7. Atmospheric neutrinos and Neutrino oscillations

(1)

7. 7. Atmospheric neutrinos and Neutrino oscillations

Corso

Corso ““Astrofisica delle particelle Astrofisica delle particelle””

Prof. Maurizio Spurio Prof. Maurizio Spurio Università di Bologna.

Università di Bologna. A.a. A.a. 2011/12 2011/12

(2)

Outlook Outlook

Some history Some history

Neutrino Oscillations Neutrino Oscillations

How do we search for neutrino oscillations How do we search for neutrino oscillations

Atmospheric neutrinos Atmospheric neutrinos

10 years of Super 10 years of Super--Kamiokande Kamiokande

Upgoing muons and MACRO Upgoing muons and MACRO

Interpretation in terms on neutrino oscillations Interpretation in terms on neutrino oscillations

Appendix: The Cherenkov lightAppendix: The Cherenkov light

(3)

At the beginning of the ’80s, some theories (GUT) predicted the proton decay with measurable livetime

The proton was thought to decay in (for instance) pe⁺π⁰ν_e

Detector size: 10³ m³, and mass 1kt (=10³¹ p)

The main background for the detection of proton decay were atmospheric neutrinos interacting inside the experiment

7.1 Some history

Proton decay γγγγγγ γγ

e Neutrino Interaction

Water Cerenkov Experiments (IMB, Kamiokande)

Tracking calorimeters (NUSEX, Frejus, KGF)

Result: NO p decay ! But some anomalies on the neutrino measurement!

(4)

7.2 Neutrino Oscillations 7.2 Neutrino Oscillations

Idea of neutrinos being massive was first suggested by B.

Pontecorvo

Prediction came from proposal of neutrino oscillations

Neutrinos are created or annihilated as

| νννν

^e

>>>> , | νννν

µµµµ

>>>> , | νννν

ττττ

>>>>

⁼Weak Interactions (WI) eigenstats

|ν

¹

> , | ν

2

> , | ν

3

>

⁼Mass (Hamiltonian) eigenstats

•Neutrinos propagate as a superposition of mass eigenstates

Neutrinos are created or annihilated as

W.I. eigenstates

(5)

Weak Weak eigenstates eigenstates ((νν

_e_e

, , νν

_µµ

, , νν

_ττ

)) are expressed are expressed as a combinations of the mass eigenstates (νννν

₁

, νννν

₂

,νννν

₃

).

These propagate with different frequencies due to their

different masses, and different phases develop with distance travelled. Let us assume two neutrino flavors only.

The time propagation: |νννν (t) >>>> = ( |ν |ν |ν |ν

¹

>>>> , |ν |ν |ν |ν

²

>>>> )







=

+

≈ +

=

0

2

2 2

2

ij

i i

ii

M

E E m

m p

M

ν ν

M =

(2x2 matrix)

(eq.2)

) (t dt M

i d ν ν

=

^(eq.1)

(6)

eq.1 becames, using eq.2)

) 2 (

2

t E v

E m dt

i d ⁱ 







 +

=

ν ν

ν

whose solution is :

t i i

i

e

i

v t

v ( ) = ( 0 )

⁻ ^ϖ

(eq.4)

(eq.5)

Time propagation Time propagation

whose solution is :

v

ⁱ

( t ) = v

ⁱ

( 0 ) e







 +

=

ν

ϖ ν

E E mⁱ

i 2

2

with

During propagation, the phase difference is:

Eν

t m

m

i 2

) ( ₂² − ₁² ⋅

=

∆Φ ^(eq.6)

(7)

|

ν

^e

> = cos θ |ν

¹

> + sin θ |ν

²

>

θ = mixing angle

Time evolution of the “physical” neutrino states:

• Let us assume two neutrino flavors only (i.e. the electon and the muon neutrinos).

