Demetrio Macías

Gustavo Olague

Eugenio Méndez
email: emendez@cicese.mx

Mac Demetrio

Ías

Eugenio R M Éndez

Charles Delaunay

Inverse scattering with far-field intensity data: random surfaces that belong to a well-defined statistical class

Introduction

The scattering of waves from randomly rough surfaces has a long history and is a subject of importance in many fields of physics. The direct scattering problem consists of finding the scattered field or the scattered intensity from knowledge of the surface profile, its optical properties, and the conditions of illumination. Considerable progress towards an understanding of the phenomenon has been made in the last few decades [START_REF] Beckmann | The Scattering of Electromagnetic Waves from Rough Surfaces[END_REF][START_REF] Ogilvy | Theory of Wave Scattering from Random Rough Surfaces[END_REF][START_REF] Desanto | Analytical techniques for multiple scattering from rough surfaces[END_REF][START_REF] Shchegrov | Multiple scattering of light from randomly rough surfaces[END_REF].

Rough surface inverse scattering problems have also received some attention, and take a variety of forms. Most of the studies have focused on the retrieval of some basic statistical parameter of the surface, such as the standard deviation of heights, or the height correlation length [START_REF] Léger | Optical surface roughness determination using speckle correlation technique[END_REF][START_REF] Chandley | Determination of the autocorrelation function of height on a rough surface from coherent light scattering[END_REF][START_REF] Welford | Optical estimation of statistics of surface roughness from light scattering measurements[END_REF][START_REF] Elson | Relation between the angular dependence of scattering and the statistical properties of optical surfaces[END_REF][START_REF] Stover | Calculation of surface statistics from light scatter[END_REF]. These methods are based on approximate solutions to the direct problem, like the Kirchhoff approximation [START_REF] Beckmann | The Scattering of Electromagnetic Waves from Rough Surfaces[END_REF][START_REF] Ogilvy | Theory of Wave Scattering from Random Rough Surfaces[END_REF] or small-amplitude perturbation theory [START_REF] Ogilvy | Theory of Wave Scattering from Random Rough Surfaces[END_REF], and on simple models for the statistical properties of the surface. A novel approach, based on a reverse Monte Carlo simulation, was reported more recently for the estimation of the power spectrum of surfaces in situations in which multiple scattering is significant [START_REF] Malyshkin | Determination of surface profile statistics from electromagnetic scattering data[END_REF].

A different kind of inverse problem is that of designing a surface with specified scattering properties. This field has developed rapidly in recent years. Procedures to design and fabricate one-dimensional surfaces that produce rather arbitrary angular scattering distributions have been developed [START_REF] Leskova | Design of one-dimensional band-limited uniform diffusers of light[END_REF][START_REF] Méndez | Photofabrication of random achromatic optical diffusers for uniform illumination[END_REF][START_REF] Maradudin | Design of one-dimensional Lambertian diffusers of light[END_REF][START_REF] Méndez | Design of one-dimensional random surfaces with specified scattering properties[END_REF]. Extensions to the case of general two-dimensional angular distributions are not simple, but procedures to generate such diffusers have been recently put forward [START_REF] Maradudin | Two-dimensional random surfaces that act as circular diffusers[END_REF][START_REF] Méndez | Design of two-dimensional random surfaces with specified scattering properties[END_REF].

The problem of recovering surface profiles from complex amplitude far-field data has also been the subject of some studies [START_REF] Wombel | Reconstruction of rough-surface profiles with the Kirchhoff approximation[END_REF][START_REF] Wombel | The reconstruction of shallow rough-surfaces profiles from scattered field data[END_REF][START_REF] Quartel | Surface reconstruction using an algorithm based on confocal imaging[END_REF][START_REF] Quartel | A surface reconstruction algorithm based on confocal interferometric profiling[END_REF][START_REF] Macías | Inverse scattering with a wave-front-matching algorithm[END_REF]. An important limitation of these methods is that they are based on approximate models for the interaction between the incident light and the surface, and fail when multiple scattering is important. In addition, since optical detectors are not phase sensitive, the need to have amplitude rather than intensity data constitutes a drawback that these methods share.

In the present work, we investigate the possibility of recovering the surface profile function from far-field intensity data. We approach the problem as a problem of constrained optimization. In previous communications [START_REF] Macías | Surface profile reconstruction from scattered intensity data using evolutionary strategies[END_REF][START_REF] Macías | Hybrid evolution strategy-downhill simplex algorithm for inverse light scattering problems[END_REF] we studied the performance of rough surface inversion algorithms based on evolutionary strategies using two kinds of representations of the objective variables. In [START_REF] Macías | Surface profile reconstruction from scattered intensity data using evolutionary strategies[END_REF], we employed a spectral representation of the surface and studied the performance of two selection strategies; the elitist and non-elitist strategies. The search space was reduced by considering surfaces that belong to a well-defined statistical class. This constraint did not permit the use of intermediate recombination, as it would eventually lead to departures from the assumed statistics. To circumvent this problem, in [START_REF] Macías | Hybrid evolution strategy-downhill simplex algorithm for inverse light scattering problems[END_REF] the surface was represented in terms of spline curves, permitting the use of intermediate recombination and the treatment of deterministic profiles. It was found that the use of intermediate recombination helped in the convergence of the algorithm.

In this paper, we consider the problem studied in [START_REF] Macías | Surface profile reconstruction from scattered intensity data using evolutionary strategies[END_REF] and explore the effects of using dominant recombination, or crossover, on the performance of the algorithm. Dominant recombination does not produce the kind of problems introduced by the intermediate recombination operator. In addition, we illustrate the robustness of the method presenting data corresponding to five different surfaces belonging to the same statistical class.

The methods described here, and in [START_REF] Macías | Surface profile reconstruction from scattered intensity data using evolutionary strategies[END_REF][START_REF] Macías | Hybrid evolution strategy-downhill simplex algorithm for inverse light scattering problems[END_REF], do not rely on approximate expressions for the field-surface interaction. Furthermore, our experience indicates that the convergence to the optimum improves when multiple scattering occurs. Since, in general, the use of intensity data implies that the solution to the problem is not unique, the improvement in the convergence is possibly due to a reduction in the number of solutions in the presence of multiple scattering.

The paper is organized as follows. Section 2 introduces the notation and presents a brief account of the direct scattering problem for one-dimensional surfaces. In Section 3, we formulate the inverse problem as an optimization problem, describing the evolutionary algorithm in some detail and discussing the differences between the intermediate and dominant recombination. Representative results are presented in Section 4 and, finally, in Section 5 we present our conclusions.

Scattering by one-dimensional rough surfaces

We consider the scattering of light from a one-dimensional, perfectly conducting, randomly rough surface defined by the equation x 3 = ζ (x 1 ). The region ζ (x 1 ) > x 3 is the vacuum, the region ζ (x 1 ) < x 3 is a perfect conductor, and the plane of incidence is the x 1 x 3 -plane. With reference to figure 1, the surface is illuminated from the vacuum side with a p-or s-polarized plane wave.

The single non-zero component of the electric or magnetic vector of the incident field has the form

ψ 2 (x 1 , x 3 |ω) inc = ψ 0 exp{i[kx 1 -α 0 (k)x 3 ]}, (1)
where α 0 (k) = (ω/c) 2k 2 , ω is the frequency of the field, and c is the speed of light in vacuum. A time dependence of the form exp(-iωt)isassumed, but explicit reference to it is suppressed. The field scattered by the surface is

ψ 2 (x 1 , x 3 | ω) sc = ∞ -∞ dq 2π R p,s (q | k)exp{iqx 1 + iα 0 (q)x 3 }, (2)
where the scattering amplitude R p,s (q | k) can be written in the form [24]

R p,s (q | k) = i 2α 0 (q) ∞ -∞ dx 1 exp{-iqx 1 -iα 0 (q)ζ (x 1 )}χ(k,ζ ′ (x 1 ))F(x 1 | ω). (3) The factor χ (k,ζ ′ (x 1 )) is χ (k,ζ ′ ) = -i[kζ ′ (x 1 ) -α 0 (k)] in p polarization 1i nspolarization, (4)
and F(x 1 | ω) represents a source function, given by

F(x 1 | ω) = ψ 2 (x 1 , x 3 ) | x 3 =ζ (x 1 ) in p polarization [∂ψ 2 (x 1 , x 3 )/∂ N ] | x 3 = ζ (x 1 ) in s polarization, (5)
where

∂/∂N = N•∇ = [-ζ ′ (x 1 )(∂/∂x 1 )+(∂/∂x 3 )
] is the non-unit normal derivative operator. The angles of incidence θ 0 and scattering θ s are related to the components of the wavenumbers k and q that are parallel to the mean surface through the expressions

k = ω c sin θ 0 , q = ω c sin θ s . (6)
The far-field intensity I p,s (q | k)isdefined as

I p,s (q | k) =|R p,s (q | k)| 2 . ( 7
)
In the present work, the goal is to retrieve the unknown surface profile function ζ (x 1 ) from the intensity data I p,s (q | k). With the squaring operation of the scattering amplitude R p,s (q | k) the phase information is lost, and this adds to the difficulties of inverting the data. Among other things, the solution to the problem is not unique. That is, there can be two or more surfaces that give rise to the same intensity scattering pattern. Aspects of the lack of uniqueness of the solution will be discussed in Section 4.

