A Monte Carlo C-code for calculating transmission efficiency of recoil separators and viewing residue trajectories

We present a semimicroscopic Monte Carlo code for calculating absolute transmission efficiency of recoil separators for heavy ion-induced complete fusion reactions. The code generates realistic distributions for energy, charge state and angle of evaporation residues. Residue trajectories are calculated using first order ion optical transfer matrices. Trajectory plots in the dispersive and the non-dispersive planes are generated. Using this code, we have obtained good agreement between calculated and measured transmission efficiencies for the Heavy Ion Reaction Analyzer at IUAC. The code can be adapted easily to any other electromagnetic recoil separator.

Program summary

Program title: TERSCatalogue identifier: AEBD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 6818No. of bytes in distributed program, including test data, etc.: 1 216 097Distribution format: tar.gzProgramming language: CComputer: The code has been developed and tested on a PC with Intel Pentium IV processorOperating system: LinuxRAM: About 8 MbytesClassification: 17.7External routines: pgplot graphics subroutine library [1] should be installed in the system for generating residue trajectory plots.Nature of problem: Recoil separators are employed to select and identify nuclei of interest, produced in a nuclear reaction, rejecting unreacted beam and other undesired reaction products. It is important to know what fraction of the selected nuclei, leaving the target, reaches the detection system. This information is crucial for determining absolute cross section of the studied reaction.Solution method:Interaction of projectiles with target nuclei is treated event by event, semimicroscopically. Position and angle (with respect to beam direction), energy and charge state of the reaction products are calculated by Monte Carlo method. Trajectory of each nuclei inside the separator is then calculated by ion optical transfer matrix method. Ratio of the number of trajectories completing their journey up to the detection system to the total number of trajectories is a direct measure of absolute transmission efficiency of the separator.Restrictions: The present version of the code is applicable to complete fusion reactions only. The code can be applied to other types of reactions (e.g., few nucleon transfer) as well, by suitably modifying energy and angular distribution of reaction products. Also, ion optical specifications and acceptance are unique for each recoil separator. Transmission efficiency calculation has been done for a specific recoil separator, viz. the Heavy Ion Reaction Analyzer [2,3] at IUAC. One has to make necessary changes in the code, while performing calculations for other recoil separators. Further, atomic number of the residual nucleus should not exceed 92, as the method used for calculating stopping power of ions [4] is valid for Z?92.Running time: From few seconds to several minutes depending on the reaction, number of events and separator layout.References:

[1]

http://www.astro.caltech.edu/~tjp/pgplot/.

[2]

A.K. Sinha, N. Madhavan, J.J. Das, P. Sugathan, D.O. Kataria, A.P. Patro, G.K. Mehta, Nucl. Instr. Methods A 339 (1994) 543.

[3]

S. Nath, Nucl. Instr. Methods A 576 (2007) 403.

[4]

J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, vol. I, Pergamon Press, Oxford, 1984.

     

MinFinder v2.0: An improved version of MinFinder

A new version of the “MinFinder” program is presented that offers an augmented linking procedure for Fortran-77 subprograms, two additional stopping rules and a new start-point rejection mechanism that saves a significant portion of gradient and function evaluations. The method is applied on a set of standard test functions and the results are reported.

New version program summary

Program title: MinFinder v2.0Catalogue identifier: ADWU_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWU_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC Licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 14 150No. of bytes in distributed program, including test data, etc.: 218 144Distribution format: tar.gzProgramming language used: GNU C++, GNU FORTRAN, GNU CComputer: The program is designed to be portable in all systems running the GNU C++ compilerOperating system: Linux, Solaris, FreeBSDRAM: 200 000 bytesClassification: 4.9Catalogue identifier of previous version: ADWU_v1_0Journal reference of previous version: Computer Physics Communications 174 (2006) 166-179Does the new version supersede the previous version?: YesNature of problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques can be trapped in any local minimum. Global optimization is then the appropriate tool. For example, solving a non-linear system of equations via optimization, one may encounter many local minima that do not correspond to solutions, i.e. they are far from zero.Solution method: Using a uniform pdf, points are sampled from a rectangular domain. A clustering technique, based on a typical distance and a gradient criterion, is used to decide from which points a local search should be started. Further searching is terminated when all the local minima inside the search domain are thought to be found. This is accomplished via three stopping rules: the “double-box” stopping rule, the “observables” stopping rule and the “expected minimizers” stopping rule.Reasons for the new version: The link procedure for source code in Fortran 77 is enhanced, two additional stopping rules are implemented and a new criterion for accepting-start points, that economizes on function and gradient calls, is introduced.Summary of revisions:

1.

Addition of command line parameters to the utility program make_program.

2.

Augmentation of the link process for Fortran 77 subprograms, by linking the final executable with the g2c library.

3.

Addition of two probabilistic stopping rules.

4.

Introduction of a rejection mechanism to the Checking step of the original method, that reduces the number of gradient evaluations.

Additional comments: A technical report describing the revisions, experiments and test runs is packaged with the source code.Running time: Depending on the objective function.     

New version of PLNoise: a package for exact numerical simulation of power-law noises

In a recent paper I have introduced a package for the exact simulation of power-law noises and other colored noises [E. Milotti, Comput. Phys. Comm. 175 (2006) 212]: in particular, the algorithm generates 1/fα noises with 0<α?2. Here I extend the algorithm to generate 1/fα noises with 2<α?4 (black noises). The method is exact in the sense that it produces a sampled process with a theoretically guaranteed range-limited power-law spectrum for any arbitrary sequence of sampling intervals, i.e. the sampling times may be unevenly spaced.

Program summary

Title of program: PLNoiseCatalogue identifier:ADXV_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXV_v2_0.htmlLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandProgramming language used: ANSI CComputer: Any computer with an ANSI C compiler: the package has been tested with gcc version 3.2.3 on Red Hat Linux 3.2.3-52 and gcc version 4.0.0 and 4.0.1 on Apple Mac OS X-10.4Operating system: All operating systems capable of running an ANSI C compilerRAM: The code of the test program is very compact (about 60 Kbytes), but the program works with list management and allocates memory dynamically; in a typical run with average list length 2⋅¹⁰⁴, the RAM taken by the list is 200 KbytesExternal routines: The package needs external routines to generate uniform and exponential deviates. The implementation described here uses the random number generation library ranlib freely available from Netlib [B.W. Brown, J. Lovato, K. Russell: ranlib, available from Netlib, http://www.netlib.org/random/index.html, select the C version ranlib.c], but it has also been successfully tested with the random number routines in Numerical Recipes [W.H. Press, S.A. Teulkolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, second ed., Cambridge Univ. Press., Cambridge, 1992, pp. 274-290]. Notice that ranlib requires a pair of routines from the linear algebra package LINPACK, and that the distribution of ranlib includes the C source of these routines, in case LINPACK is not installed on the target machine.No. of lines in distributed program, including test data, etc.:2975No. of bytes in distributed program, including test data, etc.:194 588Distribution format:tar.gzCatalogue identifier of previous version: ADXV_v1_0Journal reference of previous version: Comput. Phys. Comm. 175 (2006) 212Does the new version supersede the previous version?: YesNature of problem: Exact generation of different types of colored noise.Solution method: Random superposition of relaxation processes [E. Milotti, Phys. Rev. E 72 (2005) 056701], possibly followed by an integration step to produce noise with spectral index >2.Reasons for the new version: Extension to 1/fα noises with spectral index 2<α?4: the new version generates both noises with spectral with spectral index 0<α?2 and with 2<α?4.Summary of revisions: Although the overall structure remains the same, one routine has been added and several changes have been made throughout the code to include the new integration step.Unusual features: The algorithm is theoretically guaranteed to be exact, and unlike all other existing generators it can generate samples with uneven spacing.Additional comments: The program requires an initialization step; for some parameter sets this may become rather heavy.Running time: Running time varies widely with different input parameters, however in a test run like the one in Section 3 in the long write-up, the generation routine took on average about 75 μs for each sample.     

