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Gyorgy Szabo

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Papers by Gyorgy Szabo

Selection of dynamical rules in spatial Prisoner's Dilemma games

EPL (Europhysics Letters), 2009

We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy... more We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy and imitation rule from a randomly chosen neighbor with a probability dependent on the payoff difference when the player's income is collected from games with the neighbors. The players, located on the sites of a two-dimensional lattice, follow unconditional cooperation or defection and use individual strategy adoption rule described by a parameter. If the system is started from a random initial state then the present co-evolutionary rule drives the system towards a state where only one evolutionary rule remains alive even in the coexistence of cooperative and defective behaviors. The final rule is related to the optimum providing the highest level of cooperation and affected by the topology of the connectivity structure.

Phase transitions and volunteering in spatial public goods games

Physical review letters, Jan 9, 2002

We present a simple yet effective mechanism promoting cooperation under full anonymity by allowin... more We present a simple yet effective mechanism promoting cooperation under full anonymity by allowing for voluntary participation in public goods games. This natural extension leads to "rock-scissors-paper"-type cyclic dominance of the three strategies, cooperate, defect, and loner. In spatial settings with players arranged on a regular lattice, this results in interesting dynamical properties and intriguing spatiotemporal patterns. In particular, variations of the value of the public good leads to transitions between one-, two-, and three-strategy states which either are in the class of directed percolation or show interesting analogies to Ising-type models. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior.

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavi... more Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the

Congestion phenomena caused by matching pennies in evolutionary games

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

Evolutionary social dilemma games are extended by an additional matching-pennies game that modifi... more Evolutionary social dilemma games are extended by an additional matching-pennies game that modifies the collected payoffs. In a spatial version players are distributed on a square lattice and interact with their neighbors. First, we show that the matching-pennies game can be considered as the microscopic force of the Red Queen effect that breaks the detailed balance and induces eddies in the microscopic probability currents if the strategy update is analogous to the Glauber dynamics for the kinetic Ising models. The resulting loops in probability current breaks symmetry between the chessboardlike arrangements of strategies via a bottleneck effect occurring along the four-edge loops in the microscopic states. The impact of this congestion is analogous to the application of a staggered magnetic field in the Ising model; that is, the order-disorder critical transition is wiped out by noise. It is illustrated that the congestion induced symmetry breaking can be beneficial for the whole ...

Magnetic hysteresis in an Ising-like dipole-dipole model

Physical Review B, 1998

Using zero temperature Monte Carlo simulations we have studied the magnetic hysteresis in a three... more Using zero temperature Monte Carlo simulations we have studied the magnetic hysteresis in a three-dimensional Ising model with nearest neighbor exchange and dipolar interaction. The average magnetization of spins located inside a sphere on a cubic lattice is determined as a function of magnetic field varied periodically. The simulations have justified the appearance of hysteresis and allowed us to have a deeper insight into the series of metastable states developed during this process. 05.50.+q, 05.70.Ln, 75.40.Mg, 75.60.Ej Typeset using REVT E X

Hysteresis in a dipolar Ising model

Physica B: Condensed Matter, 2000

Zero-temperature Monte Carlo simulations are performed to study the magnetic hysteresis in a thre... more Zero-temperature Monte Carlo simulations are performed to study the magnetic hysteresis in a three-dimensional model with only dipolar interactions among the Ising spins located inside a sphere on a cubic lattice. Relevant di!erences are found in the site energy distributions between the up and down spins during the magnetization process. Variations of the dipolar energy and the density of incomplete columns are also discussed.

Segregation process and phase transition in cyclic predator-prey models with an even number of species

Physical review. E, Statistical, nonlinear, and soft matter physics, 2007

We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) ... more We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) that allows the formation of two defensive alliances consisting of the even and odd label species. The species are distributed on the sites of a square lattice. The evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly: if the sites are occupied by a predator-prey pair then the predator invades the prey's site; otherwise the species exchange their sites with a probability X . For low X values, a self-organizing pattern is maintained by cyclic invasions. If X exceeds a threshold value, then two types of domain grow up that are formed by the odd and even label species, respectively. Monte Carlo simulations indicate the blocking of this segregation process within a range of X for n=8.

