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Changed calculation of charge-exchange processes to a product of complex polarization functions, as in the paper of M. Senba

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This commit is contained in:
Thomas Prokscha
2016-04-14 15:28:33 +02:00
parent 2339ee9b80
commit 908f728744
3 changed files with 120 additions and 140 deletions
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250
src/tests/MuTransition/PSimulateMuTransition.cpp
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@ -3,9 +3,7 @@
PSimulateMuTransition.cpp
Author: Thomas Prokscha
Date: 25-Feb-2010
$Id$
Date: 25-Feb-2010, 14-Apr-2016
Use root macros runMuSimulation.C and testAnalysis.C to run the simulation
and to get a quick look on the data. Data are saved to a root histogram file
@ -13,26 +11,28 @@
analyze the simulated data.
Description:
Root class to simulate muon spin phase under successive Mu+/Mu0 charge-exchange
processes by a Monte-Carlo method. Consider transverse field geometry, and assume
initial muon spin direction in x, and field applied along z. For PxMu(t) in
muonium use the equation 8.22 of the muSR book of Yaounc and Dalmas de Réotier, in
slightly modified form (see Senba, J. Phys. B 23, 1545 (1990)); note that PxMu(t)
is given by a superposition of the four frequencies "nu_12", "nu_34", "nu_23", "nu_14".
These frequencies and the corresponding probabilities ("SetMuFractionState12" for
transitions 12 and 34, "SetMuFractionState23" for states 23 and 14) can be calculated
Root class to simulate muon spin polarization under successive Mu+/Mu0 charge-exchange
or Mu0 spin-flip processes by a Monte-Carlo method. Consider transverse field geometry,
and assume initial muon spin direction in x, and field applied along z. For PxMu(t) in
muonium use the complex expression of
equation (4) in the paper of M. Senba, J. Phys. B 23, 1545 (1990), or
equation (7) in the paper of M. Senba, J. Phys. B 24, 3531 (1991);
note that PxMu(t) is given by a superposition of the four frequencies "nu_12", "nu_34",
"nu_23", "nu_14". These frequencies and the corresponding probabilities ("SetMuFractionState12"
for transitions 12 and 34, "SetMuFractionState23" for states 23 and 14) can be calculated
for a given field with the root macro AnisotropicMu.C
Parameters:
1) Precession frequencies of "nu_12", "nu_34", "nu_23", "nu_14"
2) fractions of nu_12, nu_34; and nu_23 and nu_14
3) total Mu0 fraction
4) electron-capture rate
5) Mu ionization rate
6) initial muon spin phase
7) total muon decay asymmetry
8) number of muon decays to be generated.
9) debug flag: if TRUE print capture/ionization events on screen
4) Mu+ electron-capture rate
5) Mu0 ionization rate
6) Mu0 spin-flip rate
7) initial muon spin phase
9) total muon decay asymmetry
9) number of muon decays to be generated.
10) debug flag: if TRUE print capture/ionization events on screen
Output:
Two histograms ("forward" and "backward") are written to a root file.
@ -43,10 +43,13 @@
1) according to Mu+/Mu0 fraction begin either with a Mu+ state or Mu state
2) Mu+: determine next electron-capture time t_c. If t_c is larger than decay time t_d
calculate muon spin precession for t_d; else calculate spin precession for t_c.
3) Determine next ionization time t_i; calculate Px(t_i) in Muonium; calculate the
muon spin phase by acos(Px(t_i)).
4) get the next electron capture time, continue until t_d is reached; accumulate muon spin
phase.
3) Determine next ionization time t_i; calculate Px(t_i) in Muonium; calculate the total
muon spin polarization Px(t_i)*Px(t_c).
4) get the next electron capture time, continue until t_d is reached, and calculate
the resulting polarization.