• They are linear superposition of the n1,n2 eigenstaten:

| ν

µ

> = -sin θ |ν

¹

> + cos θ |ν

²

>

^(eq.3)

t i t

i

t t i

i e

i i

e v

v

e v

v

2 1

) 0 ( cos

) 0 ( sin

) 0 ( cos

2 1

ϖ µ ϖ

ϖ ϖ

θ θ

− −

+

−

=

+

=

(eq.7)

• Using eq. 5 in eq. 3, we get:

(8)

•At t=0, eq. 7 becomes:

) 0 ( cos

) 0 ( sin

) 0 ( cos

2 1

v v

v

v v

v _e

θ θ

µ = − +

+

=

) 0 ( sin

) 0 ( cos

) 0

1 ( θ ^v θ ^v µ

v = _e −

• By inversion of eq. 8:

(eq.8)

) 0 ( cos

) 0 ( sin

) 0

2 (

1

µ µ

θ

θ ^v ^v

v _e

e

+

=

• For the experimental point of view (accelerators, reactors), a pure muon (or electron) state a t=0 can be prepared. For a pure ν_µ beam, eq. 9:

) 0 ( cos

) 0 (

) 0 ( sin

) 0 (

2 1

µ µ

θ θ

v v

=

−

=

(eq.9)

(eq.10)

(9)

The time evolution of the νννν_µµµµ state of eq. 8:

t i t

i _i

e v

v_µ = sin ²

θ

_µ (0) ⁻ ^ϖ¹ + cos ²

θ

_µ (0) ⁻ ^ϖ ²

By definition, the probability that the state at a given time is a ν_µ is:

0 t 2

P

_ν _µ_ν _µ

≡ ν

_µ

ν

_µ

(eq. 12)

•Using eq. 11, the probability:

(eq.11)

(

ⁱ ^t ⁱ ^t

)

t

e e

P

) (

2 2

4 2 4

0

2 1

2

cos 1

sin

cos sin

ϖ ϖ

µ µ

ν ν

θ θ

ν

µ

ν

µ

−

− +

+

+ +

=

≡



 −

⋅

−

= 2

) sin (

2 sin

1 ² ² ¹ ² t

P_ν _µ_ν _µ

θ ϖ ϖ

i.e. using trigonometry rules:

(eq. 13)

(eq. 14)

(10)







 +

=

ν ν

ϖ E

E mⁱ

i 2

2

Finally, using eq.5:



 



 −

⋅

−

=

ν ν

ν _µ _µ θ

E

t m

P m

4

) sin (

2 sin

1

2 1 2

2 2 2

With the following substitutions in eq.15:

- the neutrino path length L=ct (in Km)

- the mass difference ∆∆∆∆m² = m₂² – m₁2 (in eV²) - the neutrino Energy Eν ν ν ν (in GeV)

(eq. 15)

 





 



 ∆ ⋅

⋅

−

=

ν ν

ν_µ _µ

θ

E

L P m

2 2

2

2 sin 1 . 27

sin 1

To see “oscillations” pattern:

27 2 . 1

0

2 π

θ

ν

 ≈









 ∆ ⋅

≠

E

L m

(eq. 16)

(11)

7.3 How do 7.3 How do

we search we search for neutrino for neutrino oscillations?

oscillations?

(12)

..with ..with

atmospheric atmospheric neutrinos

neutrinos

• ∆m², sin²2Θ from Nature;

• Eν = experimental











 ∆ ⋅

⋅

−

=

ν ν

ν_µ _µ θ

E L P m

2 2

2 2 sin 1.27 sin

1

parameter (energy

distribution of neutrino giving a particular

configuration of events)

• L = experimental

parameter (neutrino path length from production to interaction)

(13)

A p p ea ra n ce / D is a p p ea ra n ce A p p ea ra n ce / D is a p p ea ra n ce A p p ea ra n ce / D is a p p ea ra n ce A p p ea ra n ce / D is a p p ea ra n ce

(14)











 ∆ ⋅

⋅

−

=

ν ν

ν_µ _µ θ

E L P m

2 2

2 2 sin 1.27 sin

1

(15)

77..44-- Atmospheric neutrinos Atmospheric neutrinos

(16)

The recipes for the evaluation of the The recipes for the evaluation of the

atmospheric neutrino flux neutrino flux--

e ν νe

µ

ν µ

π

µ µ

+ +

→ +

→

+ +

∴

(17)

E^-3

E < 10¹⁵ eV Galactic

5. 10¹⁹ < E< 3. 10²⁰ eV

i) The primary spectrum

E^-3

spectrum

GZK cut

10¹⁵ < E< 10¹⁸ eV galactic ?

E ≥ 5. 10¹⁹ eV Extra-Galactic?

Unexpected?

(18)

ii)

ii)-- CR CR--air cross section air cross section

It needs a model of nucleus-nucleus interactions

pp Cross section versus center of mass energy.