Ordering in spatial evolutionary games for pairwise collective strategy updates

Physical review. E, Statistical, nonlinear, and soft matter physics, 2010

Evolutionary 2×2 games are studied with players located on a square lattice. During the evolution... more Evolutionary 2×2 games are studied with players located on a square lattice. During the evolution the randomly chosen neighboring players try to maximize their collective income by adopting a random strategy pair with a probability dependent on the difference of their summed payoffs between the final and initial states assuming quenched strategies in their neighborhood. In the case of the anticoordination game this system behaves like an antiferromagnetic kinetic Ising model. Within a wide region of social dilemmas this dynamical rule supports the formation of similar spatial arrangement of the cooperators and defectors ensuring the optimum total payoff if the temptation to choose defection exceeds a threshold value dependent on the sucker's payoff. The comparison of the results with those achieved for pairwise imitation and myopic strategy updates has indicated the relevant advantage of pairwise collective strategy update in the maintenance of cooperation.

Cooperation in spatial prisoner's dilemma with two types of players for increasing number of neighbors

Physical review. E, Statistical, nonlinear, and soft matter physics, 2009

We study a spatial two-strategy (cooperation and defection) prisoner's dilemma game with two ... more We study a spatial two-strategy (cooperation and defection) prisoner's dilemma game with two types ( A and B ) of players located on the sites of a square lattice. The evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor. The strategy adoption probability is reduced by a prefactor (w<1) from the players of type B . We consider the competition between two opposite effects when increasing the number of neighbors ( k=4 , 8, and 24). Within a range of the portion of influential players (type A ) the inhomogeneous activity in strategy transfer yields a relevant increase (dependent on w ) in the density of cooperators. The noise dependence of this phenomenon is also discussed by evaluating phase diagrams.

Self-organizing patterns maintained by competing associations in a six-species predator-prey model

Physical review. E, Statistical, nonlinear, and soft matter physics, 2008

Formation and competition of associations are studied in a six-species ecological model where eac... more Formation and competition of associations are studied in a six-species ecological model where each species has two predators and two prey. Each site of a square lattice is occupied by an individual belonging to one of the six species. The evolution of the spatial distribution of species is governed by iterated invasions between the neighboring predator-prey pairs with species specific rates and by site exchange between the neutral pairs with a probability X . This dynamical rule yields the formation of five associations composed of two or three species with proper spatiotemporal patterns. For large X a cyclic dominance can occur between the three two-species associations whereas one of the two three-species associations prevails in the whole system for low values of X in the final state. Within an intermediate range of X all the five associations coexist due to the fact that cyclic invasions between the two-species associations reduce their resistance temporarily against the invasio...

Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species

Physical review. E, Statistical, nonlinear, and soft matter physics, 2008

Cyclic predator-prey models with four or six species are studied on a square lattice when the inv... more Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations, the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.

Phase transition and selection in a four-species cyclic predator-prey model

Physical review. E, Statistical, nonlinear, and soft matter physics, 2004

We study a four-species ecological system with cyclic dominance whose individuals are distributed... more We study a four-species ecological system with cyclic dominance whose individuals are distributed on a square lattice. Randomly chosen individuals migrate to one of the neighboring sites if it is empty or invade this site if occupied by their prey. The cyclic dominance maintains the coexistence of all four species if the concentration of vacant sites is lower than a threshold value. Above the threshold, a symmetry breaking ordering occurs via growing domains containing only two neutral species inside. These two neutral species can protect each other from the external invaders (predators) and extend their common territory. According to our Monte Carlo simulations the observed phase transition seems to be equivalent to those found in spreading models with two equivalent absorbing states although the present model has continuous sets of absorbing states with different portions of the two neutral species. The selection mechanism yielding symmetric phases is related to the domain growth ...

Three-state cyclic voter model extended with Potts energy

Physical review. E, Statistical, nonlinear, and soft matter physics, 2002

The cyclically dominated voter model on a square is extended by taking into consideration the var... more The cyclically dominated voter model on a square is extended by taking into consideration the variation of Potts energy during the nearest neighbor invasions. We have investigated the effect of surface tension on the self-organizing patterns maintained by the cyclic invasions. A geometrical analysis is also developed to study the three-color patterns. These investigations clearly indicate that in the "voter model" limit the pattern evolution is governed by the loop creation due to the overhanging during the interfacial roughening. Conversely, in the presence of surface tension the evolution is governed by spiral formation whose geometrical parameters depend on the strength of cyclic dominance.