GDF v2.0, an enhanced version of GDF

An improved version of the function estimation program GDF is presented. The main enhancements of the new version include: multi-output function estimation, capability of defining custom functions in the grammar and selection of the error function. The new version has been evaluated on a series of classification and regression datasets, that are widely used for the evaluation of such methods. It is compared to two known neural networks and outperforms them in 5 (out of 10) datasets.

Program summary

Title of program: GDF v2.0Catalogue identifier: ADXC_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXC_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 98 147No. of bytes in distributed program, including test data, etc.: 2 040 684Distribution format: tar.gzProgramming language: GNU C++Computer: The program is designed to be portable in all systems running the GNU C++ compilerOperating system: Linux, Solaris, FreeBSDRAM: 200000 bytesClassification: 4.9Does the new version supersede the previous version?: YesNature of problem: The technique of function estimation tries to discover from a series of input data a functional form that best describes them. This can be performed with the use of parametric models, whose parameters can adapt according to the input data.Solution method: Functional forms are being created by genetic programming which are approximations for the symbolic regression problem.Reasons for new version: The GDF package was extended in order to be more flexible and user customizable than the old package. The user can extend the package by defining his own error functions and he can extend the grammar of the package by adding new functions to the function repertoire. Also, the new version can perform function estimation of multi-output functions and it can be used for classification problems.Summary of revisions: The following features have been added to the package GDF:

•

Multi-output function approximation. The package can now approximate any function . This feature gives also to the package the capability of performing classification and not only regression.

•

User defined function can be added to the repertoire of the grammar, extending the regression capabilities of the package. This feature is limited to 3 functions, but easily this number can be increased.

•

Capability of selecting the error function. The package offers now to the user apart from the mean square error other error functions such as: mean absolute square error, maximum square error. Also, user defined error functions can be added to the set of error functions.

•

More verbose output. The main program displays more information to the user as well as the default values for the parameters. Also, the package gives to the user the capability to define an output file, where the output of the gdf program for the testing set will be stored after the termination of the process.

Additional comments: A technical report describing the revisions, experiments and test runs is packaged with the source code.Running time: Depending on the train data.     

A C-code for the double folding interaction potential of two spherical nuclei

We present a C-code designed to obtain the nucleus-nucleus potential by using the double folding model (DFM) and in particular to find the Coulomb barrier. The program calculates the nucleus-nucleus potential as a function of the distance between the centers of mass of colliding nuclei. The most important output parameters are the Coulomb barrier energy and the radius. Since many researchers use a Woods-Saxon profile for the nuclear term of the potential we provide an option in our code for fitting the DFM potential by such a profile.

Program summary

Program title: DFMSPHCatalogue identifier: AEFH_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFH_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5929No. of bytes in distributed program, including test data, etc.: 115 740Distribution format: tar.gzProgramming language: CComputer: PCOperating system: Windows XP (with the GCC-compiler version 2)RAM: Below 10 MbyteClassification: 17.9Nature of problem: The code calculates in a semimicroscopic way the bare interaction potential between two colliding spherical nuclei as a function of the center of mass distance. The height and the position of the Coulomb barrier are found. The calculated potential is approximated by a conventional Woods-Saxon profile near the barrier. Dependence of the barrier parameters upon the characteristics of the effective NN forces (like, e.g. the range of the exchange part of the nuclear term) can be investigated.Solution method: The nucleus-nucleus potential is calculated using the double folding model with the Coulomb and the effective M3Y NN interactions. For the direct parts of the Coulomb and the nuclear terms, the Fourier transform method is used. In order to calculate the exchange parts the density matrix expansion method is applied.Running time: Less than 1 minute using a PC with a 1.60 GHz processor.     

