Third Harmonics and Defects in Modulated Phases of Structural Quantum Order-Disorder Systems

TL;DR

This paper investigates how defects acting as random fields affect modulated phases in order-disorder systems, revealing that such defects can suppress third harmonics of the ground state modulation under certain conditions.

Contribution

It demonstrates the impact of defects as random fields on the modulation harmonics in structural quantum order-disorder systems, highlighting a suppression mechanism.

Findings

Defects can suppress third harmonics in modulated phases.

Random fields influence the stability of ground state modulations.

Suppression occurs under specific defect conditions.

Abstract

Influences of defects generating random fields on modulated phases in materials with the order-disorder type of phase transition are studied. It is shown that under appropriate conditions third harmonics of the ground state modulation may be suppressed by these defects.

Equations

$$H_1=-\frac{1}{2}\sum _{i,j}{J}_{ij}{S}_i^z{S}_j^z.$$

$$H_R=-\sum _n{h}_n{S}_n^z.$$

$$\sum _n{h}_n=0.$$

$$J(\mathbf{q})=J_1\cos (\mathbf{qa})+J_2\cos (2\mathbf{qa}).$$

$$<S_n^x>=<S_n^y>=0,$$

$$<S_n^z>=\frac{1}{2}\tanh (\beta \frac{H_n}{2}),$$

$$H_n=\sum _j{J}_{nj}<S_j^z>+h_n.$$

$$k{T}_c\equiv \frac{J(Q)}{2},$$

$$<S_n^x>=<S_n^y>=0,$$

$$<S_n^z>=\cos (Qn+\varphi ),$$

$$\cos (Q)\equiv -\frac{J_1}{4J_2}.$$

$$H_n=2SJ(Q)\cos (Qn+\varphi ).$$

$$E_{GS}=-N_{latt}J(Q)\frac{S^2}{2},$$

$$S^2\equiv \frac{T^3}{T_c^3}(\frac{T_c}{T}-1).$$

$$S_3\equiv S^3.\frac{\frac{-T_c^3}{3T^3}}{1-\frac{J(3Q)}{2k_{B}T}}.$$

$$E_R=-\sum _n{h}_n<S_n^z>.$$

$$<S_n^z>=S.cos(QD+\varphi ).$$

$$<S_n^x>=<S_n^y>=0,$$

$$<S_n^z>=\frac{1}{2}\tanh (\beta \frac{H_n}{2}),$$

$$H_n=\sum _j{J}_{nj}<S_j^z>+h_n.$$

$$<S_n^x>=<S_n^y>=0,$$

$$<S_n^z>=\cos (Qn+\varphi ),$$

$$\cos (Q)\equiv -\frac{J_1}{4J_2}.$$

$$H_n=2SJ(Q)\cos (Qn+\varphi )+h_D\delta (n,D),$$

$$E_R^s=-h.S.cos(QD+\varphi ).$$

$$\varphi =-QD+2\pi k,$$

$$E_R^s=+h.S.cos(QD+\varphi )-h.S.cos(\varphi ).$$

$$\varphi =\frac{\pi }{2}-\frac{QD}{2}+2\pi k,$$

$$E_R^s=-h.S.|\cos (\frac{QD}{2})|.$$

$$<S_n^z>=S\cos (Qn+\varphi )+S_3\cos (3Qn+3\varphi ),$$

Full text

Condensed paper title:

Third Harmonics - Defects in Modulated Phases - Structural Quantum Order-Disorder Systems.

Contents

1. 1 Introduction.

2. 2 Sinusoidaly modulated incommensurate structure in quantum structural order-disorder type materials as idealized crystals.

3. 3 Model.

4. 4 Ground state in pure systems.

5. 5 Single plane wave ground state and impurities.

6. 5.1 A single impurity.

7. 5.2 A pair of impurities.

8. 6 Third harmonics of the ground state and impurities.

9. 6.1 A single impurity.

10. 6.2 A pair of impurities.

11. 7 Discussion.