Universality change in stochastic cellular automaton with applied site exchange

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1994

ABSTRACT Extended mean-field theory is used for studying the effect of nonlocal site exchange on ... more ABSTRACT Extended mean-field theory is used for studying the effect of nonlocal site exchange on the critical transition of the one-dimensional, stochastic cellular automaton of Rule 18 defined by Wolfram [Rev. Mod. Phys. 55, 6011 (1993)]. The analysis is carried out at different levels taking n-point correlations explicitly into consideration. Extrapolation to the n--&gt;∞ (exact) limit gives results, which agree well with simulation. By applying nonlocal site-exchange mixing the universality of the transition changes and a continuous crossover of critical point and beta exponent is observed.

Extended mean-field study of a stochastic cellular automaton

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1994

ABSTRACT Extended mean-field theory is used for studying the one-dimensional stochastic cellular ... more ABSTRACT Extended mean-field theory is used for studying the one-dimensional stochastic cellular automaton with Rule 18 defined by Wolfram [Rev. Mod. Phys. 55, 601 (1983)]. The analysis is carried out at different levels taking n-point and n-pair correlations explicitly into consideration. The pair approximations reproduce the exact results in the deterministic limit. The critical behavior is studied by the Padé approximant method and the predicted critical probability and exponent agree with previous data within a few percent.

Simulation of Czochralski bulk flow under vertical magnetic field

An experimental model of magnetlc Czochralski growth was constructed to invest/gate the hydrodyna... more An experimental model of magnetlc Czochralski growth was constructed to invest/gate the hydrodynamic behaviour of the melt. The time dependence of temperature in molten gallium was measured and analysed asa functlon of heating, rotation and magnetic field. The temperature fluctuations were damped when the magnetic field exceeded a threshold value dependent on heating temperature and rotation rate.

PHASES IN THE BI-CU-CA-SR-O SUPERCONDUCTIVE SYSTEM ANALYZED BY ELECTRON-BEAM X-RAY-MICROANALYSIS

SEM INVESTIGATIONS ON THE PRODUCTS OF THE GROWTH OF SMBA2CU3O7-X AND YBA2CU3O7-X SINGLE-CRYSTALS

Growth properties of YBa 2 Cu 3 O 7 single crystals

ABSTRACT

Thermodynamic aspects of chemically curved crystals

The thermodynamic stability of an elastic material with a constant concentration gradient is anal... more The thermodynamic stability of an elastic material with a constant concentration gradient is analysed in linear approximation within the framework of the Ginzburg-Landau theory including an external inhomogeneous field. The result suggests ways to produce stress-free, paraboloidically curved finite size crystals.

Selection of dynamical rules in spatial Prisoner's Dilemma games

EPL (Europhysics Letters), 2009

We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy... more We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy and imitation rule from a randomly chosen neighbor with a probability dependent on the payoff difference when the player's income is collected from games with the neighbors. The players, located on the sites of a two-dimensional lattice, follow unconditional cooperation or defection and use individual strategy adoption rule described by a parameter. If the system is started from a random initial state then the present co-evolutionary rule drives the system towards a state where only one evolutionary rule remains alive even in the coexistence of cooperative and defective behaviors. The final rule is related to the optimum providing the highest level of cooperation and affected by the topology of the connectivity structure.

Phase transitions and volunteering in spatial public goods games

Physical review letters, Jan 9, 2002

We present a simple yet effective mechanism promoting cooperation under full anonymity by allowin... more We present a simple yet effective mechanism promoting cooperation under full anonymity by allowing for voluntary participation in public goods games. This natural extension leads to "rock-scissors-paper"-type cyclic dominance of the three strategies, cooperate, defect, and loner. In spatial settings with players arranged on a regular lattice, this results in interesting dynamical properties and intriguing spatiotemporal patterns. In particular, variations of the value of the public good leads to transitions between one-, two-, and three-strategy states which either are in the class of directed percolation or show interesting analogies to Ising-type models. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior.

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavi... more Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the

Congestion phenomena caused by matching pennies in evolutionary games

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

Evolutionary social dilemma games are extended by an additional matching-pennies game that modifi... more Evolutionary social dilemma games are extended by an additional matching-pennies game that modifies the collected payoffs. In a spatial version players are distributed on a square lattice and interact with their neighbors. First, we show that the matching-pennies game can be considered as the microscopic force of the Red Queen effect that breaks the detailed balance and induces eddies in the microscopic probability currents if the strategy update is analogous to the Glauber dynamics for the kinetic Ising models. The resulting loops in probability current breaks symmetry between the chessboardlike arrangements of strategies via a bottleneck effect occurring along the four-edge loops in the microscopic states. The impact of this congestion is analogous to the application of a staggered magnetic field in the Ising model; that is, the order-disorder critical transition is wiped out by noise. It is illustrated that the congestion induced symmetry breaking can be beneficial for the whole ...

