T ) = - ∂ 2 X 2 -1 + ψ † (X, T ) ψ(X, T ) ψ(X, T ). (6)
We will solve Eq. ( 6) at γ 1, corresponding to the regime of weak interaction. In this case, one can expand the field operator as [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF][START_REF] Sykes | Drag Force on an Impurity below the Superfluid Critical Velocity in a Quasi-One-Dimensional Bose-Einstein Condensate[END_REF] ψ(X, T ) = ψ 0 (X) + α ψ1 (X, T ) + α 2 ψ2 (X, T ) + . . . , [START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF] such that [ ψ(X, T ), ψ † (X , T )] = α 2 δ(X -X ). Here the small parameter is α = (γgn 0 /µ) 1/4 ≈ γ 1/4 1. In the latter estimate we used the expression µ = gn 0 , which is valid at weak interaction, γ 1. The function ψ 0 (X) describes the time-independent wave function of the system in the absence of fluctuations. The field operators ψ1 and ψ2 account for its first and the second quantum correction. Note that the Bose-Einstein condensate does not exist in one dimension in the thermodynamic limit due to strong effect of long-wavelength fluctuations. However, in finite-size systems, the inverse system size provides an infrared cutoff. The perturbative expansion [START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF] is justified as long as [START_REF] Sykes | Drag Force on an Impurity below the Superfluid Critical Velocity in a Quasi-One-Dimensional Bose-Einstein Condensate[END_REF][START_REF] Petrov | Lowdimensional trapped gases[END_REF] ln(L/ξ µ ) 1/ √ γ where L is the length of the system. The latter inequality shows that for weakly interaction bosons [i.e., at γ 1] the system size can actually be huge. We point out that the density n(x) [see Eq. ( 3)] which is to be calculated, is well defined (i.e., cutoff-independent) in the thermodynamic limit [START_REF] Reichert | The Casimirlike effect in a one-dimensional Bose gas[END_REF].

III. SOLUTION OF THE EQUATION OF MOTION A. Wave function in the absence of fluctuations

The form (7) substituted into Eq. ( 6) leads to a hierarchy of equations controlled by the small parameter α 1. The equation of motion for ψ 0 (X) is obtained at order α 0 . It reads

L 1 (X)ψ 0 (X) = 0, (8)
where we have introduced the operator

L j (X) = - ∂ 2 X 2 + j|ψ 0 (X)| 2 -1. (9)
The expression [START_REF] Reichert | Exact Results for the Boundary Energy of One-Dimensional Bosons[END_REF] is known as the Gross-Pitaevskii equation [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF]. This second order differential equation should be supplemented by two boundary conditions. One of them follows from Eq. ( 2) and becomes ψ 0 (0) = 0. The other condition arises from the physical requirement that the density, which is proportional to |ψ 0 (X)| 2 , is unaffected by the boundary at long separations from it and thus becomes a constant. The real solution [32] of Eq. ( 8) satisfying such boundary conditions is

ψ 0 (X) = tanh X. ( 10
)
At weak interaction the chemical potential of the Bose gas is µ = gn 0 . This gives the density (3) at the mean-field level to be

n(x) = n 0 tanh 2 (x/ξ), ξ = 1/n 0 √ γ. ( 11
)
This expression shows that the density quickly saturates at distances beyond the healing length ξ (which is ξ µ in the limit γ 1), provided one takes only the leading order term from the expansion [START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF].

B. First quantum correction

We now consider the effects of quantum fluctuations and determine ψ1 . Its equation of motion is

i∂ T ψ1 (X, T ) = L 2 (X) ψ1 (X, T ) + ψ 0 (X) 2 ψ † 1 (X, T ). (12)
We seek a solution for ψ1 (X, T ) as a superposition of excitations of energy k using the ansatz based on Bogoliubov transformation [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF] ψ1 (X,

T ) = k>0 N k u k (X) bk e -i k T -v * k (X) b † k e i k T . (13)
Since Eq. ( 12) is linear and homogeneous, we must account for the normalization factor N k in Eq. ( 13), which should be determined in such a way to satisfy the proper commutation relations between ψ1 fields. This is discussed further below.

The bosonic operators bk and b † k satisfy the standard commutation relations

[ bk , b † k ] = δ k,k and [ bk , bk ] = 0. The bound- ary condition (2) at order α is given in terms of u k and v k by u k (0) = v k (0) = 0. (14)
Substitution of Eq. ( 13) into Eq. ( 12) leads to two coupled equations for u k and v k . They are known as the Bogoliubovde Gennes equations and are given by

k u k (X) = L 2 (X)u k (X) -ψ 2 0 (X)v k (X), (15a)
-k v k (X) = L 2 (X)v k (X) -ψ 2 0 (X)u k (X). (15b)
In order to simplify them, we introduce the functions

S(k, X) = u k (X) + v k (X) and D(k, X) = u k (X) -v k (X).
This enables us to decouple the equations and lead to the fourth order differential equation

2 k S(k, X) = L 3 (X) L 1 (X)S(k, X). (16a)
Note that D is given in terms of S as

D(k, X) = 1 k L 1 (X)S(k, X). (16b)
Four independent solutions of the forth-order equations ( 16) are [START_REF] Kovrizhin | Exact form of the Bogoliubov excitations in one-dimensional nonlinear Schrödinger equation[END_REF] S

n (X) = (-ik n + 2 tanh X)e iknX , (17a)
D n (X) = - ik n kn 1 cosh 2 X e iknX + k 2 n 2 kn S n (X), (17b)
where n ∈ {1, 2, 3, 4}, while k = k 2 + k 4 /4 is the energy dispersion. The four roots entering S n are k 1,2 = ±k and

k 3,4 = ±i √ 4 + k 2 in terms of k = √ 2 2 k + 1 -1.
The general solution of Eqs. ( 16) is a linear combination

S(k, X) = AS 1 (k, X) + S 2 (k, X) + BS 3 (k, X), (18a) D(k, X) = AD 1 (k, X) + D 2 (k, X) + BD 3 (k, X). (18b)
Note that S 4 and D 4 do not appear since they diverge at large X. The two unknown coefficient A and B are determined using the boundary condition ( 14) that is terms of S and D becomes S(k, 0) = D(k, 0) = 0. We find A = 1 and B = 0. Therefore the general solution of Eqs. ( 15) that satisfy the boundary condition ( 14) are the real functions

u k (X) = k 1 + k 2 + 2 cosh -2 X 2 k sin(kX) + 2 1 + k 2 2 k cos(kX) tanh X, (19a)
v k (X) = k 1 - k 2 + 2 cosh -2 X 2 k sin(kX) + 2 1 - k 2 2 k cos(kX) tanh X. (19b)
The normalization in Eq. ( 13) is obtained by requiring [13]

N k N q L ξµ 0 dX[u k (X)u q (X) -v k (X)v q (X)] = δ k,q . (20)
This leads to

N k = (ξ µ /4L k ) 1/2 at L ξ µ . One can then verify the equal time commutation relation [ ψ1 (X), ψ1 (Y )] = 0. The evaluation of the other commutation relation is more involved: [ ψ1 (X), ψ † 1 (Y )] = k>0 N 2 k [u k (X)u k (Y ) -v k (X)v k (Y )] = ξ µ L k>0 [cos (k(X -Y )) -cos (k(X + Y ))] = δ(X -Y ), (21)
since δ(X + Y ) always equals zero for X, Y > 0. The second equality in Eq. ( 21) is obtained after performing the integral over k of the function

[u k (X)u k (Y ) -v k (X)v k (Y )]/4 k - 2
sin(kX) sin(kY ), which turns out to be zero. The remaining part after integration over k then gives the delta function in Eq. ( 21). There we use k>0 (

• • • ) = (L/πξ µ ) ∞ 0 dk(• • • ).

C. Second quantum correction

The first quantum contribution to the density (3) is proportional to α 2 and thus is determined by the first two corrections of the field operator in Eq. [START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF]. We thus now consider the second quantum correction to the field operator ψ denoted by ψ2 . Its equation of motion is obtained from Eq. ( 6) at order α 2 . Since ψ 0 (X) is real [see Eq. ( 10)], it is sufficient to consider the real part of the expectation value ψ2 which enters into n(x). For this purpose we introduce the notation ψ 2 = Re ψ2 . The equation of motion for ψ 2 is

L 3 (X)ψ 2 (X) = f (X), (22)
where

f (X) = -2ψ 0 ψ † 1 ψ 1 -ψ 0 ψ 2 1 . (23)
At zero temperature one has

ψ † 1 ψ 1 = k>λ N 2 k v 2 k , ψ 2 1 = - k>λ N 2 k u k v k . ( 24
)
Note that the source term f (X) in Eq. ( 22) is timeindependent and thus ψ 2 is only a function of X. We note that evaluation of f (X) requires a small-k cutoff (λ

∼ ξ µ /L) since it is divergent at k → 0.
The solution of the linear equation ( 22) can be expressed as

ψ 2 (X) = ∞ 0 dY G(X, Y )f (Y ), ( 25
)
where G is the Green's function of the operator L 3 (X). It satisfies

L 3 (X)G(X, Y ) = δ(X -Y ). (26)
Moreover, the Green's function is symmetric G(X, Y ) = G(Y, X) and satisfies

G(Y + , Y ) = G(Y -, Y ), (27a)
∂ X G(X = Y + , Y ) -∂ X G(X = Y -, Y ) = -2, ( 27b
)
to account for the δ(X -Y ) on the right hand side of Eq. ( 26).

Here Y ± = lim δ→0 + Y ± δ. Also, the Green's function obeys G(0, Y ) = 0 due to the boundary condition imposed by the end of the system, i.e., ψ 2 (0) = 0. For the solution of Eq. ( 26) we find

G(X, Y ) = g(Y )θ(X -Y ) + g(X)θ(Y -X) cosh 2 X cosh 2 Y , (28a)
where

g(Y ) = 12Y + 8 sinh(2Y ) + sinh(4Y ) 16 . (28b)
Therefore, the solution of Eq. ( 22) for ψ 2 (X) is obtained from Eq. ( 25), where one should substitute the Green's function and the source function [START_REF] Huber | In-medium bound states of two bosonic impurities in a one-dimensional Fermi gas[END_REF]. Since the latter depends on the infrared cutoff, ψ 2 has also this feature. In Appendix A we derive ψ 2 (X) in the regime X 1.

