Dalitz analysesDalitz analyses Introduction - IJSgolob/sola/Dalitz_BAS09_20090913.pdf · Dalitz...

Dalitz analysesDalitz analyses IntroductionBelle Analysis School

October 1-2 2009

Hi, could you talk b t D lit

Sure...A couple of months ago, somwehere on

about Dalitzat the BAS?

g ,the net....

A couple of weeks ago, somwehere on

Yes...?

WHO did you saythe net.... WHO did you say I should talk about...?

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 1 BAS, KEK, October 2009

Dalitz analysesDalitz analyses IntroductionBelle Analysis School

Boštjan GolobBelle & Belle II

University of Ljubljana Jožef Stefan Institute

October 1-2 2009

University of Ljubljana, Jožef Stefan Institute

1. Introduction2. Kinematics3. Physics4. Parametrization4. Parametrization5. Experimental issues6. Specifics, Outlook

IntroductionHistoryHistory

Richard Henry Dalitz (28 February 1925 – 13 January 2006);Australian physicist;Australian physicist;

@ Cornell introduced “phase space plotsphase space plots”, i.e. Dalitz technique (as called today), e a t tec que (as ca ed today),to study 3π “tau” (kaon) decays;

On the analysis of τ meson data and the nature of the τ mesonOn the analysis of τ-meson data and the nature of the τ-meson Author: R. H. Dalitz a

Affiliation: a Department of Mathematical Physics, University of Birmingham, y gDOI: 10.1080/14786441008520365 Published in: Philosophical Magazine Series 7, Volume 44, Issue 357 October 1953 , pages 1068 - 1080

Kinematicsn-body decayJ.D. Jackson, D.R. Tovey, Kinematics, in RPP

( )Φ=Γ nn

),;(22π

∏∑==

−=Φni i

niinn E

pdpPppPd,1

41 2)2(

)(),;(π

δK LISP:Lorent InvariantPhase Sace

independent variables: 4-vectors: 4 n +

ti l 4 3 7n=3 2n=3 2

conservation laws: 4 = 3 n - 7final state masses: n -arbitrary rotations: 3

n=4 5n=5 8

Kinematics3 body decay

J.D. Jackson, D.R. Tovey, Kinematics, in RPP,

3-body decay

take two inv. masses asindependent variables decaying particle:independent variables

233 32

1 dmdmM

d M=Γ

decaying particle:scalar or averagingover spin states33 32)2( Mπ p

mij: inv. mass of part. i,j.2

212 constmmmMmmm =+++=++

dΓ332

13212 32)2(

MMdmdm π

= standard form of Dalitz plot

if |M|2 const ⇒ dΓ/dm 2dm 2 const

if |M|2 const. ⇒ dΓ/dm122dm13

2 const.

Kinematics Belle, PRL 99, 131803 (2007)

3-body decay

example: D0 → Ksπ -π+ m2 (

non-uniformity of Dalitz plot ⇒contribution of intermediate states

Ki ti li itKinematic limitsD0 → π+K*- m2(KSπ-)J.D. Jackson, D.R. Tovey, Kinematics, in RPP

θ23*=π

( ) ( ) 222 2mmm +θ23*=0

( ) ( )( ) ( )min

223max

23max23min23*23

−−+

=θ 34

KinematicsM1

3-body decay

various intermediate states Mk

contributing to same final state ⇒ interfereinterferenncece

|M|2 is not incoherent sum, (|M|2 ≠ |M1|2 + |M2|2 + ... ) ,

but a coherent sum, ( |M|2 = |M1+ M2 + ...|2 )

example: pp → 3π0

PhysicsCleo-c, arXiv:0903.1301

New states and propertiesof known states

Dalitz analysis usually not needed for narrow, non-overlapping

resonances(negligible interference)example: Ds → K+K-π+

(but interf. φπ/KKπ, φπ/f0πimportant in precise Br(Ds → φπ)determination)

determination)

another example: 37

PhysicsB → KZ+(4430)

Belle, PRD 80, 031104(R) (2009)

New states and propertiesof known states

KZ (4430)

In most cases intermediate states strongly interfere

Z+(4430)example: B → Kπ+ψ’

by fitting fitting DalitzDalitz distributiondistributionobtain evidence of newobtain evidence of new states, measure properties(mass width spin)

