Vol.: II Issue: 3 - Physical Research Laboratoryisamp/isamp_files/newsletter/Vol2_issue3.pdf · Vol.: II Issue: 3 August 21, 2006 CONTENTS Editorial 1 Abstracts 2-3 News 4 The ADAS

Vol.: II Issue: 3August 21, 2006

CONTENTS

Editorial 1 Abstracts 2-3 News 4 The ADAS software 5-6 Dimensional Analysis 7-12

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Creation of atomic coherence, which leads to interest-ing phenomena in quantum optics, till date relies on mono-chromatic lasers. This may change in future, the news ar-ticle reports theoretical studies demonstrating the possibil-ity of producing multicolour atomic coherence.

There are two abstracts on the studies of oxygen mol-ecule. These perhaps compensate for the hazardous cock-tail of molecules covered in the previous issue of the news-letter: carbondioxide and nitrogen dioxide. We then have anabstract on the trace element content in antidiabetic medici-nal plants. However this issue is not free of lethal systems.There are two abstracts on the studies of positronium, asystem containing anti-matter. There is an abstract of a theo-retical paper demonstrating the precision of relativisticcoupled-cluster calculation in atoms, and another abstracton the possibility of generating single attosecond extremeultraviolet pulses.

The issue ends with a very interesting and useful ar-ticle on the use of dimensional analysis. The authors high-lights the dimensionless quantities we encounter in physicsand extends the Szirtes algorithm to physics problems.

K.P. Subramanian Dilip AngomEDITOR, ISAMP Newsletter Guest Editor

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mailto:[email protected]

http://www.prl.res.in/~isamp

Vol. II I S A M P News Letter Issue: 3

2

ABSTRAABSTRAABSTRAABSTRAABSTRACTSCTSCTSCTSCTS

Abstract#1Abstract#1Abstract#1Abstract#1Abstract#1

PPPPParararararticticticticticle induced X-rale induced X-rale induced X-rale induced X-rale induced X-ray emissiony emissiony emissiony emissiony emissionstudies of anti-diabeticstudies of anti-diabeticstudies of anti-diabeticstudies of anti-diabeticstudies of anti-diabetic

medicinal plantsmedicinal plantsmedicinal plantsmedicinal plantsmedicinal plants

DDDDD.K..K..K..K..K. Ra Ra Ra Ra Rayyyyy1,1,1,1,1, 3 3 3 3 3,,,,, P P P P P.K..K..K..K..K. Na Na Na Na Nayyyyyakakakakak22222*,*,*,*,*, S.R. S.R. S.R. S.R. S.R. P P P P Pandaandaandaandaanda11111,,,,,

TTTTT.R..R..R..R..R. Rautra Rautra Rautra Rautra Rautrayyyyy33333,,,,, VVVVV..... VijaVijaVijaVijaVijayyyyyananananan33333,,,,,

C.C. ChristopherC.C. ChristopherC.C. ChristopherC.C. ChristopherC.C. Christopher22222 and S. Jena and S. Jena and S. Jena and S. Jena and S. Jena11111

1. Department of Chemistry, Utkal University, Vani Vihar,Bhubaneswar-751004, India

2. Department of Basic Sciences, A.K. College of Engineer-ing, Krishnankoil-626190, India

3. Institute of Physics, Bhubaneswar-751 005, India

*Email:*Email:*Email:*Email:*Email: [email protected]

Selected number of anti-diabetic medicinal plantleaves has been characterized by acceleratorbased particle-induced X-ray emission (PIXE)technique. Validity of the technique was assuredby analyzing certified plant reference materials(CRMs). A large number of trace elements likeTi, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Rb, Sr and Pbare found to be present in these studied leafsamples with variable proportions. The concen-trations of elements like K and Ca are quanti-fied in percentage level whereas other elementsare found to be in parts per million levels. Amongthe studied samples, the leaves of Methi arefound to be containing maximum amount oftrace elements.

International Journal of PIXE, 16, Nos. 1-2 (2006) 47


Determination of the nDetermination of the nDetermination of the nDetermination of the nDetermination of the nucucucucuclearlearlearlearlearquadrupole moment of quadrupole moment of quadrupole moment of quadrupole moment of quadrupole moment of 8787878787SrSrSrSrSr

BijaBijaBijaBijaBijayyyyya Ka Ka Ka Ka Kumar Sahooumar Sahooumar Sahooumar Sahooumar Sahoo

Atomphysik, Gesellschaft für Schwerionenforschun gmbH,Planckstraße 1, 64291 Darmstadt, Germany

Email:Email:Email:Email:Email: [email protected]

A new nuclear quadrupole moment,Q=0.305(2) b, of 87Sr is determined by combin-ing precise measurement of the electric qua-drupole hyperfine structure constant of the4d 2D5/2 state of its ion with calculation. This is asignificant improvement over the previous val-ues 0.335(20), 0.327(24), and 0.323(20) b.

Relativistic coupled-cluster theory is employedin the calculations and electron correlationeffects are included using the single, double, andan important subset of triple excitations. Themagnetic dipole and electric quadrupole hyper-fine structure constants of a few low-lying statesare calculated to a high accuracy. The role ofdifferent electron correlation effects in the4d 2D5/2 state is investigated.