IV. LOCAL DENSITY

Having solved the equation of motion for the terms ψ 0 , ψ1 , and ψ2 of the single-particle operator [START_REF] Alcaraz | Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models[END_REF], we are now in position to evaluate the spatial density profile of the semi-infinite Bose gas. It is given by the expansion

n(x) = µ g n (0) (X) + α 2 n (1) (X) + O(α 3 ) X= x ξµ . (29)
Here the mean-field contribution is

n (0) (X) = |ψ 0 (X)| 2 = tanh 2 X, (30)
while the quantum contribution to the density has the form

n (1) (X) = ψ † 1 (X, T ) ψ1 (X, T ) + 2ψ 0 (X)ψ 2 (X), (31)
or more explicitly

n (1) (X) = ∞ 0 dk 4π k v k (X) 2 -2 tanh X × ∞ 0 dY G(X, Y )v k (Y ) [2v k (Y ) -u k (Y )] tanh Y . (32)
In Eq. (32) one should use Eqs. ( 19) and ( 28), while we recall k = k 2 + k 4 /4. Unlike ψ 2 that requires an infrared cutoff, the density contribution (32) does not. Each of the two terms in Eq. ( 31) is divergent, however their sum is finite. This should be the case since the density fluctuations are expected to be finite unlike the fluctuations of the phase of the single particle operator [34].

The evaluation of Eq. ( 32) is rather tedious. We first perform the integration over Y using various trigonometric identities and the integration by parts. As a result, we obtain a cumbersome expression involving several hypergeometric functions. Such expression is an integrable function of k.

To perform the integration we split the integrand into several summands. We should stress that, unlike the whole integrand, the summands can be nonintegrable due to singularities and one should regularize them by adding and subtracting some functions. As a result of such procedure, we obtain the local density that can be expressed in the form

n(x) = µ g tanh 2 (x/ξ µ ) + α 2 tanh 2 (x/ξ µ ) π + (x/ξ µ ) tanh(x/ξ µ ) π cosh 2 (x/ξ µ ) + h(x/ξ µ ) , (33)
where ξ µ = / √ mµ and

h(z) = 2 π - 3 -z tanh z π cosh 2 z + πz(2 -cosh 2z) 8 cosh 4 z 8z π 2 2 F 3 1, 1; 1/2, 3/2, 2; 4z 2 -I 1 (4z)L 0 (4z) + I 0 (4z)L 1 (4z) + tanh z(1 -4z tanh z) [I 0 (4z) -L 0 (4z)] - 1 2 [1 -z tanh z(3 + 5 tanh 2 z)] [I 1 (4z) -L -1 (4z)] . (34)
Here 2 F 3 is the hypergeometric function while I ν (z) and L ν (z) are the modified Bessel function of the first kind and the modified Struve function, respectively. The density [START_REF] Kovrizhin | Exact form of the Bogoliubov excitations in one-dimensional nonlinear Schrödinger equation[END_REF] is obtained for a fixed value of the chemical potential. We eliminate µ by using the mean density n 0 that is given by n 0 = L 0 dx n(x)/L. In the thermodynamic limit, the mean density of the Bose gas is n 0 = (µ/g)(1 + α 2 /π + . . .). Inverting this relation, one obtains the chemical potential as a function of the mean density [START_REF] Lieb | Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State[END_REF], µ = gn 0 (1 -√ γ/π + . . .), where γ = mg/ 2 n 0 1. Substituting µ as a function of n 0 in Eq. ( 29) yields the density profile of the weakly-interacting semi-infinite Bose gas

n(x) = n 0 tanh 2 (x/ξ) + √ γh(x/ξ) , (35)
where ξ = 1/n 0 √ γ, while h is given by Eq. (34).

We now analyze the behavior of the local density [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF]. At short separations, x ξ, the quantum fluctuation contributions to the density can be neglected and the leading contribution is given by

n(x) = n 0 tanh 2 (x/ξ). (36)
This classical result originates from the mean-field contribution [START_REF] Petrov | Lowdimensional trapped gases[END_REF] to the density. At longer separations, x ξ, the classical result [START_REF] Pöschl | Bemerkungen zur Quantenmechanik des anharmonischen Oszillators[END_REF] quickly saturates to a constant n 0 . However, the approach of the local density toward n 0 is actually much slower than one obtains from the mean-field result [START_REF] Pöschl | Bemerkungen zur Quantenmechanik des anharmonischen Oszillators[END_REF]. At x ξ the quantum contribution [START_REF] Reichert | The Casimirlike effect in a one-dimensional Bose gas[END_REF] is the dominant term in the density deviation n(x)-n 0 since it decays algebraically with the distance. This follows from the asymptotic expansion

h(z) = - 1 16πz 2 1 + 1 z + 15 16z 2 + O(z -3 ) . (37)
Therefore, at x ξ the leading correction to the local density is given by

n(x) = n 0 1 - √ γ 16π ξ 2 x 2 . ( 38
)
The crossover distance x c between the two regimes can be estimated by equating the two expressions [START_REF] Pöschl | Bemerkungen zur Quantenmechanik des anharmonischen Oszillators[END_REF] and [START_REF]Unlike the potential -λ(λ -1)/ cosh 2 z that has bound states whenever is negative[END_REF]. As a result we get

x c ≈ ξ ln 8 √ π γ 1/4 . ( 39
)
Note that the crossover distance depends only logarithmically on γ and practically is on the order of few ξ. This is illustrated in Fig. 1 where the density profile of the Bose gas is shown as a function of the distance from the end of the system at several values of γ. The density in the crossover regime between the classical and the quantum one is given by Eq. [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF]. A simplified derivation of n (1) (X) at X 1 is given in Appendix A. We finally notice that the ground-state energy calculation of Ref. [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF] leads to the limiting forms [START_REF] Pöschl | Bemerkungen zur Quantenmechanik des anharmonischen Oszillators[END_REF] and [START_REF]Unlike the potential -λ(λ -1)/ cosh 2 z that has bound states whenever is negative[END_REF].

V. INTERACTION-INDUCED LOCALIZATION OF A SINGLE IMPURITY

In this section we study the problem of a quantum impurity in the semi-infinite Bose gas. We consider the impurity that is locally coupled to the boson density [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF]. In the case of repulsion, the density profile of the Bose gas forms an attractive potential for the impurity, which can lead to the bound states in the spectrum. In that case the particle becomes localized by the surrounding medium.

In the case of a weakly coupled impurity, one can find its wave function in the approximation of unperturbed Bose gas density (as further discussed in Sec. VI). The impurity wave function is a function of x ant t and is governed by the Schrödinger equation. After separation of variables it reduces to the eigenvalue problem on the positive semi-axis x > 0

- 2 2M d 2 dx 2 + Gn(x) -E ψ imp (x) = 0, (40a)
ψ imp (0) = 0. ( 40b
)
Here M is the impurity mass, G > 0 denotes the coupling constant, while the density of the Bose gas n(x) is given by Eq. ( 35). An alternative formulation of the eigenvalue problem ( 40) is to consider unrestricted x with the symmetric potential Gn(|x|), and to account only for odd eigenfunctions. They have a node at x = 0 and thus automatically satisfy the boundary condition (40b).

In order to simplify the notations, we rewrite Eq. ( 40) in the form

- d 2 dz 2 - λ(λ -1) cosh 2 z + √ γλ(λ -1)h(z) + κ 2 f (z) = 0, (41a)
f (0) = 0, (41b)
where we have introduced

f (z) = ψ imp (zξ), (42)
λ = 1 2 + 1 2 1 + 8GM gm , (43)
κ 2 = 2GM gm 1 - E Gn 0 . (44)
In the following we will find the bound state spectrum for the eigenvalue problem [START_REF] Schecter | Phonon-Mediated Casimir Interaction between Mobile Impurities in One-Dimensional Quantum Liquids[END_REF].

A. Bound states of Pöschl-Teller potential

In the limit where the quantum correction to the density h(z) is neglected, the potential of Eq. (41a) describes a hole of modified Pöschl-Teller type which admits an exact solution [START_REF] Pöschl | Bemerkungen zur Quantenmechanik des anharmonischen Oszillators[END_REF][START_REF] Flügge | Practical Quantum Mechanics[END_REF]. We now derive the bound states for this special eigenproblem defined by

- d 2 dz 2 - λ(λ -1) cosh 2 z + κ 2 f (z) = 0, (45a)
f (0) = 0. ( 45b
)
From the definition (43) follows λ > 1 and thus the potential -λ(λ -1)/ cosh 2 z in Eq. (45a) is negative. We also notice that for unrestricted z, the eigenstates in such symmetric potential can be even or odd functions of z. For the problem [START_REF] Edri | Observation of Spin-Spin Fermion-Mediated Interactions between Ultracold Bosons[END_REF] only odd solutions have a node at z = 0. They are of the form [START_REF] Flügge | Practical Quantum Mechanics[END_REF] f (z) = cosh λ z sinh z

× 2 F 1 λ -κ + 1 2 , λ + κ + 1 2 ; 3 2 ; -sinh 2 z , (46)
where 2 F 1 is the Gauss's hypergeometric function.