B → K*(892)ψ’

B → K0*(1430)ψ’

(mass, width, spin)

projection of Dalitz disribution and fit to

disribution and fit to M(π+ψ’)

Physics20 )( tAyixqAefDd t +

+=→Γ −

D0 Mixing and CPV

x, y: mixing parameters;

dt ff +=

due to mixing, D0 → D0 → fx, y << 1

ΓΓ−Γ

=Γ−

;;' 2121 ymmxttτ

t-dependent pDalitz analyses

different types of interm states;different types of interm. states;example: D0 → π+π- KS

CF: D0 → K*-π+

DCS: D0 → K*+π-

CP: D0 → ρ0 KS

if f = f ⇒ populate same Dalitz plot; l ti h d t i dl ti h d t i d

relative phases determined relative phases determined 28

Physics Belle, PRL 99, 131803 (2007)

t dependentt

t-dependent Dalitz analyses

t-dependence:t-dependence:regions of Dalitz plane →specific t dependence F(x, y);

time evolution of Dalitz distribution⇒⇒ x, y x, y

[ ]titi

tDKtmm

)(),,(

−−

−++−

MD0→f

[ ]titi eemmpq

21),(21

22 λλ −−+−

AD0→f

p2λ1,2=F(x,y); m±

2 = m2(KSπ±),

PhysicsBelle, arXiv:0803.3375

φ measurementφ3 measurement

B- →K- D0(→ f)B- →K- D0(→ f)B →K D0(→ f)interference ⇒ |M|2 = F(φ3) f= π+π- KS

),(),())(( 22220)(

−±± +=→→ mmAremmAfDKB ii

mmδϕM(

A: D0 decays; r: ratio of two B amplitudes; δ: strong phase diff. of two B amplitudes

A from fit to Dalitz fit to Dalitz distribution of D0 decays;

r, φ3, δ (in principle) from fit to Dalitz fit to Dalitz distribution of B± decays30

sin 2φ eff in b → sqq

Physics SM:

sin 2φ1eff in b → sqq

NP contrib. ⇒ sin 2φ1eff ≠ sin 2φ1

effVtbVts* : no weak phase

quasi twoquasi two--bodybody, B → ρ0(770)KS, f0(980)KSBelle, PRD76, 091103(R) (2007)BaBar, PRL99, 161802 (2007)B B PRL98 051803 (2007)

B→BaBar, PRL98, 051803 (2007)

Vtd* 2: (mixing): sin2φ1

[ ]tmqStmqAetP CPCP

∆∆−∆∆+=∆∆−

sincos14

interf. between various states and non-resonant contrib. ⇒ Dalitz analysis

sη’

+(δ 23

B0 gs~R

∆S=sin2φ1eff -sin2φ1

Physicssin 2φ eff in b → sqqsin 2φ1

eff in b → sqq

[ ]tmqStmqAetP CPCP

∆∆−∆∆+=∆∆−

sincos14

),( 22mmmAA ±=

each point in Dalitz space has a specific time evolution depending onevolution, depending on |A|2-|A|2 (direct CPV)(direct CPV)and ℑ(AA*) (indirect CPVindirect CPV corresponding to

ℑ(AA ) (indirect CPVindirect CPV, corresponding to specific two-body contribution sin 2sin 2φφ11

effeff (i)(i))

Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution

a matter of statistics102 pp → 3π0 events10 pp → 3π events

Adopted from K. Peters, talk at Charm 2006, Beijing

a matter of statistics103 pp → 3π0 events10 pp → 3π events

a matter of statistics104 pp → 3π0 events104 pp → 3π0 events

a matter of statistics105 pp → 3π0 events

larger stat. ⇒ larger sensitivity to model details

Breit-Wigner resonances

10 pp → 3π events

Breit Wigner resonancessimple consideration of spin 0 elastic scatteringleads to the Breit-Wigner amplitude for a b → r → a b

4/)(4/

Γ+−Γ

=Γ−−

|T|2 |T|2

mr=1.0Γ=0.3

mr=0.5Γ=0.2

ParametrizationBreit Wigner resonances

|T1+ T2 |2

Breit-Wigner resonances

several intermediate statesb ba b → r1 → a b

r2model amplitude as sum of BW amplitudes;

sum of BW’s amplitudes; - the approach violates unitarity for wide overlapping resonances;