Physical Review A73, 062501 (2006)DOI: 10.1103/Phys Rev A.73.06250


ExcExcExcExcExchanghanghanghanghange on ionisation ine on ionisation ine on ionisation ine on ionisation ine on ionisation inPs-atom scatteringPs-atom scatteringPs-atom scatteringPs-atom scatteringPs-atom scattering

Hasi RaHasi RaHasi RaHasi RaHasi Rayyyyy

Department of Physics, Indian Institute of TechnologyRoorkee, Roorkee 247887, Uttranchal, India

Email:Email:Email:Email:Email: [email protected],[email protected]

An improvement of the Coulomb-Born approxi-mation is made including the effect of exchangein positronium (Ps) and atom scattering. Theexchange amplitude for the simplest Ps-H sys-tem is a nine-dimensional integral. We devel-oped a methodology of evaluating the Coulomb-Born-Oppenheimer amplitude for Ps-atom scat-tering and apply it to calculate the Ps-ionisationcross sections in Ps-H collisions. We presentthe first and preliminary results using the presentmythology.

Status: Europhys. Lett. 73, 21 (2006)


Coulomb-Born-OppenheimerCoulomb-Born-OppenheimerCoulomb-Born-OppenheimerCoulomb-Born-OppenheimerCoulomb-Born-Oppenheimerapprapprapprapprapproooooximation in Ps-H scatteringximation in Ps-H scatteringximation in Ps-H scatteringximation in Ps-H scatteringximation in Ps-H scattering

Hasi RaHasi RaHasi RaHasi RaHasi Rayyyyy

Department of Physics, Indian Institute of TechnologyRoorkee, Roorkee 247887, Uttranchal, India

Email:Email:Email:Email:Email: [email protected],[email protected]

To improve the Coulomb-Born approximation(CBA) theory of ionisation in positronium (Ps)and atom scattering, the effect of exchange isintroduced. The nine-dimensional exchangeamplitude for ionisation of Ps in Ps-H scatteringis reduced to a two-dimensional integral usingthe present Coulomb-Born-Oppenheimer ap-






3

proximation (CBOA). The methodology is ex-tremely useful to evaluate ionisation parametersfor different target systems and different typesof ionisation processes. It is then applied toevaluate the Ps-ionisation cross section and toestimate the effect of exchange on Ps-ionisationin Ps-H system. We establish the importance ofexchange at lower energy region.


Single attosecond pulses frSingle attosecond pulses frSingle attosecond pulses frSingle attosecond pulses frSingle attosecond pulses from highom highom highom highom highharmonics driven bharmonics driven bharmonics driven bharmonics driven bharmonics driven byyyyy

self-compressed filamentsself-compressed filamentsself-compressed filamentsself-compressed filamentsself-compressed filaments

Himadri S.Himadri S.Himadri S.Himadri S.Himadri S. Chakrabor Chakrabor Chakrabor Chakrabor Chakrabortytytytyty11111� and Arnaud and Arnaud and Arnaud and Arnaud and ArnaudCouairCouairCouairCouairCouairononononon22222

1 Department of Chemistry & Physics, Northwest MissouriState University, Maryville, MO 64468; Mette B. Gaarde,

Department of Physics & Astronomy, Louisiana StateUniversity, Baton Rouge, Louisiana 70803

2 Centre de Physique Theorique, Ecole Polytechnique, CNRSUMR 7644, F-91128, Palaiseau, France

� Email:Email:Email:Email:Email: [email protected]

We show that isolated sub-femtosecond, ex-treme ultraviolet (XUV) pulses can be gener-ated via harmonic generation in argon, by few-cycle infrared pulses formed throughfilamentation-induced self-compression in neon.Our calculations show that the time structure ofthe XUV pulses depends sensitively on both theamplitude and the phase modulation that areinduced in the driving pulse during the self-com-pression process.

Status: Accepted in the Optics Letters (2006)


On the presence of On the presence of On the presence of On the presence of On the presence of 44444ΣΣΣΣΣuuuuu−−−−− resonance inresonance inresonance inresonance inresonance in

dissociative electrdissociative electrdissociative electrdissociative electrdissociative electron attacon attacon attacon attacon attachment to Ohment to Ohment to Ohment to Ohment to O22222

VVVVVaibhaaibhaaibhaaibhaaibhav S.v S.v S.v S.v S. Prabhudesai Prabhudesai Prabhudesai Prabhudesai Prabhudesai�,,,,, Dhananja Dhananja Dhananja Dhananja Dhananjay Nandiy Nandiy Nandiy Nandiy Nandiand E. Krishnakumarand E. Krishnakumarand E. Krishnakumarand E. Krishnakumarand E. Krishnakumar�

Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai 400005, India

Email:Email:Email:Email:Email: � [email protected]� [email protected]

The evidence for the presence of the 4Σu

− reso-nance in dissociative electron attachment to O

2

is obtained for the first time from the angulardistribution measurement of O− ions in the en-tire 2π angles using a novel experimental tech-nique employing the Velocity Map Imaging. Thisobservation, while settling the question of thepresence of this state observed in inelastic

vibrational excitation of O2, calls for fresh calcu-

lations on the lifetime of the resonance. It mayalso impact the interpretations of the negativeion formation from O

2 in clusters and in con-

densed state.