1 Introduction.

Incommensurate phases in ferroelectrics attracted interest of physicist due to many interesting properties [1]. As concerning theoretical and experimental methods [2] Handbook of Physics is a complete reference for scientists, engineers, and students for everyday use. Incommensurate-commensurate states phenomena are interesting phenomena, for example in these days they are concerning topological quantum matter [3] in studying commensurate and incommensurate states of topological quantum matter. Defects in general play important role in many systems. For example in [4] Lansac, Glaser and Clark study discrete elastic model for two-dimensional melting. They study both geometrical defects (localized, large-amplitude deviations from ordering) and topological defects (dislocations and disclinations), as they play a role in 2D melting. Their model exhibits a rich phase behavior including hexagonal and square crystals, expanded crystal, dodecagonal quasicrystal, and liquid structure. In [5] ordered and disordered defect chaos is studied numerically in coupled Ginzburg-Landau equations for parametrically driven waves. The motion of the defects is traced in detail yielding their life-times, annihilation partners, and distances traveled. Rubtsov and Janssen [6] study thermodynamic properties of the discommensuration point for incommensurate structures. The consequences of the opening of a phason gap in incommensurate systems are studied on a simple model, the discrete frustrated $\phi^{4}$-model.

In general together with the first harmonic response there may be higher order harmonics in the response to external fields. For exmple in ferroelectrics there exist glass states. Third and fifth harmonic responses are reviewed in [7] glasses, namely $\chi_{3}$ and $\chi_{5}$. Authors explain why these nonlinear responses are especially well adapted to test whether or not some amorphous correlations develop upon cooling. They show that the experimental frequency and temperature dependences of $\chi_{3}$ and of $\chi_{5}$ have anomalous features, since their behavior is qualitatively different to that of an ideal state. Most of the works have interpreted this anomalous behavior as reflecting the growth, upon cooling, of amorphously ordered domains, as predicted by the general framework. Finally, the comparison of the anomalous features of $\chi_{5}$ and of $\chi_{3}$ shows that the amorphously ordered domains are compact, i.e., the fractal dimension $d_{f}$ is close to the dimension $d$ of space. This suggests that the glass transition for example of molecular liquids corresponds to a new universality class of critical phenomena. The same type of study may be realized for glass states in ferroelectrics. That this study would be useful for ferroelectrics where highly-efficient second and third harmonic generation in a monocrystalline lithium niobate microresonator is used [8], and where it would be interesting to study how defects this generation change.

In this type of materials and its solid solutions [9] the structure and dielectric properties of

the $Li_{1+x-y}Nb_{1-x-3y}T_{x+4y}O_{3}$ system have been investigated. Detailed studies of the lattice parameters of these phases agree well with structure models based on intergrowths of $LiNbO_{3}$ slabs with a Ti-rich corundum-type layer. The relative permittivity ranges from $-80$ to $-55$. The temperature coefficient of the resonant frequency, $τ_{f}$ , changes sign within the solid solution region thus permitting tunability to a zero value. $LiNbO_{3}$ is interesting because of its ferroelectric and opto-electronic properties. Neither of the previous studies was able to refine the structures in detail, but two models were proposed. In one model, developed by Smith and West, the solid solutions were described as incommensurate structures comprised of random intergrowths of $LiNbO_{3}$-type and rocksalt-type $Li_{2}TiO_{3}$ slabs. This model also predicted that random stacking faults in the anion packing are associated with the interfaces between the two blocks. In a second model Roth et al. proposed the phase field is comprised of a homologous series of commensurate structures, based on ordered intergrowths of $LiNbO_{3}$ and a spinel-type end-member $Li_{4}Ti_{5}O_{12}$. As we can see there exist incommensurate phases in which defects may occur. It would be interesting to know whether measurements of the third order susceptibilities are possible to measure.