A handy tool for history keeping of Geant4 tracks - J4HistoryKeeper

The Particle Flow Analysis (PFA) is currently under intense studies as the most promising way to achieve precision jet energy measurements required at the future linear e⁺e⁻ collider. In order to optimize detector configurations and to tune up the PFA it is crucial to identify factors that limit the PFA performance and clarify the fundamental limits on the jet energy resolution that remain even with the perfect PFA and an infinitely granular calorimeter. This necessitates a tool to connect each calorimeter hit in particle showers to its parent charged track, if any, and eventually all the way back to its corresponding primary particle, while identifying possible interactions and decays along the way. In order to realize this with a realistic memory space, we have developed a set of C++ classes that facilitates history keeping of particle tracks within the framework of Geant4. This software tool, hereafter called J4HistoryKeeper, comes in handy in particular when one needs to stop this history keeping for memory space economy at multiple geometrical boundaries beyond which a particle shower is expected to start. In this paper this software tool is described and applied to a generic detector model to demonstrate its functionality.     

PLNoise: a package for exact numerical simulation of power-law noises

Many simulations of stochastic processes require colored noises: here I describe a small program library that generates samples with a tunable power-law spectral density: the algorithm can be modified to generate more general colored noises, and is exact for all time steps, even when they are unevenly spaced (as may often happen in the case of astronomical data, see e.g. [N.R. Lomb, Astrophys. Space Sci. 39 (1976) 447]. The method is exact in the sense that it reproduces a process that is theoretically guaranteed to produce a range-limited power-law spectrum 1/f1+β with −1<β?1. The algorithm has a well-behaved computational complexity, it produces a nearly perfect Gaussian noise, and its computational efficiency depends on the required degree of noise Gaussianity.

Program summary

Title of program: PLNoiseCatalogue identifier:ADXV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneProgramming language used: ANSI CComputer: Any computer with an ANSI C compiler: the package has been tested with gcc version 3.2.3 on Red Hat Linux 3.2.3-52 and gcc version 4.0.0 and 4.0.1 on Apple Mac OS X-10.4Operating system: All operating systems capable of running an ANSI C compilerNo. of lines in distributed program, including test data, etc.:6238No. of bytes in distributed program, including test data, etc.:52 387Distribution format:tar.gzRAM: The code of the test program is very compact (about 50 Kbytes), but the program works with list management and allocates memory dynamically; in a typical run (like the one discussed in Section 4 in the long write-up) with average list length 2⋅¹⁰⁴, the RAM taken by the list is 200 Kbytes.External routines: The package needs external routines to generate uniform and exponential deviates. The implementation described here uses the random number generation library ranlib freely available from Netlib [B.W. Brown, J. Lovato, K. Russell, ranlib, available from Netlib, http://www.netlib.org/random/index.html, select the C version ranlib.c], but it has also been successfully tested with the random number routines in Numerical Recipes [W.H. Press, S.A. Teulkolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, second ed., Cambridge Univ. Press, Cambridge, 1992, pp. 274-290]. Notice that ranlib requires a pair of routines from the linear algebra package LINPACK, and that the distribution of ranlib includes the C source of these routines, in case LINPACK is not installed on the target machine.Nature of problem: Exact generation of different types of Gaussian colored noise.Solution method: Random superposition of relaxation processes [E. Milotti, Phys. Rev. E 72 (2005) 056701].Unusual features: The algorithm is theoretically guaranteed to be exact, and unlike all other existing generators it can generate samples with uneven spacing.Additional comments: The program requires an initialization step; for some parameter sets this may become rather heavy.Running time: Running time varies widely with different input parameters, however in a test run like the one in Section 4 in this work, the generation routine took on average about 7 ms for each sample.     

Non-Markovian melting: a novel procedure to generate initial liquid like phases for small molecules for use in computer simulation studies

Computer simulations of liquid phases require an initial configuration from which to begin. The preparation of such an initial configuration or ‘snapshot’ often involves the melting of a solid phase. This melting is usually undertaken by heating the system at low pressure, followed by a lengthy re-compression and cooling once the melt has formed. This note looks at a novel technique to produce a liquid phase from a perfect crystal using a standard Monte Carlo simulation code.     