In order to discuss the appearance of bound states, we consider the asymptotic expansion of Eq. ( 46) in the limit z → ∞, which is given by

f (z) 2 -κ Γ(κ)Γ(3/2)e κz Γ λ+κ+1 2 Γ 2-λ+κ 2 + 2 κ Γ(-κ)Γ(3/2)e -κz Γ λ-κ+1 2 Γ 2-λ-κ 2 . ( 47
)
We notice that there are two exponential terms ∝ e ±κz at z → ∞. Taking into account the definition of κ given by Eq. ( 44), we conclude that a bound state can be realized only if κ 2 > 0, i.e., if Gn 0 > E. In the opposite case, Gn 0 < E, the solution at long separations from the boundary is an oscillating trigonometric function since κ is purely imaginary. In the following we focus on the localized (bound) states and derive their energy spectrum. We assume without loss of generality that κ > 0. Thus, the coefficient in front of the term e κz must vanish in order to obtain a nondivergent (normalizable) wave function at large z. This occurs when the arguments of the Gamma functions in the denominators are 0 or negative integers. Since λ > 1 and by assumption κ > 0, only the argument of Γ 2-λ+κ 2 matters. This yields

κ = λ -2 -2η, (48)
where η = 0, 1, 2, . . .. From the condition κ > 0 we obtain that integer η satisfies

0 ≤ η < λ 2 -1. (49)
Equation ( 49) determines the condition λ > 2 necessary for the appearance of the first bound state [START_REF]Unlike the potential -λ(λ -1)/ cosh 2 z that has bound states whenever is negative[END_REF]. The number η denotes the number of nodes of the odd bound-state eigenfunction in the region of interest z > 0. Thus, the localized solutions of the eigenvalue problem Eq. ( 45) are given by Eq. ( 46) where κ satisfies Eq. ( 48), while η is a nonnegative integer satisfying the condition [START_REF] Gaunt | Bose-Einstein Condensation of Atoms in a Uniform Potential[END_REF]. The latter enables us to reexpress the eigenfunction [START_REF] Cheon | Fermion-Boson Duality of One-Dimensional Quantum Particles with Generalized Contact Interactions[END_REF] in the form

f η (z) = η! Γ(λ -η -1) sinh z cosh λ-1 z × η j=0 (-4) j sinh 2j z (2j + 1)! (η -j)! Γ(λ -η -1 -j) ( 50
)
using 2 F 1 (c -a, c -b; c; z) = (1 -z) a+b-c
2 F 1 (a, b; c; z). The preceding expressions enable us to find the spectrum of bound states of the impurity defined by Eq. ( 40) in the case when the quantum fluctuations of the density are neglected. The eigenfunctions are ψ imp,η (x) = f η (x/ξ) [see Eq. ( 50)], where one should substitute Eq. ( 43) for λ. The corresponding energies [cf. Eq. [START_REF] Hao | Ground-state properties of one-dimensional ultracold Bose gases in a hardwall trap[END_REF]] are

E η = gn 0 m M 3 4 + η 1 + 8GM gm - 5 4 -3η -2η 2 , (51)
provided the system has bound states. The condition for that is sufficiently strong coupling between the impurity and the Bose gas, which should satisfy

GM gm > (η + 1)(2η + 1). (52)
The latter follows from Eq. ( 49). The set of allowed values for η (non-negative integers) in the spectrum (51) are determined by the inequality [START_REF] Garratt | From single-particle excitations to sound waves in a box-trapped atomic Bose-Einstein condensate[END_REF]. In particular, the first bound state (corresponding to η = 0) occurs for GM /gm > 1. The second one (corresponding to η = 1) exists for GM /gm > 6, etc.

We introduce an important quantity

E b η = Gn 0 -E η , (53)
which denotes the binding energy, E b η > 0. In Fig. 2 we show the bound state energies for the first five levels. We notice that the specific bound state energy expressed in units of Gn 0 , which is the value of the potential at infinity, is a function of the single parameter GM/gm. In Fig. 3 we show the normalized wave functions for the bound state levels for the specific value GM/gm = 30. We notice that the wave functions for energies near the top of the potential, Gn 0 , are weakly localized as their spatial extension increases. The level (quantum) number η corresponds to the number of nodes of the wave function. For a given wave function, its spatial extension decreases with increasing GM/gm, since in that case the potential becomes deeper. This is reflected in the mean distance of the particle from the origin,

x η = ξ ∞ 0 dxx|f η (x)| 2 ∞ 0 dx|f η (x)| 2 . ( 54
)
We plot this dependence in Fig. 4. Close to the threshold for the appearance of the bound state, the mean distance diverges according to the law

x η ξ = 1 2κ = η + 3 4 GM gm -(η + 1)(2η + 1) + O(1). (55)
The denominator of Eq. (55) denotes the distance from the threshold for the appearance of the bound state f η measured from the "localized" side where the particle is in the bound state. Equation (55) can be also expressed as function of the binding energy x η /ξ = mµ/8M E b η using Eqs. ( 44) and [START_REF] Essin | Quantum mechanics of the 1/x 2 potential[END_REF]. Indeed, x η determines the localization length. This easily follows from the asymptotic expansion of the wave function f η (z) ∼ exp (-κz) [cf. Eq. ( 47)], where the localization length is 1/κ close to the "transition". The localization length diverges with the first power of the distance from the "transition", which physically denotes the disappearance of the bound state as GM/gm is decreased (and the binding energy approaches zero). More generally, the universal form of the wave function enables us to find any moment as x j η /ξ j = j!/(2κ) j close to the threshold. In the preceding part we considered a localized impurity, i.e., its bound states which exist when the eigenenergies are smaller than Gn 0 (corresponding to the value of the potential at infinity). We should stress that the impurity has a continuum of scattering states with energies higher than Gn 0 . They are characterized by the oscillating wave functions [cf. Eq. ( 47)].

B. Quantum correction to the potential

In the previous subsection we have solved the eigenproblem for the impurity using the mean-field contribution to the po-Figure 2. The first five bound state energies Eη/Gn0 as a function of the dimensionless parameter GM/gm. They occur, respectively, at ratios GM/gm = 1, 6, 15, 28, 45. Notice that the energies start from Gn0, which is the value of the potential at infinity. Figure 3. Normalized wave functions for the first four states at GM/gm = 30. Notice that the wave function corresponding to η = 3 is spatially less localized since the value of GM/gm is close to the threshold for its appearance. 4. Mean distance x η /ξ of the impurity from the origin for the first four bound states as a function of GM/gm. At the threshold value for a given η, the mean distance diverges according to Eq. ( 55), signaling the disappearance of the bound state. tential. We now account for the quantum correction h(z), see Eq. [START_REF] Schecter | Phonon-Mediated Casimir Interaction between Mobile Impurities in One-Dimensional Quantum Liquids[END_REF]. We were not able to solve analytically the whole eigenproblem, since the form of h(z) is very complicated. However we can take advantage of the small parameter √ γ that controls it and find the corrections of the bound state energy (51) using perturbation theory. The energy correction is given by

δE η = √ γ Gn 0 ∞ 0 dx|f η (x)| 2 h(x) ∞ 0 dx|f η (x)| 2 , ( 56
)
where h is defined in Eq. (34), while f η by Eq. [START_REF] Chomaz | Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas[END_REF]. The quantum correction √ γ Gn 0 h(x/ξ) to the effective potential [see, e.g., Eq. (40a)] becomes practically important at distances x ξ (see Fig. 1) with respect to the classical one -Gn 0 / cosh 2 (x/ξ). Contrary to the function h(x/ξ) that has a long-range tail ∝ -1/x 2 , the wave function is localized and exponentially small at x ξ. As a result the numerator in Eq. ( 56) is small thus the quantum correction to the bound state energy is negligible. Notice that very near the threshold for the bound state appearance (see, e.g., the curve for η = 3 in Fig. 3) the wave function may have some overlap with the tail of h(x/ξ), and the correction δE η can be somewhat important with respect to E η -Gn 0 . Nevertheless, δE η is negative since the quantum correction to the potential broadens the well. The quantum corrections to the different energy levels of the bound states are numerically evaluated using Eq. ( 56) and shown in Fig. 5 [39].

VI. DISCUSSION

We have calculated the bound states of the impurity in a semi-infinite Bose gas. We assumed that the density of the Bose gas it not affected by the presence of the impurity [cf. Eq. [START_REF] Recati | Casimir forces between defects in one-dimensional quantum liquids[END_REF]]. This approach is valid in cases of weak coupling G between the impurity and the particles of the Bose gas. A more general model would be the Hamiltonian (1) sup-plemented by the impurity part

L 0 dx -Ψimp 2 ∂ 2 x 2M Ψimp + G Ψ † imp Ψimp Ψ † Ψ . (57)
From the solution of the whole problem one could obtain the spectrum of the bound states at any G. However this is in practice difficult to do analytically. The parameter region where our approach applies can be obtained by calculating the correction to the Bose gas density [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF] due to the coupling with the impurity. For a heavy impurity such correction is small at G g/ √ γ [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF]. In this regime is therefore justified to study the simplified problem [START_REF] Edri | Observation of Spin-Spin Fermion-Mediated Interactions between Ultracold Bosons[END_REF]. Notice that for weakly interacting bosons we have γ 1 and thus our work covers a big range of values for G.

The quantum contribution to the boson density that universally behaves as 1/x 2 at x ξ [see, e.g., Eq. ( 38)] leads to a small energy correction to the bound state levels of the localized particle. However, the knowledge of such spatial dependence of the density of particles far from the boundary is fundamentally important for quantities where the density gradient matters. An example is the Casimir-like interaction [START_REF] Reichert | The Casimirlike effect in a one-dimensional Bose gas[END_REF][START_REF] Recati | Casimir forces between defects in one-dimensional quantum liquids[END_REF][START_REF] Schecter | Phonon-Mediated Casimir Interaction between Mobile Impurities in One-Dimensional Quantum Liquids[END_REF][START_REF] Reichert | Field-theoretical approach to the Casimir-like interaction in a one-dimensional Bose gas[END_REF][START_REF] Pavlov | Phononmediated Casimir interaction between finite-mass impurities[END_REF] between the impurity and the boundary of the system mediated by density fluctuations of the medium. In the regime x ξ its form was found in Ref. [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF], while our result [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF] determines the interaction law at all distances. It takes the form

U (x) = G[n(x) -n 0 ] = -Gn 0 1 cosh 2 (x/ξ) - √ γh(x/ξ) . (58)
It is expected that under the influence of the long-range potential (58), a heavy impurity immersed in the system far from the boundary slowly drifts towards it and eventually get localized. During this process, the impurity energy that equals Gn 0 at long distances is dissipated by exciting the Bose gas.

The boundary in our system can exist naturally as the system's end. Alternatively, it can be created by a heavy impurity strongly coupled to the Bose gas that creates impenetrable potential and leads to the complete depletion of the boson density at its position. Thus the effective interaction between the boundary and the impurity can also be interpreted as the Casimir-like interaction between two very different impurities. We notice two related very recent experimenal works [START_REF] Desalvo | Observation of fermion-mediated interactions between bosonic atoms[END_REF][START_REF] Edri | Observation of Spin-Spin Fermion-Mediated Interactions between Ultracold Bosons[END_REF], which have demonstrated the existence of the induced interaction between quantum particles mediated by the surrounding quantum gas.