Defects presence leads to such effects like global hysteresis [10] and memory effects [11]. Order-disorder systems, like dipolar glasses of the $RADP$ and $Rb_{1-x}(NH_{4})_{x}H_{2}PO_{4}$ type contain defects conveniently described by the random fields and random interactions [12]. Spin models in weak random fields are good representations of a large number of impure compounds which undergo magnetic or structural transitions. From a theoretical point of view these systems exhibit frustration on a mesoscopic scale and take therefore an indermediate position between (spin) glasses and unfrustrated (diluted) disordered systems.

There are two types of frustration in the dipolar glasses: the frustration due to competing between ferroelectric-antiferroelectric structural ordering and the frustration due to competing between different local random fields acting on the collective ordering of the fluctuating quantum units (hydrogen bonds, deuteron bonds, …). The phase diagram of the mixed crystals $RADP$ and $DRADP$ showing ferroelectric, antiferroelectric and dipolar glass phases is the best example of real physical system with mentioned two types of frustration. There are randomly distributed ferro- and antiferro- electric local interactions due to distribution of $Rb^{+}$ and $NH_{4}^{+}$ cations. Two different orderings are associated with the two types of cations. In the glass region this type of competition would lead to long range modulated ordering, [13]. In real systems short-range correlations freeze only. A model of the hydrogen (deuteron) bonding [14] was developed, it is able to describe the incommensurate wavevector behavior as well as some other properties of the glass phase. Far-infrared and near-millimetre spectroscopy, [15] has shown that above the glass transition temperature the low frequency dynamics is similar to that of an averaged pure crystal. Below this temperature a broad background absorption in addition to phonon peaks is present. The appearance of new forbidden modes below the glass transition temperature gives, according to the authors, evidence of polar microregions. These modes $B_{2}$ are forbidden in the paraelectric phase but, at least below the glass transition temperature, they are observed. Another example is the influence of partial bromination on the $(p,T)$-diagrams. Dielectric and optical measurements, [16], have shown that an increase of the $Br$ concentration leads to decrease and finally to the disappearance of the stability regions of more and more longperiod phases in $BCCD$. It would be interesting to study qantitative changes in modulation of the dissappearing structures.

A method for measuring the nonlinear response in dielectric spectroscopy through third harmonic detection is described in [17]. This is a high sensitivity method allowing the measurement of the nonlinear dielectric susceptibility of an insulating material at finite frequency.

To interpret these and other observations, one should understand better influence of defects on modulated phases in materials with the order-disorder type. This understanding should be based on the microscopic level detailed models of c qualitative considerations based on simplified models are sufficient to discuss existing experimental evidences for $\chi_{3}$ and of $\chi_{5}$ without defects.

The ideal systems without defects are described by the quantum Ising-like Hamiltonian in the simplest cases. The main aim of this paper is to study influence of a few on-site defects on the ground state properties of incommensurate systems in quantum order-disorder type materials using expansions in a small parameter - the pseudospin amplitude.

In the next section we describe sinusoidaly modulated incommensurate structure in quantum structural order-disorder type materials as idealized crystals. Then we introduce the model, which we study. In the fourth section a description of the modulated ground state in the idealized medium without defects is presented for reference purposes. The fifth section contains description of defects and their influence on the groud state properties in modulated phases. We discuss the simplest cases of a single defect and of a pair of defects. The sixth section is devoted to the incommensurate phase properties changed due to presence of defects. In the last section we discuss our results in connection to a general theory and to related experimental systems.

2 Sinusoidaly modulated incommensurate structure in quantum structural order-disorder type materials as idealized crystals.

Below we studied interaction of a single defect and of a pair of defects, characterized by associated random fields, with ordering tendencies of fluctuating units described by an Ising type interactions between nearest neighbors and next nearest neighbors. We neglect tunneling phenomena in our approach to simplify calculation as much as possible. Incorporated are two types of frustration: due to competition between ferroelectric and antiferroelectric interactions and due to local random fields of defects. For recent review on random field systems see in [12]. Our basic questions are: how presence of defects influence ground state ordering and dynamics above the order-disorder transition temperatures, how the incommensurate wave developed below this transition temperature will be modified by defects, is it true that defect main effect is to lock-in the phase of the wave or there are changes of the wave amplitude and wave vector in the neighborhood of defects.