J. Phys. B: At. Mol. Opt. Phys. 39, No. 14 (28 July 2006) L277-283.


Excitation of AExcitation of AExcitation of AExcitation of AExcitation of Autoionization States ofutoionization States ofutoionization States ofutoionization States ofutoionization States ofOOOOO22222 b b b b by using High-Ory using High-Ory using High-Ory using High-Ory using High-Order Harmonicsder Harmonicsder Harmonicsder Harmonicsder Harmonics

KKKKKyung yung yung yung yung TTTTTaec Kimaec Kimaec Kimaec Kimaec Kim11111,,,,, K K K K Kyung Sik Kangyung Sik Kangyung Sik Kangyung Sik Kangyung Sik Kang11111,,,,, Mi Na P Mi Na P Mi Na P Mi Na P Mi Na Parkarkarkarkark11111,,,,,TTTTTaaaaayyyyyyyyyyab Imranab Imranab Imranab Imranab Imran11111,,,,, Changjun Zhu Changjun Zhu Changjun Zhu Changjun Zhu Changjun Zhu11111,,,,, Chang HeeChang HeeChang HeeChang HeeChang Hee

NamNamNamNamNam11111�, E. Krishnakumar, E. Krishnakumar, E. Krishnakumar, E. Krishnakumar, E. Krishnakumar22222� and G. Umesh and G. Umesh and G. Umesh and G. Umesh and G. Umesh33333�

1 Coherent X-ray Research Center, Department of Physics,Korean Advanced Institute for Science and Technology,

Daejeon, Republic of Korea2 Tata Institute of Fundamental Research, Homi Bhabha

Road, Mumbai 400005, India3 Department of Physics, NITK, Srinivasanagar,

Surathkal 574157, India

Email: Email: Email: Email: Email: � [email protected]� [email protected]

� [email protected]

Photoionization of oxygen molecules by high-order harmonics was investigated. High-orderharmonics of a Ti:sapphire laser produced in aKr gas cell were used to excite autoionizationstates of O2. since the high-order harmonicsource used contains several harmonic orders,the resulting photoelectron spectrum alsoshowed multiple peaks coming from differentorders of harmonics. A subtraction method us-ing known photoionization cross-sections wasemployed to separate out individual contribu-tions from the harmonics. The photoelectronspectrum from the 11th harmonic shows a cleancontribution from an autoionizing state.

Journal of Korean Physical Society, 49, No. 1, July 2006, pp309-313.

XVI NCAMP Update

Registration : 1st September,2006Abstract Submission : 1st September,2006

Contact Address :PrPrPrPrProfofofofof..... E Krishnakumar E Krishnakumar E Krishnakumar E Krishnakumar E KrishnakumarConvener,XVI-NCAMP

Tata Institute of Fundamental ResearchHomi Bhabha Road,

MUMBAI - 400 005 IndiaEmail : [email protected]

Website : www.tifr.res.in/~xvincampConf. Sec. : Vinay Bhonsle ([email protected])







http://www.tifr.res.in/~xvincamp



4

N E N E N E N E N E W SW SW SW SW S

Multicolored AtomicMulticolored AtomicMulticolored AtomicMulticolored AtomicMulticolored AtomicCoherenceCoherenceCoherenceCoherenceCoherence

HarHarHarHarHarshashashashashawarwarwarwarwardhan dhan dhan dhan dhan WWWWWanareanareanareanareanare

Department of Physics, Indian Institute of Technology,Kanpur, India


Atomic coherence has revolutionized the areaof quantum optics, with dramatic effects like ren-dering an atomic gas medium transparent, la-ser action without population inversion, super-slow and superluminal pulse propagation andmany more such effects [1]. At the heart of allthese lies creation of coherence between vari-ous atomic states. Such coherence is gener-

-30 -20 -10 0 10 20 30-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2 Excited state population

Atomic coherence

Laser detuning (units of decay)

cies are harmonically modulated. The spectrumof such a field contains a comb of frequenciesand when two such combs of fields matchacross atomic transitions it leads to multicoloredcoherent population trapping state, as seen inthe figure. The usual coherent population trap-ping state (denoted by dashed lines) involvesmatching of two monochromatic fields in Ramanresonance across atomic states. The multicol-ored counterpart, on the other hand, involvessimultaneous matching of Raman resonance forthe whole comb of frequencies. Such matchingof colored fields leads to occurrence of largeatomic coherence accompanied with reducedexcited state population at a series of equidis-tant frequencies. We have also shown coherentcontrol wherein we can selectively switch on/offcoherent population trapping at specific frequen-cies by judicious choice of modulation param-

Coherent population trapping occurs simultaneously at aseries of frequencies (integer multiples of the modulationfrequency of 10 along the x axis) wherein there is a dip inthe excited state population accompanied by enhancedatomic coherence

above and much more, particularly in laser cool-ing wherein the reduced spontaneous emissionassociated with this state leads to reduced re-heating arising from random emission of pho-tons. Until now all these atomic coherence ef-fects have relied heavily on the monochromaticnature of light. These coherent effects degradesignificantly as the frequency content of the lightfields is increased because it destroys the defi-nite phase relationship between atomic states.We have recently demonstrated the possibilityof multicolored atomic coherence [3], where weobtain atomic coherence for excitation with aspecifically structured field. We propose excita-tion of the atom by light fields whose frequen-

ties from multicolored transparency and lasercooling to possible entanglement of light fieldsacross a series of distinct frequencies.

RefRefRefRefReferences:erences:erences:erences:erences:

[1] M. O. Scully and M. S. Zubairy, "QuantumOptics" (Cambridge University Press, Cam-bridge, England, 1997) and referencestherein.

[2] E. Arimondo, in "Progress in Optics", ed-ited by E. Wolf (Elsevier, Amsterdam, 1996),Vol. 3535353535, p. 257.