Inverse scattering as a least-squares approximation problem

From the previous discussion, it is clear that establishing an inversion scheme to retrieve the profile of a rough surface from scattered intensity data is not an easy task. In this section we reformulate the problem of inverse scattering in terms of a nonlinear least-squares boundsconstrained approximation problem. It is assumed that we have access to far-field angleresolved scattered intensity data corresponding to several angles of incidence. The goal is to retrieve the unknown surface profile function from these data. Some constraints on the kind of surface that we seek are introduced in order to reduce the search space.

Definition of the fitness functional

The inverse scattering problem can be reformulated in terms of the fitness (objective) functional f (z(x 1 )) = min

N ang i = 1 I (m) (q | k i ) -I (c) (q | k i ; z(x 1 )) 2 2 , (8)
where the symbol • 2 represents the Euclidean norm of the intensity vectors as functions of the scattering wavevector components q, N ang is the number of angles of incidence considered, and the k ′ i s are related to those angles through equation [START_REF] Chandley | Determination of the autocorrelation function of height on a rough surface from coherent light scattering[END_REF]. Then, I (m) (q | k i ) represents an angle-resolved far-field scattered intensity pattern of the surface of interest (measured or calculated) and I (c) (q | k i ; z(x 1 )) is a calculated intensity pattern obtained by solving the direct problem with a trial surface profile z(x 1 ). The functional f (z(x 1 )) can be interpreted as an assessment of the closeness between the angular distributions of intensity I (m) (q | k) and I (c) (q | k; z(x 1 )). The goal would then be to find a surface for which the condition

I (c) (q | k) = I (m) (q | k)issatisfied.
When this happens, and if the solution to the problem is unique, the best approximation to the original profile has been retrieved.

Note that in our definition of the fitness functional we require that the proposed surface reproduces the 'measured' scattering data for several angles of incidence. The satisfaction of this constraint should reduce the number of possible solutions and, hopefully, produce a unique one. The inverse scattering problem can be viewed, now, as the problem of minimizing f (ζ c (x 1 )).

Representation of the objective variables

To deal with the scattering problem numerically, the surface must be sampled. From the preceding discussion it seems natural to choose, as the parameters of interest, the surface heights evaluated at the sampling points. Changing these numbers independently, however, would lead to surfaces with abrupt height changes, which does not correspond to the physical situation of interest. One way to avoid this problem is to restrict the search space to randomly rough surfaces that belong to a certain class. We are, thus, faced with a problem of constrained optimization.

In our case, we have chosen the target surface as a realization of a stationary, zero-mean one-dimensional Gaussian random process. With this assumption, the random process is completely characterized by its two-point correlation function, which we also assume to be Gaussian:

ζ (x 1 )ζ (x ′ 1 ) =δ 2 exp{(x 1 -x ′ 1 ) 2 /a 2 } . (9)
Here, the angle brackets denote an average over the ensemble of realizations of the surface profile function,

δ = ζ 2 (x 1 )
Surfaces belonging to this class can be generated numerically with the spectral method described in [START_REF] Maradudin | Enhanced backscattering of light from a random grating[END_REF]. Correlated random numbers that represent the surface heights at the sampling points can be obtained through the expression

ζ n = δ √ L N /2-1 j =-N /2 [M j + iN j ] √ 2 √ πa exp - aq j 2 2 1/2 exp{iq j χ n }. (10)
Here, N represents the total number of points on the surface, L represents its length, χ n = -L/2 + (n -0.5) x are the sampling points spaced by x along x 1 , q j =-π/ x + 2π ( j -0.5)/L are the sampling points in Fourier space, and

ζ n = ζ (χ n ).
The random sets {M j } and {N j } contain statistically independent random Gaussian variables with zero mean and unit standard deviation. In order to produce a set of real random numbers {ζ n },itisrequired that the complex array {M j + iN j } be Hermitian. The first and second derivatives of the surface profile function, which are required for the direct scattering calculations, can be obtained by differentiation of equation ( 10).

Evolutionary inversion procedure

At least in principle, any of the optimization techniques reported in the literature could be employed to minimize equation [START_REF] Elson | Relation between the angular dependence of scattering and the statistical properties of optical surfaces[END_REF]. However, as discussed in [START_REF] Macías | Surface profile reconstruction from scattered intensity data using evolutionary strategies[END_REF], the form of equation ( 8) and the constraint imposed by the representation scheme of equation [START_REF] Malyshkin | Determination of surface profile statistics from electromagnetic scattering data[END_REF] suggest the use of an algorithm belonging to the class of 'direct search methods' [START_REF] Kold | Optimization by direct search: new perspectives on some classical and modern methods[END_REF]. The main characteristic of this kind of technique is that, throughout the entire optimization process, one only needs to know the values of the fitness function and not its derivatives [START_REF] Nelder | A simplex method for function optimization[END_REF][START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF][START_REF] Bäck | Evolutionary computation: comments on the history and current state[END_REF].

Evolutionary algorithms are a relatively recent set of 'direct search methods' that have been successful in the solution of ill-posed inverse problems in different scientific disciplines [START_REF] Weile | Genetic algorithm optimization applied to electromagnetics: a review[END_REF][START_REF] Hageman | A hybrid genetic algorithmtabu search approach for optimising multilayer optical coatings[END_REF][START_REF] Hodgson | Genetic algorithm approach to particle identification by light scattering[END_REF][START_REF] Djurisic | Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment[END_REF]. Examples of these heuristic population-based techniques are the genetic algorithms [START_REF] Holland | Adaption in Natural and Artificial Systems[END_REF], the evolution strategies [START_REF] Schwefel | Evolution and Optimum Seeking[END_REF], and genetic [START_REF] Koza | Genetic programming[END_REF] and evolutionary programming [START_REF] Fogel | Autonomous automata[END_REF]. Notwithstanding their differences, all evolutionary algorithms are inspired on the Darwinian principles of variation and selection [START_REF] Salomon | Evolutionary algorithms and gradient search: similarities and differences[END_REF]. Given the characteristics of the inverse problem studied here, it was considered that the evolutionary strategies were the best-suited evolutionary algorithms for this task.

Several variations of evolutionary strategies have been proposed [START_REF] Beyer | The Theory of Evolution Strategies[END_REF]. All of them, however, follow the canonical structure shown in the flow diagram of figure 2. The starting point of the optimization process is the generation of a random set P g µ | g = 0 of µ possible solutions to the problem which, in the present context, are the Gaussian randomly rough one-dimensional surfaces {z n } generated through equation [START_REF] Malyshkin | Determination of surface profile statistics from electromagnetic scattering data[END_REF]. A secondary population P g λ of λ elements is generated through the application of the 'genetic' operations of recombination and mutation over the elements of the initial population P g µ . This represents the start of the main evolutionary loop.

At this point, it is pertinent to mention that we have previously used λ to denote the wavelength of the light, which is the usual notation in optical work. It is believed that due to the different context in which the two quantities are employed, use of the same symbol to denote both should not lead to much confusion.

Recombination exploits the search space through the exchange of information between different elements of the population. Mutation, on the other hand, explores the search space through the introduction of random variations in the newly recombined elements. Depending on the problem studied, it is possible to exclude the recombination operation from the evolutionary loop, as it is indicated by the broken line in figure 2. That is, in the search for z (1) the optimum, mutation could be the only operator employed. Once the secondary population P g λ has been generated, one needs to evaluate the quality of its elements. For this, the direct problem must be solved for each one of the newly generated surfaces of the secondary population. With this, a fitness value is associated to each trial surface. This is done through the comparison between the calculated scattering pattern I (c) (q | k i ; z n ) and the 'measured' data I (m) (q | k i ), on the basis of equation [START_REF] Elson | Relation between the angular dependence of scattering and the statistical properties of optical surfaces[END_REF]. Only those elements of the secondary population P g λ leading to promising regions of the search space will be retained, through some selection scheme, as part of the population P g + 1 µ

for the next iteration of the evolutionary loop. The procedure is repeated until a defined termination criterion has been achieved. The respective sizes of the initial and the secondary populations, P g µ and P g λ , remain constant throughout the entire search process.

Recombination and mutation.

In the search for the optimum, new trial surfaces are generated from the initial set of surfaces (population). Surfaces can be generated by selectively In the intermediate recombination, new surfaces are generated through linear combinations of the parent population. In our case, the first time that this operator is used the new surfaces are the result of linear combinations of independent Gaussian random processes. The result is also a Gaussian process and one can adjust the parameters of the combination in such a way that the new surfaces belong to the required statistical class. In successive iterations, however, the surfaces become more and more correlated, and the same operation generates surfaces that do not belong to the assumed statistical class. Intermediate recombination is, thus, not compatible with the representation and constraints imposed on the problem.

In the dominant recombination, or crossover, the surfaces exchange elements of the Hermitian arrays (see equation 10). The resulting surfaces are Gaussian and belong to the same statistical class as the parent population. With this scheme, the degree of correlation between the surfaces is unimportant and the operator itself does not introduce any bias.