Average number of charged hadrons produced in pp (andpp) collisions versus center of mass energy

(19)

iii) Model of the atmosphere

ATMOSPHERIC NEUTRINO PRODUCTION:

•high precision 3D calculations,

•refined geomagnetic cut-off treatment (also geomagnetic field in atmosphere)

•elevation models of the Earth

•different atmospheric profiles

•geometry of detector effects

(20)

Output: the neutrino ( νννν _e , νννν _µµµµ ) flux

See for instance the FLUKA MC:

http://www.mi.infn.it/~battist/ne utrino.html

(21)

iv) The Detector response

Fully

Contained

νννν ^Partially

Contained

νννν

µµµµ

Through going µµµµ Stopping µµµµ

νννν

µµµµ

νννν

µµµµ











 ∆ ⋅

⋅

−

=

ν ν

ν_µ _µ θ

E L P m

2 2

2 2 sin 1.27 sin

1

Energy spectrum of νννν for each event categoryνννν νννν

Energy spectrum (from Monte Carlo) of

atmospheric neutrinos seen with different

event topologies (SuperKamiokande)

0 250 500 750 1000

10 ⁻² 10 ⁻¹ 1 10 10² 10³ 10⁴ 10⁵

FC ν_µ FC ν_e PC ν_µ

E_ν (GeV)

up-stop µµµµ up-thru µµµµ



 ν

µ

µ E

(22)

Rough estimate: how many ‘Contained Rough estimate: how many ‘Contained

events’ in 1 kton detector events’ in 1 kton detector

1. Flux: Φ_ν ~ 1 cm^-2 s^-1

2. Cross section (@ 1GeV):

σ_ν~0.5 10^-38 cm² νµ

σ_ν~0.5 10^-38 cm²

3. Targets M= 6 10³² (nucleons/kton) 4. Time t= 3.1 10⁷ s/y

Nint = Φ_ν (cm^-2 s^-1) x σ_ν(cm²)x M (nuc/kton) x t (s/y) ~

~ 100 interactions/ (kton y) νe

(23)

7.5 10 years of Super-Kamiokande

1996.4 Start data taking

1999.6 K2K started

2001.7 data taking was stopped

2001 Evidence of solar ν oscillation (SNO+SK) 1998 Evidence of atmospheric ν oscillation (SK)

SK-I

2001.7 data taking was stopped for detector upgrade 2001.11 Accident

partial reconstruction

2002.10 data taking was resumed

2005.10 data taking stopped for full reconstruction 2006.7 data taking was resumed

2005 Confirm ν oscillation by accelerator ν (K2K)

SK-II

SK-III

SK-IV 2009 data taking

(24)

Measurement of contained events and Measurement of contained events and

SuperKamiokande (Japan) SuperKamiokande (Japan)

1000 m Deep Underground 1000 m Deep Underground

50.000 ton of Ultra 50.000 ton of Ultra--Pure Water Pure Water

11000 +2000 PMTs 11000 +2000 PMTs

(25)

As a charged particle travels, it disrupts the local electromagnetic field (EM) in a medium.

Electrons in the atoms of the medium will be displaced and polarized by the passing EM field of a charged particle.

Photons are emitted as an insulator's

electrons restore themselves to equilibrium after the disruption has passed.

Cherenkov Radiation

In a conductor, the EM disruption can be restored without emitting a photon.

In normal circumstances, these photons destructively interfere with each other and no radiation is detected.

However, when the disruption travels faster than light is propagating through the medium, the photons constructively interfere and

intensify the observed Cerenkov radiation.

(26)

Cherenkov Radiation

One of the 13000

PMTs of SK

(27)

How to tell a νννν _µµµµ from a νννν _e :

Pattern recognition

(28)

νν _µµ

(29)

νν _e _e

(30)

e or µµµµ

Fully Contained (FC)

µµµµ

No hit in Outer Detector One cluster in Outer Detector

Partially Contained (PC)

Reduction Automatic ring fitter

Contained event in SuperKamiokande Contained event in SuperKamiokande

Automatic ring fitter Particle ID

Energy reconstruction Fiducial volume (>2m from wall, 22 ktons)

E_vis > 30 MeV (FC), > 3000 p.e. (~350 MeV) (PC) Fully Contained

8.2 events/day

E_vis<1.33 GeV : Sub-GeV E_vis>1.33 GeV : Multi-GeV

Partially Contained 0.58 events/day

(31)

Contained events.