- the BW shape is distorted close

(“isobar model”)most commonly used most commonly used to model Dalitz distributionsthe BW shape is distorted close

to the thresholds;

parametrization of Dalitz distribution is

distributions

D0, B0 → ABCpa model, phenomenological object;

it should provide adequate description adequate description ∑ Φ

mmr ),(

of dataof data; caution needed in interpretation

ParametrizationBW resonancesBW resonances

parametrization usually follows Cleo PRD63 092001 (2001) J 0Cleo, PRD63, 092001 (2001) JA,B,C=0

resonance spin

parametrization usually follows Cleo PRD63 092001 (2001)Cleo, PRD63, 092001 (2001)

JA,B,C=0

mass dependent width;pAB: momentum of A/B in AB rest frame;pr : momentum of A/B in rest frame of r;

pr ;(note: MAB may not equal Mr)

Ji=1 form factors (Blatt-Weisskopf)

Ji=2 R: phenomenological factor(0 GeV-1 – 10 GeV-1 for D0, 0 GeV-1 – 3 GeV-1 for r)

Refinements - FlattFlattéé parametrizationFlattFlattéé parametrization

if 2nd channel for r decays opens close to mr;example f0(980) → ππ, +−−

KKKKfff ggimmm

Aρρ ππππππ

S.M. Flatté, PLB63, 224 (1976)

p 0( ) ,KK threshold close to mf0

- parametrization of nonnon resonantresonant contribution

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +−= 2

nonnon--resonantresonant contribution in D0 decays usually a constantterm; in B0 charmless decays phase

gij: couplings r/ij(note: mismatch in several papers on Flatté parameters of f0(980) and a (980); contact A Zupanc Zhao Li orin B charmless decays phase

space larger, some variations of ANR; example B0 →KSπ+π- ANR ∝ exp(-αmij

2) Belle, PRD79, 072004 (2009)

a0(980); contact A. Zupanc, Zhao Li orB.G.)

p S NR p( ij )B → Dππ ANR = const.

, , ( )

Belle, PRD76, 012006 (2007)

ParametrizationK matrix

E.P. Wigner, Phys. Rev. 70, 15 (1946);S.U. Chung et al., Ann. Phys. 4, 404 (1995)

K-matrix

- sum of BW’s violates unitaritybroad S wave ππ Kπ states are

ρρ TiIS

iSfS fi

=2/12/12

- broad S-wave ππ, Kπ states are not simple BW’s

scattering operator Sρ

iTKKiKIT

−=−−

scattering operator Stransition operator TK matrix

++ =⇔= KKISS

iijij = ρδρoriginally developed for description of scattering

can be adopted to production processproduction process, smm ii

221 )(1 +

−=ρ

e.g. D, B decays, using scattering data scattering data

kfkf KiKIT 1)( −−= ρ D

I.J.R. Aitchison, Nucl. Phys. A189, 514 (1972); kfkf PiKIF 1)( −−= ρ P (I-iKρ)-1

ParametrizationK matrix PiKIF 1)( −−= ρK-matrix

assume we are parametrizing a 2 body

Drescattering of k→ ππ

production of k K

kk PiKIF ,)(= ππππ ρ

parametrizing a 2-body intermediate state of a 3-body final state(example: ππ in KSπ+π-):

P (I-iKρ)-1of k→ ππ of k KS

k=ππ, KK, ηη, ...

(example: ππ in KSπ π ):Belle, PRL99, 131803 (2007)BaBar, hep-ex/0507101 scatt

scattprod

k sssf

1−−

= ∑ ππβ

s=m2ππ

scattrr 1 ⎪⎫⎪⎧

resonances - poles;(may not coincide withphysical states)

slow-varying part;

factorkinemsssf

K scatt

scattscatt

02 ⋅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−

gir, mr, fijscatt, s0

scatt : from compilation of ππ scattering datascattering data

ParametrizationK matrix gg ρρ D. Asner, Dalitz plot analysis K-matrix

for single channel (e g ππ)