Mutations are introduced by changing some of the elements of the Hermitian array employed in the generation of a given surface (see figure 3). In this case, provided that the new numbers, M j and N j , are zero-mean Gaussian-distributed random numbers with unit standard deviation, and the hermiticity of the array is conserved, the new surface will belong to the statistical class specified for the search space.

Selection.

The selection operator generates, through a deterministic process, the set of surfaces P g µ that will serve as the population for the next iteration of the algorithm. There are two selection procedures employed in evolution strategies. The first one is known as the 'elitist' or (µ + λ) strategy, whereas the second one is called the 'non-elitist' or (µ, λ) strategy. In the (µ, λ) scheme, the elements to be selected belong, exclusively, to the secondary population P g λ .A nimportant consequence of this is the possibility that the best elements of the new population P g+1 λ are less fit than the best element of the previous population P g λ . This possible deterioration of the fitness values helps the algorithm avoid regions of attraction that could lead to premature convergence to a local minimum [START_REF] Bäck | Evolutionary computation: comments on the history and current state[END_REF]. Of course, if the deterioration persists, the algorithm diverges.

In the elitist or (µ + λ) scheme, the new population P g+1 µ is drawn from the two sets; that is, from the initial population P g µ and the secondary one, P g λ . Only the best elements are kept. In this case, there is never a deterioration of the fitness value and, for the case of minimization, the elitist scheme guaranties a monotonic decrease of the fitness values. This fact, however, makes the algorithm prone to premature convergence, as it can be trapped in regions of attraction associated with local minima.

Results

Ultimately, the data that serves as input to the algorithm should be obtained experimentally. However, in order to study and optimize the performance of the algorithms, in these studies we use data obtained through a rigorous numerical solution of the direct scattering problem [START_REF] Maradudin | Enhanced backscattering of light from a random grating[END_REF].

For the two strategies explored, (µ + λ) and (µ, λ), each element of the initial population consisted of a realization of a zero-mean stationary Gaussian-correlated Gaussian random process with a 1/e-value of the correlation function a = 2λ and standard deviation of heights δ = 0.5λ. Also, we chose the typical values µ = 10 and λ = 100 [START_REF] Schwefel | Evolution and Optimum Seeking[END_REF]. In both cases, the maximum number of iterations was g = 300, which also provided the termination criterion. To illustrate the procedure, we now present a detailed discussion of the recovery of a profile.

The surface profile used to generate the original scattering data is shown in figure 4. The surface was sampled at intervals of λ/10. Since the time of computation required to find the optimum increases with the number of sampling points on the surface, and the direct problem needs to be solved many times, in order to keep the problem to a manageable size we chose a surface with N = 128 sampling points.

The data from which the profile is to be recovered were obtained by illuminating the surface in figure 4 For reference, we first present results without the use of recombination. The target profile was searched starting from 30 different, and randomly chosen initial states. As it can be expected when using a heuristic method, not in all of these attempts to recover the profile the algorithms converged to the target surface. We found, however, that in most cases a low value of f (z n ) corresponded to a profile that was close to the original one (some of the exceptions are discussed below). So, the final value of f (z n )was used as the main criterion to choose the final reconstructed profile.

In figure 6 we present results obtained with the two evolution strategies studied considering the same initial population. To facilitate the visualization of the results, the original profile is shown with a dotted curve. The profile retrieved with the elitist strategy is shown with a solid curve, whereas the profile retrieved with the non-elitist strategy is drawn with a dashed curve. The vertical and horizontal displacements of the profile are understandable, as the far-field intensity is insensitive to such shifts. On the other hand, such displacements are unimportant for practical profilometric applications. Qualitatively at least, it is considered that in this case the two algorithms were able to retrieve the profile.

Since the Euclidean norm of the difference between {z n } and {ζ n } is sensitive to vertical and lateral displacements of the profiles, it would not constitute an appropriate measure of the accuracy of the solution. For this, let us first define, respectively, the sample mean and (µ+λ) (µ,λ) Figure 6. Reconstruction of the surface profile using the (µ + λ) strategy (solid curve) and the (µ, λ) strategy (dashed curve). The original profile is plotted with a dotted line.

variance of the original profile as

ζ = 1 N N n-1 ζ n , (11)
σ 2 ζ = 1 N -1 N n-1 (ζ n -ζ ) 2 , ( 12
)
and similarly for the trial profile. Defining the new profiles

z n = z n -z, ( 13
)
ζ n = ζ n -ζ, ( 14
)
we propose the following metric for the resemblance of the profiles

ǫ = 1 2σ 2 ζ min 1 (N -|l |) n ( z n+l -ζ n ) 2 . ( 15
)
The subtraction of the sample means and the free subindex l in equation ( 15) reduce the sensitivity of the error parameter ǫ to vertical and lateral displacements of the recovered profile. For perfectly correlated surfaces, this parameter is zero. On the other hand, for uncorrelated random surfaces that belong to the same statistical class, the average ǫ should be of order one (forgetting about the shift). The recovered profiles shown in figure 6 have error parameters ǫ (µ+λ) = 0.006 and ǫ (µ,λ) = 0.004 for the elitist and non-elitist strategies, respectively. The curves in figure 7 represent the convergence behaviour associated with the reconstructions of figure 6. The solid curve shows a monotonic decrease of the fitness value, which is typical of the elitist strategy. The dashed curve shows the convergence of the non-elitist strategy towards the stationary point. One can see that the fitness value can increase in some iterations and that the convergence of the algorithm is slower. However, after 300 iterations, the respective fitness values f (z n ) (µ+λ) = 10.2 and f (z n ) (µ,λ) = 13.7 reached with the elitist and non-elitist strategies are similar.

An interesting result that demonstrates the lack of uniqueness of the solution when intensity data are used is shown in figure 8. The dotted curve represents the original profile, while the reconstructions obtained with the strategies (µ + λ) and (µ, λ) are represented with solid and broken lines, respectively. For this numerical experiment, the initial population was the same for the two strategies, but different from the one used in the examples of figure 6. One can see that, in this case, the recovered profiles do not resemble the original one. They have (µ+λ) (µ,λ) Figure 8. Reconstruction of the surface profile using the (µ + λ) (solid line) and (µ, λ) (dashed line) strategies starting from a different initial population than in figure 6. The original profile is plotted with a dotted line. associated error parameters and fitness values ǫ (µ+λ) = 0.205 and f (z n ) (µ+λ) = 23.5 (elitist) and ǫ (µ,λ) = 0.291 and f (z n ) (µ,λ) = 21.8 (non-elitist). As explained below, these results illustrate a curious symmetry property of the scattering problem for situations in which the Kirchhoff approximation is valid.

The profiles recovered in figure 8 are presented in figure 9, but reflected with respect to the x 1 and x 3 axes; that is, we replace z(x 1 )b y-z(-x 1 ). Surprisingly, one observes that the resulting profiles resemble the sought one. The error parameters are now ǫ (µ+λ) = 0.027 (elitist) and ǫ (µ,λ) = 0.007 (non-elitist). To better understand this result, let us consider the direct scattering problem in the Kirchhoff approximation. The far-field scattering amplitude R p,s (q | k) can be written in the form [

2] R (K ) p,s (q | k) = F p,s (θ 0 ,θ s ) ∞ -∞ dx 1 exp{-iv x x 1 -iv z ζ (x 1 )}, (16)
where F p,s (θ 0 ,θ s )isanangular factor given by

F p,s (θ 0 ,θ s ) =± 1 + cos θ 0 cos θ s -sin θ 0 sin θ s cos θ 0 + cos θ s , (17)
v x = kq, and v z = α 0 (k) + α 0 (q), with k and q defined as in equation [START_REF] Chandley | Determination of the autocorrelation function of height on a rough surface from coherent light scattering[END_REF]. The '+' sign in equation ( 17) is for p polarization, and the '-' sign for s polarization. If we now consider the mirror profile ζ m (x 1 ), such that ζ m (x 1 ) =-ζ (-x 1 ), we can write its scattering amplitude as

R (K ) p,s m (q | k) = F p,s (θ 0 ,θ s ) ∞ -∞ dx 1 exp{-iv x x 1 + iv z ζ (-x 1 )}. ( 18
)
With the change of variable u =-x 1 ,itcan be readily shown that the two intensity patterns are equal:

I (K ) p,s (q | k) = [I (K ) p,s (q | k)] m . (19)
Thus, the validity of the Kirchhoff approximation leads to multiple solutions of the inverse scattering problem. It should be mentioned that, in general, the rigorous solution of the direct problem does not have this kind of symmetry. It is thus tempting to think that multiple scattering effects, which invalidate the results obtained with the Kirchhoff approximation, reduce the number of possible solutions of the inverse problem.

To complete this example, we discuss some results obtained when dominant recombination is included within the evolutionary loop. The number of elements to be combined is ρ d = µ. Furthermore, we consider the same illumination conditions, number of realizations, and initial states as those employed in the studies that led to figure 6.

With recombination, we observed a deterioration in the convergence behaviour of the algorithms. The reconstructed profiles corresponding to the lowest fitness values, f (z n ) (µ/ρ d +λ) = 30.3 and f (z n ) (µ/ρ d ,λ) = 29.4, retrieved with the (µ/ρ d +λ) and (µ/ρ d ,λ) strategies are shown in figure 10 with solid and broken lines, respectively. To facilitate the comparison, the original profile is drawn with a dotted curve. Although at first sight, there seem to be important differences between the original and recovered profiles, lateral and vertical translations show that the main features are preserved in the reconstructions. This is verified by the calculated error parameters, which are ǫ (µ/ρ d +λ) = 0.042 and ǫ (µ/ρ d ,λ) = 0.043.