The up/down The up/down

symmetry in SK and symmetry in SK and

νννννννν µµµµµµµµ /ν /ν /ν /ν /ν /ν /ν /ν e e ratio. ratio.

Up/Down asymmetry interpreted as neutrino oscillations

Expectations:

events inside the detector.

For E_νννν > a few GeV,

Upward /

downward = 1

Eνννν=0.5GeV Eνννν=3 GeV Eνννν=20 GeV

(32)

Zenith angle distribution

SK:1289 days (79.3 kty)

• Electron neutrinos = DATA and MC

(almost) OK!

µµµµ ^{/ e}

DATA

µµµµ ^{/ e} ^{M C} = 0.638 ±±±±

0.017 ±±±± 0.050

Data

• Muon neutrinos = Large deficit of DATA w.r.t. MC !

Zenith angle distributions for e-like and µ-like contained atmospheric

neutrino events in SK. The lines show the best fits with (red) and without (blue) oscillations; the best-fit is ∆m² = 2.0 × 10⁻³ eV² and sin² 2θ = 1.00.

(33)

SK-I + SK-II

0 50 100 150

-1 -0.8 -0.6 -0.4 -0.2 0 cosΘ

Upward through-going showering µ

SK-I + SK-II

0 50 100 150 200

-1 -0.5 0 0.5 1 multi-ring e-like

0 100 200 300

-1 -0.5 0 0.5 1 multi-ring µ-like

0 20 40

-1 -0.5 0 0.5 1 PC stop

0 100 200 300

-1 -0.5 0 0.5 1 PC through

0 50 100 150

-1 -0.8 -0.6 -0.4 -0.2 0 cosΘ

Upward stopping µ

0 200 400 600

-1 -0.8 -0.6 -0.4 -0.2 0 cosΘ

Upward through-going non-showering µ

Zenith Angle Distributions (SK

Zenith Angle Distributions (SK--I + SK I + SK--II) II)

ν_µ–ν_τ oscillation (best fit) null oscillation

SK-I + SK-II

0 100 200 300 400

-1 -0.5 0 0.5 1 Sub-GeV e-like

P < 400 MeV/c

0 100 200 300 400

-1 -0.5 0 0.5 1 Sub-GeV µ-like P < 400 MeV/c

0 100 200 300 400

-1 -0.5 0 0.5 1

Number of Events

Sub-GeV e-like P > 400 MeV/c

0 200 400 600

-1 -0.5 0 0.5 1 Sub-GeV µ-like P > 400 MeV/c

0 50 100 150 200

-1 -0.5 0 0.5 1 cosΘ

Multi-GeV e-like

0 50 100 150 200

-1 -0.5 0 0.5 1 cosΘ

Multi-GeV µ-like

µ-like

P<400MeV/c

P>400MeV/c

P<400MeV/c

P>400MeV/c

Livetime

•SK-I

1489d (FCPC) 1646d (Upmu)

•SK-II

804d (FCPC) 828d(Upmu)

SK-I + SK-II

0 50 100 150

-1 -0.8 -0.6 -0.4 -0.2 0 cosΘ

Upward through-going showering µ

SK-I + SK-II

0 50 100 150 200

-1 -0.5 0 0.5 1 multi-ring e-like

0 100 200 300

-1 -0.5 0 0.5 1 multi-ring µ-like

0 20 40

-1 -0.5 0 0.5 1 PC stop

0 100 200 300

-1 -0.5 0 0.5 1 PC through

0 50 100 150

-1 -0.8 -0.6 -0.4 -0.2 0 cosΘ

Upward stopping µ

0 200 400 600

-1 -0.8 -0.6 -0.4 -0.2 0 cosΘ

Upward through-going non-showering µ

SK-I + SK-II

0 100 200 300 400

-1 -0.5 0 0.5 1 Sub-GeV e-like

P < 400 MeV/c

0 100 200 300 400

-1 -0.5 0 0.5 1 Sub-GeV µ-like P < 400 MeV/c

0 100 200 300 400

-1 -0.5 0 0.5 1

Number of Events

Sub-GeV e-like P > 400 MeV/c

0 200 400 600

-1 -0.5 0 0.5 1 Sub-GeV µ-like P > 400 MeV/c

0 50 100 150 200

-1 -0.5 0 0.5 1 cosΘ

Multi-GeV e-like

0 50 100 150 200

-1 -0.5 0 0.5 1 cosΘ

Multi-GeV µ-like

µ-like e-like

NOTE: All topologies, last results (September 2007)