The Hamiltonian (1) can be studied within the dual model of attractive fermions [START_REF] Cheon | Fermion-Boson Duality of One-Dimensional Quantum Particles with Generalized Contact Interactions[END_REF][START_REF] Yukalov | Fermi-Bose mapping for onedimensional Bose gases[END_REF]. The density of bosons in the original model is equal to the density of fermions in the dual one. Such mapping is particularly useful in the regime of strong interaction between bosons, corresponding to γ 1 and the Luttinger liquid parameter K 1 + 4/γ. In this case the fermions are weakly-attractive, which enables one to study their density, obtain the characteristic form with Friedel oscillations and go beyond the Tomonaga-Luttinger liquid description. It is interesting to study the density and the fate of Fridel oscillations as the value of K increases and find a connection with our result [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF]. A numerical study of finite number of bosons [START_REF] Hao | Ground-state properties of one-dimensional ultracold Bose gases in a hardwall trap[END_REF] also suggests the above picture. Note that the model ( 1) is integrable in the box geometry [START_REF] Gaudin | Boundary Energy of a Bose Gas in One Dimension[END_REF] and thus it is in principle possible to use the exact Bethe ansatz solution to find the exact density profile in the thermodynamic limit and answer the above question. We are not aware of such study in the literature.

Recent experimental realizations of cold bosons trapped in a box potential [START_REF] Gaunt | Bose-Einstein Condensation of Atoms in a Uniform Potential[END_REF][START_REF] Chomaz | Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas[END_REF][START_REF] Rauer | Recurrences in an isolated quantum manybody system[END_REF][START_REF] Garratt | From single-particle excitations to sound waves in a box-trapped atomic Bose-Einstein condensate[END_REF] provide a possibility to observe our results. In order to see the characteristic long-range tail of the density [cf. Eq. ( 38)], the system length has to be much longer than the crossover distance (39). In addition, the constraint on the temperature comes from the fact that the thermal length T = v/2πT has to be longer than the the crossover distance (39). Here v = gn 0 /m denotes the sound velocity of weakly interacting bosons. Thus we obtain T /µ log (γ -1 ). Concerning the observation of the bound states, the condition (52) could be realized for a given mass ratio M/m by tuning the interaction strengths using the Feshbach resonance. Temperature should be sufficiently small to avoid smearing of the energy levels.

VII. SUMMARY

In this paper we first calculated the density profile of the weakly-interacting semi-infinite one-dimensional Bose gas. We solved the equation of motion for the field operator. At the mean-field level, it leads to the density profile [START_REF] Haldane | Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids[END_REF]. Taking into account the effect of quantum fluctuations around the mean-field solution we calculated the quantum contribution to the density, see Eqs. (34) and [START_REF] Reichert | Fluctuationinduced potential for an impurity in a semi-infinite onedimensional Bose gas[END_REF]. It shows the universal 1/x 2 behavior at long distances.

We then studied the spectrum of the bound states of a quantum impurity which experiences the potential created by the surrounding Bose gas. We were able to exactly solve this problem accounting for the mean-field boson density. The discrete spectrum is given by Eq. ( 51), while the corresponding wave function are written in Eq. [START_REF] Chomaz | Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas[END_REF]. Bound states exist for sufficiently strong repulsion between the impurity and particles of the Bose gas, see Eq. [START_REF] Garratt | From single-particle excitations to sound waves in a box-trapped atomic Bose-Einstein condensate[END_REF]. We also found how the mean particle distance from the boundary behaves, and in particular its divergence (55) as one approaches the threshold for the appearance of the bound state levels. We showed that the quantum contribution to the density gives rise to the negligible corrections of the bound state levels. However, the quantum contribution to the density is important since it leads to the long-range Casimir-like interaction between the impurity and the boundary.

that are negligible at X 1, the last two terms in Eq. (A5) correspond to the second line of Eq. (34) taken at large argu-ment. They lead to the power-law decay of the density correction at long separations [cf. Eq. ( 37)].

Figure 1 .

 1 Figure 1. Density deviation from the mean density [n0 -n(x)]/n0 of the Bose gas for three values of the dimensionless interaction parameter γ. The dashed curve denotes the mean-field result.

Figure 5 .

 5 Figure 5.Quantum correction to the bound state energies δEη/ √ γ Gn0 for different η as a function of GM/gm for the first five levels.

ACKNOWLEDGMENTS

This study has been partially supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir".

Appendix A: The density contribution at long distances Here we present derivation of the quantum contribution to the density at long separation from the boundary. We begin with the expression for ψ 2 (X) that simplifies at X 1. In order to obtain it we first notice that Eq. ( 25) leads to the firstorder differential equation

In the regime X 1 one can set tanh X to 1 and cosh -2 X to 0 in Eqs. [START_REF] Ardila | Strong coupling Bose polarons in a two-dimensional gas[END_REF], such that the source term becomes

One can then perform the integration in Eq. (A1) for which it suffices to approximate cosh 2 Y by e 2Y /4 and solve an elementary integral. Solving the differential equation we then find

at X 1 (corresponding to x ξ). The quantum contribution to the density (31) can now be found more easily using Eq. (A3). It leads to n (1)

The fist two terms in braces give a constant after the integration. The next two terms contain poles in the complex plane and (after proper regularization) lead to exponentially small result. Finally the term with the denominator with the power exponent 3/2 is responsible for the power-law decay of n (1) (X). The actual evaluation yields

Equation (A5) can be recognized in the density correction given by Eqs. ( 33) and (34). Apart from the exponential terms


Antony Van Der Ent

Kathryn Spiers

Dennis Brueckner

Guillaume Echevarria

Mark G M Aarts

Emmanuelle Montargès-Pelletier

Spatially-resolved localization and chemical speciation of nickel and zinc in Noccaea tymphaea and Bornmuellera emarginata

Keywords: capsule, cotyledons, hyperaccumulator, seed, translocation

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Hyperaccumulators are unusual plants that accumulate particular metals or metalloids in their living tissues to levels that may be hundreds to thousands of times greater than is normal for most plants [START_REF] Jaffre | Sebertia acuminata: A Hyperaccumulator of Nickel from New Caledonia[END_REF][START_REF] Reeves | Tropical hyperaccumulators of metals and their potential for phytoextraction[END_REF] . Hyperaccumulator plants have the unique ability to take up and detoxify exceptional concentrations of metals without any signs of toxicity. Plants have been found to hyperaccumulate a wide range of elements, including nickel (Ni), manganese (Mn), cadmium (Cd), copper (Cu), cobalt (Co), selenium (Se), arsenic (As), thallium (Tl) and Zn [START_REF] Reeves | Tropical hyperaccumulators of metals and their potential for phytoextraction[END_REF][START_REF] Van Der Ent | Hyperaccumulators of metal and metalloid trace elements: Facts and fiction[END_REF] . This contrasts with 'normal plants' that have a tightly controlled regulation of essential transition elements (Cu, Fe, Ni, Mn, Zn) to avoid either deficiency or toxicity. Hyperaccumulator plants represent the most extreme example of adaptation to a surplus of metal transition elements in their environment, and are therefore ideal model systems for understanding the physiological regulation of essential and potentially toxic, non-essential transition elements [START_REF] Pollard | The Genetic Basis of Metal Hyperaccumulation in Plants[END_REF][START_REF] Krämer | Metal hyperaccumulation in plants[END_REF][START_REF] Deng | Nickel hyperaccumulation mechanisms: a review on the current state of knowledge[END_REF] . In many Ni hyperaccumulator species, Ni occurs in mixtures of citrate and malate complexes that vary in different parts of the plants [START_REF] Montargès-Pelletier | Identification of nickel chelators in three hyperaccumulating plants: an X-ray spectroscopic study[END_REF][START_REF] Callahan | Elemental and metabolite profiling of nickel hyperaccumulators from New Caledonia[END_REF][START_REF] Van Der Ent | Nickel biopathways in tropical nickel hyperaccumulating trees from Sabah (Malaysia)[END_REF] . Hyperaccumulation results from adaptations of the metal regulation mechanisms shared by all higher plants [START_REF] Clemens | How metal hyperaccumulating plants can advance Zn biofortification[END_REF] . Hence, insights into the mechanisms of hyperaccumulation may be applied to improve the uptake and accumulation of deficient elements, such as iron (Fe) and zinc (Zn), in economically important food crops. These insights may also be applied to limit uptake of potentially phytotoxic elements, such as nickel (Ni) in food crops. The extreme metal accumulation capability of hyperaccumulator plants spawned the concept of phytoextraction for remediating contaminated soils, which has attracted much research effort [START_REF] Chaney | Phytoremediation of soil metals[END_REF] . Hyperaccumulator plants also have potential for use in phytotechnologies such as biofortication, phytoremediation and phytomining, the latter utilizes hyperaccumulators as 'metal crops' to sequester Ni or other metals in harvestable biomass that can then be used to produce fine Ni chemicals [START_REF] Li | Development of a technology for commercial phytoextraction of nickel: economic and technical considerations[END_REF][START_REF] Van Der Ent | Agromining: farming for metals in the future?[END_REF] .

Nickel is the most recent element shown to be essential for higher plants [START_REF] Dixon | Jack bean urease (EC 3.5.1.5). Metalloenzyme. Simple biological role for nickel[END_REF][START_REF] Brown | Nickel: a micronutrient essential for higher plants[END_REF] , due to its key role for the activity of urease, an enzyme widely distributed in higher plants [START_REF] Hogan | Urease assay and ammonia release from leaf tissues[END_REF] , and playing a crucial role in nitrogen remobilization from senescing leaves and during seed germination. However, excess Ni induces oxidative and genotoxic stresses that are deleterious to plant growth [START_REF] Seregin | Physiological role of nickel and its toxic effects on higher plants[END_REF] . Therefore, every plant species needs to regulate Ni homeostasis according to its needs. Apart from its function in urease activation, other physiological functions of Ni are poorly understood in higher plants [START_REF] Deng | Nickel hyperaccumulation mechanisms: a review on the current state of knowledge[END_REF][START_REF] Welch | The biological significance of nickel[END_REF] .