We close this section with a summary of more extensive studies of the retrieval of surface profile functions. First, in figure 11, we present four profiles belonging to the same statistical class as the surface of figure 4 (δ = 0.5λ and a = 2λ), together with the reconstructed ones using the elitist and non-elitist strategies without recombination. If, in all cases, we keep only the reconstruction with the lowest value of the fitness functional, the error parameters associated with the reconstructions are in the range 0.002 ≤ ǫ ≤ 0.014. So, at least for this statistical class and the chosen geometry, it can be said that the profile can be retrieved with a high degree of confidence.

One should be careful, however, in trying to extrapolate on these results. As an example of this, we present reconstructions of a surface belonging to different statistical class (δ = λ and From the error parameters and the appearance of the curves, it is clear that the reconstruction obtained with the elitist strategy is a much better approximation to the original profile. However, the fitness value obtained with the non-elitist strategy is lower than the one obtained with the elitist strategy. This is probably an indication of the lack of uniqueness of the solution.

Conclusions

We have presented a study of two evolutionary algorithms to solve inverse scattering problems. Starting from far-field intensity data and using both the (µ + λ) and (µ, λ) strategies, we have successfully retrieved the surface profile that generated the scattering data. The time of computation is similar, but the elitist strategy (µ + λ) uses twice as much memory as the non-elitist strategy (µ, λ).

Although it appears that including the recombination operation degrades the performance of the inversion procedures, the partial results obtained are far from being conclusive. Moreover, apart from the works of Beyer [START_REF] Beyer | The Theory of Evolution Strategies[END_REF] on the (µ/ρ d ,λ) strategy, we are not aware of other systematic studies on the subject, and in particular with the (µ/ρ d + λ) strategy.

The numerical evidence presented here suggests that the evolutionary strategies are well adapted for inverse scattering applications. Further work is required to understand the influence of the dominant recombination on the dynamics of the inversion scheme. Another important consideration is the effect that noisy data have on the reconstructions. Some results (not presented here for brevity) have been obtained with input data contaminated with noise; they indicate that the inversion schemes are fairly insensitive to the presence of Gaussian additive noise [START_REF] Macías | Estudios numéricos de esparcimiento inverso de ondas electromagnéticas[END_REF]. More systematic studies would be necessary to assess the tolerances of the inversion method to the presence of more general kinds of noise.

Since the solution of the inverse problem is not necessarily unique, it seems appropriate to use a stochastic method to search for the solution. The fitness function has many local minima, and the initialization of the algorithm plays an important role in its convergence. In most of the cases we have studied, the stationary point at the end of the evolutionary loop not only seems to be associated with a low value of the fitness functional, but it also provides the best approximation to the problem.

We have also presented cases in which the problem has more than one solution and cases in which the best solution does not correspond to the surface with the lowest fitness value. In such situations, one would have to perform further tests to decide on the best one. Although the problem has many facets and can be rather complex, the results obtained so far are encouraging.

The success of an inversion scheme based on intensity information opens the possibility of implementing such a procedure experimentally. However, further work is needed, not just regarding the physical aspects of the problem, which have been simplified by our assumptions, but also regarding aspects related to the performance of the evolutionary algorithms.

3 Figure 1 .

 31 Figure 1. Geometry of the scattering problem.

Figure 2 .

 2 Figure 2. Flow diagram of the evolutionary loop.

Figure 3 .

 3 Figure 3. Schematic representation of the mutation operation.

3 Figure 4 .

 34 Figure 4. Profile used in the generation of the scattering data.

Figure 5 .

 5 Figure 5. Scattered intensity produced by the surface depicted in figure 4 for the case of normal incidence.

3

 3

Figure 7 .

 7 Figure 7. Convergence behaviour of the (µ + λ) strategy (solid curve) and the (µ, λ) strategy (dashed curve).

3

 3

Figure 9 .

 9 Figure 9. Curves representing the profiles shown in figure8reflected with respect to both axes (circles). The original profile is shown with a dotted curve. The reconstructions obtained with the (µ+λ) and (µ, λ) strategies are represented with a solid and a dashed curve, respectively.

Figure 10 .

 10 Figure10. Reconstruction of the surface profile using the (µ/ρ d + λ) strategy (solid curve) and, the (µ/ρ d ,λ) strategy (dashed curve). The original profile is plotted with a dotted line.

Figure 11 .Figure 12 .

 1112 Figure 11. Reconstruction of four different surface profiles using the (µ + λ) (solid curve) and (µ, λ) (dashed curve) strategies without recombination. The original profile is plotted with a dotted line. (a) f (z n ) (µ+λ) = 17.1, ǫ (µ+λ) = 0.003, and f (z n ) (µ,λ) = 27.1, ǫ (µ,λ) = 0.008; (b) f (z n ) (µ+λ) = 29.3, ǫ (µ+λ) = 0.015, and f (z n ) (µ,λ) = 25.4, ǫ (µ,λ) = 0.003; (c) f (z n ) (µ+λ) = 10.7, ǫ (µ+λ) = 0.005, and f (z n ) (µ,λ) = 5.7, ǫ (µ,λ) = 0.002; (d) f (z n ) (µ+λ) = 14.0, ǫ (µ+λ) = 0.014, and f (z n ) (µ,λ) = 22.2, ǫ (µ,λ) = 0.048.a = 2λ). The surface is in fact the same one shown in figure4, but expanded in the vertical direction by a factor of two. The surface and the attempted reconstructions are shown in figure12. The reconstructed profiles have fitness values, f (z n ) (µ+λ) = 33.6 and f (z n ) (µ,λ) = 26.0, and error parameters ǫ (µ+λ) = 0.051 and ǫ (µ,λ) = 0.339 for the elitist and non-elitist strategies, respectively.

is the rms height of the surface, and a its correlation length.

Acknowledgments

DM gratefully acknowledges financial support of the Conseil Régional de Champagne-Ardenne. The work of ERM was supported by CONACYT, under grant 47712-F.


Laurence J Kershaw Cook

Helena J Shepherd

Tim P Comyn

Chérif Baldé

Oscar Cespedes

Guillaume Chastanet
email: guillaume.chastanet@icmcb.cnrs.fr

Malcolm A Halcrow
email: m.a.halcrow@leeds.ac.uk

Decoupled spin crossover and structural phase transition in a molecular iron(II) complex

2][BF4]2 (1; bppSMe=4-(methylsulfanyl)-2,6-di(pyrazol-1-yl)pyridine) undergoes an abrupt spin-crossover (SCO) event at 265±5 K. The crystals also undergo a separate phase transition near 205 K, involving a contraction of the unit-cell a axis to one-third of its original value (high-temperature phase 1; Pbcn, Z=12; low-temperature phase 2; Pbcn, Z=4). The SCO-active phase 1 contains two unique molecular environments, one of which appears to undergo SCO more gradually than the other. In contrast, powder samples of 1 retain phase 1 between 140-300 K, although their SCO behaviour is essentially identical to the single crystals. The compounds [Fe(bppBr)2][BF4]2 (2; bppBr=4-bromo-2,6-di(pyrazol-1-yl)pyridine) and [Fe(bppI)2][BF4]2 (3; bppI=4-iodo-2,6-di(pyrazol-1-yl)-pyridine) exhibit more gradual SCO near room temperature, and adopt phase 2 in both spin states. Comparison of 1-3 reveals that the more cooperative spin transition in 1, and its separate crystallographic phase transition, can both be attributed to an intermolecular steric interaction involving the methylsulfanyl substituents. All three compounds exhibit the light-induced excited-spin-state trapping (LIESST) effect with T(LIESST=70-80 K), but show complicated LIESST relaxation kinetics involving both weakly cooperative (exponential) and strongly cooperative (sigmoidal) components.

Introduction

The structural chemistry of molecular spin-crossover (SCO) materials1-4 continues to be of great interest, for two reasons.5, 6 One, is to elucidate the factors underlying cooperative spin-state switching so that new materials with useful switching properties can be designed for nanoscience and device applications.3 However, the study of SCO by multiple techniques also affords more general information about the underlying chemistry of crystallographic phase transitions, which is important for the crystal engineering of other types of functional material. Increasingly complex structural behaviour is being reported in spin-transition crystals,5 including: stepwise spin transitions coupled to multiple crystallographic phase changes;7, 8 incomplete SCO reflecting symmetry-breaking transitions generating a mixture of active and inactive sites in a material;9 and, thermal hysteresis induced by large changes in molecular conformation during SCO, rather than a change in crystallographic symmetry. 10 We report here [Fe(bppSMe)2][BF4]2 (1; bppSMe=4-(methyl-sulfanyl)-2,6-di(pyrazol-1-yl)pyridine), whose crystals exhibit a thermal spin transition near room temperature that is decoupled from a crystallographic phase change at lower temperature. Only two other SCO materials show comparable behaviour: [{Fe(NCS)2(3-bpp)}2(μ-4,4′-bipy)] (3-bpp=2,6-di(pyrazol-3-yl)pyridine; 4,4′-bipy=4,4′-bipyridyl) 11 and [FeL2][BF4]2 (L=2, 6-di(5-methylpyrazol-3yl)pyridine),12 which both undergo one or more crystallographic phase changes above their spin-transition P a g e | 2 temperatures. Two other complexes [Fe(bppBr)2][BF4]2 [START_REF]Spin-Crossover Materials-Properties and Applications[END_REF]pyridine) and [Fe(bppI)2][BF4]2 (3; bppI=4-iodo-2,6-di(pyrazol-1-yl)pyridine)13 are also described, which are isostructural to 1 at low temperature, but do not exhibit the crystallographic phase change. Comparison of these compounds has allowed us to identify the intermolecular interactions in 1 that cause its more complicated SCO and phase behaviour, and has thus afforded a structure-function relationship for SCO in this type of crystal lattice.

