LETTER FROM THE EDITOR

Dear Readers of Virtual Physics,

This month sees the introduction of several new changes to the format and structure of Virtual Physics. Firstly, JavaScript is now being incorporated into the formatting of each edition beginning with this one. In order for you to view the content written with JavaScript language, it will be necessary to use Netscape 3.0 or newer. The email version will continue to remain in the standard format. We also thank Monika Trochimczuk , Istvan Vajda and Laszlo Farkas for their assistance with preparation of Virtual Physics.

Personally, I would like to introduce a WEB site I found recently, which offers some excellent material which impacts on many areas of contemporary Physics:

The Center for Polymer Studies, located in the Physics Department at Boston University has an excellent Web site which deserves attention. The site describes the Center's research focus, and most importantly offers extensive online information and downloadable simulation software. This site is a must-see for those interested in: polymer interactions, randomness, percolation and general fractal systems, as well as the experimental methods used to study such systems.

Importantly there is also extensive interactive information concerning current and planned Research Projects which are suitable for Undergraduate and High School Science Education. For example, one project investigates: "The Random Universe: An Interdisciplinary Approach to Investigating Patterns in Nature.

For those of you planning on teaching a course on Fractals in Physics or even introducing your students to stochastic behaviour and non-linear systems then you should take a look at a forthcoming book currently available in a pilot edition called: Fractals in Science: An Introductory Course. This is planned for release in the Summer of 1997, and includes course material, software and a guided hands-on introduction to natural fractals including such topics as: fractal behaviour in nerve cells, forest fires, mountain ranges, and cloud shapes.

There is also a regular newsletter called the: "Patterns In Nature Newsletter" aimed at informing High School and early Undergraduate students and their teachers about new experiments, and teaching tools and materials. The Winter 1996 "Patterns in Nature Newsletter" is available online.

There is also substantial cutting-edge Research progress reports which detail current work carried out by the researchers from the Center for Polymer Studies. For example, one report concerns the Theories of Diffusion -Limited Aggregation, while another explores Long-Range Correlations in DNA Base Pair Sequences.

For those students intersted in getting involved with fractals, and the analysis of random phenomena - there are patterns in Nature Student Activity Guides. These demonstrate an actual experiment, and include fully stated aims, hypotheses, methods, results and conclusions. Typical examples include: the generation of silver fractals using a straightforward chemical reaction which produces ramified, branching fractal shapes. Another example, investigates the fractal patterns in leaves.

A stunning set of Java-Powered Simulations are also available for viewing, and selected simulations can be downloaded for both Windows or Macintosh platforms. These include: one and two dimensional random walks, molecular motion, a sandpile simulation, diffusion-limited aggregation and a polymer simulation.

Check out the set of Simulation Software for visualizing aspects of Molecular Dynamics. Note however that a UNIX operating environment is required.

Finally, I urge all of you to take a look at the following FTP directory which provides downloadable simulation software for both the Windows and Macintosh platforms. Many of the programs (i.e. OGAFSAMP.ZIP) offer an excellent, graphic introduction to forest fires simulations, and generation of diffusion-limited aggregates; while another (DP.ZIP) provides software tools to calculate the mass fractal dimension of a two dimensional digital object in the plane.

ftp://erato.bu.edu/pub/OGAF/pc WINDOWS SOFTWARE
ftp://erato.bu.edu/pub/OGAF/mac MACINTOSH SOFTWARE

Sincerely,
Cameron L. Jones
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Precise Pulse Calibration for Relaxation Experiments

Krzysztof Wróblewskia and Andrzej Ejchartb

aUniversity of Pennsylvania School of Medicine, Department of Biochemistry and Biophysics, Philadelphia, PA 19104-6089, U.S.A.
bLehrstuhl für Struktur und Chemie der Biopolymere, Universität Bayreuth, D-8580 Bayreuth, Germany.

(Submitted to Virtual Physics: January 2, 1997)

Abstract:

Proper calibration of 180o pulse is important for the fast and precise measurement of spin-lattice relaxation times. A fast and convenient method for 180o pulse calibration and the complete Fortran program necessary for calculations is presented.

Introduction

The traditional two-pulse inversion-recovery sequence (IRFT) [1,2] and its later modifications, the fast inversion recovery sequence (FIRFT) [3], modified inversion-recovery sequence (MFIRFT) [4] and extended inversion-recovery (EIRFT) [5] are still the most widely used for spin-lattice relaxation time measurements. The intensity of the NMR signal under steady- state condition in any two-pulse sequence (-ti--Di), with no consideration given to pulse inhomogeneity, offset effects, and refocusing effects, is described by the equation (1) [4]. The intensities Si are a function of the evolution time ti but are also dependent upon other sequence parameters: relaxation delay, Di, perturbing pulse , and observing pulse as well as on the signal intensity S0 corresponding to magnetization in thermodynamic equilibrium. Data reduction commonly involves a iterative fit of the parameter values which correspond to a minimum of function E: where N is the number of data points corresponding to different ti values. The calculated intensities Si should differ as little as possible from the experimental ones Ii. Comparison of equations (1) and (2) indicates that four unknown parameters , , S0, and T1 need to be calculated. While S0 and T1 depend on the particular sample, and are calibrated before the experiment. Even though the precision of pulse calibration (especially of ) may not directly affect the precision of T1 measurements because these values will be calculated anyway, [5,6] these values should be as close as possible to 90o and 180o respectively, to guarantee a maximal signal to noise ratio. We present here a fast and convenient method of pulse calibration. This method reduces experimental errors related to phase distortion by utilizing power spectra.