[3] Harshawardhan Wanare, Physical ReviewLetters, 9696969696, 183601 (2006).

ated by ingeniouscoupling of differentatomic levels by afew extremely stablesingle frequencylight fields. The co-herent populationtrapping state is onesuch state which re-sults from an inter-play of light andatom dynamics re-sulting in maximumatomic coherenceand significantly re-duced spontaneousemission [2]. Itforms the basis ofmost of the amazingeffects mentioned

eters. Moreover, wedeveloped a math-ematical scheme thatallows us to calculatethe response of suchhighly coupled interac-tion. Specifically, wecalculate non-perturbative responseof atom-field interac-tion in the strongmodulation regimewherein we use thematrix-continued frac-tion technique to nu-merically calculatet w o - d i m e n s i o n a ltridiagonal matrix rela-tions. This result opensup immense possibili-



5

Atomic data and analAtomic data and analAtomic data and analAtomic data and analAtomic data and analysisysisysisysisysisstructure (ADstructure (ADstructure (ADstructure (ADstructure (ADAS) pacAS) pacAS) pacAS) pacAS) packakakakakagggggeeeee

Hem Chandra JoshiHem Chandra JoshiHem Chandra JoshiHem Chandra JoshiHem Chandra Joshi

Institute for Plasma Research, Bhat, Gandhinagar-382428,India


Atomic Data and Analysis Structure(ADAS) [1] is a comprehensive package thatincludes atomic database for calculating variousproperties of atomic and plasma emission. Ithas wide ranging applications fromastrophysical and thermonuclear plasmas toindustrial applications of plasma. It helps in theinterpretation of emission of plasmas andsupports various models.

ADAS was originated by H. Summers ofUniversity of Strathclde (Glasgow)/ and JETJoint Undertaking. Initially it was based on localmainframe IBM PC and access was restrictedto JET site. In the ear ly nineties manylaboratories got interested in ADAS and it wasagreed to form a UNIX based ADAS with IDLas the graphical user interface. Now it is knownas IDL-ADAS and can be maintained over theInternet.

PrinciplePrinciplePrinciplePrinciplePrinciple

Electrons impart their kinetic energy to animpurity ion in plasma which either reaches toits excited state or gets further ionized. Thisprocess then results in the emission of a photon.Symbolically it can be depicted as

A + e ⎯→ A ∗

A ∗ ⎯→ A + hν

A + e ⎯→ A+∗ + e + e

A+∗ ⎯→ A + + hν

where A+ is the next ionization state of A.

At high electron density, because ofcollisional plasma equilibrium, electron and ionpopulations follow Boltznman distribution andthe situation is known as local thermalequilibrium (LTE). On the other hand, for lowelectron density it can be described by coronamodel (balance between electron impactexcitation and spontaneous decay). In the

intermediate density limit a collisional radiativemodel (a model which balances all excitationand de-excitation processes for an atom) isconsidered. The coronal model had been thebasis of description of plasma in many fusiondevices in the past. However, the progress madein ignition and high-density plasma suggeststhat a better description can be achieved byusing collisional-radiative model [2]. ADAS iscentered on collisional-radiative theory in whichvarious relaxation times; metastable states andsecondary collisional times are taken intoconsideration.

Structure of ADStructure of ADStructure of ADStructure of ADStructure of ADASASASASAS

It has four main basic components:

1. A graphical interactive system

2. A very large database of fundamentaland derived atomic data.

3. Subroutine libraries

4. Capability of calculating fundamentalatomic data e.g. transition probabilities

ADAS data and ADAS generated data areused in many plasma codes e.g. STRAHL.B2-EIRENE, DIVIMP, SANCO etc.

Applications of ADApplications of ADApplications of ADApplications of ADApplications of ADASASASASAS

ADAS can be used for the modeling ofvarious atomic and plasma parameters e.g.photon emissivity, charge exchange cross-section, radiated power etc. Moreover, it can beused for spectral line fitting and Zeeman/Paschen-Back and Stark splitting analysis. Adetailed overview of applications can be foundat ADAS site [1]. Various ADAS routines are oftwo types-interrogative on the ADAS databaseand those which execute atomic modeling. Ininterrogative codes, the objective is to allowgraphical display and cubic spline interpolationis performed on the source data from data inthe database. On the other hand in atomicmodeling codes output data sets as well asnormal tabular output are created. Anotherimportant aspect of ADAS is error consideration.In any theoretical model the calculated valuesare under certain approximations. ADAS suite



6

has the ability to route text and graphical outputafter execution. In table 1, a list of the series ofADAS codes for various purposes is given.

SeriesSeriesSeriesSeriesSeries Contents Contents Contents Contents Contents

ADADADADADAS1AS1AS1AS1AS1 Entry and validation offundamental atomic data

ADADADADADAS2AS2AS2AS2AS2 Excited state population in aplasma

AD AD AD AD ADAS3AS3AS3AS3AS3 Charge exchange relatedparameters

AD AD AD AD ADAS4AS4AS4AS4AS4 Recombination, ionization andradiated power

AD AD AD AD ADAS5AS5AS5AS5AS5 General interrogation programs

AD AD AD AD ADAS6AS6AS6AS6AS6 Data Analysis and curve fitting

AD AD AD AD ADAS7AS7AS7AS7AS7 Dielectronic data

ADADADADADAS8AS8AS8AS8AS8 Structure and excitationcalculations

TTTTTababababable 1.le 1.le 1.le 1.le 1. AD AD AD AD ADAS series and its contentsAS series and its contentsAS series and its contentsAS series and its contentsAS series and its contents

Another important aspect in the analysis isthe effect of opacity of the plasma on the

resonance lines [3]. Opacity can not only affectthe line shape/width but also affect theionization/ excitation rates. ADAS incorporatesopacity factor to correct for this.