Results and Discussion

The ligands bppBr and bppI14 were both obtained from 4-amino-2,6-di(pyrazol-1-yl)pyridine (bppNH2)14 by a diazotisation procedure. Treatment of bppI with NaSH in DMF affords 4-mercapto-2,6-di(pyrazol-1-yl)pyridine (bppSH),15 which was methylated with MeI to give bppSMe, together with the disulfides bppSSbpp15 and bppSSMe (Scheme 1). Pure bppSMe can be isolated from the mixture by vacuum sublimation, whereas the disulfido products were purified from the involatile residue by silica column chromatography. Complexation of hydrated Fe[BF4]2 with 2 equivalents of the appropriate ligand in nitromethane yielded the complex salts 1-3, which all form solvent-free yellow crystals from nitromethane/diethyl ether mixtures.16

The variable-temperature magnetic susceptibility of a powder sample of 1 demonstrated a thermal spin transition with a small thermal hysteresis width of 2 K (T1/2↓=268 K, T1/2↑=270 K; Figure 1). Although the transition has no discontinuities at first glance, only about 80 % of the sample is in the low-spin state at 250 K (χMT=0.7 cm 3 mol -1 K) with the remainder of the material becoming low spin more gradually on further cooling. The first derivative of the susceptibility curve shows the transition is not symmetrical, and contains small discontinuities near 220 and 330 K (Figure 1).17 Therefore, SCO in 1 appears to contain two components: an abrupt, hysteretic transition undergone by about 70 % of the sample; and, a more gradual crossover with a similar midpoint temperature, involving the remaining 30 %. That suggestion is also supported by the differential scanning calorimetry (DSC) data described below. In contrast, 2 and 3 both exhibit more gradual SCO, with no masked features in their δ(χMT)/δT curves and midpoint temperatures of T1/2=307 K ( 2) and 332 K (3). The SCO cooperativity, as measured by the maximum value P a g e | 3 of δ(χMT)/δT, follows the order 1>3>2 (Figure 1). The SCO transition for 3 is essentially identical to that shown by the perchlorate salt of the same complex.13 Initial structure refinements showed that 1-3 are isostructural in the single crystal at 100 K, in the space group Pbcn. Their asymmetric units contain half a complex cation with crystallographic C2 symmetry, and one unique BF4 - ion (Figure 2). The iron centres are all low spin at this temperature, and show only small deviations from the idealised D2d symmetry that is expected with this ligand type (Table 1). The complex cations pack in layers in the crystals, related to the "terpyridine embrace" lattice that is often adopted by salts of [Fe(bpp)2] 2+ derivatives.12, 18-20 The cation layers in the full embrace structure have strict or approximate four-fold symmetry, with each molecule interdigitating to four nearest neighbours through face-to-face and edge-to-face π-π interactions between their pyrazolyl rings.21 In 1-3, although the layered structure is retained, this interdigitation is disrupted by the pyridyl substituents, which protrude into the adjacent layers (Figure 3). Thus, each [Fe(bppR)2] 2+ cation (R=SMe, Br or I) experiences a partial terpyridine embrace, forming just two intra-layer π-π interactions to face-toface pyrazolyl rings.

Unit-cell measurements showed that, in addition to SCO at 260-270 K, crystals of 1 undergo a second phase transition between 210 and 200 K, involving a contraction of the unit-cell a axis from 53-54 Å in the higher temperature phase (phase 1) to 18 Å in the low-temperature phase (phase 2). The spin transition near 270 K is accompanied by a sharp decrease in a, but has little effect on b or c (Figure 4). The resultant contraction in unit-cell volume (V), 260(27) Å 3 or 2.8 %, is typical for an abrupt thermal spin transition in an iron(II) complex salt. 6 The phase 1→phase 2 change around 205 K causes a small contraction in b as well as the aforementioned reduction in a, and is also reflected in a 23(6) Å 3 reduction in V (normalised to Z=4, Figure 4). Thus, the molecular packing in phase 2 of 1 is significantly more compact than the low-spin form of phase 1. Further cooling to 100 K leads to a small increase in a, which is offset by decreases in b and c to give the expected reduction in V with decreasing temperature. Investigation of multiple crystals revealed that the formation of phase 1 on warming above 200-210 K was always apparent in the diffraction images, from the ingrowth of additional diffraction peaks along a* (Figure 5). However, ingrowth of the additional reflections from phase 1 was not abrupt, and often occurred over a temperature range of 20-30 K.25 The relative intensities of the super-cell reflections, compared with the parent reflections, differed significantly between crystals, and upon repeated cycling of the same crystal.

Although crystals of 1 always diffract well at 100 K, they often suffer from twinning above the phase 2→phase 1 transition temperature. This twinning is reversed upon re-cooling the crystals below 205 K. Hence, the formation of the phase 1 super-cell along [100] may take place in an incoherent or disordered manner in those crystals. Several attempts were required to obtain good refinements of all the phases of 1 from the same crystal. One refinement of each phase was ultimately achieved at 290 K (HS-1), 240 K (LS-1) and 100 K (LS-2); HS-1 and LS-1 are isostructural in phase 1, but in the high-spin and low-spin states, respectively. Interestingly, crystals that diffracted well at 100 K had a much higher mosaicity at 30 K, which prevented us from obtaining a useful structure of 1 at that lower temperature.

The HS-1 and LS-1 phases both contain 1.5 molecules per asymmetric unit, with a half-molecule spanning a twofold rotation axis (molecule A) and a whole molecule on a general crystallographic site (molecule B). The iron centres are high spin in HS-1 at 290 K, and predominantly or fully low spin in LS-1 at 240 K, although there may be a residual high-spin fraction for molecule A in the latter structure (Table 1). That would be consistent with the magnetic data, which imply that about 15 % of the sample remains high spin at 240 K (Figure 1). The molecules differ in that molecule A has a more regular coordination geometry, closer to ideal D2d symmetry, than molecule B. That is reflected in the trans-N{pyridine}-Fe-N{pyridine} angle (φ in Table 1), which is close to the expected value of 180° in molecule A (178.6(3)° in both HS-1 and LS-1), but is significantly smaller in molecule B (168.3(2) and 175.2(2)°, respectively).23, 24 Notably, φ in HS-1 molecule B is unusually low for an SCO-active complex of this type, and the change in φ during SCO for molecule B (Δφ=6.9(3)°) is correspondingly large. 18 The greater change in shape during SCO for molecule B compared to molecule A (Δφ=0.0(4)°), implies that molecule B should undergo SCO more cooperatively than molecule A.5 That is consistent with the mixture of abrupt and gradual SCO evident in the susceptibility and DSC data for 1.

The A and B cation sites in phase 1 are segregated into distinct layers in the lattice, with an ABBABB stacking pattern along the crystallographic a axis. Nearest-neighbour cations in the A layers are related by an inversion centre and are aligned strictly co-parallel, whereas in the B layers they are related by mirror symmetry and are slightly canted with respect to each other. This leads to an undulation of the complex molecules in adjacent layers down the [100] vector.25 In LS-2, the cation layers are all identical and equivalent to the A layers in HS-1/LS-1, and the complex molecules are all aligned linearly along [100].

The π-π interactions within the layers involve pyrazolyl rings that are strictly or approximately co-planar by symmetry and separated by 3.5 Å. The dimensions of these interactions change only slightly between the different phases of 1.25 More variation occurs in intermolecular contacts involving the methylsulfanyl groups, however, which occupy cavities bounded by four pyrazolyl groups from two different cations in an adjacent layer. In HS-1 the intermolecular methyl⋅⋅⋅pyrazolyl distances are C⋅⋅⋅C≥3.640( 14) Å, close to the sum of the van der Waals radii of a methyl group and an aromatic ring (3.7 Å).26 However, in LS-1 there is a general contraction of these interactions, and the three unique methyl groups in this structure each form one close intermolecular C-H⋅⋅⋅π contact of C⋅⋅⋅C=3.452( 12), 3.531( 12) and 3.573( 15) Å. The shortest of these is associated with the unique methylsulfanyl group in molecule A, which exhibits a degree of disorder in the LS-1 phase. In LS-2 at 100 K, this ligand disorder is no longer apparent although the short C-H⋅⋅⋅π contact to the unique methylsulfanyl group is retained, at C⋅⋅⋅C=3.445( 5) Å (Figure 3).25 Relief of this steric clash may contribute to the driving force for the LS-2→LS-1 phase change. Even though there are also several cation⋅⋅⋅anion C H⋅⋅⋅F contacts in HS-1 and LS-1, these are less likely to contribute to the spin transition because the anions are comparably disordered in both structures.