Ferroelectrics and antiferroelectrics, in which quantum motion of some units is responsible for an order-disorder type phase transition are described by using pseudospin formalism. The modulated single plane wave (sinusoidal) ground state structure may be stabilized below some critical temperature $T_{c}$, [18]. At nonzero wavevector ${bfq}_{0}$ there exists maximizing the quantity $J_{{bfq}}=\sum_{j}J_{ij}\exp(i{bfq}.({bfR}_{i}-{bfR}_{j})$ in this case. Here $J_{ij}$ are the interaction energies between two units localized at lattice sites ${bfR}_{i}$ and ${bfR}_{j}$ in the most simple theories. Dynamics of such a system is described by the quantum Ising-like Hamiltonian. The excitation spectrum and response of the later model was recently studied for example in [19], [20] and by the authors of this paper [21].

The freezing phenomena of reorienting moments in solids leading to an orientational glass phase were studied thoroughly recently, [22]. Clustering phenomena are observed in some materials, like in $Rb_{1-x}(NH_{4})_{x}H_{2}PO_{4}$, which are solid solutions of hydrogen-bonded ferroelectric and antiferroelectric compounds. Satellites occurring in X-ray scattering at some specific points of the Brillouin zone reflect tendency of these systems towards a superstructure, [23]. This superstructure may be characterized by the wavevector $q=0.15a^{*}$ for $x\geq x_{c},$ here $x_{c}$ is a material constant, characteristic value of which is $\approx 0.2$. With increasing concentration x of the ammonium group the modulation wavevector q slightly increases, [14]: for $x=0.35$ it is $q\approx 0.25a^{*}$, for x=0.49 it is $q\approx 0.32a^{*}$ and for x=0.68 and x=0.75 it is $q\approx 0.35a^{*}.$ At 160K and above this temperature the modulation disappears. Deuterated isostructural compound, $Rb_{1-x}(ND_{4})_{x}D_{2}PO_{4},$ also shows modulation at a wavevector of approximately $0.3a^{*},$ see [24]. A pseudospin model appropriate for this later compound is formulated in [24]. An incommensurate phase between the ferroelectric (x=0) and the antiferroelectric (x=1) phases is predicted to be stable around $q\approx 0.35a^{*}$ for a wide range of concentrations of $ND_{4}$ group. Outside this value of concentrations the wavevector q of the resulting structure tends to the uniform modulation q=0 for x=0 and to the antiferroelectric for x=1. The final glass behavior follows from the presence of random fields due to random distribution of lattice sites in which $ND_{4}$ substituents are present. Deuterated compounds are satisfactorily described neglecting quantum tunneling phenomena. Thermal neutron scattering experiments on this later compounds are characterized by the presence of the cigar-shaped diffuse peak on $\Sigma$ line, [13]. A freezing temperature of about 100K is found. Below this temperature the spatial correlations stop to increase. Cigar-shape peaks arise from correlations which are different in different directions. For x=0.62 these correlations are larger in the (1,0,0) direction and peaked at about $q=\frac{1}{3}a^{*}$, [25]. A broad $(\nu\leq 0.5$ THz ) inelastic contribution is observed at an arbitrary point of the Brillouin zone. The dynamics of the relaxation may be characterized by two components with different ways to the frozen state, [26]. The effective medium would condense at an (incommensurate) wave vector q. However the condensation of short wave-length fluctuations in regions separated by substituents lead to the state which is determined by conflicting boundary conditions due to substitutional disorder. As a consequence the glassy state results with regions slowing down of which is described by the Vogel-Fulcher type of dependence, [22].

