The First Observation of Memory Effects in the InfraRed (FT-IR) Measurements: Do Successive Measurements Remember Each Other?

Over the past couple of decades there have been major advances in the field of nanoscience and nanotechnology. Many applications have sprouted from these fields of research. It is essential, given the scale of the materials, to attain accurate, valid and reproducible measurements. Material properties have shown to be a function of their size and composition. Physiochemical properties of the nanomaterials can significantly alter material behavior compared to bulk counterparts. For example, metal oxide nanoparticles have found broad applications ranging from photo-catalysis to antibacterial agents. In our study, we synthesized CuO nanoparticles using well established sol-gel based methods with varying levels of Ni doping. However, upon analysis of measured infrared data, we discovered the presence of quasi-periodic (QP) processes. Such processes have previously been reported to be tightly associated with measurement memory effects. We were able to detect the desired QP process in these measurements from three highly accurate repetitive experiments performed on each Ni (1–7%) doped CuO sample. In other words, successive measurements performed in a rather short period of time remember each other at least inside a group of neighboring measurements.


Introduction
Copper oxide is a technologically important material which is mostly found in two oxidation states, cupric oxide (CuO) and cuprous oxide (Cu 2 O). Both are p-type semiconductors and the preferred structure of CuO is monoclinic, while that of Cu 2 O is cubic. The energy band gap of CuO ranges between 1.2-1.9 eV [1] and Cu 2 O has an energy band gap between 1.8-2.5 eV [2], [3]. As a result of some of the fascinating properties exhibited by copper oxide, it finds its application in high T c superconductors [4], lithium ion electrode [5], gas sensors [6], [7], solar cells [8][9][10], field emission emitters [11][12][13], catalysis [14], [15], antibacterial agents [16] etc. Because CuO is a semiconductor, having a low band gap, it has also been applied for photoconductive and photothermal applications [9]. Nanomaterials, which have attracted major interest in recent years, possess a large amount of surfaces and interfaces and exhibit enhanced properties in comparison to their bulk counterparts. The properties of these materials can be further fine-tuned by controlling the particle sizes in the nano range or by playing with the crystal morphologies of the final product. Moreover, the doping of the semiconductor nanoparticles with impurity metal ions is one of the most important methods to modify the characteristics of the material.
The engineering of band gap and influencing physical, chemical, and electronic properties of the semiconductors are possible by the use of the right amount of dopants. Several authors have reported the improvement in a material's properties using various dopants. The band gap narrowing of Ni doped SnO 2 nanoparticles was observed by Ahmed et al. [17]. Das et al. reported the tuning of emission properties of Mn doped Cu 2 O nanoparticles [18]. Various methods have been developed for the synthesis of both pure and doped nanocrystalline CuO such as sol-gel method [19], one-step solid state reaction method [20], sonochemical method [21], electrochemical method [22], thermal decomposition of precursors [23] etc.. Out of these methods, sol-gel method is widely used for the synthesis of nanomaterials because of the various advantages associated with this method such as low cost, low temperatures processing, short annealing times, as well as higher purity of produced materials.
Inorganic metal oxides have also gained interest for applications as antimicrobial agents [16]. Advantages of using inorganic oxides such as TiO2, ZnO, and CuO are their stability, robustness, and long shelf life when compared with their organic counterparts [24]. A comprehensive analysis of the physiochemical properties of nanoparticles is essential for their characterization and for evaluating their activity. The determination of these properties is necessary to predict the fate, accumulation, and transport of nanoparticles based on surface and physiochemical characteristics [25]. FT-IR is a unique technique that offers a broad range of information about chemical structure and bonding of materials [26]. Many nanomaterial measurements, amongst them FT-IR, are sprinkled with fluctuations of different kinds that could actually be playing an important role. Conventional mathematical statistics could therefore be modified and tuned to analyze and overcome these unexpected properties providing more accurate and reliable information from different nanomaterial measurements. Previous methods such as procedure of the optimal linear smoothing (POLS) and probability distribution function (PDF) found that random sequences could help in data fitting parameters of wellknown beta-distributions without a trend. [27][28][29][30].
In paper [27] we discovered that quasi-periodic (QP) processes are tightly associated with previous measured processes, thus having a memory. In other words, successive measurements performed in a rather short period of time remember each other at least inside a group of neighboring measurements. The detection of the QP processes in real measurements can significantly alter the existing conception of measurements. Experimentalists were accustomed to think that a group of repeated measurements should be independent from each other and, as a consequence of this supposition, the increasing number of experiments N decreases the dispersion as s N~N {1=2 s 0 and thereby suppresses the influence of a ''noise'' (that are frequently associated with highfrequency fluctuations). The situation is completely changed if successive (repeatable) measurements have a memory. In accordance with a new conception considered in paper [27], the presence of memory between measurements mathematically can be expressed as The solution of this functional equation has a form [27] The fitting function F(t) was defined in [27] as the generalized Prony's decomposition and actually forms the generalized Prony's spectrum (GPS) because it can replace, approximately, at certain conditions the initial random function y(t) by a possible fitting function F(t) satisfying to equation (1). This function F(t) is determined by K(L+1)+2 number of parameters (Ac (s) m , As (s) m ,l s ,B,T) that form, in turn, the amplitude-frequency response (AFR) of the random function y(t) analyzed. The angle Q depends on the character of roots. In particular, for positive, negative and the complex-conjugated roots, the phase angle Q accepts the following values: 0, p, Im(l)?0, accordingly. For degenerated roots, this decomposition becomes more complicated. The view of decomposition function F(t) for this case is given in [27]. So, based on peculiarities of each experiment one can receive an alternative model, which admits the fitting of experimental functions to the function F(t) with the reduced set of fitting parameters. Therefore, in this paper we want to demonstrate new possibilities for analysis of the IR spectra that were recorded for pure CuO with different concentrations of Ni (c = 1%,2%,3%,4%,5% and 7%). Each experiment was performed only three times. In spite of the minimal number of repetitions (equaled 3), the accuracy of each measurement was rather high that allowed detecting the desired QP process in these measurements.