(34)

Atmospheric Neutrino Anomaly

Summary results since the mid-1980's:

µµµµ ^{/ e} ^{D a t a} µµµµ ^{/ e} ^{M C}

R’= Double ratio between the number of

detected and expected νµ and νe

Water Cherenkov

Calorimetric

(35)

7.6 Upgoing muons and MACRO 7.6 Upgoing muons and MACRO

(Italy) (Italy)

• Large acceptance (~10000 m ²sr for an isotropic flux)

• Low downgoing µ rate (~10 ^-6of the surface rate )

• ~600 tons of liquid scintillator to measure T.O.F.

(time resolution ~500psec)

• ~20000 m ² of streamer tubes (3cm cells) for tracking (angular resolution < 1° )

R.I.P December 2000

(36)

The Gran Sasso National Labs The Gran Sasso National Labs

http://www.lngs.infn.it/

(37)

Neutrino event topologies in MACRO Neutrino event topologies in MACRO

In up Up throughgoing

Absorber Streamer Scintillator

•

Liquid scintillator counters

, (3 planes) for the measurement of time and dE/dx.

•

Streamer tubes

(14 planes), for the measurement of the track

Up stop

In down

1) 2)

4) 3)

Scintillator

position;

• Detector mass: 5.3 kton

• Atmospheric muon neutrinos produce upward going muons

• Downward going muons ~ 10⁶ upward going muons

• Different neutrino topologies

(38)

Energy spectra of

Energy spectra of νννννννν _{µµµµµµµµ} events in MACRO events in MACRO

• <E>~ 50 GeV throughgoing µ

• <E>~ 5 GeV, Internal Upgoing (IU) µ;

• <E>~ 4 GeV , internal downgoing (ID) µ and for upgoing stopping (UGS) µ;

(39)

+1 µµµµ -1 µµµµ T₁

T₂

Streamer tube track

Neutrino induced

Neutrino induced events are events are upwardupward throughgoing throughgoing muons

muons, , Identified by the timeIdentified by the time--ofof--flight methodflight method

Atmospheric µµµµ:

downgoing µ

µ µ

µ from ν:ν:ν:ν: upgoing

( − )⋅ ₌

= L

c T

T ₁ ₂ 1

β

( ⁻ )^⋅ ₌

= L

c T

T ₁ ₂ 1

β

(40)

MACRO Results: event deficit and distortion of the angular distribution

- - - - No oscillations

____ Best fit ∆∆∆∆m²= 2.2x10^-3eV² sin²2θθθθ=1.00

Observed= 809 events

Expected= 1122 events (Bartol) Observed/Expected

= 0.721±±±0.050± _(stat+sys)±±±±0.12_(th)

(41)

MACRO Partially contained events MACRO Partially contained events

Obs. 154 events Exp. 285 events

Obs./Exp. = 0.54±±±±0.15

IU

MC with oscillations

consistent with up

throughgoing muon results

Obs. 262 events Exp. 375 events

Obs./Exp. = 0.70±±±0.19± ₎ ID+UGS

(42)

underground detector

Effects of νννν

_µµµµ

oscillations on upgoing events

νννν Earth

µ











 ∆ ⋅

⋅

−

ν =

ν_µ _µ θ

E L P m

2 2

22 sin 1.27 sin

1

θ

• If θ is the zenith angle and D= Earth diameter L=Dcosθ

• For throughgoing neutrino-induced muons in MACRO, Eν = 50 GeV (from Monte Carlo)



 

 ν

ν ν_µ _µ

E

θθθθ cos(θθθθ) ∆∆∆∆m2=0.0002 ∆∆∆∆m2=0.0001

0 -1,000 0,62 0,89

-10 -0,985 0,63 0,90

-20 -0,940 0,66 0,91

-30 -0,866 0,71 0,92

-40 -0,766 0,77 0,94

-50 -0,643 0,83 0,96

-60 -0,500 0,89 0,97

-70 -0,342 0,95 0,99

-80 -0,174 0,99 1,00

-90 0,000 1,00 1,00

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

-1,00 -0,80 -0,60 -0,40 -0,20 0,00

cosθ

µ µν

Pν

(43)

Oscillation Parameters Oscillation Parameters

• The value of the “oscillation parameters” sin²θ and ∆m² correspond to the values which provide the best fit to the data

• Different experiments different values of sin²θ and ∆m²

• The experimental data have an associated error. All the values of (sin²θ, ∆m²) which are compatible with the experimental data are

“allowed”.