The phase behaviour of bulk samples of 1 was probed by variable-temperature X-ray powder diffraction. Although the peaks are broad, the data show only minor changes on cooling from 300 to 140 K (Figure 6). The data at all temperatures are a better match for the simulated patterns based on phase 1 than for phase 2, particularly in the region 20≤2≤30° which shows the greatest difference between the two forms. Hence, in contrast to the single crystals, powder samples of 1 do not exhibit the LS-1/LS-2 transition and retain phase 1 at all accessible temperatures. This observation was supported by a DSC measurement, which showed peaks corresponding to the SCO event that are in excellent agreement with the susceptibility curve (T↓=267 K, T↑=270 K, ΔH=10.1(2) kJ mol -1 ). However, no peak was observed near 205 K that could be attributed to the phase 1/phase 2 conversion, confirming that this transition does not occur in the powder sample. The measured SCO enthalpy is approximately 50 % smaller than for other compounds of the [Fe(bpp)2] 2+ type,20, 23, 27 which is consistent with the suggestion that the abrupt part of the spin transition in 1 only involves a fraction of the iron centres in the sample (see above).

Higher-temperature crystal-structure determinations of 2 and 3 were also performed, to probe the structure changes during SCO in those materials (Table 1). In contrast to 1, 2 and 3, both adopt phase 2 at all the temperatures examined. Structures of 2 at 250 and 350 K, either side of the spin transition, showed it to be low spin at 250 K and high spin at 350 K, as expected from the susceptibility data (Figure 1). The metric parameters of 3 at 350 K imply a predominantly, but not fully, high-spin iron centre. That is again consistent with the susceptibility data, which predict an approximate 85 % high-spin population in the material at that temperature (Figure 1). X-ray powder diffraction confirmed that powder samples of 2 and 3 are phase pure, and adopt phase 2 as in the single crystals.

Salts of [Fe(bpp)2] 2+ derivatives are well known to exhibit photomagnetic effects at low temperatures. 7,20,[28][29][30] The low-spin→high-spin photo-conversion was investigated on 1-3 in bulk conditions by using a superconducting quantum interference device (SQUID) magnetometer coupled to a continuous waveoptical source. The samples were irradiated at the following wavelengths: 405, 510, 640, 830 and 980 nm. In each case, the most efficient wavelength to induce the LIESST effect was 510 nm, leading to a strong increase of the magnetic signal at 10 K. No reverse-LIESST was observed upon irradiation at longer wavelengths, however. By using the standardised T(LIESST) procedure31, 32 we monitored the direct magnetic response on a thin layer of powders of compounds 1-3 upon irradiation of the 1 A1→ 1 T1 absorption band. The T(LIESST) curves were then recorded for each compound (Figure 7) to determine the stabilisation of the photo-induced HS state.

For all complexes, a drastic increase in the magnetic signal under green-light irradiation was observed at 10 K. However, unusual for [Fe(bpp)2] 2+ derivatives,7, 20, 28-30 the photo-conversion efficiency was not quantitative. The maximum χMT values indicate photo-conversion efficiencies of 70 % (for 1), 57 % ( 2) and 60 % (3). Following the irradiation procedure, an increase in χMT occurs upon heating from 10 K in the dark, reflecting zero-field splitting of the HS iron(II) centres.33 For 1 and 2 the plateau reached at 40 K remains stable until 60 K, beyond which the metastable light-induced HS state rapidly decreases, reaching the baseline above 80 K. The T(LIESST) values can be extracted from the first derivative of the χMT versus T curves (Figure 7, inset) affording T(LIESST)=70 P a g e | 10 and 80 K, respectively. In contrast, the T(LIESST) curve for 3 clearly exhibits two steps, with two minima being resolved in the δ(χMT)/δT versus T plot at 65 and 75 K.

These T(LIESST) values all agree well with the predicted values from Equation (1) (T0=150 K),31, 32 similarly to T(LIESST) values of most other [Fe(bpp)2] 2+ derivatives that have been measured by this procedure.20, 28 However, although the differences are small, it is noteworthy that within this series 1 has both the lowest T1/2 and the lowest T(LIESST) values, in apparent violation of Equation ( 1) and Hauser's inverse energy gap law.34

The dynamics of LIESST relaxation of the photo-induced HS fraction, γHS, were investigated for all three complexes in the 50-80 K temperature range where the HS-LS relaxation is thermally activated. The value of γHS was deduced from the equation [(χMT)hv-(χMT)LS]/[(χMT)HS-(χMT)LS], in which (χMT)hv is the magnetic value reached after irradiation, (χMT)LS is the magnetic value of the initial LS state, and (χMT)HS is the magnetic value recorded at room temperature for a fully HS state.

The relaxation behaviour for 1 deviates strongly from a single-exponential. However, the relaxation curves are not perfectly sigmoidal; whereas a fast exponential decay occurs at short timescales, a much slower process is also observed as a long tail at the end of the relaxation (Figure 8). The sigmoidal shape of the relaxation curves is consistent with the cooperative thermal spin transition, whereas the superimposed exponential decay reflects either the presence of short-range interactions in the lattice, or the presence of more than one relaxing species. The latter suggestion is consistent with the presence of two different iron centres in phase 1 of the sample, as predicted by the powder-diffraction data, but it could also imply a crystallographic phase 1↔2 transition occurring during the relaxation process. Therefore, the relaxation curves were simulated according to Hauser's selfacceleration model, which reflects the change in the energy barrier as a function of γHS in cooperative SCO materials.35 [Eq. ( 2) and ( 3), with α=Ea*/kBT and kHL=k∞exp(-Ea/kBT)]:

This procedure yielded the solid lines shown in Figure 8. From these simulations, an Arrhenius plot can be drawn to extract the dynamical parameters Ea=1040 cm -1 , Ea*=120 cm -1 and k∞=6.9 10 5 s -1 . However the second, minor exponential component of the decay curves is clearly not reproduced by the simulation.

P a g e | 11

The HS-LS relaxation curves for 2 exhibit a clear multistep character (Figure 9). On the basis of the gradual, singlestep, thermal SCO exhibited by 2 (Figure 1), and the single minimum in its T(LIESST) curve (Figure 7), a singleexponential or even stretched-exponential decay might have been expected. This is clearly not the case, however. Some of the relaxation steps exhibit a sigmoidal shape, indicating the presence of cooperative phases in the powder, which can be explained by the formation of domains during the relaxation process.36 The simultaneous, coincident, relaxation processes exhibited by 2 could not be accurately simulated, and we were unable to describe its global LIESST relaxation.

Compound 3 clearly exhibits a two-step relaxation behaviour, from the shape of the T(LIESST) curve (Figure 7). The first step follows a single-exponential decay, whereas the second step exhibits a sigmoidal shape. If the relaxation process is [HS-HS]→[HS-LS]→[LS-LS], as observed in dinuclear systems,37 the second relaxation step depends on the lifetime of the first one, and the relaxation kinetics should account for this dependence. On the other hand, if the relaxation processes arise from two independent metastable sites, the two relaxation rates are not correlated.38 This latter situation seems more appropriate to describe the two-step relaxation of 3, especially, because the two processes have sufficiently different timescales to be treated separately. By inspection, the P a g e | 12 relaxation processes have approximately the following proportions: 40 % of exponential decay and 60 % of selfaccelerating process. The first component was modelled with an exponential model, and the second part was simulated by Equation ( 2) and ( 3). Figure 10 presents the simulation obtained from this approach, leading to the dynamic parameters extracted from the ln(kHL(T)) versus 1/T plot (Figure 10, inset). For the exponential relaxation Ea=840 cm -1 and k∞=4.6×10 5 s -1 ; and for the sigmoidal simulation Ea=1140 cm -1 , Ea*=190 cm -1 and k∞=2.2×10 6 s -1 . The latter values are comparable to those obtained for 1, reflecting the similar structural environment observed for both complexes.

An elegant way to validate the parameters obtained from the simulation of the relaxation kinetics is to reproduce the experimental T(LIESST) curves. The procedure takes careful account of the time and temperature dependencies of the relaxation, and combines the quantum mechanical tunnelling and the thermally activated regions [Eq. ( 4)].32, 39

The rate constant k0 characterises the relaxation in the quantum tunnelling region, and is estimated as an upper limit from the last complete kinetic recorded at lowest temperature. k0 is of the order of 2×10 -5 s -1 for 1 and 4×10 -5 s -1 for 3.

The simulation of the double step in the T(LIESST) curve of 3 was obtained by considering 40 % of the sites with exponential decay and 60 % of the sites with sigmoidal relaxation, as above. The calculated T(LIESST) curves (Figure 7) show excellent agreement with the experimental T(LIESST) data, thereby supporting the validity of the derived parameters and the simulation procedure. The two-step character for 3 is well reproduced, and the position of the two T(LIESST) values is also well simulated.