Magnetic hysteresis in an Ising-like dipole-dipole model

Physical Review B, 1998

Using zero temperature Monte Carlo simulations we have studied the magnetic hysteresis in a three... more Using zero temperature Monte Carlo simulations we have studied the magnetic hysteresis in a three-dimensional Ising model with nearest neighbor exchange and dipolar interaction. The average magnetization of spins located inside a sphere on a cubic lattice is determined as a function of magnetic field varied periodically. The simulations have justified the appearance of hysteresis and allowed us to have a deeper insight into the series of metastable states developed during this process. 05.50.+q, 05.70.Ln, 75.40.Mg, 75.60.Ej Typeset using REVT E X

Hysteresis in a dipolar Ising model

Physica B: Condensed Matter, 2000

Zero-temperature Monte Carlo simulations are performed to study the magnetic hysteresis in a thre... more Zero-temperature Monte Carlo simulations are performed to study the magnetic hysteresis in a three-dimensional model with only dipolar interactions among the Ising spins located inside a sphere on a cubic lattice. Relevant di!erences are found in the site energy distributions between the up and down spins during the magnetization process. Variations of the dipolar energy and the density of incomplete columns are also discussed.

Segregation process and phase transition in cyclic predator-prey models with an even number of species

Physical review. E, Statistical, nonlinear, and soft matter physics, 2007

We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) ... more We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) that allows the formation of two defensive alliances consisting of the even and odd label species. The species are distributed on the sites of a square lattice. The evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly: if the sites are occupied by a predator-prey pair then the predator invades the prey's site; otherwise the species exchange their sites with a probability X . For low X values, a self-organizing pattern is maintained by cyclic invasions. If X exceeds a threshold value, then two types of domain grow up that are formed by the odd and even label species, respectively. Monte Carlo simulations indicate the blocking of this segregation process within a range of X for n=8.

Ordering in spatial evolutionary games for pairwise collective strategy updates

Physical review. E, Statistical, nonlinear, and soft matter physics, 2010

Evolutionary 2×2 games are studied with players located on a square lattice. During the evolution... more Evolutionary 2×2 games are studied with players located on a square lattice. During the evolution the randomly chosen neighboring players try to maximize their collective income by adopting a random strategy pair with a probability dependent on the difference of their summed payoffs between the final and initial states assuming quenched strategies in their neighborhood. In the case of the anticoordination game this system behaves like an antiferromagnetic kinetic Ising model. Within a wide region of social dilemmas this dynamical rule supports the formation of similar spatial arrangement of the cooperators and defectors ensuring the optimum total payoff if the temptation to choose defection exceeds a threshold value dependent on the sucker's payoff. The comparison of the results with those achieved for pairwise imitation and myopic strategy updates has indicated the relevant advantage of pairwise collective strategy update in the maintenance of cooperation.

Cooperation in spatial prisoner's dilemma with two types of players for increasing number of neighbors

Physical review. E, Statistical, nonlinear, and soft matter physics, 2009

We study a spatial two-strategy (cooperation and defection) prisoner's dilemma game with two ... more We study a spatial two-strategy (cooperation and defection) prisoner's dilemma game with two types ( A and B ) of players located on the sites of a square lattice. The evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor. The strategy adoption probability is reduced by a prefactor (w<1) from the players of type B . We consider the competition between two opposite effects when increasing the number of neighbors ( k=4 , 8, and 24). Within a range of the portion of influential players (type A ) the inhomogeneous activity in strategy transfer yields a relevant increase (dependent on w ) in the density of cooperators. The noise dependence of this phenomenon is also discussed by evaluating phase diagrams.

Self-organizing patterns maintained by competing associations in a six-species predator-prey model

Physical review. E, Statistical, nonlinear, and soft matter physics, 2008

Formation and competition of associations are studied in a six-species ecological model where eac... more Formation and competition of associations are studied in a six-species ecological model where each species has two predators and two prey. Each site of a square lattice is occupied by an individual belonging to one of the six species. The evolution of the spatial distribution of species is governed by iterated invasions between the neighboring predator-prey pairs with species specific rates and by site exchange between the neutral pairs with a probability X . This dynamical rule yields the formation of five associations composed of two or three species with proper spatiotemporal patterns. For large X a cyclic dominance can occur between the three two-species associations whereas one of the two three-species associations prevails in the whole system for low values of X in the final state. Within an intermediate range of X all the five associations coexist due to the fact that cyclic invasions between the two-species associations reduce their resistance temporarily against the invasio...

Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species

Physical review. E, Statistical, nonlinear, and soft matter physics, 2008

Cyclic predator-prey models with four or six species are studied on a square lattice when the inv... more Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations, the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.

Phase transition and selection in a four-species cyclic predator-prey model

Physical review. E, Statistical, nonlinear, and soft matter physics, 2004

We study a four-species ecological system with cyclic dominance whose individuals are distributed... more We study a four-species ecological system with cyclic dominance whose individuals are distributed on a square lattice. Randomly chosen individuals migrate to one of the neighboring sites if it is empty or invade this site if occupied by their prey. The cyclic dominance maintains the coexistence of all four species if the concentration of vacant sites is lower than a threshold value. Above the threshold, a symmetry breaking ordering occurs via growing domains containing only two neutral species inside. These two neutral species can protect each other from the external invaders (predators) and extend their common territory. According to our Monte Carlo simulations the observed phase transition seems to be equivalent to those found in spreading models with two equivalent absorbing states although the present model has continuous sets of absorbing states with different portions of the two neutral species. The selection mechanism yielding symmetric phases is related to the domain growth ...

Three-state cyclic voter model extended with Potts energy

Physical review. E, Statistical, nonlinear, and soft matter physics, 2002

The cyclically dominated voter model on a square is extended by taking into consideration the var... more The cyclically dominated voter model on a square is extended by taking into consideration the variation of Potts energy during the nearest neighbor invasions. We have investigated the effect of surface tension on the self-organizing patterns maintained by the cyclic invasions. A geometrical analysis is also developed to study the three-color patterns. These investigations clearly indicate that in the "voter model" limit the pattern evolution is governed by the loop creation due to the overhanging during the interfacial roughening. Conversely, in the presence of surface tension the evolution is governed by spiral formation whose geometrical parameters depend on the strength of cyclic dominance.

Universality change in stochastic cellular automaton with applied site exchange

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1994

ABSTRACT Extended mean-field theory is used for studying the effect of nonlocal site exchange on ... more ABSTRACT Extended mean-field theory is used for studying the effect of nonlocal site exchange on the critical transition of the one-dimensional, stochastic cellular automaton of Rule 18 defined by Wolfram [Rev. Mod. Phys. 55, 6011 (1993)]. The analysis is carried out at different levels taking n-point correlations explicitly into consideration. Extrapolation to the n--&gt;∞ (exact) limit gives results, which agree well with simulation. By applying nonlocal site-exchange mixing the universality of the transition changes and a continuous crossover of critical point and beta exponent is observed.

Extended mean-field study of a stochastic cellular automaton

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1994

ABSTRACT Extended mean-field theory is used for studying the one-dimensional stochastic cellular ... more ABSTRACT Extended mean-field theory is used for studying the one-dimensional stochastic cellular automaton with Rule 18 defined by Wolfram [Rev. Mod. Phys. 55, 601 (1983)]. The analysis is carried out at different levels taking n-point and n-pair correlations explicitly into consideration. The pair approximations reproduce the exact results in the deterministic limit. The critical behavior is studied by the Padé approximant method and the predicted critical probability and exponent agree with previous data within a few percent.

Simulation of Czochralski bulk flow under vertical magnetic field

An experimental model of magnetlc Czochralski growth was constructed to invest/gate the hydrodyna... more An experimental model of magnetlc Czochralski growth was constructed to invest/gate the hydrodynamic behaviour of the melt. The time dependence of temperature in molten gallium was measured and analysed asa functlon of heating, rotation and magnetic field. The temperature fluctuations were damped when the magnetic field exceeded a threshold value dependent on heating temperature and rotation rate.

PHASES IN THE BI-CU-CA-SR-O SUPERCONDUCTIVE SYSTEM ANALYZED BY ELECTRON-BEAM X-RAY-MICROANALYSIS

SEM INVESTIGATIONS ON THE PRODUCTS OF THE GROWTH OF SMBA2CU3O7-X AND YBA2CU3O7-X SINGLE-CRYSTALS

Growth properties of YBa 2 Cu 3 O 7 single crystals

ABSTRACT

Thermodynamic aspects of chemically curved crystals

The thermodynamic stability of an elastic material with a constant concentration gradient is anal... more The thermodynamic stability of an elastic material with a constant concentration gradient is analysed in linear approximation within the framework of the Ginzburg-Landau theory including an external inhomogeneous field. The result suggests ways to produce stress-free, paraboloidically curved finite size crystals.