Although Ni is an essential micronutrient, its physiological requirement is extremely low. It is shown that 0.1 mg Ni kg -1 is sufficient for seed germination and plant growth [START_REF] Brown | Nickel: a micronutrient essential for higher plants[END_REF][START_REF] Gerendás | Significance of nickel for plant growth and metabolism[END_REF] . Hence, Ni deficiency in naturally-grown plants rarely occurs, and the only known case is for pecan [START_REF] Wood | Mouse-ear of pecan: A nickel deficiency[END_REF] . The molecular mechanisms involved in the regulation of Ni homeostasis are not well known even in model species such as Arabidopsis thaliana. Nickel can be transported from the soil and inside plants by several families of metal transporters (e.g. ZIP/IRT, IREG, YSL [START_REF] Schaaf | AtIREG2 Encodes a Tonoplast Transport Protein Involved in Iron-dependent Nickel Detoxification in Arabidopsis thaliana Roots[END_REF][START_REF] Gendre | TcYSL3, a member of the YSL gene family from the hyper-accumulator Thlaspi caerulescens, encodes a nicotianamine-Ni/Fe transporter[END_REF][START_REF] Nishida | AtIRT1, the Primary Iron Uptake Transporter in the Root, Mediates Excess Nickel Accumulation in Arabidopsis thaliana[END_REF] ). Since plants normally only require minute amounts of Ni, no Ni specific transporter has so far been identified to account for enhanced Ni uptake from soil. In non-accumulator species, this function is most likely performed by one of the ZIP family Zn/Fe/Mn uptake transporters e.g. the Zn-deficiency induced expression of AtZIP4 can be repressed by supplying Ni 2+ . Preliminary results suggest that one of these transporters has developed more affinity for Ni 2+ in Ni hyperaccumulators than in nonaccumulators [START_REF] Schaaf | AtIREG2 Encodes a Tonoplast Transport Protein Involved in Iron-dependent Nickel Detoxification in Arabidopsis thaliana Roots[END_REF][START_REF] Nishida | AtIRT1, the Primary Iron Uptake Transporter in the Root, Mediates Excess Nickel Accumulation in Arabidopsis thaliana[END_REF] . The transporter TgMTP1 of the CDF family was originally suspected to mediate Ni storage in the vacuole of the Ni hyperaccumulator Noccaea goesingense [START_REF] Persans | Functional activity and role of cation-efflux family members in Ni hyperaccumulation in Thlaspi goesingense[END_REF] . More recent studies suggested that transporters of the IREG/Ferroportin family, localized on the vacuolar membrane, are involved in the storage of Ni in non-accumulators and hyperaccumulators [START_REF] Schaaf | AtIREG2 Encodes a Tonoplast Transport Protein Involved in Iron-dependent Nickel Detoxification in Arabidopsis thaliana Roots[END_REF][START_REF] Merlot | The metal transporter PgIREG1 from the hyperaccumulator Psychotria gabriellae is a candidate gene for nickel tolerance and accumulation[END_REF] . Studies on Zn and Cd hyperaccumulation in Brassicaceae species (e.g. Arabidopsis halleri and N. caerulescens) revealed that Zn and Cd hyperaccumulation traits are correlated with high and constitutive expression of genes involved in metal transport, in the biosynthesis of metal chelators and in cellular defences to oxidative stresses [START_REF] Assunção | Elevated expression of metal transporter genes in three accessions of the metal hyperaccumulator Thlaspi caerulescens[END_REF][START_REF] Hammond | A comparison of the Thlaspi caerulescens and Thlaspi arvense shoot transcriptomes[END_REF][START_REF] Hanikenne | Metal hyperaccumulation and hypertolerance: a model for plant evolutionary genomics[END_REF] . These changes in gene expression are often the combined effect of gene duplication and altered promoter activity [START_REF] Hanikenne | Evolution of metal hyperaccumulation required cis-regulatory changes and triplication of HMA4[END_REF][START_REF] Craciun | Variation in HMA4 gene copy number and expression among Noccaea caerulescens populations presenting different levels of Cd tolerance and accumulation[END_REF] .

The genus Noccaea has at least 23 species that hyperaccumulate Ni, a further 10 that hyperaccumulate Zn, three that hyperaccumulate Cd and one that hyperaccumulates Pb [START_REF] Reeves | Brooks Hyperaccumulation of lead and zinc by two metallophytes from mining areas of Central-Europe[END_REF][START_REF] Brooks | Geobotany and hyperaccumulators[END_REF][START_REF] Lombi | Cadmium accumulation in populations of Thlaspi caerulescens and Thlaspi goesingense[END_REF][START_REF] Vogel-Mikuš | Zn, Cd and Pb accumulation and arbuscular mycorrhizal colonisation of pennycress Thlaspi praecoxWulf. (Brassicaceae) from the vicinity of a lead mine and smelter in Slovenia[END_REF][START_REF] Mohtadi | A comparison of lead accumulation and tolerance among heavy metal hyperaccumulating and non-hyperaccumulating metallophytes[END_REF] .

Noccaea caerulescens (J.Presl & C.Presl) F.K.Mey. (Thlaspi caerulescens J.Presl & C.Presl) is unique in consisting of different ecotypes with distinct metal tolerance and hyperaccumulation abilities [START_REF] Assunção | Differential metal-specific tolerance and accumulation patterns among Thlaspi caerulescens populations originating from different soil types[END_REF] . While calamine, ultramafic and non-metallicolous populations can hyperaccumulate Zn and Ni, or Cd, when supplied [START_REF] Assunção | Differential metal-specific tolerance and accumulation patterns among Thlaspi caerulescens populations originating from different soil types[END_REF] , they differ in their ability to tolerate these metals, often depending on the metal concentrations at their site of origin [START_REF] Deng | Nickel and Zinc Isotope Fractionation in Hyperaccumulating and Nonaccumulating Plants[END_REF][START_REF] Gonneau | Variation of trace metal accumulation, major nutrient uptake and growth parameters and their correlations in 22 populations of Noccaea caerulescens[END_REF] . Zinc is taken up by ZIP-like plasma membrane located Zn-transporters. Rather than storing excess Zn in vacuoles of root cells, which most non-accumulator species do, the Zn 2+ is loaded into the xylem, by HMA4, as in Arabidopsis halleri (L.) O'Kane & Al-Shehbaz [START_REF] Hanikenne | Evolution of metal hyperaccumulation required cis-regulatory changes and triplication of HMA4[END_REF] , thus translocated to the leaves, where it is stored in mesophyll and epidermal vacuoles. Ultramafic populations are known [START_REF] Gonneau | Demographic history of the trace metal hyperaccumulator Noccaea caerulescens (J. Presl and C. Presl) F. K. Mey[END_REF] and converted into genetically homogeneous lines by recurrent inbreeding for characterization of Zn, Ni and Cd accumulation and tolerance properties. The variation in metal tolerance and hyperaccumulation is heritable and independent of each other [START_REF] Deniau | QTL analysis of cadmium and zinc accumulation in the heavy metal hyperaccumulator Thlaspi caerulescens[END_REF][START_REF] Assunção | Construction of a genetic linkage map of Thlaspi caerulescens and quantitative trait loci analysis of zinc accumulation[END_REF][START_REF] Xing | Variation in root-to-shoot translocation of cadmium and zinc among different accessions of the hyperaccumulators Thlaspi caerulescens and Thlaspi praecox[END_REF] . The calamine and non-metallicolous populations have been investigated in more detail than the ultramafic populations. So far most of the analysis of ultramafic N. caerulescens involved the accession from Monte Prinzera, in the Italian Apennine mountain range [START_REF] Assunção | Differential metal-specific tolerance and accumulation patterns among Thlaspi caerulescens populations originating from different soil types[END_REF] . This accession has been subject to several proteomics studies, trying to correlate local adaptation to specific protein expression [START_REF] Visioli | Correlation between phenotype and proteome in the Ni hyperaccumulator Noccaea caerulescens subsp. caerulescens[END_REF] and was included in transcriptome studies as well [START_REF] Halimaa | Gene expression differences between Noccaea caerulescens ecotypes help to identify candidate genes for metal phytoremediation[END_REF] . Ni, and also Zn, xylem loading in N. caerulescens is facilitated by high histidine levels [START_REF] Kozhevnikova | Histidine-mediated xylem loading of zinc is a species-wide character in Noccaea caerulescens[END_REF] although the genes involved have not yet been identified.

Noccaea tymphaea (Hausskn.) F.K.Mey. (synonyms: Thlaspi tymphaeum Hausskn. and Thlaspi goesingense Halácsy) is distributed in Albania, Bosnia and Herzegovina, Greece, Macedonia. It occurs on montane ultramafic soils, where it can accumulate high foliar Ni (up to 11 800 µg g -1 ) and relatively low foliar Zn (up to 179 µg g -1 ) [START_REF] Reeves | Brooks Hyperaccumulation of lead and zinc by two metallophytes from mining areas of Central-Europe[END_REF] . Bornmuellera emarginata (Boiss.) Rešetnik (synonyms: Leptoplax emarginata (Boiss.) O. E. Schulz, Peltaria emarginata (Boiss.) Hausskn.) [START_REF] Resetnik | Two new combinations in the genus Bornmuellera (Brassicaceae)[END_REF] is endemic to ultramafic soils in Greece, with a discontinuous distribution from Pindus mountains, Mt. Smolikas, and the island of Euboea. Some specimens were also sampled in Syria and are kept at Paris Herbarium (P) showing a disjunct distribution pattern across the Eastern Mediterranean. It is a strong Ni hyperaccumulator that can accumulate up to 34 400 µg g -1 foliar Ni [START_REF] Reeves | Nickel accumulation by species of Peltaria Jacq. (Cruciferae)[END_REF] .

Currently little is known on storage and acquisition of Ni in seeds and during germination. In Noccaea praecox (Wulfen) F.K.Mey. (synonym: Thlaspi praecox Wulfen.) Cd was mobilised to the shoots during germination, but not to the roots [START_REF] Vogel-Mikuš | Localisation and quantification of elements within seeds of Cd/Zn hyperaccumulator Thlaspi praecox by micro-PIXE[END_REF] . Scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS) was undertaken on the seeds of Noccaea pindica (Hausskn.) Holub (synonym of Thlaspi pindicum Hausskn.) and the results showed that Ni accumulated in the micropylar area opposite the radicle and in the epidermis of cotyledons [START_REF] Psaras | Nickel localization in the seeds of the hyperaccumulator Thlaspi pindicum Hauskn[END_REF] . Little information is available about the elemental distribution in other hyperaccumulating genera of the Brassicaceae, and no study yet has focussed on B. emarginata except the SEM-EDS observation of herbarium specimen air-dried leaves which showed accumulation of Ni in the epidermis cells except in the vicinity of stomata [START_REF] Psaras | Relative abundance of nickel in the leaf epidermis of eight hyperaccumulators: Evidence that the metal is excluded from both guard cells and trichomes[END_REF] . Knowledge about the ecophysiology of Ni and Zn in reproductive organs of hyperaccumulator plants is especially scare. In order to provide a more general view about the ecophysiology of hyperaccumulation, including the reproduction organs and first phases of life of these plants, this study aimed to elucidate the distribution and chemical speciation of Ni and Zn in the seeds and siliques of N. tymphaea and B. emarginata originating from their native habitats in Greece. X-ray Fluorescence Microscopy (XFM) has substantial explanatory power for advancing the understanding of the ecophysiology of hyperaccumulation [START_REF] Van Der Ent | Nickel biopathways in tropical nickel hyperaccumulating trees from Sabah (Malaysia)[END_REF] . In order to determine the distribution and spatially-resolved chemical speciation of Ni and Zn in both species, we make use of the singular ability of the Maia detector system [START_REF] Ryan | Elemental X-ray imaging using the Maia detector array: The benefits and challenges of large solid-angle[END_REF][START_REF] Siddons | Maia X-ray Microprobe Detector Array System[END_REF] to perform ultra-rapid X-ray elemental mapping and spatially resolved X-ray Absorption Spectroscopy (XAS) on live/fresh samples.