UserUserUserUserUsersssss

Presently ADAS users are not only from thefusion community worldwide but also from SolarHeliospheric Observatory (SOHO) and ChandraX-Ray Observatory. Due to its wide-rangingcapability it has been to be useful for otherplasma sources also.

In short IDL-ADAS is a user-friendly packagefor various atomic plasma parameters and hasbeen found to be an important tool for modelingof atomic, plasma and astrophysicalparameters.

RefRefRefRefReferences:erences:erences:erences:erences:

1. ADAS site, http//adas.strath.ac.uk

2. Bates et al., Proc. Royal Soc. London,Ser A, 267267267267267 297 (1962).

3. K. Behringer and U. Fantz, New Journalof Physics, 22222 23.1-23.19 (2000)

International Symposium on Quantum Optics-2006International Symposium on Quantum Optics-2006International Symposium on Quantum Optics-2006International Symposium on Quantum Optics-2006International Symposium on Quantum Optics-2006The curtain raiser for the Diamond Jubilee Year Celebration of PRL

As the curtain raiser for the Diamond JubileeYear celebrations of Physical Research

abroad to interact with their Indiancolleagues. It also gave us an opportunity

Laboratory (PRL),a four-day Interna-tional Symposiumon QuantumOptics was held inPRL during July24-27. Lectureswere delivered onmany interestingtopics of currentand sustainedinterest such asentanglement, phase space distributions,quantum communication, nonlinear optics,BEC, coherent states and quantum-classical correspondence. The symposiumprovided a platform for scientists from

to discuss andappreciate thework of one ofthe most creativeand prolificQuantum Opti-cians of our time-Prof. G. S.Agarwal. Thesymposium coin-cided with hissixtieth birthday.

It was a pleasure and a privilege for all ofus to wish him a long, productive and healthylife.

J. BanerjiQuantum Optics & Quantum Information Division

PRL, Ahmedabad, [email protected]

http//adas.strath.ac.uk



7

Modeling Modeling Modeling Modeling Modeling ThrThrThrThrThrougougougougoughhhhhDimensional AnalDimensional AnalDimensional AnalDimensional AnalDimensional Analysis:ysis:ysis:ysis:ysis:An Interesting An Interesting An Interesting An Interesting An Interesting TTTTTool Forool Forool Forool Forool For

Inter Inter Inter Inter InterdisciplinardisciplinardisciplinardisciplinardisciplinaryyyyyApplicationApplicationApplicationApplicationApplication

PPPPP..... C. C. C. C. C. VinodkumarVinodkumarVinodkumarVinodkumarVinodkumar11111 and K.P and K.P and K.P and K.P and K.P..... Subramanian Subramanian Subramanian Subramanian Subramanian22222

1 Department of Physics, Sardar Patel University, Vallabh Vidyanagar - [email protected]

2 Physical Research Laboratory, Navrangpura, Ahmedabad- [email protected]

A simple algorithm developed by Szirtes for modeling based on Buckingham - Π theorem isillustrated through examples from different areas of physics. It is found to have applicationsfrom electronic devices to quantum mechanical systems.

IntrIntrIntrIntrIntroduction:oduction:oduction:oduction:oduction:

Dimensional analysis is a useful tool whichevery one of us has studied in our high schoolcurriculum. In an advanced physics course, wecheck the correctness of equations and formu-lae containing large number of terms by check-ing their dimension reduced to the basic MLTrepresentation. We are more familiar with theMLT system of dimension; however it is pos-sible to have a dimension system based on otherparameters.

We also encounter a variety of dimension-less numbers in physics. Often, we observe thatthese numbers signify the behaviour of thephysical system under the variation of a rangeof parameters that describe the system. A simpleexample is the famous ‘Chandrasekhar Limit’ inastrophysics, which in simple terms is the ratioof the mass of the stellar body to the mass ofour Sun. Without going into any complicateddetails of stellar evolution, the final course of astar would be decided by the mass it holds andthe ‘critical mass’ is predicted by theChandrasekhar Limit.

As we saw, Chandrasekhar Limit is ratio ofmasses, which obviously is dimensionless.There exist other classes of dimensionless num-bers, which are derived from more than onephysical quantity. A couple of them will be dis-cussed below.

The flow of an incompressible fluid through

a tube will be either streamlined or turbulent de-pending upon whether the Reynold’s numberassociated with the flow of fluid is below or abovea certain value. For a fluid of density ‘ρ’ and vis-cosity ‘η’ flowing through a tube having crosssection area ‘a’, the Reynold’s number is givenby

Rau= ρη (1)

where, u is the speed of fluid in the pipe. Thecritical Reynold’s number RC, where the transi-tion in flow would occur, is generally in the re-gion about 2000.

The fine structure constant

α π= =2 1

137 037

2e

hc .(2)

is another important dimensionless constant inphysics, which evolved from the spin-orbit in-teraction leading to fine-structure splitting ofeigenenergy levels of atoms. It was immediatelyrecognized that the fine-structure constant, in-troduced in somewhat a special context had amore general significance. As a pure numberwhich is independent of the units chosen, it hasan absolute meaning. Soon it was observed thatit measures the magnitudes of Bohr radius,Compton wavelength and classical radius ofelectron as shown in the relation below.

h

e

h

c

e

c

2

2 2 22

4 21

2

π µ π µ µα α:: :: :: ::= (3)




8

Further, the inverse of fine structure constant(~137) is recognized as the limit of atomic num-ber of natural elements. This means that thequest for super heavy elements will stop at anelement whose atomic number is 137.