)()1(2

ρππ

ππππ

formalism, in RPP

(e.g. ππ) and single pole(e.g. ρ0): )(

)()1( 22

gismKiKT

ρρρππ

ρππρ

−−=−=

KK--matrix = single BWmatrix = single BW

for multiple channels )(

)(2 simsm

ρρρ

Γ−−Γ

pand/or multiple poles:

KK--matrix matrix ≠≠ multiple BW multiple BW

S. Malvezzi, talk at Charm07, Cornell

Experimental issues

Inv. mass resolutiontypically σ << Γ;e g B → Dππ: σ ~3 MeV;e.g. B → Dππ: σππ 3 MeV;non-negligible for φ(K+K-), ω(π+π-)→ ABW⊗G(σ) numerically (or included in syst.)

also non-negligible at Dalitz boundaries→ mass constraint fit mass constraint fit of ABC to D0, B0;also same Dalitz region signal/sideband (background description)

EfficiencyABW(mAB

2,mAC2) → ABW (mAB

2,MAC2) • ε (mAB

2,mAC2);

ε (mAB2,mAC

2) from MC generated uniformly over phase space;parametrization (factorize mAB

2,mAC2 dependence)

Experimental issues

BackgroundD0, B0 mass sidebands 2222

ACABside

ACABsig

dmdmdN

dmdmdNdmdmdN

dmdmdN

Fitmany issues (talk by E. White)(talk by E. White);- unbinned sometimes binned;

Dalitz model testunbinned, sometimes binned;

- fit quality bin Dalitz plot, χ2;model appropriateness

pp p(hypothesis testing)→ compare lnL for various models;significances of ind. contributions

Dalitz + decay tfit test(talk by B. (talk by B. YabsleyYabsley););

- biastest using MC with generated Dalitz model,

l D0 K +

fit test

example D0 →KSπ+π- Belle, PRL99, 131803 (2007)Belle Note 9799

Experimental issuesFit

Belle, PRD79, 072004 (2009)

Fit- multiple solutions,

example B0 →KSπ+π-

test for multiple solutions by performing numerous fits with varying starting values of solution 1 solution 2y g gparameters;

toy MC to checks if some solutions due to stat fluctuations;

K0*(1430)+π- fit fraction:~ 65% ~ 17%

due to stat. fluctuations;

- individual contribution∑ Φ= 2222 ),(),(

irACAB mmAeamm rA

- individual contribution → fit fraction, fi

Σfi ≠ 100%

∫ ∑

∫ Φ

22222 ),( ACABACABBWi

dmdmmmAeaf

∫ ∑ Φ 2222 ),( ACABr

ACABBWr

ir dmdmmmAea r

Experimental issuesFitFit

- systematic uncertainty due to Dalitz model

variation of resonance parameters (width mass );variation of resonance parameters (width, mass, ...);variation of form factors, dependence of width on mass; inclusion/exclusion of intermediate states with low signif.; difference isobar / K-matrix model;;possible biases (MC)

Specifics, outlookZ+(4430)

Belle, PRD 80, 031104(R) (2009)

Z (4430)New states and propertiesof known states

Larger statistics important for significance;

ibilit t di ti i hpossibility to distinguish JZ=1 and JZ=0

(hypothesis testing)(hypothesis testing)

Outlook:Outlook:

more surprises

BaBar, PRD 79, 112001 (2009)

Specifics, outlookD0 Mi i d CPV

Belle, PRL 99, 131803 (2007), 540fb-1

D0 Mixing and CPVK*X(1400)+

)%290800( 130±±x

K*(892)+

)%24.033.0(

)%29.080.0(

14.010.0

16.013.0

K*(892)- ρ/ωpq 09.010.0

29.030.086.0/ ±±=

rad)09.030.028.024.0( ±±−=ϕ

relative sign of x, y; K-matrix used for systematics

Specifics, outlookBelle PRL 96 151801 (2006)D0 Mixing and CPV

direct determination of x, y0

∑ +−Φ

+− =r

ir mmAeamm r ),(),( 2222A

Belle, PRL 96, 151801 (2006)

instead of x’, y’ as in D0 →K+π-

(rotated for unknown δ between CF and DCS)

a1=1, Φ1=0relative phases determined from the fit to Dalitz distribution

2⎤⎡

Outlook:

fit to Dalitz distribution

2 08030.0⎥⎤

⎢⎡

2 06.0/

30.0 radrad+

⎥⎥⎦

⎢⎢⎣

LLϕσ need to reduce Dalitzmodel systematics;

2|/| 08.0

+⎥⎥⎦⎢

⎢⎣

=LLpqσ other modes: KSKK, πππ0,...