• The “allowed” values span a region in the parameter space of (sin²θ,

∆m²)











 ∆ ⋅

⋅

−

=

ν ν

ν_µ _µ θ

E L P m

2 2

2 2 sin 1.27 sin

1

1.9 x 10^-3 eV² < ∆∆∆∆m² < 3.1 x 10^-3 eV² sin²2θθθθ > 0.93 (90% CL)

(44)

“Allowed” parameters region

90% C. L. allowed regions for ν_µ → ν_τ oscillations of atmospheric neutrinos for Kamiokande, SuperK, Soudan-2 and MACRO.

(45)

Why not

Why not νν _µ _µ → → νν _e _e ??

Apollonio et al., CHOOZ Coll., Phys.Lett.B466,415

(46)

νννννννν _{µµµµµµµµ} disappearance: History disappearance: History

Anomaly in R=(Anomaly in R=(µµ/e)/e)_observed_observed/(/(µµ/e)/e)_predicted_predicted

Kamiokande: PLB 1988, 1992Kamiokande: PLB 1988, 1992

Discrepancies in various experimentsDiscrepancies in various experiments

Kamiokande: ZenithKamiokande: Zenith--angle distributionangle distribution

Kamiokande: PLB 1994Kamiokande: PLB 1994

SuperSuper--Kamiokande/MACRO: Kamiokande/MACRO:

Discovery of

Discovery of νν oscillation in 1998oscillation in 1998 Discovery of

Discovery of νν_µµ oscillation in 1998oscillation in 1998

SuperSuper--Kamiokande: PRL 1998 Kamiokande: PRL 1998

MACRO, PRL 1998MACRO, PRL 1998

K2K: First acceleratorK2K: First accelerator--based long based long baseline experiment: 1999

baseline experiment: 1999 –– 2004 2004

Confirmed atmospheric neutrino results Confirmed atmospheric neutrino results

Final result 4.3Final result 4.3σσ: PRL 2005, PRD 2006: PRL 2005, PRD 2006

MINOS: Precision measurement: 2005 MINOS: Precision measurement: 2005 --

First result: PRL2006First result: PRL2006

(47)

See for review:

The “Neutrino Industry” The “Neutrino Industry”

http://www.hep.anl.gov/ndk/hypertext/http://www.hep.anl.gov/ndk/hypertext/

Janet Conrad web pages: Janet Conrad web pages:

http://www.nevis.columbia.edu/~conrad/nupage.htmlhttp://www.nevis.columbia.edu/~conrad/nupage.html

Fermilab and KEK “Neutrino Summer School” Fermilab and KEK “Neutrino Summer School”

http://projects.fnal.gov/nuss/http://projects.fnal.gov/nuss/

Torino web Pages: Torino web Pages:

http://www.nu.to.infn.it/Neutrino_Lectures/http://www.nu.to.infn.it/Neutrino_Lectures/

Progress in the physics of massive neutrinos, hep-

ph/0308123

(48)

Appendice:

La radiazione Cerenkov

(49)

Effetto Cerenkov

Per una trattazione classica dell’effetto Cerenkov:

Jackson : Classical Electrodynamics, cap 13 e par. 13.4 e 13.5

La radiazione Cerenkov e’ emessa ogniqualvolta una particella carica attraversa un mezzo La radiazione Cerenkov e’ emessa ogniqualvolta una particella carica attraversa un mezzo (dielettrico) con velocita’

(dielettrico) con velocita’ ββc=v>c/nc=v>c/n, dove, dove v v e’ la velocita’ della particella ee’ la velocita’ della particella e nn l’indice di l’indice di rifrazione del mezzo.

rifrazione del mezzo.

Intuitivamente: la particella incidente polarizza il dielettrico

Intuitivamente: la particella incidente polarizza il dielettrico gli atomi diventano dei gli atomi diventano dei dipoli.

dipoli. Se Se ββ>1/n >1/n momento di dipolo elettrico momento di dipolo elettrico emissione di radiazione.emissione di radiazione.

dipoli.

dipoli. Se Se ββ>1/n >1/n momento di dipolo elettrico momento di dipolo elettrico emissione di radiazione.emissione di radiazione.