The complex dielectric constants in $RADP$ ($x=0.35$) and $DRADP$ ($x=0.35$) have been described also by the Vogel-Fulcher law of the relaxation time [27]. Considerable dielectric dispersions are observed at low temperatures for both $RADP$ and $DRADP$ samples, deuteration shifts the dispersion region to lower frequencies at the same temperature. The dispersion is observed in a wide range of frequencies (from 11Hz to 1THz in [27] ). Such unusual low temperature and low frequency behavior reflects also in the thermal properties of the system. Glass-like excitations are observed in low temperature thermal conductivity $\kappa\approx T^{3}$ and in the specific heat $C\approx T$ in $RADP$ ($x=0.32$, $x=0.72$) below 100K, [28]. In $DRADP$ ($x=0.62$) the same authors did not observe the glasslike behavior in the thermal conductivity while it was still present in the specific heat at very long times. Such an isotope effect is attributed in [28] to the low energy excitations in these crystals related to tunneling of protons or deuterons. A broad background absorption in addition to phonon peaks remain present below the freezing (100K) temperature down to lowest temperatures in the $RADP$ ($x=0.5$), [15], which is probably caused by breaking of the quasi-momentum conservation selection rules due to incommensurability of this system. In the isostructural $DRADP$ crystal ($x=0.5$) no background absorption is observed in the temperature range $15K-300K$, [29]. The absence of the background absorption in $DRADP$ in this temperature range may be explained, [29], by higher diffusivity of the walls between clusters in this compound than that in $RADP$. It is noticed in [29] that the mode damping is still high at such low temperatures at which the anharmonic deuteron (proton) motion (hopping) is believed to be frozen in. This behavior is in sharp contrast with that for pure $RDP$ and pure $ADP$ compounds where the damping constants are characterized by considerably lower values. In contrast to pure $RDP$ and $ADP$ the soft mode in $RADP$ and $DRADP$ exhibits continuous softening on cooling down to $5K$. The soft relaxation frequency in $DRADP$ is one order lower than in $RADP$, confirming the proton (deuteron) origin of this mode. Dielectric dispersion measurements [30] do not see any obvious x dependence in the dynamic properties of $RADP$. Not all of the mentioned low temperature characteristics are explained within the existing theories, see in [22].

Recent theoretical results were found for phases in which there are no glass like properties. However, one may expect their validity in a modified form also for the orientational glasses of the $RADP$ and $DRADP$ type in which clusters with modulated structure are observed as we discussed above. The presence of a frozen-in characteristic modulation wave vector $q_{m}$ at a given concentration x gives lower boundary $(q>q_{m})$ for those wavelengths for which the response of the $RADP$ ($DRADP$) compounds may be expected to be similar to that of the incommensurate phase. On the other hand one may expect that response of long wavelength excitations of the glass state gives information about an almost homogeneous effective medium with properties averaged over all clusters. The transition region between these two characteristic lengths behaves depending on interactions between clusters. Note that the external probing perturbation characterized by long wavelengths induces response of internal modes with very short wavelengths due to coupling between long wavelength and short wavelength modes in the incommensurate phase. One may then expect that mentioned above glass response may be partially the same as that of the incommensurate phase in corresponding ( averaged over impurities ) effective medium.

While the effective medium theory enables discussion of physical properties on the macroscopic level, considerations how the ground state modifies due to presence of defects on the microscopic level are necessary to perform to understand glassy state more thoroughly.

It is natural to extend the studies of the sinusoidaly modulated incommensurate structure in quantum structural order-disorder type materials as idealized crystals, as in [20], to more realistic crystals with a few deffects. Results of such studies may serve to be a starting point to describe materials with nonzero concentration of defects and to understand better limitations of the effective medium approach to such systems.

The ideal systems without defects are described by the quantum Ising-like Hamiltonian in the simplest cases.

3 Model.

Our description of the quantum order-disorder type of phase transitions is based on the pseudospin formalism. The basic part, $H_{1}$, of our model Hamiltonian, $H_{T}$, described in [18] for a lattice without defects, is given by

[TABLE]

Here $S^{z}_{i}$ are quantum operators describing pseudospin variables. Their eigenvalues are $S^{z}_{i}=\pm\frac{1}{2}$, at lattice sites i, where $i=1,...,N_{latt}$. For simplicity we study a s.c. lattice with $N_{latt}$ sites, in which each site represents a quantum fluctuating unit (for example an hydrogen or deuteron bond). The tunneling energy [18] is neglected in our model.