Specifics, outlookφ3 measurement

B- →K- D0(→ f) ))(( 0)(−

±± =→→ fDKBM(

Belle, arXiv:0803.3375

( )B- →K- D0(→ f)interference ⇒ |M|2 = F(φ3) f= π+π- KS

),(),(

))((2222 3±

+±± +

→→

mmAremmA

fDKBii

mmδϕ

use x±, y± instead of r, φ3, δ;→ more free parameters; )sin(

δφδφ

+±=+±=

lower bias, simple distributions

φ3=76o (+12o -13o) ± 4o ± 9o

Outlook:model independent determination;

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 33 BAS, KEK, October 2009from Cleo-c

)),(),((|| 22222±± −= mmmm DD mm δδFM

Specifics, outlooksin 2φ eff in b → sqq BaBar PRD 79 112001 (2009)

example B →J/ψKπ

Belle, PRD79, 072004 (2009)

change of variables →rectangular Dalitz plot

BaBar, PRD 79, 112001 (2009)

122arccos1'

''det θ

πππ ⎞⎜⎜⎛

−−

ddmJdmdm m

⎠⎜⎝ −− mmm KsB

a oid concentration of e ents

avoid concentration of events at Dalitz boundaries 39

Specifics, outlooksin 2φ eff in b → sqq

Belle, PRD79, 072004 (2009)

φ1eff(f0)=(12.7 ±6.9

6.5±2.8±3.3)o

Outlook:

⎨⎧≤∆ sKfS 03.0)( 0

⎩⎨⎧≤∆

fS ϕσ 05.003.0)( 0

@ 50ab-1 for ∆S=0

b D lit ln.b.: Dalitz analyses measuring φ1

⇒σ(S) = 2 cosφ1eff σ(φ1

eff)∆S

SuperBelle, 50 ab-1

Summaryi t d f D lit l i flinstead of summary.... Dalitz analysis flow....

improved methodsimproved methodsgive you larger luminosity!

is Belle note on t-dependent Dalitzready?

NP!NP!

Bckups

Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,

3-body decay

decaying particle:scalar or averaging

i t tover spin states⇒integration over dΩ1*, dΩ3

212dmdmd

233 32

3-body decay

2/)()( 222*2**2 MEEM + v

2/)(;2/)(),(

mmmMEppEEmmmmMEpEmM

+−=⇒=+=−−=⇒+=

)cos,(cos22

mfmmEmEEEmmm

=−−−++=

3-body decay

( ) ( )( ) ( )22

223max

223min

mmm−

−+=ϑ ( ) ( )min23max23 mm −

Physics moreB → K ψ(2S)

Belle, PRL 91, 262001 (2003), 140fb-1

New states

Dalitz analysis not needed B → K X(3872)

B → K ψ(2S)(→ J/ψ π -π+)

for narrow, non-overlaping resonances(negligible interference)example: B → K J/ψ π -π+

( )(→ J/ψ π -π+)

Physics moret d d t

Belle, PRL 99, 131803 (2007)

t-dependent Dalitz analyses

t d d [ ]titi

S tDKtmm

)(),,(

ππ −++− =≡M

t-dependence:regions of Dalitz plane →specific t dependence F(x, y);

[ ]titi

−−+

−−+−

time evolution of Dalitz distribution⇒⇒ x, y x, y

[ ]eemmp

),(2 +−+ A

2,12121

Γ−= imλ

22,12,1

),(),(

+−Φ

∑mmAeamm

mmAeamm

),(),( −++−

Φ=Φ=⎯⎯⎯ →⎯

= ∑aa

mmAeamm

rrrrDCPVno

),(),( 2222−++− = mmmm AA 28

Physics moreBelle, PRD79, 072004 (2009), , ( )

ci determines amount of DCPVci determines amount of DCPV

Physics moreBelle, PRD79, 072004 (2009), , ( )

vtx reconstr (non-scaling)vtx reconstr. (non-scaling)improved with better tracking

bkg model (scaling)CPV in bkg; improved with measurements of contributing modes e g ’Ks

⎩⎨⎧≤

sKKfS ϕσ 06.0

04.0)( 0 @ 50 ab-1

of contributing modes – e.g. η Kssig. model (non-scaling)misreconstructed events; comparison with MC with correct reconstr.