MATERIALS AND METHODS

Collection of plant tissues and soils

Whole, live plants of N. tymphaea were collected in Greece (at the Katara Pass, 39°48'00.5"N 21°10'60.0"E, altitude 1690 m. a.s.l.) growing in natural ultramafic soils. Intact seeds capsules were collected from B. emarginata in their native habitat Greece (near Trigona, 39°47'29"N, 21°25'32"E, altitude 830 m. a.s.l.). The soils of the collection locality are described in detail eslwhere [START_REF] Bani | Nickel Hyperaccumulation by Brassicaceae in Serpentine Soils of Albania and Northwestern Greece[END_REF] . The plants were potted in natural soil from the habitat and brought alive to the P06 beamline (PETRA III Synchrotron, DESY Campus, Hamburg Germany) for the experiments described below.

Chemical bulk analysis of tissue samples

Plant tissue samples for bulk chemical analysis were first dried on silica gel and then dried at 70°C for five days in a drying oven. They were subsequently ground and digested using 4 mL HNO 3 (70%) in a microwave oven (Milestone Start D) for a 45-minute programme and diluted to 30 mL with ultrapure water (Millipore 18.2 MΩ•cm at 25°C). Finally, they were analysed with ICP-AES (Thermo iCAP 7400) for Cd, Ni, Co, Cr, Cu, Zn, Mn, Fe, Mg, Ca, Na, K, S, P.

Preparation of tissue samples for X-ray fluorescence microscopy

The seeds and seed capsules could be investigated in their native state without any sample preparation. The intact seeds were mounted between two sheets of Ultralene thin film (4 µm) stretched over a Perspex frame magnetically attached to the x-y motion stage at atmospheric temperature (~20°C). However, in order to reveal the internal distribution of Ni, Zn and other elements inside roots, stems and leaves, cross-sections were prepared. The samples were hand cut with a stainless-steel razor blade ('dry knife'), mounted between two sheets of 4 µm Ultralene thin film in a tight sandwich to limit evaporation, and analysed within 15 minutes after excision. X-ray micro-fluorescence was performed at high speed to keep the scan time to a minimum. Since the penetration depth of the X-rays is greater than the thickness of a cell layer, the information obtained from thick sections is a combined distribution originating from more or less superimposed cell layers. The semi-thick sections (~200 µm) correspond to 3-4 cell layers. As such the obtained data do not reveal subcellular distribution, but nevertheless show the tissue-level distribution (e.g. epidermal cells, mesophyll, vascular bundles, etc.).

X-ray fluorescence microscopy

The X-ray fluorescence microscopy (XFM) experiments were undertaken at beamline P06 at PETRA III at DESY (Deutsches Elektronen-Synchrotron). The undulator beam was monochromatised using either a Si(111) channel-cut crystal or a double-crystal monochromator, depending on beamline mode for each part of the experiment. A Kirkpatrick-Baez mirror pair was used to focus the incident beam. The X-ray flux of the focussed beam was in the order of 10 10 photons/s 54 . X-ray fluorescence was detected using the Maia detector system in backscatter geometry [START_REF] Siddons | Maia X-ray Microprobe Detector Array System[END_REF][START_REF] Kirkham | The Maia Spectroscopy Detector System: Engineering for Integrated Pulse Capture, Low-Latency Scanning and Real-Time Processing[END_REF] . The large solid-angle (1.2 steradian) of the Maia detector is particularly suited to biological samples such as these as it allows detection of a good proportion of the fluoresced signal, allowing a reduction of the radiation dose and thus reducing potential damage to a specimen [START_REF] Ryan | Elemental X-ray imaging using the Maia detector array: The benefits and challenges of large solid-angle[END_REF] .

The 2D µXRF measurements carried out in the microprobe of P06 at DESY were performed with a beam size of 720 × 780 nm at an incident energy of 11 keV. The single-slice tomography measurements of the N. tymphaea seed were carried out with a beamsize of 400 × 450 nm at a photon energy of 15 keV using the same beamline endstation and general setup. Scanning parameters were a step size of 2 µm, a dwell time of 1 ms and an angular range of 452 projections covering 360° in 2 subscans. As the seed was naturally dehydrated a cryo-stream was not employed.

For the XRF 2D and tomography scans, the pixel size chosen was larger than the focused beam size as a result of necessary compromises due to time constraints.

Synchrotron X-ray Absorption Spectroscopy (XAS)

Ni and Zn K-edge XAS spectra of the plant tissue samples and standards were recorded in fluorescence mode with the Maia detector. The X-ray beam energy was calibrated using either a Ni or Zn metal foil recorded in transmission, where the first peak of the first derivative was assumed to be 8333 or 9659 eV, respectively. The seeds (~1.5 × 0.3 mm) were scanned at 1.6 µm pixels with a 20 ms per pixel dwell time. In addition to these µXRF elemental images on these seeds, spatially resolved XANES spectra were collected as image 'stacks' of µXRF maps, each with 15 µm pixels and a 12 ms per pixel dwell time, at 170 increasing energies, spanning the energy range 8183-9082 eV over the Ni K-edge at 8.333 keV, and spanning the energy range 9486-9858 eV over the Zn Kedge at 9659 eV.

Several Ni 2+ and Zn 2+ standards were prepared by adding organic ligands in calculated molar excess

Data processing and analysis

The XRF event stream was analysed using the Dynamic Analysis method [START_REF] Ryan | Dynamic analysis: on-line quantitative PIXE microanalysis and its use in overlap-resolved elemental mapping[END_REF][START_REF] Ryan | Quantitative trace element imaging using PIXE and the nuclear microprobe[END_REF] as implemented in GeoPIXE [START_REF] Ryan | Quantitative analysis of PIXE spectra in geoscience applications[END_REF][START_REF] Ryan | Nuclear microprobe -synchrotron synergy: Towards integrated quantitative real-time elemental imaging using PIXE and SXRF[END_REF] . GeoPIXE provides quantitative first-order self-absorption corrected maps of projected areal elemental density -maps of elemental content. Conversion of X-ray counts to concentration was performed through analysis of Ni and Co XRF reference foil scans (Micromatter, Canada). The samples were each considered of a uniform thickness and either hydrated or dry, with respective thicknesses and compositions of 1000 µm and C 7.3 O 33 H 59 N 0.7 S 0.15 for the (hydrated) whole leaf and leaf, stem and root cross-sections, and 1500 µm and C 31 O 15 H 51 N 2 S 0.8 for the (dry) whole seeds and capsules. Assuming a uniform thickness for the seeds introduces further approximations to the measurements of seeds and seed capsules, however, as these have a generally flattened oblate cross-section, the approximation was considered appropriate.

Reconstruction of the single-slice tomographic data was performed using a maximum-likelihood expectation-maximization (MLEM) algorithm.

PCA analysis was performed on the XANES stacks using the MANTiS package. The extracted XANES data were reduced using standard normalization procedures performed with Bruce Ravel and Matthew Newville programs ATHENA and ARTEMIS [START_REF] Newville | EXAFS analysis using FEFF and FEFFIT[END_REF][START_REF] Ravel | HEPHAESTUS: Data Analysis for x-ray absorption spectroscopy using IFEFFIT[END_REF] . Spectra were background subtracted and normalized. The XANES signals obtained were fitted as linear combinations of the standard spectra collected on solutions to evidence the main organic ligands involved in metal complexation. The number of standards was constrained to be at a maximum of three. The sum of components was released and not forced to be equal to 1. The selection of the linear combination was made on the basis of the indicators of fitting quality (chi 2 , r-factor and reduced chi 2 ), and the number of components was finally set to 2 as a third component did not improve the fitting quality.

RESULTS

Localization of Ni and Zn in N. tymphaea root, stem and leaf cross-sections

Elemental maps of the root cross-sections (Fig. 1) show that Ni is concentrated mainly in the epidermis, in phloem bundles and pericycle. Zinc is mainly enriched in the phloem bundles. In the stem cross-section (Fig. 2) Ni is enriched in the collenchyma and the phloem bundles, as well as in the xylem.

In the whole leaves of N. tymphaea, Ni is mainly localised in the leaf blade, with increasing concentrations towards the margins. The whole leaf elemental map of Ni (Fig. 3 and Suppl. . Whereas Ni is present in vacuoles of epidermal cells, Zn occurs mainly in the mesophyll cells, especially on either side of the central vascular bundles (Fig. 4). Calcium is depleted in the vascular bundles, but strongly enriched in the mesophyll and also in the epidermal region.

Potassium is enriched in the epidermal cells and in the phloem bundles of the primary vein and also in secondary veinlets.

Localization of Ni and Zn in seed capsules of N. tymphaea and B. emarginata

The intact N. tymphaea siliques contain 8-10 seeds attached to the central ovary (Fig. 5). Nickel is strongly enriched in vascular bundles of the ovary connected via the hilum onto the seeds. Nickel occurs also at the base of the style (micropyle). Zinc is highest in the micropylar region, in the vascular bundles, and in the seeds in radicles. Nickel and Zn appear to be similarly highly enriched in the cotyledons (Fig. 7). Calcium is highest in the margins of the seed capsules and in the style, whereas K is highest in the central vascular bundles of the ovary. These patterns are similar in another N. tymphaea silique (Supp. Fig. 3), but Ni and Zn differ in that Zn is particularly accumulated in the radicles of the seeds, whereas Ni is more broadly enriched in the seeds. The distribution of Co (not shown) again mirrors that of Ni with enrichment mainly in the vascular bundles of the ovary, and in connecting tissues to the seeds.

The intact silique of B. emarginata contains just a single seed (Fig. 6 and Suppl. Figs. 4 and5).