In astrophysics, there are a host of dimen-sionless number derived from universal con-stants such as gravitational constant, Hubble’sconstant, Planck scale etc., which serves as lim-its and transition phases of many astrophysicaland cosmological processes.

In engineering, dimensionless numbers findapplication in establishing the scaling laws ofvarious performance factors in a prototype(model) and its real-life version.

The early work on dimensionless numberdates back to early twentieth century. A generaltheory of dimensional analysis and its implica-tions was developed and presented in a semi-nal paper by Buckingham in 1914 [1]. The mainresult of his work is known today as theBuckingham−Theorem. This theorem statesthat, “A complete set of dimensionless variablesunambiguously describes the behaviour of aphysical system”. Buckingham used Π to rep-resent the dimensionless variables, and there-fore this theorem is popularly known asBuckingham Π−Theorem.

According to the statement of the theorem,there exists a complete set of dimensionlessquantities (Π’s) which are constructed by vari-ous quantities under consideration. (Earlier wesaw that ρ, a, u and η together made theReynold’s number associated with the fluid flow.There may be other such numbers in fluid flow.Buckingham called such numbers as Π’s). How-ever, there has been no unique or universallyapplicable method to actually construct suchquantities explicitly.

In recent times, Thomas Szirtes of SPARAerospace, Canada has introduced a rathersimple procedure for deriving the dimensionlessnumbers associated with various physical pro-cesses [2]. The procedure described by him issuited for direct application to a number of physi-cal processes and problems in engineering.Here, we illustrate the applicability of this pro-

cedure to a variety of problems including pureand advanced physics.

The idea of writing this article to this newsbulletin originated from many discussions theauthors had with our colleagues. These discus-sions were mainly about the extension of theSzirtes algorithm beyond engineering systems;to the realm of physicists which include quan-tum systems, electronics and many other disci-plines of physics. The scaling laws that one de-rives here have wide applications in modelingand provides insights to the inter-relationshipsamong the dependable physical parameters ofthe problem under study. In many cases, it canbe found that it provides solution without solv-ing complicated differential or integro-differen-tial equations.

The scaling laws for fluid flow have alreadybeen discussed by N.N. Rao (1996) [3]. Similarscaling laws for the various physical problemslike(i) FET characteristics(ii) binding energy of electrons in atomic orbit,(iii) the skin depth of electromagnetic waves in

a conducting medium, and(iv) electron scattering

will be discussed here. Extension of this al-gorithm to extract scaling laws for quantum sys-tems has very interesting prospects.

The basic algorithm and its background arebriefly discussed here for the benefit of freshreaders.

SzirSzirSzirSzirSzirtes Prtes Prtes Prtes Prtes Procedure/Algorithm:ocedure/Algorithm:ocedure/Algorithm:ocedure/Algorithm:ocedure/Algorithm:

A physical equation is said to be dimension-ally homogeneous if every term in the equationreduces to the same algebraic quantity whenexpressed in terms of the reference dimensions.

Suppose there are p−physical quantities(P

i ; i = 1 to p) [e.g, Mass, Temperature, Pres-

sure, Ionisation potential and Electric charge,making p = 5] expressed in terms of d−refer-ence dimensions (R

i ; i = 1 to d) [e.g M,L and

T, so that d = 3], then there is a set of (p−d)[2 in this example] dimensionless quantities(called Πs) which describes the system com-pletely.


9

The procedure adopted by Szirtes is simple.He used matrix method to arrive at the set ofdimensionless numbers (Πs). For this, a matrix(we call it Szirtes dimension matrix) was formed,by arranging the parameters (P

i ; i = 1 to p)

along the columns and the reference dimensions(R

i ; i = 1 to d) along the rows (Table 1). Usu-

ally, no. of parameters are larger than dimen-sions. Enough numbers of Πs are included inthe rows, such that the Szirtes matrix becomesa square matrix. [Therefore, the difference be-tween no. of parameters and no. of dimensionsis the no. of dimensionless constants that canbe arrived at in the physical process.]

The elements of the matrix are the powersof the dimension of the corresponding physicalquantity P

i (columnwise). However, the matrix

elements corresponding to Πs are not knowna-priori and the Szirtes algorithm is all about amethod to find these elements.

Following 8 steps are used by Szirtes to ob-tain the dimensionless Π values.

1) For the given problem whose model law isto be established, list all the physicalvariables/parameters that are supposed toinfluence the behaviour of the system. It isimportant to note that if a relevant entry isomitted, the model law will be erroneous,but if an irrelevant/doubtful entry is included,the algorithm rejects it automatically. So asa rule of thumb, if particular variables/parameters are in doubt, list them. Let no.of parameters be ‘p’.

2) Write the reference dimensions with whichthe listed parameters are represented/measured. For example, if energy is arelevant variable in the list, then itsdimension ML2T-2 or in eV decides thenumber of reference dimension ‘d’ as three(M,L,T) in the first case or just eV as asingle reference dimension to representenergy.

3) Construct a [d × p] dimensional matrix, asshown in Table 1.