D lit d l ( li )

some syst. errors cancel in ∆S = S(sqq)-S(J/ψKs)

⎧ Kf030Dalitz model (scaling)parametr. of NR; inclusion of other resonances;

⎩⎨⎧≤∆

sKKfS ϕσ 05.0

03.0)( 0

for ∆S=0

n.b.: Dalitz analyses measuring φ1eff ⇒σ(S) = 2 cosφ1

eff σ(φ1eff)

Dalitz analysesDalitz analyses Introduction - IJSgolob/sola/Dalitz_BAS09_20090913.pdf · Dalitz...

Documents

Transcript of Dalitz analysesDalitz analyses Introduction - IJSgolob/sola/Dalitz_BAS09_20090913.pdf · Dalitz...

Veneziano model and application to Dalitz plot analysisjpac/Resources/veneziano.pdfVeneziano model and application to Dalitz plot analysis Adam Szczepaniak, Indiana U./JLab Motivation

KLiC Activity Scenario Template - Επετειακό Video ...€¦ · · 2012-03-19KLiC Activity Scenario Template ... Gymnastics, motion analyses , Handstand – forward roll

Structural and biophysical analyses of the skeletal ...

Genomic and expression analyses of Tursiops truncatus T ... · RESEARCH ARTICLE Open Access Genomic and expression analyses of Tursiops truncatus T cell receptor gamma (TRG) and alpha/delta

Dalitz Plot Analyses of B D B D at BABAR...of interference eﬀects. The uncertainty on each fk is evaluated by propagating the full covariance matrix obtained from the ﬁt. 3. Results

The Stochastic Geometry Analyses of Cellular Networks With ...IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 3, MARCH 2019 2487 The Stochastic Geometry Analyses of Cellular Networks

A Dalitz Plot Analysis of 3 Decay · A Dalitz Plot Analysis of !!3ˇDecay Chris Zeoli Florida State University cpz11@my.fsu.edu June 4, 2015 Chris Zeoli (FSU) Dalitz Analysis of !!

Numerical Analyses for ElectronagneticsNumerical Analyses for Electronagnetics Takuichi Hirano (RA) Ando & Hirokawa lab. June 5, 2002 Tokyo Institute of Technology MoM No. 2 Differential

TED Thermomechanical Analyses Esposito Raffaele EN-STI-TCD.

Detection of the mode: NOMAD - Harvard Physicsusers.physics.harvard.edu/~feldman/Lecture_2.pdf · NOMAD: Analysis l Blind analyses l Early results with non-blind analyses indicated

Update on the DB Girders FE Analyses

Analyses of light higgs decays for the CLIC CDR

CFD Analyses of Innovative Electric Aircraft Designsmdx2.plm.automation.siemens.com/.../SGC2015_JobyAviation_Stoll.… · CFD Analyses of Innovative Electric Aircraft Designs Alex

High-content imaging analyses of γH2AX-foci and ...

Josu e Danilo Molina Rodriguez Amplitude analysis of the ...cds.cern.ch/record/2265095/files/fulltext.pdfpouco conhecidas. O estudo do Dalitz plot atrav es da an alise de amplitudes

Biochemical and Functional Analyses of Chromatin Changes ... · Biochemical and Functional Analyses of Chromatin Changes at the TCR-bGene Locus During CD42CD82 to CD41CD81 Thymocyte

In situ δ O and Mg/Ca analyses of diagenetic and planktic ...wiscsims/pdfs/Kozdon_Palo2013.pdf · In situ δ18O and Mg/Ca analyses of diagenetic and planktic foraminiferal calcite

Control Flow Analyses

Paleoprecipitation Reconstructions for the Brady Soil Based on Rock-Magnetic Analyses

Analyses of Multimode Forming Process in a Microwave ...