Real materials (for example of the $KDP$ type) are characterized by more complicated structures and are described by more complicated models, [24]. We expect that our study of the simplified model may give results which will qualitatively demonstrate effects expected to occur in real materials.

To formulate our model Hamiltonian we note that in the glassy materials of the $RADP$ and $DRADP$ type one should consider a total Hamiltonian, $H_{T}$, which contains not only the basic part given above but also the part $H_{R}$ describing random fields $h_{n}$ acting on pseudospin degrees of freedom at those sites n at which substituents $ND_{4}$ are localized, [24],

[TABLE]

In real systems it is necessary to assume that resulting total field vanishes in the sample

[TABLE]

However for study purposes it will be instructive to understand how presence of any field influences the ground state. In the next sections we consider the case of a single impurity and a pair of impurities characterized by a nonzero field.

In [24] the average lattice approximation is used as a first step in study of these materials. It is assumed that there exists a contribution to $J_{ij}$ proportional to the concentration of $ND_{4}^{+}$ ions. In the present paper we will not specify an explicit form of such contribution. It is however important that we assume presence of a competition between the ferroelectric (for x=0) and antiferroelectric (x=1) interactions leading to frustration and resulting in modulation in the pure lattice (without defects) ground state.

We assume that the modulation occurs in the direction $(1,0,0)$ only. The modulation wavevector amplitude $Q\equiv{\bf Q.a}$, where ${\bf a}$ is the basic lattice vector in the direction $(1,0,0)$, is determined by the free energy minimization. The interaction energy Fourier transform $J(q)$ is assumed to has the form:

[TABLE]

We neglect its dependence on wavevectors in directions perpendicular to the direction ${\bf a}$. Such a step is valid assuming weak modulations in these directions. We take the interaction energies in perpendicular directions to be zero. The interaction energy constants $J_{1}$ and $J_{2}$ describe interaction between the nearest and next nearest neighbors respectively.

4 Ground state in pure systems.

The random-phase-approximation type analysis, [18], of the basic part (1) of the total Hamiltonian gives description of the modulated state. We perform in this section such an analysis for reference purposes. The free energy is minimized at a given temperature T for those values of the pseudospin operators which satisfy

[TABLE]

[TABLE]

[TABLE]

Note, that a random field influence due to defects is taken into account in the internal field definition in (3). Let us now consider thermodynamic properties of our model system for pure systems without defects. The ground state equations (3) give zero mean values of the operators above some critical temperature $T_{c}$ for zero random fields, $h_{n}=0.$ Below this temperature, which is given by

[TABLE]

the single plane wave modulated state

[TABLE]

[TABLE]

realizes in pure systems without random fields. The modulation vector $Q$ is found in this case to be given by

[TABLE]

The internal mean field at the site n, $H_{n}$, from (3) is found to be given by

[TABLE]

We see that the amplitude of the internal field is proportional to the amplitude of the pseudospin variable at a given site in the single plane wave state. This later property is lost when higher harmonics terms are included in the modulated state (5).

The wavevector $Q$ is incommensurate if $Q$ is different from $2\pi\frac{M}{N}$, here M, N are any integers. Note that the case of $DRADP$ corresponds to $\frac{Q}{2\pi}\approx 0.35$. The case of $Rb_{2}ZnBr_{4}$ may be characterized by $\frac{Q}{2\pi}\approx\frac{5}{17}$, see in [31].

The ground state energy is given by

[TABLE]

where the amplitude of the wave, S, is found from (3) - (5)

[TABLE]

The validity of the expression (8) is restricted to temperatures not very low with respect to the transition temperature, the region where the single plane wave dominates. One should modify (8) whenever higher harmonics become essential. In some systems the basic harmonics may remain the most important harmonics also at low temperatures. Glassy materials in which freezing temperature is nearby the transition temperature given above are one example. Another example are systems in which some of higher harmonics are suppressed due to symmetry even at low temperatures. In systems described by the Hamiltonian given above and in which there are no reasons to suppress higher harmonics the third harmonic amplitude becomes very soon comparable to the basic harmonic one. Its amplitude $S_{3}$ is given by

[TABLE]

Here and in the following we assume that the modulation wavevector amplitude $Q$ leads to the incommensurate modulated phase. Thus we will not discuss the effects of the lock-in energies.