The Mu0 spin-flip processes are calculated in GTSpinFlip(), using eq. (17) of
M. Senba, J. Phys. B 24, 3531 (1991).
***************************************************************************/
@ -138,15 +141,21 @@ PSimulateMuTransition::~PSimulateMuTransition()
*/
void PSimulateMuTransition::PrintSettings() const
{
cout << endl << "Mu precession frequency 12 (MHz) = " << fMuPrecFreq12;
cout << endl << "Mu precession frequency 34 (MHz) = " << fMuPrecFreq34;
cout << endl << "Mu precession frequency 23 (MHz) = " << fMuPrecFreq23;
cout << endl << "Mu precession frequency 14 (MHz) = " << fMuPrecFreq14;
cout << endl << "Mu0 precession frequency 12 (MHz) = " << fMuPrecFreq12;
cout << endl << "Mu0 precession frequency 34 (MHz) = " << fMuPrecFreq34;
cout << endl << "Mu0 precession frequency 23 (MHz) = " << fMuPrecFreq23;
cout << endl << "Mu0 precession frequency 14 (MHz) = " << fMuPrecFreq14;
cout << endl << "Mu+ precession frequency (MHz) = " << fMuonGyroRatio * fBfield;
cout << endl << "B field (T) = " << fBfield;
cout << endl << "Mu+ electron capture rate (MHz) = " << fCaptureRate;
cout << endl << "Mu0 ionizatioan rate (MHz) = " << fIonizationRate;
cout << endl << "Mu0 spin-flip rate (MHz) = " << fSpinFlipRate;
cout << endl << "!!! Note: if spin-flip rate > 0.001 only spin-flip process is considered!!!";
if (fSpinFlipRate > 0.001)
cout << endl << "!!! Note: spin-flip rate > 0.001 only spin-flip processes are considered!!!";
else{
cout << endl << "!!! spin-flip rate <= 0.001: only charge-exchange cycles are considered!!!";
cout << endl << "!!! if spin-flip rate > 0.001, only spin-flip processes are considered!!!";
}
cout << endl << "Decay asymmetry = " << fAsymmetry;
cout << endl << "Muonium fraction = " << fMuFraction;
cout << endl << "Muonium fraction state12 = " << fMuFractionState12;
@ -181,23 +190,26 @@ void PSimulateMuTransition::SetSeed(UInt_t seed)
*/
void PSimulateMuTransition::Run(TH1F *histoForward, TH1F *histoBackward)
{
// Double_t muoniumPolX = 1.0; //polarization in x direction
Int_t i;
if (histoForward == 0 || histoBackward == 0)
return;
fMuonPrecFreq = fMuonGyroRatio * fBfield;
for (i = 0; i<fNmuons; i++){
fMuonPhase = TMath::TwoPi() * fInitialPhase/360.; // transform to radians
fMuonDecayTime = NextEventTime(fMuonDecayRate);
if (fSpinFlipRate > 0.001){// consider only Mu0 spin-flip in this case
fMuonPhase = TMath::ACos(GTSpinFlip(fMuonDecayTime));
fMuonPhase += TMath::ACos(GTSpinFlip(fMuonDecayTime));
}
else{
// initial muon state Mu+ or Mu0?