KEWPIE: A dynamical cascade code for decaying exited compound nuclei

A new dynamical cascade code for decaying hot nuclei is proposed and specially adapted to the synthesis of super-heavy nuclei. For such a case, the interesting channel is of the tiny fraction that will decay through particles emission, thus the code avoids classical Monte-Carlo methods and proposes a new numerical scheme. The time dependence is explicitely taken into account in order to cope with the fact that fission decay rate might not be constant. The code allows to evaluate both statistical and dynamical observables. Results are successfully compared to experimental data.     

Sampling using a ‘bank’ of clues

An easy-to-implement form of the Metropolis Algorithm is described which, unlike most standard techniques, is well suited to sampling from multi-modal distributions on spaces with moderate numbers of dimensions (order ten) in environments typical of investigations into current constraints on Beyond-the-Standard-Model physics. The sampling technique makes use of pre-existing information (which can safely be of low or uncertain quality) relating to the distribution from which it is desired to sample. This information should come in the form of a “bank” or “cache” of parameter space points of which at least some may be expected to be near regions of interest in the desired distribution. In practical circumstances such “banks of clues” are easy to assemble from earlier work, aborted runs, discarded burn-in samples from failed sampling attempts, or from prior scouting investigations. The technique equilibrates between disconnected parts of the distribution without user input. The algorithm is not lead astray by “bad” clues, but there is no free lunch: performance gains will only be seen where clues are helpful.     

Reduction of the self-forces in Monte Carlo simulations of semiconductor devices on unstructured meshes

When using an unstructured mesh for device geometry, the ensemble Monte Carlo simulations of semiconductor devices may be affected by unwanted self-forces resulting from the particle-mesh coupling. We report on the progress in minimisation of the self-forces on arbitrary meshes by showing that they can be greatly reduced on a finite element mesh with proper interpolation functions. The developed methodology is included into a self-consistent finite element 3D Monte Carlo device simulator. Minimising of the self-forces using the proper interpolation functions is tested by simulating the electron transport in a 10 nm gate length, 6.1 nm body thick, double gate metal-oxide-semiconductor field-effect transistor (MOSFET). We demonstrate the reduction in the self-force and illustrate the practical distinction by showing I-V characteristics for the device.     

Application of Wang-Landau sampling to a protein model using SMMP

We apply Wang-Landau sampling to the continuous energy model of a protein using Simple Molecular Mechanics for Protein (SMMP). We also tried to parallelize the Wang-Landau sampling method and compare our results with previous results derived from the multicanonical and parallel tempering methods.     

The simulation of spin glass model of neural networks by the Wang-Landau algorithm

In the present work, the Hopfield neural network model with infinite-range interactions was simulated by using the Wang-Landau algorithm. All simulations and measurements were done in the replica symmetric and broken regions of the model with discrete ?1 values of random variables. The physical quantities such as energy density, specific heat were evaluated at all temperatures. Our results are in good agreement with the replica symmetric mean field solutions.     

ExHuME 1.3: A Monte Carlo event generator for exclusive diffraction

We have written the Exclusive Hadronic Monte Carlo Event (ExHuME) generator. ExHuME is based around the perturbative QCD calculation of Khoze, Martin and Ryskin of the process pp→p+X+p, where X is a centrally produced colour singlet system.