The observation of distinct strongly and weakly cooperative LIESST relaxation regimes in 1 is consistent with the mixture of strongly and weakly cooperative thermal SCO observed in the susceptibility data, and the presence of two distinct molecular environments in the phase 1 crystal lattice. However, neither of those observations is true P a g e | 13 for 2 or 3, which still exhibit similarly complicated LIESST relaxation. Hence, that interpretation of the LIESST properties of 1 remains to be confirmed.

Conclusions

Compounds 1-3 are isostructural at 100 K, however, crystals of 1 undergo a crystallographic phase transition near 205 K. Thermal SCO then occurs upon further warming, centred at 270 K, without a further crystallographic phase change. The high-temperature (phase 1) and low-temperature (phase 2) structures have the same space group, but phase 1 is expanded along the unit-cell a direction and contains one-and-a-half unique molecules in its asymmetric unit. The expansion of the unit cell along [100] often occurs incoherently, as evidenced by frequent twinning of crystals in phase 1, which is reversed upon re-formation of phase 2 on cooling below 205 K. The phase transition may be driven by an intermolecular interaction between a ligand methyl substituent and a neighbouring pyrazolyl ring. In phase 2, all these C⋅⋅⋅C contacts are 0.25 Å shorter than the sum of the van der Waals radii for these groups. In phase 1, there are three unique equivalent contacts, which are 0.25, 0.17 and 0.13 Å shorter than the van der Waals sum at 240 K (just above the transition temperature). Hence, the phase 2→phase 1 transition leads to a partial relief of this steric crowding. Consistent with that suggestion, 2 and 3 do not exhibit comparable intermolecular steric clashes in their crystal lattices, and remain in phase 2 in both spin states between 100-350 K.

Interestingly, the crystallographic phase transition was only observed in single-crystal samples of 1. Powder samples of 1 retain the room-temperature phase 1 between 140-300 K, according to DSC and powder diffraction. Particle size-dependent phase behaviour is well known at the nanoscale, where the structure of the bulk becomes influenced by the particle surface energy.40, 41 Such phenomena are more unusual in macroscopic materials, however, where defect structures and domain sizes can influence the stability of crystal phases as well as surface effects.42, 43 Notably the variable-temperature unit-cell, DSC and susceptibility data all imply that SCO proceeds similarly in single-crystalline and powdered 1. That is reasonable, because both samples adopt phase 1 in the temperature range of the spin transition.

The SCO-active phase 1 contains one-and-a-half unique complex dications. The whole molecule undergoes a significant change in shape during SCO, as evidenced by an increase of nearly 7° in the trans-N(pyridine)-Fe-N(pyridine) angle (φ, Table 1), which is not exhibited by the other half-molecule in the unit cell (see figure in the Supporting Information). We have previously demonstrated that a change in molecular shape between the spin states can be correlated with cooperative SCO in [Fe(bpp)2] 2+ derivatives.5 Hence, this is consistent with the observation of abrupt (molecule B) and gradual (molecule A) SCO fractions, in an approximate 2:1 ratio, in the susceptibility data of 1.

The literature compound [Fe(bppCCH)2][BF4]2 (4;pyridine) also adopts phase 2 in the high-spin state, but undergoes a phase change during SCO to a new low-spin phase in a different orthorhombic space group.30, 44 Even though it occurs at a similar temperature to the compounds in this work, the spin transition in 4 is more cooperative with a 8 K hysteresis loop, reflecting the involvement of the phase change in its SCO. It is difficult to draw further comparison between 1-3 and 4; this emphasises the structural complexity of the [Fe(bpp)2] 2+ system in this particular lattice type.

The T(LIESST) values from 1-3 are in line with previous expectation,20, 28 but the relaxation kinetics of their metastable high-spin states unexpectedly involve a mixture of weakly (exponential) and strongly (sigmoidal) cooperative regimes. The explanation for this behaviour is uncertain, given that 1, and 2 and 3, exhibit different phase behaviour in the temperature range that could be examined (above 100 K). One potential explanation, that bulk samples of 2 and 3 contain a mixture of crystal phases, is ruled out by the X-ray powder diffraction data (the same samples were used for both measurements). Alternatively, the complexes may undergo crystallographic phase changes on cooling below 100 K, and/or during photo-excitation at very low temperatures. The poor diffraction exhibited by crystals of 1 at 30 K is intriguing in that regard, and we are currently examining these possibilities in more detail.

Experimental Section

P a g e | 14

Instrumentation

Elemental microanalyses were performed by the University of Leeds School of Chemistry microanalytical service. Infrared spectra were obtained as Nujol mulls pressed between NaCl windows, between 600-4000 cm -1 , with a Nicolet Avatar 360 spectrophotometer. 1 H NMR spectra employed a Bruker DPX300 spectrometer operating at 300.2 MHz. UV/Vis/NIR measurements were performed with a PerkinElmer Lambda900 spectrophotometer in 1 cm quartz solution cells, between 200-3000 nm. Electrospray mass spectra (ESI MS) were obtained on a Waters ZQ4000 spectrometer, from MeCN feed solutions. All mass peaks have the correct isotopic distributions for the proposed assignments. X-ray powder diffraction measurements employed a Bruker D8 Advance A25 diffractometer, with CuKα radiation (λ=1.5418 Å). DSC measurements used a TA Instruments DSC Q20 calorimeter, heating at a rate of 10 K min -1 . The magnetic susceptibility, powder diffraction, DSC and LIESST measurements were all performed on the same samples of 1-3.

Magnetic susceptibility measurements were performed on a Quantum Design SQUID/VSM magnetometer, in an applied field of 1000 or 5000 G. A diamagnetic correction for the sample was estimated from Pascal's constants;45 a previously measured diamagnetic correction for the sample holder was also applied. Photomagnetic measurements were performed with a set of photo-diodes coupled through an optical fibre to the cavity of a MPMS-55 Quantum Design SQUID magnetometer operating at 2000 G. The powder sample was prepared in a thin layer (∼0.1 mg) to promote full penetration of the irradiated light. The sample mass was obtained by comparison with the thermal spin transition curve measured on a larger, accurately weighed, polycrystalline sample. The sample was first slow cooled to 10 K, ensuring that potential trapping of HS species at low temperatures did not occur. Irradiation was carried out at a set wavelength and the power of the sample surface was adjusted to 5 mW cm -2 . Once photo-saturation was reached, irradiation was ceased and the temperature increased at a rate of 0.3 K min -1 to about 100 K and the magnetisation measured every 1 K to determine the T(LIESST) value given by the minimum of the δ(χMT)/δT versus T curve for the relaxation. [START_REF] Létard | [END_REF] The T(LIESST) value describes the limiting temperature above which the light-induced magnetic high-spin information is erased in a SQUID cavity. In the absence of irradiation, the magnetisation was also measured over the temperature range 10-290 K to follow the thermal spin transition and to obtain a low temperature baseline. Kinetic studies of LIESST relaxation were performed by irradiating the sample at 10 K until photo-saturation, then, under constant irradiation the sample was warmed to a desired temperature around the T(LIESST) region. At the desired temperature, irradiation is stopped and the decay of the magnetisation signal was followed for several hours or until complete relaxation back to the low-spin baseline.

Materials and methods

Unless otherwise stated, all reactions were carried out in air using non-pre-dried AR-grade solvents. 4-Amino-2,6di(pyrazol-1-yl)pyridine (bppNH2), 14 4-iodo-2,6-di(pyrazol-1-yl)pyridine (bppI) 14 and 4-mercapto-2,6-di(pyrazol-1yl)pyridine (bppSH)15 were synthesised by the literature methods, whereas all other reagents were used as commercially supplied.

4-(Methylsulfanyl)-2,6-di(pyrazol-1-yl)pyridine (bppSMe) and 4-(methyldisulfanyl)-2,6-di(pyrazol-1-yl)pyridine (bppSSMe):

Under dry anhydrous conditions, bppSH (0.20 g, 0.82 mmol) was added to a Schlenk tube containing K2CO3 (0.23 g, 1.64 mmol), iodomethane (0.33 g, 2.31 mmol) and activated molecular sieves (10 granules) in MeCN (25 cm 3 ). The mixture was heated to 85 °C for 18 h, during which the suspension became pale yellow in colour. The contents, once cooled, were diluted to 50 cm 3 with CHCl3 (caution: strong odour), passed over a short silica pad and the silica washed further with 1:1 CHCl3:MeCN (100 cm 3 ). The solvent was removed in vacuo and the yellow solid residue was redissolved in 1:2 hexane/EtOAc and filtered, before again being evaporated to dryness. The white solid bppSMe was isolated by sublimation under reduced pressure at 130 °C. Yield 78 mg, 37 %; m.p. 140-142 °C; H 4.31, N 27.2; found: C 55.6, H 4.35, N 27.3. The involatile residue from the sublimation contains the symmetric disulfides bppSSbpp15 and bppSSMe, which were separated by silica column chromatography (eluent 99:1 CH2Cl2/MeOH). bppSSMe (Rf 0.66) was obtained as a white solid. Yield 73 mg, 20 %; m.p. 115-117 °C; 1 H NMR (300 MHz, CDCl3); δ=2.54 (s, 3 H, SCH3), 6.51 (dd, 1.9 and 2.6 Hz, 2 H, pz H 4 ), 7.79 (d, 1.6 Hz, 2 H, pz H 3 ), 8.04 (s, 2 H, py H 3/5 ), 8.57 ppm (d, 2.6 Hz, 2 H, pz H 5 ); 13 8, H 3.83, N 24.2; found: C 49.9, H 3.80, N 24.0.16 4-Bromo-2,6-di(pyrazol-1-yl)pyridine (bppBr): bppNH2 (0.39 g, 1.73 mmol) and NaNO2 (2.51 mg, 3.64 mmol) were suspended in MeCN (25 cm 3 ) under an atmosphere of N2. Addition of KBr (0.49 g, 4.15 mmol) and 48 % aqueous HBr (10 cm 3 , 59.3 mmol) caused immediate darkening of the mixture. The mixture was stirred for 1 h at room temperature, then heated to 80 °C for 1 h. The cooled mixture was poured into saturated aqueous Na2S2O3 (100 cm 3 ) and shaken, precipitating a pale solid. The solid was then collected on a glass frit and washed with H2O. 44.2, H 3.03, N 23.4; found: C 44.6, H 2.95, N 22.8.