if (fRandom->Rndm() <= 1.-fMuFraction)
Event("Mu+");
fMuonPhase += TMath::ACos(Event("Mu+"));
else
Event("");
fMuonPhase += TMath::ACos(Event("Mu0"));
}
// fill 50% in "forward", and 50% in "backward" detector to get independent
// events in "forward" and "backward" histograms. This allows "normal" uSR
@ -233,28 +245,29 @@ Double_t PSimulateMuTransition::NextEventTime(const Double_t &EventRate)
//--------------------------------------------------------------------------
// Phase (private)
//--------------------------------------------------------------------------
/**
* <p>Determines phase of the muon spin
// /**
/* * <p>Determines phase of the muon spin
*
* \param time duration of precession (us);
* \param chargeState charge state of Mu ("Mu+" or "Mu0")
*/
Double_t PSimulateMuTransition::PrecessionPhase(const Double_t &time, const TString chargeState)
{
Double_t muonPhaseX;
Double_t muoniumPolX = 0;
if (chargeState == "Mu+")
muonPhaseX = TMath::TwoPi()*fMuonPrecFreq*time;
else if (chargeState == "Mu0"){
muoniumPolX = GTFunction(time).Re();
muonPhaseX = TMath::ACos(muoniumPolX);
}
else
muonPhaseX = 0.;
return muonPhaseX;
}
// Double_t PSimulateMuTransition::PrecessionPhase(const Double_t &time, const TString chargeState)
// {
// Double_t muonPhaseX;
// Double_t muoniumPolX = 0;
//
// if (chargeState == "Mu+")
// muonPhaseX = TMath::TwoPi()*fMuonPrecFreq*time;
// else if (chargeState == "Mu0"){
// muoniumPolX = GTFunction(time).Re();
// if (fDebugFlag) cout << "muoniumPolX = " << muoniumPolX << endl;
// muonPhaseX = TMath::ACos(muoniumPolX);
// }
// else
// muonPhaseX = 0.;
//
// return muonPhaseX;
// }
//--------------------------------------------------------------------------
// Mu0 transverse field polarization function (private)
@ -264,29 +277,26 @@ Double_t PSimulateMuTransition::PrecessionPhase(const Double_t &time, const TStr
*
* \param time (us);
*/
TComplex PSimulateMuTransition::GTFunction(const Double_t &time)
TComplex PSimulateMuTransition::GTFunction(const Double_t &time, const TString chargeState)
{
Double_t twoPi = TMath::TwoPi();
TComplex complexPol = 0;
complexPol =
0.5 * fMuFractionState12 *
(TComplex::Exp(TComplex::I()*twoPi*fMuPrecFreq12*time) +
TComplex::Exp(-TComplex::I()*twoPi*fMuPrecFreq34*time))
+
0.5 * fMuFractionState23 *
(TComplex::Exp(TComplex::I()*twoPi*fMuPrecFreq23*time) +
TComplex::Exp(TComplex::I()*twoPi*fMuPrecFreq14*time));
if (chargeState == "Mu+")
complexPol = TComplex::Exp(-TComplex::I()*twoPi*fMuonPrecFreq*time);
else{
complexPol =
0.5 * fMuFractionState12 *
(TComplex::Exp(TComplex::I()*twoPi*fMuPrecFreq12*time) +
TComplex::Exp(-TComplex::I()*twoPi*fMuPrecFreq34*time))
+
0.5 * fMuFractionState23 *
(TComplex::Exp(TComplex::I()*twoPi*fMuPrecFreq23*time) +
TComplex::Exp(TComplex::I()*twoPi*fMuPrecFreq14*time));
}
return complexPol;
// Double_t muoniumPolX = 0;
// muoniumPolX = 0.5 *
// (fMuFractionState12 * (TMath::Cos(twoPi*fMuPrecFreq12*time) + TMath::Cos(twoPi*fMuPrecFreq34*time)) +
// fMuFractionState23 * (TMath::Cos(twoPi*fMuPrecFreq23*time) + TMath::Cos(twoPi*fMuPrecFreq14*time)));
//
// return muoniumPolX;
}
//--------------------------------------------------------------------------
@ -308,18 +318,18 @@ Double_t PSimulateMuTransition::GTSpinFlip(const Double_t &time)
eventTime += NextEventTime(fSpinFlipRate);
if (eventTime >= time){
muoniumPolX = GTFunction(time).Re();
muoniumPolX = GTFunction(time, "Mu0").Re();
}
else{
while (eventTime < time){
eventDiffTime = eventTime - lastEventTime;
complexPolX = complexPolX * GTFunction(eventDiffTime);
complexPolX = complexPolX * GTFunction(eventDiffTime, "Mu0");
lastEventTime = eventTime;
eventTime += NextEventTime(fSpinFlipRate);
}
// calculate for the last collision
eventDiffTime = time - lastEventTime;
complexPolX = complexPolX * GTFunction(eventDiffTime);
complexPolX = complexPolX * GTFunction(eventDiffTime, "Mu0");
muoniumPolX = complexPolX.Re();
}
@ -330,130 +340,100 @@ Double_t PSimulateMuTransition::GTSpinFlip(const Double_t &time)
// Event (private)
//--------------------------------------------------------------------------
/**
* <p> Generates "muon event": simulate muon spin phase under charge-exchange with
* <p> Generates "muon event": simulate muon spin polarization under charge-exchange with
* a neutral muonium state in transverse field, where the polarization evolution
* PxMu(t) of the muon spin in muonium is determined by a superposition of the
* four "Mu transitions" nu_12, nu_34, nu_23, and nu_14.