The Method

A power spectrum is defined as a sum of squares of absorption and dispersion mode components. The advantages of power spectra over the absorption ones have been already demonstrated in the case of the line-shape analysis [7]. Power spectra retain their Lorentzian shape and are not affected by phase distortions. The area of a signal is proportional to the square of number of spins.
The inhomogeneity parameter I has been defined by Hansum et al. [8] as shown in the equation (3) where A(3/2) and A(/2) are the amplitudes of the signals obtained for the pulses 3/2 and /2 respectively. The sinusoidal dependence of the amplitudes of calibrated signals [8] is used to calculate 180o pulse width as shown in the Appendix. The least-squares fit algorithm described by MacDonald [9] has been used in this approach. The Fortran program that can be used to calculate a 180o pulse width is presented in the appendix.

Results

An ethylene glycol sample was used to calibrate 13C 90o pulse on the Bruker AM-500 machine at the 13C resonance frequency 125.75 MHz. 32 noise-decoupled spectra were obtained covering flip angle range of 0-360o. The spin-lattice relaxation time of the ethylene glycol at room temperature is ca. 2s. The total acquisition time is less than 10min. The pulse width was changed in increments of 2s over a range from 2s to 64s. The phase of absorption spectra was corrected for the first spectrum and the same values were used for subsequent processed data. The power spectra were also calculated. Changes of signal amplitudes and normalized power amplitudes presented in the Table 1 are shown in the Figure 1. Amplitudes of power spectra were used to calculate 90o pulse using the program SINMAC. The results are shown in the Table 2. The values of 180o pulse and 90o amplitude can be easily estimated from the graph in Fig.1. Calculated values obtained from the program SINMAC do not differ much from the estimated values.

[ Table 1 ]

Table 1. Amplitudes of absorption and power spectra obtained from the ethylene glycol sample.

[ Table 2 ]

Table 2. Calculated values of 180o pulse, 90o amplitude, and pulse inhomogeneity.

Appendix

       PROGRAM SINMAC
       REAL C(100),S(100),F(100),K(100),P(100)
       COMMON C,S,F,K,P,N,IFLAG,C1,C2,C3,C4,C5,C6,R1,R2,R3
       WRITE(*,1001)
       READ(*,*)N
       WRITE(*,1002)
       READ(*,*) A0
       WRITE(*,1003)
       READ(*,*)B0
       WRITE(*,1004)
       READ(*,*)T0
       WRITE(*,1005)
       READ(*,*)(C(I),I=1,N)
       WRITE(*,1006)
       READ(*,*)(S(I),I=1,N)
1001   FORMAT(2X,'Number of experimental points           = '\)
1002   FORMAT(2X,'Estimated amplitude of signal at 90deg. = '\)
1003   FORMAT(2X,'Correction parameter B (usually B=0)    = '\)
1004   FORMAT(2X,'Estimated duration of 180 deg. pulse    = '\)
1005   FORMAT(2X,'Pulse durations')
1006   FORMAT(2X,'Amplitudes')
C Estimated values
       E1=0.1
       E2=1.0E-4
       E3=0.1
       DA=0.0
       DB=0.0
       DT=0.0
       A=A0
       B=B0
       T=T0
       L=1
C Iteration
200    CONTINUE
       A=A+DA
       B=B+DB
       T=T+DT
       CALL POWER(A,B,T)
       D=2.0*C2*C3*C5-C3*C3*C4-C1*C5*C5+C1*C4*C6-C2*C2*C6
       D=1.0/D
       DA=D*(C3*C5*R2-C3*C4*R3+C2*C5*R3-C2*C6*R2+R1*(C4*C6-C5*C5))
       DB=D*(C2*C3*R3+R2*(C1*C6-C3*C3)+C3*C5*R1-C1*C5*R3-C2*C6*R1)
       DT=D*(C2*C5*R1-C3*C4*R1+C2*C3*R2-C1*C5*R2+R3*(C1*C4-C2*C2))
       IF(DA.GT.E1.OR.DB.GT.E2.OR.DT.GT.E3)GOTO 200
C Calculated values
       SI=0.0
       DO 10 I=1,N
       SI=SI+P(I)**2
10     CONTINUE
       SI=SQRT(SI/FLOAT(N-3))
       SA=SI*SQRT(D*(C4*C6-C5*C5))
       SB=SI*SQRT(D*(C1*C6-C3*C3))
       T9=SI*SQRT(D*(C1*C4-C2*C2))
C Correlation parameters
       AT=C3/SQRT(C1*C6)
       AB=C2/SQRT(C1*C4)
       BT=C5/SQRT(C4*C6)
       A=A*A
       PN=(1.0+(1.0-B*2.25*T**2))
       PN=0.5*(PN/(1.0-B*0.25*T*T))
       PNPLS=ACOS(PN)*180.0/ACOS(-1.0)
C Results
       WRITE(*,2001)
       WRITE(*,2002)A,SA,AT
       WRITE(*,2003)B,SB,BT
       WRITE(*,2004)T,T9,BT
       WRITE(*,2005)PNPLS
       STOP
2001   FORMAT(//2X,'Results of calculation:')
2002   FORMAT(2X,'A,SA,AT: ',3F16.8)
2003   FORMAT(2X,'B,SB,AB: ',3F16.8)
2004   FORMAT(2X,'T,ST,BT: ',3F16.8)
2005   FORMAT(2X,'PN:      ',F16.8)
       END