5 Single plane wave ground state and impurities.

The Hamiltonian part $H_{R}$ describing random fields $h_{n}$ acting on pseudospin degrees of freedom at sites n (at which substituents $ND_{4}$ are localized) has form (2), see [24]. In the mean field approximation the energy $E_{R}$ associated with this part of the total Hamiltonian has the form

[TABLE]

While in real systems it is necessary to assume that resulting total field in the sample vanishes $\sum_{n}h_{n}=0$, it is not the case of a single defect case study. We further assume that the modulated ground state has a single plane wave form

[TABLE]

This assumption is correct when random fields have a very small amplitude. As it will be discussed later, in this case the equations determining the form of the ground state have the same form as those in pure material in the highest of H order. In the same order the solution to these equations describing the ground state is of the same order. Thus we can use (9) to study influence of defects on the ground state in the highest in h order. Our aim is to determine which values of the wave amplitude S, the wave vector $Q$ and the phase variable $\phi$ minimize the free energy at given thermodynamic conditions. The free energy is minimized (at a temperature T) for those values of the pseudospin operators which satisfy

[TABLE]

[TABLE]

[TABLE]

In this paper we would like to consider systems with one and two defects. Then we may proceed our calculations as in the pure systems taking $h_{n}=0$ in (10). Influence of defects may be done in such case by considering the energy terms arising from interactions of pure system with defects.

The ground state equations (10) give zero mean values of the operators above the critical temperature $T_{c}$. Below this temperature, given by (4), which has the same value as in the pure case, the single plane wave modulated state

[TABLE]

[TABLE]

realizes again. The modulation vector $Q$ is found to be again, as in the pure case, given by

[TABLE]

The internal mean field at the site n, $H_{n}$, from (3) is found to be given by

[TABLE]

where D are sites where defects are localized.

5.1 A single impurity.

Let us assume that the lattice site D contains a defect and that the field associated with this defect is given by $h_{D}\equiv h.$ The energy $E_{R}^{s}$ associated with this part of the total Hamiltonian has the form

[TABLE]

Let us first consider influence of this energy term when the amplitude of the field h is very small and positive. Turning on slowly this field we see from (13) that the phase variable $\phi$ locks its value even in the incommensurate phase to

[TABLE]

where k is any integer. The amplitude S remains the same as in the pure case in the first order of the (small) field h. From (13) it is clear that such a defect will have tendency to suppress the amplitude S to lower values.

5.2 A pair of impurities.

Let us assume that the lattice sites [math] and $D$ contain a pair of defects. The fields associated with these defects are given by $h_{0}\equiv h,$ $h_{D}\equiv-h.$ We assume that the sum of the fields on defects is zero. The energy $E_{R}^{s}$ associated with this part of the total Hamiltonian has the form

[TABLE]

This term is the only term in the total energy which is dependent on the phase variable $\phi.$ Thus its minimization with respect to this phase variable gives description about its thermodynamic state.

Let us consider influence of this energy term when the amplitude of the field h is very small and positive. Turning on slowly this field we see from (14) that the phase variable $\phi$ locks its value even in the incommensurate phase to

[TABLE]

where k is any integer. The amplitude S remains the same as in the pure case in the first order of the (small) field h. From (14) it is clear that such a pair of defects will have tendency to suppress the amplitude S to lower values:

[TABLE]

This energy (15) is always a negative quantity for h positive a D nonzero, if $Q$ takes an incommensurate value.

6 Third harmonics of the ground state and impurities.

6.1 A single impurity.

When the third harmonics in the formation of the ground state in pure system is taken into account, we obtain the ground state

[TABLE]

where temperature dependence of amplitudes S and $S_{3}$ was shown in the Introduction.