* four "Mu transitions" nu_12, nu_34, nu_23, and nu_14. Use complex polarization
* functions.
* 1) according to Mu+/Mu0 fraction begin either with a Mu+ state or Mu state
* 2) Mu+: determine next electron-capture time t_c. If t_c is larger than decay time t_d
* calculate muon spin precession for t_d; else calculate spin precession for t_c.
* 3) Determine next ionization time t_i; calculate Px(t_i) in Muonium; calculate the
* muon spin phase by acos(Px(t_i)).
* calculate muon spin precession for t_d, Px(t_i); else calculate spin precession for t_c.
* 3) Determine next ionization time t_i+1; calculate Px(t_i+1) in Muonium. Polarization
* after ionization process is given by Px(t_i+1)*Px(t_i).
* 4) get the next electron capture time, continue until t_d is reached.
*
* <p> For isotropic muonium, TF:
* nu_12 and nu_34 with equal probabilities, probability for both states fMuFractionState12
* ni_23 and nu_14 with equal probabilities, probability for both states fMuFractionState23
*
* <p>Calculates Mu0 polarization in x direction during cyclic charge exchange.
* See M. Senba, J.Phys. B23, 1545 (1990), equations (9), (11)
* \param muonString if eq. "Mu+" begin with Mu+ precession
*/
void PSimulateMuTransition::Event(const TString muonString)
Double_t PSimulateMuTransition::Event(const TString muonString)
{
TComplex complexPolX = 1.0;
Double_t muoniumPolX = 1.0; //initial polarization in x direction
Double_t eventTime, eventDiffTime, captureTime, ionizationTime;
// Double_t muonPrecessionFreq, muoniumPrecessionFreq; // MHz
// Double_t rndm, frac1, frac2;
fMuonPrecFreq = fMuonGyroRatio * fBfield;
// charge-exchange loop until muon decay
eventTime = 0.;
eventDiffTime = 0.;
if (fDebugFlag) cout << "Decay time = " << fMuonDecayTime << endl;
//cout << muonString << endl;
// charge-exchange loop until muon decays
while (1) {
if (muonString == "Mu+"){
// Mu+ initial state; get next electron capture time
if (muonString == "Mu+"){// Mu+ initial state; get next electron capture time
captureTime = NextEventTime(fCaptureRate);
eventTime += captureTime;
if (fDebugFlag) cout << "Capture time = " << captureTime << " Phase = " << fMuonPhase << endl;
if (fDebugFlag) cout << "Capture time = " << captureTime << " PolX = " << complexPolX.Re() << endl;
if (eventTime < fMuonDecayTime)
fMuonPhase += PrecessionPhase(captureTime, "Mu+");
complexPolX *= GTFunction(captureTime, "Mu+");
else{ //muon decays; handle precession prior to muon decay
eventDiffTime = fMuonDecayTime - (eventTime - captureTime);
fMuonPhase += PrecessionPhase(eventDiffTime, "Mu+");
complexPolX *= GTFunction(eventDiffTime, "Mu+");
break;
}
// now, we have Mu0; get next ionization time
ionizationTime = NextEventTime(fIonizationRate);
eventTime += ionizationTime;
// determine Mu state
// rndm = fRandom->Rndm();
// frac1 = 1. - fMuFractionState1 - fMuFractionState2; // non-precessing Mu states
// frac2 = 1. - fMuFractionState2;
// if ( rndm < frac1 )
// muoniumPrecessionFreq = 0.;
// else if (rndm >= frac1 && rndm <= frac2){
// if (fRandom->Rndm() <= 0.5)