ββββ<1/n β>β>β>β>1/n

(50)

L’ angolo di emissione

L’ angolo di emissione θθ_cc puo’ essere interpretato qualitativamente puo’ essere interpretato qualitativamente come un’onda d’urto come succede per una barca od un aereo

come un’onda d’urto come succede per una barca od un aereo supersonico.

supersonico.

β l_light=(c/n)∆t

θ

wave front

1 ) ( 1 with

cosθ = ⁿ= ⁿ λ ≥ θ_C

Esiste una velocita’ di soglia

Esiste una velocita’ di soglia ββ_ss = 1/n = 1/n θθ_cc ~ 0 ~ 0 Esiste un angolo massimo

Esiste un angolo massimo θθ_max_max= arcos(1/n)= arcos(1/n) La cos(

La cos(θθ) =1/) =1/ββn e’ valida solo per un radiatore infinito, e’ comunque n e’ valida solo per un radiatore infinito, e’ comunque una buona approssimazione ogniqualvolta il radiatore e’ lungo L>>

una buona approssimazione ogniqualvolta il radiatore e’ lungo L>>λ λ essendo

essendo λλ la lunghezza d’onda della luce emessala lunghezza d’onda della luce emessa

l_part=βc_∆t

θ 1 with ( ) 1

cos = = λ ≥

θ β ⁿ ⁿ

C n

C

(51)

Numero di fotoni emessi per unita’ di percorso e intervallo unitario di

lunghezza d’onda. Osserviamo che decresce al crescere della λ

2 sin 1 1

2 ₂

2 2 2

2 2

2

2N z z

d

= C







 −

= θ

λ α π β

λ α π λ

dN/dλ

λ

. with

1

sin 1

2 2

2

2 2

const dxdE

N d E

hc c

dxd N d

n

dxd ^C

=

∝

 =

 

 −

=

λ ν λ

λ

λ θ β

λ λ

dN/dE

Ε

Il numero di fotoni emessi per unita’ di percorso non dipende dalla frequenza

(52)

L’ energia persa per radiazione Cerenkov cresce con

L’ energia persa per radiazione Cerenkov cresce con ββ. Comunque . Comunque anche con

anche con ββ 1 e’ molto piccola.1 e’ molto piccola.

Molto piu’ piccola di quella persa per collisione (Bethe Block), al Molto piu’ piccola di quella persa per collisione (Bethe Block), al

( ) ^ω ^ω

ω β

α

^d

n z c

dx

dE

∫

^



 



 −

=

− ² ₂ 1₂

h 1

Molto piu’ piccola di quella persa per collisione (Bethe Block), al Molto piu’ piccola di quella persa per collisione (Bethe Block), al massimo 1%

massimo 1% ..

medium n θmax (β=1) Nph (eV^-1 cm^-1) air 1.000283 1.36 0.208

isobutane 1.00127 2.89 0.941

water 1.33 41.2 160.8

quartz 1.46 46.7 196.4

(53)

1)

1) EsisteEsiste unauna sogliasoglia per per emissioneemissione di di luceluce CerenkovCerenkov

2)

2) La La luceluce e’ e’ emessaemessa ad un ad un angoloangolo particolareparticolare

Facile Facile utilizzareutilizzare l’effettol’effetto Cerenkov per Cerenkov per identificareidentificare le le particelleparticelle..

Con 1)

Con 1) possoposso sfruttaresfruttare la la sogliasoglia Cerenkov a Cerenkov a sogliasoglia..

Con 2)

Con 2) misuraremisurare l’angolol’angolo DISC, RICH etc.DISC, RICH etc.

La

La luceluce emessaemessa e e rivelabilerivelabile e’ e’ pocapoca..

La

La luceluce emessaemessa e e rivelabilerivelabile e’ e’ pocapoca..

Consideriamo

Consideriamo un un radiatoreradiatore spessospesso 1 cm un 1 cm un angoloangolo θθ_cc = 30= 30^o^o eded un un

∆∆E = 1 E = 1 eVeV eded unauna particellaparticella di di caricacarica 11..

5 . 92 25

. 0 370 sin

370

sin

2

2 2

=

×

=

∆

⋅

=

⇒

=

E L

N

c z dEdx

dN

c ph

c

ϑ α ϑ h

Considerando inoltre che l’efficienza quantica di un fotomoltiplicatore e’

~20% N_pe=18 fluttuazioni alla Poisson