       SUBROUTINE POWER(A,B,T)
       REAL C(100),S(100),P(100),K(100),F(100)
       COMMON C,S,F,K,P,N,IFLAG,C1,C2,C3,C4,C5,C6,R1,R2,R3
       C1=0.0
       C2=0.0
       C3=0.0
       C4=0.0
       C5=0.0
       C6=0.0
       R1=0.0
       R2=0.0
       R3=0.0
       DO 10 I=1,N
       AL=ACOS(-1.0)*C(I)/T
       K(I)=1.0-B*C(I)**2
       F(I)=(A*K(I)*SIN(AL))**2
       FB=-2.0*A*A*C(I)**2*K(I)*(SIN(AL))**2
       FT=-ACOS(-1.0)*C(I)*(A*K(I)/T)**2*SIN(2.0*AL)
       FA=2.0*A*K(I)**2*(SIN(AL))**2
       C1=C1+FA*FA
       C2=C2+FA*FB
       C3=C3+FA*FT
       C4=C4+FB*FB
       C5=C5+FB*FT
       C6=C6*FT*FT
       P(I)=S(I)-F(I)
       R1=R1+P(I)*FA
       R2=R2+P(I)*FB
       R3=R3+P(I)*FT
10     CONTINUE
       RETURN
       END

_____________________________

Diffusion and localization in chaotic billiards

Fausto Borgonovi, borgonov@galileo.bs.unicatt.it a,c,d, Giulio Casatib,c,e Baowen Lif,g

aDipartimento di Matematica, Universita Cattolica, via Trieste 17, 25121 Brescia, Italy, bUniversita di Milano, sede di Como, Via Lucini 3, Como, Italy, c Istituto Nazionale di Fisica della Materia, Unita di Milano, via Celoria 16, 22100 Milano, Italy, fDepartment of Physics and Centre for Nonlinear and Complex Systems, Hong Kong Baptist University, Hong Kong, gCenter for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, 2000 Maribor, Slovenia

A PostScript version is available at LANL server, http://xxx.lanl.gov/abs/cond-mat/9610158

One of the main modifications that quantum mechanics introduces in our classical picture of deterministic chaos is "quantum dynamical localization" which results e.g. in the suppression of chaotic diffusive-like process which may take place in systems under external periodic perturbations. This phenomenon, first pointed out in the model of quantum kicked rotator, is now firmly established and observed in several laboratory experiments.
For conservative Hamiltonian systems the question of localization is much less investigated. The situation here is much more intriguing: from one hand, in a conservative system, one may argue that there is always localization due to the finite number of unperturbed basis states effectively coupled by the perturbation; on the other hand a large amount of numerical evidence indicates that quantization of classically chaotic systems leads to results which appear in agreement with the predictions of Random Matrix Theory (RMT).
Recently the problem of localization in conservative systems has been explicitely investigated. In particular, on the base of Wigner band random matrix model, condi- tions for localization were explicitely given together with the relation between localization and level spacing distribution.
Billiards are very important models in the study of conservative dynamical systems since they provide clear mathematical examples of classical chaos and their quantum properties have been extensively studied theoretically and experimentally. Moreover they are becoming increasingly relevant for the study of optical processes in microcavities which may lead to possible applications such as the design of novel microlasers or other optical devices.

_____________________________

Lagrange Mechanics in Spaces with Curvature and Torsion

H. Kleinert and A. Pelster

Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

Abstract:

We present a new and simpler version of the recently discovered action principle for the classical motion of a point particle in spaces with curvature and torsion. Although the action involves only the metric, torsion enters the equation of motion via a noncommutativity of time derivative and variation. As a consequence, classical spinless point particles move along autoparallels, not geodesics as commonly believed.