Nickel is strongly enriched on the outer margins of the capsules, as well as in the micropylar region and style. Nickel appears enriched in the whole of the seeds, as well as in the hilum. As B. emarginata is not a Zn hyperaccumulator, the Zn content is low and its distribution is unremarkable. Calcium is strongly enriched in a peripheral region around the seed margin, likely the wings. Finally, K is especially high in the hilum and vascular bundles leading into the seed capsule and seed.

Localization of Ni and Zn in the seeds of N. tymphaea and B. emarginata

The distribution of Ni and Zn is similar and strongly enriched in the cotyledons in N. tymphaea as shown in the 2D maps (Fig. 7 and Suppl. Fig. 6). In contrast, in B. emarginata they are strongly enriched in the tip of the radicle (Fig. 8). The enrichment of Fe in the provascular strands of the cotyledons and in the hypocotyl) is clearly visible in the network, and Fe is highest in hotspots in the hilum of the micropylar area.

The tomographic reconstructions of the N. tymphaea seeds confirm the observations from the 2D maps and show that Ni is localised in the vacuoles (the round solid outlines of vacuoles are clearly visible) of the cotyledon epidermal cells, and similarly in the epidermal cells of the hypocotyl (Fig. 9). The virtual slice is looking tangentially showing the two cotyledons on either side (i.e. from the narrow plane) and the hypocotyl on the top right (Suppl Figs. 7, 8 and9). Nickel is also enriched in the testa (seed coat). Nickel is depleted in the vascular bundles in the cotyledons. In contrast, Zn is enriched more or less evenly throughout the cotyledons, albeit slightly higher in the hypocotyl. It cannot be ascertained whether Zn is present on the vacuoles. The distribution of Fe in the provascular strands (note 'hollow' features) of the hypocotyl and cotyledons is clearly visible.

Spatially-resolved chemical speciation of Ni and Zn in the seeds of N. tymphaea

The chemical speciation of Ni in the intact seeds of N. tymphaea is unequivocally associated with carboxylic acids (Fig. 10). The spectra extracted from the different regions of the seed (regions determined on the basis of PCA) were strictly identical to each other. Qualitative comparison with reference spectra suggested the predominance of Ni-malate species, confirmed by the linear combination results (Ni evidenced being at 80% complexed with malate). A smaller contribution of Ni-histidine complex can be discerned with the fitting. In the case of Zn (Fig. 11), Zn-phytate species were dominating Zn XANES spectra. In both Ni and Zn, spatially resolved spectroscopy did not reveal any spatial variation in chemical speciation within the seed.

DISCUSSION

Until recently detection systems for synchrotron XFM were not sufficiently fast to analyse fresh and hydrated plant tissue because the long dwell times caused excessive radiation damage [START_REF] Van Der Ent | Mesjasz-Przybyłowicz, X-ray elemental mapping techniques for elucidating the ecophysiology of hyperaccumulator plants[END_REF] . The unparalleled ability of the Maia X-ray detection system to undertake very fast measurements (perpixel dwell times as low as 1 ms and total scan times of less than 20 minutes for leaf cross-sections) makes it possible to analyse live plants and fresh plant materials [START_REF] Kopittke | Synchrotron-Based X-Ray Fluorescence Microscopy as a Technique for Imaging of Elements in Plants[END_REF] . There are limitations to this approach, however, including the fact that the elemental maps give information from different depths combined into one plane (in the case of whole plant leaves). The only approach that avoids most or all sample preparation artefacts are cryotechniques which preserve both the distribution, the chemical form and the concentration of all elements in situ [START_REF] Van Der Ent | Mesjasz-Przybyłowicz, X-ray elemental mapping techniques for elucidating the ecophysiology of hyperaccumulator plants[END_REF] . However, such techniques are not always available or not operable due to technical constraints, for example a cryo-stream can only cool samples smaller than 2 mm in diameter, and only one synchrotron facility (the BioNano Probe [START_REF] Chen | The Bionanoprobe: hard X-ray fluorescence nanoprobe with cryogenic capabilities[END_REF] ) has a fully enclosed cryogenic chamber for large samples. In order to map elemental distribution within tissues, cross-sectioning is necessary, either physical or virtual by using tomographic methods. Sectioning of fresh plant material using a 'dry knife' method (as done in this study) avoids the loss of water-soluble ions (Ni 2+ , Zn 2+ ), but may result in smearing of cell sap over the sample surface. Such artefacts were, however, not observed in this study and the elemental maps show intact inflated vacuoles (interpreted from the K maps). Seeds and seed capsules are unique among plant organs/tissues in that they are inherently dehydrated, and therefore, can be analyses "as is" in microanalytical experiments.

Previous studies on the distribution of Ni and Zn in hyperaccumulator plants have shown that in

Hybanthus floribundus subsp. adpressus (Violaceae) seeds the highest Ni concentrations were in the cotyledons, followed by the embryonic axis. In Pimelea leptospermoides (Thymelaeaceae) seeds Ni was preferentially localised in the embryonic axis, and in N. caerulescens, Zn was highest in cotyledons [START_REF] Kachenko | Nickel, Zn and Cd localisation in seeds of metal hyperaccumulators using µ-PIXE spectroscopy[END_REF] . Nickel was also concentrated in the epidermis of the cotyledons in N. caerulescens seeds [START_REF] Mattarozzi | ESEM-EDS: In vivo characterization of the Ni hyperaccumulator Noccaea caerulescens[END_REF] , whereas in N. pindica seeds Ni was concentrated in the micropylar area and in the epidermis of cotyledons [START_REF] Psaras | Relative abundance of nickel in the leaf epidermis of eight hyperaccumulators: Evidence that the metal is excluded from both guard cells and trichomes[END_REF] . In Stackhousia tryonii (Celastraceae) seeds the highest Ni concentrations were in the pericarp [START_REF] Bhatia | Elemental mapping using PIXE shows the main pathway of nickel movement is principally symplastic within the fruit of the hyperaccumulator Stackhousia tryonii[END_REF] . In Pycnandra acuminata (Sapotaceae) seeds the highest Ni concentration were in the endosperm and mesocarp [START_REF] Sagner | Hyperaccumulation, complexation and distribution of nickel in Sebertia acuminata[END_REF] . Similarly, in Biscutella laevigata (Brassicaceae) seeds, the highest concentration of Zn was in the endosperm [START_REF] Mesjasz-Przybyłowicz | Nuclear microprobe studies of elemental distribution in seeds of Biscutella laevigata L. from zinc wastes in Olkusz, Poland[END_REF] . Finally, in Berkheya coddii (Asteraceae) Ni was localised in the lower epidermis, margins of cotyledons, and the pericarp in the micropylar area [START_REF] Przybyłowicz | Investigation of Ni hyperaccumulation by true elemental imaging[END_REF][START_REF] Groeber | Fate of nickel and calcium in seedlings of the hyperaccumulator Berkheya coddii during germination[END_REF] . The diversity in location of Ni and Zn seeds of various hyperaccumulator plants reflects the variety of phylogenetic origins and distinct physiologies of hyperaccumulator plants. In this study we observed Ni to be localised in the vacuoles of the cotyledon and hypocotyl epidermal cells in N. tymphaea seeds, which agrees with the findings for other Noccaea species previously studied.

During germination the seedling relies primarily on its Fe stores before it develops a root to take up Fe from the soil [START_REF] Bastow | Vacuolar Iron Stores Gated by NRAMP3 and NRAMP4 Are the Primary Source of Iron in Germinating Seeds[END_REF] . Arabidopsis thaliana stores Fe in vacuoles of the root endodermis and around the pro-vasculature in the cotyledons [START_REF] Kim | Localization of iron in Arabidopsis seed requires the vacuolar membrane transporter VIT1[END_REF] . The Fe-rich provascular strand network of the cotyledons occurs in N. tymphaea too.

The European Noccaea caerulescens is among the most intensively-studied hyperaccumulator taxa globally, and used as a model for the genetics, ecology and molecular biology of metal hyperaccumulation [START_REF] Gonneau | Variation of trace metal accumulation, major nutrient uptake and growth parameters and their correlations in 22 populations of Noccaea caerulescens[END_REF][START_REF] Gonneau | Demographic history of the trace metal hyperaccumulator Noccaea caerulescens (J. Presl and C. Presl) F. K. Mey[END_REF][START_REF] Halimaa | Gene expression differences between Noccaea caerulescens ecotypes help to identify candidate genes for metal phytoremediation[END_REF][START_REF] Assunção | Thlaspi caerulescens, an attractive model species to study heavy metal hyperaccumulation in plants[END_REF] . The other taxa in the genus, such as N. goesingense and N. praecox, but also the various taxa in the Alyssum genus, are much less studied [START_REF] Reeves | Studies on metal uptake by plants from serpentine and nonserpentine populations of Thlaspi goesingense Hálácsy (Cruciferae)[END_REF][START_REF] Krämer | The Role of Metal Transport and Tolerance in Nickel Hyperaccumulation by Thlaspi goesingense Halacsy[END_REF][START_REF] Salt | Nickel hyperaccumulation in Thlaspi goesingense: a scientific travelogue[END_REF][START_REF] Küpper | Cellular compartmentation of nickel in the hyperaccumulators Alyssum lesbiacum, Alyssum bertolonii and Thlaspi goesingense[END_REF] . N. tymphaea, and taxa from other genera (B. emarginata), have so far obtained little attention, mostly because they grow in a rather confined and remote area in the Balkan region of Europe. Most Noccaea species can hyperaccumulate Zn, whereas many taxa also hyperaccumulate Ni [START_REF] Krämer | Metal hyperaccumulation in plants[END_REF][START_REF] Koch | Taxonomy and systematics are key to biological information: Arabidopsis, Eutrema (Thellungiella), Noccaea and Schrenkiella (Brassicaceae) as examples[END_REF][START_REF] Taylor | Within and between population variation for zinc and nickel accumulation in two species of Thlaspi (Brassicaceae)[END_REF] , but since most species have only been sampled in the field and not been re-grown on Zn or Ni containing soil, this issue is far from resolved. We set out to determine whether the distribution and chemical speciation of Ni and Zn differed in N. tymphaea and B. emarginata, both sampled from ultramafic soil. Although we expected important differences, due to the differing physiological functions (potentially toxic for Ni and essential for Zn) of these elements, the results show that Ni and Zn behave remarkably similar in N. tymphaea and B. emarginata. The chemical speciation of Ni is univocally associated with low molecular weight carboxylic ligands (likely malate), as in most hyperaccumulator species studied to date [START_REF] Montargès-Pelletier | Identification of nickel chelators in three hyperaccumulating plants: an X-ray spectroscopic study[END_REF][START_REF] Van Der Ent | Nickel biopathways in tropical nickel hyperaccumulating trees from Sabah (Malaysia)[END_REF][START_REF] Mcnear | The hyperaccumulator Alyssum murale uses complexation with nitrogen and oxygen donor ligands for Ni transport and storage[END_REF][START_REF] Callahan | Metal ion ligands in hyperaccumulating plants[END_REF] . Specifically, in N. caerulescens and B. emarginata X-ray absorption spectroscopy showed that citrate was found as the predominant ligand for Ni in stems, whereas in the leaves malate was predominant [START_REF] Montargès-Pelletier | Identification of nickel chelators in three hyperaccumulating plants: an X-ray spectroscopic study[END_REF] . In contrast, Zn was associated with phytates in the seeds. In the ultramafic soils of which N. tymphaea and B. emarginata grow, Ni is present at 20-50-fold higher concentrations, which explains concentrations differences in the plant shoots. Under these conditions N. tymphaea is not a Zn hyperaccumulator (foliar Zn reaches up to 362 µg g -1 ). Based on the predominant Ni and Zn hyperaccumulation properties found in the current Noccaea spp., we hypothesize that the genus evolved from a Ni adapted and probably Ni hyperaccumulating ancestor. Some species managed to escape from ultramafic soil and develop as Zn hyperaccumulators on non-metallicolous soils, with a few, e.g. N. caerulescens later adapting to and (re-)colonizing calamine and ultramafic soils. Noccaea tymphaea may represent a taxon that never left the