Let us assume that the lattice site D contains a defect and that the field associated with this defect is given by $h_{D}\equiv h.$ The energy $E_{R}^{s}$ associated with this part of the total Hamiltonian has the form

[TABLE]

This energy is small with respect to the energy of the whole lattice. This later energy, however, determines both amplitudes. Thus we assume that the amplitudes and the wave vector $Q$ are unchanged with respect to the pure case. Let us first consider influence of this energy term when the amplitude of the field h is very small and positive. Turning on slowly this field we see from (16) that the phase variable $\phi$ locks its value even in the incommensurate phase to

[TABLE]

where k is any integer. This later statement is true for temperature near the critical temperature, where

[TABLE]

However, there may exist metastable (a local minimum of free energy) states given by

[TABLE]

where temperature dependence of this phase is given by

[TABLE]

These states exist below some temperature lower than the critical. We will not discuss them further in this paper.

The amplitude S remains the same as in the pure case in the first order of the (small and positive) field h. It is clear that while the basic harmonics amplitude should increase when the single impurity is present, the third harmonics amplitude will have tendency to decrease its value

[TABLE]

due to fact that the sign of both amplitudes is reversed.

6.2 A pair of impurities.

Let us assume that the lattice sites 0 and D contain a pair of defects. The fields associated with these defects are given by $h_{0}\equiv h$ and $h_{D}\equiv-h.$ The energy $E_{R}^{s}$ associated with this part of the total Hamiltonian has the form

[TABLE]

[TABLE]

Here we assume that the ground state is described by the same phase variable in the whole crystal. This energy is again small with respect to the energy of the whole lattice. We assume that the amplitudes and the wave vector $Q$ are unchanged with respect to the pure case. Let us first consider influence of this energy term when the amplitude of the field h is very small and positive. Turning on slowly this field we see from (13) that the phase variable $\phi$ locks its value even in the incommensurate phase to

[TABLE]

where k is any integer. This later statement is true for temperature near the critical temperature, where

[TABLE]

and where the energy of the ground state modulated by basic and the third harmonics and originated from interaction with defects, found to be given by

[TABLE]

is negative. We see that this happens for those distances between two defets for which

[TABLE]

[TABLE]

due to fact that the sign of both amplitudes is reversed. Also the case

[TABLE]

[TABLE]

gives the same energy, the phase variable is now shifted by $\pi$. There is however region of ditances between defects for which the second term in (19) is positive. In case of two defects this does not suppress the third harmonics.

7 Discussion.

The energy $E_{R}^{s}$ associated with interaction of two defects with modulated ground state wave consisting of the basic and the third harmonics has the form given by (19). Let us consider a medium in whic such defects are distributed more-less homogeneously in all directions. If x is their density and D is their average distance in the modulation direction, then the total excess energy due to interaction of considered defects with the ground state modulated by the basic and the third harmonics is found to be given by

[TABLE]

where the last term is due to the third harmonics contribution to the total internal energy. Let us compare this energy with the energy of the same state in which the third harmonics is absent, $S_{3}=0$. This later energy excess will be

[TABLE]

Here we assume that the ground state is described by the same basic harmonics in the whole crystal.

There exists relationship between the defects average distance in the modulation direction D, the defects average distance in the perpendicular to the modulation direction $D_{p}$, the lattice constan a and the defects concentration x:

[TABLE]

The third harmonics becomes suppressed when the difference of these two energies (20) and (21) becomes comparable.

We see that this happens for those distances between two defets for which either basic harmonics (which still dominates the third harmonics) in the presence of defects and the third harmonics contributes a large amount of energy when

[TABLE]

either when

[TABLE]

and simultaneously

[TABLE]

When the inequalities just displayed are not satisfied for both cases the third harmonics is present in the system.

We hope that our theoretical discussion of possibility to find the above mentioned type of modifications of the ground state structure due to disturbancies in materials of the quantum order-disorder type ($KDP$-like materials) in which incommensurably modulated phase may occur enable to increase the number of types of materials in which studied phenomena may be confronted with reality.