Synthesis of pure and Ni doped CuO nanoparticles
Ni doped CuO nanoparticles with doping concentrations varying from 0% to7% were synthesized using previously established sol-gel based method [16]. For a typical synthesis procedure in distilled water dissolution of Cu(NO3)2.3H2O, Ni(NO3)2.6H2O and citric acid was carried out based on a molar ratio of 1-X:X:1 (where X represents the desired level of doping ranging 0.01-0.07). The solution was stirred at 80uC until gel formation (,1 hr). Next, the gel was washed 2-3 times with ethanol using a microcentrifuge system at 19090 rcf to remove any organic impurities. Afterwards, the washed material in the form of dense precipitate was allowed to combust at 200uC for 10 minutes producing a light fluffy mass. After that, it was grinded for half an hour and further annealed at 350uC to obtain the highly crystalline Ni doped CuO nanoparticles. XDR, SEM and EDX analysis was carried out to confirm the presence of single phase materials and validated the level of doping present in each sample (supplementary material in Figures S1, S2, and S3).

FT-IR Collection
FT-IR experiment was done at room temperature by an instrument from Thermo Scientific company (model name: Smart i TR). The instrument uses DTGS-KBr detector with beam splitter of KBr. Transmittance was measured with spectral data spacing of 0.964 cm 21 from 400 to 1000 cm 21 . An atmospheric background was collected first followed by measuring each sample in triplicate before cleaning and measuring the next sample. The Ni doping level ranged from 0 to 7% within CuO nanoparticles.