Synthesis of complexes:

The same basic method, described here for [Fe(L 2 SMe)2][BF4]2 (1), was used for all the complexes. A solution of bppSMe (70 mg, 0.27 mmol) and Fe[BF4]2⋅6 H2O (45 mg, 0.13 mmol) in nitromethane (20 cm 3 ) was stirred at room temperature for 1 h. The solution was filtered and concentrated to about 5 cm 3 , then the product was precipitated by addition of diethyl ether. Isolated yields ranged from 87-95 %. Microanalytical data for the complexes are listed below.

[Fe(bppSMe)2][BF4]2 (1): Calcd (%) for C24H22B2F8FeN10S2 (744.09): C 38.7, H 2.98, N 18.8; found: C 38.4, H 2.95, N 18.5.

[Fe(bppBr)2][BF4]2 (2): Calcd (%) for C22H16B2Br2F8FeN10 (809.70): C 32.6, H 1.99, N 17.3; found: C 32.5, H 1.90, N 16.9.

[Fe(bppI)2][BF4]2 (3): Calcd (%) for C22H16B2F8FeI2N10 (903.69): C 29.2, H 1.78, N 15.5; found: C 29.3, H 1.70, N 15.5. Single-crystal structure analyses: Single crystals of 1-3 were all obtained by slow diffusion of diethyl ether vapour into nitromethane solutions of the compounds. Diffraction data were collected with an Agilent Supernova dualsource diffractometer fitted with Oxford Cryosystems cryostat, or with a HELIX cryostat for the measurements at 30 K (the Agilent diffractometer is modified to accept the HELIX device). Diffraction data from 1 and 2 were measured by using monochromated CuKα radiation (λ=1.54184 Å). Several crystals of 1 were screened to find one that diffracted sufficiently well at room temperature. That crystal was then used for all three data collections, measured in a 290→240→100 K temperature sequence. Two different crystals were used for 2, one for the 100 K structure and the other for the two higher temperature measurements. Data for 3 were collected with the same diffractometer, but with monochromated MoKα radiation (λ=0.71073 Å) to avoid absorption problems. Data sets were corrected for absorption by an empirical multi-scan method, based on spherical harmonics. 46 The structures were solved by direct methods (SHELXS-97),47 then developed by least-squares refinement on F 2 (SHELXL-97).47 Crystallographic figures were prepared using XSEED.48 Experimental data for the structure determinations are collected in Tables 2 and3. The unit-cell data in Figure 4 were collected under the same conditions, with MoKα radiation.

Structure refinement of 1 (HS-1):

The asymmetric unit contains 1.5 formula units, with half a complex cation spanning the C2 axis 1/2, y, 3/4 (molecule A); and, a whole complex dication (molecule B) and three unique BF4 - anions lying on general crystallographic sites. All three anions are disordered over two or three sites, each sharing a common wholly occupied B atom, which results in some significant deviations from a tetrahedral structure. The refined restraints B-F=1.37(2) and F⋅⋅⋅F=2.24(2) Å were applied to these residues. All wholly occupied non-H atoms were refined anisotropically, and all H atoms were placed in calculated positions and refined by using a riding model.

Structure refinement of 1 (LS-1):

This structure is isostructural with HS-1, and its refinement showed only two significant differences from the previous one. First, there is a slightly different pattern of disorder in the three unique anions. The second difference is the resolution of a minor, 15 %-occupied, disorder site for the methylsulfanyl group of half-molecule A, which was refined by using the fixed restraint C S=1.78(2) Å. All wholly occupied non-H atoms plus the major methylsulfanyl disorder orientation were refined anisotropically, and all H atoms were placed in calculated positions and refined using a riding model.

Structure refinement of 1 (LS-2):

This asymmetric unit contains half a formula unit, with Fe(1) lying on the crystallographic C2 axis 1/2, y, 3/4 and one unique BF4 -ion on a general crystallographic site. No disorder was detected during refinement, and no restraints were applied. All non-H atoms were refined anisotropically, whereas H atoms were placed in calculated positions and refined by using a riding model. The highest residual Fourier peak of +1.5 e Å -3 is 1.2 Å from S (18), and may reflect a remnant of the disorder in that group that is present in molecule A of the LS-1 refinement.

Structure refinements of 2:

The asymmetric unit contains half a formula unit, with Fe(1) spanning the crystallographic C2 axis 0, y, 3/4. The unique BF4 -ion is badly disordered at 250 and 350 K, and was refined over four equally occupied orientations that share a common full-occupancy B atom. The resultant partial anions deviate significantly from a tetrahedral structure. The refined restraints B F=1.39(2) and F⋅⋅⋅F=2.27(2) Å were applied to the disordered anion. At 100 K this disorder was no longer apparent, and the anion was modelled without restraints P a g e | 18 at that temperature. All non-H atoms except the disordered F atoms (where applicable) were refined anisotropically, and H atoms were placed in calculated positions and refined by using a riding model.

Structure refinements of 3:

The asymmetric unit contains half a formula unit, with Fe(1) lying on the crystallographic C2 axis 1/2, y, 3/4. No disorder was present in the model at 100 K, but at 350 K the unique BF4 -site is disordered and was refined over four equally occupied orientations that share a common full-occupancy B atom. The resultant partial anions deviate significantly from a tetrahedral geometry. The refined restraints B-F=1.40 [START_REF]Spin-Crossover Materials-Properties and Applications[END_REF] and F⋅⋅⋅F=2.28(2) Å were applied to the disordered anion. All non-H atoms except the disordered F atoms (where applicable) were refined anisotropically, and H atoms were placed in calculated positions and refined by using a riding model.

  C NMR (75.5 MHz, CDCl3); δ=23.0 (SCH3), 105.3 (2 C, py C 3/5 ), 108.1 (2 C, pz C 4 ), 127.3 (2 C, pz C 5 ), 142.5 (2 C, pz C 3 ), 150.3 (2 C, pz C 2/6 ), 155.1 ppm (1 C, py C 4 ); MS (ES): m/z: 290.1 [H(bppSSMe)] + , 312.0 [Na(bppSSMe)] + ; elemental analysis calcd (%) for C12H11N5S2 (289.18): C 49.

  NMR (300 MHz, CDCl3); δ=2.64 (s, 3 H, SCH3), 6.50 (br s, 2 H, pz H 4 ), 7.71 (s, 2 H, py H 3/5 ), 7.76 (br s, 2 H, pz H 3 ), 8.56 ppm (d, 2.1 Hz, 2 H, pz H 5 );13 C NMR (75.5 MHz, CDCl3); δ=14.2 (SCH3), 105.1 (2 C, py C 3/5 ), 108.0 (2 C, pz C 4 ), 127.3 (2 C, pz C 5 ), 142.3 (2 C, pz C 3 ), 149.8 (2 C, pz C 2/6 ), 156.5 ppm (1 C, py C 4 ); MS (ES): m/z: 280.1 [Na(bppSMe)] + ; elemental analysis calcd (%) for C12H11N5S (257.32): C 56.0,

1 H

  Silica gel column chromatography (3:2 CH2Cl2/hexane eluent; Rf 0.19) yielded the product as a white solid, which was recrystallised from n-hexane. Yield 0.16 g, 32 %; m.p. 148-150 °C; 1 H NMR (300 MHz, CDCl3); δ=6.52 (dd, 1.7 and 2.6 Hz, 2 H, pz H 4 ), 7.77 (d, 1.7 Hz, 2 H, pz H 3 ), 8.06 (s, 2 H, py H 3/5 ), 8.53 ppm (d, 2.6 Hz, 2 H, pz H 5 ); 13 C NMR (75.5 MHz, CDCl3); δ=108.4 (2 C, pz C 4 ), 112.7 (2 C, py C 3/5 ), 127.3 (2 C, pz C 5 ), 136.7 (1 C, py C 4 ), 142.9 (2 C, pz C 5 ), 150.4 ppm (2 C, py C 2/6 ). MS (ES): m/z 290.0 [H(bppBr)] + ; elemental analysis calcd (%) for C11H8BrN5⋅1/2H2O (299.13): C

Acknowledgements

This work was funded by the EPSRC (EP/H015639/1, EP/K012576/1 and EP/K00512X/1). The Aquitaine Region is acknowledged for the development of the International Centre of Photomagnetism in Aquitaine (ICPA) platform.