Program summary

Title of program:ExHuMECatalogue identifier:ADYA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYA_v1_0Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:NoneProgramming language used:C++, some FORTRANComputer:Any computer with UNIX capability. Users should refer to the README file distributed with the source code for further detailsOperating system:Linux, Mac OS XNo. of lines in distributed program, including test data, etc.:111 145No. of bytes in distributed program, including test data, etc.: 791 085Distribution format:tar.gzRAM:60 MBExternal routines/libraries:LHAPDF [http://durpdg.dur.ac.uk/lhapdf/], CLHEP v1.8 or v1.9 [L. Lönnblad, Comput. Phys. Comm. 84 (1994) 307; http://wwwinfo.cern.ch/asd/lhc++/clhep/]Subprograms used:Pythia [T. Sjostrand et al., Comput. Phys. Comm. 135 (2001) 238], HDECAY [A. Djouadi, J. Kalinowski, M. Spira, HDECAY: A program for Higgs boson decays in the standard model and its supersymmetric extension, Comput. Phys. Comm. 108 (1998) 56, hep-ph/9704448]. Both are distributed with the source codeNature of problem:Central exclusive production offers the opportunity to study particle production in a uniquely clean environment for a hadron collider. This program implements the KMR model [V.A. Khoze, A.D. Martin, M.G. Ryskin, Prospects for New Physics observations in diffractive processes at the LHC and Tevatron, Eur. Phys. J. C 23 (2002) 311, hep-ph/0111078], which is the only fully perturbative model of exclusive production.Solution method:Monte Carlo techniques are used to produce the central exclusive parton level system. Pythia routines are then used to develop a realistic hadronic system.Restrictions:The program is, at present, limited to Higgs, di-gluon and di-quark production. However, in principle it is not difficult to include more.Running time:Approximately 10 minutes for 10000 Higgs events on an Apple 1 GHz G4 PowerPC.     

A Wang-Landau study of the phase transitions in a flexible homopolymer

Using Wang-Landau sampling we study the characteristic behavior of a flexible homopolymer (off-lattice) for chain lengths up to N=300. The Hamiltonian consists of a Lennard-Jones potential between all monomers, and an additional FENE potential between bonded monomers. From the resultant density of states, we calculate thermodynamic properties for a wide range of temperatures, including low temperatures that are inaccessible to traditional Monte Carlo algorithms. Peaks in the specific heat and radius of gyration indicate the coil-globule and solid-liquid transitions. With a careful implementation of the algorithm, we find no evidence of a liquid-liquid transition.     

Evolutionary fitting methods for the extraction of mass spectra in lattice field theory

We present an application of evolutionary algorithms to the curve-fitting problems commonly encountered when trying to extract particle masses from correlators in lattice QCD. Harnessing the flexibility of evolutionary methods in global optimization allows us to dynamically adapt the number of states to be fitted along with their energies so as to minimize overall χ²/(d.o.f.), leading to a promising new way of extracting the mass spectrum from measured correlation functions.     

OOF3D: An image-based finite element solver for materials science

Recent advances in experimental techniques (micro-CT scans, automated serial sectioning, electron back-scatter diffraction, and synchrotron radiation X-rays) have made it possible to characterize the full, three-dimensional structure of real materials. Such new experimental techniques have created a need for software tools that can model the response of these materials under various kinds of loads. OOF (Object Oriented Finite Elements) is a desktop software application for studying the relationship between the microstructure of a material and its overall mechanical, dielectric, or thermal properties using finite element models based on real or simulated micrographs. OOF provides methods for segmenting images, creating meshes of complex geometries, solving PDE's using finite element models, and visualizing 3D results. We discuss the challenges involved in implementing OOF in 3D and create a finite element mesh of trabecular bone as an illustrative example.     

Phase diagram of a two-dimensional large-Q Potts model in an external field

We use a two-dimensional Wang-Landau sampling algorithm to map out the phase diagram of a Q-state Potts model with Q?10 in an external field H that couples to one state. Finite-size scaling analyses show that for large Q the first-order phase transition point at H=0 is in fact a triple point at which three first-order phase transition lines meet. One such line is restricted to H=0; another line has H?0. The third line, which starts at the H=0 triple point, ends at a critical point (Tc,Hc) which needs to be located in a two-dimensional parameter space. The critical field Hc(Q) is positive and decreases with decreasing Q, which is in qualitative agreement with previous predictions.