4) Select a nonsingular square matrix ofdimension (d × p). Let it be [A][A][A][A][A], the rightmost as shown above. The remaining part

of the d × p matrix is the matrix [B][B][B][B][B] withdimension d × (p−d).

5) Now, the number of dimensionlessvariables; i.e. no. of Π’s in the problem areobtained as NΠ = p − d

6) Construct a unit matrix [U][U][U][U][U] of dimension(p−d)×(p−d) below the matrix [B][B][B][B][B] andanother matrix, say [C][C][C][C][C] of dimension(p−d) × d below the matrix [A][A][A][A][A] such that the

matrix shown above becomes a matrix ofdimension (p × p) as shown Table 2.

TTTTTababababable 1:le 1:le 1:le 1:le 1: Step-1 for the formation of Szirtesdimension matrix. The elements ofmatrices [A][A][A][A][A] and [B][B][B][B][B] are the powersof dimension appearing in the param-eters of the proces.

TTTTTababababable 2:le 2:le 2:le 2:le 2: Step-2: Szir tes prescription forcompletion of dimension matrix by in-cluding [C[C[C[C[C] and [U][U][U][U][U] matrices. (see textfor details regarding [C] [C] [C] [C] [C]).


10

7) According to Szirtes algorithm [2] thematrix [C][C][C][C][C] can be obtained as

[C] = −[ [A]−1�[B] ]Trans (4)

8) Construction of Πs: Let ai j

(j=1 to p) bethe elements along the Π

i row. Raise the

power of P j by corresponding a

i j and

multiply all of them. This product is Πi.

Πi ji jP

a

j

= ∏ (5)

One can verify that the dimension of thisproduct (in the chosen referencedimension) is zero.

i.e. Dim [ Πi] = 0; for i=1, (p−d).

Illustrations:Illustrations:Illustrations:Illustrations:Illustrations:

Applications of this procedure to find theaverage speed of persons with different heightson different planets, self bending of beams, dragforce on a balloon in air, and a mechanicalsystem of damped oscillation are illustrated bySzirtes [2], while application to fluid flow as beendone by N N Rao[3]. Here, we illustrate itsapplications to other areas of pure physics.

1.1.1.1.1. Scaling la Scaling la Scaling la Scaling la Scaling laws fws fws fws fws for FET cor FET cor FET cor FET cor FET characteristics:haracteristics:haracteristics:haracteristics:haracteristics:

List below are the relevant FET parametersand their physical dimensions in Gauss units[4]:

1. Drain Current ID

⇒ M1/2L3/2T−2

2. Carrier mobility µ ⇒ M−1/2L3/2

3. Thickness of the h ⇒ Lsemiconductor

4. Density of the ND ⇒ L−3

doped material

5. Gate Capacitance Cg ⇒ L

6. Gate Voltage Vg

⇒ M1/2L1/2T−1

7. Drain Voltage VD ⇒ M1/2L1/2T−1

With this seven parameters (p = 7) expressedin terms of the basic dimensional quantities likemass, length and time (d = 3), we require four(p − d) dimensionless Π’s.

The Szirtes procedure discussed aboveprovide the following 7×7 matrix for the systemof FET characteristics.

Here, the matrix [A][A][A][A][A] is the top left cornersquare matrix and the right top matrix is [B][B][B][B][B].[This demonstrate that one need not always takematrix [A][A][A][A][A] on the right side.] Using the Szirtesalgorithm, the matrix [C][C][C][C][C] is computed using eqn.(4) and is listed below [A][A][A][A][A]. Below [B] is [U] [U] [U] [U] [U] asshown in Table 3. The model law now can bewritten by expressing the Π‘s in terms of theparameters as

Π

Π

Π

Π

1

23

3

1 2

4

1 2

=

=

=�

��

��

=�

��

��

h C

N C

I CV

I CV

g

D g

D gg

D gD

/

/

/

µ

µ

(6)

The scaling laws for the system can be foundby maintaining the constancy of the Πs asΠ

1 = Π

1; Π

2 = Π

2; Π

3 = Π

3 and Π

4 = Π

4‘.

By taking product of Π3 and Π

4, we get an

expression for the carrier mobility as

µ = Π Π3 4

C

V

I

Vg

g

D

D(7)

2.2.2.2.2. Binding ener Binding ener Binding ener Binding ener Binding energy of electrgy of electrgy of electrgy of electrgy of electron in an orbiton in an orbiton in an orbiton in an orbiton in an orbit ar ar ar ar around its nound its nound its nound its nound its nucucucucucleus of cleus of cleus of cleus of cleus of charharharharharggggge Z:e Z:e Z:e Z:e Z:

The dependent parameters and theirphysical dimensions are :

TTTTTababababable 3:le 3:le 3:le 3:le 3: Szirtes’ dimensional matrix for find-ing dimensionless numbers in FETcharacteristics.


11

result for the binding energy of the electron inan hydrogen atom by identifying Π1 = 1 andΠ2 as −2.