// muoniumPrecessionFreq = fMuPrecFreq12;
// else
// muoniumPrecessionFreq = fMuPrecFreq34;
// }
// else{
// if (fRandom->Rndm() <= 0.5)
// muoniumPrecessionFreq = fMuPrecFreq23;
// else
// muoniumPrecessionFreq = fMuPrecFreq14;
// }
if (fDebugFlag) cout << "Ioniza. time = " << ionizationTime << " Phase = " << fMuonPhase << endl;
if (fDebugFlag) cout << "Ioniza. time = " << ionizationTime << " PolX = " << complexPolX.Re() << endl;
if (eventTime < fMuonDecayTime)
fMuonPhase += PrecessionPhase(ionizationTime, "Mu0");
complexPolX *= GTFunction(ionizationTime, "Mu0");
else{ //muon decays; handle precession prior to muon decay
eventDiffTime = fMuonDecayTime - (eventTime - ionizationTime);
fMuonPhase += PrecessionPhase(eventDiffTime, "Mu0");
complexPolX *= GTFunction(eventDiffTime, "Mu0");
break;
}
}
else{
// Mu0 as initial state; get next ionization time
else{// Mu0 as initial state; get next ionization time
ionizationTime = NextEventTime(fIonizationRate);
eventTime += ionizationTime;
// determine Mu state
// rndm = fRandom->Rndm();
// frac1 = 1. - fMuFractionState1 - fMuFractionState2; // non-precessing Mu states
// frac2 = 1. - fMuFractionState2;
// if ( rndm < frac1 )
// muoniumPrecessionFreq = 0.;
// else if (rndm >= frac1 && rndm <= frac2){
// if (fRandom->Rndm() <= 0.5)
// muoniumPrecessionFreq = fMuPrecFreq12;
// else
// muoniumPrecessionFreq = fMuPrecFreq34;
// }
// else{
// if (fRandom->Rndm() <= 0.5)
// muoniumPrecessionFreq = fMuPrecFreq23;
// else
// muoniumPrecessionFreq = fMuPrecFreq14;
// }
if (fDebugFlag)
cout << "Mu Ioniza. time = " << ionizationTime << " Phase = " << fMuonPhase << endl;
cout << "Mu Ioniza. time = " << ionizationTime << " PolX = " << complexPolX.Re() << endl;
if (eventTime < fMuonDecayTime)
fMuonPhase += PrecessionPhase(ionizationTime, "Mu0");
complexPolX *= GTFunction(ionizationTime, "Mu0");
else{ //muon decays; handle precession prior to muon decay
eventDiffTime = fMuonDecayTime - (eventTime - ionizationTime);
fMuonPhase += PrecessionPhase(eventDiffTime, "Mu0");
complexPolX *= GTFunction(eventDiffTime, "Mu0");
break;
}
// Mu+ state; get next electron capture time
captureTime = NextEventTime(fCaptureRate);
eventTime += captureTime;
if (fDebugFlag) cout << "Capture time = " << captureTime << " Phase = " << fMuonPhase << endl;
if (fDebugFlag) cout << "Capture time = " << captureTime << " PolX = " << complexPolX.Re() << endl;
if (eventTime < fMuonDecayTime)
fMuonPhase += PrecessionPhase(captureTime, "Mu+");
complexPolX *= GTFunction(captureTime, "Mu+");
else{ //muon decays; handle precession prior to muon decay
eventDiffTime = fMuonDecayTime - (eventTime - captureTime);
fMuonPhase += PrecessionPhase(eventDiffTime, "Mu+");
complexPolX *= GTFunction(eventDiffTime, "Mu+");
break;
}
}
}
muoniumPolX = complexPolX.Re();
if (fDebugFlag) cout << " Final PolX = " << muoniumPolX << endl;
if (fDebugFlag) cout << " Final Phase = " << fMuonPhase << endl;
//fMuonPhase = TMath::ACos(TMath::Cos(fMuonPhase))*360./TMath::TwoPi(); //transform back to [0, 180] degree interval
return;
return muoniumPolX;
}