( 1 :

 1 5) to Ni2+ and Zn 2+ to ensure the formation of organo-metallic complexes. The selection of the ligands was based on previous reports of Ni and Zn complexation in hyperaccumulator plants[START_REF] Montargès-Pelletier | Identification of nickel chelators in three hyperaccumulating plants: an X-ray spectroscopic study[END_REF][START_REF] Van Der Ent | Nickel biopathways in tropical nickel hyperaccumulating trees from Sabah (Malaysia)[END_REF][START_REF] Tappero | Microspectroscopic study of cobalt speciation and localization in hyperaccumulator Alyssum murale[END_REF][START_REF] Mcnear | The hyperaccumulator Alyssum murale uses complexation with nitrogen and oxygen donor ligands for Ni transport and storage[END_REF] .Aqueous standards were prepared from Ni(NO 3 ) 2 or Zn(NO 3 ) 2 salts respectively in ultrapure water (Millipore) with the following ligands: malate, citrate, oxalate, phytate, and histidine. Supplementary references from previous experiments were also used to increase the number of reference spectra, such as Ni in aqueous solution (Ni-aqueous), Ni-citrate with a metal:ligand ratio equal to 1 (Ni-citrate) and Zn sulfate. The solutions were diluted to 10 mM [Ni or Zn 2+ ] before analysis. The pH of the standards was checked, and adjusted to 6. The aqueous standards were then applied to filter paper (Whatmann), allowed to dry and enveloped in Kapton tape before scanning.

  Fig 1) provides intriguing insight in the distribution of Ni at the tissue-level. The circular features are suggestive of major enrichment in the apoplast surrounding large epidermal cells. In contrast, Zn is distributed primarily surrounding the secondary vasculature across the leaf blade. Calcium is clearly depleted in the veins, but high in the interveinal regions. The distribution of Co (not shown) mirrors that of Ni, with enrichment in the leaf margins, and towards the leaf apex. Potassium is enriched throughout the leaf blade, but highest near the petiole. In the leaf cross-section of N. tymphaea Ni and Zn have contrasting localizations (Fig 4 and Suppl.

Fig 2 )

 2 Fig 2). Whereas Ni is present in vacuoles of epidermal cells, Zn occurs mainly in the mesophyll

  ultramafic conditions and remained adapted to Ni hyperaccumulation. This study has shown that XFM can successfully be applied to help answering questions about the mechanisms of trace element hyperaccumulation, providing elemental distribution and chemical speciation in fresh/live hyperaccumulator plant tissues. . The dissimilar ecophysiological behaviour of Ni and Zn in N. tymphaea and B. emarginata raises questions about the evolution of hyperaccumulation in these species. Given that Zn accumulation is constitutive in Noccaea spp. occurring in non-metalliferous populations, Ni hyperaccumulation may have evolved as an adaptation when plants colonised ultramafic soils. In comparison, B. emarginata is not a Zn hyperaccumulator, and only hyperaccumulates (Ni) on metalliferous soils.

FIGURE CAPTIONS Figure 1 .

 CAPTIONS1 FIGURE CAPTIONS

Figure 2 .

 2 Figure 2. Elemental µXRF maps of fresh Noccaea tymphaea stem hand cut section. The maps measure 1.72 × 1.78 mm (430 × 444 pixels). The elemental image was acquired in 4-µm step size with 7 ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps. Abbreviations annotated of anatomical features: C cortex, Xy xylem, Ph phloem.

Figure 3 .

 3 Figure 3. Elemental µXRF maps of fresh Noccaea tymphaea whole mature leaf. The maps measure 12.55 × 9.28 mm (502 × 371 pixels). The elemental image was acquired in 25-µm step size with 10 ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps.

Figure 4 .

 4 Figure 4. Elemental µXRF maps of fresh Noccaea tymphaea leaf hand cut section. The maps measure 4.45 × 0.91 mm (890 × 181 pixels). The elemental image was acquired in 5-µm step size with 12m ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps. Abbreviations annotated of anatomical features: UE epidermis, LE epidermis, PM palisade mesophyll, SM spongy mesophyll, Xy xylem, Ph phloem.

Figure 5 .

 5 Figure 5. Elemental µXRF maps of Noccaea tymphaea intact silique. The maps measure 4.91 × 10.02 mm (327 × 668 pixels). The elemental image was acquired in 15-µm step size with 10 ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps.

Figure 6 . 5 × 6 .

 656 Figure 6. Elemental µXRF maps of Bornmuellera emarginata intact silique. The maps measure 7.5 × 6.52 mm (375 × 326 pixels). The elemental image was acquired in 20-µm step size with 20 ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps.

Figure 7 .

 7 Figure 7. Elemental µXRF maps of Noccaea tymphaea intact whole seed. The maps measure 1.83 × 1.15 mm (1143 × 717 pixels). The elemental image was acquired in 1-µm step size with 20 ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps.

Figure 8 .

 8 Figure 8. Elemental µXRF maps of Bornmuellera emarginata intact whole seed. The maps measure 4.52 × 3.73 mm (903 × 746 pixels). The elemental image was acquired in 5-µm step size with 20 ms dwell per pixel, 11.0 keV, incident beam, showing K, Ca, Ni and Zn maps.

Figure 9 .

 9 Figure 9. Single-slice tomography µXRF maps of Noccaea tymphaea intact whole seed. The elemental image was acquired in 2-µm step size with 1 ms dwell per pixel, with 15.0 keV as the energy of the incident beam, showing Compton, Fe, Ni and Zn K maps.

Figure 10 .

 10 Figure 10. Nickel speciation within the Noccaea tymphaea seed. A Principal Component Analysis (PCA) was performed on the stack of fluorescence scans, deciphering 4 regions of interest (A and B) from which XANES spectra were extracted. 2 supplementary spectra were extracted from the whole seed (white dotted line on picture A) and from the tip of the hypocotyl (black dotted line on picture A). Panel C shows the corresponding XANES, compared to Ni-malate and Ni-histidine spectra. Panel D displays the linear combination fitting (red dotted line) for one spectrum.

Figure 11 .

 11 Figure 11. Zinc speciation within the Noccaea tymphaea seed. A PCA was performed on the stack of fluorescence scans, deciphering 4 regions of interest (A and B) from which XANES spectra were extracted. 2 supplementary spectra were extracted from the whole seed (white dotted line on picture A) and from the tip of the hypocotyl (black dotted line on picture A). Panel C presents the different spectra and compares them to Zn-phytate spectrum recorded in the same conditions on P06. Spectrum 3 displays a high background level, preventing a correct interpretation and was discarded. Panel D shows the linear combination fitting (red dotted line) and the fitting residual (green line) for one spectrum.

Table 1 .

 1 Macro and trace element concentrations in roots, stems and flowers of Bornmuellera emarginata (values in µg g -1 dry weight) with ICP-AES. <LQ is lower than the limit for quantification.

	Species	Part	Al	Ca	K	Mg	Mn	Na	P	S
		Roots	<LQ	7900	11 325	1818	32	5570	1068	3559
		Roots	875	6349	10 070	3335	73	5416	1871	6492
	Bornmuellera emarginata	Stems	<LQ	14 430	23 535	3576	51	3037	2806	10 735
		Stems	1336	4583	13 350	4565	212	3669	1632	4531
		Flower	<LQ	21 010	15 350	6154	95	<LQ	2372	13870
	Species	Part	Fe	Mn	Zn	Co	Cr	Cu	Ni	Zn
		Roots	62	32	193	2.1	1.1	18	539	200
		Roots	1743	73	177	8	43	35	289	166
	Bornmuellera emarginata	Stems	43	51	251	4.9	1.0	20	1030	259
		Stems	2936	212	<LQ	16	66	21	499	90
		Flower	<LQ	95	311	43	<LQ	47	3641	306

Table 2 .

 2 Macro and trace element concentrations in leaves, fruits, and seeds of the actual Noccaea tymphaea and Bornmuellera emarginata samples used for XFM elemental mapping (values in µg g -1 dry weight) with ICP-AES. <LQ is lower than the limit for quantification.

	Species	Organ	P	S	Mg	K	Ca	Fe	Mn	Zn	Ni
	Noccaea tymphaea	Leaves	616	2284	3496	4059	9549	218	26	362	12 410
	Bornmuellera emarginata	Fruits	1639	8564	2996	7742	6317	38	7.8	96	10 345
	Noccaea tymphaea	Seeds	3834	5738	1628	6087	3701	<LQ	50	112	2504
	Bornmuellera emarginata	Seeds	2926	15 159	1775	7800	4419	24	12	69	6242

Acknowledgements

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were carried out at PETRA III, including beamtime granted within the in-house research program of DESY, and we would like to thank Jan Garrevoet and Gerald Falkenberg for assistance in using P06. The research leading to this result has been supported by the project CALIPSOplus under the Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. A. van der Ent was the recipient of a Discovery Early Career Researcher Award (DE160100429) from the Australian Research Council.