Description of treatment procedure
In this section we want to demonstrate how to detect the QP process when the number of repetitions is small but accuracy of the reproduction of a current recording is rather high. In our specific case, the functional equation can have the following form St1. The first step of the proposed algorithm began with the testing of equation (3). Let us consider these three independent measurements more attentively. The initial measurements depicted on Fig.1(a), and corresponding to pure CuO IR spectrum, look strongly-fluctuated. These ''fluctuations'' (located on the left-hand side) can hamper a possible link between measurements. A quite different picture is observed for their integrated curves, which are obtained from the initial ones by integration with respect to its mean value. The mathematical expression that helps to realize this procedure numerically can be expressed as We want to stress here that x-variable coincides with the normalized value of wavenumber l(cm 21 )/100 and y-variable coincides with the relative intensity of the measured IR spectrum. One can pose two questions: (a) why the integration used as a smoothing procedure is chosen and (b) why is it necessary to integrate with respect to its mean value? Our answers are the following: (a). Any smoothing procedure (in spite of their varieties) and accuracy in extraction of the desired trend (see, for example, the most justified procedure designed for this purpose and defined as the optimal linear smoothing (POLS) procedure) [28][29][30]) creates some error. The integration does not disturb the initial fluctuations and extract only large-scale fluctuations, decreasing the value of the initial error without any uncontrollable The result of application of expression (6) is shown on Fig. 1(b). We obtain three similar integrated curves, and after that one can try to fit the third curve by a linear combination of two previous curves that belong to the first and second measurements, correspondingly. The result of this attempt is shown on Fig.1(c) As one can notice from this figure, the fitting curve describes the third integration curve with relatively high accuracy (the relative error is about 3.5%), and the supposition (3) is therefore correct. With the same success using the procedure outlined above, we checked the other 6 files corresponding to different concentrations of Ni (1%-5% and 7%). They also satisfy hypothesis (3) with different values of the constants (a 0,1 , b) figuring in (3). The calculation of the desired roots from equation shows that one root k 1 is always positive and another one k 2 is always negative for all 7 data files studied. In accordance with the general solution (2), these roots dictate the following structure of the solution F(x) that can be used as a fitting function for the integrated curves calculated Here the functions yc (1,2) k (x), ys (1,2) k (x) depend on the character of the roots l 1,2 = ln(k 1,2 ) (k 1 .0, k 2 ,0). The calculated values of roots for all 7 files (pure CuO, (CuO+1%Ni)-(CuO+5%Ni) and CuO+7%Ni) are collected in Table 1.
We should note that the independent variable x that enters into the last expressions does not coincide with real time (it coincides with the normalized wavenumber x = l 21 /100 as mentioned above). In this case, the value of a period T in expressions (6) and (7) cannot be fixed in experimental measurements and should be considered as unknown fitting parameter.
St2. For calculation of the unknown value T, it is necessary to find the limits of T. One can find the limits of this parameter from the following speculations. The minimal value of a ''frequency'' (in indirect sense) is defined from the following expression f min = h?(length(x)), where h is the minimal value of discretization. So, the value of T opt should be located in the vicinity of this value T max = 1/f min and can be less or it exceeds this value maximum at two times. So, we make a supposition that T opt is located in the interval [0.5T max , 2T max ]. The optimal value should correspond to the value of the minimal error and it is calculated from expression  The values of the roots k 2,1 of equation (5) combined with the value of the relative error that fits equation (3), where the third integrated curve is presented as a linear combination of two previous integrated curves. doi:10.1371/journal.pone.0094305.t001 where the integrated curve is calculated from expression (4) and the fitting function is taken from expression (6). The direct calculations confirm this supposition. Figure 2 demonstrates the dependence of the relative error (8)  St3. This step can be considered as the final stage. After calculation of the nonlinear fitting parameter T opt other linear fitting parameters (B,E 0 , Ac (1,2) k , As (1,2) k ) that enter to decomposi-  figure. (b). This figure shows that integrated curves demonstrate the monotone behavior with respect to increasing of the doped Ni (1%R7%). More accurate behavior of this minimal point with respect to concentration is given in the small figure located on the right-hand side. We deliberately do not show the fitting curves to the function (6), which practically coincide with these curves. The value of the fitting error is located in the interval 0.1%,RelErr,1%. doi:10.1371/journal.pone.0094305.g003 tion (6) are found by the linear-least square method (LLSM). The fit of the third integrated curve (corresponding to pure CuO) is shown in Fig. 3(a). Figure 3(b) shows the monotone decreasing of the minimal value of the integrated curves with increasing of concentration of Ni (1%R7%). This property can be used for calibration purposes. We deliberately do not show the perfect fit (that practically coincides with these curves) avoiding excess information. The value of the fitting error is collected in the last column of Table 2. The amplitude-frequency responses for the integrated curve CuO are shown in Figs 4(a, b). In the same manner we treated the remaining 6 files corresponding to different concentrations of Ni. The necessary parameters are collected in Table 2. Analysis of these preliminary results exhibits the following peculiarities that can be used for further analysis.  (6). This function reproduced the integrated curves with high accuracy. 3. The dependencies of the amplitudes Ac (1,2) k , As (1,2) k with respect to mode of k (k = 1,2,…,K) forms the generalized Prony's spectrum (GPS) and this spectrum can be read and compared with other data that are subjected to similar procedures. However, the third point cannot be realized because the detection of the QP processes is at the beginning.
We want to stress in conclusion the following important point. The presence of memory discovered in reproducible measurements can be considered a new direction of research in the near future. In any ideal experiment, all successive measurements do not remember each other. In other words, if we have a control variable x and Pr(x) represents a perfect response of the complex system studied then from the mathematical point of view the concept of ideal experiment should be expressed as Pr (xzmT)~Pr (xz(m{1)T), m~1,2,:::,M: Here a letter m = 1,2,…,M determines the number of successive measurements. As we can see from (9), the conventional Fourier transform receives a new and rather general interpretation and the decomposition coefficients of the periodic function A 0 , Ac k , As k (k = 1,2,…,K) can be used for a quantitative expression of ideal measurements without memory. In reality we do not observe this situation. The strongly-correlated measurements (the integrated data in our case) demonstrate a memory, and instead of perfectly reproducible measurements without memory, we observe the opposite situation. Part of measurements remember each other and mathematical expression of this observation can be written as In our case we confirmed the partial case (3) of this general expression. Here the parameter L defines a length of a memory that can exist between neighboring measurements and the memory coefficients a l and b determine simultaneously the influence of the equipment and the object studied. This new and unexpected discovery that is contained in real measurements could help us to differentiate a device from an object. In other words, it will help to separate the Prony's decomposition (11) (aggravated by the uncontrollable factors from the equipment used) and be closer to the perfect case (9), where the ideal reproduction of data from the object studied could be solely present. These fine peculiarities discovered in real data will be important in experiments associated with nanotechnologies, where many ''small'' factors should be taken into account in order to provide stable and accurate measurements. In column 1 we show the optimal values of the nonlinear fitting parameter T opt . The identical values are underlined and bolded. In