3.3.3.3.3. Skin depth f Skin depth f Skin depth f Skin depth f Skin depth for a lossy/conductingor a lossy/conductingor a lossy/conductingor a lossy/conductingor a lossy/conducting medium: medium: medium: medium: medium:

The dependent parameters and theirphysical dimensions in SI system of units [4]are:

1. The skin depth δ ⇒ L

2. Frequency of elec- f ⇒ T−1

tromagnetic signal

3. The magnetic per- Ζ ⇒ MLQ2

miability of medium

3. The conductivity σ ⇒ M−1Q2L−3Tof medium

5. Mass of medium m ⇒ M

Here, p = 5, p = 4 (M,L,T and charge Q), sothe number of Π ’s is just one. The 5x5dimensional matrix for the problem becomes:

1. Binding energy E ⇒ ML2T−2

2. Orbital angular l ⇒ ML2T−1

momentum of e−

3. Charge of nucleus Ζ ⇒ M1/2L3/2T−1

4. Charge of bound e ⇒ M1/2L3/2T−1

electron

5. Mass of bound m ⇒ Melectron

Here, p = 5 and d =3. So the number of Π’srequired for this problem is only two. The modelmatrix for the problem can be constructed asbelow.

TTTTTababababable 4:le 4:le 4:le 4:le 4: Szirtes’ algorithm applied to Bohrquantisation of atomic orbits. Theabove matrix is the Szirtes matrix forthe problem.

The model law for the problem is given by

Π1

1 2= δ µ σf� � / (11)

The expression for the skin depth of a lossy/conducting medium thus we get,

δµ σ

= Π11 2

f� � / . Identifying Π11 2= −π / , it is the

same expression that one finds in text books.

The model law can be written as

Π

Π

1

2

4

2

=

=

Z e

Z m

E l

/

(8)

Using the Π1 relation , we write Z = Π

1e and

on substituting in Π2 we get

Π Π2 14

4

2= e m

E l(9)

Hence, we get the energy expression as

Ee m

l= Π

Π14 4

22 (10)

It is interesting to see that by consideringthe quantization of the orbital angularmomentum, i.e., l = nh/2π, we get the text book

TTTTTababababable 5:le 5:le 5:le 5:le 5: Szirtes’ dimensional matrix for find-ing dimensionless numbers associ-ated with estimation of skin depth forlossy medium.


12

4.4.4.4.4. Scattering cr Scattering cr Scattering cr Scattering cr Scattering cross sections of Atoms/oss sections of Atoms/oss sections of Atoms/oss sections of Atoms/oss sections of Atoms/ Molecules with c Molecules with c Molecules with c Molecules with c Molecules with charharharharharggggged pared pared pared pared particticticticticles:les:les:les:les:

Here, we explore the power of this methodto find the scaling laws of scattering crosssection of atoms/molecules with electrons. It willenable us to find the inter-dependence ofvarious parameters of the system (Target –Projectile) with the scattering cross section. Therelevant parameters of the problem and theirgenerally used dimensions in atomic units arelisted below.

1. Scattering X section σ ⇒ a02

2. Ionisation potential I ⇒ eV

3. Projectile energy E ⇒ eV

4. Ave. ex. ener. of target ∆ ⇒ eV

4. Charge den. of target ρ ⇒ (eV)1/2/a05/2

5. Atomic dimension D ⇒ a0

4. Atomic polarisibility α ⇒ a03

Here, p = 7 and d = 2, so there are five Π’sto describes the system. The model Matrix anbe obtained as

It is important to note Π3, Π 4, and Π5 andtheir dependence on the parameters of thetarget and the projectile energy E. One of theplausible dependence of the scattering crosssection that can be deduced from the abovelaws is given by

σ ρ α∝ −14 3 2∆� � D E

// (13)

It is worth verifying some of these model lawsfrom the known scattering data for selectedsystems. One would also try to see thedependence of the various parameters by takingthe logarithm of the model laws given in termsof the Π’s. It is left to the interested readers totry out or to modify by incorporating any otherrelevant dependable parameters of thescattering system.

ConcConcConcConcConclusion:lusion:lusion:lusion:lusion:

So far, we have seen many cases where theSzirtes algorithm is applied to derive thedimensionless numbers. Their use to scalinglaws is simple and a few examples could befound in two articles mentioned previously [2,3].Interested readers may find more details in abook by Thomas Szirtes [5].

We notice that the dimensionless numbersso far discussed are made out of ratios of theparameters. The next desirable step would beto arrive at a more general algorithm which canaccommodate various functions (exponential,logarithm, trigonometric function etc.) in thedimensionless numbers. In such scenario, thescope of this technique would enlarge manyfold.We also note that this algorithm does notproduce the constants (2π, 2/π1/2 etc.) in thedimensionless numbers. Though not serious innature, this is one of the limitation of the Szirtesalgorithm.

RefRefRefRefReferences :erences :erences :erences :erences :1. E. Buckingham, Phys. Rev. vol. IV (1914)

345.2. T. Szirtes, SPAR Journal of Engineering

and Technology, vol. 1 (1992) 37.3. N. N. Rao, Resonance, vol. 1, No. 11 (1996)

29.4. David L Book, NRL Plasma Formulary

(1980 revised), p 10-125. Thomas Szirtes, “Applied Dimensional

Analysis and Modeling”, McGraw-Hill(1997)

TTTTTababababable 6:le 6:le 6:le 6:le 6: Determination of scaling laws in scat-tering problem. The above Szirtes’ di-mensional matrix was used to find thedimensionless numbers in in theproblem.

The model laws can be written as

ΠΠ ∆

Π

Π

Π

1

2

35 4 1 2

412

53 2

1==

=

=

=

−

−

−

/

// /

/

E

E

E

D

σ ρ

σ

σ α

(12)

Documents

Vol.: II Issue: 3 - Physical Research Laboratoryisamp/isamp_files/newsletter/Vol2_issue3.pdf · Vol.: II Issue: 3 August 21, 2006 CONTENTS Editorial 1 Abstracts 2-3 News 4 The ADAS